# kb

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


id: encephalitis-lethargica-20260505-1
file: data/en.wikipedia.org/wiki/Encephalitis_lethargica-1.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:
In 1580, Europe was swept by a serious febrile and lethargic illness that led to Parkinsonism and other neurological sequelae. From 1673 to 1675, a similar serious epidemic occurred in London, which Thomas Sydenham described as "febris comatosa". In 1695, a 20-year-old woman in Germany experienced oculogyric crises, Parkinsonism, diplopia, strabismus, and other symptoms following an attack of somnolent brain fever, as described by Albrecht of Hildesheim. In 1712–1713, a severe epidemic of Schlafkrankheit ('sleep sickness') occurred in Tübingen, Germany, followed in many cases by persistent slowness of movement and lack of initiative (aboulia). Between 1750 and 1800, France and Germany experienced minor epidemics of "coma somnolentum" with features of Parkinsonism, including hyperkinetic hiccup, myoclonus, chorea, and tics. Between 1848 and 1882, Paris-based neurologist Jean-Martin Charcot documented many isolated cases of juvenile Parkinsonism, associated with diplopia, oculogyria, tachypnoea, retropulsion, and obsessional disorders, which were almost certainly post-encephalitic in origin. In 1890 in Italy, following the influenza epidemic of 1889–1890, a severe epidemic of somnolent illnesses (nicknamed the "Nona") appeared. For the few survivors of the Nona, Parkinsonism and other sequelae developed in almost all cases. Between 1915 and 1927, a world-wide encephalitis lethargica pandemic occurred, impacting nearly 5 million people


id: facial-onset-sensory-and-motor-neuronopathy-20260505-0
file: data/en.wikipedia.org/wiki/Facial_onset_sensory_and_motor_neuronopathy-0.md
category: reference
title: Facial onset sensory and motor neuronopathy
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Facial_onset_sensory_and_motor_neuronopathy
date_saved: 2026-05-05T11:04:26.390586+00:00
instance: kb-cron
parent: facial-onset-sensory-and-motor-neuronopathy-20260505-0
summary:
Facial onset sensory and motor neuronopathy, often abbreviated FOSMN, is a rare disorder of the nervous system in which sensory and motor nerves of the face and limbs progressively degenerate over a period of months to years. This degenerative process, the cause of which is unknown, eventually results in sensory and motor symptoms — the former consisting mainly of paresthesia followed by numbness, and the latter in muscle weakness, atrophy, and eventual paralysis. FOSM is characterized by sensory and motor loss beginning in the face and spreading to involve an increasingly larger area including the scalp, upper arms and trunk. The muscles or respiration and swallowing are commonly affected. In many ways, it is reminiscent of the much better known condition amyotrophic lateral sclerosis, with which it is closely related. There is no cure; treatment is supportive. Life expectancy may be shortened by respiratory complications arising from weakness of the muscles that aid breathing and swallowing. It was first described in four patients by Vucic and colleagues working at the Massachusetts General Hospital in the United States; subsequent reports from the United Kingdom, Europe and Asia point to a global incidence of the disease. It is thought to be exceptionally


id: fatal-insomnia-20260505-0
file: data/en.wikipedia.org/wiki/Fatal_insomnia-0.md
category: reference
title: Fatal insomnia
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Fatal_insomnia
date_saved: 2026-05-05T11:04:27.568044+00:00
instance: kb-cron
parent: fatal-insomnia-20260505-0
summary:
Fatal insomnia is a neurodegenerative disease that results in trouble sleeping as its hallmark symptom. The majority of cases are familial (fatal familial insomnia [FFI]), stemming from a mutation in the PRNP gene, with the remainder of cases occurring sporadically (sporadic fatal insomnia [sFI]). The problems with sleeping typically start out gradually and worsen over time. Eventually, the patient will succumb to total insomnia (agrypnia excitata), most often leading to other symptoms such as speech problems, coordination problems, and dementia. It results in death within a few months to a few years, and there is no known disease-modifying treatment. == Signs and symptoms == The disease has four stages: Characterized by worsening insomnia, resulting in panic attacks, paranoia, and phobias. This stage lasts for about four months. Hallucinations and panic attacks become noticeable, continuing for about five months. Complete inability to sleep followed by rapid loss of weight. This lasts for about three months. Dementia, during which the person becomes unresponsive or mute over the course of six months, is the final stage of the disease, followed by death. Clinically, Fatal insomnia manifests with a disordered sleep-wake cycle, dysautonomia, motor disturbances, and neuropsychiatric disorders. Other symptoms include profuse sweating, miosis


id: fatal-insomnia-20260505-1
file: data/en.wikipedia.org/wiki/Fatal_insomnia-1.md
category: reference
title: Fatal insomnia
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Fatal_insomnia
date_saved: 2026-05-05T11:04:27.568044+00:00
instance: kb-cron
parent: fatal-insomnia-20260505-0
summary:
=== Differential diagnosis === Other diseases involving the mammalian prion protein are known. Some are transmissible (TSEs, including FFI) such as kuru, bovine spongiform encephalopathy (BSE, also known as mad cow disease) in cattle and chronic wasting disease in American deer and American elk in some areas of the United States and Canada, as well as Creutzfeldt–Jakob disease (CJD). == Treatments == Treatment involves palliative care. There is conflicting evidence over the use of sleeping pills, including barbiturates, as a treatment for the disease. Symptoms of fatal familial insomnia may be treated with medications. Clonazepam may be prescribed to treat muscle spasms, and eszopiclone or zolpidem may be prescribed to help treat insomnia. However, these drugs do not work in the long term. == Prognosis == Like all prion diseases, FFI is invariably fatal. Life expectancy ranges from seven months to six years, with an average of 18 months. == Epidemiology and history == Fatal insomnia was first described by Elio Lugaresi et al. in 1986. In 1998, 40 families were known to carry the gene for FFI globally: eight German, five Italian, four American, two French, two Australian, two British, one Japanese and one Austrian. In the Basque Country


id: form-perception-20260505-0
file: data/en.wikipedia.org/wiki/Form_perception-0.md
category: reference
title: Form perception
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Form_perception
date_saved: 2026-05-05T11:04:28.772828+00:00
instance: kb-cron
parent: form-perception-20260505-0
summary:
Form perception is the recognition of visual elements of objects, specifically those to do with shapes, patterns and previously identified important characteristics. An object is perceived by the retina as a two-dimensional image, but the image can vary for the same object in terms of the context with which it is viewed, the apparent size of the object, the angle from which it is viewed, how illuminated it is, as well as where it resides in the field of vision. Despite the fact that each instance of observing an object leads to a unique retinal response pattern, the visual processing in the brain is capable of recognizing these experiences as analogous, allowing invariant object recognition. Visual processing occurs in a hierarchy with the lowest levels recognizing lines and contours, and slightly higher levels performing tasks such as completing boundaries and recognizing contour combinations. The highest levels integrate the perceived information to recognize an entire object. Essentially object recognition is the ability to assign labels to objects in order to categorize and identify them, thus distinguishing one object from another. During visual processing information is not created, but rather reformatted in a way that draws out the most detailed information of


id: form-perception-20260505-1
file: data/en.wikipedia.org/wiki/Form_perception-1.md
category: reference
title: Form perception
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Form_perception
date_saved: 2026-05-05T11:04:28.772828+00:00
instance: kb-cron
parent: form-perception-20260505-0
summary:
=== Injury and illness === Potential injuries to the brain include but are not limited to stroke, oxygen deprivation, blunt force trauma, and surgical injuries. When patients have lesions on their brain that develop as a result of injury or illness, such as multiple sclerosis or epilepsy, it is possible that they may have impaired object recognition which can manifest in the form of many different agnosias. Similar deficits have also been observed adults that have suffered blunt force trauma, strokes, severe carbon monoxide poisoning as well as in adults that have surgical damage following removal of tumors. Deficits have also been observed in children with types of epilepsy that do not lead to the formation of lesions. It is believed that in these cases the seizures cause a functional disruption that is capable of interfering with the processing of objects. Regions that specifically lead to deficits in object recognition when a lesion is present include the right lateral fusiform gyrus and the ventrolateral or ventromedial occipito-temporal cortex. These structures have all been identified as being crucial to the processing of shape and contour information, which is the basis for object recognition. Although people with damage to these structures are


id: memory-consolidation-20260505-0
file: data/en.wikipedia.org/wiki/Memory_consolidation-0.md
category: reference
title: Memory consolidation
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Memory_consolidation
date_saved: 2026-05-05T11:04:29.928717+00:00
instance: kb-cron
parent: memory-consolidation-20260505-0
summary:
Memory consolidation is a process in the brain that stabilizes newly learned information, allowing the memory to be stored long-term. A memory trace is the biological change in neural circuits that occurs during learning. Memory consolidation is essential for learning as it allows experiences to be transformed into stable long-term memories that can guide future behavior. Consolidation is divided into two main processes, synaptic consolidation and systems consolidation. Synaptic consolidation occurs rapidly on a small scale in the individual synapses within the first few hours of learning. Systems consolidation occurs on a larger scale and involves gradual reorganization and reduced reliance on the hippocampus and increased involvement of cortical networks over a period of weeks to years. These processes involve molecular and cellular changes that stabilize neural representations, referred to as a neural ensemble (engram). A third process, reconsolidation, has been proposed, in which previously consolidated memories can become unstable again after retrieval and may need to be updated before being re-stabilized. == History == Memory consolidation was first referred to in the writings of the renowned Roman teacher of rhetoric Quintillian, who noted that memory improves over time. In the late 19th century, German researchers Müller and Alfons Pilzecker


id: memory-consolidation-20260505-1
file: data/en.wikipedia.org/wiki/Memory_consolidation-1.md
category: reference
title: Memory consolidation
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Memory_consolidation
date_saved: 2026-05-05T11:04:29.928717+00:00
instance: kb-cron
parent: memory-consolidation-20260505-0
summary:
=== Protein synthesis === Protein synthesis has shown to play an important role in the formation of new memories. Studies have shown that protein synthesis inhibitors administered after learning weaken memory, suggesting that protein synthesis is required for memory consolidation. Additionally, reports have suggested that the effects of protein synthesis inhibitors also inhibit LTP. However, other results have shown that protein synthesis may not in fact be necessary for memory consolidation, as it has been found that the formation of memories can withstand vast amounts of protein synthesis inhibition, suggesting that its role may not be absolute. == Systems consolidation == Systems consolidation is the later phase of memory consolidation in which memories are reorganized across brain regions over time. Newly formed memories are first encoded in the hippocampus and then are gradually reorganized across cortical networks in a more permanent form of storage. Systems consolidation is a slow dynamic process which occurs over days to years as a gradual shift from hippocampus-dependent to hippocampus-independent memory storage. Systems consolidation is proposed to be involved in repeated reactivation of memory traces especially during sleep. The reactivations help strengthen the connections between cortical regions thereby gradually reducing reliance on the hippocampus. Over


id: memory-inhibition-20260505-0
file: data/en.wikipedia.org/wiki/Memory_inhibition-0.md
category: reference
title: Memory inhibition
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Memory_inhibition
date_saved: 2026-05-05T11:04:31.129532+00:00
instance: kb-cron
parent: memory-inhibition-20260505-0
summary:
In psychology, memory inhibition is the ability not to remember irrelevant information. The scientific concept of memory inhibition should not be confused with everyday uses of the word "inhibition". Scientifically speaking, memory inhibition is a type of cognitive inhibition, which is the stopping or overriding of a mental process, in whole or in part, with or without intention. Memory inhibition is a critical component of an effective memory system. While some memories are retained for a lifetime, most memories are forgotten. According to evolutionary psychologists, forgetting is adaptive because it facilitates selectivity of rapid, efficient recollection. For example, a person trying to remember where they parked their car would not want to remember every place they have ever parked. In order to remember something, therefore, it is essential not only to activate the relevant information, but also to inhibit irrelevant information. There are many memory phenomena that seem to involve inhibition, although there is often debate about the distinction between interference and inhibition. == History == In the early days of psychology, the concept of inhibition was prevalent and influential (e.g., Breese, 1899; Pillsbury, 1908; Wundt, 1902). These psychologists applied the concept of inhibition (and interference) to early theories of


id: memory-inhibition-20260505-1
file: data/en.wikipedia.org/wiki/Memory_inhibition-1.md
category: reference
title: Memory inhibition
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Memory_inhibition
date_saved: 2026-05-05T11:04:31.129532+00:00
instance: kb-cron
parent: memory-inhibition-20260505-0
summary:
=== "Think/no-think" paradigm and intentional inhibition === During the recovered memory debate of the 1990s, cognitive psychologists were dubious about whether specific memories could be repressed. One stumbling block was that repression had not been demonstrated in a research study. In 2001, researchers Anderson and Green claimed to have found laboratory evidence of suppression. They trained their participants with a list of unrelated word pairs (such as ordeal-roach), so they could respond with the second member of the pair (roach) when they saw the other member (ordeal). The more frequently participants had tried to not think about a particular word, the less likely they were to retrieve it on a final memory test. This impairment even occurred when participants were given an "independent probe" test, i.e. given a similar category (insect) instead of the original cue (roach), and asked to fill in the blank on the memory test: insect-r_____. According to Anderson and Green, the fact that participants had a decreased ability to recall items they were told to forget strongly supports the existence of an inhibitory control mechanism and the idea that people have the ability to suppress unwanted memories. Though Anderson & Green's (2001) results have been replicated


id: phantom-vibration-syndrome-20260505-0
file: data/en.wikipedia.org/wiki/Phantom_vibration_syndrome-0.md
category: reference
title: Phantom vibration syndrome
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Phantom_vibration_syndrome
date_saved: 2026-05-05T11:04:32.333444+00:00
instance: kb-cron
parent: phantom-vibration-syndrome-20260505-0
summary:
Phantom vibration syndrome or phantom ringing syndrome is the perception that one's mobile phone is vibrating or ringing when it is not. Other terms for this concept include ringxiety (a portmanteau of ring and anxiety), fauxcellarm (a portmanteau of "faux" /foʊ/ meaning "fake" or "false" and "cellphone" and "alarm" pronounced similarly to "false alarm"), phonetom (a portmanteau of phone and phantom) and phantom phone signals. According to Michael Rothberg, it is not a syndrome, but is better characterised as a tactile hallucination since the brain perceives a sensation that is not actually present. WebMD published an article on phantom vibration syndrome with Rothberg as a source. Several other articles have been published in 2010s, including in NPR, Bustle, CBS News, and Psychology Today. Phantom ringing may be experienced while taking a shower, watching television, or using a noisy device. Humans are particularly sensitive to auditory tones between 1,000 and 6,000 hertz, and basic mobile phone ringtones often fall within this range. Phantom vibrations develop after carrying a cell phone set to use vibrating alerts. Researcher Michelle Drouin found that almost 9 out of 10 undergraduates at her college experienced phantom vibrations. == History == In the comic strip Dilbert, cartoonist


id: posterior-cortical-atrophy-20260505-0
file: data/en.wikipedia.org/wiki/Posterior_cortical_atrophy-0.md
category: reference
title: Posterior cortical atrophy
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Posterior_cortical_atrophy
date_saved: 2026-05-05T11:04:33.570892+00:00
instance: kb-cron
parent: posterior-cortical-atrophy-20260505-0
summary:
Posterior cortical atrophy (PCA), also called Benson's syndrome, is a rare form of dementia which is considered a visual variant or an atypical variant of Alzheimer's disease (AD). The disease causes atrophy of the posterior part of the cerebral cortex, resulting in the progressive disruption of complex visual processing. PCA was first described by D. Frank Benson in 1988. PCA usually affects people at an earlier age than typical cases of Alzheimer's disease, with initial symptoms often experienced in people in their mid-fifties or early sixties. This was the case with writer Terry Pratchett (1948–2015), who went public in 2007 about being diagnosed with PCA. In rare cases, PCA can be caused by dementia with Lewy bodies and Creutzfeldt–Jakob disease. == Symptoms == The main symptom resulting from posterior cortical atrophy is a decrease in visuospatial and visuoperceptual capabilities, since the area of atrophy involves the occipital lobe responsible for visual processing. The atrophy is progressive; early symptoms include difficulty reading, blurred vision, light sensitivity, issues with depth perception, and trouble navigating through space. Additional symptoms include apraxia, a disorder of movement planning, alexia, an impaired ability to read, and visual agnosia, an object recognition disorder. In the two-streams hypothesis,


id: postvaccinal-encephalitis-20260505-0
file: data/en.wikipedia.org/wiki/Postvaccinal_encephalitis-0.md
category: reference
title: Postvaccinal encephalitis
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Postvaccinal_encephalitis
date_saved: 2026-05-05T11:04:34.702052+00:00
instance: kb-cron
parent: postvaccinal-encephalitis-20260505-0
summary:
Postvaccinal encephalitis (PVE) is a rare complication which was associated with vaccination with vaccinia virus during the worldwide smallpox eradication campaign. With mortality ranging between 25 – 30% it is the most severe adverse event associated with this vaccination. The mechanism of how it happens is unknown. == Symptoms and signs == PVE symptoms start to appear between 8th and 14th day after vaccination. Amongst the first are fever, headache, confusion and nausea. With passing time lethargy, seizures, short and long term memory dysfunctions, localized paralysis, hemiplegia, polyneuritis and convulsions. In extreme cases PVE can lead to coma and death. == Histology == Inflammatory extra-adventitial lesions are found not only in the brain but in the spinal cord as well. Lesions might be uniform in acute phase or disseminated in subacute phase. Unlike in cases of encephalitis lethargica the main damage is found in white brain matter. Meninges are infiltrated with T cells, plasmatic cells and phagocytic cells. Polymorfonuclear cells were found only in severe lesions. Apart from cellular infiltrate in perivascular space there is a tissue rarefication in spaces close to damaged blood vessels. Accumulated small nuclei are found in places of such rarefication. Strong demyelination with rapid clearance


id: progressive-muscular-atrophy-20260505-0
file: data/en.wikipedia.org/wiki/Progressive_muscular_atrophy-0.md
category: reference
title: Progressive muscular atrophy
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Progressive_muscular_atrophy
date_saved: 2026-05-05T11:04:35.927722+00:00
instance: kb-cron
parent: progressive-muscular-atrophy-20260505-0
summary:
Progressive muscular atrophy (PMA), also called Duchenne–Aran disease and Duchenne–Aran muscular atrophy, is a disorder characterized by the degeneration of lower motor neurons, resulting in generalized, progressive loss of muscle function. PMA is classified among motor neuron diseases (MND) and it sporadically appears during adulthood. It is speculated that about 2.5%-11% of adult onset MND is PMA, making it less common than ALS. PMA affects only the lower motor neurons, in contrast to amyotrophic lateral sclerosis (ALS), the most common MND, which affects both the upper and lower motor neurons, or primary lateral sclerosis, another MND, which affects only the upper motor neurons. The history of PMA involved determining if it was a separate disease from ALS or if it was just the earlier stages of ALS. Eventually, scientists concluded that PMA was its own disease under the same category as ALS. The signs and symptoms of PMA are similar to ALS, but are located in a different region of the body. Additionally, the diagnosis can be more complex due to the nature of PMA and ALS. The prognosis of PMA can be longer than ALS because it affects less of the motor neurons. The options for treatment are small


id: progressive-muscular-atrophy-20260505-1
file: data/en.wikipedia.org/wiki/Progressive_muscular_atrophy-1.md
category: reference
title: Progressive muscular atrophy
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Progressive_muscular_atrophy
date_saved: 2026-05-05T11:04:35.927722+00:00
instance: kb-cron
parent: progressive-muscular-atrophy-20260505-0
summary:
The prognosis is a little better. A recent study found the 5-year survival rate in PMA to be 33% (vs 20% in ALS) and the 10-year survival rate to be 12% (vs 6% in ALS). Patients with PMA do not have the cognitive change identified in certain groups of patients with MND. Because PMA patients do not have UMN signs, they usually do not meet the World Federation of Neurology El Escorial Research Criteria for "Definite" or "Probable" ALS and so are ineligible to participate in the majority of clinical trials conducted in ALS. Because of its rarity (even compared to ALS) and confusion about the condition, some insurance policies or local healthcare policies may not recognize PMA as being the life-changing illness that it is. In cases where being classified as being PMA rather than ALS is likely to restrict access to services, it may be preferable to be diagnosed as "slowly progressive ALS" or "lower motor neuron predominant" ALS. An initial diagnosis of PMA could turn out to be slowly progressive ALS many years later, sometimes even decades after the initial diagnosis. The occurrence of upper motor neuron symptoms such as brisk reflexes, spasticity, or a Babinski sign


id: qualia-20260505-0
file: data/en.wikipedia.org/wiki/Qualia-0.md
category: reference
title: Qualia
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Qualia
date_saved: 2026-05-05T11:04:37.131418+00:00
instance: kb-cron
parent: qualia-20260505-0
summary:
In philosophy of mind, qualia (; singular: quale ) are defined as instances of subjective experience. The term qualia derives from the Latin neuter plural form (qualia) of the Latin adjective quālis (Latin pronunciation: [ˈkʷaːlɪs]) meaning "of what sort" or "of what kind" in relation to a specific instance, such as "what it is like to taste a specific apple — this particular apple now". Examples of qualia include the perceived sensation of pain of a headache, the taste of wine, and the redness of an evening sky. As qualitative characteristics of sensations, qualia stand in contrast to propositional attitudes, where the focus is on beliefs about experience rather than what it is directly like to be experiencing. C.S. Peirce introduced the term quale in philosophy in 1866, and in 1929 C. I. Lewis was the first to use the term "qualia" in its generally agreed-upon modern sense. Frank Jackson later defined qualia as "...certain features of the bodily sensations especially, but also of certain perceptual experiences, which no amount of purely physical information includes". Philosopher and cognitive scientist Daniel Dennett suggested that qualia was "an unfamiliar term for something that could not be more familiar to each of us:


id: qualia-20260505-1
file: data/en.wikipedia.org/wiki/Qualia-1.md
category: reference
title: Qualia
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Qualia
date_saved: 2026-05-05T11:04:37.131418+00:00
instance: kb-cron
parent: qualia-20260505-0
summary:
The inverted spectrum thought experiment, originally developed by John Locke, invites us to imagine two individuals who perceive colors differently: where one person sees red, the other sees green, and vice versa. Despite this difference in their subjective experiences, they behave and communicate as if their perceptions are the same, and no physical or behavioral test can reveal the inversion. Critics of functionalism, and of physicalism more broadly, argue that if we can imagine this happening without contradiction, it follows that we are imagining a change in a property that determines the way things look to us, but that has no physical basis. The idea that an inverted spectrum would be undetectable in practice is also open to criticism on more scientific grounds, by C. L. Hardin, among others. As Alex Byrne puts it: ...there are more perceptually distinguishable shades between red and blue than there are between green and yellow, which would make red-green inversion behaviorally detectable. And there are yet further asymmetries. Dark yellow is brown (qualitatively different from yellow), whereas dark blue is blue[...] Similarly, desaturated bluish-red is pink (qualitatively different from saturated bluish-red), whereas desaturated greenish-yellow is similar to saturated greenish-yellow. Again, red is a "warm"


id: qualia-20260505-2
file: data/en.wikipedia.org/wiki/Qualia-2.md
category: reference
title: Qualia
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Qualia
date_saved: 2026-05-05T11:04:37.131418+00:00
instance: kb-cron
parent: qualia-20260505-0
summary:
Joseph Levine's paper Conceivability, Identity, and the Explanatory Gap takes up where the criticisms of conceivability arguments (such as the inverted spectrum argument and the zombie argument) leave off. Levine agrees that conceivability is a flawed means of establishing metaphysical realities, but points out that even if we come to the metaphysical conclusion that qualia are physical, there is still an explanatory problem. While I think this materialist response is right in the end, it does not suffice to put the mind-body problem to rest. Even if conceivability considerations do not establish that the mind is in fact distinct from the body, or that mental properties are metaphysically irreducible to physical properties, still they do demonstrate that we lack an explanation of the mental in terms of the physical. However, such an epistemological or explanatory problem might indicate an underlying metaphysical issue, as even if not proven by conceivability arguments, the non-physicality of qualia is far from ruled out. In the end, we are right back where we started. The explanatory gap argument doesn't demonstrate a gap in nature, but a gap in our understanding of nature. Of course a plausible explanation for there being a gap in our understanding


id: qualia-20260505-3
file: data/en.wikipedia.org/wiki/Qualia-3.md
category: reference
title: Qualia
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Qualia
date_saved: 2026-05-05T11:04:37.131418+00:00
instance: kb-cron
parent: qualia-20260505-0
summary:
==== E. J. Lowe ==== E. J. Lowe denies that indirect realism, wherein which we have access only to sensory features internal to the brain, necessarily implies a Cartesian dualism. He agrees with Bertrand Russell that the way images are received by our retinas, our "retinal images", are connected to "patterns of neural activity in the cortex". He defends a version of the causal theory of perception in which a causal path can be traced between the external object and the perception of it. He is careful to deny that we do any inferring from the sensory field; he believes this allows us to build an access to knowledge on that causal connection. In a later work he moves closer to the non-epistemic argument in that he postulates "a wholly non-conceptual component of perceptual experience". ==== J. B. Maund ==== John Barry Maund, an Australian philosopher of perception, argues that qualia can be described on two levels, a fact that he refers to as "dual coding". Maund extended his argument with reference to color. Color he sees as a dispositional property, not an objective one. Colors are "virtual properties", which means they are as if things possessed them. Although the


id: qualia-20260505-4
file: data/en.wikipedia.org/wiki/Qualia-4.md
category: reference
title: Qualia
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Qualia
date_saved: 2026-05-05T11:04:37.131418+00:00
instance: kb-cron
parent: qualia-20260505-0
summary:
Vilayanur S. Ramachandran and William Hirstein proposed three laws of qualia (with a fourth later added), which are "functional criteria that need to be fulfilled in order for certain neural events to be associated with qualia" by philosophers of the mind: Qualia are irrevocable and indubitable. You don't say 'maybe it is red but I can visualize it as green if I want to'. An explicit neural representation of red is created that invariably and automatically 'reports' this to higher brain centres. Once the representation is created, what can be done with it is open-ended. You have the luxury of choice, e.g., if you have the percept of an apple you can use it to tempt Adam, to keep the doctor away, bake a pie, or just to eat. Even though the representation at the input level is immutable and automatic, the output is potentially infinite. This isn't true for, say, a spinal reflex arc where the output is also inevitable and automatic. Indeed, a paraplegic can even have an erection and ejaculate without an orgasm. Short-term memory. The input invariably creates a representation that persists in short-term memory – long enough to allow time for choice of output. Without


id: qualia-20260505-5
file: data/en.wikipedia.org/wiki/Qualia-5.md
category: reference
title: Qualia
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Qualia
date_saved: 2026-05-05T11:04:37.131418+00:00
instance: kb-cron
parent: qualia-20260505-0
summary:
Dennett attempts to show that we cannot satisfy (a) either through introspection or through observation, and that qualia's very definition undermines its chances of satisfying (b). Supporters of qualia point out that in order for you to notice a change in qualia, you must compare your current qualia with your memories of past qualia. Arguably, such a comparison would involve immediate assessment of your current qualia and your memories of past qualia, but not of the past qualia themselves. Furthermore, modern functional brain imaging has increasingly suggested that the memory of an experience is processed in similar ways, and in similar zones of the brain, as the original perception. This may mean that there would be asymmetric outcomes between altering the mechanism of perception of qualia and altering the memory of that qualia. If the diabolical neurosurgery altered the immediate perception of qualia, the inversion might not be noticed directly, since the brain zones which re-process the memories would invert the remembered qualia. On the other hand, alteration of the qualia memories themselves would be processed without inversion, and thus you would perceive them as an inversion. Thus, you might know immediately if memory of your qualia had been altered,


id: qualia-20260505-6
file: data/en.wikipedia.org/wiki/Qualia-6.md
category: reference
title: Qualia
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Qualia
date_saved: 2026-05-05T11:04:37.131418+00:00
instance: kb-cron
parent: qualia-20260505-0
summary:
=== Paul Churchland === According to Paul Churchland, Mary might be considered akin to a feral child who suffered extreme isolation during childhood. Technically when Mary leaves the room, she would not have the ability to see or know what the color red is, as a brain has to learn and develop how to see colors. Patterns need to form in the V4 section of the visual cortex, which occurs via exposure to wavelengths of light. This exposure needs to occur during the early stages of brain development. In Mary's case, the identifications and categorizations of color will only be in respect to representations of black and white. === Gary Drescher === In his book Good and Real, Gary Drescher compares qualia with "gensyms" (generated symbols) in Common Lisp. These are objects that Lisp treats as having no properties or components, which can only be identified as equal or not equal to other objects. Drescher explains, "we have no introspective access to whatever internal properties make the red gensym recognizably distinct from the green [...] even though we know the sensation when we experience it." Under this interpretation of qualia, Drescher responds to the Mary thought experiment by noting that


id: qualia-20260505-7
file: data/en.wikipedia.org/wiki/Qualia-7.md
category: reference
title: Qualia
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Qualia
date_saved: 2026-05-05T11:04:37.131418+00:00
instance: kb-cron
parent: qualia-20260505-0
summary:
"[T]he purportedly metaphysical "problem of consciousness" is no more and no less than the epistemological "problem of privileged access," and that once this is seen questions about dualism versus materialism lose their interest." Rorty suggests that the only answer that one can give as to why qualia are ontologically distinct from physical processes is that the lack of an appearance/reality distinction is an essential feature of their being (i.e. it is an essential feature of a quale that it be incorrigibly known), but he notes that in making this move, qualia have been hypostatized such that they can take on the property of being pure appearances. In other words, they have been turned into non-physical, property-bearing particulars, meaning they are no longer properties of the person who has them, but rather particulars that exist in addition to that person. Thus, the dualistic intuition that an explanation of, say, pain in terms of physical functions and behaviors leaves out the raw feel of pain itself is the result of hypostatizing pain into something non-physical which accompanies those physical processes: "The neo-dualist who identifies a pain with how it feels to be in pain is hypostatizing a property—painfulness—into a special sort of


id: qualia-20260505-8
file: data/en.wikipedia.org/wiki/Qualia-8.md
category: reference
title: Qualia
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Qualia
date_saved: 2026-05-05T11:04:37.131418+00:00
instance: kb-cron
parent: qualia-20260505-0
summary:
"Poised" - the phenomenal experience is always present to the understanding, whether or not the agent is able to apply a concept to it. "Abstract" - it is unclear whether you are in touch with a concrete object (for example, someone may feel a pain in an amputated limb). "Nonconceptual" - phenomenon can exist although one does not have the concept by which to recognize it. "Intentional (Content)" - it represents something, whether or not the observer is taking advantage of that fact. Tye adds that the experience is like a map in that, in most cases, it goes beyond the shapes, edges, volumes, etc. in the world – you may not be reading the a map but, as with an actual map there is a reliable match with what it is mapping. This is why Tye calls his theory representationalism, makes it plain that Tye believes that he has retained a direct contact with what produces the phenomena and is therefore not hampered by any trace of a "veil of perception". == See also == == References == === Citations === === Other references === Various authors (19 March 2007). Chalmers, David (ed.). "Online collection of papers on qualia".


id: sleep-onset-20260505-0
file: data/en.wikipedia.org/wiki/Sleep_onset-0.md
category: reference
title: Sleep onset
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Sleep_onset
date_saved: 2026-05-05T11:04:38.285135+00:00
instance: kb-cron
parent: sleep-onset-20260505-0
summary:
Sleep onset is the transition from wakefulness into sleep. Sleep onset usually transits into non-rapid eye movement sleep (NREM sleep) but under certain circumstances (e.g. narcolepsy) it is possible to transit from wakefulness directly into rapid eye movement sleep (REM sleep). == History == During the 1920s an obscure disorder that caused encephalitis and attacked the part of the brain that regulates sleep influenced Europe and North America. Although the virus that caused this disorder was never identified, the psychiatrist and neurologist Constantin von Economo decided to study this disease and identified a key component in the sleep-wake regulation. He identified the pathways that regulated wakefulness and sleep onset by studying the parts of the brain that were affected by the disease and the consequences it had on the circadian rhythm. He stated that the pathways that regulated sleep onset are located between the brain stem and the basal forebrain. His discoveries were not appreciated until the last two decades of the 20th century when the pathways of sleep were found to reside in the exact place that Constantin von Economo stated. == Neural circuit == Sleep electrophysiological measurements can be made by attaching electrodes to the scalp to measure


id: subacute-sclerosing-panencephalitis-20260505-0
file: data/en.wikipedia.org/wiki/Subacute_sclerosing_panencephalitis-0.md
category: reference
title: Subacute sclerosing panencephalitis
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Subacute_sclerosing_panencephalitis
date_saved: 2026-05-05T11:04:39.501994+00:00
instance: kb-cron
parent: subacute-sclerosing-panencephalitis-20260505-0
summary:
Subacute sclerosing panencephalitis (SSPE), also known as Dawson disease, is a rare form of progressive brain inflammation caused by a persistent infection with the measles virus. The condition primarily affects children, teens, and young adults. It has been estimated that about 2 in 10,000 people who get measles will eventually develop SSPE. However, a 2016 study estimated that the rate for unvaccinated infants under 15 months was as high as 1 in 609. No cure for SSPE exists, and the condition is almost always fatal. SSPE is sometimes confused with acute disseminated encephalomyelitis, which can also be caused by the measles virus, but has a very different timing and course. SSPE is caused by some strains of the wild-type (naturally occurring) measles virus, such as the B3 strain, but not by the strains used in measles vaccines. == Signs and symptoms == SSPE is characterized by a history of primary measles infection, followed by a normal, unremarkable recovery. Symptoms of SSPE appear later. On average, the first symptoms appear about 10 years after the initial infection, though this varies significantly, as some people have developed SSPE symptoms as soon as 1 month after infection, and others as long as 27


id: thought-20260505-0
file: data/en.wikipedia.org/wiki/Thought-0.md
category: reference
title: Thought
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Thought
date_saved: 2026-05-05T11:04:40.951647+00:00
instance: kb-cron
parent: thought-20260505-0
summary:
In their most common sense, thought and thinking refer to cognitive processes that occur independently of direct sensory stimulation. Core forms include judging, reasoning, concept formation, problem solving, and deliberation. Other processes, such as entertaining an idea, memory, or imagination, are also frequently considered types of thought. Unlike perception, these activities can occur without immediate input from the sensory organs. In a broader sense, any mental event—including perception and unconscious processes—may be described as a form of thought. The term can also denote not the process itself, but the resulting mental states or systems of ideas. A variety of theories attempt to explain the nature of thinking. Platonism holds that thought involves discerning eternal forms and their interrelations, distinguishing these pure entities from their imperfect sensory imitations. Aristotelianism interprets thinking as instantiating the universal essence of an object within the mind, derived from sense experience rather than a changeless realm. Conceptualism, closely related to Aristotelianism, identifies thinking with the mental evocation of concepts. Inner speech theories suggest that thought takes the form of silent verbal expression, sometimes in a natural language and sometimes in a specialized "mental language," or Mentalese, as proposed by the language of thought hypothesis. Associationism views


id: thought-20260505-1
file: data/en.wikipedia.org/wiki/Thought-1.md
category: reference
title: Thought
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Thought
date_saved: 2026-05-05T11:04:40.951647+00:00
instance: kb-cron
parent: thought-20260505-0
summary:
=== Aristotelianism and conceptualism === According to Aristotelianism, the mind thinks about an object by instantiating its essence. For example, when thinking about trees, the mind instantiates "tree-ness." Unlike actual trees, this instantiation does not occur in matter but in the mind, though the universal essence is the same in both cases. In contrast with Platonism, universals are not seen as timeless forms existing in a separate intelligible realm. Instead, they exist only insofar as they are instantiated. The mind comes to recognize universals through abstraction from experience, a view that avoids some objections directed against Platonism. Conceptualism is closely related. It holds that thinking consists in mentally evoking concepts. While some concepts may be innate, most are acquired through abstraction from sensory experience before they can be used in thought. Critics argue that both theories face difficulties. One problem is explaining the logical structure of thought. For instance, to think that it will either rain or snow, it is not enough to instantiate the essences of rain and snow or to evoke the relevant concepts. The disjunctive relation between them is not captured in this way. Another challenge lies in providing a clear account of how the mind acquires


id: thought-20260505-2
file: data/en.wikipedia.org/wiki/Thought-2.md
category: reference
title: Thought
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Thought
date_saved: 2026-05-05T11:04:40.951647+00:00
instance: kb-cron
parent: thought-20260505-0
summary:
=== Associationism === An important view in the empiricist tradition has been associationism, the view that thinking consists in the succession of ideas or images. This succession is seen as being governed by laws of association, which determine how the train of thought unfolds. These laws are different from logical relations between the contents of thoughts, which are found in the case of drawing inferences by moving from the thought of the premises to the thought of the conclusion. Various laws of association have been suggested. According to the laws of similarity and contrast, ideas tend to evoke other ideas that are either very similar to them or their opposite. The law of contiguity, on the other hand, states that if two ideas were frequently experienced together, then the experience of one tends to cause the experience of the other. In this sense, the history of an organism's experience determines which thoughts the organism has and how these thoughts unfold. But such an association does not guarantee that the connection is meaningful or rational. For example, because of the association between the terms "cold" and "Idaho", the thought "this coffee shop is cold" might lead to the thought "Russia should


id: thought-20260505-3
file: data/en.wikipedia.org/wiki/Thought-3.md
category: reference
title: Thought
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Thought
date_saved: 2026-05-05T11:04:40.951647+00:00
instance: kb-cron
parent: thought-20260505-0
summary:
=== Computationalism === Computationalist theories of thinking, often found in the cognitive sciences, understand thinking as a form of information processing. These views developed with the rise of computers in the second part of the 20th century, when various theorists saw thinking in analogy to computer operations. On such views, the information may be encoded differently in the brain, but in principle, the same operations take place there as well, corresponding to the storage, transmission, and processing of information. But while this analogy has some intuitive attraction, theorists have struggled to give a more explicit explanation of what computation is. A further problem consists in explaining the sense in which thinking is a form of computing. The traditionally dominant view defines computation in terms of Turing machines, though contemporary accounts often focus on neural networks for their analogies. A Turing machine is capable of executing any algorithm based on a few very basic principles, such as reading a symbol from a cell, writing a symbol to a cell, and executing instructions based on the symbols read. This way it is possible to perform deductive reasoning following the inference rules of formal logic as well as simulating many other functions of


id: thought-20260505-4
file: data/en.wikipedia.org/wiki/Thought-4.md
category: reference
title: Thought
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Thought
date_saved: 2026-05-05T11:04:40.951647+00:00
instance: kb-cron
parent: thought-20260505-0
summary:
=== Entertaining, judging, and reasoning === Thinking is often identified with the act of judging. A judgment is a mental operation in which a proposition is evoked and then either affirmed or denied. It involves deciding what to believe and aims at determining whether the judged proposition is true or false. Various theories of judgment have been proposed. The traditionally dominant approach is the combination theory. It states that judgments consist in the combination of concepts. On this view, to judge that "all men are mortal" is to combine the concepts "man" and "mortal". The same concepts can be combined in different ways, corresponding to different forms of judgment, for example, as "some men are mortal" or "no man is mortal". Other theories of judgment focus more on the relation between the judged proposition and reality. According to Franz Brentano, a judgment is either a belief or a disbelief in the existence of some entity. In this sense, there are only two fundamental forms of judgment: "A exists" and "A does not exist". When applied to the sentence "all men are mortal", the entity in question is "immortal men", of whom it is said that they do not exist. Important


id: thought-20260505-5
file: data/en.wikipedia.org/wiki/Thought-5.md
category: reference
title: Thought
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Thought
date_saved: 2026-05-05T11:04:40.951647+00:00
instance: kb-cron
parent: thought-20260505-0
summary:
=== Problem solving === In the case of problem solving, thinking aims at reaching a predefined goal by overcoming certain obstacles. This process often involves two different forms of thinking. On the one hand, divergent thinking aims at coming up with as many alternative solutions as possible. On the other hand, convergent thinking tries to narrow down the range of alternatives to the most promising candidates. Some researchers identify various steps in the process of problem solving. These steps include recognizing the problem, trying to understand its nature, identifying general criteria the solution should meet, deciding how these criteria should be prioritized, monitoring the progress, and evaluating the results. An important distinction concerns the type of problem that is faced. For well-structured problems, it is easy to determine which steps need to be taken to solve them, but executing these steps may still be difficult. For ill-structured problems, on the other hand, it is not clear what steps need to be taken, i.e. there is no clear formula that would lead to success if followed correctly. In this case, the solution may sometimes come in a flash of insight in which the problem is suddenly seen in a new light.


id: thought-20260505-6
file: data/en.wikipedia.org/wiki/Thought-6.md
category: reference
title: Thought
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Thought
date_saved: 2026-05-05T11:04:40.951647+00:00
instance: kb-cron
parent: thought-20260505-0
summary:
=== Unconscious thought === Conscious thought is the paradigmatic form of thinking and is often the focus of the corresponding research. But it has been argued that some forms of thought also happen on the unconscious level. Unconscious thought is thought that happens in the background without being experienced. It is therefore not observed directly. Instead, its existence is usually inferred by other means. For example, when someone is faced with an important decision or a difficult problem, they may not be able to solve it straight away. But then, at a later time, the solution may suddenly flash before them even though no conscious steps of thinking were taken towards this solution in the meantime. In such cases, the cognitive labor needed to arrive at a solution is often explained in terms of unconscious thoughts. The central idea is that a cognitive transition happened and we need to posit unconscious thoughts to be able to explain how it happened. It has been argued that conscious and unconscious thoughts differ not just concerning their relation to experience but also concerning their capacities. According to unconscious thought theorists, for example, conscious thought excels at simple problems with few variables but is


id: thought-20260505-7
file: data/en.wikipedia.org/wiki/Thought-7.md
category: reference
title: Thought
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Thought
date_saved: 2026-05-05T11:04:40.951647+00:00
instance: kb-cron
parent: thought-20260505-0
summary:
=== Phenomenology === Phenomenology is the science of the structure and contents of experience. The term "cognitive phenomenology" refers to the experiential character of thinking or what it feels like to think. Some theorists claim that there is no distinctive cognitive phenomenology. On such a view, the experience of thinking is just one form of sensory experience. According to one version, thinking just involves hearing a voice internally. According to another, there is no experience of thinking apart from the indirect effects thinking has on sensory experience. A weaker version of such an approach allows that thinking may have a distinct phenomenology but contends that thinking still depends on sensory experience because it cannot occur on its own. On this view, sensory contents constitute the foundation from which thinking may arise. An often-cited thought experiment in favor of the existence of a distinctive cognitive phenomenology involves two persons listening to a radio broadcast in French, one who understands French and the other who does not. The idea behind this example is that both listeners hear the same sounds and therefore have the same non-cognitive experience. In order to explain the difference, a distinctive cognitive phenomenology has to be posited: only


id: thought-20260505-8
file: data/en.wikipedia.org/wiki/Thought-8.md
category: reference
title: Thought
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Thought
date_saved: 2026-05-05T11:04:40.951647+00:00
instance: kb-cron
parent: thought-20260505-0
summary:
=== Metaphysics === The mind–body problem concerns the explanation of the relationship that exists between minds, or mental processes, and bodily states or processes. The main aim of philosophers working in this area is to determine the nature of the mind and mental states/processes, and how—or even if—minds are affected by and can affect the body. Human perceptual experiences depend on stimuli which arrive at one's various sensory organs from the external world and these stimuli cause changes in one's mental state, ultimately causing one to feel a sensation, which may be pleasant or unpleasant. Someone's desire for a slice of pizza, for example, will tend to cause that person to move his or her body in a specific manner and in a specific direction to obtain what he or she wants. The question, then, is how it can be possible for conscious experiences to arise out of a lump of gray matter endowed with nothing but electrochemical properties. A related problem is to explain how someone's propositional attitudes (e.g. beliefs and desires) can cause that individual's neurons to fire and his muscles to contract in exactly the correct manner. These comprise some of the puzzles that have confronted epistemologists


id: thought-20260505-9
file: data/en.wikipedia.org/wiki/Thought-9.md
category: reference
title: Thought
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Thought
date_saved: 2026-05-05T11:04:40.951647+00:00
instance: kb-cron
parent: thought-20260505-0
summary:
The "id", "ego" and "super-ego" are the three parts of the "psychic apparatus" defined in Sigmund Freud's structural model of the psyche; they are the three theoretical constructs in terms of whose activity and interaction mental life is described. According to this model, the uncoordinated instinctual trends are encompassed by the "id", the organized realistic part of the psyche is the "ego", and the critical, moralizing function is the "super-ego". For psychoanalysis, the unconscious does not include all that is not conscious, rather only what is actively repressed from conscious thought or what the person is averse to knowing consciously. In a sense this view places the self in relationship to their unconscious as an adversary, warring with itself to keep what is unconscious hidden. If a person feels pain, all he can think of is alleviating the pain. Any of his desires, to get rid of pain or enjoy something, command the mind what to do. For Freud, the unconscious was a repository for socially unacceptable ideas, wishes or desires, traumatic memories, and painful emotions put out of mind by the mechanism of psychological repression. However, the contents did not necessarily have to be solely negative. In the psychoanalytic


id: thought-20260505-10
file: data/en.wikipedia.org/wiki/Thought-10.md
category: reference
title: Thought
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Thought
date_saved: 2026-05-05T11:04:40.951647+00:00
instance: kb-cron
parent: thought-20260505-0
summary:
=== Counterfactual thinking === Counterfactual thinking involves mental representations of non-actual situations and events, i.e. of what is "contrary to the facts". It is usually conditional: it aims at assessing what would be the case if a certain condition had obtained. In this sense, it tries to answer "What if"-questions. For example, thinking after an accident that one would be dead if one had not used the seatbelt is a form of counterfactual thinking: it assumes, contrary to the facts, that one had not used the seatbelt and tries to assess the result of this state of affairs. In this sense, counterfactual thinking is normally counterfactual only to a small degree since just a few facts are changed, like concerning the seatbelt, while most other facts are kept in place, like that one was driving, one's gender, the laws of physics, etc. When understood in the widest sense, there are forms of counterfactual thinking that do not involve anything contrary to the facts at all. This is the case, for example, when one tries to anticipate what might happen in the future if an uncertain event occurs and this event actually occurs later and brings with it the anticipated consequences.


id: thought-20260505-11
file: data/en.wikipedia.org/wiki/Thought-11.md
category: reference
title: Thought
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Thought
date_saved: 2026-05-05T11:04:40.951647+00:00
instance: kb-cron
parent: thought-20260505-0
summary:
=== Critical thinking === Critical thinking is a form of thinking that is reasonable, reflective, and focused on determining what to believe or how to act. It holds itself to various standards, like clarity and rationality. In this sense, it involves not just cognitive processes trying to solve the issue at hand but at the same time meta-cognitive processes ensuring that it lives up to its own standards. This includes assessing both that the reasoning itself is sound and that the evidence it rests on is reliable. This means that logic plays an important role in critical thinking. It concerns not just formal logic, but also informal logic, specifically to avoid various informal fallacies due to vague or ambiguous expressions in natural language. No generally accepted standard definition of "critical thinking" exists but there is significant overlap between the proposed definitions in their characterization of critical thinking as careful and goal-directed. According to some versions, only the thinker's own observations and experiments are accepted as evidence in critical thinking. Some restrict it to the formation of judgments but exclude action as its goal. A concrete everyday example of critical thinking, due to John Dewey, involves observing foam bubbles moving in


id: unihemispheric-slow-wave-sleep-20260505-0
file: data/en.wikipedia.org/wiki/Unihemispheric_slow-wave_sleep-0.md
category: reference
title: Unihemispheric slow-wave sleep
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Unihemispheric_slow-wave_sleep
date_saved: 2026-05-05T11:04:43.626925+00:00
instance: kb-cron
parent: unihemispheric-slow-wave-sleep-20260505-0
summary:
Unihemispheric slow-wave sleep (USWS) is sleep where one half of the brain rests while the other half remains alert. This is in contrast to normal sleep where both eyes are shut and both halves of the brain show unconsciousness. In USWS, also known as asymmetric slow-wave sleep, one half of the brain is in deep sleep, a form of non-rapid eye movement sleep and the eye corresponding to this half is closed while the other eye remains open. When examined by electroencephalography (EEG), the characteristic slow-wave sleep tracings are seen from one side while the other side shows a characteristic tracing of wakefulness. The phenomenon has been observed in a number of terrestrial, aquatic and avian species. Unique physiology, including the differential release of the neurotransmitter acetylcholine, has been linked to the phenomenon. USWS offers a number of benefits, including the ability to rest in areas of high predation or during long migratory flights. The behaviour remains an important research topic because USWS is possibly the first animal behaviour which uses different regions of the brain to simultaneously control sleep and wakefulness. The greatest theoretical importance of USWS is its potential role in elucidating the function of sleep by challenging


id: unihemispheric-slow-wave-sleep-20260505-1
file: data/en.wikipedia.org/wiki/Unihemispheric_slow-wave_sleep-1.md
category: reference
title: Unihemispheric slow-wave sleep
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Unihemispheric_slow-wave_sleep
date_saved: 2026-05-05T11:04:43.626925+00:00
instance: kb-cron
parent: unihemispheric-slow-wave-sleep-20260505-0
summary:
=== Eye opening === In domestic chicks and other species of birds exhibiting USWS, one eye remained open contra-lateral (on the opposite side) to the "awake" hemisphere. The closed eye was shown to be opposite the hemisphere engaging in slow-wave sleep. Learning tasks, such as those including predator recognition, demonstrated the open eye could be preferential. This has also been shown to be the favored behavior of belugas, although inconsistencies have arisen directly relating the sleeping hemisphere and open eye. Keeping one eye open aids birds in engaging in USWS while mid-flight as well as helping them observe predators in their vicinity. Also crocodilians have been shown to sleep with one eye open. Given that USWS is preserved also in blind animals or during a lack of visual stimuli, it cannot be considered as a consequence of keeping an eye open while sleeping. Furthermore, the open eye in dolphins does not forcibly activate the contralateral hemisphere. Although unilateral vision plays a considerable role in keeping active the contralateral hemisphere, it is not the motive power of USWS. Consequently, USWS might be generated by endogenous mechanisms. === Thermoregulation === Brain temperature has been shown to drop when a sleeping EEG is


id: unihemispheric-slow-wave-sleep-20260505-2
file: data/en.wikipedia.org/wiki/Unihemispheric_slow-wave_sleep-2.md
category: reference
title: Unihemispheric slow-wave sleep
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Unihemispheric_slow-wave_sleep
date_saved: 2026-05-05T11:04:43.626925+00:00
instance: kb-cron
parent: unihemispheric-slow-wave-sleep-20260505-0
summary:
==== Regulation based on surroundings ==== Birds can sleep more efficiently with both hemispheres sleeping simultaneously (bihemispheric slow-wave sleep) when in safe conditions, but will increase the usage of USWS if they are in a potentially more dangerous environment. It is more beneficial to sleep using both hemispheres; however, the positives of unihemispheric slow-wave sleep prevail over its negatives under extreme conditions. While in unihemispheric slow-wave sleep, birds will sleep with one open eye towards the direction from which predators are more likely to approach. When birds do this in a flock, it's called the "group edge effect". The mallard is one bird that has been used experimentally to illustrate the "group edge effect". Birds positioned at the edge of the flock are most alert, scanning often for predators. These birds are more at risk than the birds in the center of the flock and are required to be on the lookout for both their own safety and the safety of the group as a whole. They have been observed spending more time in unihemispheric slow-wave sleep than the birds in the center. Since USWS allows for the one eye to be open, the cerebral hemisphere that undergoes slow-wave sleep


id: unihemispheric-slow-wave-sleep-20260505-3
file: data/en.wikipedia.org/wiki/Unihemispheric_slow-wave_sleep-3.md
category: reference
title: Unihemispheric slow-wave sleep
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Unihemispheric_slow-wave_sleep
date_saved: 2026-05-05T11:04:43.626925+00:00
instance: kb-cron
parent: unihemispheric-slow-wave-sleep-20260505-0
summary:
== Future research == Recent studies have illustrated that the white-crowned sparrow, as well as other passerines, have the capability of sleeping most significantly during the migratory season while in flight. However, the sleep patterns in this study were observed during migratory restlessness in captivity and might not be analogous to those of free-flying birds. Free-flying birds might be able to spend some time sleeping while in non-migratory flight as well when in the unobstructed sky as opposed to in controlled captive conditions. To truly determine if birds can sleep in flight, recordings of brain activity must take place during flight instead of after landing. A method of recording brain activity in pigeons during flight has recently proven promising in that it could obtain an EEG of each hemisphere but for relatively short periods of time. Coupled with simulated wind tunnels in a controlled setting, these new methods of measuring brain activity could elucidate the truth behind whether or not birds sleep during flight. Additionally, based on research elucidating the role of acetylcholine in control of USWS, additional neurotransmitters are being researched to understand their roles in the asymmetric sleep model. == See also == Sleep in animals == References


id: list-of-unsolved-problems-in-physics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_unsolved_problems_in_physics-0.md
category: reference
title: List of unsolved problems in physics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_physics
date_saved: 2026-05-05T11:04:45.237201+00:00
instance: kb-cron
parent: list-of-unsolved-problems-in-physics-20260505-0
summary:
The following is a list of notable unsolved problems grouped into broad areas of physics. Some of the major unsolved problems in physics are theoretical, meaning that existing theories are currently unable to explain certain observed phenomena or experimental results. Others are experimental, involving challenges in creating experiments to test proposed theories or to investigate specific phenomena in greater detail. A number of important questions remain open in the area of physics beyond the Standard Model, such as the strong CP problem, determining the absolute mass of neutrinos, understanding matter–antimatter asymmetry, and identifying the nature of dark matter and dark energy. Another significant problem lies within the mathematical framework of the Standard Model itself, which remains inconsistent with general relativity. This incompatibility causes both theories to break down under extreme conditions, such as within known spacetime gravitational singularities like those at the Big Bang and at the centers of black holes beyond their event horizons. == Historical problems and lists == Isaac Newton in his book Opticks (first edition 1704), published a list of 31 queries about light. In 1900, Lord Kelvin in a speech identified two main issues in classical physics that represented "two clouds" for the physics of


id: list-of-unsolved-problems-in-physics-20260505-1
file: data/en.wikipedia.org/wiki/List_of_unsolved_problems_in_physics-1.md
category: reference
title: List of unsolved problems in physics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_physics
date_saved: 2026-05-05T11:04:45.237201+00:00
instance: kb-cron
parent: list-of-unsolved-problems-in-physics-20260505-0
summary:
== Quantum gravity == Quantum gravity: Can quantum mechanics and general relativity be realized as a fully consistent theory (perhaps as a quantum field theory)? Is spacetime fundamentally continuous or discrete? Would a consistent theory involve a force mediated by a hypothetical graviton, or be a product of a discrete structure of spacetime itself (as in loop quantum gravity)? Are there deviations from the predictions of general relativity at very small or very large scales or in other extreme circumstances that flow from a quantum gravity mechanism? Black holes, black hole information paradox, and black hole radiation: Do black holes produce thermal radiation, as expected on theoretical grounds? Does this radiation contain information about their inner structure, as suggested by gauge–gravity duality, or not, as implied by Hawking's original calculation? If not, and black holes can evaporate away, what happens to the information stored in them (since quantum mechanics does not provide for the destruction of information)? Or does the radiation stop at some point, leaving black hole remnants? Is there another way to probe their internal structure somehow, if such a structure even exists? The cosmic censorship hypothesis and the chronology protection conjecture: Can singularities not hidden behind an


id: list-of-unsolved-problems-in-physics-20260505-2
file: data/en.wikipedia.org/wiki/List_of_unsolved_problems_in_physics-2.md
category: reference
title: List of unsolved problems in physics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_physics
date_saved: 2026-05-05T11:04:45.237201+00:00
instance: kb-cron
parent: list-of-unsolved-problems-in-physics-20260505-0
summary:
Dark matter: What is the identity of dark matter? Is it a particle? If so, is it a WIMP, axion, the lightest superpartner (LSP), or some other particle? Or, are the phenomena attributed to dark matter the result of an alternate theory of gravity separate from general relativity altogether? Despite extensive research, the exact composition of dark matter remains unknown. It is inferred from gravitational effects on visible matter, radiation, and the universe's large-scale structure. Understanding its properties is crucial for a comprehensive understanding of the universe. Dark energy: What is the cause of the observed accelerating expansion of the universe (the de Sitter phase)? Are the observations rightly interpreted as the accelerating expansion of the universe, or are they evidence that the cosmological principle is false? Why is the energy density of the dark energy component of the same magnitude as the density of matter at present when the two evolve quite differently over time; could it be simply that we are observing at exactly the right time? Is dark energy a pure cosmological constant or are models of quintessence such as phantom energy applicable? Dark flow: Is a non-spherically-symmetric gravitational pull from outside the observable universe responsible for


id: list-of-unsolved-problems-in-physics-20260505-3
file: data/en.wikipedia.org/wiki/List_of_unsolved_problems_in_physics-3.md
category: reference
title: List of unsolved problems in physics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_physics
date_saved: 2026-05-05T11:04:45.237201+00:00
instance: kb-cron
parent: list-of-unsolved-problems-in-physics-20260505-0
summary:
The QCD vacuum: Many of the equations in non-perturbative QCD are currently unsolved. These energies are the energies sufficient for the formation and description of atomic nuclei. How thus does low energy /non-pertubative QCD give rise to the formation of complex nuclei and nuclear constituents? Generations of matter: Why are there three generations of quarks and leptons? Is there a theory that can explain the masses of particular quarks and leptons in particular generations from first principles (a theory of Yukawa couplings)? Neutrino mass: What is the absolute mass of neutrinos? Are they Dirac or Majorana fermions? Is the mass hierarchy normal or inverted? Is the CP violating phase equal to 0? Reactor antineutrino anomaly: There is an anomaly in the existing body of data regarding the antineutrino flux from nuclear reactors around the world. Measured values of this flux appears to be only 94% of the value expected from theory. It is unknown whether this is due to unknown physics (such as sterile neutrinos), experimental error in the measurements, or errors in the theoretical flux calculations. Strong CP problem and axions: Why is the strong nuclear interaction invariant to parity and charge conjugation? Is Peccei–Quinn theory the solution to


id: list-of-unsolved-problems-in-physics-20260505-4
file: data/en.wikipedia.org/wiki/List_of_unsolved_problems_in_physics-4.md
category: reference
title: List of unsolved problems in physics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_physics
date_saved: 2026-05-05T11:04:45.237201+00:00
instance: kb-cron
parent: list-of-unsolved-problems-in-physics-20260505-0
summary:
Solar cycle: How does the Sun generate its periodically reversing large-scale magnetic field? How do other solar-like stars generate their magnetic fields, and what are the similarities and differences between stellar activity cycles and that of the Sun? What caused the Maunder Minimum and other grand minima, and how does the solar cycle recover from a minima state? Coronal heating problem: Why is the Sun's corona (atmosphere layer) so much hotter than the Sun's surface? Why is the magnetic reconnection effect many orders of magnitude faster than predicted by standard models? Astrophysical jet: Why do only certain accretion discs surrounding certain astronomical objects emit relativistic jets along their polar axes? Why are there quasi-periodic oscillations in many accretion discs? Why does the period of these oscillations scale as the inverse of the mass of the central object? Why are there sometimes overtones, and why do these appear at different frequency ratios in different objects? Diffuse interstellar bands: What is responsible for the numerous interstellar absorption lines detected in astronomical spectra? Are they molecular in origin, and if so which molecules are responsible for them? How do they form? Supermassive black holes: What is the origin of the M–sigma relation between


id: list-of-unsolved-problems-in-physics-20260505-5
file: data/en.wikipedia.org/wiki/List_of_unsolved_problems_in_physics-5.md
category: reference
title: List of unsolved problems in physics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_physics
date_saved: 2026-05-05T11:04:45.237201+00:00
instance: kb-cron
parent: list-of-unsolved-problems-in-physics-20260505-0
summary:
Quantum chromodynamics: What are the phases of strongly interacting matter, and what roles do they play in the evolution of the cosmos? What is the detailed partonic structure of the nucleons? What does QCD predict for the properties of strongly interacting matter? What determines the key features of QCD, and what is their relation to the nature of gravity and spacetime? Does QCD truly lack CP violations? Quark–gluon plasma: Where is the onset of deconfinement: 1) as a function of temperature and chemical potentials? 2) as a function of relativistic heavy-ion collision energy and system size? What is the mechanism of energy and baryon-number stopping leading to creation of quark-gluon plasma in relativistic heavy-ion collisions? Why is sudden hadronization and the statistical-hadronization model a near-to-perfect description of hadron production from quark–gluon plasma? Is quark flavor conserved in quark–gluon plasma? Are strangeness and charm in chemical equilibrium in quark–gluon plasma? Does strangeness in quark–gluon plasma flow at the same speed as up and down quark flavours? Why does deconfined matter show ideal flow? Specific models of quark–gluon plasma formation: Do gluons saturate when their occupation number is large? Do gluons form a dense system called colour glass condensate? What are the


id: list-of-unsolved-problems-in-physics-20260505-6
file: data/en.wikipedia.org/wiki/List_of_unsolved_problems_in_physics-6.md
category: reference
title: List of unsolved problems in physics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_physics
date_saved: 2026-05-05T11:04:45.237201+00:00
instance: kb-cron
parent: list-of-unsolved-problems-in-physics-20260505-0
summary:
== Plasma physics == Plasma physics and fusion power: Fusion energy may potentially provide power from an abundant resource (e.g. hydrogen) without the type of radioactive waste that fission energy currently produces. However, can ionized gases (plasma) be confined long enough and at a high enough temperature to create fusion power? What is the physical origin of H-mode? The injection problem: Fermi acceleration is thought to be the primary mechanism that accelerates astrophysical particles to high energy. However, it is unclear what mechanism causes those particles to initially have energies high enough for Fermi acceleration to work on them. Alfvénic turbulence: In the solar wind and the turbulence in solar flares, coronal mass ejections, and magnetospheric substorms are major unsolved problems in space plasma physics. Ball lightning: the exact physical nature of this mystery in atmospheric electricity. == Biophysics == Abiogenesis: can life be created from physical processes alone? Stochasticity and robustness to noise in gene expression: How do genes govern our body, withstanding different external pressures and internal stochasticity? Certain models exist for genetic processes, but we are far from understanding the whole picture, in particular in development where gene expression must be tightly regulated. Quantitative study of the


id: list-of-unsolved-problems-in-physics-20260505-7
file: data/en.wikipedia.org/wiki/List_of_unsolved_problems_in_physics-7.md
category: reference
title: List of unsolved problems in physics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_physics
date_saved: 2026-05-05T11:04:45.237201+00:00
instance: kb-cron
parent: list-of-unsolved-problems-in-physics-20260505-0
summary:
=== High-energy physics/particle physics === Existence of pentaquarks (1964–2015): In July 2015, the LHCb collaboration at CERN identified pentaquarks in the Λ0b→J/ψK−p channel, which represents the decay of the bottom lambda baryon (Λ0b) into a J/ψ meson (J/ψ), a kaon (K−) and a proton (p). The results showed that sometimes, instead of decaying directly into mesons and baryons, the Λ0b decayed via intermediate pentaquark states. The two states, named P+c(4380) and P+c(4450), had individual statistical significances of 9 σ and 12 σ, respectively, and a combined significance of 15 σ—enough to claim a formal discovery. The two pentaquark states were both observed decaying strongly to J/ψp, hence must have a valence quark content of two up quarks, a down quark, a charm quark, and an anti-charm quark (uudcc), making them charmonium-pentaquarks. Existence of quark-gluon plasma, a new phase of matter was discovered and confirmed in experiments at CERN-SPS (2000), BNL-RHIC (2005) and CERN-LHC (2010). Higgs boson and electroweak symmetry breaking (1963–2012): The mechanism responsible for breaking the electroweak gauge symmetry, giving mass to the W and Z bosons, was solved with the discovery of the Higgs boson of the Standard Model, with the expected couplings to the weak bosons. No


id: accretion-disk-20260505-0
file: data/en.wikipedia.org/wiki/Accretion_disk-0.md
category: reference
title: Accretion disk
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Accretion_disk
date_saved: 2026-05-05T11:04:46.406689+00:00
instance: kb-cron
parent: accretion-disk-20260505-0
summary:
An accretion disk is a structure (often a circumstellar disk) formed by diffuse material in orbital motion around a massive central body. The central body is most frequently a star. Friction, uneven irradiance, magnetohydrodynamic effects, and other forces induce instabilities causing orbiting material in the disk to spiral inward toward the central body. Gravitational and frictional forces compress and raise the temperature of the material, causing the emission of electromagnetic radiation. The frequency range of that radiation depends on the central object's mass. Accretion disks of young stars and protostars radiate in the infrared; those around neutron stars and black holes in the X-ray part of the spectrum. The study of oscillation modes in accretion disks is referred to as diskoseismology. == Manifestations == Accretion disks are a ubiquitous phenomenon in astrophysics; active galactic nuclei, protoplanetary disks, and gamma ray bursts all involve accretion disks. These disks very often give rise to astrophysical jets coming from the vicinity of the central object. Jets are an efficient way for the star-disk system to shed angular momentum without losing too much mass. The most prominent accretion disks are those of active galactic nuclei and of quasars, which are thought to be massive


id: accretion-disk-20260505-1
file: data/en.wikipedia.org/wiki/Accretion_disk-1.md
category: reference
title: Accretion disk
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Accretion_disk
date_saved: 2026-05-05T11:04:46.406689+00:00
instance: kb-cron
parent: accretion-disk-20260505-0
summary:
=== α-Disk model === Shakura and Sunyaev (1973) proposed turbulence in the gas as the source of an increased viscosity. Assuming subsonic turbulence and the disk height as an upper limit for the size of the eddies, the disk viscosity can be estimated as ν = α c s H {\displaystyle \nu =\alpha c_{\rm {s}}H} where c s {\displaystyle c_{\rm {s}}} is the sound speed, H {\displaystyle H} is the scale height of the disk, and α {\displaystyle \alpha } is a free parameter between zero (no accretion) and approximately one. In a turbulent medium ν ≈ v t u r b l t u r b {\displaystyle \nu \approx v_{\rm {turb}}l_{\rm {turb}}} , where v t u r b {\displaystyle v_{\rm {turb}}} is the velocity of turbulent cells relative to the mean gas motion, and l t u r b {\displaystyle l_{\rm {turb}}} is the size of the largest turbulent cells, which is estimated as l t u r b ≈ H = c s / Ω {\displaystyle l_{\rm {turb}}\approx H=c_{\rm {s}}/\Omega } and v t u r b ≈ c s {\displaystyle v_{\rm {turb}}\approx c_{\rm {s}}} , where Ω = ( G M ) 1 / 2 r


id: accretion-disk-20260505-2
file: data/en.wikipedia.org/wiki/Accretion_disk-2.md
category: reference
title: Accretion disk
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Accretion_disk
date_saved: 2026-05-05T11:04:46.406689+00:00
instance: kb-cron
parent: accretion-disk-20260505-0
summary:
Most astrophysical disks do not meet this criterion and are therefore prone to this magnetorotational instability. The magnetic fields present in astrophysical objects (required for the instability to occur) are believed to be generated via dynamo action. === Magnetic fields and jets === Accretion disks are usually assumed to be threaded by the external magnetic fields present in the interstellar medium. These fields are typically weak (about few micro-Gauss), but they can get anchored to the matter in the disk, because of its high electrical conductivity, and carried inward toward the central star. This process can concentrate the magnetic flux around the centre of the disk giving rise to very strong magnetic fields. Formation of powerful astrophysical jets along the rotation axis of accretion disks requires a large scale poloidal magnetic field in the inner regions of the disk. Such magnetic fields may be advected inward from the interstellar medium or generated by a magnetic dynamo within the disk. Magnetic fields strengths at least of order 100 Gauss seem necessary for the magneto-centrifugal mechanism to launch powerful jets. There are problems, however, in carrying external magnetic flux inward toward the central star of the disk. High electric conductivity dictates that


id: accretion-disk-20260505-3
file: data/en.wikipedia.org/wiki/Accretion_disk-3.md
category: reference
title: Accretion disk
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Accretion_disk
date_saved: 2026-05-05T11:04:46.406689+00:00
instance: kb-cron
parent: accretion-disk-20260505-0
summary:
== Decretion disk == The opposite of an accretion disk is an decretion disk where instead of material accreting from a disk on to a central object, material is excreted from the center outward onto the disk. They are formed from matter ejected from a star, and are often assosicated with Be stars. == See also == == Notes == == References == == Further reading == Frank, Juhan; Andrew King; Derek Raine (2002), Accretion power in astrophysics (Third ed.), Cambridge University Press, ISBN 978-0-521-62957-7 Krolik, Julian H. (1999), Active Galactic Nuclei, Princeton University Press, ISBN 978-0-691-01151-6 Blaes, Omer (2014). "General Overview of Black Hole Accretion Theory". Space Science Reviews. 183 (1–4): 21–41. arXiv:1304.4879. Bibcode:2014SSRv..183...21B. doi:10.1007/s11214-013-9985-6. A modern review focused on models for turbulence required by alpha process. Abramowicz, M.A. (December 22, 2005). "Super-Eddington Black Hole Accretion:: Polish Doughnuts and Slim Disks". In Merloni, Andrea; Nayakshin, Sergei; Sunyaev, Rashid A. (eds.). Growing Black Holes: Accretion in a Cosmological Context. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 257–273. doi:10.1007/11403913_49. ISBN 978-3-540-25275-7. Retrieved 2026-04-13. Review with a good introduction to the topic. == External links == Professor John F. Hawley homepage, University of Virginia (archived 2015) The Dynamical Structure of Nonradiative Black Hole


id: ball-lightning-20260505-0
file: data/en.wikipedia.org/wiki/Ball_lightning-0.md
category: reference
title: Ball lightning
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Ball_lightning
date_saved: 2026-05-05T11:04:47.852283+00:00
instance: kb-cron
parent: ball-lightning-20260505-0
summary:
Ball lightning is a rare and unexplained phenomenon described as luminescent spherical objects that vary from pea-sized to several meters in diameter. Though usually associated with thunderstorms, the observed phenomenon is reported to last considerably longer than the split-second flash of a lightning bolt and is distinct from St. Elmo's fire and will-o'-the-wisp. Some 20th-century reports describe balls that eventually explode and leave behind an odor of sulfur. Descriptions of ball lightning appear in a variety of accounts over the centuries and have received attention from scientists. An optical spectrum of what appears to have been a ball lightning event was published in January 2014 and included a video at high frame rate. Nevertheless, scientific data on ball lightning remain scarce. Although laboratory experiments have produced effects that are visually similar to reports of ball lightning, how these relate to the phenomenon remains unclear. == Characteristics == Descriptions of ball lightning vary widely. It has been described as moving up and down, sideways or in unpredictable trajectories, hovering and moving with or against the wind; attracted to, unaffected by, or repelled from buildings, people, cars and other objects. Some accounts describe it as moving through solid masses of wood or


id: ball-lightning-20260505-1
file: data/en.wikipedia.org/wiki/Ball_lightning-1.md
category: reference
title: Ball lightning
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Ball_lightning
date_saved: 2026-05-05T11:04:47.852283+00:00
instance: kb-cron
parent: ball-lightning-20260505-0
summary:
=== The sloop Catherine and Mary === In December 1726, a number of British newspapers printed an extract of a letter from John Howell of the sloop Catherine and Mary: As we were coming thro' the Gulf of Florida on 29th of August, a large ball of fire fell from the Element and split our mast in Ten Thousand Pieces, if it were possible; split our Main Beam, also Three Planks of the Side, Under Water, and Three of the Deck; killed one man, another had his Hand carried of [sic], and had it not been for the violent rains, our Sails would have been of a Blast of Fire. === HMS Montague === One particularly large example was reported "on the authority of Dr. Gregory" in 1749: Admiral Chambers on board the HMS Montague, 4 November 1749, was taking an observation just before noon ... he observed a large ball of blue fire about three miles [5 km] distant from them. They immediately lowered their topsails, but it came up so fast upon them, that, before they could raise the main tack, they observed the ball rise almost perpendicularly, and not above forty or fifty yards [35 or 45


id: ball-lightning-20260505-2
file: data/en.wikipedia.org/wiki/Ball_lightning-2.md
category: reference
title: Ball lightning
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Ball_lightning
date_saved: 2026-05-05T11:04:47.852283+00:00
instance: kb-cron
parent: ball-lightning-20260505-0
summary:
Once my parents were away, and I was at the all-night vigil with my grandfather in the small church in Alexandria. During the service there was a powerful thunderstorm, streaks of lightning flashed one after the other, and it seemed as if the peals of thunder would shake even the church and the whole world to its foundations. Suddenly it became quite dark, a blast of wind from the open door blew out the flame of the candles which were lit in front of the iconostasis, there was a long clap of thunder, louder than before, and I suddenly saw a fiery ball flying from the window straight towards the head of the Emperor. The ball (it was of lightning) whirled around the floor, then passed the chandelier and flew out through the door into the park. My heart froze, I glanced at my grandfather – his face was completely calm. He crossed himself just as calmly as he had when the fiery ball had flown near us, and I felt that it was unseemly and not courageous to be frightened as I was. I felt that one had only to look at what was happening and believe in the


id: ball-lightning-20260505-3
file: data/en.wikipedia.org/wiki/Ball_lightning-3.md
category: reference
title: Ball lightning
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Ball_lightning
date_saved: 2026-05-05T11:04:47.852283+00:00
instance: kb-cron
parent: ball-lightning-20260505-0
summary:
Willy Ley discussed a sighting in Paris on 5 July 1852 "for which sworn statements were filed with the French Academy of Sciences". During a thunderstorm, a tailor living next to Church of the Val-de-Grâce saw a ball the size of a human head come out of the fireplace. It flew around the room, reentered the fireplace, and exploded in and destroyed the top of the chimney. On 30 April 1877, a ball of lightning entered the Golden Temple at Amritsar, India, and exited through a side door. Several people observed the ball, and the incident is inscribed on the front wall of Darshani Deori. On 22 November 1894, an unusually prolonged instance of natural ball lightning occurred in Golden, Colorado, which suggests it could be artificially induced from the atmosphere. The Golden Globe newspaper reported:A beautiful yet strange phenomenon was seen in this city on last Monday night. The wind was high and the air seemed to be full of electricity. In front of, above and around the new Hall of Engineering of the School of Mines, balls of fire played tag for half an hour, to the wonder and amazement of all who saw the display. In this


id: ball-lightning-20260505-4
file: data/en.wikipedia.org/wiki/Ball_lightning-4.md
category: reference
title: Ball lightning
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Ball_lightning
date_saved: 2026-05-05T11:04:47.852283+00:00
instance: kb-cron
parent: ball-lightning-20260505-0
summary:
== Direct measurements of natural ball lightning == In January 2014, scientists from Northwest Normal University in Lanzhou, China, published the results of recordings made in July 2012 of the optical spectrum of what was thought to be natural ball lightning made by chance during the study of ordinary cloud–ground lightning on the Tibetan Plateau. At a distance of 900 m (3,000 ft), a total of 1.64 seconds of digital video of the ball lightning and its spectrum was made, from the formation of the ball lightning after the ordinary lightning struck the ground, up to the optical decay of the phenomenon. Additional video was recorded by a high-speed (3000 frames/sec) camera, which captured only the last 0.78 seconds of the event, due to its limited recording capacity. Both cameras were equipped with slitless spectrographs. The researchers detected emission lines of neutral atomic silicon, calcium, iron, nitrogen, and oxygen—in contrast with mainly ionized nitrogen emission lines in the spectrum of the parent lightning. The ball lightning traveled horizontally across the video frame at an average speed equivalent of 8.6 m/s (28 ft/s). It had a diameter of 5 m (16 ft) and covered a distance of about 15 m (49


id: ball-lightning-20260505-5
file: data/en.wikipedia.org/wiki/Ball_lightning-5.md
category: reference
title: Ball lightning
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Ball_lightning
date_saved: 2026-05-05T11:04:47.852283+00:00
instance: kb-cron
parent: ball-lightning-20260505-0
summary:
=== Vaporized silicon hypothesis === This hypothesis suggests that ball lightning consists of vaporized silicon burning through oxidation. Lightning striking Earth's soil could vaporize the silica contained within it, and somehow separate the oxygen from the silicon dioxide, turning it into pure silicon vapor. As it cools, the silicon could condense into a floating aerosol, bound by its charge, glowing due to the heat of silicon recombining with oxygen. An experimental investigation of this effect, published in 2007, reported producing "luminous balls with lifetime in the order of seconds" by evaporating pure silicon with an electric arc. Videos and spectrographs of this experiment have been made available. This hypothesis got significant supportive data in 2014, when the first ever recorded spectra of natural ball lightning were published. The theorized forms of silicon storage in soil include nanoparticles of Si, SiO, and SiC. Matthew Francis has dubbed this the "dirt clod hypothesis", in which the spectrum of ball lightning shows that it shares chemistry with soil. === Electrically charged solid-core model === In this model, ball lightning is assumed to have a solid, positively charged core. According to this underlying assumption, the core is surrounded by a thin electron layer with


id: ball-lightning-20260505-6
file: data/en.wikipedia.org/wiki/Ball_lightning-6.md
category: reference
title: Ball lightning
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Ball_lightning
date_saved: 2026-05-05T11:04:47.852283+00:00
instance: kb-cron
parent: ball-lightning-20260505-0
summary:
=== Buoyant plasma hypothesis === The declassified Project Condign report concludes that buoyant charged plasma formations similar to ball lightning are formed by novel physical, electrical, and magnetic phenomena, and that these charged plasmas are capable of being transported at enormous speeds under the influence and balance of electrical charges in the atmosphere. These plasmas appear to originate due to more than one set of weather and electrically charged conditions, the scientific rationale for which is incomplete or not fully understood. One suggestion is that meteoroids breaking up in the atmosphere and forming charged plasmas as opposed to burning completely or impacting as meteorites could explain some instances of the phenomena, in addition to other unknown atmospheric events. However, according to Stenhoff, this explanation is considered insufficient to explain the ball lightning phenomenon, and would likely not withstand peer review. === Hallucinations induced by magnetic field === Cooray and Cooray (2008) stated that the features of hallucinations experienced by patients having epileptic seizures in the occipital lobe are similar to the observed features of ball lightning. The study also showed that the rapidly changing magnetic field of a close lightning flash is strong enough to excite the neurons in the


id: ball-lightning-20260505-7
file: data/en.wikipedia.org/wiki/Ball_lightning-7.md
category: reference
title: Ball lightning
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Ball_lightning
date_saved: 2026-05-05T11:04:47.852283+00:00
instance: kb-cron
parent: ball-lightning-20260505-0
summary:
Spinning electric dipole hypothesis. A 1976 article by V. G. Endean postulated that ball lightning could be described as an electric field vector spinning in the microwave frequency region. Electrostatic Leyden jar models. Stanley Singer discussed (1971) this type of hypothesis and suggested that the electrical recombination time would be too short for the ball lightning lifetimes often reported. Smirnov proposed (1987) a fractal aerogel hypothesis. M. I. Zelikin proposed (2006) an explanation (with a rigorous mathematical foundation) based on the hypothesis of plasma superconductivity (see also). A. Meessen presented a theory at the 10th International Symposium on Ball Lightning (21–27 June 2010, Kaliningrad, Russia) explaining all known properties of ball lightning in terms of collective oscillations of free electrons. The simplest case corresponds to radial oscillations in a spherical plasma membrane. These oscillations are sustained by parametric amplification, resulting from regular "inhalation" of charged particles that are present at lower densities in the ambient air. Ball lightning vanishes thus by silent extinction when the available density of charged particles is too low, while it disappears with a loud and sometimes very violent explosion when this density is too high. Electronic oscillations are also possible as stationary waves in a


id: baryogenesis-20260505-0
file: data/en.wikipedia.org/wiki/Baryogenesis-0.md
category: reference
title: Baryogenesis
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Baryogenesis
date_saved: 2026-05-05T11:04:49.171800+00:00
instance: kb-cron
parent: baryogenesis-20260505-0
summary:
In physical cosmology, baryogenesis (also known as baryosynthesis) is the physical process that is hypothesized to have taken place during the early universe to produce baryonic asymmetry, the observation that only matter (baryons) and not antimatter (antibaryons) is detected in the universe (other than in cosmic ray collisions). Since it is assumed in cosmology that the particles we see were created using the same physics we measure today, and in particle physics experiments today matter and antimatter are always symmetric, the dominance of matter over antimatter is unexplained. A number of theoretical mechanisms are proposed to account for this discrepancy, namely identifying conditions that favour symmetry breaking and the creation of normal matter (as opposed to antimatter). This imbalance has to be exceptionally small, on the order of 1 in every 1630000000 (≈2×109) particles a small fraction of a second after the Big Bang. After most of the matter and antimatter was annihilated, what remained was all the baryonic matter in the current universe, along with a much greater number of bosons. Experiments reported in 2010 at Fermilab, however, seem to show that this imbalance is much greater than previously assumed. These experiments involved a series of particle collisions and


id: baryogenesis-20260505-1
file: data/en.wikipedia.org/wiki/Baryogenesis-1.md
category: reference
title: Baryogenesis
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Baryogenesis
date_saved: 2026-05-05T11:04:49.171800+00:00
instance: kb-cron
parent: baryogenesis-20260505-0
summary:
== In the Standard Model == The Standard Model can incorporate baryogenesis, though the amount of net baryons (and leptons) thus created may not be sufficient to account for the present baryon asymmetry. There is a required one excess quark per billion quark-antiquark pairs in the early universe in order to provide all the observed matter in the universe. This insufficiency has not yet been explained, theoretically or otherwise. Baryogenesis within the Standard Model requires the electroweak symmetry breaking to be a first-order cosmological phase transition, since otherwise sphalerons wipe out any baryon asymmetry that happened up to the phase transition. Beyond this, the remaining amount of baryon non-conserving interactions is negligible. The phase transition domain wall breaks the P-symmetry spontaneously, allowing for CP-symmetry violating interactions to break C-symmetry on both its sides. Quarks tend to accumulate on the broken phase side of the domain wall, while anti-quarks tend to accumulate on its unbroken phase side. Due to CP-symmetry violating electroweak interactions, some amplitudes involving quarks are not equal to the corresponding amplitudes involving anti-quarks, but rather have opposite phase (see CKM matrix and Kaon); since time reversal takes an amplitude to its complex conjugate, CPT-symmetry is conserved in this


id: baryon-asymmetry-20260505-0
file: data/en.wikipedia.org/wiki/Baryon_asymmetry-0.md
category: reference
title: Baryon asymmetry
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Baryon_asymmetry
date_saved: 2026-05-05T11:04:50.353706+00:00
instance: kb-cron
parent: baryon-asymmetry-20260505-0
summary:
In physical cosmology, the baryon asymmetry problem, also known as the matter asymmetry problem or the matter–antimatter asymmetry problem, is the observed imbalance in baryonic matter and antibaryonic matter in the observable universe. As the two form and behave in nearly identical ways, it is expected that they would have been created in near equal portions by the Big Bang, where in reality matter makes up the vast majority of the universe (including the Earth and humanity). Neither the Standard Model of particle physics nor the theory of general relativity provides a known explanation for why this should be so, and it is a natural assumption that the universe is neutral with all conserved charges. Since this does not seem to have been the case, it is likely some physical laws must have acted differently or did not exist for matter and/or antimatter. Several competing hypotheses exist to explain the imbalance of matter and antimatter that resulted in baryogenesis. However, there is as of yet no consensus theory to explain the phenomenon, which has been described as "one of the great mysteries in physics". == Sakharov conditions == In 1967, Andrei Sakharov proposed a set of three necessary conditions that


id: baryon-asymmetry-20260505-1
file: data/en.wikipedia.org/wiki/Baryon_asymmetry-1.md
category: reference
title: Baryon asymmetry
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Baryon_asymmetry
date_saved: 2026-05-05T11:04:50.353706+00:00
instance: kb-cron
parent: baryon-asymmetry-20260505-0
summary:
=== Regions of the universe where antimatter dominates === Another possible explanation of the apparent baryon asymmetry is that matter and antimatter are essentially separated into different, widely distant regions of the universe. The formation of antimatter galaxies was originally thought to explain the baryon asymmetry, as from a distance, antimatter atoms are indistinguishable from matter atoms; both produce light (photons) in the same way. Along the boundary between matter and antimatter regions, however, annihilation (and the subsequent production of gamma radiation) would be detectable, depending on its distance and the density of matter and antimatter. Such boundaries, if they exist, would likely lie in deep intergalactic space. The density of matter in intergalactic space is reasonably well established at about one atom per cubic meter. Assuming this is a typical density near a boundary, the gamma ray luminosity of the boundary interaction zone can be calculated. No such zones have been detected, but 30 years of research have placed bounds on how far they might be. On the basis of such analyses, it is now deemed unlikely that any region within the observable universe is dominated by antimatter. === Mirror anti-universe === The state of the universe, as it


id: bgs-conjecture-20260505-0
file: data/en.wikipedia.org/wiki/BGS_conjecture-0.md
category: reference
title: BGS conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/BGS_conjecture
date_saved: 2026-05-05T11:04:51.490503+00:00
instance: kb-cron
parent: bgs-conjecture-20260505-0
summary:
The Bohigas–Giannoni–Schmit (BGS) conjecture also known as the random matrix conjecture) for simple quantum mechanical systems (ergodic with a classical limit) few degrees of freedom holds that spectra of time reversal-invariant systems whose classical analogues are K-systems show the same fluctuation properties as predicted by the GOE (Gaussian orthogonal ensembles). Alternatively, the spectral fluctuation measures of a classically chaotic quantum system coincide with those of the canonical random-matrix ensemble in the same symmetry class (unitary, orthogonal, or symplectic). That is, the Hamiltonian of a microscopic analogue of a classical chaotic system can be modeled by a random matrix from a Gaussian ensemble as the distance of a few spacings between eigenvalues of a chaotic Hamiltonian operator generically statistically correlates with the spacing laws for eigenvalues of large random matrices. A simple example of the unfolded quantum energy levels in a classically chaotic system correlating like that would be Sinai billiards: Energy levels: − ℏ 2 2 m ▽ 2 ψ + V ( x ) ψ = E i ψ {\displaystyle -{\frac {\hbar ^{2}}{2{\mathit {m}}}}\bigtriangledown ^{2}\psi +{\mathit {V}}({\mathit {x}})\psi ={{\mathit {E}}_{\mathit {i}}}\psi } Spectral density: ρ ( x ) = ∑ i δ ( x − E i ) {\displaystyle


id: color-confinement-20260505-0
file: data/en.wikipedia.org/wiki/Color_confinement-0.md
category: reference
title: Color confinement
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Color_confinement
date_saved: 2026-05-05T11:04:52.706223+00:00
instance: kb-cron
parent: color-confinement-20260505-0
summary:
In quantum chromodynamics (QCD), color confinement or infrared slavery, often simply called confinement, is the phenomenon that color-charged particles (such as quarks and gluons) cannot be isolated, and therefore cannot be directly observed in normal conditions below the Hagedorn temperature of approximately 2 terakelvin (corresponding to energies of approximately 130–140 MeV per particle). Quarks and gluons must clump together to form hadrons. The two main types of hadron are the mesons (one quark, one antiquark) and the baryons (often three quarks or antiquarks, though other exotic variants exist). In addition, colorless glueballs formed only of gluons are also consistent with confinement, though difficult to identify experimentally. Quarks and gluons cannot be separated from their parent hadron without producing new hadrons. == Origin == There is not yet an analytic proof of color confinement in any non-abelian gauge theory. The phenomenon can be understood qualitatively by noting that the force-carrying gluons of QCD have color charge, unlike the photons of quantum electrodynamics (QED). Whereas the electric field between electrically charged particles decreases rapidly as those particles are separated, the gluon field between a pair of color charges forms a narrow flux tube (or string) between them. Because of this behavior of


id: cosmic-coincidence-20260505-0
file: data/en.wikipedia.org/wiki/Cosmic_coincidence-0.md
category: reference
title: Cosmic coincidence
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Cosmic_coincidence
date_saved: 2026-05-05T11:04:53.889377+00:00
instance: kb-cron
parent: cosmic-coincidence-20260505-0
summary:
In cosmology, the cosmic coincidence is the observation that at the present epoch of the universe's evolution, the energy densities associated with dark matter and dark energy are of the same order of magnitude, leading to their comparable effects on the dynamics of the cosmos. This coincidence is puzzling because these energies have vastly different effects on the universe's expansion—dark matter tends to slow down expansion through gravitational attraction, while dark energy seems to accelerate it. The observed similarity in the magnitudes of these two components' energy densities at this particular epoch in the universe's history raises questions about whether there might be some underlying physical connection or shared origin between dark matter and dark energy. Indeed, some theories attempt to explain this coincidence by proposing that they are different manifestations of the same fundamental force or field. == References == == See also == Fine-tuned universe


id: cuspy-halo-problem-20260505-0
file: data/en.wikipedia.org/wiki/Cuspy_halo_problem-0.md
category: reference
title: Cuspy halo problem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Cuspy_halo_problem
date_saved: 2026-05-05T11:04:55.069610+00:00
instance: kb-cron
parent: cuspy-halo-problem-20260505-0
summary:
The cuspy halo problem (also known as the core-cusp problem) is a discrepancy between the inferred dark matter density profiles of low-mass galaxies and the density profiles predicted by cosmological N-body simulations. Nearly all simulations form dark matter halos which have "cuspy" dark matter distributions, with density increasing steeply at small radii, while the rotation curves of most observed dwarf galaxies suggest that they have flat central dark matter density profiles ("cores"). Several possible solutions to the core-cusp problem have been proposed. Many recent studies have shown that including baryonic feedback (particularly feedback from supernovae and active galactic nuclei) can "flatten out" the core of a galaxy's dark matter profile, since feedback-driven gas outflows produce a time-varying gravitational potential that transfers energy to the orbits of the collisionless dark matter particles. Other works have shown that the core-cusp problem can be solved outside of the most widely accepted Lambda-Cold Dark Matter paradigm (ΛCDM): simulations with warm or self-interacting dark matter also produce dark matter cores in low-mass galaxies. It is also possible that the distribution of dark matter that minimizes the system energy has a flat central dark matter density profile. == Simulation results == According to W.J.G. de Blok


