# kb

id: extraterrestrial-life-20260505-10
file: data/en.wikipedia.org/wiki/Extraterrestrial_life-10.md
category: reference
title: Extraterrestrial life
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Extraterrestrial_life
date_saved: 2026-05-05T11:01:43.119211+00:00
instance: kb-cron
parent: extraterrestrial-life-20260505-0
summary:
The search and study of extraterrestrial life became a science of its own, astrobiology. Also known as exobiology, this discipline is studied by the NASA, the ESA, the INAF, and others. Astrobiology studies life from Earth as well, but with a cosmic perspective. For example, abiogenesis is of interest to astrobiology, not because of the origin of life on Earth, but for the chances of a similar process taking place in other celestial bodies. Many aspects of life, from its definition to its chemistry, are analyzed as either likely to be similar in all forms of life across the cosmos or only native to Earth. Astrobiology, however, remains constrained by the current lack of extraterrestrial life-forms to study, as all life on Earth comes from the same ancestor, and it is hard to infer general characteristics from a group with a single example to analyse. The 20th century came with great technological advances, speculations about future hypothetical technologies, and an increased basic knowledge of science by the general population thanks to science divulgation through the mass media. The public interest in extraterrestrial life and the lack of discoveries by mainstream science led to the emergence of pseudosciences that provided affirmative,


id: extraterrestrial-life-20260505-11
file: data/en.wikipedia.org/wiki/Extraterrestrial_life-11.md
category: reference
title: Extraterrestrial life
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Extraterrestrial_life
date_saved: 2026-05-05T11:01:43.119211+00:00
instance: kb-cron
parent: extraterrestrial-life-20260505-0
summary:
The 1967 Outer Space Treaty and the 1979 Moon Agreement define rules of planetary protection against potentially hazardous extraterrestrial life. COSPAR also provides guidelines for planetary protection. A committee of the United Nations Office for Outer Space Affairs had in 1977 discussed for a year strategies for interacting with extraterrestrial life or intelligence. The discussion ended without any conclusions. As of 2010, the UN lacks response mechanisms for the case of an extraterrestrial contact. One of the NASA divisions is the Office of Safety and Mission Assurance (OSMA), also known as the Planetary Protection Office. A part of its mission is to "rigorously preclude backward contamination of Earth by extraterrestrial life." In 2016, the Chinese Government released a white paper detailing its space program. According to the document, one of the research objectives of the program is the search for extraterrestrial life. It is also one of the objectives of the Chinese Five-hundred-meter Aperture Spherical Telescope (FAST) program. In 2020, Dmitry Rogozin, the head of the Russian space agency, said the search for extraterrestrial life is one of the main goals of deep space research. He also acknowledged the possibility of existence of primitive life on other planets of the


id: missing-heritability-problem-20260505-0
file: data/en.wikipedia.org/wiki/Missing_heritability_problem-0.md
category: reference
title: Missing heritability problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Missing_heritability_problem
date_saved: 2026-05-05T11:01:44.300327+00:00
instance: kb-cron
parent: missing-heritability-problem-20260505-0
summary:
In genetics, the missing heritability problem refers to a difference between heritability estimates obtained from early genome-wide association studies (GWAS) and heritability estimates from twin and family data across many physical and mental traits, including diseases, behaviors, and other phenotypes. An influential review article in 2008 noted that the amount of phenotypic variance explained by significant loci in GWAS studies up to that point was usually far less than expected based on family studies. This gap was referred to as "missing heritability". Using height as a model trait, a paper in 2010 showed that most of the missing heritability can be explained by the presence of large numbers of low variants whose effect sizes were too small to detect at the sample sizes that were then available. This conclusion has subsequently been confirmed using much larger sample sizes, including a study of 5.4 million individuals that identified around 12,000 independent variants that affect human height. While studies of height have particularly large power due to their very large sample size, other complex traits likely have similar genetic architecture. Thus, the missing heritability problem is largely resolved by the presence of tens of thousands of variants of small effects that could


id: nanobacterium-20260505-0
file: data/en.wikipedia.org/wiki/Nanobacterium-0.md
category: reference
title: Nanobacterium
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Nanobacterium
date_saved: 2026-05-05T11:01:45.528380+00:00
instance: kb-cron
parent: nanobacterium-20260505-0
summary:
Nanobacterium ( NAN-oh-bak-TEER-ee-əm, pl. nanobacteria NAN-oh-bak-TEER-ee-ə) is the unit or member name of a former proposed class of living organisms, specifically cell-walled microorganisms, now discredited, with a size much smaller than the generally accepted lower limit for life (about 200 nm for bacteria, like mycoplasma). Originally based on observed nano-scale structures in geological formations (including the Martian meteorite Allan Hills 84001), the status of nanobacteria was controversial, with some researchers suggesting they are a new class of living organism capable of incorporating radiolabeled uridine, and others attributing to them a simpler, abiotic nature. One skeptic dubbed them "the cold fusion of microbiology", in reference to a notorious episode of supposed erroneous science. The term "calcifying nanoparticles" (CNPs) has also been used as a conservative name regarding their possible status as a life form. Research tends to agree that these structures exist, and appear to replicate in some way. However, the idea that they are living entities has now largely been discarded, and the particles are instead thought to be nonliving crystallizations of minerals and organic molecules. == 1981–2000 == In 1981 Francisco Torella and Richard Y. Morita described very small cells called ultramicrobacteria. Defined as being smaller than 300 nm,


id: dining-philosophers-problem-20260505-0
file: data/en.wikipedia.org/wiki/Dining_philosophers_problem-0.md
category: reference
title: Dining philosophers problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Dining_philosophers_problem
date_saved: 2026-05-05T11:01:47.089416+00:00
instance: kb-cron
parent: dining-philosophers-problem-20260505-0
summary:
In computer science, the dining philosophers problem is an example problem often used in concurrent algorithm design to illustrate synchronization issues and techniques for resolving them. It was originally formulated in 1965 by Edsger Dijkstra as a student exam exercise, presented in terms of computers competing for access to tape drive peripherals. Soon after, Tony Hoare gave the problem its present form. == Problem statement == Five philosophers dine together at the same table. Each philosopher has their own plate at the table. There is a fork between each pair of adjacent plates. The dish served is a kind of spaghetti which has to be eaten with two forks. Each philosopher can only alternately think and eat. Moreover, a philosopher can only eat their spaghetti when they have both a left and a right fork. Thus, two forks will only be available when their two nearest neighbors are thinking, not eating. After an individual philosopher finishes eating, they will put down both forks. The problem is how to design a regimen (a concurrent algorithm) such that any philosopher will not starve; i.e., each can forever continue to alternate between eating and thinking, assuming that no philosopher can know when others


id: dining-philosophers-problem-20260505-1
file: data/en.wikipedia.org/wiki/Dining_philosophers_problem-1.md
category: reference
title: Dining philosophers problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Dining_philosophers_problem
date_saved: 2026-05-05T11:01:47.089416+00:00
instance: kb-cron
parent: dining-philosophers-problem-20260505-0
summary:
=== Arbitrator solution === Another approach is to guarantee that a philosopher can only pick up both forks or none by introducing an arbitrator to replace circular waiting, e.g., a waiter. In order to pick up the forks, a philosopher must ask permission of the waiter. The waiter gives permission to only one philosopher at a time until the philosopher has picked up both of his forks. Putting down a fork is always allowed. The waiter can be implemented as a mutex. In addition to introducing a new central entity (the waiter), this approach can result in reduced parallelism: if a philosopher is eating and one of his neighbors is requesting the forks, all other philosophers must wait until this request has been fulfilled, even if forks for them are still available. === Limiting the number of diners in the table === A solution presented by William Stallings is to allow a maximum of n-1 philosophers to sit down at any time. The last philosopher would have to wait (for example, using a semaphore) for someone to finish dining before he "sits down" and requests access to any fork. This negates circular wait, guaranteeing that at least one philosopher may


id: pagh-s-problem-20260505-0
file: data/en.wikipedia.org/wiki/Pagh's_problem-0.md
category: reference
title: Pagh's problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Pagh's_problem
date_saved: 2026-05-05T11:01:48.275852+00:00
instance: kb-cron
parent: pagh-s-problem-20260505-0
summary:
Pagh's problem is a data structure problem often used when studying lower bounds in computer science named after Rasmus Pagh. Mihai Pătrașcu was the first to give lower bounds for the problem. In 2021 it was shown that, given popular conjectures, the naive linear time algorithm is optimal. == Definition == We are given as inputs k {\displaystyle k} subsets X 1 , X 2 , … , X k {\displaystyle X_{1},X_{2},\dots ,X_{k}} over a universe U = { 1 , … , k } {\displaystyle U=\{1,\dots ,k\}} . We must accept updates of the following kind: Given a pointer to two subsets X 1 {\displaystyle X_{1}} and X 2 {\displaystyle X_{2}} , create a new subset X 1 ∩ X 2 {\displaystyle X_{1}\cap X_{2}} . After each update, we must output whether the new subset is empty or not. == References ==


id: producer-consumer-problem-20260505-0
file: data/en.wikipedia.org/wiki/Producer–consumer_problem-0.md
category: reference
title: Producer–consumer problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Producer–consumer_problem
date_saved: 2026-05-05T11:01:49.467920+00:00
instance: kb-cron
parent: producer-consumer-problem-20260505-0
summary:
In computing, the producer-consumer problem (also known as the bounded-buffer problem) is a family of problems described by Edsger W. Dijkstra since 1965. Dijkstra found the solution for the producer-consumer problem as he worked as a consultant for the Electrologica X1 and X8 computers: "The first use of producer-consumer was partly software, partly hardware: The component taking care of the information transport between store and peripheral was called 'a channel' ... Synchronization was controlled by two counting semaphores in what we now know as the producer/consumer arrangement: the one semaphore indicating the length of the queue, was incremented (in a V) by the CPU and decremented (in a P) by the channel, the other one, counting the number of unacknowledged completions, was incremented by the channel and decremented by the CPU. [The second semaphore being positive would raise the corresponding interrupt flag.]" Dijkstra wrote about the unbounded buffer case: "We consider two processes, which are called the 'producer' and the 'consumer' respectively. The producer is a cyclic process and each time it goes through its cycle it produces a certain portion of information, that has to be processed by the consumer. The consumer is also a cyclic process and each


id: producer-consumer-problem-20260505-1
file: data/en.wikipedia.org/wiki/Producer–consumer_problem-1.md
category: reference
title: Producer–consumer problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Producer–consumer_problem
date_saved: 2026-05-05T11:01:49.467920+00:00
instance: kb-cron
parent: producer-consumer-problem-20260505-0
summary:
The Go producer-consumer solution uses the main Go routine for consumer and creates a new, unnamed Go routine for the producer. The two Go routines are connected with channel ch. This channel can queue up to three int values. The statement ch := make(chan int, 3) creates the channel, the statement ch <- produceMessage() sends a value into the channel and the statement recvMsg := range ch receives a value from the channel. The allocation of memory resources, the allocation of processing resources, and the synchronization of resources are done by the programming language automatically. == Without semaphores or monitors == Leslie Lamport documented a bounded buffer producer-consumer solution for one producer and one consumer: We assume that the buffer can hold at most b messages, b >= 1. In our solution, we let k be a constant greater than b, and let s and r be integer variables assuming values between 0 and k-1. We assume that initially s=r and the buffer is empty. By choosing k to be a multiple of b, the buffer can be implemented as an array B [0: b - 1]. The producer simply puts each new message into B[s mod b], and the


id: resource-starvation-20260505-0
file: data/en.wikipedia.org/wiki/Resource_starvation-0.md
category: reference
title: Resource starvation
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Resource_starvation
date_saved: 2026-05-05T11:01:50.668301+00:00
instance: kb-cron
parent: resource-starvation-20260505-0
summary:
In computer science, resource starvation is a problem encountered in concurrent computing where a process is perpetually denied necessary resources to process its work. Starvation may be caused by errors in a scheduling or mutual exclusion algorithm, but can also be caused by resource leaks, and can be intentionally caused via a denial-of-service attack such as a fork bomb. When starvation is impossible in a concurrent algorithm, the algorithm is called starvation-free, lockout-freed or said to have finite bypass. This property is an instance of liveness, and is one of the two requirements for any mutual exclusion algorithm; the other being correctness. The name "finite bypass" means that any process (concurrent part) of the algorithm is bypassed at most a finite number times before being allowed access to the shared resource. == Scheduling == Starvation is usually caused by an overly simplistic scheduling algorithm. For example, if a (poorly designed) multi-tasking system always switches between the first two tasks while a third never gets to run, then the third task is being starved of CPU time. The scheduling algorithm, which is part of the kernel, is supposed to allocate resources equitably; that is, the algorithm should allocate resources so that


id: sleeping-barber-problem-20260505-0
file: data/en.wikipedia.org/wiki/Sleeping_barber_problem-0.md
category: reference
title: Sleeping barber problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Sleeping_barber_problem
date_saved: 2026-05-05T11:01:51.847483+00:00
instance: kb-cron
parent: sleeping-barber-problem-20260505-0
summary:
In computer science, the sleeping barber problem is a classic inter-process communication and synchronization problem that illustrates the complexities that arise when there are multiple operating system processes. The problem was originally proposed in 1965 by computer science pioneer Edsger Dijkstra, who used it to make the point that general semaphores are often superfluous. == Problem statement == Imagine a hypothetical barbershop with one barber, one barber chair, and a waiting room with n chairs (n may be 0) for waiting customers. The following rules apply: If there are no customers, the barber falls asleep in the chair A customer must wake the barber if he is asleep If a customer arrives while the barber is working, the customer leaves if all chairs are occupied and sits in an empty chair if it's available When the barber finishes a haircut, he inspects the waiting room to see if there are any waiting customers and falls asleep if there are none There are two main complications. First, there is a risk that a race condition, where the barber sleeps while a customer waits for the barber to get them for a haircut, arises because all of the actions—checking the waiting room,


id: list-of-unsolved-problems-in-computer-science-20260505-0
file: data/en.wikipedia.org/wiki/List_of_unsolved_problems_in_computer_science-0.md
category: reference
title: List of unsolved problems in computer science
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_computer_science
date_saved: 2026-05-05T11:01:53.389450+00:00
instance: kb-cron
parent: list-of-unsolved-problems-in-computer-science-20260505-0
summary:
This article is a list of notable unsolved problems in computer science. A problem in computer science is considered unsolved when no solution is known or when experts in the field disagree about proposed solutions. == Computational complexity == P versus NP problem – The P vs NP problem is a major unsolved question in computer science that asks whether every problem whose solution can be quickly verified by a computer (NP) can also be quickly solved by a computer (P). This question has profound implications for fields such as cryptography, algorithm design, and computational theory. What is the relationship between BQP and NP? NC = P problem NP = co-NP problem P = BPP problem P = PSPACE problem L = NL problem PH = PSPACE problem L = P problem L = RL problem Unique games conjecture Is the exponential time hypothesis true? Is the strong exponential time hypothesis (SETH) true? Do one-way functions exist? Is public-key cryptography possible? Log-rank conjecture Hartmanis–Stearns conjecture == Polynomial versus nondeterministic-polynomial time for specific algorithmic problems == Can integer factorization be done in polynomial time on a classical (non-quantum) computer? Can the discrete logarithm be computed in polynomial time on a classical


id: 3sum-20260505-0
file: data/en.wikipedia.org/wiki/3SUM-0.md
category: reference
title: 3SUM
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/3SUM
date_saved: 2026-05-05T11:01:54.553179+00:00
instance: kb-cron
parent: 3sum-20260505-0
summary:
In computational complexity theory, the 3SUM problem asks if a given set of n {\displaystyle n} real numbers contains three elements that sum to zero. A generalized version, k {\displaystyle k} -SUM, asks the same question on k {\displaystyle k} elements, rather than simply 3. 3SUM can be easily solved in O ( n 2 ) {\displaystyle O(n^{2})} time, and matching Ω ( n ⌈ k / 2 ⌉ ) {\displaystyle \Omega (n^{\lceil k/2\rceil })} lower bounds are known in some specialized models of computation (Erickson 1999). It was conjectured that any deterministic algorithm for the 3SUM requires Ω ( n 2 ) {\displaystyle \Omega (n^{2})} time. In 2014, the original 3SUM conjecture was refuted by Allan Grønlund and Seth Pettie who gave a deterministic algorithm that solves 3SUM in O ( n 2 / ( log ⁡ n / log ⁡ log ⁡ n ) 2 / 3 ) {\displaystyle O(n^{2}/({\log n}/{\log \log n})^{2/3})} time. Additionally, Grønlund and Pettie showed that the 4-linear decision tree complexity of 3SUM is O ( n 3 / 2 log ⁡ n ) {\displaystyle O(n^{3/2}{\sqrt {\log n}})} . These bounds were subsequently improved. The current best known algorithm for 3SUM runs in O


id: 3sum-20260505-1
file: data/en.wikipedia.org/wiki/3SUM-1.md
category: reference
title: 3SUM
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/3SUM
date_saved: 2026-05-05T11:01:54.553179+00:00
instance: kb-cron
parent: 3sum-20260505-0
summary:
=== Three different arrays === Instead of searching for the 3 numbers in a single array, we can search for them in 3 different arrays. I.e., given three arrays X, Y and Z, find three numbers a∈X, b∈Y, c∈Z, such that ⁠ a + b + c = 0 {\displaystyle a+b+c=0} ⁠. Call the 1-array variant 3SUM×1 and the 3-array variant 3SUM×3. Given a solver for 3SUM×1, the 3SUM×3 problem can be solved in the following way (assuming all elements are integers): For every element in X, Y and Z, set: ⁠ X [ i ] ← X [ i ] ∗ 10 + 1 {\displaystyle X[i]\gets X[i]*10+1} ⁠, ⁠ Y [ i ] ← Y [ i ] ∗ 10 + 2 {\displaystyle Y[i]\gets Y[i]*10+2} ⁠, ⁠ Z [ i ] ← Z [ i ] ∗ 10 − 3 {\displaystyle Z[i]\gets Z[i]*10-3} ⁠. Let S be a concatenation of the arrays X, Y and Z. Use the 3SUM×1 oracle to find three elements ⁠ a ′ ∈ S , b ′ ∈ S , c ′ ∈ S {\displaystyle a'\in S,\ b'\in S,\ c'\in S} ⁠ such that ⁠ a ′ + b ′ + c ′ =


id: 3sum-20260505-2
file: data/en.wikipedia.org/wiki/3SUM-2.md
category: reference
title: 3SUM
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/3SUM
date_saved: 2026-05-05T11:01:54.553179+00:00
instance: kb-cron
parent: 3sum-20260505-0
summary:
If there is a solution for 3SUM: z = x + y {\displaystyle z=x+y} , then: T [ h ( z ) ] = T [ h ( x ) ] + T [ h ( y ) ] {\displaystyle T[h(z)]=T[h(x)]+T[h(y)]} and h ( z ) = h ( x ) + h ( y ) {\displaystyle h(z)=h(x)+h(y)} , so this solution will be found by the Conv3SUM solver on T. Conversely, if a Conv3SUM is found on T, then obviously it corresponds to a 3SUM solution on S since T is just a permutation of S. This idealized solution doesn't work, because any hash function might map several distinct elements of S to the same cell of T. The trick is to create an array ⁠ T ∗ {\displaystyle T^{*}} ⁠ by selecting a single random element from each cell of T, and run Conv3SUM on ⁠ T ∗ {\displaystyle T^{*}} ⁠. If a solution is found, then it is a correct solution for 3SUM on S. If no solution is found, then create a different random ⁠ T ∗ {\displaystyle T^{*}} ⁠ and try again. Suppose there are at most R elements in each cell of T. Then


id: aanderaa-karp-rosenberg-conjecture-20260505-0
file: data/en.wikipedia.org/wiki/Aanderaa–Karp–Rosenberg_conjecture-0.md
category: reference
title: Aanderaa–Karp–Rosenberg conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Aanderaa–Karp–Rosenberg_conjecture
date_saved: 2026-05-05T11:01:55.807273+00:00
instance: kb-cron
parent: aanderaa-karp-rosenberg-conjecture-20260505-0
summary:
In theoretical computer science, the Aanderaa–Karp–Rosenberg conjecture (also known as the Aanderaa–Rosenberg conjecture or the evasiveness conjecture) is a group of related conjectures about the number of questions of the form "Is there an edge between vertex u {\displaystyle u} and vertex v {\displaystyle v} ?" that have to be answered to determine whether or not an undirected graph has a particular property such as planarity or bipartiteness. They are named after Stål Aanderaa, Richard M. Karp, and Arnold L. Rosenberg. According to the conjecture, for a wide class of properties, no algorithm can guarantee that it will be able to skip any questions: any algorithm for determining whether the graph has the property, no matter how clever, might need to examine every pair of vertices before it can give its answer. A property satisfying this conjecture is called evasive. More precisely, the Aanderaa–Rosenberg conjecture states that any deterministic algorithm must test at least a constant fraction of all possible pairs of vertices, in the worst case, to determine any non-trivial monotone graph property. In this context, a property is monotone if it remains true when edges are added; for example, planarity is not monotone, but non-planarity is monotone. A


id: aanderaa-karp-rosenberg-conjecture-20260505-1
file: data/en.wikipedia.org/wiki/Aanderaa–Karp–Rosenberg_conjecture-1.md
category: reference
title: Aanderaa–Karp–Rosenberg conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Aanderaa–Karp–Rosenberg_conjecture
date_saved: 2026-05-05T11:01:55.807273+00:00
instance: kb-cron
parent: aanderaa-karp-rosenberg-conjecture-20260505-0
summary:
== Definitions == In the context of this article, all graphs will be simple and undirected, unless stated otherwise. This means that the edges of the graph form a set (and not a multiset) and each edge is a pair of distinct vertices. Graphs are assumed to have an implicit representation in which each vertex has a unique identifier or label and in which it is possible to test the adjacency of any two vertices, but for which adjacency testing is the only allowed primitive operation. Informally, a graph property is a property of a graph that is independent of labeling. More formally, a graph property is a mapping from the class of all graphs to { 0 , 1 } {\displaystyle \{0,1\}} such that isomorphic graphs are mapped to the same value. For example, the property of containing at least one vertex of degree two is a graph property, but the property that the first vertex has degree two is not, because it depends on the labeling of the graph (in particular, it depends on which vertex is the "first" vertex). A graph property is called non-trivial if it does not assign the same value to all graphs. For


id: aanderaa-karp-rosenberg-conjecture-20260505-2
file: data/en.wikipedia.org/wiki/Aanderaa–Karp–Rosenberg_conjecture-2.md
category: reference
title: Aanderaa–Karp–Rosenberg conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Aanderaa–Karp–Rosenberg_conjecture
date_saved: 2026-05-05T11:01:55.807273+00:00
instance: kb-cron
parent: aanderaa-karp-rosenberg-conjecture-20260505-0
summary:
== Deterministic query complexity == For deterministic algorithms, Rosenberg (1973) originally conjectured that for all nontrivial graph properties on n {\displaystyle n} vertices, deciding whether a graph possesses this property requires Ω ( n 2 ) {\displaystyle \Omega (n^{2})} The non-triviality condition is clearly required because there are trivial properties like "is this a graph?" which can be answered with no queries at all. The conjecture was disproved by Aanderaa, who exhibited a directed graph property (the property of containing a "sink") which required only O ( n ) {\displaystyle O(n)} queries to test. A sink, in a directed graph, is a vertex of indegree n − 1 {\displaystyle n-1} and outdegree zero. The existence of a sink can be tested with less than 3 n {\displaystyle 3n} queries. An undirected graph property which can also be tested with O ( n ) {\displaystyle O(n)} queries is the property of being a scorpion graph, first described in Best, van Emde Boas & Lenstra (1974). A scorpion graph is a graph containing a three-vertex path, such that one endpoint of the path is connected to all remaining vertices, while the other two path vertices have no incident edges other than the


id: aanderaa-karp-rosenberg-conjecture-20260505-3
file: data/en.wikipedia.org/wiki/Aanderaa–Karp–Rosenberg_conjecture-3.md
category: reference
title: Aanderaa–Karp–Rosenberg conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Aanderaa–Karp–Rosenberg_conjecture
date_saved: 2026-05-05T11:01:55.807273+00:00
instance: kb-cron
parent: aanderaa-karp-rosenberg-conjecture-20260505-0
summary:
== Quantum query complexity == For bounded-error quantum query complexity, the best known lower bound is Ω ( n 2 / 3 ( log ⁡ n ) 1 / 6 ) {\displaystyle \Omega {\bigl (}n^{2/3}(\log n)^{1/6}{\bigr )}} as observed by Andrew Yao. It is obtained by combining the randomized lower bound with the quantum adversary method. The best possible lower bound one could hope to achieve is Ω ( n ) {\displaystyle \Omega (n)} , unlike the classical case, due to Grover's algorithm which gives an O ( n ) {\displaystyle O(n)} -query algorithm for testing the monotone property of non-emptiness. Similar to the deterministic and randomized case, there are some properties which are known to have an Ω ( n ) {\displaystyle \Omega (n)} lower bound, for example non-emptiness (which follows from the optimality of Grover's algorithm) and the property of containing a triangle. There are some graph properties which are known to have an Ω ( n 3 / 2 ) {\displaystyle \Omega (n^{3/2})} lower bound, and even some properties with an Ω ( n 2 ) {\displaystyle \Omega (n^{2})} lower bound. For example, the monotone property of nonplanarity requires Θ ( n 3 / 2 ) {\displaystyle


id: artificial-general-intelligence-20260505-0
file: data/en.wikipedia.org/wiki/Artificial_general_intelligence-0.md
category: reference
title: Artificial general intelligence
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Artificial_general_intelligence
date_saved: 2026-05-05T11:01:56.982213+00:00
instance: kb-cron
parent: artificial-general-intelligence-20260505-0
summary:
Artificial general intelligence (AGI) is a hypothetical type of artificial intelligence that matches or surpasses human capabilities across virtually all cognitive tasks. Beyond AGI, artificial superintelligence (ASI) would outperform the best human abilities across every domain by a wide margin. Unlike artificial narrow intelligence (ANI), whose competence is confined to well‑defined tasks, an AGI system can generalise knowledge, transfer skills between domains, and solve novel problems without task‑specific reprogramming. Creating AGI is a stated goal of technology companies such as OpenAI, Google, xAI, and Meta. A 2020 survey identified 72 active AGI research and development projects across 37 countries. AGI is a common topic in science fiction and futures studies. Contention exists over whether AGI represents an existential risk. Some AI experts and industry figures have stated that mitigating the risk of human extinction posed by AGI should be a global priority. Others find the development of AGI to be in too remote a stage to present such a risk. == Terminology == AGI is also known as strong AI, full AI, human-level AI, human-level intelligent AI, or general intelligent action. The term "artificial general intelligence" was used in 1997 by Mark Gubrud in a discussion of the implications of


id: artificial-general-intelligence-20260505-1
file: data/en.wikipedia.org/wiki/Artificial_general_intelligence-1.md
category: reference
title: Artificial general intelligence
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Artificial_general_intelligence
date_saved: 2026-05-05T11:01:56.982213+00:00
instance: kb-cron
parent: artificial-general-intelligence-20260505-0
summary:
==== Ikea test ==== The "Ikea test", also known as the Flat Pack Furniture Test, involves an AI controls a robot which attempts to assemble a Ikea flat-pack furniture product after having been shown the parts and instructions. As early as 2013, MIT's IkeaBot demonstrated fully autonomous multi-robot assembly of an IKEA Lack table in ten minutes, with no human intervention and no pre-programmed assembly instructions. The robots inferred the assembly sequence from the geometry of the parts alone. In December 2025, MIT researchers demonstrated a "speech-to-reality" system combining large language models with vision-language models and robotic assembly: a user says "I want a simple stool" and a robotic arm constructs the furniture from modular components within five minutes, using generative AI to reason about geometry, function, and assembly sequence from natural language alone. The FurnitureBench benchmark, published in the International Journal of Robotics Research in 2025, now provides a standardised real-world furniture assembly benchmark with over 200 hours of demonstration data for training and evaluating autonomous assembly systems. ==== Coffee test ==== Steve Wozniak proposed a test where a machine is required to enter an average American home and figure out how to make coffee. It must find the


id: artificial-general-intelligence-20260505-2
file: data/en.wikipedia.org/wiki/Artificial_general_intelligence-2.md
category: reference
title: Artificial general intelligence
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Artificial_general_intelligence
date_saved: 2026-05-05T11:01:56.982213+00:00
instance: kb-cron
parent: artificial-general-intelligence-20260505-0
summary:
At the turn of the century, many mainstream AI researchers hoped that strong AI could be developed by combining programs that solve various sub-problems. Hans Moravec wrote in 1988: I am confident that this bottom-up route to artificial intelligence will one day meet the traditional top-down route more than halfway, ready to provide the real-world competence and the commonsense knowledge that has been so frustratingly elusive in reasoning programs. Fully intelligent machines will result when the metaphorical golden spike is driven, uniting the two efforts. However, even at the time, this was disputed. For example, Stevan Harnad of Princeton University concluded his 1990 paper on the symbol grounding hypothesis by stating: The expectation has often been voiced that "top-down" (symbolic) approaches to modeling cognition will somehow meet "bottom-up" (sensory) approaches somewhere in between. If the grounding considerations in this paper are valid, then this expectation is hopelessly modular and there is really only one viable route from sense to symbols: from the ground up. A free-floating symbolic level like the software level of a computer will never be reached by this route (or vice versa) – nor is it clear why we should even try to reach such a level,


id: artificial-general-intelligence-20260505-3
file: data/en.wikipedia.org/wiki/Artificial_general_intelligence-3.md
category: reference
title: Artificial general intelligence
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Artificial_general_intelligence
date_saved: 2026-05-05T11:01:56.982213+00:00
instance: kb-cron
parent: artificial-general-intelligence-20260505-0
summary:
Progress in artificial intelligence has historically gone through periods of rapid progress separated by periods when progress appeared to stop. Ending each hiatus were fundamental advances in hardware, software or both to create space for further progress. For example, the computer hardware available in the twentieth century was not sufficient to implement deep learning, which requires large numbers of GPU-enabled CPUs. In the introduction to his 2006 book, Goertzel says that estimates of the time needed before a truly flexible AGI is built vary from 10 years to over a century. As of 2007, the consensus in the AGI research community seemed to be that the timeline discussed by Ray Kurzweil in 2005 in The Singularity is Near (i.e. between 2015 and 2045) was plausible. Mainstream AI researchers have given a wide range of opinions on whether progress will be this rapid. A 2012 meta-analysis of 95 such opinions found a bias towards predicting that the onset of AGI would occur within 16–26 years for modern and historical predictions alike. That paper has been criticized for how it categorized opinions as expert or non-expert. In 2012, Alex Krizhevsky, Ilya Sutskever, and Geoffrey Hinton developed a neural network called AlexNet, which


id: artificial-general-intelligence-20260505-4
file: data/en.wikipedia.org/wiki/Artificial_general_intelligence-4.md
category: reference
title: Artificial general intelligence
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Artificial_general_intelligence
date_saved: 2026-05-05T11:01:56.982213+00:00
instance: kb-cron
parent: artificial-general-intelligence-20260505-0
summary:
=== Early estimates === For low-level brain simulation, a very powerful cluster of computers or GPUs would be required, given the enormous quantity of synapses within the human brain. Each of the 1011 (one hundred billion) neurons has on average 7,000 synaptic connections (synapses) to other neurons. The brain of a three-year-old child has about 1015 synapses (1 quadrillion). This number declines with age, stabilizing by adulthood. Estimates vary for an adult, ranging from 1014 to 5×1014 synapses (100 to 500 trillion). An estimate of the brain's processing power, based on a simple switch model for neuron activity, is around 1014 (100 trillion) synaptic updates per second (SUPS). In 1997, Kurzweil looked at various estimates for the hardware required to equal the human brain and adopted a figure of 1016 computations per second. (For comparison, if a "computation" was equivalent to one "floating-point operation" – a measure used to rate current supercomputers – then 1016 "computations" would be equivalent to 10 petaFLOPS, achieved in 2011, while 1018 was achieved in 2022.) He used this figure to predict that the necessary hardware would be available sometime between 2015 and 2025, if the exponential growth in computer power at the time of


id: artificial-general-intelligence-20260505-5
file: data/en.wikipedia.org/wiki/Artificial_general_intelligence-5.md
category: reference
title: Artificial general intelligence
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Artificial_general_intelligence
date_saved: 2026-05-05T11:01:56.982213+00:00
instance: kb-cron
parent: artificial-general-intelligence-20260505-0
summary:
Sentience (or "phenomenal consciousness"): The ability to "feel" perceptions or emotions subjectively, as opposed to the ability to reason about perceptions. Some philosophers, such as David Chalmers, use the term "consciousness" to refer exclusively to phenomenal consciousness, which is roughly equivalent to sentience. Determining why and how subjective experience arises is known as the hard problem of consciousness. Thomas Nagel explained in 1974 that it "feels like" something to be conscious. If we are not conscious, then it doesn't feel like anything. Nagel uses the example of a bat: we can sensibly ask "what does it feel like to be a bat?" However, we are unlikely to ask "what does it feel like to be a toaster?" Nagel concludes that a bat appears to be conscious (i.e., has consciousness) but a toaster does not. In 2022, a Google engineer claimed that the company's AI chatbot, LaMDA, had achieved sentience, though this claim was widely disputed by other experts. Self-awareness: To have conscious awareness of oneself as a separate individual, especially to be consciously aware of one's own thoughts. This is opposed to simply being the "subject of one's thought"—an operating system or debugger can be "aware of itself" (that is,


id: artificial-general-intelligence-20260505-6
file: data/en.wikipedia.org/wiki/Artificial_general_intelligence-6.md
category: reference
title: Artificial general intelligence
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Artificial_general_intelligence
date_saved: 2026-05-05T11:01:56.982213+00:00
instance: kb-cron
parent: artificial-general-intelligence-20260505-0
summary:
=== Revitalising environmental conservation and biodiversity === AGI could significantly contribute to preserving the natural environment and protecting endangered species. By analyzing satellite imagery, climate data, and wildlife patterns, AGI systems could identify environmental threats earlier and recommend targeted conservation strategies. AGI could help optimize land use, monitor illegal activities like poaching or deforestation in real-time, and support global efforts to restore ecosystems. Advanced predictive models developed by AGI could also assist in reversing biodiversity loss, ensuring the survival of critical species and maintaining ecological balance. === Enhancing space exploration and colonization === AGI could revolutionize humanity's ability to explore and settle beyond Earth. With its advanced problem-solving skills, AGI could autonomously manage complex space missions, including navigation, resource management, and emergency response. It could accelerate the design of life support systems, habitats, and spacecraft optimized for extraterrestrial environments. Furthermore, AGI could support efforts to colonize planets like Mars by simulating survival scenarios and helping humans adapt to new worlds, expanding the possibilities for interplanetary civilization. == Risks == === Existential risks === AGI may represent multiple types of existential risk, which are risks that threaten "the premature extinction of Earth-originating intelligent life or the permanent and drastic destruction of


id: artificial-general-intelligence-20260505-7
file: data/en.wikipedia.org/wiki/Artificial_general_intelligence-7.md
category: reference
title: Artificial general intelligence
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Artificial_general_intelligence
date_saved: 2026-05-05T11:01:56.982213+00:00
instance: kb-cron
parent: artificial-general-intelligence-20260505-0
summary:
=== Mass unemployment === Researchers from OpenAI estimated in 2023 that "80% of the U.S. workforce could have at least 10% of their work tasks affected by the introduction of LLMs, while around 19% of workers may see at least 50% of their tasks impacted". They consider office workers to be the most exposed, for example mathematicians, accountants or web designers. AGI could have a better autonomy, ability to make decisions, to interface with other computer tools, but also to control robotized bodies. A common belief among top AI company insiders is that most workers will face technological unemployment from AGI, starting with white-collar jobs and, as robotics improves, extending to blue-collar jobs. Critics of the idea argue that AGI will complement rather than replace humans, and that automation displaces work in the short term but not in the long term. According to Stephen Hawking, the outcome of automation on the quality of life will depend on how the wealth will be redistributed: Everyone can enjoy a life of luxurious leisure if the machine-produced wealth is shared, or most people can end up miserably poor if the machine-owners successfully lobby against wealth redistribution. So far, the trend seems to be


id: artificial-intelligence-content-detection-20260505-0
file: data/en.wikipedia.org/wiki/Artificial_intelligence_content_detection-0.md
category: reference
title: Artificial intelligence content detection
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Artificial_intelligence_content_detection
date_saved: 2026-05-05T11:01:58.213819+00:00
instance: kb-cron
parent: artificial-intelligence-content-detection-20260505-0
summary:
Artificial intelligence detection software aims to determine whether some content (text, image, video or audio) was generated using artificial intelligence (AI). This software is often unreliable. == Accuracy issues == Many AI detection tools have been shown to be unreliable in detecting AI-generated text. In a 2023 study conducted by Weber-Wulff et al., researchers evaluated 14 detection tools including Turnitin and GPTZero and found that "all scored below 80% of accuracy and only 5 over 70%." They also found that these tools tend to have a bias for classifying texts more as human than as AI, and that accuracy of these tools worsens upon paraphrasing. === False positives === In AI content detection, a false positive is when human-written work is incorrectly flagged as AI-written. Many AI detection platforms claim to have a minimal level of false positives, with Turnitin claiming a less than 1% false positive rate. However, later research by The Washington Post produced much higher rates of 50%, though they used a smaller sample size. False positives in an academic setting frequently lead to accusations of academic misconduct, which can have serious consequences for a student's academic record. Additionally, studies have shown evidence that many AI detection


id: artificial-intelligence-content-detection-20260505-1
file: data/en.wikipedia.org/wiki/Artificial_intelligence_content_detection-1.md
category: reference
title: Artificial intelligence content detection
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Artificial_intelligence_content_detection
date_saved: 2026-05-05T11:01:58.213819+00:00
instance: kb-cron
parent: artificial-intelligence-content-detection-20260505-0
summary:
== Anti text detection == There is software available designed to bypass AI text detection. In practice, evasion may not require specialized bypass tools. Paraphrasing, style editing, and removal of repeated discourse markers can substantially reduce the effectiveness of detectors that rely on recognizable surface patterns. A study published in August 2023 analyzed 20 abstracts from papers published in the Eye Journal, which were then paraphrased using GPT-4.0. The AI-paraphrased abstracts were examined for plagiarism using QueText and for AI-generated content using Originality.AI. The texts were then re-processed through an adversarial software called Undetectable.ai in order to reduce the AI-detection scores. The study found that the AI detection tool, Originality.AI, identified text generated by GPT-4 with a mean accuracy of 91.3%. However, after reprocessing by Undetectable.ai, the detection accuracy of Originality.ai dropped to a mean accuracy of 27.8%. Some experts also believe that techniques like digital watermarking are ineffective because they can be removed or added to trigger false positives. "A Watermark for Large Language Models" paper by Kirchenbauer et al. (2023) also addresses potential vulnerabilities of watermarking techniques. The authors outline a range of adversarial tactics, including text insertion, deletion, and substitution attacks, that could be used to bypass


id: artificial-wisdom-20260505-0
file: data/en.wikipedia.org/wiki/Artificial_wisdom-0.md
category: reference
title: Artificial wisdom
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Artificial_wisdom
date_saved: 2026-05-05T11:01:59.482721+00:00
instance: kb-cron
parent: artificial-wisdom-20260505-0
summary:
Artificial wisdom (AW) is an artificial intelligence (AI) system which is able to display the human traits of wisdom and morals while being able to contemplate its own “endpoint”. Artificial wisdom can be described as artificial intelligence reaching the top-level of decision-making when confronted with the most complex challenging situations. The term artificial wisdom is used when the "intelligence" is based on more than by chance collecting and interpreting data, but by design enriched with smart and conscience strategies that wise people would use. == Overview == The goal of artificial wisdom is to create artificial intelligence that can successfully replicate the “uniquely human trait[s]” of having wisdom and morals as closely as possible. Thus, artificial wisdom, must “incorporate [the] ethical and moral considerations” of the data it uses. There are also many significant ethical and legal implications of AW which are compounded by the rapid advances in AI and related technologies alongside the lack of the development of ethics, guidelines, and regulations without the oversight of any kind of overarching advisory board. Additionally, there are challenges in how to develop, test, and implement AW in real world scenarios. Existing tests do not test the internal thought process by which


id: berman-hartmanis-conjecture-20260505-0
file: data/en.wikipedia.org/wiki/Berman–Hartmanis_conjecture-0.md
category: reference
title: Berman–Hartmanis conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Berman–Hartmanis_conjecture
date_saved: 2026-05-05T11:02:01.730158+00:00
instance: kb-cron
parent: berman-hartmanis-conjecture-20260505-0
summary:
In structural complexity theory, the Berman–Hartmanis conjecture is an unsolved conjecture named after Leonard C. Berman and Juris Hartmanis. Informally, it states that all NP-complete languages look alike, in the sense that they can be related to each other by polynomial time isomorphisms. == Statements == === Statement using p-isomorphism === An isomorphism between formal languages L1 and L2 is a bijective map f from strings in the alphabet of L1 to strings in the alphabet of L2, with the property that a string x belongs to L1 if and only if f(x) belongs to L2. A polynomial-time isomorphism, or p-isomorphism for short, is an isomorphism f where both f and its inverse function can be computed in an amount of time polynomial in the lengths of their arguments. Berman and Hartmanis conjectured that all NP-complete languages are p-isomorphic to each other. === Statement using paddable languages === A formal language L is paddable if there is a polynomial time function f(x,y), with a polynomial time inverse, such that for every x and every y, the string x belongs to L if and only if f(x,y) belongs to L. That is, it is possible to pad the input x with


id: computational-complexity-of-mathematical-operations-20260505-0
file: data/en.wikipedia.org/wiki/Computational_complexity_of_mathematical_operations-0.md
category: reference
title: Computational complexity of mathematical operations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Computational_complexity_of_mathematical_operations
date_saved: 2026-05-05T11:02:05.185210+00:00
instance: kb-cron
parent: computational-complexity-of-mathematical-operations-20260505-0
summary:
The following tables list the computational complexity of various algorithms for common mathematical operations. Here, complexity refers to the time complexity of performing computations on a multitape Turing machine. See big O notation for an explanation of the notation used. Note: Due to the variety of multiplication algorithms, M ( n ) {\displaystyle M(n)} below stands in for the complexity of the chosen multiplication algorithm. == Arithmetic functions == This table lists the complexity of mathematical operations on integers. On stronger computational models, specifically a pointer machine and consequently also a unit-cost random-access machine it is possible to multiply two n-bit numbers in time O(n). == Algebraic functions == Here we consider operations over polynomials and n denotes their degree; for the coefficients we use a unit-cost model, ignoring the number of bits in a number. In practice this means that we assume them to be machine integers. For this section M ( n ) {\displaystyle M(n)} indicates the time needed for multiplying two polynomials of degree at most n {\displaystyle n} . == Special functions == Many of the methods in this section are given in Borwein & Borwein. === Elementary functions === The elementary functions are constructed by


id: computational-complexity-of-matrix-multiplication-20260505-0
file: data/en.wikipedia.org/wiki/Computational_complexity_of_matrix_multiplication-0.md
category: reference
title: Computational complexity of matrix multiplication
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Computational_complexity_of_matrix_multiplication
date_saved: 2026-05-05T11:02:06.334401+00:00
instance: kb-cron
parent: computational-complexity-of-matrix-multiplication-20260505-0
summary:
In theoretical computer science, the computational complexity of matrix multiplication dictates how quickly the operation of matrix multiplication can be performed. Matrix multiplication algorithms are a central subroutine in theoretical and numerical algorithms for numerical linear algebra and optimization, so finding the fastest algorithm for matrix multiplication is of major practical relevance. Directly applying the mathematical definition of matrix multiplication gives an algorithm that requires n3 field operations to multiply two n × n matrices over that field (Θ(n3) in big O notation). Surprisingly, algorithms exist that provide better running times than this straightforward "schoolbook algorithm". The first to be discovered was Strassen's algorithm, devised by Volker Strassen in 1969 and often referred to as "fast matrix multiplication". The optimal number of field operations needed to multiply two square n × n matrices up to constant factors is still unknown. This is a major open question in theoretical computer science. As of January 2024, the best bound on the asymptotic complexity of a matrix multiplication algorithm is O(n2.371339). However, this and similar improvements to Strassen are not used in practice, because they are galactic algorithms: the constant coefficient hidden by the big O notation is so large that they are


id: computational-complexity-of-matrix-multiplication-20260505-1
file: data/en.wikipedia.org/wiki/Computational_complexity_of_matrix_multiplication-1.md
category: reference
title: Computational complexity of matrix multiplication
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Computational_complexity_of_matrix_multiplication
date_saved: 2026-05-05T11:02:06.334401+00:00
instance: kb-cron
parent: computational-complexity-of-matrix-multiplication-20260505-0
summary:
=== Group theory reformulation of matrix multiplication algorithms === Henry Cohn, Robert Kleinberg, Balázs Szegedy and Chris Umans put methods such as the Strassen and Coppersmith–Winograd algorithms in an entirely different group-theoretic context, by utilising triples of subsets of finite groups which satisfy a disjointness property called the triple product property (TPP). They also give conjectures that, if true, would imply that there are matrix multiplication algorithms with essentially quadratic complexity. This implies that the optimal exponent of matrix multiplication is 2, which most researchers believe is indeed the case. One such conjecture is that families of wreath products of Abelian groups with symmetric groups realise families of subset triples with a simultaneous version of the TPP. Several of their conjectures have since been disproven by Blasiak, Cohn, Church, Grochow, Naslund, Sawin, and Umans using the Slice Rank method. Further, Alon, Shpilka and Chris Umans have recently shown that some of these conjectures implying fast matrix multiplication are incompatible with another plausible conjecture, the sunflower conjecture, which in turn is related to the cap set problem. === Lower bounds for ω === There is a trivial lower bound of ⁠ ω ≥ 2 {\displaystyle \omega \geq 2} ⁠. Since any


id: computational-complexity-of-matrix-multiplication-20260505-2
file: data/en.wikipedia.org/wiki/Computational_complexity_of_matrix_multiplication-2.md
category: reference
title: Computational complexity of matrix multiplication
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Computational_complexity_of_matrix_multiplication
date_saved: 2026-05-05T11:02:06.334401+00:00
instance: kb-cron
parent: computational-complexity-of-matrix-multiplication-20260505-0
summary:
provided that A and D − C A − 1 B {\displaystyle {D}-{CA}^{-1}{B}} are invertible. Thus, the inverse of a 2n×2n matrix may be computed with two inversions, six multiplications and four additions or additive inverses of n×n matrices. It follows that, denoting respectively by I(n), M(n) and A(n) = n2 the number of operations needed for inverting, multiplying and adding n×n matrices, one has I ( 2 n ) ≤ 2 I ( n ) + 6 M ( n ) + 4 A ( n ) . {\displaystyle I(2n)\leq 2I(n)+6M(n)+4A(n).} If n = 2 k , {\displaystyle n=2^{k},} one may apply this formula recursively: I ( 2 k ) ≤ 2 I ( 2 k − 1 ) + 6 M ( 2 k − 1 ) + 4 A ( 2 k − 1 ) ≤ 2 2 I ( 2 k − 2 ) + 6 ( M ( 2 k − 1 ) + 2 M ( 2 k − 2 ) ) + 4 ( A ( 2 k − 1 ) + 2 A ( 2 k − 2 ) ) ⋮ {\displaystyle {\begin{aligned}I(2^{k})&\leq 2I(2^{k-1})+6M(2^{k-1})+4A(2^{k-1})\\&\leq 2^{2}I(2^{k-2})+6(M(2^{k-1})+2M(2^{k-2}))+4(A(2^{k-1})+2A(2^{k-2}))\\&\,\,\,\vdots \end{aligned}}} If M ( n ) ≤


id: discrete-logarithm-20260505-0
file: data/en.wikipedia.org/wiki/Discrete_logarithm-0.md
category: reference
title: Discrete logarithm
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Discrete_logarithm
date_saved: 2026-05-05T11:02:07.615506+00:00
instance: kb-cron
parent: discrete-logarithm-20260505-0
summary:
In mathematics, for given real numbers a {\displaystyle a} and b {\displaystyle b} , the logarithm log b ⁡ ( a ) {\displaystyle \log _{b}(a)} is a number x {\displaystyle x} such that b x = a {\displaystyle b^{x}=a} . The discrete logarithm generalizes this concept to a cyclic group. A simple example is the group of integers modulo a prime number (such as 5) under modular multiplication of nonzero elements. For instance, take b = 2 {\displaystyle b=2} in the multiplicative group modulo 5, whose elements are 1 , 2 , 3 , 4 {\displaystyle {1,2,3,4}} . Then: 2 1 = 2 , 2 2 = 4 , 2 3 = 8 ≡ 3 ( mod 5 ) , 2 4 = 16 ≡ 1 ( mod 5 ) . {\displaystyle 2^{1}=2,\quad 2^{2}=4,\quad 2^{3}=8\equiv 3{\pmod {5}},\quad 2^{4}=16\equiv 1{\pmod {5}}.} The powers of 2 modulo 5 cycle through all nonzero elements, so discrete logarithms exist and are given by: log 2 ⁡ 1 = 4 , log 2 ⁡ 2 = 1 , log 2 ⁡ 3 = 3 , log 2 ⁡ 4 = 2. {\displaystyle \log _{2}1=4,\quad \log _{2}2=1,\quad \log _{2}3=3,\quad \log _{2}4=2.} More generally, in any group


id: discrete-logarithm-20260505-1
file: data/en.wikipedia.org/wiki/Discrete_logarithm-1.md
category: reference
title: Discrete logarithm
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Discrete_logarithm
date_saved: 2026-05-05T11:02:07.615506+00:00
instance: kb-cron
parent: discrete-logarithm-20260505-0
summary:
=== Modular arithmetic === One of the simplest settings for discrete logarithms is the group Zp×. This is the group of multiplication modulo the prime p {\displaystyle p} . Its elements are non-zero congruence classes modulo p {\displaystyle p} , and the group product of two elements may be obtained by ordinary integer multiplication of the elements followed by reduction modulo p {\displaystyle p} . The k {\displaystyle k} th power of one of the numbers in this group may be computed by finding its ' k {\displaystyle k} th power as an integer and then finding the remainder after division by p {\displaystyle p} . When the numbers involved are large, it is more efficient to reduce modulo p {\displaystyle p} multiple times during the computation. Regardless of the specific algorithm used, this operation is called modular exponentiation. For example, consider Z17×. To compute 3 4 {\displaystyle 3^{4}} in this group, compute 3 4 = 81 {\displaystyle 3^{4}=81} , and then divide 81 {\displaystyle 81} by 17 {\displaystyle 17} , obtaining a remainder of 13 {\displaystyle 13} . Thus 3 4 = 13 {\displaystyle 3^{4}=13} in the group Z17×. The discrete logarithm is just the inverse operation. For example,


id: discrete-logarithm-20260505-2
file: data/en.wikipedia.org/wiki/Discrete_logarithm-2.md
category: reference
title: Discrete logarithm
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Discrete_logarithm
date_saved: 2026-05-05T11:02:07.615506+00:00
instance: kb-cron
parent: discrete-logarithm-20260505-0
summary:
Baby-step giant-step Function field sieve Index calculus algorithm Number field sieve Pohlig–Hellman algorithm Pollard's rho algorithm for logarithms Pollard's kangaroo algorithm (aka Pollard's lambda algorithm) There is an efficient quantum algorithm due to Peter Shor. Efficient classical algorithms also exist in certain special cases. For example, in the group of the integers modulo p {\displaystyle p} under addition, the power b k {\displaystyle b^{k}} becomes a product b ⋅ k {\displaystyle b\cdot k} , and equality means congruence modulo p {\displaystyle p} in the integers. The extended Euclidean algorithm finds k {\displaystyle k} quickly. With Diffie–Hellman, a cyclic group modulo a prime p {\displaystyle p} is used, allowing an efficient computation of the discrete logarithm with Pohlig–Hellman if the order of the group (being p − 1 {\displaystyle p-1} ) is sufficiently smooth, i.e. has no large prime factors. == Comparison with integer factorization == While computing discrete logarithms and integer factorization are distinct problems, they share some properties: both are special cases of the hidden subgroup problem for finite abelian groups, both problems seem to be difficult (no efficient algorithms are known for non-quantum computers), for both problems efficient algorithms on quantum computers are known, algorithms from one problem


id: expression-problem-20260505-0
file: data/en.wikipedia.org/wiki/Expression_problem-0.md
category: reference
title: Expression problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Expression_problem
date_saved: 2026-05-05T11:02:10.988150+00:00
instance: kb-cron
parent: expression-problem-20260505-0
summary:
The expression problem is a problem in programming languages that concerns the extensibility and modularity of statically typed data abstractions. The goal is to define a data abstraction that is extensible both in its representations and its behaviors, where one can add new representations and new behaviors to the data abstraction, without recompiling existing code, and while retaining static type safety (e.g., no casts). The statement of the problem exposes deficiencies in programming paradigms and programming languages. Philip Wadler, one of the co-authors of Haskell, has originated the term. == History == Philip Wadler formulated the challenge and named it "The Expression Problem" in response to a discussion with Rice University's Programming Languages Team (PLT). He also cited three sources that defined the context for his challenge: The problem was first observed by John Reynolds in 1975. Reynolds discussed two forms of Data Abstraction: User-defined Types, which are now known as Abstract Data Types (ADTs) (not to be confused with Algebraic Data Types), and Procedural Data Structures, which are now understood as a primitive form of Objects with only one method. He argued that they are complementary, in that User-defined Types could be extended with new behaviors, and Procedural Data


id: generalized-star-height-problem-20260505-0
file: data/en.wikipedia.org/wiki/Generalized_star-height_problem-0.md
category: reference
title: Generalized star-height problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Generalized_star-height_problem
date_saved: 2026-05-05T11:02:12.169658+00:00
instance: kb-cron
parent: generalized-star-height-problem-20260505-0
summary:
The generalized star-height problem in formal language theory is the open question whether all regular languages can be expressed using generalized regular expressions with a limited nesting depth of Kleene stars. Here, generalized regular expressions are defined like regular expressions, but they have a built-in complement operator. For a regular language, its generalized star height is defined as the minimum nesting depth of Kleene stars needed in order to describe the language by means of a generalized regular expression, hence the name of the problem. More specifically, it is an open question whether a nesting depth of more than 1 is required, and if so, whether there is an algorithm to determine the minimum required star height. Regular languages of generalized star-height 0 are also known as star-free languages. The theorem of Schützenberger provides an algebraic characterization of star-free languages by means of aperiodic syntactic monoids. In particular star-free languages are a proper decidable subclass of regular languages. == See also == Eggan's theorem and Generalized star height sections of the Star height article Star height problem == References == Janusz A. Brzozowski (1980). "Open problems about regular languages". In Ronald V. Book (ed.). Formal Language Theory: Perspectives and Open


id: gilbert-pollak-conjecture-20260505-0
file: data/en.wikipedia.org/wiki/Gilbert–Pollak_conjecture-0.md
category: reference
title: Gilbert–Pollak conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Gilbert–Pollak_conjecture
date_saved: 2026-05-05T11:02:13.415449+00:00
instance: kb-cron
parent: gilbert-pollak-conjecture-20260505-0
summary:
In mathematics, the Gilbert–Pollak conjecture is an unproven conjecture on the ratio of lengths of Steiner trees and Euclidean minimum spanning trees for the same point sets in the Euclidean plane. It was proposed by Edgar Gilbert and Henry O. Pollak in 1968. Specifically, the conjecture states that for every finite set of points in the Euclidean plane, the Euclidean minimum spanning tree is no longer than 2 / 3 {\displaystyle 2/{\sqrt {3}}} times the length of the Steiner minimum tree. == Statement == For a set of points in the plane, the shortest network of line segments that connects the points, having only the given points as endpoints, is the Euclidean minimum spanning tree of the set. It may be possible to construct a shorter network by using additional endpoints, not present in the given point set. These additional points are called Steiner points and the shortest network that can be constructed using them is called a Steiner minimum tree. The Steiner ratio is the supremum, over all point sets, of the ratio of lengths of the Euclidean minimum spanning tree to the Steiner minimum tree. Because the Steiner minimum tree is shorter, this ratio is always greater than


id: graph-isomorphism-problem-20260505-0
file: data/en.wikipedia.org/wiki/Graph_isomorphism_problem-0.md
category: reference
title: Graph isomorphism problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Graph_isomorphism_problem
date_saved: 2026-05-05T11:02:14.671887+00:00
instance: kb-cron
parent: graph-isomorphism-problem-20260505-0
summary:
The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. The problem is not known to be solvable in polynomial time nor to be NP-complete, and therefore may be in the computational complexity class NP-intermediate. It is known that the graph isomorphism problem is in the low hierarchy of class NP, which implies that it is not NP-complete unless the polynomial time hierarchy collapses to its second level. At the same time, isomorphism for many special classes of graphs can be solved in polynomial time, and in practice graph isomorphism can often be solved efficiently. This problem is a special case of the subgraph isomorphism problem, which asks whether a given graph G contains a subgraph that is isomorphic to another given graph H; this problem is known to be NP-complete. It is also known to be a special case of the non-abelian hidden subgroup problem over the symmetric group. In the area of image recognition it is known as the exact graph matching problem. == State of the art == In November 2015, László Babai announced a quasi-polynomial time algorithm for all graphs, that is, one with running time 2 O ( (


id: graph-isomorphism-problem-20260505-1
file: data/en.wikipedia.org/wiki/Graph_isomorphism_problem-1.md
category: reference
title: Graph isomorphism problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Graph_isomorphism_problem
date_saved: 2026-05-05T11:02:14.671887+00:00
instance: kb-cron
parent: graph-isomorphism-problem-20260505-0
summary:
digraphs labelled graphs, with the proviso that an isomorphism is not required to preserve the labels, but only the equivalence relation consisting of pairs of vertices with the same label "polarized graphs" (made of a complete graph Km and an empty graph Kn plus some edges connecting the two; their isomorphism must preserve the partition) 2-colored graphs explicitly given finite structures multigraphs hypergraphs finite automata Markov Decision Processes commutative class 3 nilpotent (i.e., xyz = 0 for every elements x, y, z) semigroups finite rank associative algebras over a fixed algebraically closed field with zero squared radical and commutative factor over the radical. context-free grammars normal-form games balanced incomplete block designs Recognizing combinatorial isomorphism of convex polytopes represented by vertex-facet incidences. ==== GI-complete classes of graphs ==== A class of graphs is called GI-complete if recognition of isomorphism for graphs from this subclass is a GI-complete problem. The following classes are GI-complete: connected graphs graphs of diameter 2 and radius 1 directed acyclic graphs regular graphs bipartite graphs without non-trivial strongly regular subgraphs bipartite Eulerian graphs bipartite regular graphs line graphs split graphs chordal graphs regular self-complementary graphs polytopal graphs of general, simple, and simplicial convex polytopes in arbitrary dimensions.


id: integer-factorization-20260505-0
file: data/en.wikipedia.org/wiki/Integer_factorization-0.md
category: reference
title: Integer factorization
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Integer_factorization
date_saved: 2026-05-05T11:02:17.162295+00:00
instance: kb-cron
parent: integer-factorization-20260505-0
summary:
In mathematics, integer factorization is the decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors greater than 1, in which case it is a composite number, or it is not, in which case it is a prime number. For example, 15 is a composite number because 15 = 3 · 5, but 7 is a prime number because it cannot be decomposed in this way. If one of the factors is composite, it can in turn be written as a product of smaller factors, for example 60 = 3 · 20 = 3 · (5 · 4). Continuing this process until every factor is prime is called prime factorization; the result is always unique up to the order of the factors by the prime factorization theorem. To factorize a small integer n using mental or pen-and-paper arithmetic, the simplest method is trial division: checking if the number is divisible by prime numbers 2, 3, 5, and so on, up to the square root of n. For larger numbers, especially when using a computer, various more sophisticated factorization algorithms are more efficient. A prime


id: integer-factorization-20260505-1
file: data/en.wikipedia.org/wiki/Integer_factorization-1.md
category: reference
title: Integer factorization
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Integer_factorization
date_saved: 2026-05-05T11:02:17.162295+00:00
instance: kb-cron
parent: integer-factorization-20260505-0
summary:
For current computers, GNFS is the best published algorithm for large n (more than about 400 bits). For a quantum computer, however, Peter Shor discovered an algorithm in 1994 that solves it in polynomial time. Shor's algorithm takes only O(b3) time and O(b) space on b-bit number inputs. In 2001, Shor's algorithm was implemented for the first time, by using NMR techniques on molecules that provide seven qubits. In order to talk about complexity classes such as P, NP, and co-NP, the problem has to be stated as a decision problem. It is known to be in both NP and co-NP, meaning that both "yes" and "no" answers can be verified in polynomial time. An answer of "yes" can be certified by exhibiting a factorization n = d(⁠n/d⁠) with d ≤ k. An answer of "no" can be certified by exhibiting the factorization of n into distinct primes, all larger than k; one can verify their primality using the AKS primality test, and then multiply them to obtain n. The fundamental theorem of arithmetic guarantees that there is only one possible string of increasing primes that will be accepted, which shows that the problem is in both UP and co-UP.


id: integer-factorization-20260505-2
file: data/en.wikipedia.org/wiki/Integer_factorization-2.md
category: reference
title: Integer factorization
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Integer_factorization
date_saved: 2026-05-05T11:02:17.162295+00:00
instance: kb-cron
parent: integer-factorization-20260505-0
summary:
To obtain an algorithm for factoring any positive integer, it is necessary to add a few steps to this algorithm such as trial division, and the Jacobi sum test. === Expected running time === The algorithm as stated is a probabilistic algorithm as it makes random choices. Its expected running time is at most Ln[⁠1/2⁠, 1+o(1)]. == See also == Aurifeuillean factorization Bach's algorithm – For generation of random factorized numbers Canonical representation of a positive integer Factorization Multiplicative partition – Way to write a number as a product of other numbers p-adic valuation Integer partition – Decomposition of an integer as a sum of positive integers == Notes == == References == Richard Crandall and Carl Pomerance (2001). Prime Numbers: A Computational Perspective. Springer. ISBN 0-387-94777-9. Chapter 5: Exponential Factoring Algorithms, pp. 191–226. Chapter 6: Subexponential Factoring Algorithms, pp. 227–284. Section 7.4: Elliptic curve method, pp. 301–313. Donald Knuth. The Art of Computer Programming, Volume 2: Seminumerical Algorithms, Third Edition. Addison-Wesley, 1997. ISBN 0-201-89684-2. Section 4.5.4: Factoring into Primes, pp. 379–417. Samuel S. Wagstaff Jr. (2013). The Joy of Factoring. Providence, RI: American Mathematical Society. ISBN 978-1-4704-1048-3.. Warren, Henry S. Jr. (2013). Hacker's Delight (2 ed.). Addison Wesley -


id: k-server-problem-20260505-0
file: data/en.wikipedia.org/wiki/K-server_problem-0.md
category: reference
title: K-server problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/K-server_problem
date_saved: 2026-05-05T11:02:18.322111+00:00
instance: kb-cron
parent: k-server-problem-20260505-0
summary:
The k-server problem is a problem of theoretical computer science in the category of online algorithms, one of two abstract problems on metric spaces that are central to the theory of competitive analysis (the other being metrical task systems). In this problem, an online algorithm must control the movement of a set of k servers, represented as points in a metric space, and handle requests that are also in the form of points in the space. As each request arrives, the algorithm must determine which server to move to the requested point. The goal of the algorithm is to keep the total distance all servers move small, relative to the total distance the servers could have moved by an optimal adversary who knows in advance the entire sequence of requests. The problem was first posed by Mark Manasse, Lyle A. McGeoch and Daniel Sleator (1988). The most prominent open question concerning the k-server problem is the so-called k-server conjecture, also posed by Manasse et al. This conjecture states that there is an algorithm for solving the k-server problem in an arbitrary metric space and for any number k of servers that has competitive ratio exactly k. Manasse et al. were


id: log-rank-conjecture-20260505-0
file: data/en.wikipedia.org/wiki/Log-rank_conjecture-0.md
category: reference
title: Log-rank conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Log-rank_conjecture
date_saved: 2026-05-05T11:02:19.507086+00:00
instance: kb-cron
parent: log-rank-conjecture-20260505-0
summary:
In theoretical computer science, the log-rank conjecture states that the deterministic communication complexity of a two-party Boolean function is polynomially related to the logarithm of the rank of its input matrix. Let D ( f ) {\displaystyle D(f)} denote the deterministic communication complexity of a function, and let rank ⁡ ( f ) {\displaystyle \operatorname {rank} (f)} denote the rank of its input matrix M f {\displaystyle M_{f}} (over the reals). Since every protocol using up to c {\displaystyle c} bits partitions M f {\displaystyle M_{f}} into at most 2 c {\displaystyle 2^{c}} monochromatic rectangles, and each of these has rank at most 1, D ( f ) ≥ log 2 ⁡ rank ⁡ ( f ) . {\displaystyle D(f)\geq \log _{2}\operatorname {rank} (f).} The log-rank conjecture states that D ( f ) {\displaystyle D(f)} is also upper-bounded by a polynomial in the log-rank: for some constant C {\displaystyle C} , D ( f ) = O ( ( log ⁡ rank ⁡ ( f ) ) C ) . {\displaystyle D(f)=O((\log \operatorname {rank} (f))^{C}).} Lovett proved the upper bound D ( f ) = O ( rank ⁡ ( f ) log ⁡ rank ⁡ ( f ) )


id: one-way-function-20260505-0
file: data/en.wikipedia.org/wiki/One-way_function-0.md
category: reference
title: One-way function
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/One-way_function
date_saved: 2026-05-05T11:02:24.099877+00:00
instance: kb-cron
parent: one-way-function-20260505-0
summary:
In computer science, a one-way function is a function that is easy to compute on every input, but hard to invert given the image of a random input. Here, "easy" and "hard" are to be understood in the sense of computational complexity theory, specifically the theory of polynomial time problems. This has nothing to do with whether the function is one-to-one; finding any one input with the desired image is considered a successful inversion. (See § Theoretical definition, below.) The existence of such one-way functions is still an open conjecture. Their existence would prove that the complexity classes P and NP are not equal, thus resolving the foremost unsolved question of theoretical computer science. The converse is not known to be true, i.e. the existence of a proof that P ≠ NP would not directly imply the existence of one-way functions. In applied contexts, the terms "easy" and "hard" are usually interpreted relative to some specific computing entity; typically "cheap enough for the legitimate users" and "prohibitively expensive for any malicious agents". One-way functions, in this sense, are fundamental tools for cryptography, personal identification, authentication, and other data security applications. While the existence of one-way functions in this sense is


id: one-way-function-20260505-1
file: data/en.wikipedia.org/wiki/One-way_function-1.md
category: reference
title: One-way function
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/One-way_function
date_saved: 2026-05-05T11:02:24.099877+00:00
instance: kb-cron
parent: one-way-function-20260505-0
summary:
=== Multiplication and factoring === The function f takes as inputs two prime numbers p and q in binary notation and returns their product. This function can be "easily" computed in O(b2) time, where b is the total number of bits of the inputs. Inverting this function requires finding the factors of a given integer N. The best factoring algorithms known run in O ( exp ⁡ 64 9 b ( log ⁡ b ) 2 3 ) {\displaystyle O\left(\exp {\sqrt[{3}]{{\frac {64}{9}}b(\log b)^{2}}}\right)} time, where b is the number of bits needed to represent N. This function can be generalized by allowing p and q to range over a suitable set of semiprimes. Note that f is not one-way for randomly selected integers p, q > 1, since the product will have 2 as a factor with probability 3/4 (because the probability that an arbitrary p is odd is 1/2, and likewise for q, so if they're chosen independently, the probability that both are odd is therefore 1/4; hence the probability that p or q is even, is 1 − 1/4 = 3/4). === The Rabin function (modular squaring) === The Rabin function, or squaring modulo N = p q


id: online-matrix-vector-multiplication-problem-20260505-0
file: data/en.wikipedia.org/wiki/Online_matrix-vector_multiplication_problem-0.md
category: reference
title: Online matrix-vector multiplication problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Online_matrix-vector_multiplication_problem
date_saved: 2026-05-05T11:02:25.285814+00:00
instance: kb-cron
parent: online-matrix-vector-multiplication-problem-20260505-0
summary:
In computational complexity theory, the online matrix-vector multiplication problem (OMv) asks an online algorithm to return, at each round, the product of an n × n {\displaystyle n\times n} matrix and a newly-arrived n {\displaystyle n} -dimensional vector. OMv is conjectured to require roughly cubic time. This conjectured hardness implies lower bounds on the time needed to solve various dynamic problems and is of particular interest in fine-grained complexity. == Definition == In OMv, an algorithm is given an integer n {\displaystyle n} and an n × n {\displaystyle n\times n} Boolean matrix M {\displaystyle M} . The algorithm then runs for n {\displaystyle n} rounds, and at each round i {\displaystyle i} receives an n {\displaystyle n} -dimensional Boolean vector v i {\displaystyle v_{i}} and must return the product M v i {\displaystyle Mv_{i}} (before continuing to round i + 1 {\displaystyle i+1} ). An algorithm is said to solve OMv if, with probability at least 2 / 3 {\displaystyle 2/3} over the randomness of the algorithm, it returns the product M v i {\displaystyle Mv_{i}} at every round i {\displaystyle i} . === Variants of OMv === The online vector-matrix-vector problem (OuMv) is a variant of OMv where


id: separating-words-problem-20260505-0
file: data/en.wikipedia.org/wiki/Separating_words_problem-0.md
category: reference
title: Separating words problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Separating_words_problem
date_saved: 2026-05-05T11:02:29.955919+00:00
instance: kb-cron
parent: separating-words-problem-20260505-0
summary:
In theoretical computer science, the separating words problem is the problem of finding the smallest deterministic finite automaton that behaves differently on two given strings, meaning that it accepts one of the two strings and rejects the other string. It is an open problem how large such an automaton must be, in the worst case, as a function of the length of the input strings. == Example == The two strings 0010 and 1000 may be distinguished from each other by a three-state automaton in which the transitions from the start state go to two different states, both of which are terminal in the sense that subsequent transitions from these two states always return to the same state. The state of this automaton records the first symbol of the input string. If one of the two terminal states is accepting and the other is rejecting, then the automaton will accept only one of the strings 0010 and 1000. However, these two strings cannot be distinguished by any automaton with fewer than three states. == Simplifying assumptions == For proving bounds on this problem, it may be assumed without loss of generality that the inputs are strings over a two-letter alphabet.


id: solving-chess-20260505-0
file: data/en.wikipedia.org/wiki/Solving_chess-0.md
category: reference
title: Solving chess
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Solving_chess
date_saved: 2026-05-05T11:02:31.151898+00:00
instance: kb-cron
parent: solving-chess-20260505-0
summary:
Solving chess consists of finding an optimal strategy for the game of chess; that is, one by which one of the players (White or Black) can always force either a victory or a draw (see solved game). It is also related to more generally solving chess-like games (i.e. combinatorial games of perfect information) such as Capablanca chess and infinite chess. In a weaker sense, solving chess may refer to proving which one of the three possible outcomes (White wins; Black wins; draw) is the result of two perfect players, without necessarily revealing the optimal strategy itself (see indirect proof). No complete solution for chess in either of the two senses is known, nor is it expected that chess will be solved in the near future (if ever). Progress to date is extremely limited; there are tablebases of perfect endgame play with a small number of pieces (up to seven), and some chess variants have been solved at least weakly. Calculated estimates of game-tree complexity and state-space complexity of chess exist which provide a bird's eye view of the computational effort that might be required to solve the game. == Partial results == === Endgame tablebases === Endgame tablebases are computerized


id: solving-chess-20260505-1
file: data/en.wikipedia.org/wiki/Solving_chess-1.md
category: reference
title: Solving chess
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Solving_chess
date_saved: 2026-05-05T11:02:31.151898+00:00
instance: kb-cron
parent: solving-chess-20260505-0
summary:
== Predictions on when or if chess will be solved == In 1950, Shannon calculated, based on a game tree complexity of 10120 and a computer operating at one megahertz (a big stretch at that time: the UNIVAC 1 introduced in 1951 could perform ~2000 operations per second or 2 kilohertz) that could evaluate a terminal node in 1 microsecond would take 1090 years to make its first move. Even allowing for technological advances, solving chess within a practical time frame would therefore seem beyond any conceivable technology. Hans-Joachim Bremermann, a professor of mathematics and biophysics at the University of California at Berkeley, further argued in a 1965 paper that the "speed, memory, and processing capacity of any possible future computer equipment are limited by specific physical barriers: the light barrier, the quantum barrier, and the thermodynamical barrier. These limitations imply, for example, that no computer, however constructed, will ever be able to examine the entire tree of possible move sequences of the game of chess." Nonetheless, Bremermann did not foreclose the possibility that a computer would someday be able to solve chess. He wrote, "In order to have a computer play a perfect or nearly perfect game, it will


id: superintelligence-ban-20260505-0
file: data/en.wikipedia.org/wiki/Superintelligence_ban-0.md
category: reference
title: Superintelligence ban
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Superintelligence_ban
date_saved: 2026-05-05T11:02:32.351937+00:00
instance: kb-cron
parent: superintelligence-ban-20260505-0
summary:
Superintelligence ban refers to proposed legal, ethical, or policy measures intended to restrict or prohibit the development of artificial superintelligence, AI systems that would surpass human cognitive abilities in nearly all domains. The idea arises from concerns that such systems could become uncontrollable, potentially posing existential threats to humanity or causing severe social and economic disruption. == Background == The concept of limiting or banning superintelligence research has roots in early 21st-century debates on artificial general intelligence (AGI) safety. Thinkers such as Nick Bostrom and Eliezer Yudkowsky warned that self-improving AI could rapidly exceed human oversight. As advanced models like large-scale language models and autonomous agents began demonstrating complex reasoning abilities, policymakers and ethicists increasingly discussed the need for legal constraints on the creation of systems capable of recursive self-improvement. In October 2025, the Future of Life Institute published a statement calling for "a prohibition on the development of superintelligence, not lifted before there is broad scientific consensus that it will be done safely and controllably, and strong public buy-in." This statement was signed by various public personalities, such as Richard Branson and Steve Wozniak, and AI experts, such as Yoshua Bengio and Geoffrey Hinton. == Rationale == Supporters of


id: synchronizing-word-20260505-0
file: data/en.wikipedia.org/wiki/Synchronizing_word-0.md
category: reference
title: Synchronizing word
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Synchronizing_word
date_saved: 2026-05-05T11:02:33.627503+00:00
instance: kb-cron
parent: synchronizing-word-20260505-0
summary:
In computer science, more precisely, in the theory of deterministic finite automata (DFA), a synchronizing word or reset sequence is a word in the input alphabet of the DFA that sends any state of the DFA to one and the same state. That is, if an ensemble of copies of the DFA are each started in different states, and all of the copies process the synchronizing word, they will all end up in the same state. Not every DFA has a synchronizing word; for instance, a DFA with two states, one for words of even length and one for words of odd length, can never be synchronized. == Existence == Given a DFA, the problem of determining if it has a synchronizing word can be solved in polynomial time using a theorem due to Ján Černý. A simple approach considers the power set of states of the DFA, and builds a directed graph where nodes belong to the power set, and a directed edge describes the action of the transition function. A path from the node of all states to a singleton state shows the existence of a synchronizing word. This algorithm is exponential in the number of states. A


id: unique-games-conjecture-20260505-0
file: data/en.wikipedia.org/wiki/Unique_games_conjecture-0.md
category: reference
title: Unique games conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Unique_games_conjecture
date_saved: 2026-05-05T11:02:34.810219+00:00
instance: kb-cron
parent: unique-games-conjecture-20260505-0
summary:
In computational complexity theory, the unique games conjecture (often referred to as UGC) is a conjecture made by Subhash Khot in 2002. The conjecture postulates that the problem of determining the approximate value of a certain type of game, known as a unique game, has NP-hard computational complexity. It has broad applications in the theory of hardness of approximation. If the unique games conjecture is true and P ≠ NP, then for many important problems it is not only impossible to get an exact solution in polynomial time (as postulated by the P versus NP problem), but also impossible to get a good polynomial-time approximation. The problems for which such an inapproximability result would hold include constraint satisfaction problems, which crop up in a wide variety of disciplines. The conjecture is unusual in that the academic world seems about evenly divided on whether it is true or not. == Formulations == The unique games conjecture can be stated in a number of equivalent ways. === Unique label cover === The following formulation of the unique games conjecture is often used in hardness of approximation. The conjecture postulates the NP-hardness of the following promise problem known as label cover with unique


id: unique-games-conjecture-20260505-1
file: data/en.wikipedia.org/wiki/Unique_games_conjecture-1.md
category: reference
title: Unique games conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Unique_games_conjecture
date_saved: 2026-05-05T11:02:34.810219+00:00
instance: kb-cron
parent: unique-games-conjecture-20260505-0
summary:
=== Two-prover proof systems === A unique game is a special case of a two-prover one-round (2P1R) game. A two-prover one-round game has two players (also known as provers) and a referee. The referee sends each player a question drawn from a known probability distribution, and the players each have to send an answer. The answers come from a set of fixed size. The game is specified by a predicate that depends on the questions sent to the players and the answers provided by them. The players may decide on a strategy beforehand, although they cannot communicate with each other during the game. The players win if the predicate is satisfied by their questions and their answers. A two-prover one-round game is called a unique game if for every question and every answer by the first player, there is exactly one answer by the second player that results in a win for the players, and vice versa. The value of a game is the maximum winning probability for the players over all strategies. The unique games conjecture states that for every sufficiently small pair of constants ε, δ > 0, there exists a constant k such that the following promise


id: unique-games-conjecture-20260505-2
file: data/en.wikipedia.org/wiki/Unique_games_conjecture-2.md
category: reference
title: Unique games conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Unique_games_conjecture
date_saved: 2026-05-05T11:02:34.810219+00:00
instance: kb-cron
parent: unique-games-conjecture-20260505-0
summary:
== Results == Marek Karpinski and Warren Schudy have constructed linear time approximation schemes for dense instances of unique games problem. In 2008, Prasad Raghavendra has shown that if the unique games conjecture is true, then for every constraint satisfaction problem the best approximation ratio is given by a certain simple semidefinite programming instance, which is in particular polynomial. In 2010, Prasad Raghavendra and David Steurer defined the gap-small-set expansion problem, and conjectured that it is NP-hard. The resulting small set expansion hypothesis implies the unique games conjecture. It has also been used to prove strong hardness of approximation results for finding complete bipartite subgraphs. In 2010, Sanjeev Arora, Boaz Barak and David Steurer found a subexponential time approximation algorithm for the unique games problem. A key ingredient in their result was the spectral algorithm of Alexandra Kolla (see also the earlier manuscript of A. Kolla and Madhur Tulsiani). The latter also re-proved that unique games on expander graphs could be solved in polynomial time, and was one of (if not the) first graph algorithms to take advantage of the full spectrum of a graph rather than just its first two eigenvalues. In 2012, it was shown that distinguishing instances


id: x-y-sorting-20260505-0
file: data/en.wikipedia.org/wiki/X_+_Y_sorting-0.md
category: reference
title: X + Y sorting
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/X_+_Y_sorting
date_saved: 2026-05-05T11:02:36.037820+00:00
instance: kb-cron
parent: x-y-sorting-20260505-0
summary:
In computer science, X + Y sorting is the problem of sorting pairs of numbers by their sums. Applications of the problem include transit fare minimisation, VLSI design, and sparse polynomial multiplication. As with comparison sorting and integer sorting more generally, algorithms for this problem can be based only on comparisons of these sums, or on other operations that work only when the inputs are small integers. It is unknown whether this problem has a comparison-based solution whose running time is asymptotically faster than sorting an unstructured list of equally many items. Therefore, research on the problem has focused on two approaches to settle the question of whether such an improvement is possible: the development of algorithms that improve on unstructured sorting in their number of comparisons rather than in their total running time, and lower bounds for the number of comparisons based on counting cells in subdivisions of high-dimensional spaces. Both approaches are historically tied together, in that the first algorithms that used few comparisons were based on the weakness of the cell-counting lower bounds. == Problem statement and history == The input to the X + Y {\displaystyle X+Y} sorting problem consists of two finite collections of numbers


id: x-y-sorting-20260505-1
file: data/en.wikipedia.org/wiki/X_+_Y_sorting-1.md
category: reference
title: X + Y sorting
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/X_+_Y_sorting
date_saved: 2026-05-05T11:02:36.037820+00:00
instance: kb-cron
parent: x-y-sorting-20260505-0
summary:
( x + x 2 + x 9 ) ( 1 + x 4 + x 9 ) = x + x 2 + x 5 + x 6 + x 9 + x 10 + x 11 + x 13 + x 18 . {\displaystyle {\begin{aligned}(&x+x^{2}+x^{9})(1+x^{4}+x^{9})\\&=x+x^{2}+x^{5}+x^{6}+x^{9}+x^{10}+x^{11}+x^{13}+x^{18}.\\\end{aligned}}} The degrees are always integers, so integer-based algorithms for X + Y {\displaystyle X+Y} sorting may be applied. However, for polynomials whose number of terms is comparable to their degree, FFT-based polynomial multiplication algorithms may be significantly more efficient than term-by-term multiplication. == Number of orderings == A well-known lower bound for unstructured sorting, in the decision tree model, is based on the factorial number of sorted orders that an unstructured list may have. Because each comparison can at best reduce the number of possible orderings by a factor of two, sorting requires a number of comparisons at least equal to the binary logarithm of the factorial, which is n log 2 ⁡ n − O ( n ) {\displaystyle n\log _{2}n-O(n)} . Early work on X + Y {\displaystyle X+Y} sorting followed a similar approach by asking how many different sorted orderings are possible for this problem, and proving that this number


id: x-y-sorting-20260505-2
file: data/en.wikipedia.org/wiki/X_+_Y_sorting-2.md
category: reference
title: X + Y sorting
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/X_+_Y_sorting
date_saved: 2026-05-05T11:02:36.037820+00:00
instance: kb-cron
parent: x-y-sorting-20260505-0
summary:
== Number of comparisons == For the classical comparison sorting problem, the time to sort and the number of comparisons needed to sort are within constant factors of each other. But for X + Y {\displaystyle X+Y} sorting, the number of comparisons is smaller than the best time bound known: Michael Fredman showed in 1976 that X + Y {\displaystyle X+Y} sorting can be done using only O ( n 2 ) {\displaystyle O(n^{2})} comparisons. More generally, he showed that any set of N {\displaystyle N} elements, whose sorted ordering has already been restricted to a family Γ {\displaystyle \Gamma } of orderings, can be sorted using log 2 ⁡ | Γ | + O ( N ) {\displaystyle \log _{2}|\Gamma |+O(N)} comparisons, by a form of binary insertion sort. For the X + Y {\displaystyle X+Y} sorting problem, N = n 2 {\displaystyle N=n^{2}} , and | Γ | = O ( n 8 n ) {\displaystyle |\Gamma |=O(n^{8n})} , so log 2 ⁡ | Γ | = O ( n log ⁡ n ) {\displaystyle \log _{2}|\Gamma |=O(n\log n)} and Fredman's bound implies that only O ( n 2 ) {\displaystyle O(n^{2})} comparisons are needed. However, in Fredman's


id: list-of-unsolved-problems-in-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics-0.md
category: reference
title: List of unsolved problems in mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics
date_saved: 2026-05-05T11:02:37.770898+00:00
instance: kb-cron
parent: list-of-unsolved-problems-in-mathematics-20260505-0
summary:
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, mathematical logic, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations. Some problems belong to more than one discipline and are studied using techniques from different areas. Prizes are often awarded for the solution to a long-standing problem, and some lists of unsolved problems, such as the Millennium Prize Problems, receive considerable attention. This list is a composite of notable unsolved problems mentioned in previously published lists, including but not limited to lists considered authoritative, and the problems listed here vary widely in both difficulty and importance. == Notable lists == Various mathematicians and organizations have published and promoted lists of unsolved mathematical problems. In some cases, the lists have been associated with prizes for the discoverers of solutions. === Millennium Prize Problems === Of the original seven Millennium Prize Problems listed by the Clay Mathematics Institute in 2000, six remain unsolved to date: Birch and Swinnerton-Dyer conjecture Hodge conjecture Navier–Stokes existence and smoothness P versus NP Riemann hypothesis Yang–Mills


id: list-of-unsolved-problems-in-mathematics-20260505-1
file: data/en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics-1.md
category: reference
title: List of unsolved problems in mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics
date_saved: 2026-05-05T11:02:37.770898+00:00
instance: kb-cron
parent: list-of-unsolved-problems-in-mathematics-20260505-0
summary:
Birch–Tate conjecture on the relation between the order of the center of the Steinberg group of the ring of integers of a number field to the field's Dedekind zeta function. Casas-Alvero conjecture: if a polynomial of degree d {\displaystyle d} defined over a field K {\displaystyle K} of characteristic 0 {\displaystyle 0} has a factor in common with its first through d − 1 {\displaystyle d-1} -th derivative, then must f {\displaystyle f} be the d {\displaystyle d} -th power of a linear polynomial? Connes embedding problem in Von Neumann algebra theory Crouzeix's conjecture: the matrix norm of a complex function f {\displaystyle f} applied to a complex matrix A {\displaystyle A} is at most twice the supremum of | f ( z ) | {\displaystyle |f(z)|} over the field of values of A {\displaystyle A} . Determinantal conjecture on the determinant of the sum of two normal matrices. Eilenberg–Ganea conjecture: a group with cohomological dimension 2 also has a 2-dimensional Eilenberg–MacLane space K ( G , 1 ) {\displaystyle K(G,1)} . Farrell–Jones conjecture on whether certain assembly maps are isomorphisms. Bost conjecture: a specific case of the Farrell–Jones conjecture Finite lattice representation problem: is every finite lattice isomorphic to


id: list-of-unsolved-problems-in-mathematics-20260505-2
file: data/en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics-2.md
category: reference
title: List of unsolved problems in mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics
date_saved: 2026-05-05T11:02:37.770898+00:00
instance: kb-cron
parent: list-of-unsolved-problems-in-mathematics-20260505-0
summary:
The Brennan conjecture: estimating the integral powers of the moduli of the derivative of conformal maps into the open unit disk, on certain subsets of C {\displaystyle \mathbb {C} } Fuglede's conjecture on whether nonconvex sets in R {\displaystyle \mathbb {R} } and R 2 {\displaystyle \mathbb {R} ^{2}} are spectral if and only if they tile by translation. Goodman's conjecture on the coefficients of multivalued functions Invariant subspace problem – does every bounded operator on a complex Banach space send some non-trivial closed subspace to itself? Kung–Traub conjecture on the optimal order of a multipoint iteration without memory Lehmer's conjecture on the Mahler measure of non-cyclotomic polynomials The mean value problem: given a complex polynomial f {\displaystyle f} of degree d ≥ 2 {\displaystyle d\geq 2} and a complex number z {\displaystyle z} , is there a critical point c {\displaystyle c} of f {\displaystyle f} such that | f ( z ) − f ( c ) | ≤ | f ′ ( z ) | | z − c | {\displaystyle |f(z)-f(c)|\leq |f'(z)||z-c|} ? The Pompeiu problem on the topology of domains for which some nonzero function has integrals that vanish over every congruent copy Sendov's conjecture:


id: list-of-unsolved-problems-in-mathematics-20260505-3
file: data/en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics-3.md
category: reference
title: List of unsolved problems in mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics
date_saved: 2026-05-05T11:02:37.770898+00:00
instance: kb-cron
parent: list-of-unsolved-problems-in-mathematics-20260505-0
summary:
Sudoku: How many puzzles have exactly one solution? How many puzzles with exactly one solution are minimal? What is the maximum number of givens for a minimal puzzle? Tic-tac-toe variants: Given the width of a tic-tac-toe board, what is the smallest dimension such that X is guaranteed to have a winning strategy? (See also Hales–Jewett theorem and nd game) Chess: What is the outcome of a perfectly played game of chess? (See also first-move advantage in chess) Go: What is the perfect value of Komi? Set: What is the largest possible cap set, as a function of n {\displaystyle n} in the n {\displaystyle n} -dimensional affine space over the three-element field? Are the nim-sequences of all finite octal games eventually periodic? Is the nim-sequence of Grundy's game eventually periodic? ==== Games with imperfect information ==== Rendezvous problem === Geometry === ==== Algebraic geometry ==== Abundance conjecture: if the canonical bundle of a projective variety with Kawamata log terminal singularities is nef, then it is semiample. Bass conjecture on the finite generation of certain algebraic K-groups. Bass–Quillen conjecture relating vector bundles over a regular Noetherian ring and over the polynomial ring A [ t 1 , … , t n


id: list-of-unsolved-problems-in-mathematics-20260505-4
file: data/en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics-4.md
category: reference
title: List of unsolved problems in mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics
date_saved: 2026-05-05T11:02:37.770898+00:00
instance: kb-cron
parent: list-of-unsolved-problems-in-mathematics-20260505-0
summary:
The spherical Bernstein's problem, a generalization of Bernstein's problem Carathéodory conjecture: any convex, closed, and twice-differentiable surface in three-dimensional Euclidean space admits at least two umbilical points. Cartan–Hadamard conjecture: can the classical isoperimetric inequality for subsets of Euclidean space be extended to spaces of nonpositive curvature, known as Cartan–Hadamard manifolds? Chern's conjecture (affine geometry) that the Euler characteristic of a compact affine manifold vanishes. Chern's conjecture for hypersurfaces in spheres, a number of closely related conjectures. Closed curve problem: find (explicit) necessary and sufficient conditions that determine when, given two periodic functions with the same period, the integral curve is closed. The filling area conjecture, that a hemisphere has the minimum area among shortcut-free surfaces in Euclidean space whose boundary forms a closed curve of given length The Hopf conjectures relating the curvature and Euler characteristic of higher-dimensional Riemannian manifolds Osserman conjecture: that every Osserman manifold is either flat or locally isometric to a rank-one symmetric space Yau's conjecture on the first eigenvalue that the first eigenvalue for the Laplace–Beltrami operator on an embedded minimal hypersurface of S n + 1 {\displaystyle S^{n+1}} is n {\displaystyle n} . ==== Discrete geometry ==== The big-line-big-clique conjecture on the existence of either


id: list-of-unsolved-problems-in-mathematics-20260505-5
file: data/en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics-5.md
category: reference
title: List of unsolved problems in mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics
date_saved: 2026-05-05T11:02:37.770898+00:00
instance: kb-cron
parent: list-of-unsolved-problems-in-mathematics-20260505-0
summary:
Falconer's conjecture: sets of Hausdorff dimension greater than d / 2 {\displaystyle d/2} in R d {\displaystyle \mathbb {R} ^{d}} must have a distance set of nonzero Lebesgue measure The values of the Hermite constants for dimensions other than 1–8 and 24 What is the lowest number of faces possible for a holyhedron? Inscribed square problem, also known as Toeplitz' conjecture and the square peg problem – does every Jordan curve have an inscribed square? The Kakeya conjecture – do n {\displaystyle n} -dimensional sets that contain a unit line segment in every direction necessarily have Hausdorff dimension and Minkowski dimension equal to n {\displaystyle n} ? The Kelvin problem on minimum-surface-area partitions of space into equal-volume cells, and the optimality of the Weaire–Phelan structure as a solution to the Kelvin problem Lebesgue's universal covering problem on the minimum-area convex shape in the plane that can cover any shape of diameter one Mahler's conjecture on the product of the volumes of a centrally symmetric convex body and its polar. In Meissner body): Are the two Meissner tetrahedra the minimum-volume three-dimensional shapes of constant width? Moser's worm problem – what is the smallest area of a shape that can cover every


id: list-of-unsolved-problems-in-mathematics-20260505-6
file: data/en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics-6.md
category: reference
title: List of unsolved problems in mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics
date_saved: 2026-05-05T11:02:37.770898+00:00
instance: kb-cron
parent: list-of-unsolved-problems-in-mathematics-20260505-0
summary:
==== Graph drawing and embedding ==== The Albertson conjecture: the crossing number can be lower-bounded by the crossing number of a complete graph with the same chromatic number Conway's thrackle conjecture that thrackles cannot have more edges than vertices The GNRS conjecture on whether minor-closed graph families have ℓ 1 {\displaystyle \ell _{1}} embeddings with bounded distortion Harborth's conjecture: every planar graph can be drawn with integer edge lengths Negami's conjecture on projective-plane embeddings of graphs with planar covers The strong Papadimitriou–Ratajczak conjecture: every polyhedral graph has a convex greedy embedding Turán's brick factory problem – Is there a drawing of any complete bipartite graph with fewer crossings than the number given by Zarankiewicz? Guy's conjecture on the crossing number for complete graphs – Is there a drawing of any complete graph with fewer crossings than the number given by his upper bound? Universal point sets of subquadratic size for planar graphs ==== Restriction of graph parameters ==== Does there exist a conference graph for every number of vertices v > 1 {\displaystyle v>1} where v ≡ 1 mod 4 {\displaystyle v\equiv 1{\bmod {4}}} and v {\displaystyle v} is an odd sum of two squares? Conway's 99-graph problem: does there


id: list-of-unsolved-problems-in-mathematics-20260505-7
file: data/en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics-7.md
category: reference
title: List of unsolved problems in mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics
date_saved: 2026-05-05T11:02:37.770898+00:00
instance: kb-cron
parent: list-of-unsolved-problems-in-mathematics-20260505-0
summary:
==== Miscellaneous graph theory ==== Delta-conjecture (1978): consider a complete graph ( V ; E 1 , … , E d ) {\displaystyle (V;E_{1},\dots ,E_{d})} each edge of which is colored by one of d {\displaystyle d} colors such that there exists a triangle Δ {\displaystyle \Delta } colored in three pairwise distinct colors. Then, in each chromatic component ( V , E i ) , i = 1 , … , d {\displaystyle (V,E_{i}),i=1,\dots ,d} , one can choose a maximal independent vertex-set such that the intersection of the obtained d {\displaystyle d} sets is empty. The imbalance conjecture: If the imbalance for each edge of a graph is at least 1, is the multiset of all edge imbalances always graphic? The implicit graph conjecture on the existence of implicit representations for slowly-growing hereditary families of graphs Ryser's conjecture relating the maximum matching size and minimum transversal size in hypergraphs The second neighborhood problem: does every oriented graph contain a vertex for which there are at least as many other vertices at distance two as at distance one? Sidorenko's conjecture on homomorphism densities of graphs in graphons Teschner's bondage number conjecture: is the bondage number of a graph always


id: list-of-unsolved-problems-in-mathematics-20260505-8
file: data/en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics-8.md
category: reference
title: List of unsolved problems in mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics
date_saved: 2026-05-05T11:02:37.770898+00:00
instance: kb-cron
parent: list-of-unsolved-problems-in-mathematics-20260505-0
summary:
Büchi's problem on sufficiently large sequences of square numbers with constant second difference. Carmichael's totient function conjecture: do all values of Euler's totient function have multiplicity greater than 1 {\displaystyle 1} ? Catalan–Dickson conjecture on aliquot sequences: no aliquot sequences are infinite but non-repeating. Exponent pair conjecture: for all ε > 0 {\displaystyle \varepsilon >0} , is the pair ( ε , 1 / 2 + ε ) {\displaystyle (\varepsilon ,1/2+\varepsilon )} an exponent pair? The Gauss circle problem: how far can the number of integer points in a circle centered at the origin be from the area of the circle? Grimm's conjecture: each element of a set of consecutive composite numbers can be assigned a distinct prime number that divides it. Hall's conjecture: for any ε > 0 {\displaystyle \varepsilon >0} , there is some constant c ( ε ) {\displaystyle c(\varepsilon )} such that either y 2 = x 3 {\displaystyle y^{2}=x^{3}} or | y 2 − x 3 | > c ( ε ) x 1 / 2 − ε {\displaystyle |y^{2}-x^{3}|>c(\varepsilon )x^{1/2-\varepsilon }} . Lehmer's totient problem: if ϕ ( n ) {\displaystyle \phi (n)} divides n − 1 {\displaystyle n-1} , must n {\displaystyle n}


id: list-of-unsolved-problems-in-mathematics-20260505-9
file: data/en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics-9.md
category: reference
title: List of unsolved problems in mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics
date_saved: 2026-05-05T11:02:37.770898+00:00
instance: kb-cron
parent: list-of-unsolved-problems-in-mathematics-20260505-0
summary:
Leopoldt's conjecture: a p-adic analogue of the regulator of an algebraic number field does not vanish. Stark conjectures (including Brumer–Stark conjecture) Characterize all algebraic number fields that have some power basis. ==== Analytic number theory ==== Beilinson's conjectures about special values of L-functions. Do Siegel zeros exist? Find the value of the De Bruijn–Newman constant. Is Selberg class of Dirichlet series equal to class of automorphic L-functions? Hardy–Littlewood zeta function conjectures Keating–Snaith conjecture concerning the asymptotics of an integral involving the Riemann zeta function Hilbert–Pólya conjecture: the nontrivial zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator. Lindelöf hypothesis that for all ε > 0 {\displaystyle \varepsilon >0} , ζ ( 1 / 2 + i t ) = o ( t ε ) {\displaystyle \zeta (1/2+it)=o(t^{\varepsilon })} The density hypothesis for zeroes of the Riemann zeta function. Location of nontrivial zeros of L-functions, especially conjectures concerning their critical lines: Grand Riemann hypothesis: do the nontrivial zeros of all automorphic L-functions lie on the critical line 1 / 2 + i t {\displaystyle 1/2+it} with real t {\displaystyle t} ? Generalized Riemann hypothesis for Selberg class: do the nontrivial zeros of all functions in Selberg class


id: list-of-unsolved-problems-in-mathematics-20260505-10
file: data/en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics-10.md
category: reference
title: List of unsolved problems in mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics
date_saved: 2026-05-05T11:02:37.770898+00:00
instance: kb-cron
parent: list-of-unsolved-problems-in-mathematics-20260505-0
summary:
Littlewood conjecture: for any two real numbers α , β {\displaystyle \alpha ,\beta } , lim inf n → ∞ n ‖ n α ‖ ‖ n β ‖ = 0 {\displaystyle \liminf _{n\rightarrow \infty }n\,\Vert n\alpha \Vert \,\Vert n\beta \Vert =0} , where ‖ x ‖ {\displaystyle \Vert x\Vert } is the distance from x {\displaystyle x} to the nearest integer. Schanuel's conjecture on the transcendence degree of certain field extensions of the rational numbers. In particular: Are π {\displaystyle \pi } and e {\displaystyle e} algebraically independent? Which nontrivial combinations of transcendental numbers (such as e + π , e π , π e , π π , e e {\displaystyle e+\pi ,e\pi ,\pi ^{e},\pi ^{\pi },e^{e}} ) are themselves transcendental? The four exponentials conjecture: the transcendence of at least one of four exponentials of combinations of irrationals Are Euler's constant γ {\displaystyle \gamma } and Catalan's constant G {\displaystyle G} irrational? Are they transcendental? Is Apéry's constant ζ ( 3 ) {\displaystyle \zeta (3)} transcendental? Which transcendental numbers are (exponential) periods? How well can non-quadratic irrational numbers be approximated? What is the irrationality measure of specific (suspected) transcendental numbers such as π {\displaystyle \pi } and γ


id: list-of-unsolved-problems-in-mathematics-20260505-11
file: data/en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics-11.md
category: reference
title: List of unsolved problems in mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics
date_saved: 2026-05-05T11:02:37.770898+00:00
instance: kb-cron
parent: list-of-unsolved-problems-in-mathematics-20260505-0
summary:
Fortune's conjecture that no Fortunate number is composite. The Gaussian moat problem: is it possible to find an infinite sequence of distinct Gaussian prime numbers such that the difference between consecutive numbers in the sequence is bounded? Gillies' conjecture on the distribution of prime divisors of Mersenne numbers. Landau's problems Goldbach conjecture: all even natural numbers greater than 2 {\displaystyle 2} are the sum of two prime numbers. Legendre's conjecture: for every positive integer n {\displaystyle n} , there is a prime between n 2 {\displaystyle n^{2}} and ( n + 1 ) 2 {\displaystyle (n+1)^{2}} . Twin prime conjecture: there are infinitely many twin primes. Are there infinitely many primes of the form n 2 + 1 {\displaystyle n^{2}+1} ? Problems associated to Linnik's theorem New Mersenne conjecture: for any odd natural number p {\displaystyle p} , if any two of the three conditions p = 2 k ± 1 {\displaystyle p=2^{k}\pm 1} or p = 4 k ± 3 {\displaystyle p=4^{k}\pm 3} , 2 p − 1 {\displaystyle 2^{p}-1} is prime, and ( 2 p + 1 ) / 3 {\displaystyle (2^{p}+1)/3} is prime are true, then the third condition is also true. Polignac's conjecture: for all positive


id: list-of-unsolved-problems-in-mathematics-20260505-12
file: data/en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics-12.md
category: reference
title: List of unsolved problems in mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics
date_saved: 2026-05-05T11:02:37.770898+00:00
instance: kb-cron
parent: list-of-unsolved-problems-in-mathematics-20260505-0
summary:
Does the partition principle (PP) imply the axiom of choice (AC)? (Woodin) Does the generalized continuum hypothesis below a strongly compact cardinal imply the generalized continuum hypothesis everywhere? Does the generalized continuum hypothesis entail ♢ ( E cf ⁡ ( λ ) λ + ) {\displaystyle {\diamondsuit (E_{\operatorname {cf} (\lambda )}^{\lambda ^{+}}})} for every singular cardinal λ {\displaystyle \lambda } ? Does the generalized continuum hypothesis imply the existence of an ℵ2-Suslin tree? If ℵω is a strong limit cardinal, is 2 ℵ ω < ℵ ω 1 {\displaystyle 2^{\aleph _{\omega }}<\aleph _{\omega _{1}}} (see Singular cardinals hypothesis)? The best bound, ℵω4, was obtained by Shelah using his PCF theory. The problem of finding the ultimate core model, one that contains all large cardinals. Woodin's Ω-conjecture: if there is a proper class of Woodin cardinals, then Ω-logic satisfies an analogue of Gödel's completeness theorem. Does the consistency of the existence of a strongly compact cardinal imply the consistent existence of a supercompact cardinal? Does there exist a Jónsson algebra on ℵω? Is OCA (the open coloring axiom) consistent with 2 ℵ 0 > ℵ 2 {\displaystyle 2^{\aleph _{0}}>\aleph _{2}} ? Reinhardt cardinals: Without assuming the axiom of choice, can a


id: list-of-unsolved-problems-in-mathematics-20260505-13
file: data/en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics-13.md
category: reference
title: List of unsolved problems in mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics
date_saved: 2026-05-05T11:02:37.770898+00:00
instance: kb-cron
parent: list-of-unsolved-problems-in-mathematics-20260505-0
summary:
=== Graph theory === Kahn–Kalai conjecture (Jinyoung Park and Huy Tuan Pham, 2022) Blankenship–Oporowski conjecture on the book thickness of subdivisions (Vida Dujmović, David Eppstein, Robert Hickingbotham, Pat Morin, and David Wood, 2021) Ringel's conjecture that the complete graph K 2 n + 1 {\displaystyle K_{2n+1}} can be decomposed into 2 n + 1 {\displaystyle 2n+1} copies of any tree with n {\displaystyle n} edges (Richard Montgomery, Benny Sudakov, Alexey Pokrovskiy, 2020) Disproof of Hedetniemi's conjecture on the chromatic number of tensor products of graphs (Yaroslav Shitov, 2019) Kelmans–Seymour conjecture (Dawei He, Yan Wang, and Xingxing Yu, 2020) Goldberg–Seymour conjecture (Guantao Chen, Guangming Jing, and Wenan Zang, 2019) Babai's problem (Alireza Abdollahi, Maysam Zallaghi, 2015) Alspach's conjecture (Darryn Bryant, Daniel Horsley, William Pettersson, 2014) Alon–Saks–Seymour conjecture (Hao Huang, Benny Sudakov, 2012) Read–Hoggar conjecture (June Huh, 2009) Scheinerman's conjecture (Jeremie Chalopin and Daniel Gonçalves, 2009) Erdős–Menger conjecture (Ron Aharoni, Eli Berger 2007) Road coloring conjecture (Avraham Trahtman, 2007) Robertson–Seymour theorem (Neil Robertson, Paul Seymour, 2004) Strong perfect graph conjecture (Maria Chudnovsky, Neil Robertson, Paul Seymour and Robin Thomas, 2002) Toida's conjecture (Mikhail Muzychuk, Mikhail Klin, and Reinhard Pöschel, 2001) Harary's conjecture on the integral sum number of complete graphs (Zhibo Chen,


id: list-of-unsolved-problems-in-mathematics-20260505-14
file: data/en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics-14.md
category: reference
title: List of unsolved problems in mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics
date_saved: 2026-05-05T11:02:37.770898+00:00
instance: kb-cron
parent: list-of-unsolved-problems-in-mathematics-20260505-0
summary:
=== Books discussing unsolved problems === Chung, Fan; Graham, Ron (1999). Erdös on Graphs: His Legacy of Unsolved Problems. AK Peters. ISBN 978-1-56881-111-6. Croft, Hallard T.; Falconer, Kenneth J.; Guy, Richard K. (1994). Unsolved Problems in Geometry. Springer. ISBN 978-0-387-97506-1. Guy, Richard K. (2004). Unsolved Problems in Number Theory. Springer. ISBN 978-0-387-20860-2. Klee, Victor; Wagon, Stan (1996). Old and New Unsolved Problems in Plane Geometry and Number Theory. The Mathematical Association of America. ISBN 978-0-88385-315-3. du Sautoy, Marcus (2003). The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics. Harper Collins. ISBN 978-0-06-093558-0. Derbyshire, John (2003). Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Joseph Henry Press. ISBN 978-0-309-08549-6. Devlin, Keith (2006). The Millennium Problems – The Seven Greatest Unsolved* Mathematical Puzzles Of Our Time. Barnes & Noble. ISBN 978-0-7607-8659-8. Blondel, Vincent D.; Megrestski, Alexandre (2004). Unsolved problems in mathematical systems and control theory. Princeton University Press. ISBN 978-0-691-11748-5. Ji, Lizhen; Poon, Yat-Sun; Yau, Shing-Tung (2013). Open Problems and Surveys of Contemporary Mathematics (volume 6 in the Surveys in Modern Mathematics series) (Surveys of Modern Mathematics). International Press of Boston. ISBN 978-1-57146-278-7. Waldschmidt, Michel (2004). "Open Diophantine Problems" (PDF). Moscow Mathematical Journal. 4 (1):


id: list-of-problems-in-loop-theory-and-quasigroup-theory-20260505-0
file: data/en.wikipedia.org/wiki/List_of_problems_in_loop_theory_and_quasigroup_theory-0.md
category: reference
title: List of problems in loop theory and quasigroup theory
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_problems_in_loop_theory_and_quasigroup_theory
date_saved: 2026-05-05T11:02:39.093380+00:00
instance: kb-cron
parent: list-of-problems-in-loop-theory-and-quasigroup-theory-20260505-0
summary:
In mathematics, especially abstract algebra, loop theory and quasigroup theory are active research areas with many open problems. As in other areas of mathematics, such problems are often made public at professional conferences and meetings. Many of the problems posed here first appeared in the Loops (Prague) conferences and the Mile High (Denver) conferences. == Open problems (Moufang loops) == === Abelian by cyclic groups resulting in Moufang loops === Let L be a Moufang loop with normal abelian subgroup (associative subloop) M of odd order such that L/M is a cyclic group of order bigger than 3. (i) Is L a group? (ii) If the orders of M and L/M are relatively prime, is L a group? Proposed: by Michael Kinyon, based on (Chein and Rajah, 2000) Comments: The assumption that L/M has order bigger than 3 is important, as there is a (commutative) Moufang loop L of order 81 with normal commutative subgroup of order 27. === Embedding CMLs of period 3 into alternative algebras === Conjecture: Any finite commutative Moufang loop of period 3 can be embedded into a commutative alternative algebra. Proposed: by Alexander Grishkov at Loops '03, Prague 2003 === Frattini subloop for Moufang loops


id: list-of-problems-in-loop-theory-and-quasigroup-theory-20260505-1
file: data/en.wikipedia.org/wiki/List_of_problems_in_loop_theory_and_quasigroup_theory-1.md
category: reference
title: List of problems in loop theory and quasigroup theory
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_problems_in_loop_theory_and_quasigroup_theory
date_saved: 2026-05-05T11:02:39.093380+00:00
instance: kb-cron
parent: list-of-problems-in-loop-theory-and-quasigroup-theory-20260505-0
summary:
Let ( Q , ∗ ) {\displaystyle (Q,*)} , ( Q , + ) {\displaystyle (Q,+)} be two quasigroups defined on the same underlying set Q {\displaystyle Q} . The distance d ( ∗ , + ) {\displaystyle d(*,+)} is the number of pairs ( a , b ) {\displaystyle (a,b)} in Q × Q {\displaystyle Q\times Q} such that a ∗ b ≠ a + b {\displaystyle a*b\neq a+b} . Call a class of finite quasigroups quadratic if there is a positive real number α {\displaystyle \alpha } such that any two quasigroups ( Q , ∗ ) {\displaystyle (Q,*)} , ( Q , + ) {\displaystyle (Q,+)} of order n {\displaystyle n} from the class satisfying d ( ∗ , + ) < α n 2 {\displaystyle d(*,+)<\alpha \,n^{2}} are isomorphic. Are Moufang loops quadratic? Are Bol loops quadratic? Proposed: by Aleš Drápal at Loops '99, Prague 1999 Comments: Drápal proved in (Drápal, 1992) that groups are quadratic with α = 1 / 9 {\displaystyle \alpha =1/9} , and in (Drápal, 2000) that 2-groups are quadratic with α = 1 / 4 {\displaystyle \alpha =1/4} . === Campbell–Hausdorff series for analytic Bol loops === Determine the Campbell–Hausdorff series


id: list-of-problems-in-loop-theory-and-quasigroup-theory-20260505-2
file: data/en.wikipedia.org/wiki/List_of_problems_in_loop_theory_and_quasigroup_theory-2.md
category: reference
title: List of problems in loop theory and quasigroup theory
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_problems_in_loop_theory_and_quasigroup_theory
date_saved: 2026-05-05T11:02:39.093380+00:00
instance: kb-cron
parent: list-of-problems-in-loop-theory-and-quasigroup-theory-20260505-0
summary:
For a loop Q, let Mlt(Q) denote the multiplication group of Q, that is, the group generated by all left and right translations. Is |Mlt(Q)| < f(|Q|) for some variety of loops and for some polynomial f? Proposed: at the Milehigh conference on quasigroups, loops, and nonassociative systems, Denver 2005 === Does every finite alternative loop have 2-sided inverses? === Does every finite alternative loop, that is, every loop satisfying x(xy) = (xx)y and x(yy) = (xy)y, have 2-sided inverses? Proposed: by Warren D. Smith Comments: There are infinite alternative loops without 2-sided inverses, cf. (Ormes and Vojtěchovský, 2007) === Finite simple nonassociative automorphic loop === Find a nonassociative finite simple automorphic loop, if such a loop exists. Proposed: by Michael Kinyon at Loops '03, Prague 2003 Comments: It is known that such a loop cannot be commutative (Grishkov, Kinyon and Nagý, 2013) nor have odd order (Kinyon, Kunen, Phillips and Vojtěchovský, 2013). === Moufang theorem in non-Moufang loops === We say that a variety V of loops satisfies the Moufang theorem if for every loop Q in V the following implication holds: for every x, y, z in Q, if x(yz) = (xy)z then the subloop generated by x,


id: list-of-problems-in-loop-theory-and-quasigroup-theory-20260505-3
file: data/en.wikipedia.org/wiki/List_of_problems_in_loop_theory_and_quasigroup_theory-3.md
category: reference
title: List of problems in loop theory and quasigroup theory
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_problems_in_loop_theory_and_quasigroup_theory
date_saved: 2026-05-05T11:02:39.093380+00:00
instance: kb-cron
parent: list-of-problems-in-loop-theory-and-quasigroup-theory-20260505-0
summary:
Proposed: by Jonathan D. H. Smith and Alena Vanžurová at Loops '03, Prague 2003 Solved by: Alena Vanžurová, 2004. Solution: No, the class of cores of Bol loops is not closed under subalgebras. Furthermore, the class of cores of groups is not closed under subalgebras. Here is an outline of the proof: Cores of abelian groups are medial, by (Romanowska and Smith, 1985), (Rozskowska-Lech, 1999). The smallest nonabelian group S 3 {\displaystyle S_{3}} has core containing a submagma G {\displaystyle G} of order 4 that is not medial. If G {\displaystyle G} is a core of a Bol loop, it is a core of a Bol loop of order 4, hence a core of an abelian group, a contradiction. === Parity of the number of quasigroups up to isomorphism === Let I(n) be the number of isomorphism classes of quasigroups of order n. Is I(n) odd for every n? Proposed: by Douglas S. Stones at 2nd Mile High Conference on Nonassociative Mathematics, Denver 2009 Solved by: Douglas S. Stones, 2010. Solution: I(12) is even. In fact, I(n) is odd for all n ≤ 17 except 12. (Stones 2010) === Classification of finite simple paramedial quasigroups === Classify the finite simple


id: list-of-unsolved-problems-in-statistics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_unsolved_problems_in_statistics-0.md
category: reference
title: List of unsolved problems in statistics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_statistics
date_saved: 2026-05-05T11:02:40.300576+00:00
instance: kb-cron
parent: list-of-unsolved-problems-in-statistics-20260505-0
summary:
There are many longstanding unsolved problems in mathematics for which a solution has still not yet been found. The notable unsolved problems in statistics are generally of a different flavor; according to John Tukey, "difficulties in identifying problems have delayed statistics far more than difficulties in solving problems." A list of "one or two open problems" (in fact 22 of them) was given by David Cox. == Inference and testing == How to detect and correct for systematic errors, especially in sciences where random errors are large (a situation Tukey termed uncomfortable science). The Graybill–Deal estimator is often used to estimate the common mean of two normal populations with unknown and possibly unequal variances. Though this estimator is generally unbiased, its admissibility remains to be shown. Meta-analysis: Though independent p-values can be combined using Fisher's method, techniques are still being developed to handle the case of dependent p-values. Behrens–Fisher problem: Yuri Linnik showed in 1966 that there is no uniformly most powerful test for the difference of two means when the variances are unknown and possibly unequal. That is, there is no exact test (meaning that, if the means are in fact equal, one that rejects the null hypothesis with


id: 1-3-2-3-conjecture-20260505-0
file: data/en.wikipedia.org/wiki/1/3–2/3_conjecture-0.md
category: reference
title: 1/3–2/3 conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/1/3–2/3_conjecture
date_saved: 2026-05-05T11:02:41.499495+00:00
instance: kb-cron
parent: 1-3-2-3-conjecture-20260505-0
summary:
In order theory, a branch of mathematics, the 1/3–2/3 conjecture states that, if one is comparison sorting a set of items then, no matter what comparisons may have already been performed, it is always possible to choose the next comparison in such a way that it will reduce the number of possible sorted orders by a factor of 2/3 or better. Equivalently, in every finite partially ordered set that is not totally ordered, there exists a pair of elements x and y with the property that at least 1/3 and at most 2/3 of the linear extensions of the partial order place x earlier than y. == Example == The partial order formed by three elements a, b, and c with a single comparability relationship, a ≤ b, has three linear extensions, a ≤ b ≤ c, a ≤ c ≤ b, and c ≤ a ≤ b. In all three of these extensions, a is earlier than b. However, a is earlier than c in only two of them, and later than c in the third. Therefore, the pair of a and c have the desired property, showing that this partial order obeys the 1/3–2/3 conjecture. This example shows


id: 1-3-2-3-conjecture-20260505-1
file: data/en.wikipedia.org/wiki/1/3–2/3_conjecture-1.md
category: reference
title: 1/3–2/3 conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/1/3–2/3_conjecture
date_saved: 2026-05-05T11:02:41.499495+00:00
instance: kb-cron
parent: 1-3-2-3-conjecture-20260505-0
summary:
== Applications == In 1984 Jeff Kahn and Saks proposed the following application for the problem: suppose one wishes to comparison sort a totally ordered set X, for which some partial order information is already known in the form of a partial order on X. In the worst case, each additional comparison between a pair x and y of elements may yield as little information as possible, by resolving the comparison in a way that leaves as many linear extensions as possible compatible with the comparison result. The 1/3–2/3 conjecture states that, at each step, one may choose a comparison to perform that reduces the remaining number of linear extensions by a factor of 2/3; therefore, if there are E linear extensions of the partial order given by the initial information, the sorting problem can be completed in at most log 3 / 2 ⁡ E {\displaystyle \log _{3/2}E} additional comparisons. However, this analysis neglects the complexity of selecting the optimal pair x and y to compare. Additionally, it may be possible to sort a partial order using a number of comparisons that is better than this analysis would suggest, because it may not be possible for this worst-case behavior


id: 1-3-2-3-conjecture-20260505-2
file: data/en.wikipedia.org/wiki/1/3–2/3_conjecture-2.md
category: reference
title: 1/3–2/3 conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/1/3–2/3_conjecture
date_saved: 2026-05-05T11:02:41.499495+00:00
instance: kb-cron
parent: 1-3-2-3-conjecture-20260505-0
summary:
== References == Aigner, Martin (1985), "A note on merging", Order, 2 (3): 257–264, doi:10.1007/BF00333131, S2CID 118877843. Brightwell, Graham R. (1989), "Semiorders and the 1/3–2/3 conjecture", Order, 5 (4): 369–380, doi:10.1007/BF00353656, S2CID 86860160. Brightwell, Graham R. (1999), "Balanced pairs in partial orders", Discrete Mathematics, 201 (1–3): 25–52, doi:10.1016/S0012-365X(98)00311-2. Brightwell, Graham R.; Felsner, Stefan; Trotter, William T. (1995), "Balancing pairs and the cross product conjecture", Order, 12 (4): 327–349, CiteSeerX 10.1.1.38.7841, doi:10.1007/BF01110378, MR 1368815, S2CID 14793475. Brightwell, Graham R.; Winkler, Peter (1991), "Counting linear extensions", Order, 8 (3): 225–242, doi:10.1007/BF00383444, S2CID 119697949. Felsner, Stefan; Trotter, William T. (1993), "Balancing pairs in partially ordered sets", Combinatorics, Paul Erdős is eighty, Bolyai Society Mathematical Studies, vol. 1, Budapest: János Bolyai Mathematical Society, pp. 145–157, MR 1249709. Fredman, M. L. (1976), "How good is the information theory bound in sorting?", Theoretical Computer Science, 1 (4): 355–361, doi:10.1016/0304-3975(76)90078-5 Kahn, Jeff; Linial, Nati (1991), "Balancing extensions via Brunn-Minkowski", Combinatorica, 11 (4): 363–368, doi:10.1007/BF01275670, S2CID 206793172. Kahn, Jeff; Saks, Michael (1984), "Balancing poset extensions", Order, 1 (2): 113–126, doi:10.1007/BF00565647, S2CID 123370506. Khachiyan, Leonid (1989), "Optimal algorithms in convex programming decomposition and sorting", in Jaravlev, J. (ed.), Computers and Decision Problems (in Russian), Moscow: Nauka, pp. 161–205. As


id: andrews-curtis-conjecture-20260505-0
file: data/en.wikipedia.org/wiki/Andrews–Curtis_conjecture-0.md
category: reference
title: Andrews–Curtis conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Andrews–Curtis_conjecture
date_saved: 2026-05-05T11:02:42.701762+00:00
instance: kb-cron
parent: andrews-curtis-conjecture-20260505-0
summary:
In mathematics, the Andrews–Curtis conjecture states that every balanced presentation (i.e. a presentation with the same number of generators and relations) of the trivial group can be transformed into a trivial presentation by a sequence of Nielsen transformations on the relators together with conjugations of relators, named after James J. Andrews and Morton L. Curtis who proposed it in 1965. It is difficult to verify whether the conjecture holds for a given balanced presentation or not. It is widely believed that the Andrews–Curtis conjecture is false. While there are no counterexamples known, there are numerous potential counterexamples. It is known that the Zeeman conjecture on collapsibility implies the Andrews–Curtis conjecture. == Further reading == Andrews, J. J.; Curtis, M. L. (1965), "Free groups and handlebodies", Proceedings of the American Mathematical Society, 16 (2), American Mathematical Society: 192–195, doi:10.2307/2033843, JSTOR 2033843, MR 0173241 "Low-dimensional topology, problems in", Encyclopedia of Mathematics, EMS Press, 2001 [1994] == References ==


id: arnold-conjecture-20260505-0
file: data/en.wikipedia.org/wiki/Arnold_conjecture-0.md
category: reference
title: Arnold conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Arnold_conjecture
date_saved: 2026-05-05T11:02:43.869021+00:00
instance: kb-cron
parent: arnold-conjecture-20260505-0
summary:
The Arnold conjecture, named after mathematician Vladimir Arnold, is a mathematical conjecture in the field of symplectic geometry, a branch of differential geometry. == Strong Arnold conjecture == Let ( M , ω ) {\displaystyle (M,\omega )} be a closed (compact without boundary) symplectic manifold. For any smooth function H : M → R {\displaystyle H:M\to {\mathbb {R} }} , the symplectic form ω {\displaystyle \omega } induces a Hamiltonian vector field X H {\displaystyle X_{H}} on M {\displaystyle M} defined by the formula ω ( X H , ⋅ ) = d H . {\displaystyle \omega (X_{H},\cdot )=dH.} The function H {\displaystyle H} is called a Hamiltonian function. Suppose there is a smooth 1-parameter family of Hamiltonian functions H t ∈ C ∞ ( M ) {\displaystyle H_{t}\in C^{\infty }(M)} , t ∈ [ 0 , 1 ] {\displaystyle t\in [0,1]} . This family induces a 1-parameter family of Hamiltonian vector fields X H t {\displaystyle X_{H_{t}}} on M {\displaystyle M} . The family of vector fields integrates to a 1-parameter family of diffeomorphisms φ t : M → M {\displaystyle \varphi _{t}:M\to M} . Each individual φ t {\displaystyle \varphi _{t}} is a called a Hamiltonian diffeomorphism of


id: baum-connes-conjecture-20260505-0
file: data/en.wikipedia.org/wiki/Baum–Connes_conjecture-0.md
category: reference
title: Baum–Connes conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Baum–Connes_conjecture
date_saved: 2026-05-05T11:02:45.104924+00:00
instance: kb-cron
parent: baum-connes-conjecture-20260505-0
summary:
In mathematics, specifically in operator K-theory, the Baum–Connes conjecture suggests a link between the K-theory of the reduced C*-algebra of a group and the K-homology of the classifying space of proper actions of that group. The conjecture sets up a correspondence between different areas of mathematics, with the K-homology of the classifying space being related to geometry, differential operator theory, and homotopy theory, while the K-theory of the group's reduced C*-algebra is a purely analytical object. The conjecture, if true, would have some older famous conjectures as consequences. For instance, the surjectivity part implies the Kadison–Kaplansky conjecture for discrete torsion-free groups, and the injectivity is closely related to the Novikov conjecture. The conjecture is also closely related to index theory, as the assembly map μ {\displaystyle \mu } is a sort of index, and it plays a major role in Alain Connes' noncommutative geometry program. The origins of the conjecture go back to Fredholm theory, the Atiyah–Singer index theorem and the interplay of geometry with operator K-theory as expressed in the works of Brown, Douglas and Fillmore, among many other motivating subjects. == Formulation == Let Γ be a second countable locally compact group (for instance a countable discrete group).


id: baum-connes-conjecture-20260505-1
file: data/en.wikipedia.org/wiki/Baum–Connes_conjecture-1.md
category: reference
title: Baum–Connes conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Baum–Connes_conjecture
date_saved: 2026-05-05T11:02:45.104924+00:00
instance: kb-cron
parent: baum-connes-conjecture-20260505-0
summary:
Discrete subgroups of S O ( n , 1 ) {\displaystyle SO(n,1)} and S U ( n , 1 ) {\displaystyle SU(n,1)} . Groups with the Haagerup property, sometimes called a-T-menable groups. These are groups that admit an isometric action on an affine Hilbert space H {\displaystyle H} which is proper in the sense that lim n → ∞ g n ξ → ∞ {\displaystyle \lim _{n\to \infty }g_{n}\xi \to \infty } for all ξ ∈ H {\displaystyle \xi \in H} and all sequences of group elements g n {\displaystyle g_{n}} with lim n → ∞ g n → ∞ {\displaystyle \lim _{n\to \infty }g_{n}\to \infty } . Examples of a-T-menable groups are amenable groups, Coxeter groups, groups acting properly on trees, and groups acting properly on simply connected C A T ( 0 ) {\displaystyle CAT(0)} cubical complexes. Groups that admit a finite presentation with only one relation. Discrete cocompact subgroups of real Lie groups of real rank 1. Cocompact lattices in S L ( 3 , R ) , S L ( 3 , C ) {\displaystyle SL(3,\mathbb {R} ),SL(3,\mathbb {C} )} or S L ( 3 , Q p ) {\displaystyle SL(3,\mathbb {Q} _{p})} . It was


id: bing-borsuk-conjecture-20260505-0
file: data/en.wikipedia.org/wiki/Bing–Borsuk_conjecture-0.md
category: reference
title: Bing–Borsuk conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Bing–Borsuk_conjecture
date_saved: 2026-05-05T11:02:46.320784+00:00
instance: kb-cron
parent: bing-borsuk-conjecture-20260505-0
summary:
In mathematics, the Bing–Borsuk conjecture states that every n {\displaystyle n} -dimensional homogeneous absolute neighborhood retract space is a topological manifold. The conjecture has been proved for dimensions 1 and 2, and it is known that the 3-dimensional version of the conjecture implies the Poincaré conjecture. == Definitions == A topological space is homogeneous if, for any two points m 1 , m 2 ∈ M {\displaystyle m_{1},m_{2}\in M} , there is a homeomorphism of M {\displaystyle M} which takes m 1 {\displaystyle m_{1}} to m 2 {\displaystyle m_{2}} . A metric space M {\displaystyle M} is an absolute neighborhood retract (ANR) if, for every closed embedding f : M → N {\displaystyle f:M\rightarrow N} (where N {\displaystyle N} is a metric space), there exists an open neighbourhood U {\displaystyle U} of the image f ( M ) {\displaystyle f(M)} which retracts to f ( M ) {\displaystyle f(M)} . There is an alternate statement of the Bing–Borsuk conjecture: suppose M {\displaystyle M} is embedded in R m + n {\displaystyle \mathbb {R} ^{m+n}} for some m ≥ 3 {\displaystyle m\geq 3} and this embedding can be extended to an embedding of M × ( − ε , ε )


id: birch-tate-conjecture-20260505-0
file: data/en.wikipedia.org/wiki/Birch–Tate_conjecture-0.md
category: reference
title: Birch–Tate conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Birch–Tate_conjecture
date_saved: 2026-05-05T11:02:47.529569+00:00
instance: kb-cron
parent: birch-tate-conjecture-20260505-0
summary:
The Birch–Tate conjecture is a conjecture in mathematics (more specifically in algebraic K-theory) proposed by both Bryan John Birch and John Tate. == Statement == In algebraic K-theory, the group K2 is defined as the center of the Steinberg group of the ring of integers of a number field F. K2 is also known as the tame kernel of F. The Birch–Tate conjecture relates the order of this group (its number of elements) to the value of the Dedekind zeta function ζ F {\displaystyle \zeta _{F}} . More specifically, let F be a totally real number field and let N be the largest natural number such that the extension of F by the Nth root of unity has an elementary abelian 2-group as its Galois group. Then the conjecture states that # K 2 = | N ζ F ( − 1 ) | . {\displaystyle \#K_{2}=|N\zeta _{F}(-1)|.} == Status == Progress on this conjecture has been made as a consequence of work on Iwasawa theory, and in particular of the proofs given for the so-called "main conjecture of Iwasawa theory." == References == J. T. Tate, Symbols in Arithmetic, Actes, Congrès Intern. Math., Nice, 1970, Tome 1, Gauthier–Villars(1971), 201–211


id: bloch-s-theorem-complex-analysis-20260505-0
file: data/en.wikipedia.org/wiki/Bloch's_theorem_(complex_analysis)-0.md
category: reference
title: Bloch's theorem (complex analysis)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Bloch's_theorem_(complex_analysis)
date_saved: 2026-05-05T11:02:48.678417+00:00
instance: kb-cron
parent: bloch-s-theorem-complex-analysis-20260505-0
summary:
In complex analysis, a branch of mathematics, Bloch's theorem describes the behaviour of holomorphic functions defined on the unit disk. It gives a lower bound on the size of a disk in which an inverse to a holomorphic function exists. It is named after André Bloch. == Statement == Let f be a holomorphic function in the unit disk |z| ≤ 1 for which | f ′ ( 0 ) | = 1 {\displaystyle |f'(0)|=1} Bloch's theorem states that there is a disk S ⊂ D on which f is biholomorphic and f(S) contains a disk with radius 1/72. == Landau's theorem == If f is a holomorphic function in the unit disk with the property |f′(0)| = 1, then let Lf be the radius of the largest disk contained in the image of f. Landau's theorem states that there is a constant L defined as the infimum of Lf over all such functions f, and that L is greater than Bloch's constant L ≥ B. This theorem is named after Edmund Landau. == Valiron's theorem == Bloch's theorem was inspired by the following theorem of Georges Valiron: Theorem. If f is a non-constant entire function then there exist disks


id: brennan-conjecture-20260505-0
file: data/en.wikipedia.org/wiki/Brennan_conjecture-0.md
category: reference
title: Brennan conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Brennan_conjecture
date_saved: 2026-05-05T11:02:49.824714+00:00
instance: kb-cron
parent: brennan-conjecture-20260505-0
summary:
In mathematics, specifically complex analysis, the Brennan conjecture is a conjecture estimating (under specified conditions) the integral powers of the moduli of the derivatives of conformal maps into the open unit disk. The conjecture was formulated by James E. Brennan in 1978. Let W be a simply connected open subset of C {\displaystyle \mathbb {C} } with at least two boundary points in the extended complex plane. Let φ {\displaystyle \varphi } be a conformal map of W onto the open unit disk. The Brennan conjecture states that ∫ W | φ ′ | p d x d y < ∞ {\displaystyle \int _{W}|\varphi \ '|^{p}\,\mathrm {d} x\,\mathrm {d} y<\infty } whenever 4 / 3 < p < 4 {\displaystyle 4/3<p<4} . Brennan proved the result when 4 / 3 < p < p 0 {\displaystyle 4/3<p<p_{0}} for some constant p 0 > 3 {\displaystyle p_{0}>3} . Bertilsson proved in 1999 that the result holds when 4 / 3 < p < 3.422 {\displaystyle 4/3<p<3.422} , but the full result remains open. == References ==


id: b-chi-s-problem-20260505-0
file: data/en.wikipedia.org/wiki/Büchi's_problem-0.md
category: reference
title: Büchi's problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Büchi's_problem
date_saved: 2026-05-05T11:02:50.999583+00:00
instance: kb-cron
parent: b-chi-s-problem-20260505-0
summary:
In number theory, Büchi's problem, also known as the n squares' problem, is an open problem named after the Swiss mathematician Julius Richard Büchi. It asks whether there is a positive integer M such that every sequence of M or more integer squares, whose second difference is constant and equal to 2, is necessarily a sequence of squares of the form (x + i)2, i = 1, 2, ..., M,... for some integer x. In 1983, Douglas Hensley observed that Büchi's problem is equivalent to the following: Does there exist a positive integer M such that, for all integers x and a, the quantity (x + n)2 + a cannot be a square for more than M consecutive values of n, unless a = 0? == Statement of Büchi's problem == Büchi's problem can be stated in the following way: Does there exist a positive integer M such that the system of equations { x 2 2 − 2 x 1 2 + x 0 2 = 2 x 3 2 − 2 x 2 2 + x 1 2 = 2 ⋮ x M − 1 2 − 2 x M − 2 2 + x M − 3


id: burnside-problem-20260505-0
file: data/en.wikipedia.org/wiki/Burnside_problem-0.md
category: reference
title: Burnside problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Burnside_problem
date_saved: 2026-05-05T11:02:52.202280+00:00
instance: kb-cron
parent: burnside-problem-20260505-0
summary:
The Burnside problem asks whether a finitely generated group in which every element has finite order must necessarily be a finite group. It was posed by William Burnside in 1902, making it one of the oldest questions in group theory, and was influential in the development of combinatorial group theory. It is known to have a negative answer in general, as Evgeny Golod and Igor Shafarevich provided a counter-example in 1964. The problem has many refinements and variants that differ in the additional conditions imposed on the orders of the group elements (see bounded and restricted below). Some of these variants are still open questions. == Brief history == Initial work pointed towards the affirmative answer. For example, if a group G is finitely generated and the order of each element of G is a divisor of 4, then G is finite. Moreover, A. I. Kostrikin was able to prove in 1958 that among the finite groups with a given number of generators and a given prime exponent, there exists a largest one. This provides a solution for the restricted Burnside problem for the case of prime exponent. (Later, in 1989, Efim Zelmanov was able to solve the restricted Burnside


id: burnside-problem-20260505-1
file: data/en.wikipedia.org/wiki/Burnside_problem-1.md
category: reference
title: Burnside problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Burnside_problem
date_saved: 2026-05-05T11:02:52.202280+00:00
instance: kb-cron
parent: burnside-problem-20260505-0
summary:
The particular case of B(2, 5) remains open. The breakthrough in solving the Burnside problem was achieved by Pyotr Novikov and Sergei Adian in 1968. Using a complicated combinatorial argument, they demonstrated that for every odd number n with n > 4381, there exist infinite, finitely generated groups of exponent n. Adian later improved the bound on the odd exponent to 665. In 2015, Adian claimed to have obtained a lower bound of 101 for odd n; however, the full proof of this lower bound was never completed and never published. The case of even exponent turned out to be considerably more difficult. It was only in 1994 that Sergei Vasilievich Ivanov was able to prove an analogue of Novikov–Adian theorem: for any m > 1 and an even n ≥ 248, n divisible by 29, the group B(m, n) is infinite; together with the Novikov–Adian theorem, this implies infiniteness for all m > 1 and n ≥ 248. This was improved in 1996 by I. G. Lysënok to m > 1 and n ≥ 8000. Novikov–Adian, Ivanov and Lysënok established considerably more precise results on the structure of the free Burnside groups. In the case of the odd exponent,


id: burnside-problem-20260505-2
file: data/en.wikipedia.org/wiki/Burnside_problem-2.md
category: reference
title: Burnside problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Burnside_problem
date_saved: 2026-05-05T11:02:52.202280+00:00
instance: kb-cron
parent: burnside-problem-20260505-0
summary:
== Bibliography == S. I. Adian (1979) The Burnside problem and identities in groups. Translated from the Russian by John Lennox and James Wiegold. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 95. Springer-Verlag, Berlin-New York. ISBN 3-540-08728-1. S. I. Adian (2015). "New estimates of odd exponents of infinite Burnside groups". Trudy Matematicheskogo Instituta imeni V. A. Steklova (in Russian). 289: 41–82. doi:10.1134/S0371968515020041. Translation in Adian, S. I. (2015). "New estimates of odd exponents of infinite Burnside groups". Proceedings of the Steklov Institute of Mathematics. 289 (1): 33–71. doi:10.1134/S0081543815040045. S. V. Ivanov (1994). "The Free Burnside Groups of Sufficiently Large Exponents". International Journal of Algebra and Computation. 04: 1–308. doi:10.1142/S0218196794000026. S. V. Ivanov; A. Yu. Ol'Shanskii (1996). "Hyperbolic groups and their quotients of bounded exponents". Transactions of the American Mathematical Society. 348 (6): 2091–2138. doi:10.1090/S0002-9947-96-01510-3. A. I. Kostrikin (1990) Around Burnside. Translated from the Russian and with a preface by James Wiegold. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 20. Springer-Verlag, Berlin. ISBN 3-540-50602-0. I. G. Lysënok (1996). "Infinite Burnside groups of even exponent". Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya (in Russian). 60 (3): 3–224. doi:10.4213/im77. Translation in Lysënok,


id: crouzeix-s-conjecture-20260505-0
file: data/en.wikipedia.org/wiki/Crouzeix's_conjecture-0.md
category: reference
title: Crouzeix's conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Crouzeix's_conjecture
date_saved: 2026-05-05T11:02:53.424444+00:00
instance: kb-cron
parent: crouzeix-s-conjecture-20260505-0
summary:
Crouzeix's conjecture is an unsolved problem in matrix analysis. It was proposed by Michel Crouzeix in 2004, and it can be stated as follows: ‖ f ( A ) ‖ ≤ 2 sup z ∈ W ( A ) | f ( z ) | , {\displaystyle \|f(A)\|\leq 2\sup _{z\in W(A)}|f(z)|,} where the set W ( A ) {\displaystyle W(A)} is the field of values of a n×n (i.e. square) complex matrix A {\displaystyle A} and f {\displaystyle f} is a complex function that is analytic in the interior of W ( A ) {\displaystyle W(A)} and continuous up to the boundary of W ( A ) {\displaystyle W(A)} . Slightly reformulated, the conjecture can also be stated as follows: for all square complex matrices A {\displaystyle A} and all complex polynomials p {\displaystyle p} : ‖ p ( A ) ‖ ≤ 2 sup z ∈ W ( A ) | p ( z ) | {\displaystyle \|p(A)\|\leq 2\sup _{z\in W(A)}|p(z)|} holds, where the norm on the left-hand side is the spectral operator 2-norm. == History == Crouzeix's theorem, proved in 2007, states that: ‖ f ( A ) ‖ ≤ 11.08 sup z ∈ W ( A )


id: discrepancy-of-hypergraphs-20260505-0
file: data/en.wikipedia.org/wiki/Discrepancy_of_hypergraphs-0.md
category: reference
title: Discrepancy of hypergraphs
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Discrepancy_of_hypergraphs
date_saved: 2026-05-05T11:02:54.676918+00:00
instance: kb-cron
parent: discrepancy-of-hypergraphs-20260505-0
summary:
Discrepancy of hypergraphs is an area of discrepancy theory that studies the discrepancy of general set systems. == Definitions == In the classical setting, we aim at partitioning the vertices of a hypergraph H = ( V , E ) {\displaystyle {\mathcal {H}}=(V,{\mathcal {E}})} into two classes in such a way that ideally each hyperedge contains the same number of vertices in both classes. A partition into two classes can be represented by a coloring χ : V → { − 1 , + 1 } {\displaystyle \chi \colon V\rightarrow \{-1,+1\}} . We call −1 and +1 colors. The color-classes χ − 1 ( − 1 ) {\displaystyle \chi ^{-1}(-1)} and χ − 1 ( + 1 ) {\displaystyle \chi ^{-1}(+1)} form the corresponding partition. For a hyperedge E ∈ E {\displaystyle E\in {\mathcal {E}}} , set χ ( E ) := ∑ v ∈ E χ ( v ) . {\displaystyle \chi (E):=\sum _{v\in E}\chi (v).} The discrepancy of H {\displaystyle {\mathcal {H}}} with respect to χ {\displaystyle \chi } and the discrepancy of H {\displaystyle {\mathcal {H}}} are defined by disc ⁡ ( H , χ ) := max E ∈ E | χ ( E ) |


id: discrepancy-of-hypergraphs-20260505-1
file: data/en.wikipedia.org/wiki/Discrepancy_of_hypergraphs-1.md
category: reference
title: Discrepancy of hypergraphs
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Discrepancy_of_hypergraphs
date_saved: 2026-05-05T11:02:54.676918+00:00
instance: kb-cron
parent: discrepancy-of-hypergraphs-20260505-0
summary:
Since a random coloring with positive probability has discrepancy at most λ {\displaystyle \lambda } , in particular, there are colorings that have discrepancy at most λ {\displaystyle \lambda } . Hence disc ⁡ ( H ) ≤ λ . ◻ {\displaystyle \operatorname {disc} ({\mathcal {H}})\leq \lambda .\ \Box } 2. For any hypergraph H {\displaystyle {\mathcal {H}}} with n vertices and m edges such that m ≥ n {\displaystyle m\geq n} : disc ⁡ ( H ) ∈ O ( n ) . {\displaystyle \operatorname {disc} ({\mathcal {H}})\in O({\sqrt {n}}).} To prove this, a much more sophisticated approach using the entropy function was necessary. Of course this is particularly interesting for m = O ( n ) {\displaystyle m=O(n)} . In the case m = n {\displaystyle m=n} , disc ⁡ ( H ) ≤ 6 n {\displaystyle \operatorname {disc} ({\mathcal {H}})\leq 6{\sqrt {n}}} can be shown for n large enough. Therefore, this result is usually known to as 'Six Standard Deviations Suffice'. It is considered to be one of the milestones of discrepancy theory. The entropy method has seen numerous other applications, e.g. in the proof of the tight upper bound for the arithmetic progressions of Matoušek and


id: discrepancy-theory-20260505-0
file: data/en.wikipedia.org/wiki/Discrepancy_theory-0.md
category: reference
title: Discrepancy theory
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Discrepancy_theory
date_saved: 2026-05-05T11:02:55.884364+00:00
instance: kb-cron
parent: discrepancy-theory-20260505-0
summary:
In mathematics, discrepancy theory describes the deviation of a situation from the state one would like it to be in. It is also called the theory of irregularities of distribution. This refers to the theme of classical discrepancy theory, namely distributing points in some space such that they are evenly distributed with respect to some (mostly geometrically defined) subsets. The discrepancy (irregularity) measures how far a given distribution deviates from an ideal one. Discrepancy theory can be described as the study of inevitable irregularities of distributions, in measure-theoretic and combinatorial settings. Just as Ramsey theory elucidates the impossibility of total disorder, discrepancy theory studies the deviations from total uniformity. A significant event in the history of discrepancy theory was the 1916 paper of Weyl on the uniform distribution of sequences in the unit interval. == Theorems == Discrepancy theory is based on the following classic theorems: Geometric discrepancy theory The theorem of van Aardenne-Ehrenfest Arithmetic progressions (Roth, Sarkozy, Beck, Matousek & Spencer) Beck–Fiala theorem Six Standard Deviations Suffice (Spencer) == Major open problems == The unsolved problems relating to discrepancy theory include: Axis-parallel rectangles in dimensions three and higher (folklore) Komlós conjecture Heilbronn triangle problem on the minimum area of


id: dixmier-conjecture-20260505-0
file: data/en.wikipedia.org/wiki/Dixmier_conjecture-0.md
category: reference
title: Dixmier conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Dixmier_conjecture
date_saved: 2026-05-05T11:02:57.042829+00:00
instance: kb-cron
parent: dixmier-conjecture-20260505-0
summary:
In algebra the Dixmier conjecture, asked by Jacques Dixmier in 1968, is the conjecture that any endomorphism of a Weyl algebra is an automorphism. Tsuchimoto in 2005, and independently Belov-Kanel and Kontsevich in 2007, showed that the Dixmier conjecture is stably equivalent to the Jacobian conjecture. == References ==


id: eilenberg-ganea-conjecture-20260505-0
file: data/en.wikipedia.org/wiki/Eilenberg–Ganea_conjecture-0.md
category: reference
title: Eilenberg–Ganea conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Eilenberg–Ganea_conjecture
date_saved: 2026-05-05T11:03:00.497348+00:00
instance: kb-cron
parent: eilenberg-ganea-conjecture-20260505-0
summary:
The Eilenberg–Ganea conjecture is a claim in algebraic topology. It was formulated by Samuel Eilenberg and Tudor Ganea in 1957, in a short, but influential paper. It states that if a group G has cohomological dimension 2, then it has a 2-dimensional Eilenberg–MacLane space K ( G , 1 ) {\displaystyle K(G,1)} . For n different from 2, a group G of cohomological dimension n has an n-dimensional Eilenberg–MacLane space. It is also known that a group of cohomological dimension 2 has a 3-dimensional Eilenberg−MacLane space. In 1997, Mladen Bestvina and Noel Brady constructed a group G so that either G is a counterexample to the Eilenberg–Ganea conjecture, or there must be a counterexample to the Whitehead conjecture; in other words, it is not possible for both conjectures to be true. == References ==


id: erd-s-conjecture-on-arithmetic-progressions-20260505-0
file: data/en.wikipedia.org/wiki/Erdős_conjecture_on_arithmetic_progressions-0.md
category: reference
title: Erdős conjecture on arithmetic progressions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Erdős_conjecture_on_arithmetic_progressions
date_saved: 2026-05-05T11:03:01.649502+00:00
instance: kb-cron
parent: erd-s-conjecture-on-arithmetic-progressions-20260505-0
summary:
Erdős' conjecture on arithmetic progressions, often referred to as the Erdős–Turán conjecture, is a conjecture in arithmetic combinatorics. It states that if the sum of the reciprocals of the members of a set A of positive integers diverges, then A contains arbitrarily long arithmetic progressions. Formally, the conjecture states that if A is a large set in the sense that ∑ n ∈ A 1 n = ∞ , {\displaystyle \sum _{n\in A}{\frac {1}{n}}\ =\ \infty ,} then A contains arithmetic progressions of any given length, meaning that for every positive integer k there are an integer a and a non-zero integer c such that { a , a + c , a + 2 c , … , a + k c } ⊂ A {\displaystyle \{a,a{+}c,a{+}2c,\ldots ,a{+}kc\}\subset A} . == History == In 1936, Paul Erdős and Pál Turán made the weaker conjecture that any set of integers with positive natural density contains infinitely many three-term arithmetic progressions. This was proven by Klaus Roth in 1952, and generalized to arbitrarily long arithmetic progressions by Szemerédi in 1975 in what is now known as Szemerédi's theorem. In a 1976 talk titled "To the memory of my lifelong friend and


id: eternity-ii-puzzle-20260505-0
file: data/en.wikipedia.org/wiki/Eternity_II_puzzle-0.md
category: reference
title: Eternity II puzzle
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Eternity_II_puzzle
date_saved: 2026-05-05T11:03:02.845398+00:00
instance: kb-cron
parent: eternity-ii-puzzle-20260505-0
summary:
The Eternity II puzzle (E2 or E II) is an edge-matching puzzle launched on 28 July 2007. It was developed by Christopher Monckton and marketed and copyrighted by TOMY UK Ltd as a successor to the original Eternity puzzle. The puzzle was part of a competition in which a $2 million prize was offered for the first complete solution. The competition ended at noon on 31 December 2010, with no solution being found. == Description == The Eternity II puzzle is an edge-matching puzzle which involves placing 256 square puzzle pieces into a 16 × 16 grid, constrained by the requirement to match adjacent edges. It has been designed to be difficult to solve by brute-force computer search. Each puzzle piece has its edges on one side marked with different shape/colour combinations (collectively called "colours" here), each of which must match precisely with its neighbouring side on each adjacent piece when the puzzle is complete. The other side of each piece is blank apart from an identifying number, and is not used in the puzzle. Thus, each piece can be used in only 4 orientations. There are 22 colours, not including the grey edges. (Grey edges must face the outside


id: eternity-ii-puzzle-20260505-1
file: data/en.wikipedia.org/wiki/Eternity_II_puzzle-1.md
category: reference
title: Eternity II puzzle
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Eternity_II_puzzle
date_saved: 2026-05-05T11:03:02.845398+00:00
instance: kb-cron
parent: eternity-ii-puzzle-20260505-0
summary:
== History and design == The original Eternity puzzle was a tiling puzzle with a million-pound prize, created by Monckton. Launched in June 1999, it was solved by a computer search algorithm designed by Alex Selby and Oliver Riordan, which exploited combinatorial weaknesses of the original puzzle design. The prize money was paid out in full to Selby and Riordan. A puzzle with striking similarities to both eternity puzzles, the Diamond Dilemma, with a deadline in 1990, 10 years before the deadline of the original eternity puzzle, has fewer puzzle pieces, 160 compared with 209 and 256 for the first two eternity puzzles respectively, and yet Diamond Dilemma has not yet been solved in over 25 years. The Eternity II puzzle was designed by Monckton in 2005, this time in collaboration with Selby and Riordan, who designed a computer program that generated the final Eternity II design. According to the mathematical game enthusiast Brendan Owen, the Eternity II puzzle appears to have been designed to avoid the combinatorial flaws of the previous puzzle, with design parameters which appear to have been chosen to make the puzzle as difficult as possible to solve. In particular, unlike the original Eternity puzzle, there


id: farrell-jones-conjecture-20260505-0
file: data/en.wikipedia.org/wiki/Farrell–Jones_conjecture-0.md
category: reference
title: Farrell–Jones conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Farrell–Jones_conjecture
date_saved: 2026-05-05T11:03:04.056769+00:00
instance: kb-cron
parent: farrell-jones-conjecture-20260505-0
summary:
In mathematics, the Farrell–Jones conjecture, named after F. Thomas Farrell and Lowell E. Jones, states that certain assembly maps are isomorphisms. These maps are given as certain homomorphisms. The motivation is the interest in the target of the assembly maps; this may be, for instance, the algebraic K-theory of a group ring K n ( R G ) {\displaystyle K_{n}(RG)} or the L-theory of a group ring L n ( R G ) , {\displaystyle L_{n}(RG),} where G is some group. The sources of the assembly maps are equivariant homology theory evaluated on the classifying space of G with respect to the family of virtually cyclic subgroups of G. So assuming the Farrell–Jones conjecture is true, it is possible to restrict computations to virtually cyclic subgroups to get information on complicated objects such as K n ( R G ) {\displaystyle K_{n}(RG)} or L n ( R G ) {\displaystyle L_{n}(RG)} . The Baum–Connes conjecture formulates a similar statement, for the topological K-theory of reduced group C ∗ {\displaystyle C^{*}} -algebras K n t o p ( C ∗ r ( G ) ) {\displaystyle K_{n}^{top}(C_{*}^{r}(G))} . == Formulation == One can find for any ring R {\displaystyle R} equivariant


id: farrell-jones-conjecture-20260505-1
file: data/en.wikipedia.org/wiki/Farrell–Jones_conjecture-1.md
category: reference
title: Farrell–Jones conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Farrell–Jones_conjecture
date_saved: 2026-05-05T11:03:04.056769+00:00
instance: kb-cron
parent: farrell-jones-conjecture-20260505-0
summary:
K n Z ( p t ) = K n ( R [ Z ] ) ≅ K n ( R ) ⊕ K n − 1 ( R ) ⊕ N K n ( R ) ⊕ N K n ( R ) . {\displaystyle K_{n}^{\mathbb {Z} }(pt)=K_{n}(R[\mathbb {Z} ])\cong K_{n}(R)\oplus K_{n-1}(R)\oplus NK_{n}(R)\oplus NK_{n}(R).} Indeed one checks that the assembly map is given by the canonical inclusion. K n ( R ) ⊕ K n − 1 ( R ) ↪ K n ( R ) ⊕ K n − 1 ( R ) ⊕ N K n ( R ) ⊕ N K n ( R ) {\displaystyle K_{n}(R)\oplus K_{n-1}(R)\hookrightarrow K_{n}(R)\oplus K_{n-1}(R)\oplus NK_{n}(R)\oplus NK_{n}(R)} So it is an isomorphism if and only if N K n ( R ) = 0 {\displaystyle NK_{n}(R)=0} , which is the case if R {\displaystyle R} is a regular ring. So in this case one can really use the family of finite subgroups. On the other hand this shows that the isomorphism conjecture for algebraic K-Theory and the family of finite subgroups is not true. One has to extend the conjecture to a larger family of subgroups which contains all the counterexamples.


id: finite-lattice-representation-problem-20260505-0
file: data/en.wikipedia.org/wiki/Finite_lattice_representation_problem-0.md
category: reference
title: Finite lattice representation problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Finite_lattice_representation_problem
date_saved: 2026-05-05T11:03:05.197102+00:00
instance: kb-cron
parent: finite-lattice-representation-problem-20260505-0
summary:
In mathematics, the finite lattice representation problem, or finite congruence lattice problem, asks whether every finite lattice is isomorphic to the congruence lattice of some finite algebra. == Background == A lattice is called algebraic if it is complete and compactly generated. In 1963, Grätzer and Schmidt proved that every algebraic lattice is isomorphic to the congruence lattice of some algebra. Thus there is essentially no restriction on the shape of a congruence lattice of an algebra. The finite lattice representation problem asks whether the same is true for finite lattices and finite algebras. That is, does every finite lattice occur as the congruence lattice of a finite algebra? In 1980, Pálfy and Pudlák proved that this problem is equivalent to the problem of deciding whether every finite lattice occurs as an interval in the subgroup lattice of a finite group. For an overview of the group theoretic approach to the problem, see Pálfy (1993) and Pálfy (2001). This problem should not be confused with the congruence lattice problem. == Significance == This is among the oldest unsolved problems in universal algebra. Until it is answered, the theory of finite algebras is incomplete since, given a finite algebra, it is


id: generalized-riemann-hypothesis-20260505-0
file: data/en.wikipedia.org/wiki/Generalized_Riemann_hypothesis-0.md
category: reference
title: Generalized Riemann hypothesis
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Generalized_Riemann_hypothesis
date_saved: 2026-05-05T11:03:06.339730+00:00
instance: kb-cron
parent: generalized-riemann-hypothesis-20260505-0
summary:
The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global L-functions, which are formally similar to the Riemann zeta-function. One can then ask the same question about the zeros of these L-functions, yielding various generalizations of the Riemann hypothesis. Many mathematicians believe these generalizations of the Riemann hypothesis to be true. The only cases of these conjectures which have been proven occur in the algebraic function field case (not the number field case). Global L-functions can be associated to elliptic curves, number fields (in which case they are called Dedekind zeta-functions), Maass forms, and Dirichlet characters (in which case they are called Dirichlet L-functions). When the Riemann hypothesis is formulated for Dedekind zeta-functions, it is known as the extended Riemann hypothesis (ERH) and when it is formulated for Dirichlet L-functions, it is known as the generalised Riemann hypothesis (GRH). Another approach to generalization of Riemann hypothesis was given by Atle Selberg and his introduction of class of function satisfying certain properties rather than specific functions, nowadays known as Selberg class. These three statements will be


id: generalized-riemann-hypothesis-20260505-1
file: data/en.wikipedia.org/wiki/Generalized_Riemann_hypothesis-1.md
category: reference
title: Generalized Riemann hypothesis
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Generalized_Riemann_hypothesis
date_saved: 2026-05-05T11:03:06.339730+00:00
instance: kb-cron
parent: generalized-riemann-hypothesis-20260505-0
summary:
Taking the trivial character χ ( n ) = 1 {\textstyle \chi (n)=1} yields the ordinary Riemann hypothesis. More effective version of Dirichlet's theorem on arithmetic progressions: Let π ( x , a , d ) {\textstyle \pi (x,a,d)} where a and d are coprime denote the number of prime numbers in arithmetic progression n ⋅ d + a {\textstyle n\cdot d+a} which are less than or equal to x. If the generalized Riemann hypothesis is true, then for every ε > 0: π ( x , a , d ) = 1 φ ( d ) ∫ 2 x 1 ln ⁡ t d t + O ( x 1 / 2 + ε ) as x → ∞ , {\displaystyle \pi (x,a,d)={\frac {1}{\varphi (d)}}\int _{2}^{x}{\frac {1}{\ln t}}\,dt+O(x^{1/2+\varepsilon })\quad {\mbox{ as }}\ x\to \infty ,} where φ {\displaystyle \varphi } is Euler's totient function and O {\displaystyle O} is the Big O notation. This is a considerable strengthening of the prime number theorem. Every proper subgroup of the multiplicative group ( Z / n Z ) × {\textstyle (\mathbb {Z} /n\mathbb {Z} )^{\times }} has a set of generators less than 2 ln ⁡ ( n ) 2 {\textstyle


id: generalized-riemann-hypothesis-20260505-2
file: data/en.wikipedia.org/wiki/Generalized_Riemann_hypothesis-2.md
category: reference
title: Generalized Riemann hypothesis
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Generalized_Riemann_hypothesis
date_saved: 2026-05-05T11:03:06.339730+00:00
instance: kb-cron
parent: generalized-riemann-hypothesis-20260505-0
summary:
=== Consequences of ERH === The ordinary Riemann hypothesis follows from the extended one if one takes the number field to be Q {\displaystyle \mathbb {Q} } , whose ring of integers is: O Q = Z {\displaystyle O_{\mathbb {Q} }=\mathbb {Z} } . Generalized Riemann hypothesis for Dirichlet L-functions is equivalent to ERH for K being abelian extension of rational numbers, since for abelian extensions ζ K {\textstyle \zeta _{K}} is finite product of some Dirichlet L-functions depending on K. Conversely, all L-functions for character modulo n appears in product for K = Q ( ζ n ) {\textstyle K=\mathbb {Q} (\zeta _{n})} , where ζ n {\textstyle \zeta _{n}} is n-th primitive root of unity. For general extensions, similar role to Dirichlet L-functions is played by Artin L-functions. Then, ERH is equivalent to Riemann Hypothesis for Artin L-functions. The ERH implies an effective version of the Chebotarev density theorem: if L/K is a finite Galois extension with Galois group G, and C a union of conjugacy classes of G, the number of unramified primes of K of norm below x with Frobenius conjugacy class in C is | C | | G | ( Li ⁡ ( x


id: grand-riemann-hypothesis-20260505-0
file: data/en.wikipedia.org/wiki/Grand_Riemann_hypothesis-0.md
category: reference
title: Grand Riemann hypothesis
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Grand_Riemann_hypothesis
date_saved: 2026-05-05T11:03:07.567528+00:00
instance: kb-cron
parent: grand-riemann-hypothesis-20260505-0
summary:
In mathematics, the grand Riemann hypothesis is a generalisation of both the Riemann hypothesis and the generalized Riemann hypothesis. It states that the non-trivial zeros of all automorphic L-functions lie on the critical line 1 / 2 + i t {\displaystyle 1/2+it} with t {\displaystyle t} a real number variable and i {\displaystyle i} the imaginary unit. The modified grand Riemann hypothesis is the assertion that the nontrivial zeros of all automorphic L-functions lie on the critical line or the real line. == Notes == Robert Langlands, in his general functoriality conjectures, asserts that all global L-functions should be automorphic. The Siegel zero, conjectured not to exist, is a possible real zero of a Dirichlet L-series, rather near s = 1. L-functions of Maass cusp forms can have trivial zeros which are off the real line. == References == == Further reading == Borwein, Peter B. (2008), The Riemann hypothesis: a resource for the aficionado and virtuoso alike, CMS books in mathematics, vol. 27, Springer-Verlag, ISBN 978-0-387-72125-5


id: hadamard-matrix-20260505-0
file: data/en.wikipedia.org/wiki/Hadamard_matrix-0.md
category: reference
title: Hadamard matrix
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Hadamard_matrix
date_saved: 2026-05-05T11:03:08.746930+00:00
instance: kb-cron
parent: hadamard-matrix-20260505-0
summary:
In mathematics, a Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. In geometric terms, this means that each pair of rows in a Hadamard matrix represents two perpendicular vectors, while in combinatorial terms, it means that each pair of rows has matching entries in exactly half of their columns and mismatched entries in the remaining columns. It is a consequence of this definition that the corresponding properties hold for columns as well as rows. The n-dimensional parallelotope spanned by the rows of an n × n Hadamard matrix has the maximum possible n-dimensional volume among parallelotopes spanned by vectors whose entries are bounded in absolute value by 1. Equivalently, a Hadamard matrix has maximal determinant among matrices with entries of absolute value less than or equal to 1 and so is an extremal solution of Hadamard's maximal determinant problem. Certain Hadamard matrices can almost directly be used as an error-correcting code using a Hadamard code (generalized in Reed–Muller codes), and are also used in balanced repeated replication (BRR), used by statisticians to estimate the variance of a parameter estimator. == Properties ==


id: hadamard-matrix-20260505-1
file: data/en.wikipedia.org/wiki/Hadamard_matrix-1.md
category: reference
title: Hadamard matrix
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Hadamard_matrix
date_saved: 2026-05-05T11:03:08.746930+00:00
instance: kb-cron
parent: hadamard-matrix-20260505-0
summary:
α − β = β − α {\displaystyle \alpha -\beta =\beta -\alpha } α = β {\displaystyle \alpha =\beta } But we have as the number of 1s in row 1 the odd number n / 2 = α + β {\displaystyle n/2=\alpha +\beta } , contradiction. == Sylvester's construction == Examples of Hadamard matrices were actually first constructed by James Joseph Sylvester in 1867. Let H be a Hadamard matrix of order n. Then the partitioned matrix [ H H H − H ] {\displaystyle {\begin{bmatrix}H&H\\H&-H\end{bmatrix}}} is a Hadamard matrix of order 2n. This observation can be applied repeatedly and leads to the following sequence of matrices, also called Walsh matrices. H 1 = [ 1 ] , H 2 = [ 1 1 1 − 1 ] , H 4 = [ 1 1 1 1 1 − 1 1 − 1 1 1 − 1 − 1 1 − 1 − 1 1 ] , {\displaystyle {\begin{aligned}H_{1}&={\begin{bmatrix}1\end{bmatrix}},\\H_{2}&={\begin{bmatrix}1&1\\1&-1\end{bmatrix}},\\H_{4}&={\begin{bmatrix}1&1&1&1\\1&-1&1&-1\\1&1&-1&-1\\1&-1&-1&1\end{bmatrix}},\end{aligned}}} and H 2 k = [ H 2 k − 1 H 2 k − 1 H 2 k − 1 − H 2 k − 1 ] = H 2 ⊗ H 2 k − 1 , {\displaystyle H_{2^{k}}={\begin{bmatrix}H_{2^{k-1}}&H_{2^{k-1}}\\H_{2^{k-1}}&-H_{2^{k-1}}\end{bmatrix}}=H_{2}\otimes


id: hadamard-matrix-20260505-2
file: data/en.wikipedia.org/wiki/Hadamard_matrix-2.md
category: reference
title: Hadamard matrix
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Hadamard_matrix
date_saved: 2026-05-05T11:03:08.746930+00:00
instance: kb-cron
parent: hadamard-matrix-20260505-0
summary:
== Equivalence and uniqueness == Two Hadamard matrices are considered equivalent if one can be obtained from the other by negating rows or columns, or by interchanging rows or columns. Up to equivalence, there is a unique Hadamard matrix of orders 1, 2, 4, 8, and 12. There are 5 inequivalent matrices of order 16, 3 of order 20, 60 of order 24, and 487 of order 28. Millions of inequivalent matrices are known for orders 32, 36, and 40. Using a coarser notion of equivalence that also allows transposition, there are 4 inequivalent matrices of order 16, 3 of order 20, 36 of order 24, and 294 of order 28. Hadamard matrices are also uniquely recoverable, in the following sense: If an Hadamard matrix H {\displaystyle H} of order n {\displaystyle n} has O ( n 2 / log ⁡ n ) {\displaystyle O(n^{2}/\log n)} entries randomly deleted, then with overwhelming likelihood, one can perfectly recover the original matrix H {\displaystyle H} from the damaged one. The algorithm of recovery has the same computational cost as matrix inversion. == Special cases == Many special cases of Hadamard matrices have been investigated in the mathematical literature. === Skew Hadamard matrices


id: hadamard-matrix-20260505-3
file: data/en.wikipedia.org/wiki/Hadamard_matrix-3.md
category: reference
title: Hadamard matrix
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Hadamard_matrix
date_saved: 2026-05-05T11:03:08.746930+00:00
instance: kb-cron
parent: hadamard-matrix-20260505-0
summary:
== Practical applications == Olivia MFSK – an amateur-radio digital protocol designed to work in difficult (low signal-to-noise ratio plus multipath propagation) conditions on shortwave bands. Balanced repeated replication (BRR) – a technique used by statisticians to estimate the variance of a statistical estimator. Coded aperture spectrometry – an instrument for measuring the spectrum of light. The mask element used in coded aperture spectrometers is often a variant of a Hadamard matrix. Feedback delay networks – Digital reverberation devices which use Hadamard matrices to blend sample values Plackett–Burman design of experiments for investigating the dependence of some measured quantity on a number of independent variables. Robust parameter designs for investigating noise factor impacts on responses Compressed sensing for signal processing and under-determined linear systems (inverse problems) Quantum Hadamard gate for quantum computing and the Hadamard transform for quantum algorithms. == See also == Combinatorial design Hadamard transform Quincunx matrix Walsh matrix Weighing matrix Quantum logic gate Algebraic signal processing – framework in which the Hadamard matrix arises as the character table of the elementary Abelian group Z 2 n {\displaystyle \mathbb {Z} _{2}^{n}} == Notes == == Further reading == Baumert, L. D.; Hall, Marshall (1965). "Hadamard matrices of the


id: hadamard-s-maximal-determinant-problem-20260505-0
file: data/en.wikipedia.org/wiki/Hadamard's_maximal_determinant_problem-0.md
category: reference
title: Hadamard's maximal determinant problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Hadamard's_maximal_determinant_problem
date_saved: 2026-05-05T11:03:09.913796+00:00
instance: kb-cron
parent: hadamard-s-maximal-determinant-problem-20260505-0
summary:
Hadamard's maximal determinant problem, named after Jacques Hadamard, asks for the largest determinant of a matrix with elements equal to 1 or −1. The analogous question for matrices with elements equal to 0 or 1 is equivalent since, as will be shown below, the maximal determinant of a {1,−1} matrix of size n is 2n−1 times the maximal determinant of a {0,1} matrix of size n−1. The problem was posed by Hadamard in the 1893 paper in which he presented his famous determinant bound and remains unsolved for matrices of general size. Hadamard's bound implies that {1, −1}-matrices of size n have determinant at most nn/2. Hadamard observed that a construction of Sylvester produces examples of matrices that attain the bound when n is a power of 2, and produced examples of his own of sizes 12 and 20. He also showed that the bound is only attainable when n is equal to 1, 2, or a multiple of 4. Additional examples were later constructed by Scarpis and Paley and subsequently by many other authors. Such matrices are now known as Hadamard matrices. They have received intensive study. Matrix sizes n for which n ≡ 1, 2, or 3 (mod


id: hadamard-s-maximal-determinant-problem-20260505-1
file: data/en.wikipedia.org/wiki/Hadamard's_maximal_determinant_problem-1.md
category: reference
title: Hadamard's maximal determinant problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Hadamard's_maximal_determinant_problem
date_saved: 2026-05-05T11:03:09.913796+00:00
instance: kb-cron
parent: hadamard-s-maximal-determinant-problem-20260505-0
summary:
is an integer matrix, is symmetric, is positive-semidefinite, has constant diagonal whose value equals n. Negating rows of R or applying a permutation to them results in the same negations and permutation being applied both to the rows, and to the corresponding columns, of G. We may also define the matrix G′=RTR. The matrix G is the usual Gram matrix of a set of vectors, derived from the set of rows of R, while G′ is the Gram matrix derived from the set of columns of R. A matrix R for which G = G′ is a normal matrix. Every known maximal-determinant matrix is equivalent to a normal matrix, but it is not known whether this is always the case. === Hadamard's bound (for all n) === Hadamard's bound can be derived by noting that |det R| = (det G)1/2 ≤ (det nI)1/2 = nn/2, which is a consequence of the observation that nI, where I is the n by n identity matrix, is the unique matrix of maximal determinant among matrices satisfying properties 1–4. That det R must be an integer multiple of 2n−1 can be used to provide another demonstration that Hadamard's bound is not always attainable. When


id: hadamard-s-maximal-determinant-problem-20260505-2
file: data/en.wikipedia.org/wiki/Hadamard's_maximal_determinant_problem-2.md
category: reference
title: Hadamard's maximal determinant problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Hadamard's_maximal_determinant_problem
date_saved: 2026-05-05T11:03:09.913796+00:00
instance: kb-cron
parent: hadamard-s-maximal-determinant-problem-20260505-0
summary:
det R ≤ ( n − 3 ) ( n − s ) / 2 ( n − 3 + 4 r ) u / 2 ( n + 1 + 4 r ) v / 2 [ 1 − u r n − 3 + 4 r − v ( r + 1 ) n + 1 + 4 r ] 1 / 2 , {\displaystyle \det R\leq (n-3)^{(n-s)/2}(n-3+4r)^{u/2}(n+1+4r)^{v/2}\left[1-{\frac {ur}{n-3+4r}}-{\frac {v(r+1)}{n+1+4r}}\right]^{1/2},} where v = n−rs is the number of blocks of size r+1 and u =s−v is the number of blocks of size r. Cohn analyzed the bound and determined that, apart from n = 3, it is an integer only for n = 112t2±28t+7 for some positive integer t. Tamura derived additional restrictions on the attainability of the bound using the Hasse–Minkowski theorem on the rational equivalence of quadratic forms, and showed that the smallest n > 3 for which Ehlich's bound is conceivably attainable is 511. == Maximal determinants up to size 22 == The maximal determinants of {1, −1} matrices up to size n = 22 are given in the following table. Size 23 is the smallest open case. In the table, D(n) represents the maximal


id: hartmanis-stearns-conjecture-20260505-0
file: data/en.wikipedia.org/wiki/Hartmanis–Stearns_conjecture-0.md
category: reference
title: Hartmanis–Stearns conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Hartmanis–Stearns_conjecture
date_saved: 2026-05-05T11:03:11.202494+00:00
instance: kb-cron
parent: hartmanis-stearns-conjecture-20260505-0
summary:
In theoretical computer science and mathematics, the Hartmanis–Stearns conjecture is an open problem named after Juris Hartmanis and Richard E. Stearns, who posed it in a 1965 paper that founded the field of computational complexity theory (earning them the 1993 ACM Turing Award). An infinite word is said to be real-time computable when there exists a multitape Turing machine which (run without input) writes the successive letters of the word on its output tape, taking a bounded amount of time between two successive letters. Equivalently, there exists a multitape Turing machine which given a natural number n {\displaystyle n} in unary outputs the first n {\displaystyle n} letters of the word in time O ( n ) {\displaystyle O(n)} . The Hartmanis–Stearns conjecture states that if x {\displaystyle x} is a real number whose expansion in some base b ≥ 2 {\displaystyle b\geq 2} (e.g., the decimal expansion for b = 10 {\displaystyle b=10} ) is real-time computable, then x {\displaystyle x} is rational or transcendental. The conjecture has the deep implication that there is no integer multiplication algorithm in O ( n ) {\displaystyle O(n)} (while an O ( n log ⁡ ( n ) ) {\displaystyle O(n\log(n))} algorithm


id: herzog-sch-nheim-conjecture-20260505-0
file: data/en.wikipedia.org/wiki/Herzog–Schönheim_conjecture-0.md
category: reference
title: Herzog–Schönheim conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Herzog–Schönheim_conjecture
date_saved: 2026-05-05T11:03:12.382021+00:00
instance: kb-cron
parent: herzog-sch-nheim-conjecture-20260505-0
summary:
In mathematics, the Herzog–Schönheim conjecture is a combinatorial problem in the area of group theory, posed by Marcel Herzog and Jochanan Schönheim in 1974. Let G {\displaystyle G} be a group, and let A = { a 1 G 1 , … , a k G k } {\displaystyle A=\{a_{1}G_{1},\ \ldots ,\ a_{k}G_{k}\}} be a finite system of left cosets of subgroups G 1 , … , G k {\displaystyle G_{1},\ldots ,G_{k}} of G {\displaystyle G} . Herzog and Schönheim conjectured that if A {\displaystyle A} forms a partition of G {\displaystyle G} with k > 1 {\displaystyle k>1} , then the (finite) indices [ G : G 1 ] , … , [ G : G k ] {\displaystyle [G:G_{1}],\ldots ,[G:G_{k}]} cannot be distinct. In contrast, if repeated indices are allowed, then partitioning a group into cosets is easy: if H {\displaystyle H} is any subgroup of G {\displaystyle G} with index k = [ G : H ] < ∞ {\displaystyle k=[G:H]<\infty } then G {\displaystyle G} can be partitioned into k {\displaystyle k} left cosets of H {\displaystyle H} . == Subnormal subgroups == In 2004, Zhi-Wei Sun proved a special case of the Herzog–Schönheim conjecture in


id: hilbert-p-lya-conjecture-20260505-0
file: data/en.wikipedia.org/wiki/Hilbert–Pólya_conjecture-0.md
category: reference
title: Hilbert–Pólya conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Hilbert–Pólya_conjecture
date_saved: 2026-05-05T11:03:13.566044+00:00
instance: kb-cron
parent: hilbert-p-lya-conjecture-20260505-0
summary:
In mathematics, the Hilbert–Pólya conjecture states that the non-trivial zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator. It is a possible approach to the Riemann hypothesis, by means of spectral theory. == History == In a letter to Andrew Odlyzko, dated January 3, 1982, George Pólya said that while he was in Göttingen around 1912 to 1914 he was asked by Edmund Landau for a physical reason that the Riemann hypothesis should be true, and suggested that this would be the case if the imaginary parts t of the zeros 1 2 + i t {\displaystyle {\tfrac {1}{2}}+it} of the Riemann zeta function corresponded to eigenvalues of a self-adjoint operator. The earliest published statement of the conjecture seems to be in Montgomery (1973). David Hilbert did not work in the central areas of analytic number theory, but his name has become known for the Hilbert–Pólya conjecture due to a story told by Ernst Hellinger, a student of Hilbert, to André Weil. Hellinger said that Hilbert announced in his seminar in the early 1900s that he expected the Riemann Hypothesis would be a consequence of Fredholm's work on integral equations with a symmetric kernel. === 1950s


id: hilbert-p-lya-conjecture-20260505-1
file: data/en.wikipedia.org/wiki/Hilbert–Pólya_conjecture-1.md
category: reference
title: Hilbert–Pólya conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Hilbert–Pólya_conjecture
date_saved: 2026-05-05T11:03:13.566044+00:00
instance: kb-cron
parent: hilbert-p-lya-conjecture-20260505-0
summary:
This refinement of the Hilbert–Pólya conjecture is known as the Berry conjecture (or the Berry–Keating conjecture). As of 2008, it is still quite far from being concrete, as it is not clear on which space this operator should act in order to get the correct dynamics, nor how to regularize it in order to get the expected logarithmic corrections. Berry and Keating have conjectured that since this operator is invariant under dilations perhaps the boundary condition f(nx) = f(x) for integer n may help to get the correct asymptotic results valid for large n 1 2 + i 2 π n log ⁡ n . {\displaystyle {\frac {1}{2}}+i{\frac {2\pi n}{\log n}}.} A paper was published in March 2017, written by Carl M. Bender, Dorje C. Brody, and Markus P. Müller, which builds on Berry's approach to the problem. There the operator H ^ = 1 1 − e − i p ^ ( x ^ p ^ + p ^ x ^ ) ( 1 − e − i p ^ ) {\displaystyle {\hat {H}}={\frac {1}{1-e^{-i{\hat {p}}}}}\left({\hat {x}}{\hat {p}}+{\hat {p}}{\hat {x}}\right)\left(1-e^{-i{\hat {p}}}\right)} was introduced, which they claim satisfies a certain modified versions of the conditions of the Hilbert–Pólya conjecture. Jean


id: hilbert-arnold-problem-20260505-0
file: data/en.wikipedia.org/wiki/Hilbert–Arnold_problem-0.md
category: reference
title: Hilbert–Arnold problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Hilbert–Arnold_problem
date_saved: 2026-05-05T11:03:14.731188+00:00
instance: kb-cron
parent: hilbert-arnold-problem-20260505-0
summary:
In mathematics, particularly in dynamical systems, the Hilbert–Arnold problem is an unsolved problem concerning the estimation of limit cycles of the dynamics of a flow on a sphere, and whether the number of such cycles can be bounded. A flow on a sphere means that one can imagine that the velocities of particles are prescribed, and a limit cycle is a limit of those velocities, like the Gulf Stream on the globe. The problem asks whether in a generic family of smooth vector fields, smoothly parameterized over a compact set in finite dimensional Euclidean space, the number of limit cycles is uniformly bounded across all parameter values. Thus, under perturbations of climate conditions, it asks whether there is a bounded number of "Gulf Streams". The problem is historically related to Hilbert's sixteenth problem and was first formulated by Russian mathematicians Vladimir Arnold and Yulij Ilyashenko in the 1980s. In Arnold's Problems there are many questions related to the Hilbert–Arnold problem: 1978–6, 1979–16, 1980–1, 1983–11, 1989–17, 1990–24, 1990–25, 1994–51 and 1994–52. == Overview == The problem arises from considering modern approaches to Hilbert's sixteenth problem. While Hilbert's original question focused on polynomial vector fields, mathematical attention shifted toward properties of generic


id: hilbert-s-problems-20260505-0
file: data/en.wikipedia.org/wiki/Hilbert's_problems-0.md
category: reference
title: Hilbert's problems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Hilbert's_problems
date_saved: 2026-05-05T11:03:16.004414+00:00
instance: kb-cron
parent: hilbert-s-problems-20260505-0
summary:
Hilbert's problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time, and several proved to be very influential for 20th-century mathematics. Hilbert presented ten of the problems (1, 2, 6, 7, 8, 13, 16, 19, 21, and 22) at the Paris conference of the International Congress of Mathematicians, speaking on August 8 at the Sorbonne. The complete list of 23 problems was published later, and translated into English in 1902 by Mary Frances Winston Newson in the Bulletin of the American Mathematical Society. Earlier publications (in the original German) appeared in Archiv der Mathematik und Physik. Of the cleanly formulated Hilbert problems: 3, 6a, 7, 10, 11, 14, 17, 18, 19, and 21 have resolutions that are accepted by consensus of the mathematical community. The status of problems 1, 2, 5, 6b, 8c, 13, and 15 is controversial: there are some results, but there exists some controversy as to whether they resolve the problems. Problems 8a, 8b, 9, 12, 16, 20 and 22 are unresolved or widely agreed as unresolved despite some partial results. Problems 4 and 23 are considered as too vague to ever be described as solved;


id: hilbert-s-problems-20260505-1
file: data/en.wikipedia.org/wiki/Hilbert's_problems-1.md
category: reference
title: Hilbert's problems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Hilbert's_problems
date_saved: 2026-05-05T11:03:16.004414+00:00
instance: kb-cron
parent: hilbert-s-problems-20260505-0
summary:
== Knowability == Following Gottlob Frege and Bertrand Russell, Hilbert sought to define mathematics logically using the method of formal systems, i.e., finitistic proofs from an agreed-upon set of axioms. One of the main goals of Hilbert's program was a finitistic proof of the consistency of the axioms of arithmetic: that is his second problem. However, Gödel's second incompleteness theorem gives a precise sense in which such a finitistic proof of the consistency of arithmetic is provably impossible. Hilbert lived for 12 years after Kurt Gödel published his theorem, but does not seem to have written any formal response to Gödel's work. Hilbert's tenth problem does not ask whether there exists an algorithm for deciding the solvability of Diophantine equations, but rather asks for the construction of such an algorithm: "to devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers". That this problem was solved by showing that there cannot be any such algorithm contradicted Hilbert's philosophy of mathematics. In discussing his opinion that every mathematical problem should have a solution, Hilbert allows for the possibility that the solution could be a proof that the


id: homological-conjectures-in-commutative-algebra-20260505-0
file: data/en.wikipedia.org/wiki/Homological_conjectures_in_commutative_algebra-0.md
category: reference
title: Homological conjectures in commutative algebra
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Homological_conjectures_in_commutative_algebra
date_saved: 2026-05-05T11:03:17.244447+00:00
instance: kb-cron
parent: homological-conjectures-in-commutative-algebra-20260505-0
summary:
In mathematics, homological conjectures have been a focus of research activity in commutative algebra since the early 1960s. They concern a number of interrelated (sometimes surprisingly so) conjectures relating various homological properties of a commutative ring to its internal ring structure, particularly its Krull dimension and depth. The following list given by Melvin Hochster is considered definitive for this area. In the sequel, A , R {\displaystyle A,R} , and S {\displaystyle S} refer to Noetherian commutative rings; R {\displaystyle R} will be a local ring with maximal ideal m R {\displaystyle m_{R}} , and M {\displaystyle M} and N {\displaystyle N} are finitely generated R {\displaystyle R} -modules. The Zero Divisor Theorem. If M ≠ 0 {\displaystyle M\neq 0} has finite projective dimension and r ∈ R {\displaystyle r\in R} is not a zero divisor on M {\displaystyle M} , then r {\displaystyle r} is not a zero divisor on R {\displaystyle R} . Bass's Question. If M ≠ 0 {\displaystyle M\neq 0} has a finite injective resolution, then R {\displaystyle R} is a Cohen–Macaulay ring. The Intersection Theorem. If M ⊗ R N ≠ 0 {\displaystyle M\otimes _{R}N\neq 0} has finite length, then the Krull dimension of N


id: invariant-subspace-problem-20260505-0
file: data/en.wikipedia.org/wiki/Invariant_subspace_problem-0.md
category: reference
title: Invariant subspace problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Invariant_subspace_problem
date_saved: 2026-05-05T11:03:18.440037+00:00
instance: kb-cron
parent: invariant-subspace-problem-20260505-0
summary:
In the field of mathematics known as functional analysis, the invariant subspace problem is a partially unresolved problem asking whether every bounded operator on a complex Banach space sends some non-trivial closed subspace to itself. Many variants of the problem have been solved, by restricting the class of bounded operators considered or by specifying a particular class of Banach spaces. The problem is still open for separable Hilbert spaces (in other words, each example, found so far, of an operator with no non-trivial invariant subspaces is an operator that acts on a Banach space that is not isomorphic to a separable Hilbert space). == History == The problem seems to have been stated in the mid-20th century after work by Beurling and von Neumann, who found (but never published) a positive solution for the case of compact operators. It was then posed by Paul Halmos for the case of operators T {\displaystyle T} such that T 2 {\displaystyle T^{2}} is compact. This was resolved affirmatively, for the more general class of polynomially compact operators (operators T {\displaystyle T} such that p ( T ) {\displaystyle p(T)} is a compact operator for a suitably chosen nonzero polynomial p {\displaystyle p} ),


id: invariant-subspace-problem-20260505-1
file: data/en.wikipedia.org/wiki/Invariant_subspace_problem-1.md
category: reference
title: Invariant subspace problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Invariant_subspace_problem
date_saved: 2026-05-05T11:03:18.440037+00:00
instance: kb-cron
parent: invariant-subspace-problem-20260505-0
summary:
For finite-dimensional complex vector spaces, every operator admits an eigenvector, so it has a 1-dimensional invariant subspace. The conjecture is true if the Hilbert space H {\displaystyle H} is not separable (i.e. if it has an uncountable orthonormal basis). In fact, if x {\displaystyle x} is a non-zero vector in H {\displaystyle H} , the norm closure of the linear orbit [ x ] {\displaystyle [x]} is separable (by construction) and hence a proper subspace and also invariant. von Neumann showed that any compact operator on a Hilbert space of dimension at least 2 has a non-trivial invariant subspace. The spectral theorem shows that all normal operators admit invariant subspaces. Aronszajn & Smith (1954) proved that every compact operator on any Banach space of dimension at least 2 has an invariant subspace. Bernstein & Robinson (1966) proved using non-standard analysis that if the operator T {\displaystyle T} on a Hilbert space is polynomially compact (in other words p ( T ) {\displaystyle p(T)} is compact for some nonzero polynomial p {\displaystyle p} ) then T {\displaystyle T} has an invariant subspace. Their proof uses the original idea of embedding the infinite-dimensional Hilbert space in a hyperfinite-dimensional Hilbert space (see Non-standard


id: invariant-subspace-problem-20260505-2
file: data/en.wikipedia.org/wiki/Invariant_subspace_problem-2.md
category: reference
title: Invariant subspace problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Invariant_subspace_problem
date_saved: 2026-05-05T11:03:18.440037+00:00
instance: kb-cron
parent: invariant-subspace-problem-20260505-0
summary:
== References == Abramovich, Yuri A.; Aliprantis, Charalambos D. (2002), An Invitation to Operator Theory, Graduate Studies in Mathematics, vol. 50, Providence, RI: American Mathematical Society, doi:10.1090/gsm/050, ISBN 978-0-8218-2146-6, MR 1921782 Argyros, Spiros A.; Haydon, Richard G. (2011), "A hereditarily indecomposable L∞-space that solves the scalar-plus-compact problem", Acta Math., 206 (1): 1–54, arXiv:0903.3921, doi:10.1007/s11511-011-0058-y, MR 2784662, S2CID 119532059 Aronszajn, N.; Smith, K. T. (1954), "Invariant subspaces of completely continuous operators", Annals of Mathematics, Second Series, 60 (2): 345–350, doi:10.2307/1969637, JSTOR 1969637, MR 0065807 Atzmon, Aharon (1983), "An operator without invariant subspaces on a nuclear Fréchet space", Annals of Mathematics, Second Series, 117 (3): 669–694, doi:10.2307/2007039, JSTOR 2007039, MR 0701260 Beauzamy, Bernard (1985), "Un opérateur sans sous-espace invariant: simplification de l'exemple de P. Enflo" [An operator with no invariant subspace: simplification of the example of P. Enflo], Integral Equations and Operator Theory (in French), 8 (3): 314–384, doi:10.1007/BF01202903, MR 0792905, S2CID 121418247 Beauzamy, Bernard (1988), Introduction to operator theory and invariant subspaces, North-Holland Mathematical Library, vol. 42, Amsterdam: North-Holland, ISBN 978-0-444-70521-1, MR 0967989 Bernstein, Allen R.; Robinson, Abraham (1966), "Solution of an invariant subspace problem of K. T. Smith and P. R. Halmos", Pacific Journal of Mathematics, 16 (3): 421–431, doi:10.2140/pjm.1966.16.421,


id: inverse-galois-problem-20260505-0
file: data/en.wikipedia.org/wiki/Inverse_Galois_problem-0.md
category: reference
title: Inverse Galois problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Inverse_Galois_problem
date_saved: 2026-05-05T11:03:19.714664+00:00
instance: kb-cron
parent: inverse-galois-problem-20260505-0
summary:
In Galois theory, the inverse Galois problem concerns whether or not every finite group appears as the Galois group of some Galois extension of the rational numbers Q {\displaystyle \mathbb {Q} } . This problem, first posed in the early 19th century, is unsolved. There are some permutation groups for which generic polynomials are known, which define all algebraic extensions of Q {\displaystyle \mathbb {Q} } having a particular group as Galois group. These groups include all of degree no greater than 5. There also are groups known not to have generic polynomials, such as the cyclic group of order 8. More generally, let G be a given finite group, and K a field. If there is a Galois extension field L/K whose Galois group is isomorphic to G, one says that G is realizable over K. == Partial results == Many cases are known. It is known that every finite group is realizable over any function field in one variable over the complex numbers C {\displaystyle \mathbb {C} } , and more generally over function fields in one variable over any algebraically closed field of characteristic zero. Igor Shafarevich showed that every finite solvable group is realizable over Q


id: inverse-galois-problem-20260505-1
file: data/en.wikipedia.org/wiki/Inverse_Galois_problem-1.md
category: reference
title: Inverse Galois problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Inverse_Galois_problem
date_saved: 2026-05-05T11:03:19.714664+00:00
instance: kb-cron
parent: inverse-galois-problem-20260505-0
summary:
x n − x 2 − 1 2 − ( s − 1 2 ) ( x + 1 ) {\displaystyle x^{n}-{\tfrac {x}{2}}-{\tfrac {1}{2}}-\left(s-{\tfrac {1}{2}}\right)\!(x+1)} and f(x, 1/2) can be factored to: 1 2 ( x − 1 ) ( 1 + 2 x + 2 x 2 + ⋯ + 2 x n − 1 ) {\displaystyle {\tfrac {1}{2}}(x-1)\!\left(1+2x+2x^{2}+\cdots +2x^{n-1}\right)} whose second factor is irreducible (but not by Eisenstein's criterion). Only the reciprocal polynomial is irreducible by Eisenstein's criterion. We have now shown that the group Gal(f(x, s)/Q(s)) is doubly transitive. We can then find that this Galois group has a transposition. Use the scaling (1 − n)x = ny to get y n − { s ( 1 − n n ) n − 1 } y − { s ( 1 − n n ) n } {\displaystyle y^{n}-\left\{s\left({\frac {1-n}{n}}\right)^{n-1}\right\}y-\left\{s\left({\frac {1-n}{n}}\right)^{n}\right\}} and with t = s ( 1 − n ) n − 1 n n , {\displaystyle t={\frac {s(1-n)^{n-1}}{n^{n}}},} we arrive at: g(y, t) = yn − nty + (n − 1)t which can be arranged to yn − y − (n − 1)(y − 1) + (t − 1)(−ny + n − 1). Then g(y,


id: inverse-galois-problem-20260505-2
file: data/en.wikipedia.org/wiki/Inverse_Galois_problem-2.md
category: reference
title: Inverse Galois problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Inverse_Galois_problem
date_saved: 2026-05-05T11:03:19.714664+00:00
instance: kb-cron
parent: inverse-galois-problem-20260505-0
summary:
== Rigid groups == Suppose that C1, …, Cn are conjugacy classes of a finite group G, and A be the set of n-tuples (g1, …, gn) of G such that gi is in Ci and the product g1…gn is trivial. Then A is called rigid if it is nonempty, G acts transitively on it by conjugation, and each element of A generates G. Thompson (1984) showed that if a finite group G has a rigid set then it can often be realized as a Galois group over a cyclotomic extension of the rationals. (More precisely, over the cyclotomic extension of the rationals generated by the values of the irreducible characters of G on the conjugacy classes Ci.) This can be used to show that many finite simple groups, including the monster group, are Galois groups of extensions of the rationals. The monster group is generated by a triad of elements of orders 2, 3, and 29. All such triads are conjugate. The prototype for rigidity is the symmetric group Sn, which is generated by an n-cycle and a transposition whose product is an (n − 1)-cycle. The construction in the preceding section used these generators to establish a polynomial's


id: isomorphism-problem-of-coxeter-groups-20260505-0
file: data/en.wikipedia.org/wiki/Isomorphism_problem_of_Coxeter_groups-0.md
category: reference
title: Isomorphism problem of Coxeter groups
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Isomorphism_problem_of_Coxeter_groups
date_saved: 2026-05-05T11:03:20.894258+00:00
instance: kb-cron
parent: isomorphism-problem-of-coxeter-groups-20260505-0
summary:
It is an unresolved problem in the mathematical field of group theory to determine whether or not two Coxeter groups (specified by their Coxeter diagrams) are isomorphic as abstract groups. Equivalently, the problem asks to determine, for a given Coxeter group W {\displaystyle W} , the possible subsets S {\displaystyle S} of W {\displaystyle W} that are Coxeter generating sets for W {\displaystyle W} (that is, for which ( W , S ) {\displaystyle (W,S)} is a Coxeter system). A slight generalization of the problem can be made by asking to find to all isomorphisms from one group onto the other. In 2022, Yuri Santos Rego and Petra Schwer introduced a new framework to deal with the problem (a finite dimensional, locally finite, ranked simplicial complex to capture isomorphisms between finite rank Coxeter systems) and asked more related open questions motivated by it. == References ==


id: jacobi-bound-problem-20260505-0
file: data/en.wikipedia.org/wiki/Jacobi_bound_problem-0.md
category: reference
title: Jacobi bound problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Jacobi_bound_problem
date_saved: 2026-05-05T11:03:22.035859+00:00
instance: kb-cron
parent: jacobi-bound-problem-20260505-0
summary:
The Jacobi bound problem concerns the veracity of Jacobi's inequality which is an inequality on the absolute dimension of a differential algebraic variety in terms of its defining equations. This is one of Kolchin's Problems. The inequality is the differential algebraic analog of Bézout's theorem in affine space. Although first formulated by Jacobi, In 1936 Joseph Ritt recognized the problem as non-rigorous in that Jacobi didn't even have a rigorous notion of absolute dimension (Jacobi and Ritt used the term "order" - which Ritt first gave a rigorous definition for using the notion of transcendence degree). Intuitively, the absolute dimension is the number of constants of integration required to specify a solution of a system of ordinary differential equations. A mathematical proof of the inequality has been open since 1936. == Statement == Let ( K , ∂ ) {\displaystyle (K,\partial )} be a differential field of characteristic zero and consider Γ {\displaystyle \Gamma } a differential algebraic variety determined by the vanishing of differential polynomials u 1 , … , u n ∈ K { x 1 , … , x n } {\displaystyle u_{1},\ldots ,u_{n}\in K\lbrace x_{1},\ldots ,x_{n}\rbrace } . If Γ 1 {\displaystyle \Gamma _{1}} is an


id: jacobson-s-conjecture-20260505-0
file: data/en.wikipedia.org/wiki/Jacobson's_conjecture-0.md
category: reference
title: Jacobson's conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Jacobson's_conjecture
date_saved: 2026-05-05T11:03:23.190472+00:00
instance: kb-cron
parent: jacobson-s-conjecture-20260505-0
summary:
In abstract algebra, Jacobson's conjecture is an open problem in ring theory concerning the intersection of powers of the Jacobson radical of a Noetherian ring. It has only been proven for special types of Noetherian rings, so far. Examples exist to show that the conjecture can fail when the ring is not Noetherian on a side, so it is absolutely necessary for the ring to be two-sided Noetherian. The conjecture is named for the algebraist Nathan Jacobson who posed the first version of the conjecture. == Statement == For a ring R with Jacobson radical J, the nonnegative powers J n {\displaystyle J^{n}} are defined by using the product of ideals. Jacobson's conjecture: In a right-and-left Noetherian ring, ⋂ n ∈ N J n = { 0 } . {\displaystyle \bigcap _{n\in \mathbb {N} }J^{n}=\{0\}.} In other words: "The only element of a Noetherian ring in all powers of J is 0." The original conjecture posed by Jacobson in 1956 asked about noncommutative one-sided Noetherian rings, however Israel Nathan Herstein produced a counterexample in 1965, and soon afterwards, Arun Vinayak Jategaonkar produced a different example which was a left principal ideal domain. From that point on, the conjecture was reformulated


id: kaplansky-s-conjectures-20260505-0
file: data/en.wikipedia.org/wiki/Kaplansky's_conjectures-0.md
category: reference
title: Kaplansky's conjectures
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Kaplansky's_conjectures
date_saved: 2026-05-05T11:03:24.413902+00:00
instance: kb-cron
parent: kaplansky-s-conjectures-20260505-0
summary:
The mathematician Irving Kaplansky is notable for proposing numerous conjectures in several branches of mathematics, including a list of ten conjectures on Hopf algebras. They are usually known as Kaplansky's conjectures. == Group rings == Let K be a field, and G a torsion-free group. Kaplansky's zero divisor conjecture states: The group ring K[G] does not contain nontrivial zero divisors, that is, it is a domain. Two related conjectures are known as, respectively, Kaplansky's idempotent conjecture: K[G] does not contain any non-trivial idempotents, i.e., if a2 = a, then a = 1 or a = 0. and Kaplansky's unit conjecture (which was originally made by Graham Higman and popularized by Kaplansky): K[G] does not contain any non-trivial units, i.e., if ab = 1 in K[G], then a = kg for some k in K and g in G. The zero-divisor conjecture implies the idempotent conjecture and is implied by the unit conjecture. As of 2021, the zero divisor and idempotent conjectures are open. The unit conjecture, however, was disproved in characteristic 2 by Giles Gardam by exhibiting an explicit counterexample in a crystallographic group, namely the fundamental group of the Hantzsche–Wendt manifold; see also Fibonacci group. A later preprint by


id: k-the-conjecture-20260505-0
file: data/en.wikipedia.org/wiki/Köthe_conjecture-0.md
category: reference
title: Köthe conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Köthe_conjecture
date_saved: 2026-05-05T11:03:25.599538+00:00
instance: kb-cron
parent: k-the-conjecture-20260505-0
summary:
In mathematics, the Köthe conjecture is a problem in ring theory, open as of 2025. It is formulated in various ways. Suppose that R is a ring. One way to state the conjecture is that if R has no nil ideal, other than {0}, then it has no nil one-sided ideal, other than {0}. This question was posed in 1930 by Gottfried Köthe (1905–1989). The Köthe conjecture has been shown to be true for various classes of rings, such as polynomial identity rings and right Noetherian rings, but a general solution remains elusive. == Equivalent formulations == The conjecture has several different formulations: (Köthe conjecture) In any ring, the sum of two nil left ideals is nil. In any ring, the sum of two one-sided nil ideals is nil. In any ring, every nil left or right ideal of the ring is contained in the upper nil radical of the ring. For any ring R and for any nil ideal J of R, the matrix ideal Mn(J) is a nil ideal of Mn(R) for every n. For any ring R and for any nil ideal J of R, the matrix ideal M2(J) is a nil ideal of M2(R). For any


id: kurosh-problem-20260505-0
file: data/en.wikipedia.org/wiki/Kurosh_problem-0.md
category: reference
title: Kurosh problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Kurosh_problem
date_saved: 2026-05-05T11:03:26.737798+00:00
instance: kb-cron
parent: kurosh-problem-20260505-0
summary:
In mathematics, the Kurosh problem is one general problem, and several more special questions, in ring theory. The general problem is known to have a negative solution, since one of the special cases has been shown to have counterexamples. These matters were brought up by Aleksandr Gennadievich Kurosh as analogues of the Burnside problem in group theory. Kurosh asked whether there can be a finitely-generated infinite-dimensional algebraic algebra (the problem being to show this cannot happen). A special case is whether or not every nil algebra is locally nilpotent. For PI-algebras the Kurosh problem has a positive solution. Golod showed a counterexample to that case, as an application of the Golod–Shafarevich theorem. The Kurosh problem on group algebras concerns the augmentation ideal I. If I is a nil ideal, is the group algebra locally nilpotent? There is an important problem which is often referred as the Kurosh's problem on division rings. The problem asks whether there exists an algebraic (over the center) division ring which is not locally finite. This problem has not been solved until now. == References == Vesselin S. Drensky, Edward Formanek (2004), Polynomial Identity Rings, p. 89. Some open problems in the theory of infinite dimensional


id: m-g-k-queue-20260505-0
file: data/en.wikipedia.org/wiki/M/G/k_queue-0.md
category: reference
title: M/G/k queue
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/M/G/k_queue
date_saved: 2026-05-05T11:03:27.937939+00:00
instance: kb-cron
parent: m-g-k-queue-20260505-0
summary:
In queueing theory, a discipline within the mathematical theory of probability, an M/G/k queue is a queue model where arrivals are Markovian (modulated by a Poisson process), service times have a general distribution and there are k servers. The model name is written in Kendall's notation, and is an extension of the M/M/c queue, where service times must be exponentially distributed and of the M/G/1 queue with a single server. Most performance metrics for this queueing system are not known and remain an open problem. == Model definition == A queue represented by a M/G/k queue is a stochastic process whose state space is the set {0,1,2,3...}, where the value corresponds to the number of customers in the queue, including any being served. Transitions from state i to i + 1 represent the arrival of a new customer: the times between such arrivals have an exponential distribution with parameter λ. Transitions from state i to i − 1 represent the departure of a customer who has just finished being served: the length of time required for serving an individual customer has a general distribution function. The lengths of times between arrivals and of service periods are random variables which are


id: magic-square-20260505-0
file: data/en.wikipedia.org/wiki/Magic_square-0.md
category: reference
title: Magic square
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Magic_square
date_saved: 2026-05-05T11:03:29.149149+00:00
instance: kb-cron
parent: magic-square-20260505-0
summary:
In mathematics, especially historical and recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The order of the magic square is the number of integers along one side (n), and the constant sum is called the magic constant. If the array includes just the positive integers 1 , 2 , . . . , n 2 {\displaystyle 1,2,...,n^{2}} , the magic square is said to be normal. Many authors take magic square to mean normal magic square. Magic squares that include repeated entries do not fall under this definition and are referred to as trivial. Some well-known examples, including the Sagrada Família magic square are trivial in this sense. When all the rows and columns but not both diagonals sum to the magic constant, this gives a semimagic square (sometimes called orthomagic square). The mathematical study of magic squares typically deals with its construction, classification, and enumeration. Although completely general methods for producing all the magic squares of all orders do not exist, historically three general techniques have been discovered: by bordering, by making composite magic


id: magic-square-20260505-1
file: data/en.wikipedia.org/wiki/Magic_square-1.md
category: reference
title: Magic square
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Magic_square
date_saved: 2026-05-05T11:03:29.149149+00:00
instance: kb-cron
parent: magic-square-20260505-0
summary:
The above magic squares of orders 3 to 9 are taken from Yang Hui's treatise, in which the Luo Shu principle is clearly evident. The order 5 square is a bordered magic square, with central 3×3 square formed according to Luo Shu principle. The order 9 square is a composite magic square, in which the nine 3×3 sub squares are also magic. After Yang Hui, magic squares frequently occur in Chinese mathematics such as in Ding Yidong's Dayan suoyin (c. 1300), Cheng Dawei's Suanfa tongzong (1593), Fang Zhongtong's Shuduyan (1661) which contains magic circles, cubes and spheres, Zhang Chao's Xinzhai zazu (c. 1650), who published China's first magic square of order ten, and lastly Bao Qishou's Binaishanfang ji (c. 1880), who gave various three dimensional magic configurations. However, despite being the first to discover the magic squares and getting a head start by several centuries, the Chinese development of the magic squares are much inferior compared to the Indian, Middle Eastern, or European developments. The high point of Chinese mathematics that deals with the magic squares seems to be contained in the work of Yang Hui; but even as a collection of older methods, this work is much more primitive,


id: magic-square-20260505-2
file: data/en.wikipedia.org/wiki/Magic_square-2.md
category: reference
title: Magic square
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Magic_square
date_saved: 2026-05-05T11:03:29.149149+00:00
instance: kb-cron
parent: magic-square-20260505-0
summary:
The square of Varahamihira as given above has sum of 18. Here the numbers 1 to 8 appear twice in the square. It is a pan-diagonal magic square. Four different magic squares can be obtained by adding 8 to one of the two sets of 1 to 8 sequence. The sequence is selected such that the number 8 is added exactly twice in each row, each column and each of the main diagonals. One of the possible magic squares shown in the right side. This magic square is remarkable in that it is a 90 degree rotation of a magic square that appears in the 13th century Islamic world as one of the most popular magic squares. The construction of 4th-order magic square is detailed in a work titled Kaksaputa, composed by the alchemist Nagarjuna around 10th century CE. All of the squares given by Nagarjuna are 4×4 magic squares, and one of them is called Nagarjuniya after him. Nagarjuna gave a method of constructing 4×4 magic square using a primary skeleton square, given an odd or even magic sum. The Nagarjuniya square is given below, and has the sum total of 100. The Nagarjuniya square is a pan-diagonal magic


id: magic-square-20260505-3
file: data/en.wikipedia.org/wiki/Magic_square-3.md
category: reference
title: Magic square
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Magic_square
date_saved: 2026-05-05T11:03:29.149149+00:00
instance: kb-cron
parent: magic-square-20260505-0
summary:
The 11th century saw the finding of several ways to construct simple magic squares for odd and evenly even orders; the more difficult case of oddly even case (n = 4k + 2) was solved by Ibn al-Haytham with k even (c. 1040), and completely by the beginning of 12th century, if not already in the latter half of the 11th century. Around the same time, pandiagonal squares were being constructed. Treaties on magic squares were numerous in the 11th and 12th century. These later developments tended to be improvements on or simplifications of existing methods. From the 13th century onwards, magic squares were increasingly put to occult purposes. However, much of these later texts written for occult purposes merely depict certain magic squares and mention their attributes, without describing their principle of construction, with only some authors keeping the general theory alive. One such occultist was the Algerian Ahmad al-Buni (c. 1225), who gave general methods on constructing bordered magic squares; some others were the 17th century Egyptian Shabramallisi and the 18th century Nigerian al-Kishnawi. The magic square of order three was described as a child-bearing charm since its first literary appearances in the alchemical works of Jābir ibn


id: magic-square-20260505-4
file: data/en.wikipedia.org/wiki/Magic_square-4.md
category: reference
title: Magic square
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Magic_square
date_saved: 2026-05-05T11:03:29.149149+00:00
instance: kb-cron
parent: magic-square-20260505-0
summary:
In 1624 France, Claude Gaspard Bachet described the "diamond method" for constructing Agrippa's odd ordered squares in his book Problèmes Plaisants. During 1640 Bernard Frenicle de Bessy and Pierre Fermat exchanged letters on magic squares and cubes, and in one of the letters Fermat boasts of being able to construct 1,004,144,995,344 magic squares of order 8 by his method. An early account on the construction of bordered squares was given by Antoine Arnauld in his Nouveaux éléments de géométrie (1667). In the two treatise Des quarrez ou tables magiques and Table générale des quarrez magiques de quatre de côté, published posthumously in 1693, twenty years after his death, Bernard Frenicle de Bessy demonstrated that there were exactly 880 distinct magic squares of order four. Frenicle gave methods to construct magic square of any odd and even order, where the even ordered squares were constructed using borders. He also showed that interchanging rows and columns of a magic square produced new magic squares. In 1691, Simon de la Loubère described the Indian continuous method of constructing odd ordered magic squares in his book Du Royaume de Siam, which he had learned while returning from a diplomatic mission to Siam, which was


id: magic-square-20260505-5
file: data/en.wikipedia.org/wiki/Magic_square-5.md
category: reference
title: Magic square
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Magic_square
date_saved: 2026-05-05T11:03:29.149149+00:00
instance: kb-cron
parent: magic-square-20260505-0
summary:
The order four normal magic square Albrecht Dürer immortalized in his 1514 engraving Melencolia I, referred to above, is believed to be the first seen in European art. The square associated with Jupiter appears as a talisman used to drive away melancholy. It is very similar to Yang Hui's square, which was created in China about 250 years before Dürer's time. As with every order 4 normal magic square, the magic sum is 34. But in the Durer square this sum is also found in each of the quadrants, in the center four squares, and in the corner squares (of the 4×4 as well as the four contained 3×3 grids). This sum can also be found in the four outer numbers clockwise from the corners (3+8+14+9) and likewise the four counter-clockwise (the locations of four queens in the two solutions of the 4 queens puzzle), the two sets of four symmetrical numbers (2+8+9+15 and 3+5+12+14), the sum of the middle two entries of the two outer columns and rows (5+9+8+12 and 3+2+15+14), and in four kite or cross shaped quartets (3+5+11+15, 2+10+8+14, 3+9+7+15, and 2+6+12+14). The two numbers in the middle of the bottom row give the date of the


id: magic-square-20260505-6
file: data/en.wikipedia.org/wiki/Magic_square-6.md
category: reference
title: Magic square
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Magic_square
date_saved: 2026-05-05T11:03:29.149149+00:00
instance: kb-cron
parent: magic-square-20260505-0
summary:
This representation may not be unique in general. By Marcus-Ree theorem, however, there need not be more than k ≤ n 2 − 2 n + 2 {\displaystyle k\leq n^{2}-2n+2} terms in any decomposition. Clearly, this decomposition carries over to magic squares as well, since a magic square can be recovered from a doubly stochastic matrix by multiplying it by the magic constant. == Classification of magic squares == While the classification of magic squares can be done in many ways, some useful categories are given below. An n×n square array of integers 1, 2, ..., n2 is called: Semi-magic square when its rows and columns sum to give the magic constant. Simple magic square when its rows, columns, and two diagonals sum to give magic constant and no more. They are also known as ordinary magic squares or normal magic squares. Self-complementary magic square when it is a magic square which when complemented (i.e. each number subtracted from n2 + 1) will give a rotated or reflected version of the original magic square. Associative magic square when it is a magic square with a further property that every number added to the number equidistant, in a straight line, from


id: magic-square-20260505-7
file: data/en.wikipedia.org/wiki/Magic_square-7.md
category: reference
title: Magic square
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Magic_square
date_saved: 2026-05-05T11:03:29.149149+00:00
instance: kb-cron
parent: magic-square-20260505-0
summary:
== Transformations that preserve the magic property == === For any magic square === A magic square remains magic when its numbers are multiplied by any constant. A magic square remains magic when a constant is added or subtracted to its numbers, or if its numbers are subtracted from a constant. In particular, if every element in a normal magic square is subtracted from n2 + 1, the resulting square is the complement of the original square. In the example below, elements of 4×4 square on the left is subtracted from 17 to obtain the complement of the square on the right. The numbers of a magic square can be substituted with corresponding numbers from a set of s arithmetic progressions with the same common difference among r terms, such that r × s = n2, and whose initial terms are also in arithmetic progression, to obtain a non-normal magic square. Here either s or r should be a multiple of n. Let us have s arithmetic progressions given by a a + c a + 2 c ⋯ a + ( r − 1 ) c a + d a + c + d a + 2 c +


id: magic-square-20260505-8
file: data/en.wikipedia.org/wiki/Magic_square-8.md
category: reference
title: Magic square
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Magic_square
date_saved: 2026-05-05T11:03:29.149149+00:00
instance: kb-cron
parent: magic-square-20260505-0
summary:
A magic square remains magic when any of its non-central rows x and y are interchanged, along with the interchange of their complementary rows n − x + 1 and n − y + 1; and then interchanging like columns. This is a generalization of the above two transforms. When y = n − x + 1, this transform reduces to the first of the above two transforms. When x and y are on the same side of the center line, this transform reduces to the second of the above two transforms. In the example below, the original square is on the left side, while the final square on the right. The middle square has been obtained by interchanging rows 1 and 3, and rows 2 and 4 of the original square. The final square on the right is obtained by interchanging columns 1 and 3, and columns 2 and 4 of the middle square. In this example, this transform amounts to interchanging the quadrants diagonally. Since the original square is associative, the middle square also happens to be magic. A magic square remains magic when its quadrants are diagonally interchanged because this is another symmetric permutation of the form


id: magic-square-20260505-9
file: data/en.wikipedia.org/wiki/Magic_square-9.md
category: reference
title: Magic square
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Magic_square
date_saved: 2026-05-05T11:03:29.149149+00:00
instance: kb-cron
parent: magic-square-20260505-0
summary:
A bordered magic square remains a bordered magic square after each of its concentric borders are independently rotated or reflected with respect to the central core magic square. If there are b borders, then this transform will yield 8b equivalent squares. In the example below of the 5×5 magic square, the border has been rotated 90 degrees anti-clockwise. === For composite magic squares === A composite magic square remains a composite magic square when the embedded magic squares undergo transformations that do not disturb the magic property (e.g. rotation, reflection, shifting of rows and columns, and so on). == Special methods of construction == == Solving partially completed magic squares == Solving partially completed magic squares is a popular mathematical pastime. The techniques needed are similar to those used in Sudoku or KenKen puzzles, and involve deducing the values of unfilled squares using logic and permutation group theory (Sudoku grids are not magic squares but are based on a related idea called Graeco-Latin squares). == Variations of the magic square == === Extra constraints === Certain extra restrictions can be imposed on magic squares. If raising each number to the nth power yields another magic square, the result is a


id: magic-square-20260505-10
file: data/en.wikipedia.org/wiki/Magic_square-10.md
category: reference
title: Magic square
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Magic_square
date_saved: 2026-05-05T11:03:29.149149+00:00
instance: kb-cron
parent: magic-square-20260505-0
summary:
== Magic squares in occultism == Magic squares of order 3 through 9, assigned to the seven planets, and described as means to attract the influence of planets and their angels (or demons) during magical practices, can be found in several manuscripts all around Europe starting at least since the 15th century. Among the best known, the Liber de Angelis, a magical handbook written around 1440, is included in Cambridge Univ. Lib. MS Dd.xi.45. The text of the Liber de Angelis is very close to that of De septem quadraturis planetarum seu quadrati magici, another handbook of planetary image magic contained in the Codex 793 of the Biblioteka Jagiellońska (Ms BJ 793). The magical operations involve engraving the appropriate square on a plate made with the metal assigned to the corresponding planet, as well as performing a variety of rituals. For instance, the 3×3 square, that belongs to Saturn, has to be inscribed on a lead plate. It will, in particular, help women during a difficult childbirth. In about 1510 Heinrich Cornelius Agrippa wrote De Occulta Philosophia, drawing on the Hermetic and magical works of Marsilio Ficino and Pico della Mirandola. In its 1531 edition, he expounded on the magical


id: magic-square-of-squares-20260505-0
file: data/en.wikipedia.org/wiki/Magic_square_of_squares-0.md
category: reference
title: Magic square of squares
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Magic_square_of_squares
date_saved: 2026-05-05T11:03:30.353991+00:00
instance: kb-cron
parent: magic-square-of-squares-20260505-0
summary:
The magic square of squares is an unsolved problem in mathematics which asks whether it is possible to construct a three-by-three magic square, the elements of which are all square numbers. The problem was first posed anonymously by Martin LaBar in 1984, before being included in Richard Guy's Unsolved problems in number theory (2nd edition) in 1994. The problem has been a popular choice for recreational mathematicians following two articles Martin Gardner published in Quantum Magazine on the problem, offering a prize of US$100 in 1996. Other prizes have subsequently been offered for the first solution. == Background == A magic square is a square array of integer numbers in which each row, column, and diagonal sums to the same number. The order of the square refers to the number of integers along each side. A trivial magic square is a magic square which has at least one repeated element, and a semimagic square is a magic square in which the rows and columns, but not both diagonals, sum to the same number. == Problem == The problem asks whether it is possible to construct a third-order magic square such that every element is itself a square number. A square


id: map-folding-20260505-0
file: data/en.wikipedia.org/wiki/Map_folding-0.md
category: reference
title: Map folding
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Map_folding
date_saved: 2026-05-05T11:03:31.538980+00:00
instance: kb-cron
parent: map-folding-20260505-0
summary:
In the mathematics of paper folding, map folding and stamp folding are two problems of counting the number of ways that a piece of paper can be folded. In the stamp folding problem, the paper is a strip of stamps with creases between them, and the folds must lie on the creases. In the map folding problem, the paper is a map, divided by creases into rectangles, and the folds must again lie only along these creases. Lucas (1891) credits the invention of the stamp folding problem to Émile Lemoine. Touchard (1950) provides several other early references. == Labeled stamps == In the stamp folding problem, the paper to be folded is a strip of square or rectangular stamps, separated by creases, and the stamps can only be folded along those creases. In one commonly considered version of the problem, each stamp is considered to be distinguishable from each other stamp, so two foldings of a strip of stamps are considered equivalent only when they have the same vertical sequence of stamps. For example, there are six ways to fold a strip of three different stamps: These include all six permutations of the stamps, but for more than three stamps


id: matrix-mortality-problem-20260505-0
file: data/en.wikipedia.org/wiki/Matrix_mortality_problem-0.md
category: reference
title: Matrix mortality problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Matrix_mortality_problem
date_saved: 2026-05-05T11:03:32.800214+00:00
instance: kb-cron
parent: matrix-mortality-problem-20260505-0
summary:
In computer science, the matrix mortality problem (or mortal matrix problem) is a decision problem that asks, given a set of size m of n×n matrices with integer coefficients, whether the zero matrix can be expressed as a finite product of matrices from this set. The matrix mortality problem is known to be undecidable when n ≥ 3. In fact, it is already undecidable for sets of 6 matrices (or more) when n = 3, for 4 matrices when n = 5, for 3 matrices when n = 9, and for 2 matrices when n = 15. In the case n = 2, it is an open problem whether matrix mortality is decidable, but several special cases have been solved: the problem is decidable for sets of 2 matrices, and for sets of matrices which contain at most one invertible matrix. == References == Bell, Paul; Potapov, Igor (2008-02-04). "On undecidability bounds for matrix decision problems". Theoretical Computer Science. Combinatorics, Automata and Number Theory. 391 (1): 3–13. doi:10.1016/j.tcs.2007.10.025. ISSN 0304-3975. Halava, Vesa (August 1997). Decidable and Undecidable Problems in Matrix Theory (Report). Turku Centre for Computer Science.>


id: mean-value-problem-20260505-0
file: data/en.wikipedia.org/wiki/Mean_value_problem-0.md
category: reference
title: Mean value problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Mean_value_problem
date_saved: 2026-05-05T11:03:33.999443+00:00
instance: kb-cron
parent: mean-value-problem-20260505-0
summary:
In mathematics, the mean value problem was posed by Stephen Smale in 1981. This problem is still open in full generality. The problem asks: For a given complex polynomial f {\displaystyle f} of degree d ≥ 2 {\displaystyle d\geq 2} and a complex number z {\displaystyle z} , is there a critical point c {\displaystyle c} of f {\displaystyle f} (i.e. f ′ ( c ) = 0 {\displaystyle f'(c)=0} ) such that | f ( z ) − f ( c ) z − c | ≤ K | f ′ ( z ) | for K = 1 ? {\displaystyle \left|{\frac {f(z)-f(c)}{z-c}}\right|\leq K|f'(z)|{\text{ for }}K=1{\text{?}}} It was proved for K = 4 {\displaystyle K=4} . For a polynomial of degree d {\displaystyle d} the constant K {\displaystyle K} has to be at least d − 1 d {\displaystyle {\frac {d-1}{d}}} from the example f ( z ) = z d − d z {\displaystyle f(z)=z^{d}-dz} , therefore no bound better than K = 1 {\displaystyle K=1} can exist. == Partial results == The conjecture is known to hold in special cases; for other cases, the bound on K {\displaystyle K} could be improved depending on the degree d


id: millennium-prize-problems-20260505-0
file: data/en.wikipedia.org/wiki/Millennium_Prize_Problems-0.md
category: reference
title: Millennium Prize Problems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Millennium_Prize_Problems
date_saved: 2026-05-05T11:03:35.142675+00:00
instance: kb-cron
parent: millennium-prize-problems-20260505-0
summary:
The Millennium Prize Problems are seven well-known complex mathematical problems selected by the Clay Mathematics Institute in 2000. The Clay Institute has pledged to pay one million US dollars for the first correct solution to each problem. The Clay Mathematics Institute officially designated the title Millennium Problem for the seven unsolved mathematical problems, the Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier–Stokes existence and smoothness, P versus NP problem, Riemann hypothesis, Yang–Mills existence and mass gap, and the Poincaré conjecture at the Millennium Meeting held on May 24, 2000. Thus, on the official website of the Clay Mathematics Institute, these seven problems are officially called the Millennium Problems. As of 2026, the only Millennium Prize problem to have been solved is the Poincaré conjecture. The Clay Institute awarded the monetary prize to Russian mathematician Grigori Perelman in 2010. However, he declined the award because it was not also offered to Richard S. Hamilton, upon whose work Perelman built. == Overview == The Clay Institute was inspired by a set of twenty-three problems organized by the mathematician David Hilbert in 1900 which were highly influential in driving the progress of mathematics in the twentieth century. The seven selected problems span a number


id: millennium-prize-problems-20260505-1
file: data/en.wikipedia.org/wiki/Millennium_Prize_Problems-1.md
category: reference
title: Millennium Prize Problems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Millennium_Prize_Problems
date_saved: 2026-05-05T11:03:35.142675+00:00
instance: kb-cron
parent: millennium-prize-problems-20260505-0
summary:
− ν ∇ 2 u ⏟ Diffusion = − ∇ w ⏟ Internal source ⏞ Divergence of stress + g ⏟ External source . {\displaystyle \overbrace {\underbrace {\frac {\partial \mathbf {u} }{\partial t}} _{\begin{smallmatrix}{\text{Variation}}\end{smallmatrix}}+\underbrace {(\mathbf {u} \cdot \nabla )\mathbf {u} } _{\begin{smallmatrix}{\text{Convection}}\end{smallmatrix}}} ^{\text{Inertia (per volume)}}\overbrace {{}-\underbrace {\nu \,\nabla ^{2}\mathbf {u} } _{\text{Diffusion}}=\underbrace {-\nabla w} _{\begin{smallmatrix}{\text{Internal}}\\{\text{source}}\end{smallmatrix}}} ^{\text{Divergence of stress}}+\underbrace {\mathbf {g} } _{\begin{smallmatrix}{\text{External}}\\{\text{source}}\end{smallmatrix}}.} The Navier–Stokes equations describe the motion of fluids, and are one of the pillars of fluid mechanics. However, theoretical understanding of their solutions is incomplete, despite its importance in science and engineering. For the three-dimensional system of equations, and given some initial conditions, mathematicians have not yet proven that smooth solutions always exist. This is called the Navier–Stokes existence and smoothness problem. The problem, restricted to the case of an incompressible flow, is to prove either that smooth, globally defined solutions exist that meet certain conditions, or that they do not always exist and the equations break down. The official statement of the problem was given by Charles Fefferman. === P versus NP === The question is whether or not, for all problems for which an algorithm can verify a given solution quickly (that is, in polynomial time),


id: naimark-s-problem-20260505-0
file: data/en.wikipedia.org/wiki/Naimark's_problem-0.md
category: reference
title: Naimark's problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Naimark's_problem
date_saved: 2026-05-05T11:03:37.505239+00:00
instance: kb-cron
parent: naimark-s-problem-20260505-0
summary:
Naimark's problem is a question in functional analysis asked by Naimark (1951). It asks whether every C*-algebra that has only one irreducible ∗ {\displaystyle *} -representation up to unitary equivalence is isomorphic to the ∗ {\displaystyle *} -algebra of compact operators on some (not necessarily separable) Hilbert space. The problem has been solved in the affirmative for special cases (specifically for separable and Type-I C*-algebras). Akemann & Weaver (2004) used the diamond principle to construct a C*-algebra with ℵ 1 {\displaystyle \aleph _{1}} generators that serves as a counterexample to Naimark's problem. More precisely, they showed that the existence of a counterexample generated by ℵ 1 {\displaystyle \aleph _{1}} elements is independent of the axioms of Zermelo–Fraenkel set theory and the axiom of choice ( Z F C {\displaystyle {\mathsf {ZFC}}} ). Whether Naimark's problem itself is independent of Z F C {\displaystyle {\mathsf {ZFC}}} remains unknown. == See also == List of statements undecidable in Z F C {\displaystyle {\mathsf {ZFC}}} Gelfand–Naimark theorem == References == Akemann, Charles; Weaver, Nik (2004), "Consistency of a counterexample to Naimark's problem", Proceedings of the National Academy of Sciences of the United States of America, 101 (20): 7522–7525, arXiv:math.OA/0312135, Bibcode:2004PNAS..101.7522A, doi:10.1073/pnas.0401489101, MR 2057719,


id: nd-game-20260505-0
file: data/en.wikipedia.org/wiki/Nd_game-0.md
category: reference
title: Nd game
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Nd_game
date_saved: 2026-05-05T11:03:39.878700+00:00
instance: kb-cron
parent: nd-game-20260505-0
summary:
A nd game (or nk game) is a generalization of the combinatorial game tic-tac-toe to higher dimensions. It is a game played on a nd hypercube with 2 players. If one player creates a line of length n of their symbol (X or O) they win the game. However, if all nd spaces are filled then the game is a draw. Tic-tac-toe is the game where n equals 3 and d equals 2 (3, 2). Qubic is the (4, 3) game. The (n > 0, 0) or (1, 1) games are trivially won by the first player as there is only one space (n0 = 1 and 11 = 1). A game with d = 1 and n > 1 cannot be won if both players are playing well as an opponent's piece will block the one-dimensional line. == Game theory == An nd game is a symmetric combinatorial game. There are a total of ( n + 2 ) d − n d 2 {\displaystyle {\frac {\left(n+2\right)^{d}-n^{d}}{2}}} winning lines in a nd game. For any width n, at some dimension d (thanks to the Hales-Jewett theorem), there will always be a winning strategy for player X. There will never


id: nearby-lagrangian-conjecture-20260505-0
file: data/en.wikipedia.org/wiki/Nearby_Lagrangian_conjecture-0.md
category: reference
title: Nearby Lagrangian conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Nearby_Lagrangian_conjecture
date_saved: 2026-05-05T11:03:41.128885+00:00
instance: kb-cron
parent: nearby-lagrangian-conjecture-20260505-0
summary:
In mathematics, more specifically symplectic topology, the nearby Lagrangian conjecture, is an open mathematical problem often attributed to Vladimir Arnold. It states that every closed exact Lagrangian submanifold of a cotangent bundle T∗M (with symplectic structure) is Hamiltonian isotopic to the zero section. It is closely related to the theory of generating functions. == References ==


id: open-problems-in-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Open_Problems_in_Mathematics-0.md
category: reference
title: Open Problems in Mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Open_Problems_in_Mathematics
date_saved: 2026-05-05T11:03:42.350674+00:00
instance: kb-cron
parent: open-problems-in-mathematics-20260505-0
summary:
Open Problems in Mathematics is a book, edited by John Forbes Nash Jr. and Michael Th. Rassias, published in 2016 by Springer (ISBN 978-3-319-32160-8). The book consists of seventeen expository articles, written by outstanding researchers, on some of the central open problems in the field of mathematics. The book also features an Introduction on John Nash: Theorems and Ideas, by Mikhail Leonidovich Gromov. According to the editors’ Preface, each article is devoted to one open problem or a “constellation of related problems”. == Choice of problems == Nash and Rassias write in the preface of the book that the open problems presented “were chosen for a variety of reasons. Some were chosen for their undoubtedly importance and applicability, others because they constitute intriguing curiosities which remain unexplained mysteries on the basis of current knowledge and techniques, and some for more emotional reasons. Additionally, the attribute of a problem having a somewhat vintage flavor was also influential” in their decision process. == Table of contents == Preface, by John F. Nash Jr. and Michael Th. Rassias A Farewell to “A Beautiful Mind and a Beautiful Person”, by Michael Th. Rassias Introduction, John Nash: Theorems and Ideas, by Mikhail Leonidovich Gromov P


id: p-versus-np-problem-20260505-0
file: data/en.wikipedia.org/wiki/P_versus_NP_problem-0.md
category: reference
title: P versus NP problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/P_versus_NP_problem
date_saved: 2026-05-05T11:03:43.640123+00:00
instance: kb-cron
parent: p-versus-np-problem-20260505-0
summary:
The P versus NP problem is a major unsolved problem in theoretical computer science. Informally, it asks whether every problem whose solution can be quickly verified can also be quickly solved. Here, "quickly" means an algorithm exists that solves the task and runs in polynomial time (as opposed to, say, exponential time), meaning the task completion time is bounded above by a polynomial function on the size of the input to the algorithm. The general class of questions that some algorithm can answer in polynomial time is "P" or "class P". For some questions, there is no known way to find an answer quickly, but if provided with an answer, it can be verified quickly. The class of questions where an answer can be verified in polynomial time is "NP", standing for "nondeterministic polynomial time". An answer to the P versus NP question would determine whether problems that can be verified in polynomial time can also be solved in polynomial time. If P ≠ NP, which is widely believed, it would mean that there are problems in NP that are harder to compute than to verify: they could not be solved in polynomial time, but the answer could be verified


id: p-versus-np-problem-20260505-1
file: data/en.wikipedia.org/wiki/P_versus_NP_problem-1.md
category: reference
title: P versus NP problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/P_versus_NP_problem
date_saved: 2026-05-05T11:03:43.640123+00:00
instance: kb-cron
parent: p-versus-np-problem-20260505-0
summary:
To attack the P = NP question, the concept of NP-completeness is very useful. NP-complete problems are problems that any other NP problem is reducible to in polynomial time and whose solution is still verifiable in polynomial time. That is, any NP problem can be transformed into any NP-complete problem. Informally, an NP-complete problem is an NP problem that is at least as "tough" as any other problem in NP. NP-hard problems are those at least as hard as NP problems; i.e., all NP problems can be reduced (in polynomial time) to them. NP-hard problems need not be in NP; i.e., they need not have solutions verifiable in polynomial time. For instance, the Boolean satisfiability problem is NP-complete by the Cook–Levin theorem, so any instance of any problem in NP can be transformed mechanically into a Boolean satisfiability problem in polynomial time. The Boolean satisfiability problem is one of many NP-complete problems. If any NP-complete problem is in P, then it would follow that P = NP. However, many important problems are NP-complete, and no fast algorithm for any of them is known. From the definition alone it is unintuitive that NP-complete problems exist; however, a trivial NP-complete problem can


id: p-versus-np-problem-20260505-2
file: data/en.wikipedia.org/wiki/P_versus_NP_problem-2.md
category: reference
title: P versus NP problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/P_versus_NP_problem
date_saved: 2026-05-05T11:03:43.640123+00:00
instance: kb-cron
parent: p-versus-np-problem-20260505-0
summary:
In 1975, Richard E. Ladner showed that if P ≠ NP, then there exist problems in NP that are neither in P nor NP-complete. Such problems are called NP-intermediate problems. The graph isomorphism problem, the discrete logarithm problem, and the integer factorization problem are examples of problems believed to be NP-intermediate. They are some of the very few NP problems not known to be in P or to be NP-complete. The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. An important unsolved problem in complexity theory is whether the graph isomorphism problem is in P, NP-complete, or NP-intermediate. The answer is not known, but it is believed that the problem is at least not NP-complete. If graph isomorphism is NP-complete, the polynomial time hierarchy collapses to its second level. Since it is widely believed that the polynomial hierarchy does not collapse to any finite level, it is believed that graph isomorphism is not NP-complete. The best algorithm for this problem, due to László Babai, runs in quasi-polynomial time. The integer factorization problem is the computational problem of determining the prime factorization of a given integer. Phrased as a decision problem, it is the


id: p-versus-np-problem-20260505-3
file: data/en.wikipedia.org/wiki/P_versus_NP_problem-3.md
category: reference
title: P versus NP problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/P_versus_NP_problem
date_saved: 2026-05-05T11:03:43.640123+00:00
instance: kb-cron
parent: p-versus-np-problem-20260505-0
summary:
The main argument in favor of P ≠ NP is the total lack of fundamental progress in the area of exhaustive search. This is, in my opinion, a very weak argument. The space of algorithms is very large and we are only at the beginning of its exploration. [...] The resolution of Fermat's Last Theorem also shows that very simple questions may be settled only by very deep theories. Being attached to a speculation is not a good guide to research planning. One should always try both directions of every problem. Prejudice has caused famous mathematicians to fail to solve famous problems whose solution was opposite to their expectations, even though they had developed all the methods required. === DLIN vs NLIN === When one substitutes "linear time on a multitape Turing machine" for "polynomial time" in the definitions of P and NP, one obtains the classes DLIN and NLIN. It is known that DLIN ≠ NLIN. == Consequences of solution == One of the reasons the problem attracts so much attention is the consequences of the possible answers. Either direction of resolution would advance theory enormously, and perhaps have huge practical consequences as well. === P = NP ===


id: p-versus-np-problem-20260505-4
file: data/en.wikipedia.org/wiki/P_versus_NP_problem-4.md
category: reference
title: P versus NP problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/P_versus_NP_problem
date_saved: 2026-05-05T11:03:43.640123+00:00
instance: kb-cron
parent: p-versus-np-problem-20260505-0
summary:
=== P ≠ NP === A proof of P ≠ NP would lack the practical computational benefits of a proof that P = NP, but would represent a great advance in computational complexity theory and guide future research. It would demonstrate that many common problems cannot be solved efficiently, so that the attention of researchers can be focused on partial solutions or solutions to other problems. Due to widespread belief in P ≠ NP, much of this focusing of research has already taken place. P ≠ NP still leaves open the average-case complexity of hard problems in NP. For example, it is possible that SAT requires exponential time in the worst case, but that almost all randomly selected instances of it are efficiently solvable. Russell Impagliazzo has described five hypothetical "worlds" that could result from different possible resolutions to the average-case complexity question. These range from "Algorithmica", where P = NP and problems like SAT can be solved efficiently in all instances, to "Cryptomania", where P ≠ NP and generating hard instances of problems outside P is easy, with three intermediate possibilities reflecting different possible distributions of difficulty over instances of NP-hard problems. The "world" where P ≠ NP but


id: p-versus-np-problem-20260505-5
file: data/en.wikipedia.org/wiki/P_versus_NP_problem-5.md
category: reference
title: P versus NP problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/P_versus_NP_problem
date_saved: 2026-05-05T11:03:43.640123+00:00
instance: kb-cron
parent: p-versus-np-problem-20260505-0
summary:
// Algorithm that accepts the NP-complete language SUBSET-SUM. // // this is a polynomial-time algorithm if and only if P = NP. // // "Polynomial-time" means it returns "yes" in polynomial time when // the answer should be "yes", and runs forever when it is "no". // // Input: S = a finite set of integers // Output: "yes" if any subset of S adds up to 0. // Runs forever with no output otherwise. // Note: "Program number M" is the program obtained by // writing the integer M in binary, then // considering that string of bits to be a // program. Every possible program can be // generated this way, though most do nothing // because of syntax errors. FOR K = 1...∞ FOR M = 1...K Run program number M for K steps with input S IF the program outputs a list of distinct integers AND the integers are all in S AND the integers sum to 0 THEN OUTPUT "yes" and HALT This is a polynomial-time algorithm accepting an NP-complete language only if P = NP. "Accepting" means it gives "yes" answers in polynomial time, but is allowed to run forever when the answer is


id: p-versus-np-problem-20260505-6
file: data/en.wikipedia.org/wiki/P_versus_NP_problem-6.md
category: reference
title: P versus NP problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/P_versus_NP_problem
date_saved: 2026-05-05T11:03:43.640123+00:00
instance: kb-cron
parent: p-versus-np-problem-20260505-0
summary:
L ∈ NP; and any L′ in NP is polynomial-time-reducible to L (written as L ′ ≤ p L {\displaystyle L'\leq _{p}L} ), where L ′ ≤ p L {\displaystyle L'\leq _{p}L} if, and only if, the following two conditions are satisfied: There exists f : Σ* → Σ* such that for all w in Σ* we have: ( w ∈ L ′ ⇔ f ( w ) ∈ L ) {\displaystyle (w\in L'\Leftrightarrow f(w)\in L)} ; and there exists a polynomial-time Turing machine that halts with f(w) on its tape on any input w. Alternatively, if L ∈ NP, and there is another NP-complete problem that can be polynomial-time reduced to L, then L is NP-complete. This is a common way of proving some new problem is NP-complete. == Claimed solutions == While the P versus NP problem is generally considered unsolved, many amateur and some professional researchers have claimed solutions. Gerhard J. Woeginger compiled a list of 116 purported proofs from 1986 to 2016, of which 61 were proofs of P = NP, 49 were proofs of P ≠ NP, and 6 proved other results, e.g. that the problem is undecidable. Some attempts at resolving P versus NP


id: problems-in-latin-squares-20260505-0
file: data/en.wikipedia.org/wiki/Problems_in_Latin_squares-0.md
category: reference
title: Problems in Latin squares
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Problems_in_Latin_squares
date_saved: 2026-05-05T11:03:44.850208+00:00
instance: kb-cron
parent: problems-in-latin-squares-20260505-0
summary:
In mathematics, the theory of Latin squares is an active research area with many open problems. As in other areas of mathematics, such problems are often made public at professional conferences and meetings. Problems posed here appeared in, for instance, the Loops (Prague) conferences and the Milehigh (Denver) conferences. == Open problems == === Bounds on maximal number of transversals in a Latin square === A transversal in a Latin square of order n is a set S of n cells such that every row and every column contains exactly one cell of S, and such that the symbols in S form {1, ..., n}. Let T(n) be the maximum number of transversals in a Latin square of order n. Estimate T(n). Proposed: by Ian Wanless at Loops '03, Prague 2003 Comments: Wanless, McKay and McLeod have bounds of the form cn < T(n) < d n n!, where c > 1 and d is about 0.6. A conjecture by Rivin, Vardi and Zimmermann (Rivin et al., 1994) says that you can place at least exp(c n log n) queens in non-attacking positions on a toroidal chessboard (for some constant c). If true this would imply that T(n) > exp(c


id: prouhet-tarry-escott-problem-20260505-0
file: data/en.wikipedia.org/wiki/Prouhet–Tarry–Escott_problem-0.md
category: reference
title: Prouhet–Tarry–Escott problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Prouhet–Tarry–Escott_problem
date_saved: 2026-05-05T11:03:46.040926+00:00
instance: kb-cron
parent: prouhet-tarry-escott-problem-20260505-0
summary:
In mathematics, the Prouhet–Tarry–Escott problem asks for two disjoint multisets A and B of n integers each, whose first k power sum symmetric polynomials are all equal. That is, the two multisets should satisfy the equations ∑ a ∈ A a i = ∑ b ∈ B b i {\displaystyle \sum _{a\in A}a^{i}=\sum _{b\in B}b^{i}} for each integer i from 1 to a given k. It has been shown that n must be strictly greater than k. Solutions with k = n − 1 {\displaystyle k=n-1} are called ideal solutions. Ideal solutions are known for 3 ≤ n ≤ 10 {\displaystyle 3\leq n\leq 10} and for n = 12 {\displaystyle n=12} . No ideal solution is known for n = 11 {\displaystyle n=11} or for n ≥ 13 {\displaystyle n\geq 13} . This problem was named after Eugène Prouhet, who studied it in the early 1850s, and Gaston Tarry and Edward B. Escott, who studied it in the early 1910s. The problem originates from letters of Christian Goldbach and Leonhard Euler (1750/1751). == Examples == === Ideal solutions === An ideal solution for n = 6 is given by the two sets { 0, 5, 6, 16, 17, 22 }


id: prouhet-tarry-escott-problem-20260505-1
file: data/en.wikipedia.org/wiki/Prouhet–Tarry–Escott_problem-1.md
category: reference
title: Prouhet–Tarry–Escott problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Prouhet–Tarry–Escott_problem
date_saved: 2026-05-05T11:03:46.040926+00:00
instance: kb-cron
parent: prouhet-tarry-escott-problem-20260505-0
summary:
{ ( x 1 ′ , y 1 ′ ) , … , ( x 6 ′ , y 6 ′ ) } = { ( 1 , 2 ) , ( 2 , 5 ) , ( 5 , 7 ) , ( 7 , 6 ) , ( 6 , 3 ) , ( 3 , 1 ) } {\displaystyle \{(x'_{1},y'_{1}),\dots ,(x'_{6},y'_{6})\}=\{(1,2),(2,5),(5,7),(7,6),(6,3),(3,1)\}} . No solutions for n = k + 1 {\displaystyle n=k+1} with k ≥ 6 {\displaystyle k\geq 6} are known. --- "Pythagorean Triangle Approach" to the Prouhet-Tarry-Escott Problem for k=2, n=3 by Ali Uzel, a solution where the algebra meets geometry. a1 + b1 + c1 = d1 + e1 + f1 a2 + b2 + c2 = d2 + e2 + f2 say, a2 + b2 = d2 Pythagóras triangle A, where d is hypotenuse, a and b are sides c2 = e2 + f2 Pythagóras triangle B, where c is hypotenuse, e and f are sides if you sum both sides, a2 + b2 + c2 = d2 + e2 + f2 Thus, this equation seems to contain two Pythagorean Triangles. from equation a1 + b1 + c1 = d1 + e1 + f1


id: sendov-s-conjecture-20260505-0
file: data/en.wikipedia.org/wiki/Sendov's_conjecture-0.md
category: reference
title: Sendov's conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Sendov's_conjecture
date_saved: 2026-05-05T11:03:47.194322+00:00
instance: kb-cron
parent: sendov-s-conjecture-20260505-0
summary:
In mathematics, Sendov's conjecture, sometimes also called Ilieff's conjecture, concerns the relationship between the locations of roots and critical points of a polynomial function of a complex variable. It is named after Blagovest Sendov. The conjecture states that for a polynomial f ( z ) = ( z − r 1 ) ⋯ ( z − r n ) , ( n ≥ 2 ) {\displaystyle f(z)=(z-r_{1})\cdots (z-r_{n}),\qquad (n\geq 2)} with all roots r1, ..., rn inside the closed unit disk |z| ≤ 1, each of the n roots is at a distance no more than 1 from at least one critical point. The Gauss–Lucas theorem says that all of the critical points lie within the convex hull of the roots. It follows that the critical points must be within the unit disk, since the roots are. The conjecture has been proven for n < 9 by Brown-Xiang and for n sufficiently large by Tao. == History == The conjecture was first proposed by Blagovest Sendov in 1959; he described the conjecture to his colleague Nikola Obreshkov. In 1967 the conjecture was misattributed to Ljubomir Iliev by Walter Hayman. In 1969 Meir and Sharma proved the conjecture for polynomials with


id: serre-s-multiplicity-conjectures-20260505-0
file: data/en.wikipedia.org/wiki/Serre's_multiplicity_conjectures-0.md
category: reference
title: Serre's multiplicity conjectures
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Serre's_multiplicity_conjectures
date_saved: 2026-05-05T11:03:48.355879+00:00
instance: kb-cron
parent: serre-s-multiplicity-conjectures-20260505-0
summary:
In mathematics, Serre's multiplicity conjectures, named after Jean-Pierre Serre, are certain problems in commutative algebra, motivated by the needs of algebraic geometry. Since André Weil's initial definition of intersection numbers, around 1949, there had been a question of how to provide a more flexible and computable theory, which Serre sought to address. In 1958, Serre realized that classical algebraic-geometric ideas of multiplicity could be generalized using the concepts of homological algebra. Let R be a Noetherian, commutative, regular local ring and let P and Q be prime ideals of R. Serre defined the intersection multiplicity of R/P and R/Q by means of their Tor functors. Below, ℓ R ( M ) {\displaystyle \ell _{R}(M)} denotes the length of the module M {\displaystyle M} , and we assume for the remainder of the article that ℓ R ( ( R / P ) ⊗ R ( R / Q ) ) < ∞ . {\displaystyle \ell _{R}((R/P)\otimes _{R}(R/Q))<\infty .} Serre defined the intersection multiplicity of R/P and R/Q by the Euler characteristic-like formula: χ ( R / P , R / Q ) := ∑ i = 0 ∞ ( − 1 ) i ℓ R ( Tor i R ⁡


id: skolem-problem-20260505-0
file: data/en.wikipedia.org/wiki/Skolem_problem-0.md
category: reference
title: Skolem problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Skolem_problem
date_saved: 2026-05-05T11:03:52.059388+00:00
instance: kb-cron
parent: skolem-problem-20260505-0
summary:
In mathematics, the Skolem problem is the problem of determining whether the values of a constant-recursive sequence include the number zero. The problem can be formulated for recurrences over different types of numbers, including integers, rational numbers, and algebraic numbers. It is not known whether there exists an algorithm that can solve this problem. A linear recurrence relation expresses the values of a sequence of numbers as a linear combination of earlier values; for instance, the Fibonacci numbers may be defined from the recurrence relation F(n) = F(n − 1) + F(n − 2) together with the initial values F(0) = 0 and F(1) = 1. The Skolem problem is named after Thoralf Skolem, because of his 1933 paper proving the Skolem–Mahler–Lech theorem on the zeros of a sequence satisfying a linear recurrence with constant coefficients. This theorem states that, if such a sequence has zeros, then with finitely many exceptions the positions of the zeros repeat regularly. Skolem proved this for recurrences over the rational numbers, and Mahler and Lech extended it to other systems of numbers. However, the proofs of the theorem do not show how to test whether there exist any zeros. There does exist an algorithm


id: smale-s-problems-20260505-0
file: data/en.wikipedia.org/wiki/Smale's_problems-0.md
category: reference
title: Smale's problems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Smale's_problems
date_saved: 2026-05-05T11:03:53.279678+00:00
instance: kb-cron
parent: smale-s-problems-20260505-0
summary:
Smale's problems is a list of eighteen unsolved problems in mathematics proposed by Steve Smale in 1998 and republished in 1999. Smale composed this list in reply to a request from Vladimir Arnold, then vice-president of the International Mathematical Union, who asked several mathematicians to propose a list of problems for the 21st century. Arnold's inspiration came from the list of Hilbert's problems that had been published at the beginning of the 20th century. == Table of problems == In later versions, Smale also listed three additional problems, "that don't seem important enough to merit a place on our main list, but it would still be nice to solve them:" Mean value problem Is the three-sphere a minimal set (Gottschalk's conjecture)? Is an Anosov diffeomorphism of a compact manifold topologically the same as the Lie group model of John Franks? == See also == Millennium Prize Problems Simon problems Taniyama's problems Hilbert's problems Thurston's 24 questions == References ==


id: snake-in-the-box-20260505-0
file: data/en.wikipedia.org/wiki/Snake-in-the-box-0.md
category: reference
title: Snake-in-the-box
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Snake-in-the-box
date_saved: 2026-05-05T11:03:54.453665+00:00
instance: kb-cron
parent: snake-in-the-box-20260505-0
summary:
The snake-in-the-box problem in graph theory and coding theory deals with finding a certain kind of path along the edges of a hypercube. This path starts at one corner and travels along the edges to as many corners as it can reach. After it gets to a new corner, the previous corner and all of its neighbors must be marked as unusable. The path must never travel to a corner which has been marked unusable. In other words, a snake is a connected open path in the hypercube where each node has exactly two neighbors that are also in the path, with the exception of the first and last nodes, which each has only one neighbor in the path. The rule for generating a snake is that a node in the hypercube may be visited if it is connected to the current node and it is not a neighbor of any previously visited node in the snake, other than the current node. In graph theory terminology, this is called finding the longest possible induced path in a hypercube; it can be viewed as a special case of the induced subgraph isomorphism problem. There is a similar problem of finding long


id: snake-in-the-box-20260505-1
file: data/en.wikipedia.org/wiki/Snake-in-the-box-1.md
category: reference
title: Snake-in-the-box
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Snake-in-the-box
date_saved: 2026-05-05T11:03:54.453665+00:00
instance: kb-cron
parent: snake-in-the-box-20260505-0
summary:
== References == Abbot, H. L.; Katchalski, M. (1988), "On the snake in the box problem", Journal of Combinatorial Theory, Series B, 45: 13–24, doi:10.1016/0095-8956(88)90051-2 Abbot, H. L.; Katchalski, M. (1991), "On the construction of snake-in-the-box codes", Utilitas Mathematica, 40: 97–116 Allison, David; Paulusma, Daniel (2016), New Bounds for the Snake-in-the-Box Problem, arXiv:1603.05119, Bibcode:2016arXiv160305119A Bitterman, D. S. (2004), New Lower Bounds for the Snake-In-The-Box Problem: A Prolog Genetic Algorithm and Heuristic Search Approach (PDF) (M.S. thesis), Department of Computer Science, University of Georgia Blaum, Mario; Etzion, Tuvi (2002), Use of snake-in-the-box codes for reliable identification of tracks in servo fields of a disk drive, U.S. patent 6,496,312 Casella, D. A.; Potter, W. D. (2005), "Using Evolutionary Techniques to Hunt for Snakes and Coils", 2005 IEEE Congress on Evolutionary Computation (CEC2005), vol. 3, pp. 2499–2504 Casella, D. A. (2005), New Lower Bounds for the Snake-in-the-Box and the Coil-in-the-Box Problems (PDF) (M.S. thesis), Department of Computer Science, University of Georgia Dang, Chu; Bazgan, Cristina; Cazenave, Tristan; Chopin, Morgan; Wuillemin, Pierre-Henri (2023). "Warm-Starting Nested Rollout Policy Adaptation with Optimal Stopping". Proceedings of the AAAI Conference on Artificial Intelligence. 37 (10): 12381–12389. doi:10.1609/aaai.v37i10.26459. Danzer, L.; Klee, V. (1967), "Length of snakes in boxes", Journal


id: sofic-group-20260505-0
file: data/en.wikipedia.org/wiki/Sofic_group-0.md
category: reference
title: Sofic group
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Sofic_group
date_saved: 2026-05-05T11:03:55.665327+00:00
instance: kb-cron
parent: sofic-group-20260505-0
summary:
In mathematics, a sofic group is a group whose Cayley graph is an initially subamenable graph, or equivalently a subgroup of an ultraproduct of finite-rank symmetric groups such that every two elements of the group have distance 1. They were introduced by Gromov (1999) as a common generalization of amenable and residually finite groups. The name "sofic", from the Hebrew word סופי meaning "finite", was later applied by Weiss (2000), following Weiss's earlier use of the same word to indicate a generalization of finiteness in sofic subshifts. The class of sofic groups is closed under the operations of taking subgroups, extensions by amenable groups, and free products. A finitely generated group is sofic if it is the limit of a sequence of sofic groups. The limit of a sequence of amenable groups (that is, an initially subamenable group) is necessarily sofic, but there exist sofic groups that are not initially subamenable groups. As Gromov proved, Sofic groups are surjunctive. That is, they obey a form of the Garden of Eden theorem for cellular automata defined over the group (dynamical systems whose states are mappings from the group to a finite set and whose state transitions are translation-invariant and continuous) stating


id: szilassi-polyhedron-20260505-0
file: data/en.wikipedia.org/wiki/Szilassi_polyhedron-0.md
category: reference
title: Szilassi polyhedron
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Szilassi_polyhedron
date_saved: 2026-05-05T11:03:56.878411+00:00
instance: kb-cron
parent: szilassi-polyhedron-20260505-0
summary:
In geometry, the Szilassi polyhedron is a nonconvex polyhedron, topologically a torus, with seven hexagonal faces. The tetrahedron and the Szilassi polyhedron are the only two known polyhedra in which each face shares an edge with each other face. == Properties == === Coloring and symmetry === The 14 vertices and 21 edges of the Szilassi polyhedron form an embedding of the Heawood graph onto the surface of a torus. Each face of this polyhedron shares an edge with each other face. As a result, it requires seven colours to colour all adjacent faces. This example shows that, on surfaces topologically equivalent to a torus, some subdivisions require seven colors, providing the lower bound for the seven colour theorem. The other half of the theorem states that all toroidal subdivisions can be colored with seven or fewer colors. The Szilassi polyhedron has an axis of 180-degree symmetry. This symmetry swaps three pairs of congruent faces, leaving one unpaired hexagon that has the same rotational symmetry as the polyhedron. === Complete face adjacency === The tetrahedron and the Szilassi polyhedron are the only two known polyhedra in which each face shares an edge with each other face. If a polyhedron with


id: tarski-s-exponential-function-problem-20260505-0
file: data/en.wikipedia.org/wiki/Tarski's_exponential_function_problem-0.md
category: reference
title: Tarski's exponential function problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Tarski's_exponential_function_problem
date_saved: 2026-05-05T11:03:58.126083+00:00
instance: kb-cron
parent: tarski-s-exponential-function-problem-20260505-0
summary:
In model theory, Tarski's exponential function problem asks whether the theory of the real numbers together with the exponential function is decidable. Alfred Tarski had previously shown that the theory of the real numbers (without the exponential function) is decidable. == The problem == The ordered real field R {\displaystyle \mathbb {R} } is a structure over the language of ordered rings L or = ( + , − , < , 0 , 1 ) {\displaystyle L_{\text{or}}=(+,-,<,0,1)} , with the usual interpretation given to each symbol. It was proved by Tarski that the theory of the real field, Th ⁡ ( R ) {\displaystyle \operatorname {Th} (\mathbb {R} )} , is decidable. That is, given any L or {\displaystyle L_{\text{or}}} -sentence φ {\displaystyle \varphi } there is an effective procedure for determining whether R ⊨ φ . {\displaystyle \mathbb {R} \models \varphi .} He then asked whether this was still the case if one added a unary function exp {\displaystyle \exp } to the language that was interpreted as the exponential function on R {\displaystyle \mathbb {R} } , to get the structure R exp {\displaystyle \mathbb {R} _{\exp }} . == Conditional and equivalent results == The problem


id: thomson-problem-20260505-0
file: data/en.wikipedia.org/wiki/Thomson_problem-0.md
category: reference
title: Thomson problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Thomson_problem
date_saved: 2026-05-05T11:03:59.517890+00:00
instance: kb-cron
parent: thomson-problem-20260505-0
summary:
The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of N electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. The physicist J. J. Thomson posed the problem in 1904 after proposing an atomic model, later called the plum pudding model, based on his knowledge of the existence of negatively charged electrons within neutrally-charged atoms. Related problems include the study of the geometry of the minimum energy configuration and the study of the large N behavior of the minimum energy. == Mathematical statement == The electrostatic interaction energy occurring between each pair of electrons of equal charges ( e i = e j = e {\displaystyle e_{i}=e_{j}=e} , with e {\displaystyle e} the elementary charge of an electron) is given by Coulomb's law, U i j ( N ) = e i e j 4 π ϵ 0 r i j , {\displaystyle U_{ij}(N)={e_{i}e_{j} \over 4\pi \epsilon _{0}r_{ij}},} where ϵ 0 {\displaystyle \epsilon _{0}} is the electric constant and r i j = | r i − r j | {\displaystyle r_{ij}=|\mathbf {r} _{i}-\mathbf {r} _{j}|} is the distance between each pair of


id: thomson-problem-20260505-1
file: data/en.wikipedia.org/wiki/Thomson_problem-1.md
category: reference
title: Thomson problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Thomson_problem
date_saved: 2026-05-05T11:03:59.517890+00:00
instance: kb-cron
parent: thomson-problem-20260505-0
summary:
=== Continuous spherical shell charge === The energy of a continuous spherical shell of charge distributed across its surface is given by U shell ( N ) = N 2 2 {\displaystyle U_{\text{shell}}(N)={\frac {N^{2}}{2}}} and is, in general, greater than the energy of every Thomson problem solution. Note: Here N is used as a continuous variable that represents the infinitely divisible charge, Q, distributed across the spherical shell. For example, a spherical shell of N = 1 {\displaystyle N=1} represents the uniform distribution of a single electron's charge, − e {\displaystyle -e} , across the entire shell. === Randomly distributed point charges === The expected global energy of a system of electrons distributed in a purely random manner across the surface of the sphere is given by U rand ( N ) = N ( N − 1 ) 2 {\displaystyle U_{\text{rand}}(N)={\frac {N(N-1)}{2}}} and is, in general, greater than the energy of every Thomson problem solution. Here, N is a discrete variable that counts the number of electrons in the system. As well, U rand ( N ) < U shell ( N ) {\displaystyle U_{\text{rand}}(N)<U_{\text{shell}}(N)} . === Charge-centered distribution === For every Nth solution of the Thomson problem there


id: thomson-problem-20260505-2
file: data/en.wikipedia.org/wiki/Thomson_problem-2.md
category: reference
title: Thomson problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Thomson_problem
date_saved: 2026-05-05T11:03:59.517890+00:00
instance: kb-cron
parent: thomson-problem-20260505-0
summary:
== Notes == Whyte, L.L. (1952). "Unique arrangements of points on a sphere". Amer. Math. Monthly. 59 (9): 606–611. doi:10.2307/2306764. JSTOR 2306764. Cohn, Harvey (1956). "Stability configurations of electrons on a sphere". Math. Comput. 10 (55): 117–120. doi:10.1090/S0025-5718-1956-0081133-0. Goldberg, Michael (1969). "Stability configurations of electrons on a sphere". Math. Comp. 23 (108): 785–786. doi:10.1090/S0025-5718-69-99642-2. Erber, T.; Hockney, G. M. (1991). "equilibrium configurations of N equal charges on a sphere". J. Phys. A: Math. Gen. 24 (23): L1369. Bibcode:1991JPhA...24L1369E. doi:10.1088/0305-4470/24/23/008. S2CID 122561279. Morris, J. R.; Deaven, D. M.; Ho, K. M. (1996). "Genetic-algorithm energy minimization for point charges on a sphere". Phys. Rev. B. 53 (4): R1740–R1743. Bibcode:1996PhRvB..53.1740M. CiteSeerX 10.1.1.28.93. doi:10.1103/PhysRevB.53.R1740. PMID 9983695. Erber, T.; Hockney, G. M. (1997). "Complex Systems: Equilibrium Configurations of N {\displaystyle N} Equal Charges on a Sphere ( 2 ≤ N ≤ 112 ) {\displaystyle (2\leq N\leq 112)} ". Advances in Chemical Physics. Vol. 98. pp. 495–594. doi:10.1002/9780470141571.ch5. ISBN 978-0-470-14157-1.. Altschuler, E. L.; Williams, T. J.; Ratner, E. R.; Tipton, R.; Stong, R.; Dowla, F.; Wooten, F. (1997). "Possible global minimum lattice configurations for Thomson's problem of charges on a sphere". Phys. Rev. Lett. 78 (14): 2681–2685. Bibcode:1997PhRvL..78.2681A. doi:10.1103/PhysRevLett.78.2681. Bowick, M.; Cacciuto, A.; Nelson, D. R.;


id: thurston-s-24-questions-20260505-0
file: data/en.wikipedia.org/wiki/Thurston's_24_questions-0.md
category: reference
title: Thurston's 24 questions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Thurston's_24_questions
date_saved: 2026-05-05T11:04:00.697718+00:00
instance: kb-cron
parent: thurston-s-24-questions-20260505-0
summary:
Thurston's 24 questions are a set of mathematical problems in differential geometry posed by American mathematician William Thurston in his influential 1982 paper Three-dimensional manifolds, Kleinian groups and hyperbolic geometry published in the Bulletin of the American Mathematical Society. These questions significantly influenced the development of geometric topology and related fields over the following decades. == History == The questions appeared following Thurston's announcement of the geometrization conjecture, which proposed that all compact 3-manifolds could be decomposed into geometric pieces. This conjecture, later proven by Grigori Perelman in 2003, represented a complete classification of 3-manifolds and included the famous Poincaré conjecture as a special case. By 2012, 22 of Thurston's 24 questions had been resolved. == Table of problems == Thurston's 24 questions are: == See also == Geometrization conjecture Hilbert's problems Taniyama's problems List of unsolved problems in mathematics Poincaré conjecture Smale's problems == References ==


id: union-closed-sets-conjecture-20260505-0
file: data/en.wikipedia.org/wiki/Union-closed_sets_conjecture-0.md
category: reference
title: Union-closed sets conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Union-closed_sets_conjecture
date_saved: 2026-05-05T11:04:01.996932+00:00
instance: kb-cron
parent: union-closed-sets-conjecture-20260505-0
summary:
The union-closed sets conjecture, also known as Frankl’s conjecture, is an open problem in combinatorics posed by Péter Frankl in 1979. A family of sets is said to be union-closed if the union of any two sets from the family belongs to the family. The conjecture states: For every finite union-closed family of sets, other than the empty family, there exists an element that belongs to at least half of the sets in the family. Professor Timothy Gowers has called this "one of the best known open problems in combinatorics" and has said that the conjecture "feels as though it ought to be easy (and as a result has attracted a lot of false proofs over the years). A good way to understand why it isn't easy is to spend an afternoon trying to prove it. That clever averaging argument you had in mind doesn't work ..." == Example == The family of sets ∅ , { 1 } , { 1 , 2 } , { 2 , 3 , 4 } , { 1 , 2 , 3 , 4 } {\displaystyle \varnothing ,\{1\},\{1,2\},\{2,3,4\},\{1,2,3,4\}} consists of five different sets and is union-closed. The element 1 {\displaystyle 1} is


id: union-closed-sets-conjecture-20260505-1
file: data/en.wikipedia.org/wiki/Union-closed_sets_conjecture-1.md
category: reference
title: Union-closed sets conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Union-closed_sets_conjecture
date_saved: 2026-05-05T11:04:01.996932+00:00
instance: kb-cron
parent: union-closed-sets-conjecture-20260505-0
summary:
=== Lattice formulation === Although stated above in terms of families of sets, Frankl's conjecture has also been formulated and studied as a question in lattice theory. A lattice is a partially ordered set in which for two elements x and y there is a unique greatest element less than or equal to both of them (the meet of x and y) and a unique least element greater than or equal to both of them (the join of x and y). The family of all subsets of a set S, ordered by set inclusion, forms a lattice in which the meet is represented by the set-theoretic intersection and the join is represented by the set-theoretic union; a lattice formed in this way is called a Boolean lattice. The lattice-theoretic version of Frankl's conjecture is that in any finite lattice there exists an element x that is not the join of any two smaller elements, and such that the number of elements greater than or equal to x totals at most half the lattice, with equality only if the lattice is a Boolean lattice. As Abe (2000) shows, this statement about lattices is equivalent to the Frankl conjecture for union-closed sets:


id: union-closed-sets-conjecture-20260505-2
file: data/en.wikipedia.org/wiki/Union-closed_sets_conjecture-2.md
category: reference
title: Union-closed sets conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Union-closed_sets_conjecture
date_saved: 2026-05-05T11:04:01.996932+00:00
instance: kb-cron
parent: union-closed-sets-conjecture-20260505-0
summary:
Therefore a belongs to at most half of the sets in A {\displaystyle {\mathcal {A}}} . This proves the intersection-closed formulation. === Graph-theoretic formulation === Another equivalent formulation of the union-closed sets conjecture uses graph theory. In an undirected graph, an Independent set is a set of vertices no two of which are adjacent to each other; an independent set is maximal if it is not a subset of a larger independent set. In any graph, the "heavy" vertices that appear in more than half of the maximal independent sets must themselves form an independent set. So, if the graph is non-empty, there always exists at least one non-heavy vertex, a vertex that appears in at most half of the maximal independent sets. The graph formulation of the union-closed sets conjecture states that every finite non-empty graph contains two adjacent non-heavy vertices. It is automatically true when the graph contains an odd cycle, because the independent set of all heavy vertices cannot cover all the edges of the cycle. Therefore, the more interesting case of the conjecture is for bipartite graphs, which have no odd cycles. Another equivalent formulation of the conjecture is that, in every bipartite graph, there exist


id: vaught-conjecture-20260505-0
file: data/en.wikipedia.org/wiki/Vaught_conjecture-0.md
category: reference
title: Vaught conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Vaught_conjecture
date_saved: 2026-05-05T11:04:03.194954+00:00
instance: kb-cron
parent: vaught-conjecture-20260505-0
summary:
The Vaught conjecture is a conjecture in the mathematical field of model theory originally proposed by Robert Lawson Vaught in 1961. It states that the number of countable models of a first-order complete theory in a countable language is finite or ℵ0 or 2ℵ0. Morley showed that the number of countable models is finite or ℵ0 or ℵ1 or 2ℵ0, which solves the conjecture except for the case of ℵ1 models when the continuum hypothesis fails. For this remaining case, Robin Knight (2002, 2007) has announced a counterexample to the Vaught conjecture and the topological Vaught conjecture. As of 2021, the counterexample has not been verified. == Statement of the conjecture == Let T {\displaystyle T} be a first-order, countable, complete theory with infinite models. Let I ( T , α ) {\displaystyle I(T,\alpha )} denote the number of models of T of cardinality α {\displaystyle \alpha } up to isomorphism—the spectrum of the theory T {\displaystyle T} . Morley proved that if I(T, ℵ0) is infinite then it must be ℵ0 or ℵ1 or the cardinality of the continuum. The Vaught conjecture is the statement that it is not possible for ℵ 0 < I ( T , ℵ


id: whitehead-conjecture-20260505-0
file: data/en.wikipedia.org/wiki/Whitehead_conjecture-0.md
category: reference
title: Whitehead conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Whitehead_conjecture
date_saved: 2026-05-05T11:04:04.346031+00:00
instance: kb-cron
parent: whitehead-conjecture-20260505-0
summary:
The Whitehead conjecture (also known as the Whitehead asphericity conjecture) is a claim in algebraic topology. It was formulated by J. H. C. Whitehead in 1941. It states that every connected subcomplex of a two-dimensional aspherical CW complex is aspherical. A group presentation G = ( S ∣ R ) {\displaystyle G=(S\mid R)} is called aspherical if the two-dimensional CW complex K ( S ∣ R ) {\displaystyle K(S\mid R)} associated with this presentation is aspherical or, equivalently, if π 2 ( K ( S ∣ R ) ) = 0 {\displaystyle \pi _{2}(K(S\mid R))=0} . The Whitehead conjecture is equivalent to the conjecture that every sub-presentation of an aspherical presentation is aspherical. In 1997, Mladen Bestvina and Noel Brady constructed a group G so that either G is a counterexample to the Eilenberg–Ganea conjecture, or there must be a counterexample to the Whitehead conjecture; in other words, it is not possible for both conjectures to be true. == References == Whitehead, J. H. C. (1941). "On adding relations to homotopy groups". Annals of Mathematics. 2nd Ser. 42 (2): 409–428. doi:10.2307/1968907. JSTOR 1968907. MR 0004123. Bestvina, Mladen; Brady, Noel (1997). "Morse theory and finiteness properties of groups". Inventiones Mathematicae. 129


id: yau-s-conjecture-on-the-first-eigenvalue-20260505-0
file: data/en.wikipedia.org/wiki/Yau's_conjecture_on_the_first_eigenvalue-0.md
category: reference
title: Yau's conjecture on the first eigenvalue
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Yau's_conjecture_on_the_first_eigenvalue
date_saved: 2026-05-05T11:04:06.733095+00:00
instance: kb-cron
parent: yau-s-conjecture-on-the-first-eigenvalue-20260505-0
summary:
In mathematics, Yau's conjecture on the first eigenvalue is, as of 2018, an unsolved conjecture proposed by Shing-Tung Yau in 1982. It asks: Is it true that the first eigenvalue for the Laplace–Beltrami operator on an embedded closed minimal hypersurface of S n + 1 {\displaystyle S^{n+1}} is n {\displaystyle n} ? If true, it will imply that the area of embedded minimal hypersurfaces in S 3 {\displaystyle S^{3}} will have an upper bound depending only on the genus. The Yau's conjecture is verified for several special cases, but still open in general. Shiing-Shen Chern conjectured that a closed, minimally immersed hypersurface in S n + 1 {\displaystyle S^{n+1}} (1), whose second fundamental form has constant length, is isoparametric. If true, it would have established the Yau's conjecture for the minimal hypersurface whose second fundamental form has constant length. A possible generalization of the Yau's conjecture: Let M d {\displaystyle M^{d}} be a closed minimal submanifold in the unit sphere S N + 1 {\displaystyle S^{N+1}} (1) with dimension d {\displaystyle d} of M d {\displaystyle M^{d}} satisfying d ≥ 2 n / 3 + 1 {\displaystyle d\geq 2n/3+1} . Is it true that the first eigenvalue of M d


id: list-of-unsolved-problems-in-neuroscience-20260505-0
file: data/en.wikipedia.org/wiki/List_of_unsolved_problems_in_neuroscience-0.md
category: reference
title: List of unsolved problems in neuroscience
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_neuroscience
date_saved: 2026-05-05T11:04:08.315856+00:00
instance: kb-cron
parent: list-of-unsolved-problems-in-neuroscience-20260505-0
summary:
The following is a list of notable unsolved problems in neuroscience. A problem is considered unsolved if no answer is known or if there is significant disagreement among experts about a proposed solution. == Consciousness == Consciousness: How can consciousness be defined? What is the neural basis of subjective experience, cognition, wakefulness, alertness, arousal, and attention? Binding problem: How exactly is it that objects, background, and abstract or emotional features are combined into a single experience? What is the neural basis of self? Quantum mind: Do quantum mechanical phenomena, such as entanglement and superposition, play an important part in the brain's function and can it explain critical aspects of consciousness? Is there a "hard problem of consciousness"? If so, how is it solved? Vertiginous question: Why is it that a specific subject of experience is "live" from a given perspective? What, if anything, is the function of consciousness? Problem of mental causation: How exactly do mental states cause intentional actions to happen? What is the nature and mechanism behind near-death experiences? How can death be defined? Can consciousness exist after death? If consciousness is generated by brain activity, then how do some patients with physically deteriorated brains suddenly gain a


id: activation-synthesis-hypothesis-20260505-0
file: data/en.wikipedia.org/wiki/Activation-synthesis_hypothesis-0.md
category: reference
title: Activation-synthesis hypothesis
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Activation-synthesis_hypothesis
date_saved: 2026-05-05T11:04:09.469790+00:00
instance: kb-cron
parent: activation-synthesis-hypothesis-20260505-0
summary:
The activation-synthesis hypothesis, proposed by Harvard University psychiatrists John Allan Hobson and Robert McCarley, is a neurobiological theory of dreams first published in the American Journal of Psychiatry in December 1977. The differences in neuronal activity of the brainstem during waking and REM sleep were observed, and the hypothesis proposes that dreams result from brain activation during REM sleep. Since then, the hypothesis has undergone an evolution as technology and experimental equipment has become more precise. Currently, a three-dimensional model called AIM Model, described below, is used to determine the different states of the brain over the course of the day and night. The AIM Model introduces a new hypothesis that primary consciousness is an important building block on which secondary consciousness is constructed. == Background == With the advancement of brain imaging technology, the sleep-waking cycle can be studied as never before. The brain can be objectively quantified and identified as being in either one of three states: awake, REM sleep, and NREM sleep due to these advanced methods of measurement. It has been shown that global deactivation of the brain from waking state to NREM sleep occurs, and a subsequent reactivation during REM sleep, to a degree greater


id: activation-synthesis-hypothesis-20260505-1
file: data/en.wikipedia.org/wiki/Activation-synthesis_hypothesis-1.md
category: reference
title: Activation-synthesis hypothesis
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Activation-synthesis_hypothesis
date_saved: 2026-05-05T11:04:09.469790+00:00
instance: kb-cron
parent: activation-synthesis-hypothesis-20260505-0
summary:
=== REM === REM sleep may be a more evolutionarily recent sleep state, and is prominent in most birds and mammals, although may exist in reptiles and other vertebrates to varying degrees. REM stands for rapid eye movement. It is generally a later sleep state following non-REM (NREM) sleep. It is regulated in part by the pontine brainstem. Infants spend most of their time in REM sleep, and rather than enter stage 1 sleep they may go directly to REM sleep. Most REM sleep occurs just above stage I of sleep, and experiences different mental abilities than during NREM sleep. The thought process is sometimes non-logical or even bizarre, sensation and perception is vivid but created internally by the brain, and the body's movements are inhibited. Most REM stages last 10–15 minutes, and the average human will go through 4–6 of these stages during sleep each night. Subsequent REM stages increase in duration, so the last REM stage before awakening is the longest and thus may have the most vivid dream imagery. It has been proposed that REM sleep is necessary for preparation of many integrative functions, of which one is consciousness. It supports the idea that sleep, and dreaming,


id: activation-synthesis-hypothesis-20260505-2
file: data/en.wikipedia.org/wiki/Activation-synthesis_hypothesis-2.md
category: reference
title: Activation-synthesis hypothesis
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Activation-synthesis_hypothesis
date_saved: 2026-05-05T11:04:09.469790+00:00
instance: kb-cron
parent: activation-synthesis-hypothesis-20260505-0
summary:
==== Activation (A) ==== Large parts of the brain that are activated and sending signals during waking are inactive during NREM sleep and become reactivated during REM sleep. It is based on the fact that the brain and its neural circuitry is plastic and self-regulating, especially in its own activation and inactivation. This was observed by two experiments: development of sleepiness after dopamine neuron destruction in substantia nigra in the midbrain, and discovery of the reticular activating system, which are visual cues received through our eyes and to our brain that begin the waking process, that waking consciousness depends sleep. Following these studies, it became clear that activity levels and quality of consciousness were functions of brain activation and deactivation. ==== Input-output gating (I) ==== It has been shown that the internal activation of the brain is associated with the inhibition of both external sensory input and motor output. This implies that the brain is actively kept offline during REM, and the brainstem guarantees the coordination of factors I and A via the input-output gate control within the brainstem. PGO waves play a part in the ability of the brain to remain asleep while constituting the building blocks for perception


id: als-20260505-0
file: data/en.wikipedia.org/wiki/ALS-0.md
category: reference
title: ALS
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/ALS
date_saved: 2026-05-05T11:04:10.701513+00:00
instance: kb-cron
parent: als-20260505-0
summary:
Amyotrophic lateral sclerosis (ALS), also known as motor neuron disease (MND) or Lou Gehrig's disease, is a rare terminal neurodegenerative disease defined by the progressive loss of both upper and lower motor neurons that normally control voluntary muscle contraction. ALS is the most common of the motor neuron diseases. ALS often presents with gradual muscle stiffness, twitches, weakness, and wasting. Motor neuron loss typically continues until the ability to eat, speak, move, and breathe without mechanical support is lost. At least 50% of people with ALS experience significant changes in thinking and behavior, with 15% of individuals going on to develop frontotemporal dementia. An ALS diagnosis is made based on a person's signs and symptoms, with additional testing conducted to rule out other potential causes. Depending on which areas of the body are affected first, ALS may be classified as limb-onset (beginning with weakness in the arms or legs) or bulbar-onset (beginning with difficulty in speaking or swallowing). Respiratory onset occurs in approximately 1–3% of cases. Most cases of ALS (about 90–95%) have no known cause and are known as sporadic ALS. Genetic, autoimmune, and environmental factors are believed to be involved in the onset of ALS. Approximately 5–10% of


id: als-20260505-1
file: data/en.wikipedia.org/wiki/ALS-1.md
category: reference
title: ALS
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/ALS
date_saved: 2026-05-05T11:04:10.701513+00:00
instance: kb-cron
parent: als-20260505-0
summary:
Classical ALS accounts for about 70% of all cases of ALS and can be subdivided into cases where symptoms first appear, as these are usually focused on one region of the body at initial presentation before later spread. Limb-onset ALS (also known as spinal-onset) and bulbar-onset ALS. Limb-onset ALS begins with weakness in the hands, arms, feet, and/or legs and accounts for about two-thirds of all classical ALS cases. Bulbar-onset ALS begins with weakness in the muscles of speech, chewing, and swallowing and accounts for about 25% of classical ALS cases. A rarer type of classical ALS affecting about 3% of patients is respiratory-onset, in which the initial symptoms are difficulty breathing (dyspnea) upon exertion, at rest, or while lying flat (orthopnea). Primary lateral sclerosis (PLS) accounts for about 5% of all ALS cases and only affects the upper motor neurons in the arms, legs, and bulbar region. However, more than 75% of people with apparent PLS go on to later develop lower motor neuron signs within four years of symptom onset, meaning that a definitive diagnosis of PLS cannot be made until several years have passed. PLS has a better prognosis than classical ALS, as it progresses more slowly,


id: als-20260505-2
file: data/en.wikipedia.org/wiki/ALS-2.md
category: reference
title: ALS
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/ALS
date_saved: 2026-05-05T11:04:10.701513+00:00
instance: kb-cron
parent: als-20260505-0
summary:
=== Progression === Although the initial site of symptoms and subsequent rate of disability progression vary from person to person, the initially affected body region is usually the most affected over time, and symptoms usually spread to a neighbouring body region. For example, symptoms starting in one arm usually spread to either the opposite arm or to the leg on the same side. Bulbar-onset patients most typically experience their next symptoms in their arms rather than legs, arm-onset patients typically spread to the legs before the bulbar region, and leg-onset patients typically spread to the arms rather than the bulbar region. Over time, regardless of where symptoms began, most people eventually lose the ability to walk or use their hands and arms independently. They may also lose the ability to speak and to swallow food. It is the eventual development of weakness of the respiratory muscles, with the loss of ability to cough and to breathe without mechanical support, that is ultimately life-shortening in ALS. The rate of progression can be measured using the ALS Functional Rating Scale - Revised (ALSFRS-R), a 12-item instrument survey administered as a clinical interview or self-reported questionnaire that produces a score between 48 (normal


id: als-20260505-3
file: data/en.wikipedia.org/wiki/ALS-3.md
category: reference
title: ALS
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/ALS
date_saved: 2026-05-05T11:04:10.701513+00:00
instance: kb-cron
parent: als-20260505-0
summary:
=== Cognitive, emotional, and behavioral symptoms === Cognitive impairment or behavioral dysfunction is present in 30–50% of individuals with ALS, and can appear more frequently in later stages of the disease. Language dysfunction, executive dysfunction, and troubles with social cognition and verbal memory are the most commonly reported cognitive symptoms in ALS. Cognitive impairment is found more frequently in patients with C9orf72 gene repeat expansions, bulbar onset, bulbar symptoms, family history of ALS and/or a predominantly upper motor neuron phenotype. Pseudobulbar affect and emotional lability are a type of symptom in which patients cry, smile, yawn, or laugh, either in the absence of emotional stimuli, or when they are feeling the opposite emotion to that being expressed. Approximately half of ALS patients experience this; it is more common in those with bulbar-onset ALS. While more or less benign relative to other symptoms, it can cause increased stigma and social isolation as people around the patient struggle to react appropriately to what can be frequent and inappropriate outbursts in public. In addition to mild changes in cognition that may only emerge during neuropsychological testing, around 10–15% of individuals have signs of frontotemporal dementia (FTD). Repeating phrases or gestures, apathy, and loss


id: als-20260505-4
file: data/en.wikipedia.org/wiki/ALS-4.md
category: reference
title: ALS
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/ALS
date_saved: 2026-05-05T11:04:10.701513+00:00
instance: kb-cron
parent: als-20260505-0
summary:
=== Environmental and other factors === The multi-step hypothesis suggests the disease is caused by some interaction between an individual's genetic risk factors and their cumulative lifetime of exposures to environmental factors, termed their exposome. The most consistent lifetime exposures associated with developing ALS (other than genetic mutations) include heavy metals (e.g. lead and mercury), chemicals (e.g. pesticides and solvents), electric shock, physical injury (including head injury), and smoking (in men more than women). Generally, the effect of each exposure is relatively small. For instance, an individual's lifetime risk of developing ALS might increase from "1 in 400" without exposure to between "1 in 300" and "1 in 200" if they were exposed to heavy metals. Some industries are heavily dependent upon the use or exposure to these environmental factors, which may increase employees' susceptibility. For example, agricultural tasks may involve as many as 5 such risk factors excluding workers' smoking habits. A range of other factors have weaker evidence supporting them and include participation in professional sports, having a lower body mass index, lower educational attainment, manual occupations, military service, exposure to beta-Methylamino-L-alanine (BMAA), and viral infections. Although some personality traits, such as openness, agreeableness and conscientiousness appear remarkably


id: als-20260505-5
file: data/en.wikipedia.org/wiki/ALS-5.md
category: reference
title: ALS
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/ALS
date_saved: 2026-05-05T11:04:10.701513+00:00
instance: kb-cron
parent: als-20260505-0
summary:
Neurodegeneration in ALS is thought to involve many different cellular and molecular processes. The genes known to be involved in ALS can be grouped into three general categories based on their normal function: protein degradation, the cytoskeleton, and RNA processing. Mutant SOD1 protein forms intracellular aggregations that inhibit protein degradation. Cytoplasmic aggregations of wild-type (normal) SOD1 protein are common in sporadic ALS. It is thought that misfolded mutant SOD1 can cause misfolding and aggregation of wild-type SOD1 in neighboring neurons in a prion-like manner. Other protein degradation genes that can cause ALS when mutated include VCP, OPTN, TBK1, and SQSTM1. Three genes implicated in ALS that are important for cytoskeletal maintenance and axonal transport include DCTN1, PFN1, and TUBA4A. Several ALS genes encode RNA-binding proteins. The first to be discovered was TDP-43 protein, a nuclear protein that aggregates in the cytoplasm of motor neurons in almost all cases of ALS; however, mutations in TARDBP, the gene that codes for TDP-43, are a rare cause of ALS. FUS codes for FUS, another RNA-binding protein with a similar function to TDP-43, which can cause ALS when mutated. It is thought that mutations in TARDBP and FUS increase the binding affinity of the


id: als-20260505-6
file: data/en.wikipedia.org/wiki/ALS-6.md
category: reference
title: ALS
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/ALS
date_saved: 2026-05-05T11:04:10.701513+00:00
instance: kb-cron
parent: als-20260505-0
summary:
=== Differential diagnosis === Because symptoms of ALS can be similar to those of a wide variety of other diseases or disorders, appropriate tests must be conducted to exclude the possibility of other conditions. One of these tests is electromyography (EMG), a special recording technique that detects electrical activity in muscles. Certain EMG findings can support the diagnosis of ALS. Another common test measures nerve conduction velocity (NCV). Specific abnormalities in the NCV results may suggest, for example, that the person has a form of peripheral neuropathy (damage to peripheral nerves) or myopathy (muscle disease) rather than ALS. While a magnetic resonance imaging (MRI) is often normal in people with early-stage ALS, it can reveal evidence of other problems that may be causing the symptoms, such as a spinal cord tumor, multiple sclerosis, a herniated disc in the neck, syringomyelia, or cervical spondylosis. Based on the person's symptoms and findings from these tests, the physician may order tests on blood and urine samples to eliminate the possibility of other diseases, as well as routine laboratory tests. In some cases, for example, if a physician suspects the person may have a myopathy rather than ALS, a muscle biopsy may be performed.


id: als-20260505-7
file: data/en.wikipedia.org/wiki/ALS-7.md
category: reference
title: ALS
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/ALS
date_saved: 2026-05-05T11:04:10.701513+00:00
instance: kb-cron
parent: als-20260505-0
summary:
Riluzole has been found to modestly prolong survival by about 2–3 months. It may have a greater survival benefit for those with bulbar-onset ALS. It may work by decreasing release of the excitatory neurotransmitter glutamate from pre-synaptic neurons. The most common side effects are nausea and a lack of energy (asthenia). People with ALS should begin treatment with riluzole as soon as possible following their diagnosis. Riluzole is available as a tablet, liquid, or dissolvable oral film. Edaravone has been shown to modestly slow the decline in function in a small group of people with early-stage ALS. It may work by protecting motor neurons from oxidative stress. The most common side effects are bruising and gait disturbance. Edaravone is available as an intravenous infusion or as an oral suspension. Tofersen (Qalsody) is an antisense oligonucleotide that was approved for medical use in the United States in April 2023 for the treatment of SOD1-associated ALS. In a study of 108 patients with SOD1-associated ALS there was a non-significant trend towards a slowing of progression, as well as a significant reduction in neurofilament light chain, a putative ALS biomarker thought to indicate neuronal damage. A follow-up study and open-label extension suggested that


id: als-20260505-8
file: data/en.wikipedia.org/wiki/ALS-8.md
category: reference
title: ALS
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/ALS
date_saved: 2026-05-05T11:04:10.701513+00:00
instance: kb-cron
parent: als-20260505-0
summary:
==== Invasive ventilation ==== Invasive ventilation bypasses the nose and mouth (the upper airways) by making a cut in the trachea (tracheostomy) and inserting a tube connected to a ventilator. It is an option for people with advanced ALS whose respiratory symptoms are poorly managed despite continuous NIV use. While invasive ventilation prolongs survival, especially for those younger than 60, it does not treat the underlying neurodegenerative process. The person with ALS will continue to lose motor function, making communication increasingly difficult and sometimes leading to locked-in syndrome, in which they are completely paralyzed except for their eye muscles. About half of the people with ALS who choose to undergo invasive ventilation report a decrease in their quality of life, but most still consider it to be satisfactory. However, invasive ventilation imposes a heavy burden on caregivers and may decrease their quality of life. Attitudes toward invasive ventilation vary from country to country; about 30% of people with ALS in Japan choose invasive ventilation, versus less than 5% in North America and Europe. === Therapy === Physical therapy plays a large role in rehabilitation for individuals with ALS. Specifically, physical, occupational, and speech therapists can set goals and promote benefits


id: als-20260505-9
file: data/en.wikipedia.org/wiki/ALS-9.md
category: reference
title: ALS
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/ALS
date_saved: 2026-05-05T11:04:10.701513+00:00
instance: kb-cron
parent: als-20260505-0
summary:
=== End-of-life care === Palliative care, which relieves symptoms and improves the quality of life without treating the underlying disease, should begin shortly after someone is diagnosed with ALS. Early discussion of end-of-life issues gives people with ALS time to reflect on their preferences for end-of-life care and can help avoid unwanted interventions or procedures. Once they have been fully informed about all aspects of various life-prolonging measures, they can fill out advance directives indicating their attitude toward noninvasive ventilation, invasive ventilation, and feeding tubes. Late in the disease course, difficulty speaking due to muscle weakness (dysarthria) and cognitive dysfunction may impair their ability to communicate their wishes regarding care. Continued failure to solicit the preferences of the person with ALS may lead to unplanned and potentially unwanted emergency interventions, such as invasive ventilation. If people with ALS or their family members are reluctant to discuss end-of-life issues, it may be useful to use the introduction of gastrostomy or noninvasive ventilation as an opportunity to bring up the subject. Hospice care, or palliative care at the end of life, is especially important in ALS because it helps to optimize the management of symptoms and increases the likelihood of a peaceful


id: als-20260505-10
file: data/en.wikipedia.org/wiki/ALS-10.md
category: reference
title: ALS
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/ALS
date_saved: 2026-05-05T11:04:10.701513+00:00
instance: kb-cron
parent: als-20260505-0
summary:
Descriptions of the disease date back to at least 1824 by Charles Bell. In 1850, François-Amilcar Aran was the first to describe a disorder he named "progressive muscular atrophy", a form of ALS in which only the lower motor neurons are affected. In 1869, the connection between the symptoms and the underlying neurological problems was first described by Jean-Martin Charcot, who initially introduced the term amyotrophic lateral sclerosis in his 1874 paper. Flail arm syndrome, a regional variant of ALS, was first described by Alfred Vulpian in 1886. Flail leg syndrome, another regional variant of ALS, was first described by Pierre Marie and his student Patrikios in 1918. === Diagnostic criteria === In the 1950s, electrodiagnostic testing (EMG) and nerve conduction velocity (NCV) testing began to be used to evaluate clinically suspected ALS. In 1969, Edward H. Lambert published the first EMG/NCS diagnostic criteria for ALS, consisting of four findings he considered to strongly support the diagnosis. Since then, several diagnostic criteria have been developed, which are mostly in use for research purposes for inclusion/exclusion criteria, and to stratify patients for analysis in trials. Research diagnostic criteria for ALS include the "El Escorial" in 1994, revised in 1998. In 2006,


id: alzheimer-s-disease-20260505-0
file: data/en.wikipedia.org/wiki/Alzheimer's_disease-0.md
category: reference
title: Alzheimer's disease
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Alzheimer's_disease
date_saved: 2026-05-05T11:04:11.975566+00:00
instance: kb-cron
parent: alzheimer-s-disease-20260505-0
summary:
Alzheimer's disease (AD) is a neurodegenerative disease and is the most common form of dementia, accounting for around 60–70% of cases. The most common early symptom is difficulty in remembering recent events. As the disease advances, symptoms can include problems with language, disorientation (including easily getting lost), mood swings, loss of motivation, self-neglect, and behavioral issues. As a person's condition declines, they often withdraw from family and society. Gradually, bodily functions are lost, ultimately leading to death. Although the speed of progression can vary, the average life expectancy following diagnosis is three to twelve years. The causes of Alzheimer's disease remain poorly understood. There are many environmental and genetic risk factors associated with its development. The strongest genetic risk factor is from an allele of apolipoprotein E, a protein involved in the metabolism of fats in mammals. Other risk factors include a history of head injury, clinical depression, and high blood pressure. The progression of the disease is largely characterised by the accumulation of malformed protein deposits in the cerebral cortex, called amyloid plaques and neurofibrillary tangles. These misfolded protein aggregates interfere with normal cell function, and over time lead to irreversible degeneration of neurons and loss of synaptic connections


id: alzheimer-s-disease-20260505-1
file: data/en.wikipedia.org/wiki/Alzheimer's_disease-1.md
category: reference
title: Alzheimer's disease
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Alzheimer's_disease
date_saved: 2026-05-05T11:04:11.975566+00:00
instance: kb-cron
parent: alzheimer-s-disease-20260505-0
summary:
=== Early stage === In people with Alzheimer's disease, the increasing impairment of learning and memory eventually leads to a definitive diagnosis. In a small percentage, difficulties with language, executive functions, perception (agnosia), or execution of movements (apraxia) are more prominent than memory problems. Alzheimer's disease does not affect all memory capacities equally. Older memories of the person's life (episodic memory), facts learned (semantic memory), and implicit memory (the memory of the body on how to do things, such as using a fork to eat or how to drink from a glass) are affected to a lesser degree than new facts or memories. Language problems are mainly characterised by a shrinking vocabulary and decreased word fluency, leading to a general impoverishment of oral and written language. In this stage, the person with Alzheimer's is usually capable of communicating basic ideas adequately. While performing fine motor tasks such as writing, drawing, or dressing, certain movement coordination and planning difficulties (apraxia) may be present; however, they are commonly unnoticed. As the disease progresses, people with Alzheimer's disease can often continue to perform many tasks independently; however, they may need assistance or supervision with the most cognitively demanding activities. === Middle stage ===


id: alzheimer-s-disease-20260505-2
file: data/en.wikipedia.org/wiki/Alzheimer's_disease-2.md
category: reference
title: Alzheimer's disease
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Alzheimer's_disease
date_saved: 2026-05-05T11:04:11.975566+00:00
instance: kb-cron
parent: alzheimer-s-disease-20260505-0
summary:
Only 1–2% of Alzheimer's cases are inherited due to autosomal dominant mutations, as Alzheimer's disease is substantially polygenic. When autosomal dominant variants cause the disease, it is known as early-onset familial Alzheimer's disease, which is rarer and tends to progress more rapidly. Less than 5% of sporadic Alzheimer's disease have an earlier onset, and early-onset Alzheimer's is about 90% heritable. Familial Alzheimer's disease usually implies two or more persons affected in one or more generations. Early onset familial Alzheimer's disease can be attributed to mutations in one of three genes: those encoding amyloid-beta precursor protein (APP) and presenilins PSEN1 and PSEN2. Most mutations in the APP and presenilin genes increase the production of a small protein called amyloid beta (Aβ)42, which is the main component of amyloid plaques. Some of the mutations merely alter the ratio between Aβ42 and the other major forms—particularly Aβ40—without increasing Aβ42 levels in the brain. Two other genes associated with autosomal dominant Alzheimer's disease are ABCA7 and SORL1. Alleles in the TREM2 gene have been associated with a three to five times higher risk of developing Alzheimer's disease. A Japanese pedigree of familial Alzheimer's disease was found to be associated with a deletion mutation of


id: alzheimer-s-disease-20260505-3
file: data/en.wikipedia.org/wiki/Alzheimer's_disease-3.md
category: reference
title: Alzheimer's disease
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Alzheimer's_disease
date_saved: 2026-05-05T11:04:11.975566+00:00
instance: kb-cron
parent: alzheimer-s-disease-20260505-0
summary:
==== Infection ==== The possibility that infectious agents cause Alzheimer's disease has been considered since the early 20th century, when Oskar Fischer likened amyloid plaques to small masses (called 'Drusen') of a microbe called actinomyces. Since then, at least 15 different agents, including bacteria, viruses, fungi and protozoa, have been proposed to cause Alzheimer's disease. No definitive evidence has been presented that a specific infectious agent is necessary and sufficient to cause Alzheimer's disease. However, it is possible that microbial infections might act as risk factors for the disease. For example, human herpes viruses such as HSV-1, HHV-6, and HHV-7 have been linked to the risk of Alzheimer's disease. In addition, some pathogens have been reported to seed Aβ deposition in the brain, and aggregated Aβ has antimicrobial properties, suggesting that Aβ plaques might form when brain cells generate Aβ to fight infection. Researchers caution that brain infections can cause dementia by mechanisms unrelated to Alzheimer's disease. ==== DNA damage ==== DNA damage accumulates in affected brains; reactive oxygen species may be the major source of this DNA damage. ==== Cholinergic ==== The cholinergic hypothesis proposes that the loss of neurons in the basal forebrain, which produce the neurotransmitter acetylcholine,


id: alzheimer-s-disease-20260505-4
file: data/en.wikipedia.org/wiki/Alzheimer's_disease-4.md
category: reference
title: Alzheimer's disease
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Alzheimer's_disease
date_saved: 2026-05-05T11:04:11.975566+00:00
instance: kb-cron
parent: alzheimer-s-disease-20260505-0
summary:
The gross (macroscopic) appearance of the brain in Alzheimer's disease is variable. In many cases the cortical sulci are widened and the gyri are shrunken, but the degree of cortical atrophy varies. It can sometimes be difficult to discern, particularly in the very elderly. The areas most affected by atrophy are the medial temporal lobe including the hippocampal formation, the amygdala, the frontal lobe and the parietal lobe; the occipital lobe is relatively unaffected by atrophy. The volume of the ventricles increases in parallel with cortical shrinkage. Studies using MRI and PET have documented reductions in the size of specific brain regions in people with Alzheimer's disease as they progress from mild cognitive impairment to Alzheimer's disease, and in comparison with similar images from healthy older adults. These macroscopic changes in the brain are not specific to Alzheimer's and can occur in other disorders and to some extent in normal aging. At the microscopic level, the defining histopathologic characteristics of Alzheimer's disease are abundant amyloid plaques and neurofibrillary tangles in certain brain regions. Both of these abnormalities are clearly visible by microscopy, and amyloid imaging. In the early stages of disease, tangles are present mainly in the medial temporal lobe


id: alzheimer-s-disease-20260505-5
file: data/en.wikipedia.org/wiki/Alzheimer's_disease-5.md
category: reference
title: Alzheimer's disease
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Alzheimer's_disease
date_saved: 2026-05-05T11:04:11.975566+00:00
instance: kb-cron
parent: alzheimer-s-disease-20260505-0
summary:
=== Disease mechanism === Exactly how disturbances of production and aggregation of the Aβ peptide give rise to the pathology of Alzheimer's disease is not known. The amyloid hypothesis (also known as the 'amyloid cascade hypothesis') posits that the accumulation of abnormally shaped Aβ peptides is the central event triggering the sequence of changes that eventually lead to neurodegeneration and dementia. Misfolded Aβ accumulates in the brain because it causes normal Aβ molecules to similarly misfold by a prion-like 'seeding' mechanism. The aggregated Aβ takes the form of small oligomers (which are particularly toxic to neurons) and amyloid fibrils, the long polymers that are the main components of Aβ plaques. Some researchers have argued that the amyloid fibrils bind up smaller oligomers and thus protect brain cells from the injurious effects of the oligomers. However, the plaques are not benign inasmuch as they are associated with abnormal neuronal processes and local inflammation. Whatever the relative influence of Aβ oligomers and fibrils, the presence of aggregated Aβ is associated with the disruption of neuronal metabolism and various other changes such as inflammation. Aβ also selectively builds up in mitochondria in the cells of Alzheimer's-affected brains, and it inhibits certain enzyme functions


id: alzheimer-s-disease-20260505-6
file: data/en.wikipedia.org/wiki/Alzheimer's_disease-6.md
category: reference
title: Alzheimer's disease
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Alzheimer's_disease
date_saved: 2026-05-05T11:04:11.975566+00:00
instance: kb-cron
parent: alzheimer-s-disease-20260505-0
summary:
Alzheimer's disease (AD) can only be definitively diagnosed with autopsy findings; in the absence of autopsy, clinical diagnoses of AD are "possible" or "probable", based on other findings. Up to 23% of those clinically diagnosed with AD may be misdiagnosed and may have pathology suggestive of another condition with symptoms that mimic those of AD. AD is usually clinically diagnosed based on a person's medical history, observations from friends or relatives, and behavioral changes. The presence of characteristic neuropsychological changes with impairments in at least two cognitive domains that are severe enough to affect a person's functional abilities is required for the diagnosis. Domains that may be impaired include memory (most commonly impaired), language, executive function, visuospatial functioning, or other areas of cognition. The neurocognitive changes must be a decline from a prior level of function, and the diagnosis requires ruling out other common causes of neurocognitive decline. Advanced medical imaging with computed tomography (CT) or magnetic resonance imaging (MRI), and with single-photon emission computed tomography (SPECT) or positron emission tomography (PET), can be used to help exclude other cerebral pathology or subtypes of dementia. On MRI or CT, Alzheimer's disease usually shows a generalised or focal cortical atrophy, which


id: alzheimer-s-disease-20260505-7
file: data/en.wikipedia.org/wiki/Alzheimer's_disease-7.md
category: reference
title: Alzheimer's disease
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Alzheimer's_disease
date_saved: 2026-05-05T11:04:11.975566+00:00
instance: kb-cron
parent: alzheimer-s-disease-20260505-0
summary:
Neuropsychological tests including cognitive tests such as the mini–mental state examination (MMSE), the Montreal Cognitive Assessment (MoCA) and the Mini-Cog are widely used to aid in diagnosis of the cognitive impairments in AD. These tests may not always be accurate, as they lack sensitivity to mild cognitive impairment, and can be biased by language or attention problems; more comprehensive test arrays are necessary for high reliability of results, particularly in the earliest stages of the disease. Further neurological examinations are crucial in the differential diagnosis of Alzheimer's disease and other diseases. Interviews with family members are used in assessment, and caregivers can supply important information on daily living abilities and on the decrease in the person's mental function. A caregiver's viewpoint is particularly important, since a person with Alzheimer's disease is commonly unaware of their deficits. Many times, families have difficulties in the detection of initial dementia symptoms and may not communicate accurate information to a physician. Supplemental testing can rule out other potentially treatable diagnoses and help avoid misdiagnoses. Common supplemental tests include blood tests, thyroid function tests, as well as tests to assess vitamin B12 levels, rule out neurosyphilis and rule out metabolic problems (including tests for kidney


id: alzheimer-s-disease-20260505-8
file: data/en.wikipedia.org/wiki/Alzheimer's_disease-8.md
category: reference
title: Alzheimer's disease
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Alzheimer's_disease
date_saved: 2026-05-05T11:04:11.975566+00:00
instance: kb-cron
parent: alzheimer-s-disease-20260505-0
summary:
Certain lifestyle activities, such as physical and cognitive exercises, higher education and occupational attainment, cigarette smoking, stress, sleep, and the management of other comorbidities, including diabetes and hypertension, may affect the risk of developing AD. Physical exercise is associated with a decreased rate of dementia, and is effective in reducing symptom severity in those with AD. Memory and cognitive functions can be improved with aerobic exercises including brisk walking three times weekly for forty minutes. It may also induce neuroplasticity of the brain. Participating in mental exercises, such as reading, crossword puzzles, and chess have reported potential to be preventive. Meeting the WHO recommendations for physical activity is associated with a lower risk of AD. Higher education and occupational attainment, and participation in leisure activities, contribute to a reduced risk of developing AD, or of delaying the onset of symptoms. This is compatible with the cognitive reserve theory, which states that some life experiences result in more efficient neural functioning, providing the individual with a cognitive reserve that delays the onset of dementia. Education delays the onset of Alzheimer's disease syndrome without changing the duration of the disease. Cessation in smoking may reduce the risk of developing AD, specifically in


id: alzheimer-s-disease-20260505-9
file: data/en.wikipedia.org/wiki/Alzheimer's_disease-9.md
category: reference
title: Alzheimer's disease
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Alzheimer's_disease
date_saved: 2026-05-05T11:04:11.975566+00:00
instance: kb-cron
parent: alzheimer-s-disease-20260505-0
summary:
=== Psychosocial === Psychosocial interventions are used as an adjunct to pharmaceutical treatment and can be classified within behavior-, emotion-, cognition- or stimulation-oriented approaches. Behavioral interventions attempt to identify and reduce the antecedents and consequences of problem behaviors. This approach has not reported success in improving overall functioning, but can help to reduce some specific problem behaviors, such as incontinence. There is a lack of high quality data on the effectiveness of these techniques in other behavior problems such as wandering. Music therapy is effective in reducing behavioral and psychological symptoms. Emotion-oriented interventions include reminiscence therapy, validation therapy, supportive psychotherapy, sensory integration, also called snoezelen, and simulated presence therapy. A Cochrane review has found no evidence that this is effective. Reminiscence therapy (RT) involves the discussion of past experiences, individually or in groups, many times with the aid of photographs, household items, music and sound recordings, or other familiar items from the past. A 2018 review of the effectiveness of RT found that effects were inconsistent, small in size, and of doubtful clinical significance, and varied by setting. Simulated presence therapy (SPT) is based on attachment theories and involves playing a recording with the voices of the closest relatives of


id: alzheimer-s-disease-20260505-10
file: data/en.wikipedia.org/wiki/Alzheimer's_disease-10.md
category: reference
title: Alzheimer's disease
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Alzheimer's_disease
date_saved: 2026-05-05T11:04:11.975566+00:00
instance: kb-cron
parent: alzheimer-s-disease-20260505-0
summary:
Regarding incidence, where a disease-free population is followed over the years have shown rates between 10 and 15 per thousand person-years for all dementias and 5–8 for AD in Spain and Italy, which means that half of new dementia cases each year are Alzheimer's disease. Advancing age is a primary risk factor for the disease, and incidence rates are not equal for all ages: every 5 years after the age of 65, the risk of acquiring the disease approximately doubles, increasing from 3 to as much as 69 per thousand person-years. The prevalence of AD in populations is dependent upon factors including incidence and survival. Since the incidence of AD increases with age, prevalence depends on the mean age of the population for which prevalence is given. In the United States in 2020, AD dementia prevalence was estimated to be 5.3% for those in the 60–74 age group, with the rate increasing to 13.8% in the 74–84 group and to 34.6% in those greater than 85. Prevalence rates in some less developed regions around the globe are lower. Both the prevalence and incidence rates of AD are steadily increasing, and the prevalence rate is estimated to triple by 2050 reaching


id: alzheimer-s-disease-20260505-11
file: data/en.wikipedia.org/wiki/Alzheimer's_disease-11.md
category: reference
title: Alzheimer's disease
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Alzheimer's_disease
date_saved: 2026-05-05T11:04:11.975566+00:00
instance: kb-cron
parent: alzheimer-s-disease-20260505-0
summary:
=== Social costs === Dementia, and specifically Alzheimer's disease, may be among the most costly diseases for societies worldwide. As populations age, these costs will probably increase and become an important social problem and economic burden. Costs associated with AD include direct and indirect medical costs, which vary between countries depending on social care for a person with AD. Direct costs include doctor visits, hospital care, medical treatments, nursing home care, specialised equipment, and household expenses. Indirect costs include the cost of informal care and the loss in productivity of informal caregivers. In the United States as of 2019, informal (family) care is estimated to constitute nearly three-fourths of caregiving for people with AD at a cost of US$234 billion per year and approximately 18.5 billion hours of care. The cost to society worldwide to care for individuals with AD is projected to increase nearly ten-fold, and reach about US$9.1 trillion by 2050. Costs for those with more severe dementia or behavioral disturbances are higher and are related to the additional caregiving time to provide physical care. === Caregiving burden === Individuals with Alzheimer's will require assistance in their lifetime, and care will most likely come in the form of


id: attention-20260505-0
file: data/en.wikipedia.org/wiki/Attention-0.md
category: reference
title: Attention
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Attention
date_saved: 2026-05-05T11:04:13.138266+00:00
instance: kb-cron
parent: attention-20260505-0
summary:
Attention is the concentration of awareness directed at some task or phenomenon while mostly excluding others. Across disciplines, the nature of this directedness is conceptualized in different ways. In cognitive psychology, attention is often described as the allocation of limited cognitive processing resources to a subset of information, thoughts, or tasks. In neuropsychology, attention is understood as a set of mechanisms by which sensory cues and internal goals modulate neuronal tuning and orient behavioral and cognitive processes. Attention is not a unitary phenomenon but an umbrella term for multiple related processes, including selective attention (prioritizing some stimuli over others), sustained attention (maintaining focus), divided attention (sharing resources across tasks), and orienting (shifting focus in space or time). These processes are supported by distributed neural networks in frontal, parietal, and subcortical regions and are closely linked to working memory, executive functions, and consciousness. Patterns of attention also vary across cultures, especially in how individuals attend to context versus focal objects and how children are guided to manage attention in everyday activities. == History == === 16th century === John B. Watson called Juan Luis Vives the father of modern psychology. In his book De Anima et Vita, Vives argued that the


id: attention-20260505-1
file: data/en.wikipedia.org/wiki/Attention-1.md
category: reference
title: Attention
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Attention
date_saved: 2026-05-05T11:04:13.138266+00:00
instance: kb-cron
parent: attention-20260505-0
summary:
Modern experimental work began with investigations of the "cocktail party problem" by Colin Cherry in 1953. Cherry asked how people at a noisy party can attend to one conversation while ignoring others. He studied this through dichotic listening tasks, in which participants heard two simultaneous streams of speech through headphones and were asked to shadow (repeat) one stream while ignoring the other. These paradigms were extended by Donald Broadbent and others. By the 1990s, psychologists increasingly used positron emission tomography (PET) and functional magnetic resonance imaging (fMRI) to study attention in the brain. Because this equipment was typically housed in hospitals, psychologists collaborated with neurologists. Psychologist Michael Posner and neurologist Marcus Raichle pioneered imaging studies of selective attention. The adoption of neuroimaging, alongside long-standing techniques such as electroencephalography (EEG), led to extensive research on the neural basis of attention. A growing body of work identified a frontoparietal attention network implicated in the control of attention. == Components == Attention is constrained by both the number of elements that can be processed and the duration of exposure. Experimental studies of attention began with Wundt’s findings. He suggested the scope of attention limited to about 3–6 items. Decades of research on subitizing


id: attention-20260505-2
file: data/en.wikipedia.org/wiki/Attention-2.md
category: reference
title: Attention
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Attention
date_saved: 2026-05-05T11:04:13.138266+00:00
instance: kb-cron
parent: attention-20260505-0
summary:
=== Load === Some individuals can perform certain overlearned tasks with minimal conscious attention. For example, highly trained Morse code operators have been shown to copy messages accurately while engaging in a concurrent conversation. This reflects the development of automaticity through extensive practice: once a skill is overlearned beyond 100% accuracy, its execution can become largely autonomous and require fewer attentional resources. Perceptual load theory proposes that attentional resources are limited and must be fully used. Under high perceptual load, fewer resources remain available for processing irrelevant stimuli, reducing distraction; under low load, more resources “spill over”, increasing susceptibility to distraction. In applied settings, such as education, measurement often emphasizes accuracy and reaction time (RT), which can obscure finer-grained distributions of temporal and spatial attention. === Neural correlates === Eric Knudsen identified four fundamental components of attention: (a) working memory, (b) competitive selection, (c) top-down sensitivity control, and (d) salience filters. At different hierarchical levels, spatial maps enhance or inhibit activity in sensory areas and guide orienting behaviors such as eye movements: At higher levels, the frontal eye fields (FEF) and dorsolateral prefrontal cortex contain retinocentric spatial maps. Microstimulation in the FEF can induce saccades to specific locations, and subthreshold


id: attention-20260505-3
file: data/en.wikipedia.org/wiki/Attention-3.md
category: reference
title: Attention
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Attention
date_saved: 2026-05-05T11:04:13.138266+00:00
instance: kb-cron
parent: attention-20260505-0
summary:
Divided attention refers to attempts to allocate attention to multiple sources of information or tasks at the same time. In everyday language, this is often called multitasking. Research consistently shows that people make more errors and perform tasks more slowly when multitasking than when focusing on one task at a time. Early research examined limits on learning and perception when people performed simultaneous tasks, such as reading while listening and writing something else, or listening to two different messages through each ear (dichotic listening). Later work examined divided attention in applied settings such as driving while tuning a radio or driving while talking on a phone. Most contemporary research on multitasking investigates dual-task performance, often combining driving with secondary tasks such as texting, eating, or conversing. These studies show that driving performance deteriorates under dual-task conditions: drivers are more error-prone, brake harder and later, drift across lanes, and show reduced awareness of their surroundings. There is little difference between speaking on hands-free versus hand-held phones, suggesting that the primary limitation is cognitive rather than motor. In contrast, passengers in the car can adapt their conversation to the demands of driving (e.g., pausing when traffic becomes complex), reducing interference. Daniel Kahneman


id: attention-20260505-4
file: data/en.wikipedia.org/wiki/Attention-4.md
category: reference
title: Attention
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Attention
date_saved: 2026-05-05T11:04:13.138266+00:00
instance: kb-cron
parent: attention-20260505-0
summary:
=== Filter === Broadbent's Filter Model of Attention proposed that sensory information is briefly held in a pre-attentive store, and a filter selects input based on simple physical characteristics (e.g., location, pitch). Selected information passes into a limited capacity processing system, while unattended inputs are blocked and not semantically processed. This implies that information in the unattended channel does not reach meaning-level analysis, and that shifting the filter between channels requires time. Experiments by Gray and Wedderburn and subsequent work by Anne Treisman challenged this strict early filter model, leading to refinements. In Treisman’s attenuation model, unattended inputs are not completely blocked; instead, they are attenuated and can be processed to some extent, especially if they are highly salient (e.g., a person’s own name). Deutsch and Deutsch later proposed a late selection model in which all inputs are processed semantically, and selection occurs at the point of response or conscious access. In this view, the ‘‘attentional bottleneck’’ occurs late, after semantic analysis. === Selection === Debates between early-selection and late-selection models centered on whether unattended information is processed semantically. Early-selection models (e.g., Broadbent, Treisman) posit that attention acts before full semantic analysis, whereas late-selection models (e.g., Deutsch & Deutsch) posit


id: autosomal-dominant-nocturnal-frontal-lobe-epilepsy-20260505-0
file: data/en.wikipedia.org/wiki/Autosomal_dominant_nocturnal_frontal_lobe_epilepsy-0.md
category: reference
title: Autosomal dominant nocturnal frontal lobe epilepsy
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Autosomal_dominant_nocturnal_frontal_lobe_epilepsy
date_saved: 2026-05-05T11:04:14.343019+00:00
instance: kb-cron
parent: autosomal-dominant-nocturnal-frontal-lobe-epilepsy-20260505-0
summary:
Autosomal dominant nocturnal frontal lobe epilepsy (ADNFLE) is an epileptic disorder that causes frequent violent seizures during sleep. These seizures often involve complex motor movements, such as hand clenching, arm raising/lowering, and knee bending. Vocalizations such as shouting, moaning, or crying are also common. ADNFLE is often misdiagnosed as nightmares. Attacks often occur in clusters and typically first manifest in childhood. There are four known loci for ADNFLE, three with known causative genes. These genes, CHRNA4, CHRNB2, and CHRNA2, encode various nicotinic acetylcholine receptor α and β subunits. == Signs and symptoms == == Causes == While not well understood, it is believed that malfunction in thalamocortical loops plays a vital role in ADNFLE. The reasons for this belief are threefold. Firstly, thalamocortical loops are important in sleep and the frontal cortex is the origin of ADNFLE seizures. Secondly, both the thalamus and cortex receive cholinergic inputs and acetylcholine receptor subunits comprise the three known causative genes for ADNFLE. Thirdly, K-complex are almost invariably present at the start of seizures. == Mechanism == === CHRNA4 === The first mutation associated with ADNFLE is a serine to phenylalanine transition at position 248 (S248F), located in the second transmembrane spanning region of


id: awareness-20260505-0
file: data/en.wikipedia.org/wiki/Awareness-0.md
category: reference
title: Awareness
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Awareness
date_saved: 2026-05-05T11:04:15.557513+00:00
instance: kb-cron
parent: awareness-20260505-0
summary:
In psychology and philosophy, awareness is the perception or knowledge of something. The concept is often synonymous with consciousness; however, one can be aware of something without being explicitly conscious of it (e.g., blindsight). The states of awareness are also associated with the states of experience, so that the structure represented in awareness is mirrored in the structure of experience. == Concept == Awareness is a relative concept. It may refer to an internal state, such as a visceral feeling, or on external events by way of sensory perception. It is analogous to sensing something, a process distinguished from observing and perceiving (which involves a basic process of acquainting with the items we perceive). Awareness can be described as something that occurs when the brain is activated in certain ways, such as when the color red is seen once the retina is stimulated by light waves. This conceptualization is posited due to the difficulty in developing an analytic definition of awareness or sensory awareness. Awareness is also associated with consciousness in the sense that it denotes a fundamental experience such as a feeling or intuition that accompanies the experience of phenomena. Specifically, this is referred to as awareness of experience.


id: awareness-20260505-1
file: data/en.wikipedia.org/wiki/Awareness-1.md
category: reference
title: Awareness
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Awareness
date_saved: 2026-05-05T11:04:15.557513+00:00
instance: kb-cron
parent: awareness-20260505-0
summary:
Living systems are cognitive systems, and living as a process is a process of cognition. This statement is valid for all organisms, with or without a nervous system. This theory contributes a perspective that cognition is a process present at organic levels that people do not usually consider to be aware. Given the possible relationship between awareness and cognition, and consciousness, this theory contributes an interesting perspective in the philosophical and scientific dialogue of awareness and living systems theory. == Communications and information systems == In cooperative settings, awareness is a term used to denote "knowledge created through the interaction of an agent and its environment — in simple terms 'knowing what is going on'". In this setting, awareness is meant to convey how individuals monitor and perceive the information surrounding their colleagues and the environment they are in. This information is incredibly useful and critical to the performance and success of collaborations. Awareness can be further defined by breaking it down into a set of characteristics: Knowledge about the state of some environment Environments are continually changing, therefore awareness knowledge must be constantly maintained Individuals interact with the environment, and maintenance of awareness is accomplished through this interaction. Generally


id: binding-problem-20260505-0
file: data/en.wikipedia.org/wiki/Binding_problem-0.md
category: reference
title: Binding problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Binding_problem
date_saved: 2026-05-05T11:04:16.760782+00:00
instance: kb-cron
parent: binding-problem-20260505-0
summary:
The unity of consciousness and (cognitive) binding problem is the problem of how objects, background, and abstract or emotional features are combined into a single experience. The binding problem refers to the overall encoding of our brain circuits for the combination of decisions, actions, and perception. It is considered a "problem" because no complete model exists. The binding problem can be subdivided into the four areas of perception, neuroscience, cognitive science, and the philosophy of mind. It includes general considerations on coordination, the subjective unity of perception, and variable binding. == General considerations on coordination == === Summary of problem === Attention is crucial in determining which phenomena appear to be bound together, noticed, and remembered. This specific binding problem is generally referred to as temporal synchrony. At the most basic level, all neural firing and its adaptation depends on specific consideration to timing (Feldman, 2010). At a much larger level, frequent patterns in large scale neural activity are a major diagnostic and scientific tool. === Synchronization theory and research === A popular hypothesis mentioned by neuroscientist Peter Milner, in his 1974 article A Model for Visual Shape Recognition, has been that features of individual objects are bound/segregated via synchronization


id: binding-problem-20260505-1
file: data/en.wikipedia.org/wiki/Binding_problem-1.md
category: reference
title: Binding problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Binding_problem
date_saved: 2026-05-05T11:04:16.760782+00:00
instance: kb-cron
parent: binding-problem-20260505-0
summary:
=== Summary of problem === The visual feature binding problem refers to the question of why we do not confuse a red circle and a blue square with a blue circle and a red square. The understanding of the circuits in the brain stimulated for visual feature binding is increasing. A binding process is required for us to accurately encode various visual features in separate cortical areas. In her feature integration theory, Treisman suggested that one of the first stages of binding between features is mediated by the features' links to a common location. The second stage is combining individual features of an object that requires attention, and selecting that object occurs within a "master map" of locations. Psychophysical demonstrations of binding failures under conditions of full attention provide support for the idea that binding is accomplished through common location tags. An implication of these approaches is that sensory data such as color or motion may not normally exist in "unallocated" form. For Merker: "The 'red' of a red ball does not float disembodied in an abstract color space in V4." If color information allocated to a point in the visual field is converted directly, via the instantiation of some


id: binding-problem-20260505-2
file: data/en.wikipedia.org/wiki/Binding_problem-2.md
category: reference
title: Binding problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Binding_problem
date_saved: 2026-05-05T11:04:16.760782+00:00
instance: kb-cron
parent: binding-problem-20260505-0
summary:
=== Summary of problem === Smythies defines the combination problem, also known as the subjective unity of perception, as "How do the brain mechanisms actually construct the phenomenal object?". Revonsuo equates this to "consciousness-related binding", emphasizing the entailment of a phenomenal aspect. As Revonsuo explores in 2006, there are nuances of difference beyond the basic BP1:BP2 division. Smythies speaks of constructing a phenomenal object ("local unity" for Revonsuo) but philosophers such as René Descartes, Gottfried Wilhelm Leibniz, Immanuel Kant, and James (see Brook and Raymont) have typically been concerned with the broader unity of a phenomenal experience ("global unity" for Revonsuo) – which, as Bayne illustrates may involve features as diverse as seeing a book, hearing a tune and feeling an emotion. Further discussion will focus on this more general problem of how sensory data that may have been segregated into, for instance, "blue square" and "yellow circle" are to be re-combined into a single phenomenal experience of a blue square next to a yellow circle, plus all other features of their context. There is a wide range of views on just how real this "unity" is, but the existence of medical conditions in which it appears to be subjectively


id: binding-problem-20260505-3
file: data/en.wikipedia.org/wiki/Binding_problem-3.md
category: reference
title: Binding problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Binding_problem
date_saved: 2026-05-05T11:04:16.760782+00:00
instance: kb-cron
parent: binding-problem-20260505-0
summary:
==== fMRI work ==== Stoll and colleagues conducted an fMRI experiment to see whether participants would view a dynamic bistable stimulus globally or locally. Responses in lower visual cortical regions were suppressed when participants viewed the stimulus globally. However, if global perception was without shape grouping, higher cortical regions were suppressed. This experiment shows that higher order cortex is important in perceptual grouping. Grassi and colleagues used three different motion stimuli to investigate scene segmentation or how meaningful entities are grouped together and separated from other entities in a scene. Across all stimuli, scene segmentation was associated with increased activity in the posterior parietal cortex and decreased activity in lower visual areas. This suggests that the posterior parietal cortex is important for viewing an integrated whole. ==== EEG work ==== Mersad and colleagues used an EEG frequency tagging technique to differentiate between brain activity for the integrated whole object and brain activity for parts of the object. The results showed that the visual system binds two humans in close proximity as part of an integrated whole. These results are consistent with evolutionary theories that face-to-face bodies are one of the earliest representations of social interaction. It also supports other experimental


id: binding-problem-20260505-4
file: data/en.wikipedia.org/wiki/Binding_problem-4.md
category: reference
title: Binding problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Binding_problem
date_saved: 2026-05-05T11:04:16.760782+00:00
instance: kb-cron
parent: binding-problem-20260505-0
summary:
=== Modern theories === Daniel Dennett has proposed that we, as humans, sensing our experiences as individual single events is illusory and that, instead, at any one time there are "multiple drafts" of sensory patterns at multiple sites. Each would only cover a fragment of what we think we experience. Arguably, Dennett is claiming that consciousness is not unified and there is no phenomenal binding problem. Most philosophers have difficulty with this position (see Bayne), but some physiologists agree with it. In particular, the demonstration of perceptual asynchrony in psychophysical experiments by Moutoussis and Zeki, where color is perceived before orientation of lines and before motion by 40 and 80 ms respectively, constitutes an argument that, over these very short time periods, different attributes are consciously perceived at different times, leading to the view that at least over these brief periods of time after visual stimulation, different events are not bound to each other, leading to the view of a disunity of consciousness, at least over these brief time intervals. Dennett's view might be in keeping with evidence from recall experiments and change blindness purporting to show that our experiences are much less rich than we sense them to be


id: binding-problem-20260505-5
file: data/en.wikipedia.org/wiki/Binding_problem-5.md
category: reference
title: Binding problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Binding_problem
date_saved: 2026-05-05T11:04:16.760782+00:00
instance: kb-cron
parent: binding-problem-20260505-0
summary:
== Shared intentionality and binding == According to bioengineer Igor Val Danilov, the mother–fetus neurocognitive model—knowledge about neurophysiological processes during shared intentionality—can reveal insights into the binding problem and even the perception of object development since intentionality succeeds before organisms confront the binding problem. Indeed, at the beginning of life, the environment is the cacophony of stimuli: electromagnetic waves, chemical interactions, and pressure fluctuations. Because the environment is uncategorised for the organisms at this beginning stage of development, the sensation is too limited by the noise to solve the cue problem—the relevant stimulus cannot overcome the noise magnitude if it passes through the senses. While very young organisms need to combine objects, background and abstract or emotional features into a single experience for building the representation of the surrounded reality, they cannot distinguish relevant sensory stimuli independently to integrate them into object representations. Even the embodied dynamical system approach cannot get around the cue to noise problem. The application of embodied information requires an already categorised environment onto objects—holistic representation of reality—which occurs through (and only after the emergence of) perception and intentionality. In short, properties of the mother's heart—the electromagnetic and acoustic oscillations—converge the neuronal activity of both nervous


id: biochemistry-of-alzheimer-s-disease-20260505-0
file: data/en.wikipedia.org/wiki/Biochemistry_of_Alzheimer's_disease-0.md
category: reference
title: Biochemistry of Alzheimer's disease
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Biochemistry_of_Alzheimer's_disease
date_saved: 2026-05-05T11:04:17.929946+00:00
instance: kb-cron
parent: biochemistry-of-alzheimer-s-disease-20260505-0
summary:
The biochemistry of Alzheimer's disease, the most common cause of dementia, is not yet very well understood. Alzheimer's disease (AD) has been identified as a proteopathy: a protein misfolding disease due to the accumulation of abnormally folded amyloid beta (Aβ) protein in the brain. Amyloid beta is a short peptide that is an abnormal proteolytic byproduct of the transmembrane protein amyloid-beta precursor protein (APP), whose function is unclear but thought to be involved in neuronal development. The presenilins are components of proteolytic complex involved in APP processing and degradation. Amyloid beta monomers are soluble and contain short regions of beta sheet and polyproline II helix secondary structures in solution, though they are largely alpha helical in membranes; however, at sufficiently high concentration, they undergo a dramatic conformational change to form a beta sheet-rich tertiary structure that aggregates to form amyloid fibrils. These fibrils and oligomeric forms of Aβ deposit outside neurons in formations known as senile plaques. There are different types of plaques, including the diffuse, compact, cored or neuritic plaque types, as well as Aβ deposits in the walls of small blood vessel walls in the brain called cerebral amyloid angiopathy. AD is also considered a tauopathy due to


id: biochemistry-of-alzheimer-s-disease-20260505-1
file: data/en.wikipedia.org/wiki/Biochemistry_of_Alzheimer's_disease-1.md
category: reference
title: Biochemistry of Alzheimer's disease
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Biochemistry_of_Alzheimer's_disease
date_saved: 2026-05-05T11:04:17.929946+00:00
instance: kb-cron
parent: biochemistry-of-alzheimer-s-disease-20260505-0
summary:
== Disease mechanisms == While the gross histological features of AD in the brain have been well characterized, several different hypotheses have been advanced regarding the primary cause. Among the oldest hypotheses is the cholinergic hypothesis, which suggests that deficiency in cholinergic signaling initiates the progression of the disease. Current theories establish that both misfolding tau protein inside the cell and aggregation of amyloid beta outside the cell initiates the cascade leading to AD pathology. Newer potential hypotheses propose metabolic factors, vascular disturbance, lipid invasion and chronically elevated inflammation in the brain as contributing factors to AD. The amyloid beta hypothesis of molecular initiation have become dominant among many researchers to date. The amyloid and tau hypothesis are the most widely accepted. === Tau hypothesis === The hypothesis that tau is the primary causative factor has long been grounded in the observation that deposition of amyloid plaques does not correlate well with neuron loss. A mechanism for neurotoxicity has been proposed based on the loss of microtubule-stabilizing tau protein that leads to the degradation of the cytoskeleton. However, consensus has not been reached on whether tau hyperphosphorylation precedes or is caused by the formation of the abnormal helical filament aggregates.


id: biochemistry-of-alzheimer-s-disease-20260505-2
file: data/en.wikipedia.org/wiki/Biochemistry_of_Alzheimer's_disease-2.md
category: reference
title: Biochemistry of Alzheimer's disease
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Biochemistry_of_Alzheimer's_disease
date_saved: 2026-05-05T11:04:17.929946+00:00
instance: kb-cron
parent: biochemistry-of-alzheimer-s-disease-20260505-0
summary:
=== Historical cholinergic hypothesis === The cholinergic hypothesis of AD development was first proposed in 1976 by Peter Davies and A.J.F Maloney. It claimed that Alzheimer's begins as a deficiency in the production of acetylcholine, a vital neurotransmitter. Much early therapeutic research was based on this hypothesis, including restoration of the "cholinergic nuclei". The possibility of cell-replacement therapy was investigated on the basis of this hypothesis. All of the first-generation anti-Alzheimer's medications are based on this hypothesis and work to preserve acetylcholine by inhibiting acetylcholinesterases (enzymes that break down acetylcholine). These medications, though sometimes beneficial, have not led to a cure. In all cases, they have served to only treat symptoms of the disease and have neither halted nor reversed it. These results and other research have led to the conclusion that acetylcholine deficiencies may not be directly causal, but are a result of widespread brain tissue damage, damage so widespread that cell-replacement therapies are likely to be impractical. More recent findings center on the effects of the misfolded and aggregated proteins, amyloid beta and tau: tau protein abnormalities may initiate the disease cascade, then beta amyloid deposits progress the disease. === Glucose consumption === The human brain is one


id: biochemistry-of-alzheimer-s-disease-20260505-3
file: data/en.wikipedia.org/wiki/Biochemistry_of_Alzheimer's_disease-3.md
category: reference
title: Biochemistry of Alzheimer's disease
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Biochemistry_of_Alzheimer's_disease
date_saved: 2026-05-05T11:04:17.929946+00:00
instance: kb-cron
parent: biochemistry-of-alzheimer-s-disease-20260505-0
summary:
=== Oxidative stress === Oxidative stress is emerging as a key factor in the pathogenesis of AD. Reactive oxygen species (ROS) over-production is thought to play a critical role in the accumulation and deposition of amyloid beta in AD. Brains of AD patients have elevated levels of oxidative DNA damage in both nuclear and mitochondrial DNA, but the mitochondrial DNA has approximately 10-fold higher levels than nuclear DNA. Aged mitochondria may be the critical factor in the origin of neurodegeneration in AD. Even individuals with mild cognitive impairment, the phase between normal aging and early dementia, have increased oxidative damage in their nuclear and mitochondrial brain DNA (see Aging brain). Naturally occurring DNA double-strand breaks (DSBs) arise in human cells largely from single-strand breaks induced by various processes including the activity of reactive oxygen species, topoisomerases, and hydrolysis due to thermal fluctuations. In neurons DSBs are induced by a type II topoisomerase as part of the physiologic process of memory formation. DSBs are present in both neurons and astrocytes in the postmortem human hippocampus of AD patients at a higher level than in non-AD individuals. AD is associated with an accumulation of DSBs in neurons and astrocytes in the hippocampus


id: biochemistry-of-alzheimer-s-disease-20260505-4
file: data/en.wikipedia.org/wiki/Biochemistry_of_Alzheimer's_disease-4.md
category: reference
title: Biochemistry of Alzheimer's disease
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Biochemistry_of_Alzheimer's_disease
date_saved: 2026-05-05T11:04:17.929946+00:00
instance: kb-cron
parent: biochemistry-of-alzheimer-s-disease-20260505-0
summary:
=== Reelin hypothesis === A 1994 study showed that the isoprenoid changes in Alzheimer's disease differ from those occurring during normal aging and that this disease cannot, therefore, be regarded as a result of premature aging. During aging the human brain shows a progressive increase in levels of dolichol, a reduction in levels of ubiquinone, but relatively unchanged concentrations of cholesterol and dolichyl phosphate. In Alzheimer's disease, the situation is reversed with decreased levels of dolichol and increased levels of ubiquinone. The concentrations of dolichyl phosphate are also increased, while cholesterol remains unchanged. The increase in the sugar carrier dolichyl phosphate may reflect an increased rate of glycosylation in the diseased brain and the increase in the endogenous anti-oxidant ubiquinone an attempt to protect the brain from oxidative stress, for instance induced by lipid peroxidation. Ropren, identified previously in Russia, is neuroprotective in a rat model of Alzheimer's disease. A relatively recent hypothesis based mainly on rodent experiments links the onset of Alzheimer's disease to the hypofunction of the large extracellular protein reelin. A decrease of reelin in the human entorhinal cortex where the disease typically initiates is evident while compensatory increase of reelin levels in other brain structures of


id: catecholaldehyde-hypothesis-20260505-0
file: data/en.wikipedia.org/wiki/Catecholaldehyde_hypothesis-0.md
category: reference
title: Catecholaldehyde hypothesis
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Catecholaldehyde_hypothesis
date_saved: 2026-05-05T11:04:19.139868+00:00
instance: kb-cron
parent: catecholaldehyde-hypothesis-20260505-0
summary:
The catecholaldehyde hypothesis is a scientific theory positing that neurotoxic aldehyde metabolites of the catecholamine neurotransmitters dopamine and norepinephrine are responsible for neurodegenerative diseases involving loss of catecholaminergic neurons, for instance Parkinson's disease. The specific metabolites thought to be involved include 3,4-dihydroxyphenylacetaldehyde (DOPAL) and 3,4-dihydroxyphenylglycolaldehyde (DOPEGAL), which are formed from dopamine and norepinephrine by monoamine oxidase, respectively. These metabolites are subsequently inactivated and detoxified by aldehyde dehydrogenase (ALDH). DOPAL and DOPEGAL are monoaminergic neurotoxins in preclinical models and inhibition of and polymorphisms in ALDH are associated with Parkinson's disease. The catecholaldehyde hypothesis additionally posits that DOPAL oligomerizes with α-synuclein resulting in accumulation of oligomerized α-synuclein (i.e., synucleinopathy) and that this contributes to cytotoxicity. == See also == Amyloid hypothesis == References ==


id: cerebral-polyopia-20260505-0
file: data/en.wikipedia.org/wiki/Cerebral_polyopia-0.md
category: reference
title: Cerebral polyopia
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Cerebral_polyopia
date_saved: 2026-05-05T11:04:20.324237+00:00
instance: kb-cron
parent: cerebral-polyopia-20260505-0
summary:
Cerebral diplopia or polyopia describes seeing two or more images arranged in ordered rows, columns, or diagonals after fixation on a stimulus. The polyopic images occur monocular bilaterally (one eye open on both sides) and binocularly (both eyes open), differentiating it from ocular diplopia or polyopia. The number of duplicated images can range from one to hundreds. Some patients report difficulty in distinguishing the replicated images from the real images, while others report that the false images differ in size, intensity, or color. Cerebral polyopia is sometimes confused with palinopsia (visual trailing), in which multiple images appear while watching an object. However, in cerebral polyopia, the duplicated images are of a stationary object which are perceived even after the object is removed from the visual field. Movement of the original object causes all of the duplicated images to move, or the polyopic images disappear during motion. In palinoptic polyopia, movement causes each polyopic image to leave an image in its wake, creating hundreds of persistent images (entomopia). Infarctions, tumors, multiple sclerosis, trauma, encephalitis, migraines, and seizures have been reported to cause cerebral polyopia. Cerebral polyopia has been reported in extrastriate visual cortex lesions, which is important for detecting motion, orientation,


id: cerebral-polyopia-20260505-1
file: data/en.wikipedia.org/wiki/Cerebral_polyopia-1.md
category: reference
title: Cerebral polyopia
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Cerebral_polyopia
date_saved: 2026-05-05T11:04:20.324237+00:00
instance: kb-cron
parent: cerebral-polyopia-20260505-0
summary:
== Theories of cerebral polyopia == The preliminary theory of cerebral polyopia proposed by Bender postulated that polyopia occurs as a result of instability of fixation due to occipital lobe disease. Under this explanation, small, involuntary eye movements that accompanied normal fixation were the cause of polyopic images. These involuntary eye movements lead to the development of new retinal and corresponding cortical regions that code for central vision called false maculae. Thus, polyopic images resulted from the stimulation of both the original and acquired maculae. However, Bender's theory does not account for recent studies in which fixation did not change and no eye movements were produced while polyopia was experienced, therefore polyopic images were not a result of involuntary eye movements. Instead, Cornblath offers a possible pathophysiological mechanism in which polyopia results from the recoding of visual receptive fields in primary visual cortex (Area V1). The report of polyopic images of complex objects at fixation suggests that the disorder is not limited to lower-order visual areas in occipital cortex (in which simple features such as borders and angles are encoded), but rather it involves the interaction between lower-order visual areas in the occipital lobe and higher-order visual areas in the


id: cortisol-awakening-response-20260505-0
file: data/en.wikipedia.org/wiki/Cortisol_awakening_response-0.md
category: reference
title: Cortisol awakening response
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Cortisol_awakening_response
date_saved: 2026-05-05T11:04:21.561040+00:00
instance: kb-cron
parent: cortisol-awakening-response-20260505-0
summary:
The cortisol awakening response (CAR) is an increase between 38% and 75% in cortisol levels peaking 30–45 minutes after awakening in the morning in some people. This rise is superimposed upon the late-night rise in cortisol which occurs before awakening. While its purpose is uncertain, it may be linked to the hippocampus' preparation of the hypothalamic-pituitary-adrenal axis (HPA) in order to face anticipated stress. Other research has challenged the idea that this elevation in morning cortisol is specifically linked to the onset of waking. == Description == Shortly after awakening, a sharp 38–75% (average 50%) increase occurs in the blood level of cortisol in about 77% of healthy people of all ages. The average level of salivary cortisol upon waking is roughly 15 nmol/L; 30 minutes later it may be 23 nmol/L, though there are wide variations. The cortisol awakening response reaches a maximum approximately 30 minutes after awakening though it may still be heightened by 34% an hour after waking. The pattern of this response to waking is relatively stable for any individual. Twin studies show its pattern is largely genetically determined since there is a heritability of 0.40 for the mean cortisol increase after awakening and 0.48 for


id: decision-making-20260505-0
file: data/en.wikipedia.org/wiki/Decision-making-0.md
category: reference
title: Decision-making
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Decision-making
date_saved: 2026-05-05T11:04:22.741043+00:00
instance: kb-cron
parent: decision-making-20260505-0
summary:
In psychology, decision-making (also spelled decision making and decisionmaking) is regarded as the cognitive process resulting in the selection of a belief or a course of action among several possible alternative options. It could be either rational or irrational. The decision-making process is a reasoning process based on assumptions of values, preferences and beliefs of the decision-maker. Every decision-making process produces a final choice, which may or may not prompt action. Research about decision-making is also published under the label problem solving, particularly in European psychological research. == Overview == Decision-making can be regarded as a problem-solving activity yielding a solution deemed to be optimal, or at least satisfactory. It is therefore a process which can be more or less rational or irrational and can be based on explicit or tacit knowledge and beliefs. Tacit knowledge is often used to fill the gaps in complex decision-making processes. Usually, both of these types of knowledge, tacit and explicit, are used together in the decision-making process. Human performance has been the subject of active research from several perspectives: Psychological: examining individual decisions in the context of a set of needs, preferences and values the individual has or seeks. Cognitive: the decision-making process


id: decision-making-20260505-1
file: data/en.wikipedia.org/wiki/Decision-making-1.md
category: reference
title: Decision-making
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Decision-making
date_saved: 2026-05-05T11:04:22.741043+00:00
instance: kb-cron
parent: decision-making-20260505-0
summary:
The first is analysis process paralysis. This type of paralysis is often spoken of as a cyclical process. One is unable to make a decision because they get stuck going over the information again and again for fear of making the wrong decision. The second is decision precision paralysis. This paralysis is cyclical, just like the first one, but instead of going over the same information, the decision-maker will find new questions and information from their analysis and that will lead them to explore into further possibilities rather than making a decision. The third is risk uncertainty paralysis. This paralysis occurs when the decision-maker wants to eliminate any uncertainty but the examination of provided information is unable to get rid of all uncertainty. === Extinction by instinct === On the opposite side of analysis paralysis is the phenomenon called extinction by instinct. Extinction by instinct is the state that a person is in when they make careless decisions without detailed planning or thorough systematic processes. Extinction by instinct can possibly be fixed by implementing a structural system, like checks and balances into a group or one's life. Analysis paralysis is the exact opposite where a group's schedule could be saturated


id: decision-making-20260505-2
file: data/en.wikipedia.org/wiki/Decision-making-2.md
category: reference
title: Decision-making
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Decision-making
date_saved: 2026-05-05T11:04:22.741043+00:00
instance: kb-cron
parent: decision-making-20260505-0
summary:
=== Group === Consensus decision-making tries to avoid "winners" and "losers". Consensus requires that a majority approve a given course of action, but that the minority agree to go along with the course of action. In other words, if the minority opposes the course of action, consensus requires that the course of action be modified to remove objectionable features. Voting-based methods: Majority requires support from more than 50% of the members of the group. Thus, the bar for action is lower than with consensus. See also Condorcet method. Plurality, where the largest faction in a group decides, even if it falls short of a majority. Score voting (or range voting) lets each member score one or more of the available options, specifying both preference and intensity of preference information. The option with the highest total or average is chosen. This method has experimentally been shown to produce the lowest Bayesian regret among common voting methods, even when voters are strategic. It addresses issues of voting paradox and majority rule. See also approval voting. Quadratic voting allows participants to cast their preference and intensity of preference for each decision (as opposed to a simple for or against decision). As in score


id: decision-making-20260505-3
file: data/en.wikipedia.org/wiki/Decision-making-3.md
category: reference
title: Decision-making
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Decision-making
date_saved: 2026-05-05T11:04:22.741043+00:00
instance: kb-cron
parent: decision-making-20260505-0
summary:
Orientation. Members meet for the first time and start to get to know each other. Conflict. Once group members become familiar with each other, disputes, little fights and arguments occur. Group members eventually work it out. Emergence. The group begins to clear up vague opinions by talking about them. Reinforcement. Members finally make a decision and provide justification for it. It is said that establishing critical norms in a group improves the quality of decisions, while the majority of opinions (called consensus norms) do not. Conflicts in socialization are divided in to functional and dysfunctional types. Functional conflicts are mostly the questioning the managers assumptions in their decision making and dysfunctional conflicts are like personal attacks and every action which decrease team effectiveness. Functional conflicts are the better ones to gain higher quality decision-making caused by the increased team knowledge and shared understanding. == Rational and irrational == In economics, it is thought that if humans are rational and free to make their own decisions, then they would behave according to rational choice theory. Rational choice theory says that a person consistently makes choices that lead to the best situation for themselves, taking into account all available considerations including costs


id: decision-making-20260505-4
file: data/en.wikipedia.org/wiki/Decision-making-4.md
category: reference
title: Decision-making
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Decision-making
date_saved: 2026-05-05T11:04:22.741043+00:00
instance: kb-cron
parent: decision-making-20260505-0
summary:
=== Adults === Adults are generally better able to control their risk-taking because their cognitive-control system has matured enough to the point where it can control the socioemotional network, even in the context of high arousal or when psychosocial capacities are present. Also, adults are less likely to find themselves in situations that push them to do risky things. For example, teens are more likely to be around peers who peer pressure them into doing things, while adults are not as exposed to this sort of social setting. == Cognitive and personal biases == Biases usually affects decision-making processes. They appear more when decision task has time pressure, is done under high stress and/or are highly complex. Here is a list of commonly debated biases in judgment and decision-making: Selective search for evidence (also known as confirmation bias): People tend to be willing to gather facts that support certain conclusions but disregard other facts that support different conclusions. Individuals who are highly defensive in this manner show significantly greater left prefrontal cortex activity as measured by EEG than do less defensive individuals. Premature termination of search for evidence: People tend to accept the first alternative that looks like it might


id: decision-making-20260505-5
file: data/en.wikipedia.org/wiki/Decision-making-5.md
category: reference
title: Decision-making
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Decision-making
date_saved: 2026-05-05T11:04:22.741043+00:00
instance: kb-cron
parent: decision-making-20260505-0
summary:
The psychologist Daniel Kahneman, adopting terms originally proposed by the psychologists Keith Stanovich and Richard West, has theorized that a person's decision-making is the result of an interplay between two kinds of cognitive processes: an automatic intuitive system (called "System 1") and an effortful rational system (called "System 2"). System 1 is a bottom-up, fast, and implicit system of decision-making, while system 2 is a top-down, slow, and explicit system of decision-making. System 1 includes simple heuristics in judgment and decision-making such as the affect heuristic, the availability heuristic, the familiarity heuristic, and the representativeness heuristic. === Combinatorial vs. positional === Styles and methods of decision-making were elaborated by Aron Katsenelinboigen, the founder of predispositioning theory. In his analysis of styles and methods, Katsenelinboigen referred to the game of chess, saying that "chess does disclose various methods of operation, notably the creation of predisposition methods which may be applicable to other, more complex systems." Katsenelinboigen states that apart from the methods (reactive and selective) and sub-methods randomization, predispositions, programming), there are two major styles: positional and combinational. Both styles are utilized in the game of chess. The two styles reflect two basic approaches to uncertainty: deterministic (combinational style) and indeterministic


id: emotion-and-memory-20260505-0
file: data/en.wikipedia.org/wiki/Emotion_and_memory-0.md
category: reference
title: Emotion and memory
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Emotion_and_memory
date_saved: 2026-05-05T11:04:24.054436+00:00
instance: kb-cron
parent: emotion-and-memory-20260505-0
summary:
Emotion can have a powerful effect on humans and animals. Numerous studies have shown that the most vivid autobiographical memories tend to be of emotional events, which are likely to be recalled more often and with more clarity and detail than neutral events. The activity of emotionally enhanced memory retention can be linked to human evolution; during early development, responsive behavior to environmental events would have progressed as a process of trial and error. Survival depended on behavioral patterns that were repeated or reinforced through life and death situations. Through evolution, this process of learning became genetically embedded in humans and all animal species in what is known as flight or fight instinct. Artificially inducing this instinct through traumatic physical or emotional stimuli essentially creates the same physiological condition that heightens memory retention by exciting neuro-chemical activity affecting areas of the brain responsible for encoding and recalling memory. This memory-enhancing effect of emotion has been demonstrated in many laboratory studies, using stimuli ranging from words to pictures to narrated slide shows, as well as autobiographical memory studies. However, as described below, emotion does not always enhance memory. == Arousal and valence in memory == One of the most common frameworks


id: emotion-and-memory-20260505-1
file: data/en.wikipedia.org/wiki/Emotion_and_memory-1.md
category: reference
title: Emotion and memory
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Emotion_and_memory
date_saved: 2026-05-05T11:04:24.054436+00:00
instance: kb-cron
parent: emotion-and-memory-20260505-0
summary:
==== Prioritized processing ==== Emotional items also appear more likely to be processed when attention is limited, suggesting a facilitated or prioritized processing of emotional information. This effect was demonstrated using the attentional blink paradigm in which 2 target items are presented in close temporal proximity within a stream of rapidly presented stimuli. The typical finding is that participants often miss the second target item, as if there were a "blink" of attention following the first target's presentation, reducing the likelihood that the second target stimulus is attended. However, when the second target stimulus elicits emotional arousal (a "taboo" word), participants are less likely to miss the target's presentation, which suggests that under conditions of limited attention, arousing items are more likely to be processed than neutral items. Additional support for the prioritized processing hypothesis was provided by studies investigating the visual extinction deficit. People suffering from this deficit can perceive a single stimulus in either side visual field if it is presented alone but are unaware of the same stimulus in the visual field opposed to the lesional side, if another stimulus is presented simultaneously on the lesional side. Emotion has been found to modulate the magnitude of the


id: emotion-and-memory-20260505-2
file: data/en.wikipedia.org/wiki/Emotion_and_memory-2.md
category: reference
title: Emotion and memory
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Emotion_and_memory
date_saved: 2026-05-05T11:04:24.054436+00:00
instance: kb-cron
parent: emotion-and-memory-20260505-0
summary:
==== Contextual effects of emotion on memory ==== Contextual effects occur as a result of the degree of similarity between the encoding context and the retrieval context of an emotional dimension. The main findings are that the current mood we are in affects what is attended, encoded and ultimately retrieved, as reflected in two similar but subtly different effects: the mood congruence effect and mood-state dependent retrieval. Positive encoding contexts have been connected to activity in the right fusiform gyrus. Negative encoding contexts have been correlated to activity in the right amygdala (Lewis & Critchley, 2003). However, Lewis and Critchley (2003) claim that it is not clear whether involvement of the emotional system in encoding memory differs for positive or negative emotions, or whether moods at recall lead to activity in the corresponding positive or negative neural networks. ===== Mood congruence effect ===== The mood congruence effect refers to the tendency of individuals to retrieve information more easily when it has the same emotional content as their current emotional state. For instance, being in a depressed mood increases the tendency to remember negative events (Drace, 2013). This effect has been demonstrated for explicit retrieval as well as implicit retrieval. =====


id: emotion-and-memory-20260505-3
file: data/en.wikipedia.org/wiki/Emotion_and_memory-3.md
category: reference
title: Emotion and memory
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Emotion_and_memory
date_saved: 2026-05-05T11:04:24.054436+00:00
instance: kb-cron
parent: emotion-and-memory-20260505-0
summary:
== Neurobiological mechanisms of emotional memory enhancement == The neural mechanism underlying emotional memory enhancement involves the interaction between the amygdala and the hippocampus, as well as several other factors that prioritize the encoding of emotional experiences. When an emotional experience occurs, the amygdala becomes highly active, signaling the hippocampus to strengthen the encoding and consolidation of these memories. This process is facilitated by the release of stress hormones and neurotransmitters, which modulate synaptic plasticity and enhance neural connectivity. Multiple mechanisms have been proposed to explain this prioritized encoding, including the neuromodulatory effects on plasticity and the dynamic interplay between the amygdala and the hippocampus. Intracranial EEG studies have shown that the amygdala triggers pronounced hippocampal sharp-wave ripples after encoding emotional experiences, further reinforcing the consolidation of these memories during both awake and sleep. This coordinated activity between the amygdala and hippocampus ensures that emotionally significant events are prioritized in long-term memory storage, leveraging both immediate neurochemical changes and enduring structural adaptations in neural circuits. == Memory of felt emotion == Many researchers use self-report measures of felt emotion as a manipulation check. This raises an interesting question and a possible methodological weakness: are people always accurate when they recall


id: emotion-and-memory-20260505-4
file: data/en.wikipedia.org/wiki/Emotion_and_memory-4.md
category: reference
title: Emotion and memory
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Emotion_and_memory
date_saved: 2026-05-05T11:04:24.054436+00:00
instance: kb-cron
parent: emotion-and-memory-20260505-0
summary:
== Emotion regulation effects on memory == An interesting issue in the study of the emotion-memory relationship is whether our emotions are influenced by our behavioral reaction to them, and whether this reaction—in the form of expression or suppression of the emotion—might affect what we remember about an event. Researchers have begun to examine whether concealing feelings influences our ability to perform common cognitive tasks, such as forming memories, and found that the emotion regulation efforts do have cognitive consequences. In the seminal work on negative affect arousal and white noise, Seidner found support for the existence of a negative affect arousal mechanism through observations regarding the devaluation of speakers from other ethnic origins." In a study of Richards and Gross (1999) and Tiwari (2013), participants viewed slides of injured men that produced increases in negative emotions, while information concerning each man was presented orally with his slide. The participants were assigned to either an expressive suppression group (where they were asked to refrain from showing emotion while watching the slides) or to a control group (where they were not given regulatory instructions at all). As predicted by the researchers, suppressors showed significantly worse performance on a memory test for


id: encephalitis-lethargica-20260505-0
file: data/en.wikipedia.org/wiki/Encephalitis_lethargica-0.md
category: reference
title: Encephalitis lethargica
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Encephalitis_lethargica
date_saved: 2026-05-05T11:04:25.215156+00:00
instance: kb-cron
parent: encephalitis-lethargica-20260505-0
summary:
Encephalitis lethargica (EL) is an atypical form of encephalitis. Also known as "von Economo Encephalitis", "sleeping sickness" or "sleepy sickness" (distinct from tsetse fly–transmitted sleeping sickness), it was first described in 1917 by neurologist Constantin von Economo and pathologist Jean-René Cruchet. The disease attacks the brain, leaving some victims in a statue-like condition, speechless and motionless. Between 1915 and 1926, an epidemic of encephalitis lethargica spread around the world. The exact number of people infected is unknown, but it is estimated that more than one million people contracted the disease during the epidemic, which directly caused more than 500,000 deaths. Most of those who survived never recovered their pre-morbid vigor. == Signs and symptoms == Encephalitis lethargica is characterized by high fever, sore throat, headache, lethargy, double vision, delayed physical and mental response, sleep inversion and catatonia. In severe cases, patients may enter a coma-like state (akinetic mutism). Patients may also experience abnormal eye movements ("oculogyric crises"), Parkinsonism, upper body weakness, muscular pains, tremors, neck rigidity, and behavioral changes including psychosis. Klazomania, a vocal tic involving compulsive screaming, is sometimes present. Approximately a third of the affected children experienced change of behavior, with many of them becoming "delinquents". Boys between


