# kb

id: correlation-coefficient-20260505-0
file: data/en.wikipedia.org/wiki/Correlation_coefficient-0.md
category: reference
title: Correlation coefficient
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Correlation_coefficient
date_saved: 2026-05-05T07:23:42.530685+00:00
instance: kb-cron
parent: correlation-coefficient-20260505-0
summary:
A correlation coefficient is a numerical measure of some type of linear correlation, meaning a linear function between two variables. The variables may be two columns of a given data set of observations, often called a sample, or two components of a multivariate random variable with a known distribution. Several types of correlation coefficient exist, each with their own definition and range of usability and characteristics. They all assume values in the range from −1 to +1, where ±1 indicates the strongest possible correlation and 0 indicates no correlation. As tools of analysis, correlation coefficients present certain problems, including the propensity of some types to be distorted by outliers and the possibility of incorrectly being used to infer a causal relationship between the variables (for more, see Correlation does not imply causation). == Types == There are several different measures for the degree of correlation in data, depending on the kind of data: principally, whether the data is a measurement, ordinal, or categorical. === Pearson === The Pearson product-moment correlation coefficient, also known as r, R, or Pearson's r, is a measure of the strength and direction of the linear relationship between two variables that is defined as the covariance


id: counterexample-20260505-0
file: data/en.wikipedia.org/wiki/Counterexample-0.md
category: reference
title: Counterexample
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Counterexample
date_saved: 2026-05-05T07:23:43.842156+00:00
instance: kb-cron
parent: counterexample-20260505-0
summary:
A counterexample is a specific example that contradicts a claim, hypothesis, or generalization. In logic a counterexample disproves a universally stated claim, and does so rigorously in the fields of mathematics and philosophy. For example, the statement that "student John Smith is not lazy" is a counterexample to the generalization "students are lazy", and both a counterexample to, and disproof of, the universally quantified "all students are lazy." == In mathematics == In mathematics, counterexamples are often used to prove the boundaries of possible theorems. By using counterexamples to show that certain conjectures are false, mathematical researchers can then avoid going down blind alleys and learn to modify conjectures to produce provable theorems. It is sometimes said that mathematical development consists primarily in finding (and proving) theorems and counterexamples. === Rectangle example === Suppose that a mathematician is studying geometry and shapes, and she wishes to prove certain theorems about them. She conjectures that "All rectangles are squares", and she is interested in knowing whether this statement is true or false. In this case, she can either attempt to prove the truth of the statement using deductive reasoning, or she can attempt to find a counterexample of the statement if


id: cryptomorphism-20260505-0
file: data/en.wikipedia.org/wiki/Cryptomorphism-0.md
category: reference
title: Cryptomorphism
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Cryptomorphism
date_saved: 2026-05-05T07:23:45.040735+00:00
instance: kb-cron
parent: cryptomorphism-20260505-0
summary:
In mathematics, two objects, especially systems of axioms or semantics for them, are called cryptomorphic if they are equivalent but not obviously equivalent. In particular, two definitions or axiomatizations of the same object are "cryptomorphic" if it is not obvious that they define the same object. Examples of cryptomorphic definitions abound in matroid theory and others can be found elsewhere, e.g., in group theory the definition of a group by a single operation of division, which is not obviously equivalent to the usual three "operations" of identity element, inverse, and multiplication. This word is a play on the many morphisms in mathematics, but "cryptomorphism" is only very distantly related to "isomorphism", "homomorphism", or "morphisms". The equivalence may in a cryptomorphism, if it is not actual identity, be informal, or may be formalized in terms of a bijection or equivalence of categories between the mathematical objects defined by the two cryptomorphic axiom systems. == Etymology == The word was coined by Garrett Birkhoff before 1967, for use in the third edition of his book Lattice Theory. Birkhoff did not give it a formal definition, though others working in the field have made some attempts since. == Use in matroid theory ==


id: definition-20260505-0
file: data/en.wikipedia.org/wiki/Definition-0.md
category: reference
title: Definition
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Definition
date_saved: 2026-05-05T07:23:46.235974+00:00
instance: kb-cron
parent: definition-20260505-0
summary:
A definition is a semantic statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definitions (which try to list the objects that a term describes). Another important category of definitions is the class of ostensive definitions, which convey the meaning of a term by pointing out examples. A term may have many different senses and multiple meanings, and thus require multiple definitions. In mathematics, a definition is used to give a precise meaning to a new term, by describing a condition which unambiguously qualifies what the mathematical term is and is not. Definitions and axioms form the basis on which all of modern mathematics is to be constructed. == Basic terminology == In modern usage, a definition is something, typically expressed in words, that attaches a meaning to a word or group of words. The word or group of words that is to be defined is called the definiendum, and the word, group of words, or action that defines it is called the definiens. For example, in the definition "An elephant is a


id: definition-20260505-1
file: data/en.wikipedia.org/wiki/Definition-1.md
category: reference
title: Definition
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Definition
date_saved: 2026-05-05T07:23:46.235974+00:00
instance: kb-cron
parent: definition-20260505-0
summary:
In classical thought, a definition was taken to be a statement of the essence of a thing. Aristotle had it that an object's essential attributes form its "essential nature", and that a definition of the object must include these essential attributes. The idea that a definition should state the essence of a thing led to the distinction between nominal and real essence—a distinction originating with Aristotle. In the Posterior Analytics, he says that the meaning of a made-up name can be known (he gives the example "goat stag") without knowing what he calls the "essential nature" of the thing that the name would denote (if there were such a thing). This led medieval logicians to distinguish between what they called the quid nominis, or the "whatness of the name", and the underlying nature common to all the things it names, which they called the quid rei, or the "whatness of the thing". The name "hobbit", for example, is perfectly meaningful. It has a quid nominis, but one could not know the real nature of hobbits, and so the quid rei of hobbits cannot be known. By contrast, the name "man" denotes real things (men) that have a certain quid rei.


id: definition-20260505-2
file: data/en.wikipedia.org/wiki/Definition-2.md
category: reference
title: Definition
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Definition
date_saved: 2026-05-05T07:23:46.235974+00:00
instance: kb-cron
parent: definition-20260505-0
summary:
At least one thing is stated to be a member of the set being defined; this is sometimes called a "base set". All things bearing a certain relation to other members of the set are also to count as members of the set. It is this step that makes the definition recursive. All other things are excluded from the set For instance, we could define a natural number as follows (after Peano): "0" is a natural number. Each natural number has a unique successor, such that: the successor of a natural number is also a natural number; distinct natural numbers have distinct successors; no natural number is succeeded by "0". Nothing else is a natural number. So "0" will have exactly one successor, which for convenience can be called "1". In turn, "1" will have exactly one successor, which could be called "2", and so on. The second condition in the definition itself refers to natural numbers, and hence involves self-reference. Although this sort of definition involves a form of circularity, it is not vicious, and the definition has been quite successful. In the same way, we can define ancestor as follows: A parent is an ancestor. A parent of


id: definition-20260505-3
file: data/en.wikipedia.org/wiki/Definition-3.md
category: reference
title: Definition
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Definition
date_saved: 2026-05-05T07:23:46.235974+00:00
instance: kb-cron
parent: definition-20260505-0
summary:
=== Limitations of definition === Given that a natural language such as English contains, at any given time, a finite number of words, any comprehensive list of definitions must either be circular or rely upon primitive notions. If every term of every definiens must itself be defined, "where at last should we stop?" A dictionary, for instance, insofar as it is a comprehensive list of lexical definitions, must resort to circularity. Many philosophers have chosen instead to leave some terms undefined. The scholastic philosophers claimed that the highest genera (called the ten generalissima) cannot be defined, since a higher genus cannot be assigned under which they may fall. Thus being, unity and similar concepts cannot be defined. Locke supposes in An Essay Concerning Human Understanding that the names of simple concepts do not admit of any definition. More recently Bertrand Russell sought to develop a formal language based on logical atoms. Other philosophers, notably Wittgenstein, rejected the need for any undefined simples. Wittgenstein pointed out in his Philosophical Investigations that what counts as a "simple" in one circumstance might not do so in another. He rejected the very idea that every explanation of the meaning of a term needed itself


id: dependent-and-independent-variables-20260505-0
file: data/en.wikipedia.org/wiki/Dependent_and_independent_variables-0.md
category: reference
title: Dependent and independent variables
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Dependent_and_independent_variables
date_saved: 2026-05-05T07:23:47.442630+00:00
instance: kb-cron
parent: dependent-and-independent-variables-20260505-0
summary:
A variable is considered dependent if it depends on (or is hypothesized to depend on) an independent variable. Dependent variables are the outcome of the test they depend on, by some law or rule (e.g., by a mathematical function). Independent variables, on the other hand, are not seen as depending on any other variable in the scope of the experiment in question. Rather, they are controlled by the experimenter. == In pure mathematics == In mathematics, a function is a rule for taking an input (in the simplest case, a number or set of numbers) and providing an output (which may also be a number or set of numbers). A symbol that stands for an arbitrary input is called an independent variable, while a symbol that stands for an arbitrary output is called a dependent variable. The most common symbol for the input is x, and the most common symbol for the output is y; the function itself is commonly written y = f(x). It is possible to have multiple independent variables or multiple dependent variables. For instance, in multivariable calculus, one often encounters functions of the form z = f(x,y), where z is a dependent variable and x and


id: dependent-and-independent-variables-20260505-1
file: data/en.wikipedia.org/wiki/Dependent_and_independent_variables-1.md
category: reference
title: Dependent and independent variables
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Dependent_and_independent_variables
date_saved: 2026-05-05T07:23:47.442630+00:00
instance: kb-cron
parent: dependent-and-independent-variables-20260505-0
summary:
== Other variables == A variable may be thought to alter the dependent or independent variables, but may not actually be the focus of the experiment. So that the variable will be kept constant or monitored to try to minimize its effect on the experiment. Such variables may be designated as either a "controlled variable", "control variable", or "fixed variable". Extraneous variables are candidate independent variables which may be included in a regression analysis to aid a researcher with accurate response parameter estimation, prediction, and goodness of fit, but are not of substantive interest to the hypothesis under examination. For example, in a study examining the effect of post-secondary education on lifetime earnings, some extraneous variables might be gender, ethnicity, social class, genetics, intelligence, age, and so forth. A variable is extraneous only when it can be assumed (or shown) to influence the dependent variable. If included in a regression, it can improve the fit of the model. If it is excluded from the regression and if it has a non-zero covariance with one or more of the independent variables of interest, its omission will bias the regression's result for the effect of that independent variable of interest. This effect


id: differential-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Differential_(mathematics)-0.md
category: reference
title: Differential (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Differential_(mathematics)
date_saved: 2026-05-05T07:23:48.842935+00:00
instance: kb-cron
parent: differential-mathematics-20260505-0
summary:
In mathematics, differential refers to several related notions derived from the early days of calculus, put on a rigorous footing, such as infinitesimal differences and the derivatives of functions. The term is used in various branches of mathematics such as calculus, differential geometry, algebraic geometry and algebraic topology. == Introduction == The term differential is used nonrigorously in calculus to refer to an infinitesimal ("infinitely small") change in some varying quantity. For example, if x is a variable, then a change in the value of x is often denoted Δx (pronounced delta x). The differential dx represents an infinitely small change in the variable x. The idea of an infinitely small or infinitely slow change is, intuitively, extremely useful, and there are a number of ways to make the notion mathematically precise. Using calculus, it is possible to relate the infinitely small changes of various variables to each other mathematically using derivatives. If y is a function of x, then the differential dy of y is related to dx by the formula d y = d y d x d x , {\displaystyle dy={\frac {dy}{dx}}\,dx,} where d y d x {\displaystyle {\frac {dy}{dx}}\,} denotes not 'dy divided by dx' as


id: differential-mathematics-20260505-1
file: data/en.wikipedia.org/wiki/Differential_(mathematics)-1.md
category: reference
title: Differential (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Differential_(mathematics)
date_saved: 2026-05-05T07:23:48.842935+00:00
instance: kb-cron
parent: differential-mathematics-20260505-0
summary:
Differentials as linear maps. This approach underlies the definition of the derivative and the exterior derivative in differential geometry. Differentials as nilpotent elements of commutative rings. This approach is popular in algebraic geometry. Differentials in smooth models of set theory. This approach is known as synthetic differential geometry or smooth infinitesimal analysis and is closely related to the algebraic geometric approach, except that ideas from topos theory are used to hide the mechanisms by which nilpotent infinitesimals are introduced. Differentials as infinitesimals in hyperreal number systems, which are extensions of the real numbers that contain invertible infinitesimals and infinitely large numbers. This is the approach of nonstandard analysis pioneered by Abraham Robinson. These approaches are very different from each other, but they have in common the idea of being quantitative, i.e., saying not just that a differential is infinitely small, but how small it is. === Differentials as linear maps === There is a simple way to make precise sense of differentials, first used on the Real line by regarding them as linear maps. It can be used on R {\displaystyle \mathbb {R} } , R n {\displaystyle \mathbb {R} ^{n}} , a Hilbert space, a Banach space, or more


id: differential-mathematics-20260505-2
file: data/en.wikipedia.org/wiki/Differential_(mathematics)-2.md
category: reference
title: Differential (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Differential_(mathematics)
date_saved: 2026-05-05T07:23:48.842935+00:00
instance: kb-cron
parent: differential-mathematics-20260505-0
summary:
d f p = ∑ j = 1 n D j f ( p ) ( d x j ) p . {\displaystyle df_{p}=\sum _{j=1}^{n}D_{j}f(p)\,(dx_{j})_{p}.} The coefficients D j f ( p ) {\displaystyle D_{j}f(p)} are (by definition) the partial derivatives of f {\displaystyle f} at p {\displaystyle p} with respect to x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\ldots ,x_{n}} . Hence, if f {\displaystyle f} is differentiable on all of R n {\displaystyle \mathbb {R} ^{n}} , we can write, more concisely: d f = ∂ f ∂ x 1 d x 1 + ∂ f ∂ x 2 d x 2 + ⋯ + ∂ f ∂ x n d x n . {\displaystyle df={\frac {\partial f}{\partial x_{1}}}\,dx_{1}+{\frac {\partial f}{\partial x_{2}}}\,dx_{2}+\cdots +{\frac {\partial f}{\partial x_{n}}}\,dx_{n}.} In the one-dimensional case this becomes d f = d f d x d x {\displaystyle df={\frac {df}{dx}}dx} as before. This idea generalizes straightforwardly to functions from R n {\displaystyle \mathbb {R} ^{n}} to R m {\displaystyle \mathbb {R} ^{m}} . Furthermore, it has the decisive advantage over other definitions of the derivative that it is invariant under changes of coordinates. This means that the same idea can


id: differential-mathematics-20260505-3
file: data/en.wikipedia.org/wiki/Differential_(mathematics)-3.md
category: reference
title: Differential (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Differential_(mathematics)
date_saved: 2026-05-05T07:23:48.842935+00:00
instance: kb-cron
parent: differential-mathematics-20260505-0
summary:
=== Algebraic geometry === In algebraic geometry, differentials and other infinitesimal notions are handled in a very explicit way by accepting that the coordinate ring or structure sheaf of a space may contain nilpotent elements. The simplest example is the ring of dual numbers R[ε], where ε2 = 0. This can be motivated by the algebro-geometric point of view on the derivative of a function f from R to R at a point p. For this, note first that f − f(p) belongs to the ideal Ip of functions on R which vanish at p. If the derivative f vanishes at p, then f − f(p) belongs to the square Ip2 of this ideal. Hence the derivative of f at p may be captured by the equivalence class [f − f(p)] in the quotient space Ip/Ip2, and the 1-jet of f (which encodes its value and its first derivative) is the equivalence class of f in the space of all functions modulo Ip2. Algebraic geometers regard this equivalence class as the restriction of f to a thickened version of the point p whose coordinate ring is not R (which is the quotient space of functions on R modulo Ip) but


id: essentially-unique-20260505-0
file: data/en.wikipedia.org/wiki/Essentially_unique-0.md
category: reference
title: Essentially unique
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Essentially_unique
date_saved: 2026-05-05T07:23:50.018300+00:00
instance: kb-cron
parent: essentially-unique-20260505-0
summary:
In mathematics, the term essentially unique is used to describe a weaker form of uniqueness, where an object satisfying a property is "unique" only in the sense that all objects satisfying the property are equivalent to each other. The notion of essential uniqueness presupposes some form of "sameness", which is often formalized using an equivalence relation. A related notion is a universal property, where an object is not only essentially unique, but unique up to a unique isomorphism (meaning that it has trivial automorphism group). In general there can be more than one isomorphism between examples of an essentially unique object. == Examples == === Set theory === At the most basic level, there is an essentially unique set of any given cardinality, whether one labels the elements { 1 , 2 , 3 } {\displaystyle \{1,2,3\}} or { a , b , c } {\displaystyle \{a,b,c\}} . In this case, the non-uniqueness of the isomorphism (e.g., match 1 to a {\displaystyle a} or 1 to c {\displaystyle c} ) is reflected in the symmetric group. On the other hand, there is an essentially unique totally ordered set of any given finite cardinality that is unique up to unique isomorphism:


id: eventually-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Eventually_(mathematics)-0.md
category: reference
title: Eventually (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Eventually_(mathematics)
date_saved: 2026-05-05T07:23:51.308179+00:00
instance: kb-cron
parent: eventually-mathematics-20260505-0
summary:
In the mathematical areas of number theory and analysis, an infinite sequence or a function is said to eventually have a certain property, if it does not have the said property across all its ordered instances, but will after some instances have passed. The use of the term "eventually" can be often rephrased as "for sufficiently large numbers", and can be also extended to the class of properties that apply to elements of any ordered set (such as sequences and subsets of R {\displaystyle \mathbb {R} } ). == Notation == The general form where the phrase eventually (or sufficiently large) is found appears as follows: P {\displaystyle P} is eventually true for x {\displaystyle x} ( P {\displaystyle P} is true for sufficiently large x {\displaystyle x} ), where ∀ {\displaystyle \forall } and ∃ {\displaystyle \exists } are the universal and existential quantifiers, which is actually a shorthand for: ∃ a ∈ R {\displaystyle \exists a\in \mathbb {R} } such that P {\displaystyle P} is true ∀ x ≥ a {\displaystyle \forall x\geq a} or somewhat more formally: ∃ a ∈ R : ∀ x ∈ R : x ≥ a ⇒ P ( x ) {\displaystyle \exists


id: exceptional-object-20260505-0
file: data/en.wikipedia.org/wiki/Exceptional_object-0.md
category: reference
title: Exceptional object
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Exceptional_object
date_saved: 2026-05-05T07:23:52.521982+00:00
instance: kb-cron
parent: exceptional-object-20260505-0
summary:
In mathematics, an exceptional object is one of finitely many exceptions to some classification of objects. Many branches of mathematics study objects of a given type and prove a classification theorem. A common theme is that the classification results in a number of series of objects and a finite number of exceptions — often with desirable properties — that do not fit into any series; these are known as exceptional objects. In many cases, these exceptional objects play a further and important role in the subject. Furthermore, the exceptional objects in one branch of mathematics often relate to the exceptional objects in others. A related phenomenon is exceptional isomorphism, when two series are in general different, but agree for some small values. For example, spin groups in low dimensions are isomorphic to other classical Lie groups. == Regular polytopes == The prototypical examples of exceptional objects arise in the classification of regular polytopes: in two dimensions, there is a series of regular n-gons for n ≥ 3. In every dimension above 2, one can find analogues of the cube, tetrahedron and octahedron. In three dimensions, one finds two more regular polyhedra — the dodecahedron (12-hedron) and the icosahedron (20-hedron) —


id: exceptional-object-20260505-1
file: data/en.wikipedia.org/wiki/Exceptional_object-1.md
category: reference
title: Exceptional object
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Exceptional_object
date_saved: 2026-05-05T07:23:52.521982+00:00
instance: kb-cron
parent: exceptional-object-20260505-0
summary:
== Block designs == An exceptional block design is the Steiner system S(5,8,24) whose automorphism group is the sporadic simple Mathieu group M 24 {\displaystyle M_{24}} . The codewords of the extended binary Golay code have a length of 24 bits and have weights 0, 8, 12, 16, or 24. This code can correct up to three errors. So every 24-bit word with weight 5 can be corrected to a codeword with weight 8. The bits of a 24-bit word can be thought of as specifying the possible subsets of a 24 element set. So the extended binary Golay code gives a unique 8 element subset for each 5 element subset. In fact, it defines S(5,8,24). == Outer automorphisms == Certain families of groups often have a certain outer automorphism group, but in particular cases, they have other exceptional outer automorphisms. Among families of finite simple groups, the only example is in the automorphisms of the symmetric and alternating groups: for n ≥ 3 , n ≠ 6 {\displaystyle n\geq 3,n\neq 6} the alternating group A n {\displaystyle A_{n}} has one outer automorphism (corresponding to conjugation by an odd element of S n {\displaystyle S_{n}} ) and the symmetric group


id: exceptional-object-20260505-2
file: data/en.wikipedia.org/wiki/Exceptional_object-2.md
category: reference
title: Exceptional object
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Exceptional_object
date_saved: 2026-05-05T07:23:52.521982+00:00
instance: kb-cron
parent: exceptional-object-20260505-0
summary:
The Leech lattice is 24-dimensional. Most sporadic simple groups can be related to the Leech lattice, or more broadly the Monster. The exceptional Jordan algebra has a representation in terms of 24×24 real matrices together with the Jordan product rule. These objects are connected to various other phenomena in math which may be considered surprising but not themselves "exceptional". For example, in algebraic topology, 8-fold real Bott periodicity can be seen as coming from the octonions. In the theory of modular forms, the 24-dimensional nature of the Leech lattice underlies the presence of 24 in the formulas for the Dedekind eta function and the modular discriminant, which connection is deepened by Monstrous moonshine, a development that related modular functions to the Monster group. === Physics === In string theory and superstring theory we often find that particular dimensions are singled out as a result of exceptional algebraic phenomena. For example, bosonic string theory requires a spacetime of dimension 26 which is directly related to the presence of 24 in the Dedekind eta function. Similarly, the possible dimensions of supergravity are related to the dimensions of the division algebras. === Monstrous moonshine === Many of the exceptional objects in mathematics and


id: formulario-mathematico-20260505-0
file: data/en.wikipedia.org/wiki/Formulario_mathematico-0.md
category: reference
title: Formulario mathematico
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Formulario_mathematico
date_saved: 2026-05-05T07:23:53.723865+00:00
instance: kb-cron
parent: formulario-mathematico-20260505-0
summary:
Formulario Mathematico (Latino sine flexione: Formulary for Mathematics) is a book by Giuseppe Peano which expresses fundamental theorems of mathematics in a symbolic language developed by Peano. The author was assisted by Giovanni Vailati, Mario Pieri, Alessandro Padoa, Giovanni Vacca, Vincenzo Vivanti, Gino Fano and Cesare Burali-Forti. The Formulario was first published in 1894. The fifth and last edition was published in 1908. Nicolas Bourbaki described Peano's notation in the Formulario as "following current mathematical usage, and introducing many well-chosen abbreviating symbols, his language succeeded moreover in being fairly readable, ..." Hubert Kennedy wrote "the development and use of mathematical logic is the guiding motif of the project". He also explains the variety of Peano's publication under the title: the five editions of the Formulario [are not] editions in the usual sense of the word. Each is essentially a new elaboration, although much material is repeated. Moreover, the title and language varied: the first three, titled Formulaire de Mathématiques, and the fourth, titled, Formulaire Mathématique, were written in French, while Latino sine flexione, Peano's own invention, was used for the fifth edition, titled Formulario Mathematico. ... Ugo Cassina lists no less than twenty separately published items as being parts of


id: generalized-inverse-20260505-0
file: data/en.wikipedia.org/wiki/Generalized_inverse-0.md
category: reference
title: Generalized inverse
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Generalized_inverse
date_saved: 2026-05-05T07:23:55.090630+00:00
instance: kb-cron
parent: generalized-inverse-20260505-0
summary:
In mathematics, and in particular, algebra, a generalized inverse (or, g-inverse) of an element x is an element y that has some properties of an inverse element but not necessarily all of them. The purpose of constructing a generalized inverse of a matrix is to obtain a matrix that can serve as an inverse in some sense for a wider class of matrices than invertible matrices. Generalized inverses can be defined in any mathematical structure that involves associative multiplication, that is, in a semigroup. This article describes generalized inverses of a matrix A {\displaystyle A} . A matrix A g ∈ R n × m {\displaystyle A^{\mathrm {g} }\in \mathbb {R} ^{n\times m}} is a generalized inverse of a matrix A ∈ R m × n {\displaystyle A\in \mathbb {R} ^{m\times n}} if A A g A = A . {\displaystyle AA^{\mathrm {g} }A=A.} A generalized inverse exists for an arbitrary matrix, and when a matrix has a regular inverse, this inverse is its unique generalized inverse. == Motivation == Consider the linear system A x = y {\displaystyle Ax=y} where A {\displaystyle A} is an m × n {\displaystyle m\times n} matrix and y ∈ C ( A )


id: generalized-inverse-20260505-1
file: data/en.wikipedia.org/wiki/Generalized_inverse-1.md
category: reference
title: Generalized inverse
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Generalized_inverse
date_saved: 2026-05-05T07:23:55.090630+00:00
instance: kb-cron
parent: generalized-inverse-20260505-0
summary:
( A A g ) ∗ = A A g {\displaystyle (AA^{\mathrm {g} })^{*}=AA^{\mathrm {g} }} ( A g A ) ∗ = A g A , {\displaystyle (A^{\mathrm {g} }A)^{*}=A^{\mathrm {g} }A,} where ∗ {\displaystyle {}^{*}} denotes conjugate transpose. If A g {\displaystyle A^{\mathrm {g} }} satisfies the first condition, then it is a generalized inverse of A {\displaystyle A} . If it satisfies the first two conditions, then it is a reflexive generalized inverse of A {\displaystyle A} . If it satisfies all four conditions, then it is the pseudoinverse of A {\displaystyle A} , which is denoted by A + {\displaystyle A^{+}} and also known as the Moore–Penrose inverse, after the pioneering works by E. H. Moore and Roger Penrose. It is convenient to define an I {\displaystyle I} -inverse of A {\displaystyle A} as an inverse that satisfies the subset I ⊂ { 1 , 2 , 3 , 4 } {\displaystyle I\subset \{1,2,3,4\}} of the Penrose conditions listed above. Relations, such as A ( 1 , 4 ) A A ( 1 , 3 ) = A + {\displaystyle A^{(1,4)}AA^{(1,3)}=A^{+}} , can be established between these different classes of I {\displaystyle I} -inverses. When A


id: generalized-inverse-20260505-2
file: data/en.wikipedia.org/wiki/Generalized_inverse-2.md
category: reference
title: Generalized inverse
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Generalized_inverse
date_saved: 2026-05-05T07:23:55.090630+00:00
instance: kb-cron
parent: generalized-inverse-20260505-0
summary:
A right inverse of a non-square matrix A {\displaystyle A} is given by A R − 1 = A ⊺ ( A A ⊺ ) − 1 {\displaystyle A_{\mathrm {R} }^{-1}=A^{\intercal }\left(AA^{\intercal }\right)^{-1}} , provided A {\displaystyle A} has full row rank. A left inverse of a non-square matrix A {\displaystyle A} is given by A L − 1 = ( A ⊺ A ) − 1 A ⊺ {\displaystyle A_{\mathrm {L} }^{-1}=\left(A^{\intercal }A\right)^{-1}A^{\intercal }} , provided A {\displaystyle A} has full column rank. If A = B C {\displaystyle A=BC} is a rank factorization, then G = C R − 1 B L − 1 {\displaystyle G=C_{\mathrm {R} }^{-1}B_{\mathrm {L} }^{-1}} is a g-inverse of A {\displaystyle A} , where C R − 1 {\displaystyle C_{\mathrm {R} }^{-1}} is a right inverse of C {\displaystyle C} and B L − 1 {\displaystyle B_{\mathrm {L} }^{-1}} is left inverse of B {\displaystyle B} . If A = P [ I r 0 0 0 ] Q {\displaystyle A=P{\begin{bmatrix}I_{r}&0\\0&0\end{bmatrix}}Q} for any non-singular matrices P {\displaystyle P} and Q {\displaystyle Q} , then G = Q − 1 [ I r U W V ] P − 1 {\displaystyle G=Q^{-1}{\begin{bmatrix}I_{r}&U\\W&V\end{bmatrix}}P^{-1}} is a


id: generalized-space-20260505-0
file: data/en.wikipedia.org/wiki/Generalized_space-0.md
category: reference
title: Generalized space
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Generalized_space
date_saved: 2026-05-05T07:23:56.343111+00:00
instance: kb-cron
parent: generalized-space-20260505-0
summary:
In mathematics, a generalized space is a generalization of a topological space. Impetuses for such a generalization comes at least in two forms: A desire to apply concepts like cohomology for objects that are not traditionally viewed as spaces. For example, a topos was originally introduced for this reason. A practical need to remedy the deficiencies that some naturally occurring categories of spaces (e.g., ones in functional analysis) tend not to be abelian, a standard requirement to do homological algebra. Alexander Grothendieck's dictum says a topos is a generalized space; precisely, he and his followers write in exposé 4 of SGA I: On peut done dire que la notion de topos, dérivé naturel du point de vue faisceautique en Topologie, constitue à son tour un élargissement substantiel de la notion d'espace topologique, un grand nombre de situations qui autrefois n'étaient pas considérées comme relevant de intuition topologique However, William Lawvere argues in his 1975 paper that this dictum should be turned backward; namely, "a topos is the 'algebra of continuous (set-valued) functions' on a generalized space, not the generalized space itself." A generalized space should not be confused with a geometric object that can substitute the role of spaces. For


id: hand-waving-20260505-0
file: data/en.wikipedia.org/wiki/Hand-waving-0.md
category: reference
title: Hand-waving
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Hand-waving
date_saved: 2026-05-05T07:23:57.540774+00:00
instance: kb-cron
parent: hand-waving-20260505-0
summary:
Hand-waving is a pejorative label for attempting to be seen as effective – in word, reasoning, or deed – while actually doing nothing effective or substantial. It is often applied to debating techniques that involve fallacies, misdirection and the glossing over of details. It is also used academically to indicate unproven claims and skipped steps in proofs (sometimes intentionally, as in lectures and instructional materials), with some specific meanings in particular fields, including literary criticism, speculative fiction, mathematics, logic, science and engineering. The term can additionally be used in work situations, when attempts are made to display productivity or assure accountability without actually resulting in them. The term can also be used as a self-admission of, and suggestion to defer discussion about, an allegedly unimportant weakness in one's own argument's evidence, to forestall an opponent dwelling on it. In debate competition, certain cases of this form of hand-waving may be explicitly permitted. Hand-waving is an idiomatic metaphor, derived in part from the use of excessive gesticulation, perceived as unproductive, distracting or nervous, in communication or other effort. The term also evokes the sleight-of-hand distraction techniques of stage magic, and suggests that the speaker or writer seems to believe that if


id: hand-waving-20260505-1
file: data/en.wikipedia.org/wiki/Hand-waving-1.md
category: reference
title: Hand-waving
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Hand-waving
date_saved: 2026-05-05T07:23:57.540774+00:00
instance: kb-cron
parent: hand-waving-20260505-0
summary:
If someone starts a sentence with "Clearly..." or "Obviously..." or "It is self-evident that...", it is a good bet he is about to handwave (alternatively, use of these constructions in a sarcastic tone before a paraphrase of someone else's argument suggests that it is a handwave). The theory behind this term is that if you wave your hands at the right moment, the listener may be sufficiently distracted to not notice that what you have said is bogus [i.e., incorrect]. Failing that, if a listener does object, you might try to dismiss the objection with a wave of your hand. The implication that hand-waving is done with the specific intent to mislead has long been attached to the term, due to the use of literal waving of a hand – either natural-looking or showy, but never desperate – by illusionists to distract audiences and misdirect their attention from the mechanisms of the sleight-of-hand, gimmicked props or other trick being used in the performance. This meaning has become reinforced in recent decades by the influence of Star Wars (1977) and its sequels, in which the fictional Jedi mind trick involves a subtle hand wave with mystical powers – that only work


id: hand-waving-20260505-2
file: data/en.wikipedia.org/wiki/Hand-waving-2.md
category: reference
title: Hand-waving
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Hand-waving
date_saved: 2026-05-05T07:23:57.540774+00:00
instance: kb-cron
parent: hand-waving-20260505-0
summary:
Hand-waving arguments in engineering and other applied sciences often include order-of-magnitude estimates and dimensional analysis, especially in the use of Fermi problems in physics and engineering education. However, competent, well-intentioned researchers and professors also rely on explicitly declared hand-waving when, given a limited time, a large result must be shown and minor technical details cannot be given much attention—e.g., "it can be shown that z is an even number", as an intermediary step in reaching a conclusion. Another example of hand-waving can be found in the oversimplifications of the geologic representations commonly used in groundwater models created in support of land-development applications, especially those involving metal mining and aggregate extraction. Back-of-the-envelope calculations are approximate ways to get an answer by over-simplification, and are comparable to hand-waving in this sense. == In business == Hand-waving has been used to describe work-related situations where productivity is seemingly displayed, but deliverables are not produced, especially in the context of intentional engagement in busy work or pretend-work, vague claims of overwork or complications, impenetrably buzzword-laden rationalizations for delays or otherwise poor performance, and plausible-sounding but weak excuse-making and attention-deflecting tactics. In employment situations, as in political discourse, a hand-waving effort may seek to shift


id: harmonic-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Harmonic_(mathematics)-0.md
category: reference
title: Harmonic (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Harmonic_(mathematics)
date_saved: 2026-05-05T07:23:58.780140+00:00
instance: kb-cron
parent: harmonic-mathematics-20260505-0
summary:
In mathematics, a number of concepts employ the word harmonic. The similarity of this terminology to that of music is not accidental: the equations of motion of vibrating strings, drums and columns of air are given by formulas involving Laplacians; the solutions to which are given by eigenvalues corresponding to their modes of vibration. Thus, the term "harmonic" is applied when one is considering functions with sinusoidal variations, or solutions of Laplace's equation and related concepts. Mathematical terms whose names include "harmonic" include: Projective harmonic conjugate Cross-ratio Harmonic analysis Harmonic conjugate Harmonic form Harmonic function Harmonic mean Harmonic mode Harmonic number Harmonic series Alternating harmonic series Harmonic tremor Spherical harmonics


id: if-and-only-if-20260505-0
file: data/en.wikipedia.org/wiki/If_and_only_if-0.md
category: reference
title: If and only if
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/If_and_only_if
date_saved: 2026-05-05T07:24:00.078315+00:00
instance: kb-cron
parent: if-and-only-if-20260505-0
summary:
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its converse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, P if and only if Q means that P is true whenever Q is true, and the only case in which P is true is if Q is also true, whereas in the case of P if Q, there could be other scenarios where P is true and Q is false. In writing, phrases commonly used as alternatives to P "if and only if" Q include: Q is necessary


id: if-and-only-if-20260505-1
file: data/en.wikipedia.org/wiki/If_and_only_if-1.md
category: reference
title: If and only if
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/If_and_only_if
date_saved: 2026-05-05T07:24:00.078315+00:00
instance: kb-cron
parent: if-and-only-if-20260505-0
summary:
== More general usage == Iff is used outside the field of logic as well. Wherever logic is applied, especially in mathematical discussions, it has the same meaning as above: it is an abbreviation for if and only if, indicating that one statement is both necessary and sufficient for the other. This is an example of mathematical jargon (although, as noted above, if is more often used than iff in statements of definition). The elements of X are all and only the elements of Y means: "For any z in the domain of discourse, z is in X if and only if z is in Y." == When "if" means "if and only if" == In their Artificial Intelligence: A Modern Approach, Russell and Norvig note (page 282), in effect, that it is often more natural to express if and only if as if together with a "database (or logic programming) semantics". They give the example of the English sentence "Richard has two brothers, Geoffrey and John". In a database or logic program, this could be represented simply by two sentences: Brother(Richard, Geoffrey). Brother(Richard, John). The database semantics interprets the database (or program) as containing all and only the knowledge


id: inequality-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Inequality_(mathematics)-0.md
category: reference
title: Inequality (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Inequality_(mathematics)
date_saved: 2026-05-05T07:24:01.502525+00:00
instance: kb-cron
parent: inequality-mathematics-20260505-0
summary:
In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. The main types of inequality are less than and greater than (denoted by < and >, respectively the less-than and greater-than signs). == Notation == There are several different notations used to represent different kinds of inequalities: The notation a < b means that a is less than b. The notation a > b means that a is greater than b. In either case, a is not equal to b. These relations are known as strict inequalities, meaning that a is strictly less than or strictly greater than b. Equality is excluded. In contrast to strict inequalities, there are two types of inequality relations that are not strict: The notation a ≤ b or a ⩽ b or a ≦ b means that a is less than or equal to b (or, equivalently, at most b). The notation a ≥ b or a ⩾ b or a ≧ b means that a is greater than or equal to b (or, equivalently, at least b). In


id: inequality-mathematics-20260505-1
file: data/en.wikipedia.org/wiki/Inequality_(mathematics)-1.md
category: reference
title: Inequality (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Inequality_(mathematics)
date_saved: 2026-05-05T07:24:01.502525+00:00
instance: kb-cron
parent: inequality-mathematics-20260505-0
summary:
== Formal definitions and generalizations == A (non-strict) partial order is a binary relation ≤ over a set P which is reflexive, antisymmetric, and transitive. That is, for all a, b, and c in P, it must satisfy the three following clauses: a ≤ a (reflexivity) if a ≤ b and b ≤ a, then a = b (antisymmetry) if a ≤ b and b ≤ c, then a ≤ c (transitivity) A set with a partial order is called a partially ordered set. Those are the very basic axioms that every kind of order has to satisfy. A strict partial order is a relation < that satisfies a ≮ a (irreflexivity), if a < b, then b ≮ a (asymmetry), if a < b and b < c, then a < c (transitivity), where ≮ means that < does not hold. Some types of partial orders are specified by adding further axioms, such as: Total order: For every a and b in P, a ≤ b or b ≤ a . Dense order: For all a and b in P for which a < b, there is a c in P such that a < c < b. Least-upper-bound


id: inequality-mathematics-20260505-2
file: data/en.wikipedia.org/wiki/Inequality_(mathematics)-2.md
category: reference
title: Inequality (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Inequality_(mathematics)
date_saved: 2026-05-05T07:24:01.502525+00:00
instance: kb-cron
parent: inequality-mathematics-20260505-0
summary:
| ⟨ u , v ⟩ | 2 ≤ ⟨ u , u ⟩ ⋅ ⟨ v , v ⟩ , {\displaystyle |\langle \mathbf {u} ,\mathbf {v} \rangle |^{2}\leq \langle \mathbf {u} ,\mathbf {u} \rangle \cdot \langle \mathbf {v} ,\mathbf {v} \rangle ,} where ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is the inner product. Examples of inner products include the real and complex dot product; In Euclidean space Rn with the standard inner product, the Cauchy–Schwarz inequality is ( ∑ i = 1 n u i v i ) 2 ≤ ( ∑ i = 1 n u i 2 ) ( ∑ i = 1 n v i 2 ) . {\displaystyle {\biggl (}\sum _{i=1}^{n}u_{i}v_{i}{\biggr )}^{2}\leq {\biggl (}\sum _{i=1}^{n}u_{i}^{2}{\biggr )}{\biggl (}\sum _{i=1}^{n}v_{i}^{2}{\biggr )}.} == Power inequalities == A power inequality is an inequality containing terms of the form ab, where a and b are real positive numbers or variable expressions. They often appear in mathematical olympiads exercises. Examples: For any real x, e x ≥ 1 + x . {\displaystyle e^{x}\geq 1+x.} If x > 0 and p > 0, then 1 p ( x p − 1 ) ≥ ln ⁡ ( x


id: inequation-20260505-0
file: data/en.wikipedia.org/wiki/Inequation-0.md
category: reference
title: Inequation
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Inequation
date_saved: 2026-05-05T07:24:02.687697+00:00
instance: kb-cron
parent: inequation-20260505-0
summary:
In mathematics, an inequation is a statement that either an inequality (relations "greater than" and "less than", < and >) or a relation "not equal to" (≠) holds between two values. It is usually written in the form of a pair of expressions denoting the values in question, with a relational sign between the two sides, indicating the specific inequality relation. Some examples of inequations are: a < b {\displaystyle a<b} x + y + z ≤ 1 {\displaystyle x+y+z\leq 1} n > 1 {\displaystyle n>1} x ≠ 0 {\displaystyle x\neq 0} In some cases, the term "inequation" has a more restricted definition, reserved only for statements whose inequality relation is "not equal to" (or "distinct"). == Chains of inequations == A shorthand notation is used for the conjunction of several inequations involving common expressions, by chaining them together. For example, the chain 0 ≤ a < b ≤ 1 {\displaystyle 0\leq a<b\leq 1} is shorthand for 0 ≤ a a n d a < b a n d b ≤ 1 {\displaystyle 0\leq a~~\mathrm {and} ~~a<b~~\mathrm {and} ~~b\leq 1} which also implies that 0 < b {\displaystyle 0<b} and a < 1 {\displaystyle a<1} . In rare cases, chains


id: proof-by-infinite-descent-20260505-0
file: data/en.wikipedia.org/wiki/Proof_by_infinite_descent-0.md
category: reference
title: Proof by infinite descent
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Proof_by_infinite_descent
date_saved: 2026-05-05T07:24:03.923311+00:00
instance: kb-cron
parent: proof-by-infinite-descent-20260505-0
summary:
In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number, leading to an infinite descent and ultimately a contradiction. It is a method which relies on the well-ordering principle, and is often used to show that a given equation, such as a Diophantine equation, has no solutions. Typically, one shows that if a solution to a problem existed, which in some sense was related to one or more natural numbers, it would necessarily imply that a second solution existed, which was related to one or more 'smaller' natural numbers. This in turn would imply a third solution related to smaller natural numbers, implying a fourth solution, therefore a fifth solution, and so on. However, there cannot be an infinity of ever-smaller natural numbers, and therefore by mathematical induction, the original premise—that any solution exists—is incorrect: its correctness produces a contradiction. An alternative way to express this is to assume one or more solutions or


id: proof-by-infinite-descent-20260505-1
file: data/en.wikipedia.org/wiki/Proof_by_infinite_descent-1.md
category: reference
title: Proof by infinite descent
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Proof_by_infinite_descent
date_saved: 2026-05-05T07:24:03.923311+00:00
instance: kb-cron
parent: proof-by-infinite-descent-20260505-0
summary:
p q = 2 r 2 s = r s {\displaystyle {\frac {p}{q}}={\frac {2r}{2s}}={\frac {r}{s}}} . Therefore, if √2 could be written as a rational number, then it could always be written as a rational number with smaller parts, which itself could be written with yet-smaller parts, ad infinitum. But this is impossible in the set of natural numbers. Since √2 is a real number, which can be either rational or irrational, the only option left is for √2 to be irrational. (Alternatively, this proves that if √2 were rational, no "smallest" representation as a fraction could exist, as any attempt to find a "smallest" representation p/q would imply that a smaller one existed, which is a similar contradiction.) === Irrationality of √k if it is not an integer === For positive integer k, suppose that √k is not an integer, but is rational and can be expressed as ⁠m/n⁠ for natural numbers m and n, and let q be the largest integer less than √k (that is, q is the floor of √k). Then k = m n = m ( k − q ) n ( k − q ) = m k − m q n k


id: invariant-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Invariant_(mathematics)-0.md
category: reference
title: Invariant (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Invariant_(mathematics)
date_saved: 2026-05-05T07:24:05.176600+00:00
instance: kb-cron
parent: invariant-mathematics-20260505-0
summary:
In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used. For example, the area of a triangle is an invariant with respect to isometries of the Euclidean plane. The phrases "invariant under" and "invariant to" a transformation are both used. More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class. Invariants are used in diverse areas of mathematics such as geometry, topology, algebra and discrete mathematics. Some important classes of transformations are defined by an invariant they leave unchanged. For example, conformal maps are defined as transformations of the plane that preserve angles. The discovery of invariants is an important step in the process of classifying mathematical objects. == Examples == A simple example of invariance is expressed in our ability to count. For a finite set of objects of any kind, there is a number to which we always arrive, regardless of the order in


id: invariant-mathematics-20260505-1
file: data/en.wikipedia.org/wiki/Invariant_(mathematics)-1.md
category: reference
title: Invariant (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Invariant_(mathematics)
date_saved: 2026-05-05T07:24:05.176600+00:00
instance: kb-cron
parent: invariant-mathematics-20260505-0
summary:
The table above shows clearly that the invariant holds for each of the possible transformation rules, which means that whichever rule one picks, at whatever state, if the number of I's was not a multiple of three before applying the rule, then it will not be afterwards either. Given that there is a single I in the starting string MI, and one is not a multiple of three, one can then conclude that it is impossible to go from MI to MU (as the number of I's will never be a multiple of three). == Invariant set == A subset S of the domain U of a mapping T: U → U is an invariant set under the mapping when x ∈ S ⟺ T ( x ) ∈ S . {\displaystyle x\in S\iff T(x)\in S.} The elements of S are not necessarily fixed, even though the set S is fixed in the power set of U. (Some authors use the terminology setwise invariant, vs. pointwise invariant, to distinguish between these cases.) For example, a circle is an invariant subset of the plane under a rotation about the circle's center. Further, a conical surface is invariant as a set under


id: invariant-mathematics-20260505-2
file: data/en.wikipedia.org/wiki/Invariant_(mathematics)-2.md
category: reference
title: Invariant (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Invariant_(mathematics)
date_saved: 2026-05-05T07:24:05.176600+00:00
instance: kb-cron
parent: invariant-mathematics-20260505-0
summary:
=== Unchanged under perturbation === Thirdly, if one is studying an object that varies in a family, as is common in algebraic geometry and differential geometry, one may ask if the property is unchanged under perturbation (for example, if an object is constant on families or invariant under change of metric). == Invariants in computer science == In computer science, an invariant is a logical assertion that is always held to be true during a certain phase of execution of a computer program. For example, a loop invariant is a condition that is true at the beginning and the end of every iteration of a loop. Invariants are especially useful when reasoning about the correctness of a computer program. The theory of optimizing compilers, the methodology of design by contract, and formal methods for determining program correctness, all rely heavily on invariants. Programmers often use assertions in their code to make invariants explicit. Some object oriented programming languages have a special syntax for specifying class invariants. === Automatic invariant detection in imperative programs === Abstract interpretation tools can compute simple invariants of given imperative computer programs. The kinds of properties that can be found depend on the abstract domains used.


id: irreducibility-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Irreducibility_(mathematics)-0.md
category: reference
title: Irreducibility (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Irreducibility_(mathematics)
date_saved: 2026-05-05T07:24:06.491146+00:00
instance: kb-cron
parent: irreducibility-mathematics-20260505-0
summary:
In mathematics, the concept of irreducibility is used in several ways. A polynomial over a field may be an irreducible polynomial if it cannot be factored over that field. In abstract algebra, irreducible can be an abbreviation for irreducible element of an integral domain; for example an irreducible polynomial. In representation theory, an irreducible representation is a nontrivial representation with no nontrivial proper subrepresentations. Similarly, an irreducible module is another name for a simple module. Absolutely irreducible is a term applied to mean irreducible, even after any finite extension of the field of coefficients. It applies in various situations, for example to irreducibility of a linear representation, or of an algebraic variety; where it means just the same as irreducible over an algebraic closure. In commutative algebra, a commutative ring R is irreducible if its prime spectrum, that is, the topological space Spec R, is an irreducible topological space. A matrix is irreducible if it is not similar via a permutation to a block upper triangular matrix (that has more than one block of positive size). (Replacing non-zero entries in the matrix by one, and viewing the matrix as the adjacency matrix of a directed graph, the matrix is irreducible


id: jacobian-20260505-0
file: data/en.wikipedia.org/wiki/Jacobian-0.md
category: reference
title: Jacobian
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Jacobian
date_saved: 2026-05-05T07:24:07.753500+00:00
instance: kb-cron
parent: jacobian-20260505-0
summary:
In mathematics, a Jacobian, named for Carl Gustav Jacob Jacobi, may refer to: Jacobian matrix and determinant Jacobian elliptic functions Jacobian variety Jacobian ideal Intermediate Jacobian Generalized Jacobian Robot Jacobian


id: law-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Law_(mathematics)-0.md
category: reference
title: Law (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Law_(mathematics)
date_saved: 2026-05-05T07:24:08.972603+00:00
instance: kb-cron
parent: law-mathematics-20260505-0
summary:
In mathematics, a law is a formula that is always true within a given context. Laws describe a relationship, between two or more expressions or terms (which may contain variables), usually using equality or inequality, or between formulas themselves, for instance, in mathematical logic. For example, the formula a 2 ≥ 0 {\displaystyle a^{2}\geq 0} is true for all real numbers a, and is therefore a law. Laws over an equality are called identities. For example, ( a + b ) 2 = a 2 + 2 a b + b 2 {\displaystyle (a+b)^{2}=a^{2}+2ab+b^{2}} and cos 2 ⁡ θ + sin 2 ⁡ θ = 1 {\displaystyle \cos ^{2}\theta +\sin ^{2}\theta =1} are identities. Mathematical laws are distinguished from scientific laws which are based on observations, and try to describe or predict a range of natural phenomena. The more significant laws are often called theorems. == Notable examples == === Geometric laws === Triangle inequality: If a, b, and c are the lengths of the sides of a triangle then the triangle inequality states that c ≤ a + b , {\displaystyle c\leq a+b,} with equality only in the degenerate case of a triangle with zero area. In Euclidean geometry


id: law-mathematics-20260505-1
file: data/en.wikipedia.org/wiki/Law_(mathematics)-1.md
category: reference
title: Law (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Law_(mathematics)
date_saved: 2026-05-05T07:24:08.972603+00:00
instance: kb-cron
parent: law-mathematics-20260505-0
summary:
=== Logical laws === De Morgan's laws: In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathematician. The rules allow the expression of conjunctions and disjunctions purely in terms of each other via negation. The rules can be expressed in English as: not (A or B) = (not A) and (not B) not (A and B) = (not A) or (not B) where "A or B" is an "inclusive or" meaning at least one of A or B rather than an "exclusive or" that means exactly one of A or B. In formal language, the rules are written as ¬ ( P ∨ Q ) ⟺ ( ¬ P ) ∧ ( ¬ Q ) , and ¬ ( P ∧ Q ) ⟺ ( ¬ P ) ∨ ( ¬ Q ) , {\displaystyle {\begin{aligned}\neg (P\lor Q)&\iff (\neg P)\land (\neg Q),\quad {\text{ and }}\\\neg (P\land Q)&\iff (\neg P)\lor (\neg Q),\end{aligned}}} where P and Q are propositions, ¬ {\displaystyle \neg } is the negation logic operator (NOT), ∧ {\displaystyle \land }


id: left-and-right-algebra-20260505-0
file: data/en.wikipedia.org/wiki/Left_and_right_(algebra)-0.md
category: reference
title: Left and right (algebra)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Left_and_right_(algebra)
date_saved: 2026-05-05T07:24:10.169286+00:00
instance: kb-cron
parent: left-and-right-algebra-20260505-0
summary:
In algebra, the terms left and right denote the order of a binary operation (usually, but not always, called "multiplication") in non-commutative algebraic structures. A binary operation ∗ is usually written in the infix form: s ∗ t The argument s is placed on the left side, and the argument t is on the right side. Even if the symbol of the operation is omitted, the order of s and t does matter (unless ∗ is commutative). A two-sided property is fulfilled on both sides. A one-sided property is related to one (unspecified) of two sides. Although the terms are similar, left–right distinction in algebraic parlance is not related either to left and right limits in calculus, or to left and right in geometry. == Binary operation as an operator == A binary operation ∗ may be considered as a family of unary operators through currying: Rt(s) = s ∗ t, depending on t as a parameter – this is the family of right operations. Similarly, Ls(t) = s ∗ t defines the family of left operations parametrized with s. If for some e, the left operation Le is the identity operation, then e is called a left identity. Similarly,


id: lemma-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Lemma_(mathematics)-0.md
category: reference
title: Lemma (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Lemma_(mathematics)
date_saved: 2026-05-05T07:24:11.535656+00:00
instance: kb-cron
parent: lemma-mathematics-20260505-0
summary:
In mathematics and other fields, a lemma (pl.: lemmas or lemmata) is a generally minor, proven proposition which is used to prove a larger statement. For that reason, it is also known as a "helping theorem" or an "auxiliary theorem". In many cases, a lemma derives its importance from the theorem it aims to prove; however, a lemma can also turn out to be more important than originally thought. == Etymology == From the Ancient Greek λῆμμα, (perfect passive εἴλημμαι) something received or taken. Thus, something taken for granted in an argument. == Comparison with theorem == There is no formal distinction between a lemma and a theorem, only one of intention (see Theorem terminology). However, a lemma can be considered a minor result whose sole purpose is to help prove a more substantial theorem – a step in the direction of proof. == Well-known lemmas == Some powerful results in mathematics are known as lemmas, first named for their originally minor purpose. These include, among others: While these results originally seemed too simple or too technical to warrant independent interest, they have eventually turned out to be central to the theories in which they occur. == See also ==


id: lemniscate-20260505-0
file: data/en.wikipedia.org/wiki/Lemniscate-0.md
category: reference
title: Lemniscate
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Lemniscate
date_saved: 2026-05-05T07:24:12.760643+00:00
instance: kb-cron
parent: lemniscate-20260505-0
summary:
In algebraic geometry, a lemniscate ( or ) is any of several figure-eight or ∞-shaped curves. The word comes from the Latin lēmniscātus, meaning "decorated with ribbons", from the Greek λημνίσκος (lēmnískos), meaning "ribbon", or which alternatively may refer to the wool from which the ribbons were made. Curves that have been called a lemniscate include three quartic plane curves: the hippopede or lemniscate of Booth, the lemniscate of Bernoulli, and the lemniscate of Gerono. The hippopede was studied by Proclus (5th century), but the term "lemniscate" was not used until the work of Jacob Bernoulli in the late 17th century. == History and examples == === Lemniscate of Booth === The consideration of curves with a figure-eight shape can be traced back to Proclus, a Greek Neoplatonist philosopher and mathematician who lived in the 5th century AD. Proclus considered the cross-sections of a torus by a plane parallel to the axis of the torus. As he observed, for most such sections the cross section consists of either one or two ovals; however, when the plane is tangent to the inner surface of the torus, the cross-section takes on a figure-eight shape, which Proclus called a horse fetter (a device


id: level-logarithmic-quantity-20260505-0
file: data/en.wikipedia.org/wiki/Level_(logarithmic_quantity)-0.md
category: reference
title: Level (logarithmic quantity)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Level_(logarithmic_quantity)
date_saved: 2026-05-05T07:24:14.200490+00:00
instance: kb-cron
parent: level-logarithmic-quantity-20260505-0
summary:
In science and engineering, a power level and a field level (also called a root-power level) are logarithmic magnitudes of certain quantities referenced to a standard reference value of the same type. A power level is a logarithmic quantity used to measure power, power density or sometimes energy, with commonly used unit decibel (dB). A field level (or root-power level) is a logarithmic quantity used to measure quantities of which the square is typically proportional to power (for instance, the square of voltage is proportional to power multiplied by the conductor's resistance), with commonly used units neper (Np) or decibel (dB). The type of level and choice of units indicate the scaling of the logarithm of the ratio between the quantity and its reference value, though a logarithm may be considered to be a dimensionless quantity. The reference values for each type of quantity are often specified by international standards. Power and field levels are used in electronic engineering, telecommunications, acoustics and related disciplines. Power levels are used for signal power, noise power, sound power, sound exposure, etc. Field levels are used for voltage, current, sound pressure. == Power level == Level of a power quantity, denoted LP, is defined


id: like-terms-20260505-0
file: data/en.wikipedia.org/wiki/Like_terms-0.md
category: reference
title: Like terms
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Like_terms
date_saved: 2026-05-05T07:24:15.379757+00:00
instance: kb-cron
parent: like-terms-20260505-0
summary:
In mathematics, like terms are summands in a sum that differ only by a numerical factor. Like terms can be regrouped by adding their coefficients. Typically, in a polynomial expression, like terms are those that contain the same variables to the same powers, possibly with different coefficients. More generally, when some variable are considered as parameters, like terms are defined similarly, but "numerical factors" must be replaced by "factors depending only on the parameters". For example, when considering a quadratic equation, one considers often the expression ( x − r ) ( x − s ) , {\displaystyle (x-r)(x-s),} where r {\displaystyle r} and s {\displaystyle s} are the roots of the equation and may be considered as parameters. Then, expanding the above product and regrouping the like terms gives x 2 − ( r + s ) x + r s . {\displaystyle x^{2}-(r+s)x+rs.} == Generalization == In this discussion, a "term" will refer to a string of numbers being multiplied or divided (that division is simply multiplication by a reciprocal) together. Terms are within the same expression and are combined by either addition or subtraction. For example, take the expression: a x + b x {\displaystyle ax+bx} There


id: linearity-20260505-0
file: data/en.wikipedia.org/wiki/Linearity-0.md
category: reference
title: Linearity
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Linearity
date_saved: 2026-05-05T07:24:16.618385+00:00
instance: kb-cron
parent: linearity-20260505-0
summary:
In mathematics, the term linear is used in two distinct senses for two different properties: linearity of a function (or mapping); linearity of a polynomial. An example of a linear function is the function defined by f ( x ) = ( a x , b x ) {\displaystyle f(x)=(ax,bx)} that maps the real line to a line in the Euclidean plane R2 that passes through the origin. An example of a linear polynomial in the variables X , {\displaystyle X,} Y {\displaystyle Y} and Z {\displaystyle Z} is a X + b Y + c Z + d . {\displaystyle aX+bY+cZ+d.} Linearity of a mapping is closely related to proportionality. Examples in physics include the linear relationship of voltage and current in an electrical conductor (Ohm's law), and the relationship of mass and weight. By contrast, more complicated relationships, such as between velocity and kinetic energy, are nonlinear. Generalized for functions in more than one dimension, linearity means the property of a function of being compatible with addition and scaling, also known as the superposition principle. Linearity of a polynomial means that its degree is less than two. The use of the term for polynomials stems from the fact


id: linearity-20260505-1
file: data/en.wikipedia.org/wiki/Linearity-1.md
category: reference
title: Linearity
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Linearity
date_saved: 2026-05-05T07:24:16.618385+00:00
instance: kb-cron
parent: linearity-20260505-0
summary:
In every row in which the truth value of the function is T, there are an odd number of Ts assigned to the arguments, and in every row in which the function is F there is an even number of Ts assigned to arguments. Specifically, f(F, F, ..., F) = F, and these functions correspond to linear maps over the Boolean vector space. In every row in which the value of the function is T, there is an even number of Ts assigned to the arguments of the function; and in every row in which the truth value of the function is F, there are an odd number of Ts assigned to arguments. In this case, f(F, F, ..., F) = T. Another way to express this is that each variable always makes a difference in the truth value of the operation or it never makes a difference. Negation, Logical biconditional, exclusive or, tautology, and contradiction are linear functions. == Physics == In physics, linearity is a property of the differential equations governing many systems; for instance, the Maxwell equations or the diffusion equation. Linearity of a homogeneous differential equation means that if two functions f and g are solutions


id: mathematical-beauty-20260505-0
file: data/en.wikipedia.org/wiki/Mathematical_beauty-0.md
category: reference
title: Mathematical beauty
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Mathematical_beauty
date_saved: 2026-05-05T07:24:17.882469+00:00
instance: kb-cron
parent: mathematical-beauty-20260505-0
summary:
Mathematical beauty is a type of aesthetic value that is experienced in doing or contemplating mathematics. The testimonies of mathematicians indicate that various aspects of mathematics—including results, formulae, proofs and theories—can trigger subjective responses similar to the beauty of art, music, or nature. The pleasure in this experience can serve as a motivation for doing mathematics, and some mathematicians, such as G.H. Hardy, have characterized mathematics as an art form that seeks beauty. Beauty in mathematics has been subject to examination by mathematicians themselves and by philosophers, psychologists, and neuroscientists. Understanding beauty in general can be difficult because it is a subjective response to sense-experience but is perceived as a property of an external object, and because it may be shaped by cultural influence or personal experience. Mathematical beauty presents additional problems, since the aesthetic response is evoked by abstract ideas which can be communicated symbolically, and which may only be available to a minority of people with mathematical ability and training. The appreciation of mathematics may also be less passive than (for example) listening to music. Furthermore, beauty in mathematics may be connected to other aesthetic or non-aesthetic values. Some authors identify mathematical elegance with mathematical beauty; others distinguish


id: mathematical-beauty-20260505-1
file: data/en.wikipedia.org/wiki/Mathematical_beauty-1.md
category: reference
title: Mathematical beauty
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Mathematical_beauty
date_saved: 2026-05-05T07:24:17.882469+00:00
instance: kb-cron
parent: mathematical-beauty-20260505-0
summary:
== Properties of beautiful mathematics == Many mathematicians and philosophers who have written about mathematical beauty have tried to identify properties or criteria that are conducive to the perception of beauty in a piece of mathematics. It is debated whether beauty can be clarified or explained by such properties: Paul Erdős thought that it was no more possible to convince someone of the beauty of a piece of mathematics than to convince them of the beauty of Beethoven's Ninth Symphony, if they couldn't see it for themselves. === Results === In his 1940 essay A Mathematician's Apology, G. H. Hardy said that a beautiful result, including its proof, possesses three "purely aesthetic qualities", namely "inevitability", "unexpectedness", and "economy". He particularly excluded enumeration of cases as "one of the duller forms of mathematical argument". In 1997, Gian-Carlo Rota disagreed with unexpectedness as a sufficient condition for beauty and proposed a counterexample: A great many theorems of mathematics, when first published, appear to be surprising; thus for example some twenty years ago [from 1977] the proof of the existence of non-equivalent differentiable structures on spheres of high dimension was thought to be surprising, but it did not occur to anyone to call


id: mathematical-beauty-20260505-2
file: data/en.wikipedia.org/wiki/Mathematical_beauty-2.md
category: reference
title: Mathematical beauty
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Mathematical_beauty
date_saved: 2026-05-05T07:24:17.882469+00:00
instance: kb-cron
parent: mathematical-beauty-20260505-0
summary:
=== Information-theory model === In the 1970s, Abraham Moles and Frieder Nake analyzed links between beauty, information processing, and information theory. In the 1990s, Jürgen Schmidhuber formulated a mathematical theory of observer-dependent subjective beauty based on algorithmic information theory: the most beautiful objects among subjectively comparable objects have short algorithmic descriptions (i.e., Kolmogorov complexity) relative to what the observer already knows. Schmidhuber explicitly distinguishes between beautiful and interesting. The latter corresponds to the first derivative of subjectively perceived beauty: the observer continually tries to improve the predictability and compressibility of the observations by discovering regularities such as repetitions and symmetries and fractal self-similarity. Whenever the observer's learning process (possibly a predictive artificial neural network) leads to improved data compression such that the observation sequence can be described by fewer bits than before, the temporary interesting-ness of the data corresponds to the compression progress, and is proportional to the observer's internal curiosity reward. === Neural correlates === Brain imaging experiments conducted by Semir Zeki, Michael Atiyah and collaborators show that the experience of mathematical beauty has, as a neural correlate, activity in field A1 of the medial orbito-frontal cortex (mOFC) of the brain and that this activity is parametrically related to


id: mathematical-model-20260505-0
file: data/en.wikipedia.org/wiki/Mathematical_model-0.md
category: reference
title: Mathematical model
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Mathematical_model
date_saved: 2026-05-05T07:24:19.245253+00:00
instance: kb-cron
parent: mathematical-model-20260505-0
summary:
A mathematical model is an abstract description of a concrete system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in many fields, including applied mathematics, natural sciences, social sciences and engineering. In particular, the field of operations research studies the use of mathematical modelling and related tools to solve problems in business or military operations. A model may help to characterize a system by studying the effects of different components, which may be used to make predictions about behavior or solve specific problems. == Elements of a mathematical model == Mathematical models can take many forms, including dynamical systems, statistical models, differential equations, or game theoretic models. These and other types of models can overlap, with a given model involving a variety of abstract structures. In many cases, the quality of a scientific field depends on how well the mathematical models developed on the theoretical side agree with results of repeatable experiments. Lack of agreement between theoretical mathematical models and experimental measurements often leads to important advances as better theories are developed. In the physical sciences, a traditional mathematical model contains most of the following elements: Governing equations


id: mathematical-model-20260505-1
file: data/en.wikipedia.org/wiki/Mathematical_model-1.md
category: reference
title: Mathematical model
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Mathematical_model
date_saved: 2026-05-05T07:24:19.245253+00:00
instance: kb-cron
parent: mathematical-model-20260505-0
summary:
== Construction == In business and engineering, mathematical models may be used to maximize a certain output. The system under consideration will require certain inputs. The system relating inputs to outputs depends on other variables too: decision variables, state variables, exogenous variables, and random variables. Decision variables are sometimes known as independent variables. Exogenous variables are sometimes known as parameters or constants. The variables are not independent of each other as the state variables are dependent on the decision, input, random, and exogenous variables. Furthermore, the output variables are dependent on the state of the system (represented by the state variables). Objectives and constraints of the system and its users can be represented as functions of the output variables or state variables. The objective functions will depend on the perspective of the model's user. Depending on the context, an objective function is also known as an index of performance, as it is some measure of interest to the user. Although there is no limit to the number of objective functions and constraints a model can have, using or optimizing the model becomes more involved (computationally) as the number increases. For example, economists often apply linear algebra when using input–output models.


id: mathematical-model-20260505-2
file: data/en.wikipedia.org/wiki/Mathematical_model-2.md
category: reference
title: Mathematical model
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Mathematical_model
date_saved: 2026-05-05T07:24:19.245253+00:00
instance: kb-cron
parent: mathematical-model-20260505-0
summary:
=== Complexity === In general, model complexity involves a trade-off between simplicity and accuracy of the model. Occam's razor is a principle particularly relevant to modeling, its essential idea being that among models with roughly equal predictive power, the simplest one is the most desirable. While added complexity usually improves the realism of a model, it can make the model difficult to understand and analyze, and can also pose computational problems, including numerical instability. Thomas Kuhn argues that as science progresses, explanations tend to become more complex before a paradigm shift offers radical simplification. For example, when modeling the flight of an aircraft, we could embed each mechanical part of the aircraft into our model and would thus acquire an almost white-box model of the system. However, the computational cost of adding such a huge amount of detail would effectively inhibit the usage of such a model. Additionally, the uncertainty would increase due to an overly complex system, because each separate part induces some amount of variance into the model. It is therefore usually appropriate to make some approximations to reduce the model to a sensible size. Engineers often can accept some approximations in order to get a more robust


id: mathematical-model-20260505-3
file: data/en.wikipedia.org/wiki/Mathematical_model-3.md
category: reference
title: Mathematical model
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Mathematical_model
date_saved: 2026-05-05T07:24:19.245253+00:00
instance: kb-cron
parent: mathematical-model-20260505-0
summary:
== Significance in the natural sciences == Mathematical models are of great importance in the natural sciences, particularly in physics. Physical theories are almost invariably expressed using mathematical models. Throughout history, more and more accurate mathematical models have been developed. Newton's laws accurately describe many everyday phenomena, but at certain limits theory of relativity and quantum mechanics must be used. It is common to use idealized models in physics to simplify things. Massless ropes, point particles, ideal gases and the particle in a box are among the many simplified models used in physics. The laws of physics are represented with simple equations such as Newton's laws, Maxwell's equations and the Schrödinger equation. These laws are a basis for making mathematical models of real situations. Many real situations are very complex and thus modeled approximately on a computer, a model that is computationally feasible to compute is made from the basic laws or from approximate models made from the basic laws. For example, molecules can be modeled by molecular orbital models that are approximate solutions to the Schrödinger equation. In engineering, physics models are often made by mathematical methods such as finite element analysis. Different mathematical models use different geometries that


id: mathematical-model-20260505-4
file: data/en.wikipedia.org/wiki/Mathematical_model-4.md
category: reference
title: Mathematical model
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Mathematical_model
date_saved: 2026-05-05T07:24:19.245253+00:00
instance: kb-cron
parent: mathematical-model-20260505-0
summary:
The state S 1 {\displaystyle S_{1}} represents that there has been an even number of 0s in the input so far, while S 2 {\displaystyle S_{2}} signifies an odd number. A 1 in the input does not change the state of the automaton. When the input ends, the state will show whether the input contained an even number of 0s or not. If the input did contain an even number of 0s, M {\displaystyle M} will finish in state S 1 , {\displaystyle S_{1},} an accepting state, so the input string will be accepted. The language recognized by M {\displaystyle M} is the regular language given by the regular expression 1*( 0 (1*) 0 (1*) )*, where "*" is the Kleene star, e.g., 1* denotes any non-negative number (possibly zero) of symbols "1". Many everyday activities carried out without a thought are uses of mathematical models. A geographical map projection of a region of the earth onto a small, plane surface is a model which can be used for many purposes such as planning travel. Another simple activity is predicting the position of a vehicle from its initial position, direction and speed of travel, using the equation that distance traveled


id: metatheorem-20260505-0
file: data/en.wikipedia.org/wiki/Metatheorem-0.md
category: reference
title: Metatheorem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Metatheorem
date_saved: 2026-05-05T07:24:20.538671+00:00
instance: kb-cron
parent: metatheorem-20260505-0
summary:
In logic, a metatheorem is a statement about a formal system proven in a metalanguage. Unlike theorems proved within a given formal system, a metatheorem is proved within a metatheory, and may reference concepts that are present in the metatheory but not the object theory. A formal system is determined by a formal language and a deductive system (axioms and rules of inference). The formal system can be used to prove particular sentences of the formal language with that system. Metatheorems, however, are proved externally to the system in question, in its metatheory. Common metatheories used in logic are set theory (especially in model theory) and primitive recursive arithmetic (especially in proof theory). Rather than demonstrating particular sentences to be provable, metatheorems may show that each of a broad class of sentences can be proved, or show that certain sentences cannot be proved. == Examples == Examples of metatheorems include: The deduction theorem for first-order logic says that a sentence of the form φ→ψ is provable from a set of axioms A if and only if the sentence ψ is provable from the system whose axioms consist of φ and all the axioms of A. The class existence theorem of


id: minimal-counterexample-20260505-0
file: data/en.wikipedia.org/wiki/Minimal_counterexample-0.md
category: reference
title: Minimal counterexample
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Minimal_counterexample
date_saved: 2026-05-05T07:24:21.770105+00:00
instance: kb-cron
parent: minimal-counterexample-20260505-0
summary:
In mathematics, a minimal counterexample is the smallest example which falsifies a claim. It is also sometimes called a minimal criminal, smallest criminal, or least criminal, especially (but not exclusively) in the context of the four-color theorem. A proof by minimal counterexample (or by minimal/smallest/least criminal) is a method of proof which combines the use of a minimal counterexample with the methods of proof by induction and proof by contradiction. More specifically, in trying to prove a proposition P, one first assumes by contradiction that it is false, and that therefore there must be at least one counterexample. With respect to some idea of size (which may need to be chosen carefully), one then concludes that there is such a counterexample C that is minimal. In regard to the argument, C is generally something quite hypothetical (since the truth of P excludes the possibility of C), but it may be possible to argue that if C existed, then it would have some definite properties which, after applying some reasoning similar to that in an inductive proof, would lead to a contradiction, thereby showing that the proposition P is indeed true. If the form of the contradiction is that we can


id: modulo-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Modulo_(mathematics)-0.md
category: reference
title: Modulo (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Modulo_(mathematics)
date_saved: 2026-05-05T07:24:33.217058+00:00
instance: kb-cron
parent: modulo-mathematics-20260505-0
summary:
In mathematics, the term modulo ("with respect to a modulus of", the Latin ablative of modulus which itself means "a small measure") is often used to assert that two distinct mathematical objects can be regarded as equivalent—if their difference is accounted for by an additional factor. It was initially introduced into mathematics in the context of modular arithmetic by Carl Friedrich Gauss in 1801. Since then, the term has gained many meanings—some exact and some imprecise (such as equating "modulo" with "except for"). For the most part, the term often occurs in statements of the form: A is the same as B modulo C which is often equivalent to "A is the same as B up to C", and means A and B are the same—except for differences accounted for or explained by C. == History == Modulo is a mathematical jargon that was introduced into mathematics in the book Disquisitiones Arithmeticae by Carl Friedrich Gauss in 1801. Given the integers a, b and n, the expression "a ≡ b (mod n)", pronounced "a is congruent to b modulo n", means that a − b is an integer multiple of n, or equivalently, a and b both share the same


id: necessity-and-sufficiency-20260505-0
file: data/en.wikipedia.org/wiki/Necessity_and_sufficiency-0.md
category: reference
title: Necessity and sufficiency
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Necessity_and_sufficiency
date_saved: 2026-05-05T07:24:34.458153+00:00
instance: kb-cron
parent: necessity-and-sufficiency-20260505-0
summary:
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If P then Q", Q is necessary for P, because the truth of Q is "necessarily" guaranteed by the truth of P. (Equivalently, it is impossible to have P without Q, or the falsity of Q ensures the falsity of P.) Similarly, P is sufficient for Q, because P being true always or "sufficiently" implies that Q is true, but P not being true does not always imply that Q is not true. In general, a necessary condition is one (possibly one of several conditions) that must be present in order for another condition to occur, while a sufficient condition is one that produces the said condition. The assertion that a statement is a "necessary and sufficient" condition of another means that the former statement is true if and only if the latter is true. That is, the two statements must be either simultaneously true, or simultaneously false. In ordinary English (also natural language) "necessary" and "sufficient" often indicate relations between conditions or states of affairs, not statements. For example, being round is a


id: necessity-and-sufficiency-20260505-1
file: data/en.wikipedia.org/wiki/Necessity_and_sufficiency-1.md
category: reference
title: Necessity and sufficiency
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Necessity_and_sufficiency
date_saved: 2026-05-05T07:24:34.458153+00:00
instance: kb-cron
parent: necessity-and-sufficiency-20260505-0
summary:
Example 1 For it to be true that "John is a bachelor", it is necessary that it be also true that he is unmarried, male, adult, since to state "John is a bachelor" implies John has each of those three additional predicates. Example 2 For the whole numbers greater than two, being odd is necessary to being prime, since two is the only whole number that is both even and prime. Example 3 Consider thunder, the sound caused by lightning. One says that thunder is necessary for lightning, since lightning never occurs without thunder. Whenever there is lightning, there is thunder. The thunder does not cause the lightning (since lightning causes thunder), but because lightning always comes with thunder, we say that thunder is necessary for lightning. (That is, in its formal sense, necessity doesn't imply causality.) Example 4 Being at least 30 years old is necessary for serving in the U.S. Senate. If you are under 30 years old, then it is impossible for you to be a senator. That is, if you are a senator, it follows that you must be at least 30 years old. Example 5 In algebra, for some set S together with an operation


id: necessity-and-sufficiency-20260505-2
file: data/en.wikipedia.org/wiki/Necessity_and_sufficiency-2.md
category: reference
title: Necessity and sufficiency
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Necessity_and_sufficiency
date_saved: 2026-05-05T07:24:34.458153+00:00
instance: kb-cron
parent: necessity-and-sufficiency-20260505-0
summary:
that P is necessary for Q, P ⇐ Q {\displaystyle P\Leftarrow Q} , and that P is sufficient for Q, P ⇒ Q {\displaystyle P\Rightarrow Q} . equivalently, it may be understood to say that P and Q is necessary for the other, P ⇒ Q ∧ Q ⇒ P {\displaystyle P\Rightarrow Q\land Q\Rightarrow P} , which can also be stated as each is sufficient for or implies the other. One may summarize any, and thus all, of these cases by the statement "P if and only if Q", which is denoted by P ⇔ Q {\displaystyle P\Leftrightarrow Q} , whereas cases tell us that P ⇔ Q {\displaystyle P\Leftrightarrow Q} is identical to P ⇒ Q ∧ Q ⇒ P {\displaystyle P\Rightarrow Q\land Q\Rightarrow P} . For example, in graph theory a graph G is called bipartite if it is possible to assign to each of its vertices the color black or white in such a way that every edge of G has one endpoint of each color. And for any graph to be bipartite, it is a necessary and sufficient condition that it contain no odd-length cycles. Thus, discovering whether a graph has any odd cycles tells one


id: null-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Null_(mathematics)-0.md
category: reference
title: Null (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Null_(mathematics)
date_saved: 2026-05-05T07:24:35.707916+00:00
instance: kb-cron
parent: null-mathematics-20260505-0
summary:
In mathematics, the word null (from German: null meaning "zero", which is from Latin: nullus meaning "none") is often associated with the concept of zero, or with the concept of nothing. It is used in varying contexts from "having zero members in a set" (e.g., null set) to "having a value of zero" (e.g., null vector). In a vector space, the null vector is the neutral element of vector addition; depending on the context, a null vector may also be a vector mapped to some null by a function under consideration (such as a quadratic form coming with the vector space, see null vector, a linear mapping given as matrix product or dot product, a seminorm in a Minkowski space, etc.). In set theory, the empty set, that is, the set with zero elements, denoted "{}" or "∅", may also be called null set. In measure theory, a null set is a (possibly nonempty) set with zero measure. A null space of a mapping is the part of the domain that is mapped into the null element of the image (the inverse image of the null element). For example, in linear algebra, the null space of a linear mapping, also


id: of-the-form-20260505-0
file: data/en.wikipedia.org/wiki/Of_the_form-0.md
category: reference
title: Of the form
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Of_the_form
date_saved: 2026-05-05T07:24:37.077999+00:00
instance: kb-cron
parent: of-the-form-20260505-0
summary:
In mathematics, the phrase "of the form" indicates that a mathematical object, or (more frequently) a collection of objects, follows a certain pattern of expression. It is frequently used to reduce the formality of mathematical proofs. == Example of use == Here is a proof which should be appreciable with limited mathematical background: Statement: The product of any two even natural numbers is also even. Proof: Any even natural number is of the form 2n, where n is a natural number. Therefore, let us assume that we have two even numbers which we will denote by 2k and 2l. Their product is (2k)(2l) = 4(kl) = 2(2kl). Since 2kl is also a natural number, the product is even. Note: In this case, both exhaustivity and exclusivity were needed. That is, it was not only necessary that every even number is of the form 2n (exhaustivity), but also that every expression of the form 2n is an even number (exclusivity). This will not be the case in every proof, but normally, at least exhaustivity is implied by the phrase of the form. == References == == External links == Weisstein, Eric W. "Of the Form". MathWorld.


id: order-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Order_(mathematics)-0.md
category: reference
title: Order (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Order_(mathematics)
date_saved: 2026-05-05T07:24:38.280206+00:00
instance: kb-cron
parent: order-mathematics-20260505-0
summary:
Order in mathematics may refer to: == Set theory == Total order and partial order, a binary relation generalizing the usual ordering of numbers and of words in a dictionary Ordered set Order in Ramsey theory, uniform structures in consequence to critical set cardinality == Algebra == Order (group theory), the cardinality of a group or period of an element Order of a polynomial (disambiguation) Order of a square matrix, its dimension Order (ring theory), an algebraic structure Ordered group Ordered field == Analysis == Order (differential equation) or order of highest derivative, of a differential equation Leading-order terms NURBS order, a number one greater than the degree of the polynomial representation of a non-uniform rational B-spline Order of convergence, a measurement of convergence Order of derivation Order of an entire function Order of a power series, the lowest degree of its terms Ordered list, a sequence or tuple Orders of approximation in Big O notation Z-order (curve), a space-filling curve == Arithmetic == Multiplicative order in modular arithmetic Order of operations Orders of magnitude, a class of scale or magnitude of any amount == Combinatorics == Order in the Josephus permutation Ordered selections and partitions of the twelvefold way


id: orthomorphism-20260505-0
file: data/en.wikipedia.org/wiki/Orthomorphism-0.md
category: reference
title: Orthomorphism
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Orthomorphism
date_saved: 2026-05-05T07:24:39.493560+00:00
instance: kb-cron
parent: orthomorphism-20260505-0
summary:
In abstract algebra, an orthomorphism is a certain kind of mapping from a group into itself. Let G be a group, and let θ be a permutation of G. Then θ is an orthomorphism of G if the mapping f defined by f(x) = x−1 θ(x) is also a permutation of G. A permutation φ of G is a complete mapping if the mapping g defined by g(x) = xφ(x) is also a permutation of G. Orthomorphisms and complete mappings are closely related. == References ==


id: parameter-20260505-0
file: data/en.wikipedia.org/wiki/Parameter-0.md
category: reference
title: Parameter
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Parameter
date_saved: 2026-05-05T07:24:40.755713+00:00
instance: kb-cron
parent: parameter-20260505-0
summary:
A parameter (from Ancient Greek παρά (pará) 'beside, subsidiary' and μέτρον (métron) 'measure'), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when identifying the system, or when evaluating its performance, status, condition, etc. Parameter has more specific meanings within various disciplines, including mathematics, computer programming, engineering, statistics, logic, linguistics, and electronic musical composition. In addition to its technical uses, there are also extended uses, especially in non-scientific contexts, where it is used to mean defining characteristics or boundaries, as in the phrases 'test parameters' or 'game play parameters'. == Modelization == When a system is modeled by equations, the values that describe the system are called parameters. For example, in mechanics, the masses, the dimensions and shapes (for solid bodies), the densities and the viscosities (for fluids), appear as parameters in the equations modeling movements. There are often several choices for the parameters, and choosing a convenient set of parameters is called parametrization. For example, if one were considering the movement of an object on the surface of a sphere much larger


id: parameter-20260505-1
file: data/en.wikipedia.org/wiki/Parameter-1.md
category: reference
title: Parameter
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Parameter
date_saved: 2026-05-05T07:24:40.755713+00:00
instance: kb-cron
parent: parameter-20260505-0
summary:
[Kilpatrick quoting Woods] "Now ... the engineers ... change the lever arms of the linkage ... the speed of the car ... will still depend on the pedal position ... but in a ... different manner. You have changed a parameter" A parametric equaliser is an audio filter that allows the frequency of maximum cut or boost to be set by one control, and the size of the cut or boost by another. These settings, the frequency level of the peak or trough, are two of the parameters of a frequency response curve, and in a two-control equaliser they completely describe the curve. More elaborate parametric equalisers may allow other parameters to be varied, such as skew. These parameters each describe some aspect of the response curve seen as a whole, over all frequencies. A graphic equaliser provides individual level controls for various frequency bands, each of which acts only on that particular frequency band. If asked to imagine the graph of the relationship y = ax2, one typically visualizes a range of values of x, but only one value of a. Of course a different value of a can be used, generating a different relation between x and y.


id: parameter-20260505-2
file: data/en.wikipedia.org/wiki/Parameter-2.md
category: reference
title: Parameter
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Parameter
date_saved: 2026-05-05T07:24:40.755713+00:00
instance: kb-cron
parent: parameter-20260505-0
summary:
In statistics and econometrics, the probability framework above still holds, but attention shifts to estimating the parameters of a distribution based on observed data, or testing hypotheses about them. In frequentist estimation parameters are considered "fixed but unknown", whereas in Bayesian estimation they are treated as random variables, and their uncertainty is described as a distribution. In estimation theory of statistics, "statistic" or estimator refers to samples, whereas "parameter" or estimand refers to populations, where the samples are taken from. A statistic is a numerical characteristic of a sample that can be used as an estimate of the corresponding parameter, the numerical characteristic of the population from which the sample was drawn. For example, the sample mean (estimator), denoted X ¯ {\displaystyle {\overline {X}}} , can be used as an estimate of the mean parameter (estimand), denoted μ, of the population from which the sample was drawn. Similarly, the sample variance (estimator), denoted S2, can be used to estimate the variance parameter (estimand), denoted σ2, of the population from which the sample was drawn. (Note that the sample standard deviation (S) is not an unbiased estimate of the population standard deviation (σ): see Unbiased estimation of standard deviation.) It is


id: parameter-20260505-3
file: data/en.wikipedia.org/wiki/Parameter-3.md
category: reference
title: Parameter
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Parameter
date_saved: 2026-05-05T07:24:40.755713+00:00
instance: kb-cron
parent: parameter-20260505-0
summary:
== Engineering == In engineering (especially involving data acquisition) the term parameter sometimes loosely refers to an individual measured item. This usage is not consistent, as sometimes the term channel refers to an individual measured item, with parameter referring to the setup information about that channel. "Speaking generally, properties are those physical quantities which directly describe the physical attributes of the system; parameters are those combinations of the properties which suffice to determine the response of the system. Properties can have all sorts of dimensions, depending upon the system being considered; parameters are dimensionless, or have the dimension of time or its reciprocal." The term can also be used in engineering contexts, however, as it is typically used in the physical sciences. == Environmental science == In environmental science and particularly in chemistry and microbiology, a parameter is used to describe a discrete chemical or microbiological entity that can be assigned a value: commonly a concentration, but may also be a logical entity (present or absent), a statistical result such as a 95 percentile value or in some cases a subjective value. == Linguistics == Within linguistics, the word "parameter" is almost exclusively used to denote a binary switch in


id: parameter-space-20260505-0
file: data/en.wikipedia.org/wiki/Parameter_space-0.md
category: reference
title: Parameter space
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Parameter_space
date_saved: 2026-05-05T07:24:41.985731+00:00
instance: kb-cron
parent: parameter-space-20260505-0
summary:
The parameter space is the space of all possible parameter values that define a particular mathematical model. It is also sometimes called weight space, and is often a subset of finite-dimensional Euclidean space. In statistics, parameter spaces are particularly useful for describing parametric families of probability distributions. They also form the background for parameter estimation. In the case of extremum estimators for parametric models, a certain objective function is maximized or minimized over the parameter space. Theorems of existence and consistency of such estimators require some assumptions about the topology of the parameter space. For instance, compactness of the parameter space, together with continuity of the objective function, suffices for the existence of an extremum estimator. Sometimes, parameters are analyzed to view how they affect their statistical model. In that context, they can be viewed as inputs of a function, in which case the technical term for the parameter space is domain of a function. The ranges of values of the parameters may form the axes of a plot, and particular outcomes of the model may be plotted against these axes to illustrate how different regions of the parameter space produce different types of behavior in the model. == Examples


id: parametric-family-20260505-0
file: data/en.wikipedia.org/wiki/Parametric_family-0.md
category: reference
title: Parametric family
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Parametric_family
date_saved: 2026-05-05T07:24:43.335955+00:00
instance: kb-cron
parent: parametric-family-20260505-0
summary:
In mathematics and its applications, a parametric family or a parameterized family is a family of objects (a set of related objects) whose differences depend only on the chosen values for a set of parameters. Common examples are parametrized (families of) functions, probability distributions, curves, shapes, etc. == In probability and its applications == For example, the probability density function fX of a random variable X may depend on a parameter θ. In that case, the function may be denoted f X ( ⋅ ; θ ) {\displaystyle f_{X}(\cdot \,;\theta )} to indicate the dependence on the parameter θ. θ is not a formal argument of the function as it is considered to be fixed. However, each different value of the parameter gives a different probability density function. Then the parametric family of densities is the set of functions { f X ( ⋅ ; θ ) ∣ θ ∈ Θ } {\displaystyle \{f_{X}(\cdot \,;\theta )\mid \theta \in \Theta \}} , where Θ denotes the parameter space, the set of all possible values that the parameter θ can take. As an example, the normal distribution is a family of similarly shaped distributions parametrized by their mean and their variance. In


id: parts-per-notation-20260505-0
file: data/en.wikipedia.org/wiki/Parts-per_notation-0.md
category: reference
title: Parts-per notation
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Parts-per_notation
date_saved: 2026-05-05T07:24:44.742123+00:00
instance: kb-cron
parent: parts-per-notation-20260505-0
summary:
In science and engineering, parts-per notation is a set of pseudo-units to describe the small values of miscellaneous dimensionless quantities, e.g. mole fraction or mass fraction. Since these fractions are quantity-per-quantity measures, they are pure numbers with no associated units of measurement. Commonly used are parts-per-million – ppm, 10−6 parts-per-billion – ppb, 10−9 parts-per-trillion – ppt, 10−12 parts-per-quadrillion – ppq, 10−15 This notation is not part of the International System of Units (the SI system) and its meaning is ambiguous. In chemistry, ambiguity arises because parts-per notation is able to be used to refer to a mole fraction and a mass fraction, which are unequal depending on the molar mass of the substance. == Applications == Parts-per notation is often used describing dilute solutions in chemistry, for instance, the relative abundance of dissolved minerals or pollutants in water. The quantity "1 ppm" can be used for a mass fraction if a water-borne pollutant is present at one-millionth of a gram per gram of sample solution. When working with aqueous solutions, it is common to assume that the density of water is 1.00 g/mL. Therefore, it is common to equate 1 kilogram of water with 1 L of water. Consequently, 1


id: parts-per-notation-20260505-1
file: data/en.wikipedia.org/wiki/Parts-per_notation-1.md
category: reference
title: Parts-per notation
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Parts-per_notation
date_saved: 2026-05-05T07:24:44.742123+00:00
instance: kb-cron
parent: parts-per-notation-20260505-0
summary:
One part per billion (ppb) denotes one part per 1,000,000,000 (109) parts, and a value of 10−9. This is equivalent to about three seconds out of a century. One part per trillion (ppt) denotes one part per 1,000,000,000,000 (1012) parts, and a value of 10−12. This is equivalent to about thirty seconds out of every million years. One part per quadrillion (ppq) denotes one part per 1,000,000,000,000,000 (1015) parts, and a value of 10−15. This is equivalent to about two and a half minutes out of the age of the Earth (4.5 billion years). Although relatively uncommon in analytical chemistry, measurements at the ppq level are sometimes performed. == Criticism == Although the International Bureau of Weights and Measures (an international standards organization known also by its French-language initials BIPM) recognizes the use of parts-per notation, it is not formally part of the International System of Units (SI). Note that although "percent" (%) is not formally part of the SI, both the BIPM and the International Organization for Standardization (ISO) take the position that "in mathematical expressions, the internationally recognized symbol % (percent) may be used with the SI to represent the number 0.01" for dimensionless quantities. According to IUPAP,


id: pathological-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Pathological_(mathematics)-0.md
category: reference
title: Pathological (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Pathological_(mathematics)
date_saved: 2026-05-05T07:24:45.941558+00:00
instance: kb-cron
parent: pathological-mathematics-20260505-0
summary:
In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition, it is sometimes called well-behaved or nice. These terms are sometimes useful in mathematical research and teaching, but there is no strict mathematical definition of pathological or well-behaved. == In analysis == A classic example of a pathology is the Weierstrass function, a function that is continuous everywhere but differentiable nowhere. The sum of a differentiable function and the Weierstrass function is again continuous but nowhere differentiable; so there are at least as many such functions as differentiable functions. In fact, using the Baire category theorem, one can show that continuous functions are generically nowhere differentiable. Such examples were deemed pathological when they were first discovered. To quote Henri Poincaré: Logic sometimes breeds monsters. For half a century there has been springing up a host of weird functions, which seem to strive to have as little resemblance as possible to honest functions that are of some use. No more continuity, or else continuity but no derivatives, etc. More than this, from the point of view of logic, it is


id: pathological-mathematics-20260505-1
file: data/en.wikipedia.org/wiki/Pathological_(mathematics)-1.md
category: reference
title: Pathological (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Pathological_(mathematics)
date_saved: 2026-05-05T07:24:45.941558+00:00
instance: kb-cron
parent: pathological-mathematics-20260505-0
summary:
== Well-behaved == Mathematicians (and those in related sciences) very frequently speak of whether a mathematical object—a function, a set, a space of one sort or another—is "well-behaved". While the term has no fixed formal definition, it generally refers to the quality of satisfying a list of prevailing conditions, which might be dependent on context, mathematical interests, fashion, and taste. To ensure that an object is "well-behaved", mathematicians introduce further axioms to narrow down the domain of study. This has the benefit of making analysis easier, but produces a loss of generality of any conclusions reached. In both pure and applied mathematics (e.g., optimization, numerical integration, mathematical physics), well-behaved also means not violating any assumptions needed to successfully apply whatever analysis is being discussed. The opposite case is usually labeled "pathological". It is not unusual to have situations in which most cases (in terms of cardinality or measure) are pathological, but the pathological cases will not arise in practice—unless constructed deliberately. The term "well-behaved" is generally applied in an absolute sense—either something is well-behaved or it is not. For example: In algorithmic inference, a well-behaved statistic is monotonic, well-defined, and sufficient. In Bézout's theorem, two polynomials are well-behaved, and thus


id: pathological-mathematics-20260505-2
file: data/en.wikipedia.org/wiki/Pathological_(mathematics)-2.md
category: reference
title: Pathological (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Pathological_(mathematics)
date_saved: 2026-05-05T07:24:45.941558+00:00
instance: kb-cron
parent: pathological-mathematics-20260505-0
summary:
The Cantor set is a subset of the interval [ 0 , 1 ] {\displaystyle [0,1]} that has measure zero but is uncountable. The fat Cantor set is nowhere dense but has positive measure. The Fabius function is everywhere smooth but nowhere analytic. Volterra's function is differentiable with bounded derivative everywhere, but the derivative is not Riemann-integrable. The Peano space-filling curve is a continuous surjective function that maps the unit interval [ 0 , 1 ] {\displaystyle [0,1]} onto [ 0 , 1 ] × [ 0 , 1 ] {\displaystyle [0,1]\times [0,1]} . The Dirichlet function, which is the indicator function for rationals, is a bounded function that is not Riemann integrable. The Cantor function is a monotonic continuous surjective function that maps [ 0 , 1 ] {\displaystyle [0,1]} onto [ 0 , 1 ] {\displaystyle [0,1]} , but has zero derivative almost everywhere. The Minkowski question-mark function is continuous and strictly increasing but has zero derivative almost everywhere. Satisfaction classes containing "intuitively false" arithmetical statements can be constructed for countable, recursively saturated models of Peano arithmetic. The Osgood curve is a Jordan curve (unlike most space-filling curves) of positive area. An exotic sphere is homeomorphic but not


id: percentage-point-20260505-0
file: data/en.wikipedia.org/wiki/Percentage_point-0.md
category: reference
title: Percentage point
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Percentage_point
date_saved: 2026-05-05T07:24:47.195404+00:00
instance: kb-cron
parent: percentage-point-20260505-0
summary:
A percentage point or percent point is the description for the arithmetic difference between two percentages. For example, moving up from 40 percent to 44 percent is an increase of 4 percentage points (although it is a 10-percent increase in the quantity being measured, if the total amount remains the same). In written text, the unit (the percentage point) is usually either written out, or abbreviated as pp, p.p., or %pt. to avoid confusion with percentage increase or decrease in the actual quantity. After the first occurrence, some writers abbreviate by using just "point" or "points". == Differences between percentages and percentage points == Consider the following hypothetical example: In 1980, 50 percent of the population smoked, and in 1990 only 40 percent of the population smoked. One can thus say that from 1980 to 1990, the prevalence of smoking decreased by 10 percentage points (or by 10 percent of the population) or by 20 percent when talking about smokers only – percentages indicate proportionate part of a total. Percentage-point differences are one way to express a risk or probability. Consider a drug that cures a given disease in 70 percent of all cases, while without the drug, the disease


id: pointwise-20260505-0
file: data/en.wikipedia.org/wiki/Pointwise-0.md
category: reference
title: Pointwise
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Pointwise
date_saved: 2026-05-05T07:24:48.491777+00:00
instance: kb-cron
parent: pointwise-20260505-0
summary:
In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f ( x ) {\displaystyle f(x)} of some function f . {\displaystyle f.} An important class of pointwise concepts are the pointwise operations, that is, operations defined on functions by applying the operations to function values separately for each point in the domain of definition. Important relations can also be defined pointwise. == Pointwise operations == === Formal definition === A binary operation o: Y × Y → Y on a set Y can be lifted pointwise to an operation O: (X→Y) × (X→Y) → (X→Y) on the set X → Y of all functions from X to Y as follows: Given two functions f1: X → Y and f2: X → Y, define the function O(f1, f2): X → Y by Commonly, o and O are denoted by the same symbol. A similar definition is used for unary operations o, and for operations of other arity. === Examples === The pointwise addition f + g {\displaystyle f+g} of two functions f {\displaystyle f} and g {\displaystyle g} with the same domain and codomain is defined by: The pointwise product or


id: porism-20260505-0
file: data/en.wikipedia.org/wiki/Porism-0.md
category: reference
title: Porism
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Porism
date_saved: 2026-05-05T07:24:49.702070+00:00
instance: kb-cron
parent: porism-20260505-0
summary:
A porism is a mathematical proposition or corollary. It has been used to refer to a direct consequence of a proof, analogous to how a corollary refers to a direct consequence of a theorem. In modern usage, it is a relationship that holds for an infinite range of values but only if a certain condition is assumed, such as Steiner's porism. The term originates from three books of Euclid that have been lost. A proposition may not have been proven, so a porism may not be a theorem or true. == Origins == The book that talks about porisms first is Euclid's Porisms. What is known of it is in Pappus of Alexandria's Collection, who mentions it along with other geometrical treatises, and gives several lemmas necessary for understanding it. Pappus states: The porisms of all classes are neither theorems nor problems, but occupy a position intermediate between the two, so that their enunciations can be stated either as theorems or problems, and consequently some geometers think that they are theorems, and others that they are problems, being guided solely by the form of the enunciation. But it is clear from the definitions that the old geometers understood better the


id: porism-20260505-1
file: data/en.wikipedia.org/wiki/Porism-1.md
category: reference
title: Porism
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Porism
date_saved: 2026-05-05T07:24:49.702070+00:00
instance: kb-cron
parent: porism-20260505-0
summary:
under certain conditions a problem becomes impossible; under certain other conditions, indeterminate or capable of an infinite number of solutions. These cases could be defined separately, were in a manner intermediate between theorems and problems, and were called "porisms." Playfair defined a porism as "[a] proposition affirming the possibility of finding such conditions as will render a certain problem indeterminate or capable of innumerable solutions." Although Playfair's definition of a porism appears to be most favoured in England, Simson's view has been most generally accepted abroad, and had the support of Michel Chasles. However, in Liouville's Journal de mathematiques pures et appliquées (vol. xx., July, 1855), P. Breton published Recherches nouvelles sur les porismes d'Euclide, in which he gave a new translation of the text of Pappus, and sought to base a view of the nature of a porism that conforms more closely to Pappus's definition. This was followed in the same journal and in La Science by a controversy between Breton and A. J. H. Vincent, who disputed the interpretation given by the former of Pappus's text, and declared himself in favour of Frans van Schooten's idea, put forward in his Mathematicae exercitationes (1657). According to Schooten, if the


id: power-root-power-and-field-quantities-20260505-0
file: data/en.wikipedia.org/wiki/Power,_root-power,_and_field_quantities-0.md
category: reference
title: Power, root-power, and field quantities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Power,_root-power,_and_field_quantities
date_saved: 2026-05-05T07:24:50.916584+00:00
instance: kb-cron
parent: power-root-power-and-field-quantities-20260505-0
summary:
A power quantity is a power or a quantity directly proportional to power, e.g., energy density, acoustic intensity, and luminous intensity. Energy quantities may also be labelled as power quantities in this context. A root-power quantity is a quantity such as voltage, current, sound pressure, electric field strength, speed, or charge density, the square of which, in linear systems, is proportional to power. The term root-power quantity refers to the square root that relates these quantities to power. The term was introduced in ISO 80000-1 § Annex C; it replaces and deprecates the term field quantity. == Implications == It is essential to know which category a measurement belongs to when using decibels (dB) for comparing the levels of such quantities. A change of one bel in the level corresponds to a 10× change in power, so when comparing power quantities x and y, the difference is defined to be 10×log10(y/x) decibel. With root-power quantities, however the difference is defined as 20×log10(y/x) dB. In the analysis of signals and systems using sinusoids, field quantities and root-power quantities may be complex-valued, as in the propagation constant. == "Root-power quantity" vs. "field quantity" == In justifying the deprecation of the term "field


id: projection-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Projection_(mathematics)-0.md
category: reference
title: Projection (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Projection_(mathematics)
date_saved: 2026-05-05T07:24:52.122463+00:00
instance: kb-cron
parent: projection-mathematics-20260505-0
summary:
In mathematics, a projection is a mapping from a set to itself—or an endomorphism of a mathematical structure—that is idempotent, that is, equals its composition with itself. The image of a point or a subset ⁠ S {\displaystyle S} ⁠ under a projection is called the projection of ⁠ S {\displaystyle S} ⁠. An everyday example of a projection is the casting of shadows onto a plane (sheet of paper): the projection of a point is its shadow on the sheet of paper, and the projection (shadow) of a point on the sheet of paper is that point itself (idempotency). The shadow of a three-dimensional sphere is a disk. Originally, the notion of projection was introduced in Euclidean geometry to denote the projection of the three-dimensional Euclidean space onto a plane in it, like the shadow example. The two main projections of this kind are: The projection from a point onto a plane or central projection: If C is a point, called the centre of projection, then the projection of a point P different from C onto a plane that does not contain C is the intersection of the line CP with the plane. The points P such that the


id: projection-mathematics-20260505-1
file: data/en.wikipedia.org/wiki/Projection_(mathematics)-1.md
category: reference
title: Projection (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Projection_(mathematics)
date_saved: 2026-05-05T07:24:52.122463+00:00
instance: kb-cron
parent: projection-mathematics-20260505-0
summary:
In set theory: An operation typified by the j-th projection map, written projj, that takes an element x = (x1, ..., xj, ..., xn) of the Cartesian product X1 × ⋯ × Xj × ⋯ × Xn to the value projj(x) = xj. This map is always surjective and, when each space Xk has a topology, this map is also continuous and open. A mapping that takes an element to its equivalence class under a given equivalence relation is known as the canonical projection. The evaluation map sends a function f to the value f(x) for a fixed x. The space of functions YX can be identified with the Cartesian product ∏ i ∈ X Y {\textstyle \prod _{i\in X}Y} , and the evaluation map is a projection map from the Cartesian product. For relational databases and query languages, the projection is a unary operation written as Π a 1 , … , a n ( R ) {\displaystyle \Pi _{a_{1},\ldots ,a_{n}}(R)} where a 1 , … , a n {\displaystyle a_{1},\ldots ,a_{n}} is a set of attribute names. The result of such projection is defined as the set that is obtained when all tuples in R are restricted to


id: mathematical-proof-20260505-0
file: data/en.wikipedia.org/wiki/Mathematical_proof-0.md
category: reference
title: Mathematical proof
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Mathematical_proof
date_saved: 2026-05-05T07:24:53.330614+00:00
instance: kb-cron
parent: mathematical-proof-20260505-0
summary:
A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning that establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning that establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work. Proofs employ logic expressed in mathematical symbols, along with natural language that usually admits some ambiguity. In most mathematical literature, proofs are written in terms of rigorous informal logic. Purely formal proofs, written fully in symbolic language without the involvement of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current


id: mathematical-proof-20260505-1
file: data/en.wikipedia.org/wiki/Mathematical_proof-1.md
category: reference
title: Mathematical proof
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Mathematical_proof
date_saved: 2026-05-05T07:24:53.330614+00:00
instance: kb-cron
parent: mathematical-proof-20260505-0
summary:
In direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier theorems. For example, direct proof can be used to prove that the sum of two even integers is always even: Consider two even integers x and y. Since they are even, they can be written as x = 2a and y = 2b, respectively, for some integers a and b. Then the sum is x + y = 2a + 2b = 2(a+b). Therefore x+y has 2 as a factor and, by definition, is even. Hence, the sum of any two even integers is even. This proof uses the definition of even integers, the integer properties of closure under addition and multiplication, and the distributive property. === Proof by mathematical induction === Despite its name, mathematical induction is a method of deduction, not a form of inductive reasoning. In proof by mathematical induction, a single "base case" is proved, and an "induction rule" is proved that establishes that any arbitrary case implies the next case. Since in principle the induction rule can be applied repeatedly (starting from the proven base case), it follows that all (usually infinitely many) cases are provable. This avoids having to


id: mathematical-proof-20260505-2
file: data/en.wikipedia.org/wiki/Mathematical_proof-2.md
category: reference
title: Mathematical proof
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Mathematical_proof
date_saved: 2026-05-05T07:24:53.330614+00:00
instance: kb-cron
parent: mathematical-proof-20260505-0
summary:
In proof by exhaustion, the conclusion is established by dividing it into a finite number of cases and proving each one separately. The number of cases sometimes can become very large. For example, the first proof of the four color theorem was a proof by exhaustion with 1,936 cases. This proof was controversial because the majority of the cases were checked by a computer program, not by hand. === Closed chain inference === A closed chain inference shows that a collection of statements are pairwise equivalent. In order to prove that the statements φ 1 , … , φ n {\displaystyle \varphi _{1},\ldots ,\varphi _{n}} are each pairwise equivalent, proofs are given for the implications φ 1 ⇒ φ 2 {\displaystyle \varphi _{1}\Rightarrow \varphi _{2}} , φ 2 ⇒ φ 3 {\displaystyle \varphi _{2}\Rightarrow \varphi _{3}} , … {\displaystyle \dots } , φ n − 1 ⇒ φ n {\displaystyle \varphi _{n-1}\Rightarrow \varphi _{n}} and φ n ⇒ φ 1 {\displaystyle \varphi _{n}\Rightarrow \varphi _{1}} . The pairwise equivalence of the statements then results from the transitivity of the material conditional. === Probabilistic proof === A probabilistic proof is one in which an example is shown to exist, with certainty,


id: mathematical-proof-20260505-3
file: data/en.wikipedia.org/wiki/Mathematical_proof-3.md
category: reference
title: Mathematical proof
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Mathematical_proof
date_saved: 2026-05-05T07:24:53.330614+00:00
instance: kb-cron
parent: mathematical-proof-20260505-0
summary:
== Undecidable statements == A statement that is neither provable nor disprovable from a set of axioms is called undecidable (from those axioms). One example is the parallel postulate, which is neither provable nor refutable from the remaining axioms of Euclidean geometry. Mathematicians have shown there are many statements that are neither provable nor disprovable in Zermelo–Fraenkel set theory with the axiom of choice (ZFC), the standard system of set theory in mathematics (assuming that ZFC is consistent); see List of statements undecidable in ZFC. Gödel's (first) incompleteness theorem shows that many axiom systems of mathematical interest will have undecidable statements. == Heuristic mathematics and experimental mathematics == While early mathematicians such as Eudoxus of Cnidus did not use proofs, from Euclid to the foundational mathematics developments of the late 19th and 20th centuries, proofs were an essential part of mathematics. With the increase in computing power in the 1960s, significant work began to be done investigating mathematical objects beyond the proof-theorem framework, in experimental mathematics. Early pioneers of these methods intended the work ultimately to be resolved into a classical proof-theorem framework, e.g. the early development of fractal geometry, which was ultimately so resolved. == Related concepts == ===


id: universal-property-20260505-0
file: data/en.wikipedia.org/wiki/Universal_property-0.md
category: reference
title: Universal property
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Universal_property
date_saved: 2026-05-05T07:24:54.601173+00:00
instance: kb-cron
parent: universal-property-20260505-0
summary:
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently from the method chosen for constructing them. For example, the definitions of the integers from the natural numbers, of the rational numbers from the integers, of the real numbers from the rational numbers, and of polynomial rings from the field of their coefficients can all be done in terms of universal properties. In particular, the concept of universal property allows a simple proof that all constructions of real numbers are equivalent: it suffices to prove that they satisfy the same universal property. Technically, a universal property is defined in terms of categories and functors by means of a universal morphism (see § Formal definition, below). Universal morphisms can also be thought more abstractly as initial or terminal objects of a comma category (see § Connection with comma categories, below). Universal properties occur almost everywhere in mathematics, and the use of the concept allows the use of general properties of universal properties for easily proving some properties that would need boring verifications otherwise. For example, given


id: universal-property-20260505-1
file: data/en.wikipedia.org/wiki/Universal_property-1.md
category: reference
title: Universal property
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Universal_property
date_saved: 2026-05-05T07:24:54.601173+00:00
instance: kb-cron
parent: universal-property-20260505-0
summary:
== Connection with comma categories == Universal morphisms can be described more concisely as initial and terminal objects in a comma category (i.e. one where morphisms are seen as objects in their own right). Let F : C → D {\displaystyle F:{\mathcal {C}}\to {\mathcal {D}}} be a functor and X {\displaystyle X} an object of D {\displaystyle {\mathcal {D}}} . Then recall that the comma category ( X ↓ F ) {\displaystyle (X\downarrow F)} is the category where Objects are pairs of the form ( B , f : X → F ( B ) ) {\displaystyle (B,f:X\to F(B))} , where B {\displaystyle B} is an object in C {\displaystyle {\mathcal {C}}} A morphism from ( B , f : X → F ( B ) ) {\displaystyle (B,f:X\to F(B))} to ( B ′ , f ′ : X → F ( B ′ ) ) {\displaystyle (B',f':X\to F(B'))} is given by a morphism h : B → B ′ {\displaystyle h:B\to B'} in C {\displaystyle {\mathcal {C}}} such that the diagram commutes: Now suppose that the object ( A , u : X → F ( A ) ) {\displaystyle (A,u:X\to F(A))} in ( X ↓ F ) {\displaystyle


id: universal-property-20260505-2
file: data/en.wikipedia.org/wiki/Universal_property-2.md
category: reference
title: Universal property
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Universal_property
date_saved: 2026-05-05T07:24:54.601173+00:00
instance: kb-cron
parent: universal-property-20260505-0
summary:
such that for any other object Z {\displaystyle Z} of C {\displaystyle {\mathcal {C}}} and morphisms f : Z → X {\displaystyle f:Z\to X} and g : Z → Y {\displaystyle g:Z\to Y} there exists a unique morphism h : Z → X × Y {\displaystyle h:Z\to X\times Y} such that f = π 1 ∘ h {\displaystyle f=\pi _{1}\circ h} and g = π 2 ∘ h {\displaystyle g=\pi _{2}\circ h} . To understand this characterization as a universal property, take the category D {\displaystyle {\mathcal {D}}} to be the product category C × C {\displaystyle {\mathcal {C}}\times {\mathcal {C}}} and define the diagonal functor Δ : C → C × C {\displaystyle \Delta :{\mathcal {C}}\to {\mathcal {C}}\times {\mathcal {C}}} by Δ ( X ) = ( X , X ) {\displaystyle \Delta (X)=(X,X)} and Δ ( f : X → Y ) = ( f , f ) {\displaystyle \Delta (f:X\to Y)=(f,f)} . Then ( X × Y , ( π 1 , π 2 ) ) {\displaystyle (X\times Y,(\pi _{1},\pi _{2}))} is a universal morphism from Δ {\displaystyle \Delta } to the object ( X , Y ) {\displaystyle (X,Y)} of C × C {\displaystyle {\mathcal {C}}\times


id: universal-property-20260505-3
file: data/en.wikipedia.org/wiki/Universal_property-3.md
category: reference
title: Universal property
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Universal_property
date_saved: 2026-05-05T07:24:54.601173+00:00
instance: kb-cron
parent: universal-property-20260505-0
summary:
k : A → A ′ {\displaystyle k:A\to A'} such that u ′ = F ( k ) ∘ u {\displaystyle u'=F(k)\circ u} . This is easily seen by substituting ( A , u ′ ) {\displaystyle (A,u')} in the definition of a universal morphism. It is the pair ( A , u ) {\displaystyle (A,u)} which is essentially unique in this fashion. The object A {\displaystyle A} itself is only unique up to isomorphism. Indeed, if ( A , u ) {\displaystyle (A,u)} is a universal morphism and k : A → A ′ {\displaystyle k:A\to A'} is any isomorphism then the pair ( A ′ , u ′ ) {\displaystyle (A',u')} , where u ′ = F ( k ) ∘ u {\displaystyle u'=F(k)\circ u} is also a universal morphism. === Equivalent formulations === The definition of a universal morphism can be rephrased in a variety of ways. Let F : C → D {\displaystyle F:{\mathcal {C}}\to {\mathcal {D}}} be a functor and let X {\displaystyle X} be an object of D {\displaystyle {\mathcal {D}}} . Then the following statements are equivalent: ( A , u ) {\displaystyle (A,u)} is a universal morphism from X {\displaystyle X} to


id: universal-property-20260505-4
file: data/en.wikipedia.org/wiki/Universal_property-4.md
category: reference
title: Universal property
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Universal_property
date_saved: 2026-05-05T07:24:54.601173+00:00
instance: kb-cron
parent: universal-property-20260505-0
summary:
For each object X {\displaystyle X} in C {\displaystyle {\mathcal {C}}} , ( F ( X ) , η X ) {\displaystyle (F(X),\eta _{X})} is a universal morphism from X {\displaystyle X} to G {\displaystyle G} . That is, for all f : X → G ( Y ) {\displaystyle f:X\to G(Y)} there exists a unique g : F ( X ) → Y {\displaystyle g:F(X)\to Y} for which the following diagrams commute. For each object Y {\displaystyle Y} in D {\displaystyle {\mathcal {D}}} , ( G ( Y ) , ϵ Y ) {\displaystyle (G(Y),\epsilon _{Y})} is a universal morphism from F {\displaystyle F} to Y {\displaystyle Y} . That is, for all g : F ( X ) → Y {\displaystyle g:F(X)\to Y} there exists a unique f : X → G ( Y ) {\displaystyle f:X\to G(Y)} for which the following diagrams commute. Universal constructions are more general than adjoint functor pairs: a universal construction is like an optimization problem; it gives rise to an adjoint pair if and only if this problem has a solution for every object of C {\displaystyle {\mathcal {C}}} (equivalently, every object of D {\displaystyle {\mathcal {D}}} ). == History == Universal


id: proportion-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Proportion_(mathematics)-0.md
category: reference
title: Proportion (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Proportion_(mathematics)
date_saved: 2026-05-05T07:24:55.781739+00:00
instance: kb-cron
parent: proportion-mathematics-20260505-0
summary:
A proportion is a mathematical statement expressing equality of two ratios. a : b = c : d {\displaystyle a:b=c:d} a and d are called extremes, b and c are called means. Proportion can be written as a b = c d {\displaystyle {\frac {a}{b}}={\frac {c}{d}}} , where ratios are expressed as fractions. Such a proportion is known as geometrical proportion, not to be confused with arithmetical proportion and harmonic proportion. == Properties of proportions == Fundamental rule of proportion. This rule is sometimes called Means‐Extremes Property. If the ratios are expressed as fractions, then the same rule can be phrased in terms of the equality of "cross-products" and is called Cross‐Products Property. If a b = c d {\displaystyle \ {\frac {a}{b}}={\frac {c}{d}}} , then a d = b c {\displaystyle \ ad=bc} If a b = c d {\displaystyle \ {\frac {a}{b}}={\frac {c}{d}}} , then b a = d c {\displaystyle \ {\frac {b}{a}}={\frac {d}{c}}} If a b = c d {\displaystyle \ {\frac {a}{b}}={\frac {c}{d}}} , then a c = b d {\displaystyle \ {\frac {a}{c}}={\frac {b}{d}}} , d b = c a {\displaystyle \ {\frac {d}{b}}={\frac {c}{a}}} . If a b = c d {\displaystyle \


id: proportionality-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Proportionality_(mathematics)-0.md
category: reference
title: Proportionality (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Proportionality_(mathematics)
date_saved: 2026-05-05T07:24:56.943613+00:00
instance: kb-cron
parent: proportionality-mathematics-20260505-0
summary:
In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio. The ratio is called coefficient of proportionality (or proportionality constant) and its reciprocal is known as constant of normalization (or normalizing constant). Two sequences are inversely proportional if corresponding elements have a constant product. Two functions f ( x ) {\displaystyle f(x)} and g ( x ) {\displaystyle g(x)} are proportional if their ratio f ( x ) g ( x ) {\textstyle {\frac {f(x)}{g(x)}}} is a constant function. If several pairs of variables share the same direct proportionality constant, the equation expressing the equality of these ratios is called a proportion, e.g., ⁠a/b⁠ = ⁠x/y⁠ = ⋯ = k (for details see Ratio). Proportionality is closely related to linearity. == Direct proportionality == Given an independent variable x and a dependent variable y, y is directly proportional to x if there is a positive constant k such that: y = k x . {\displaystyle y=kx.} The relation is often denoted using the symbols ∝ (not to be confused with the Greek letter alpha) or ~, with exception of Japanese texts, where ~ is reserved for intervals: y


id: q-e-d-20260505-0
file: data/en.wikipedia.org/wiki/Q.E.D.-0.md
category: reference
title: Q.E.D.
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Q.E.D.
date_saved: 2026-05-05T07:24:58.133655+00:00
instance: kb-cron
parent: q-e-d-20260505-0
summary:
Q.E.D. or QED is an initialism of the Latin phrase quod erat demonstrandum, meaning "that which was to be demonstrated". Literally, it states "what was to be shown". Traditionally, the abbreviation is placed at the end of mathematical proofs and philosophical arguments in print publications, to indicate that the proof or the argument is complete. == Etymology and early use == The phrase quod erat demonstrandum is a translation into Latin from the Greek ὅπερ ἔδει δεῖξαι (hoper edei deixai; abbreviated as ΟΕΔ). The meaning of the Latin phrase is "that [thing] which was to be demonstrated" (with demonstrandum in the gerundive). The Greek phrase was used by many early Greek mathematicians, including Euclid and Archimedes. The Latin phrase is attested in a 1501 Euclid translation of Giorgio Valla. Its abbreviation q.e.d. is used once in 1598 by Johannes Praetorius, more in 1643 by Anton Deusing, extensively in 1655 by Isaac Barrow in the form Q.E.D., and subsequently by many post-Renaissance mathematicians and philosophers. == Modern philosophy == During the European Renaissance, scholars often wrote in Latin, and phrases such as Q.E.D. were often used to conclude proofs. Perhaps the most famous use of Q.E.D. in a philosophical argument is


id: quadrature-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Quadrature_(mathematics)-0.md
category: reference
title: Quadrature (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Quadrature_(mathematics)
date_saved: 2026-05-05T07:24:59.521729+00:00
instance: kb-cron
parent: quadrature-mathematics-20260505-0
summary:
In mathematics, quadrature is a historic term for the computation of areas and is thus used for computation of integrals. The word is derived from the Latin quadratus meaning "square". The reason is that, for Ancient Greek mathematicians, the computation of an area consisted of constructing a square of the same area. In this sense, the modern term is squaring. For example, the quadrature of the circle, (or squaring the circle) is a famous old problem that has been shown, in the 19th century, to be impossible with the methods available to the Ancient Greeks. Integral calculus, introduced in the 17th century, is a general method for computation of areas. Quadrature came to refer to the computation of any integral; such a computation is presently called more often "integral" or "integration". However, the computation of solutions of differential equations and differential systems is also called integration, and quadrature remains useful for distinguishing integrals from solutions of differential equations, in contexts where both problems are considered. This is the case in numerical analysis; see numerical quadrature. Also, reduction to quadratures and solving by quadratures means expressing solutions of differential equations in terms of integrals. The remainder of this article is devoted


id: qualitative-property-20260505-0
file: data/en.wikipedia.org/wiki/Qualitative_property-0.md
category: reference
title: Qualitative property
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Qualitative_property
date_saved: 2026-05-05T07:25:00.803798+00:00
instance: kb-cron
parent: qualitative-property-20260505-0
summary:
Qualitative properties are properties that are observed and can generally not be measured with a numerical result, unlike quantitative properties, which have numerical characteristics. == Description == Qualitative properties are properties that are observed and can generally not be measured with a numerical result. They are contrasted to quantitative properties which have numerical characteristics. === Evaluation === Although measuring something in qualitative terms is difficult, most people can (and will) make a judgement about a behaviour on the basis of how they feel treated. This indicates that qualitative properties are closely related to emotional impressions. A test method can result in qualitative data about something. This can be a categorical result or a binary classification (e.g., pass/fail, go/no go, conform/non-conform). It can sometimes be an engineering judgement. === Categorization === The data that all share a qualitative property form a nominal category. A variable which codes for the presence or absence of such a property is called a binary categorical variable, or equivalently a dummy variable. == Types == Some engineering and scientific properties are qualitative. Some important qualitative properties that concern businesses are: Human factors, human work capital is important issue that deals with qualitative properties. Some common aspects


id: reduction-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Reduction_(mathematics)-0.md
category: reference
title: Reduction (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Reduction_(mathematics)
date_saved: 2026-05-05T07:25:01.964455+00:00
instance: kb-cron
parent: reduction-mathematics-20260505-0
summary:
In mathematics, reduction refers to the rewriting of an expression into a simpler form. For example, the process of rewriting a fraction into one with the smallest whole-number denominator possible (while keeping the numerator a whole number) is called "reducing a fraction". Rewriting a radical (or "root") expression with the smallest possible whole number under the radical symbol is called "reducing a radical". Minimizing the number of radicals that appear underneath other radicals in an expression is called denesting radicals. == Algebra == In linear algebra, reduction refers to applying simple rules to a series of equations or matrices to change them into a simpler form. In the case of matrices, the process involves manipulating either the rows or the columns of the matrix and so is usually referred to as row-reduction or column-reduction, respectively. Often the aim of reduction is to transform a matrix into its "row-reduced echelon form" or "row-echelon form"; this is the goal of Gaussian elimination. == Calculus == In calculus, reduction refers to using the technique of integration by parts to evaluate integrals by reducing them to simpler forms. == Static (Guyan) reduction == In dynamic analysis, static reduction refers to reducing the number of


id: resilience-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Resilience_(mathematics)-0.md
category: reference
title: Resilience (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Resilience_(mathematics)
date_saved: 2026-05-05T07:25:03.234631+00:00
instance: kb-cron
parent: resilience-mathematics-20260505-0
summary:
In mathematical modeling, resilience refers to the ability of a dynamical system to recover from perturbations and return to its original stable steady state. It is a measure of the stability and robustness of a system in the face of changes or disturbances. If a system is not resilient enough, it is more susceptible to perturbations and can more easily undergo a critical transition. A common analogy used to explain the concept of resilience of an equilibrium is one of a ball in a valley. A resilient steady state corresponds to a ball in a deep valley, so any push or perturbation will very quickly lead the ball to return to the resting point where it started. On the other hand, a less resilient steady state corresponds to a ball in a shallow valley, so the ball will take a much longer time to return to the equilibrium after a perturbation. The concept of resilience is particularly useful in systems that exhibit tipping points, whose study has a long history that can be traced back to catastrophe theory. While this theory was initially overhyped and fell out of favor, its mathematical foundation remains strong and is now recognized as relevant


id: rigidity-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Rigidity_(mathematics)-0.md
category: reference
title: Rigidity (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Rigidity_(mathematics)
date_saved: 2026-05-05T07:25:04.445143+00:00
instance: kb-cron
parent: rigidity-mathematics-20260505-0
summary:
In mathematics, a rigid collection C of mathematical objects c (for instance sets or functions) is one in which every c ∈ C is uniquely determined by less information about c than one would expect. The above statement does not define a mathematical property; instead, it describes in what sense the adjective "rigid" is typically used in mathematics, by mathematicians. == Examples == Some examples include: Harmonic functions on the unit disk are rigid in the sense that they are uniquely determined by their boundary values. Holomorphic functions are determined by the set of all derivatives at a single point. A smooth function from the real line to the complex plane is not, in general, determined by all its derivatives at a single point, but it is if we require additionally that it be possible to extend the function to one on a neighbourhood of the real line in the complex plane. The Schwarz lemma is an example of such a rigidity theorem. By the fundamental theorem of algebra, polynomials in C are rigid in the sense that any polynomial is completely determined by its values on any infinite set, say N, or the unit disk. By the previous example,


id: scope-logic-20260505-0
file: data/en.wikipedia.org/wiki/Scope_(logic)-0.md
category: reference
title: Scope (logic)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Scope_(logic)
date_saved: 2026-05-05T07:25:06.412643+00:00
instance: kb-cron
parent: scope-logic-20260505-0
summary:
In logic, the scope of a quantifier or connective is the shortest formula in which it occurs, determining the range in the formula to which the quantifier or connective is applied. The notions of a free variable and bound variable are defined in terms of whether that formula is within the scope of a quantifier, and the notions of a dominant connective and subordinate connective are defined in terms of whether a connective includes another within its scope. == Connectives == The scope of a logical connective occurring within a formula is the smallest well-formed formula that contains the connective in question. The connective with the largest scope in a formula is called its dominant connective, main connective, main operator, major connective, or principal connective; a connective within the scope of another connective is said to be subordinate to it. For instance, in the formula ( ( ( P → Q ) ∨ ¬ Q ) ↔ ( ¬ ¬ P ∧ Q ) ) {\displaystyle (\left(\left(P\rightarrow Q\right)\lor \lnot Q\right)\leftrightarrow \left(\lnot \lnot P\land Q\right))} , the dominant connective is ↔, and all other connectives are subordinate to it; the → is subordinate to the ∨, but not to the ∧;


id: sides-of-an-equation-20260505-0
file: data/en.wikipedia.org/wiki/Sides_of_an_equation-0.md
category: reference
title: Sides of an equation
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Sides_of_an_equation
date_saved: 2026-05-05T07:25:07.638119+00:00
instance: kb-cron
parent: sides-of-an-equation-20260505-0
summary:
In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. The two sides have the same value, expressed differently, since equality is symmetric. More generally, these terms may apply to an inequation or inequality; the right-hand side is everything on the right side of a test operator in an expression, with LHS defined similarly. == Example == The expression on the right side of the "=" sign is the right side of the equation and the expression on the left of the "=" is the left side of the equation. For example, in x + 5 = y + 8 {\displaystyle x+5=y+8} x + 5 is the left-hand side (LHS) and y + 8 is the right-hand side (RHS). == Homogeneous and inhomogeneous equations == In solving mathematical equations, particularly linear simultaneous equations, differential equations and integral equations, the terminology homogeneous is often used for equations with some linear operator L on the LHS and 0 on the RHS. In contrast, an equation with a non-zero RHS is called inhomogeneous or non-homogeneous, as exemplified by Lf = g, with g a fixed function, which equation is to be solved for f.


id: significand-20260505-0
file: data/en.wikipedia.org/wiki/Significand-0.md
category: reference
title: Significand
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Significand
date_saved: 2026-05-05T07:25:08.958728+00:00
instance: kb-cron
parent: significand-20260505-0
summary:
The significand (also coefficient, sometimes argument, or more ambiguously mantissa, fraction, or characteristic) is the first (left) part of a number in scientific notation or related concepts in floating-point representation, consisting of its significant digits. For negative numbers, it does not include the initial minus sign. Depending on the interpretation of the exponent, the significand may represent an integer or a fractional number, which may cause the term "mantissa" to be misleading, since the mantissa of a logarithm is always its fractional part. Although the other names mentioned are common, significand is the word used by IEEE 754, an important technical standard for floating-point arithmetic. In mathematics, the term "argument" may also be ambiguous, since "the argument of a number" sometimes refers to the length of a circular arc from 1 to a number on the unit circle in the complex plane. == Example == The number 123.45 can be represented as a decimal floating-point number with the integer 12345 as the significand and a 10−2 power term, also called characteristics, where −2 is the exponent (and 10 is the base). Its value is given by the following arithmetic: 123.45 = 12345 × 10−2. This same value can also be


id: stochastic-20260505-0
file: data/en.wikipedia.org/wiki/Stochastic-0.md
category: reference
title: Stochastic
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Stochastic
date_saved: 2026-05-05T07:25:10.291744+00:00
instance: kb-cron
parent: stochastic-20260505-0
summary:
Stochastic (; from Ancient Greek στόχος (stókhos) 'aim, guess') is the property of being well-described by a random probability distribution. Stochasticity and randomness are technically distinct concepts. Stochasticity refers to a modeling approach, while randomness describes phenomena. These terms are often used interchangeably. In probability theory, the formal concept of a stochastic process is also referred to as a random process. Stochasticity is used in many different fields, including actuarial science, image processing, signal processing, computer science, information theory, telecommunications, chemistry, ecology, neuroscience, physics, and cryptography. It is also used in finance, medicine, linguistics, music, media, colour theory, botany, manufacturing and geomorphology. == Etymology == The word stochastic in English was originally used as an adjective with the definition "pertaining to conjecturing", and stemming from a Greek word meaning "to aim at a mark, guess", and the Oxford English Dictionary gives the year 1662 as its earliest occurrence. In his work on probability Ars Conjectandi, originally published in Latin in 1713, Jakob Bernoulli used the phrase "Ars Conjectandi sive Stochastice", which has been translated to "the art of conjecturing or stochastics". This phrase was used, with reference to Bernoulli, by Ladislaus Bortkiewicz, who in 1917 wrote in German the word


id: stochastic-20260505-1
file: data/en.wikipedia.org/wiki/Stochastic-1.md
category: reference
title: Stochastic
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Stochastic
date_saved: 2026-05-05T07:25:10.291744+00:00
instance: kb-cron
parent: stochastic-20260505-0
summary:
== Finance == The financial markets use stochastic models to represent the seemingly random behaviour of various financial assets, including the random behavior of the price of one currency compared to that of another (such as the price of US Dollar compared to that of the Euro), and also to represent random behaviour of interest rates. These models are then used by financial analysts to value options on stock prices, bond prices, and on interest rates, see Markov models. Moreover, it is at the heart of the insurance industry. == Geomorphology == The formation of river meanders has been analyzed as a stochastic process. == Language and linguistics == Non-deterministic approaches in language studies are largely inspired by the work of Ferdinand de Saussure, for example, in functionalist linguistic theory, which argues that competence is based on performance. This distinction in functional theories of grammar should be carefully distinguished from the langue and parole distinction. To the extent that linguistic knowledge is constituted by experience with language, grammar is argued to be probabilistic and variable rather than fixed and absolute. This conception of grammar as probabilistic and variable follows from the idea that one's competence changes in accordance with one's


id: stratification-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Stratification_(mathematics)-0.md
category: reference
title: Stratification (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Stratification_(mathematics)
date_saved: 2026-05-05T07:25:11.575927+00:00
instance: kb-cron
parent: stratification-mathematics-20260505-0
summary:
Stratification has several usages in mathematics. == In mathematical logic == In mathematical logic, stratification is any consistent assignment of numbers to predicate symbols guaranteeing that a unique formal interpretation of a logical theory exists. Specifically, we say that a set of clauses of the form Q 1 ∧ ⋯ ∧ Q n ∧ ¬ Q n + 1 ∧ ⋯ ∧ ¬ Q n + m → P {\displaystyle Q_{1}\wedge \dots \wedge Q_{n}\wedge \neg Q_{n+1}\wedge \dots \wedge \neg Q_{n+m}\rightarrow P} is stratified if and only if there is a stratification assignment S that fulfills the following conditions: If a predicate P is positively derived from a predicate Q (i.e., P is the head of a rule, and Q occurs positively in the body of the same rule), then the stratification number of P must be greater than or equal to the stratification number of Q, in short S ( P ) ≥ S ( Q ) {\displaystyle S(P)\geq S(Q)} . If a predicate P is derived from a negated predicate Q (i.e., P is the head of a rule, and Q occurs negatively in the body of the same rule), then the stratification number of P must be greater


id: strict-20260505-0
file: data/en.wikipedia.org/wiki/Strict-0.md
category: reference
title: Strict
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Strict
date_saved: 2026-05-05T07:25:12.769533+00:00
instance: kb-cron
parent: strict-20260505-0
summary:
In mathematical writing, the term strict refers to the property of excluding equality and equivalence and often occurs in the context of inequality and monotonic functions. It is often attached to a technical term to indicate that the exclusive meaning of the term is to be understood. The opposite is non-strict, which is often understood to be the case but can be put explicitly for clarity. In some contexts, the word "proper" can also be used as a mathematical synonym for "strict". == Use == This term is commonly used in the context of inequalities — the phrase "strictly less than" means "less than and not equal to" (likewise "strictly greater than" means "greater than and not equal to"). More generally, a strict partial order, strict total order, and strict weak order exclude equality and equivalence. When comparing numbers to zero, the phrases "strictly positive" and "strictly negative" mean "positive and not equal to zero" and "negative and not equal to zero", respectively. In the context of functions, the adverb "strictly" is used to modify the terms "monotonic", "increasing", and "decreasing". On the other hand, sometimes one wants to specify the inclusive meanings of terms. In the context of comparisons,


id: t-c-mits-20260505-0
file: data/en.wikipedia.org/wiki/T.C._Mits-0.md
category: reference
title: T.C. Mits
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/T.C._Mits
date_saved: 2026-05-05T07:25:14.020975+00:00
instance: kb-cron
parent: t-c-mits-20260505-0
summary:
T.C. Mits (acronym for "the celebrated man in the street"), is a term coined by Lillian Rosanoff Lieber to refer to an everyman. In Lieber's works, T.C. Mits was a character who made scientific topics more approachable to the public audience. The phrase has enjoyed sparse use by authors in fields such as molecular biology, secondary education, and general semantics. == The Education of T.C. MITS == Dr. Lillian Rosanoff Lieber wrote this treatise on mathematical thinking in twenty chapters. The writing took a form that resembled free-verse poetry, though Lieber included an introduction stating that the form was meant only to facilitate rapid reading, rather than emulate free-verse. Lieber's husband, a fellow professor at Long Island University, Hugh Gray Lieber, provided illustrations for the book. The title of the book was meant to emphasize that mathematics can be understood by anyone, which was further shown when a special "Overseas edition for the Armed Forces" was published in 1942, and approved by the Council on Books in Wartime to be sent to American troops fighting in World War II. == See also == John Doe The man on the Clapham omnibus == References ==


id: the-politics-of-modelling-numbers-between-science-and-policy-20260505-0
file: data/en.wikipedia.org/wiki/The_Politics_of_Modelling,_Numbers_Between_Science_and_Policy-0.md
category: reference
title: The Politics of Modelling, Numbers Between Science and Policy
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/The_Politics_of_Modelling,_Numbers_Between_Science_and_Policy
date_saved: 2026-05-05T07:25:15.216481+00:00
instance: kb-cron
parent: the-politics-of-modelling-numbers-between-science-and-policy-20260505-0
summary:
The Politics of Modelling, Numbers Between Science and Policy is a multi-authors book edited by Andrea Saltelli and Monica Di Fiore and published in August 2023 by Oxford University Press. == Synopsis == The Politics of Modelling elaborates and expands on themes of responsible modelling from a manifesto published in the journal Nature in 2020. The text is structured into three main sections: Meeting Models, The Rules, and The Rules in Practice. The combination of theory with policy relevant examples makes the book accessible to modellers, researchers of modelling, and policy makers. The volume comes with a foreword of Wendy Nelson Espeland and a preface of Daniel Sarewitz, with chapters from Andy Stirling, Wolfgang Drechsler, Philip B. Stark, Ting Xu, Paolo Vineis, Andrea Saltelli, and other scholars. == Reception == The book argues that models live in a "state of exception" provided by their access to a wealth of methodology of analysis, and by their epistemic authority borrowed from mathematics. This state allows models to better defend an appearance of neutrality that is appreciated by policymakers in search of a justification. A review published in the Science notes that the volume incorporated insights from science and technology studies to explore


id: theorem-20260505-0
file: data/en.wikipedia.org/wiki/Theorem-0.md
category: reference
title: Theorem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Theorem
date_saved: 2026-05-05T07:25:16.454658+00:00
instance: kb-cron
parent: theorem-20260505-0
summary:
In mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In mainstream mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), or of a less powerful theory, such as Peano arithmetic. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as theorems only the most important results, and use the terms lemma, proposition and corollary for less important theorems. In mathematical logic, the concepts of theorems and proofs have been formalized in order to allow mathematical reasoning about them. In this context, statements become well-formed formulas of some formal language. A theory consists of some basis statements called axioms, and some deducing rules (sometimes included in the axioms). The theorems of the theory are the statements that can be


id: theorem-20260505-1
file: data/en.wikipedia.org/wiki/Theorem-1.md
category: reference
title: Theorem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Theorem
date_saved: 2026-05-05T07:25:16.454658+00:00
instance: kb-cron
parent: theorem-20260505-0
summary:
== Epistemological considerations == Many mathematical theorems are conditional statements, whose proofs deduce conclusions from conditions known as hypotheses or premises. In light of the interpretation of proof as justification of truth, the conclusion is often viewed as a necessary consequence of the hypotheses. Namely, that the conclusion is true in case the hypotheses are true—without any further assumptions. However, the conditional could also be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol (e.g., non-classical logic). Although theorems can be written in a completely symbolic form (e.g., as propositions in propositional calculus), they are often expressed informally in a natural language such as English for better readability. The same is true of proofs, which are often expressed as logically organized and clearly worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, and from which a formal symbolic proof can in principle be constructed. In addition to the better readability, informal arguments are typically easier to check than purely symbolic ones—indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but


id: theorem-20260505-2
file: data/en.wikipedia.org/wiki/Theorem-2.md
category: reference
title: Theorem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Theorem
date_saved: 2026-05-05T07:25:16.454658+00:00
instance: kb-cron
parent: theorem-20260505-0
summary:
Nonetheless, there is some degree of empiricism and data collection involved in the discovery of mathematical theorems. By establishing a pattern, sometimes with the use of a powerful computer, mathematicians may have an idea of what to prove, and in some cases even a plan for how to set about doing the proof. It is also possible to find a single counter-example and so establish the impossibility of a proof for the proposition as-stated, and possibly suggest restricted forms of the original proposition that might have feasible proofs. For example, both the Collatz conjecture and the Riemann hypothesis are well-known unsolved problems; they have been extensively studied through empirical checks, but remain unproven. The Collatz conjecture has been verified for start values up to about 2.88 × 1018. The Riemann hypothesis has been verified to hold for the first 10 trillion non-trivial zeroes of the zeta function. Although most mathematicians can tolerate supposing that the conjecture and the hypothesis are true, neither of these propositions is considered proved. Such evidence does not constitute proof. For example, the Mertens conjecture is a statement about natural numbers that is now known to be false, but no explicit counterexample (i.e., a natural number


id: theorem-20260505-3
file: data/en.wikipedia.org/wiki/Theorem-3.md
category: reference
title: Theorem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Theorem
date_saved: 2026-05-05T07:25:16.454658+00:00
instance: kb-cron
parent: theorem-20260505-0
summary:
Theorem (name of the person who proved it, along with year of discovery or publication of the proof) Statement of theorem (sometimes called the proposition) Proof Description of proof End The end of the proof may be signaled by the letters Q.E.D. (quod erat demonstrandum) or by one of the tombstone marks, such as "□" or "∎", meaning "end of proof", introduced by Paul Halmos following their use in magazines to mark the end of an article. The exact style depends on the author or publication. Many publications provide instructions or macros for typesetting in the house style. It is common for a theorem to be preceded by definitions describing the exact meaning of the terms used in the theorem. It is also common for a theorem to be preceded by a number of propositions or lemmas which are then used in the proof. However, lemmas are sometimes embedded in the proof of a theorem, either with nested proofs, or with their proofs presented after the proof of the theorem. Corollaries to a theorem are either presented between the theorem and the proof, or directly after the proof. Sometimes, corollaries have proofs of their own that explain why they follow


id: theorem-20260505-4
file: data/en.wikipedia.org/wiki/Theorem-4.md
category: reference
title: Theorem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Theorem
date_saved: 2026-05-05T07:25:16.454658+00:00
instance: kb-cron
parent: theorem-20260505-0
summary:
=== Works cited === Boolos, George; Burgess, John; Jeffrey, Richard (2007). Computability and Logic (5th ed.). Cambridge University Press. Enderton, Herbert (2001). A Mathematical Introduction to Logic (2nd ed.). Harcourt Academic Press. Heath, Sir Thomas Little (1897). The works of Archimedes. Dover. Retrieved 2009-11-15. Hedman, Shawn (2004). A First Course in Logic. Oxford University Press. Hinman, Peter (2005). Fundamentals of Mathematical Logic. Wellesley, MA: A K Peters. Hoffman, Paul (1998). The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. Hyperion, New York. ISBN 1-85702-829-5. Hodges, Wilfrid (1993). Model Theory. Cambridge University Press. Johnstone, P. T. (1987). Notes on Logic and Set Theory. Cambridge University Press. Monk, J. Donald (1976). Mathematical Logic. Springer-Verlag. Petkovsek, Marko; Wilf, Herbert; Zeilberger, Doron (1996). A = B (PDF). A.K. Peters, Wellesley, Massachusetts. ISBN 1-56881-063-6. Rautenberg, Wolfgang (2010). A Concise Introduction to Mathematical Logic (3rd ed.). Springer. van Dalen, Dirk (1994). Logic and Structure (3rd ed.). Springer-Verlag. Wentworth, G.; Smith, D.E. (1913). Plane Geometry. Ginn & Co. == Further reading == Chiswell, Ian; Hodges, Wilfred (2007). Mathematical Logic. Oxford University Press. Hunter, Geoffrey (1996) [1971]. Metalogic: An Introduction to the Metatheory of Standard First-Order Logic. University of California


id: toy-model-20260505-0
file: data/en.wikipedia.org/wiki/Toy_model-0.md
category: reference
title: Toy model
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Toy_model
date_saved: 2026-05-05T07:25:17.746720+00:00
instance: kb-cron
parent: toy-model-20260505-0
summary:
In scientific modeling, a toy model is a deliberately simplistic model with many details removed so that it can be used to explain a mechanism concisely. It is also useful in a description of the fuller model. In toy mathematical models used in mathematical physics, this is usually done by reducing or extending the number of dimensions or reducing the number of fields/variables or restricting them to a particular symmetric form. In toy economic models, some may be only loosely based on theory, others more explicitly so. They allow for a quick first pass at some question, and present the essence of the answer from a more complicated model or from a class of models. For the researcher, they may come before writing a more elaborate model, or after, once the elaborate model has been worked out. Blanchard's list of examples includes the IS–LM model, the Mundell–Fleming model, the RBC model, and the New Keynesian model. The phrase "tinker-toy model" is also used, in reference to the Tinkertoys product used for children's constructivist learning. == Examples == Examples of toy models in physics include: the Ising model as a toy model for ferromagnetism, or lattice models more generally. It is


id: toy-theorem-20260505-0
file: data/en.wikipedia.org/wiki/Toy_theorem-0.md
category: reference
title: Toy theorem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Toy_theorem
date_saved: 2026-05-05T07:25:18.952329+00:00
instance: kb-cron
parent: toy-theorem-20260505-0
summary:
In mathematics, a toy theorem is a simplified instance (special case) of a more general theorem, which can be useful in providing a handy representation of the general theorem, or a framework for proving the general theorem. One way of obtaining a toy theorem is by introducing some simplifying assumptions in a theorem. In many cases, a toy theorem is used to illustrate the claim of a theorem, while in other cases, studying the proofs of a toy theorem (derived from a non-trivial theorem) can provide insight that would be hard to obtain otherwise. Toy theorems can also have educational value as well. For example, after presenting a theorem (with, say, a highly non-trivial proof), one can sometimes give some assurance that the theorem really holds, by proving a toy version of the theorem. == Examples == A toy theorem of the Brouwer fixed-point theorem is obtained by restricting the dimension to one. In this case, the Brouwer fixed-point theorem follows almost immediately from the intermediate value theorem. Another example of toy theorem is Rolle's theorem, which is obtained from the mean value theorem by equating the function values at the endpoints. == See also == Corollary Fundamental theorem Lemma


id: transport-of-structure-20260505-0
file: data/en.wikipedia.org/wiki/Transport_of_structure-0.md
category: reference
title: Transport of structure
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Transport_of_structure
date_saved: 2026-05-05T07:25:20.207704+00:00
instance: kb-cron
parent: transport-of-structure-20260505-0
summary:
In mathematics, particularly in universal algebra and category theory, transport of structure refers to the process whereby a mathematical object acquires a new structure and its canonical definitions, as a result of being isomorphic to (or otherwise identified with) another object with a pre-existing structure. Definitions by transport of structure are regarded as canonical. == Examples == Since mathematical structures are often defined in reference to an underlying space, many examples of transport of structure involve spaces and mappings between them. For example, if V {\displaystyle V} and W {\displaystyle W} are vector spaces with ( ⋅ , ⋅ ) {\displaystyle (\cdot ,\cdot )} being an inner product on W {\displaystyle W} , such that there is an isomorphism ϕ {\displaystyle \phi } from V {\displaystyle V} to W {\displaystyle W} , then one can define an inner product [ ⋅ , ⋅ ] {\displaystyle [\cdot ,\cdot ]} on V {\displaystyle V} by the following rule: [ v 1 , v 2 ] = ( ϕ ( v 1 ) , ϕ ( v 2 ) ) {\displaystyle [v_{1},v_{2}]=(\phi (v_{1}),\phi (v_{2}))} Although the equation makes sense even when ϕ {\displaystyle \phi } is not an isomorphism, it only defines an


id: triviality-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Triviality_(mathematics)-0.md
category: reference
title: Triviality (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Triviality_(mathematics)
date_saved: 2026-05-05T07:25:21.450294+00:00
instance: kb-cron
parent: triviality-mathematics-20260505-0
summary:
In mathematics, the adjective trivial is often used to refer to a claim or a case which can be readily obtained from context, or a particularly simple object possessing a given structure (e.g., group, topological space). The noun triviality usually refers to a simple technical aspect of some proof or definition. The origin of the term in mathematical language comes from the medieval trivium curriculum, which distinguishes from the more difficult quadrivium curriculum. The opposite of trivial is nontrivial, which is commonly used to indicate that an example or a solution is not simple, or that a statement or a theorem is not easy to prove. Triviality does not have a rigorous definition in mathematics. It is subjective, and often determined in a given situation by the knowledge and experience of those considering the case. == Trivial and nontrivial solutions == In mathematics, the term "trivial" is often used to refer to objects (e.g., groups, topological spaces) with a very simple structure. These include, among others: Empty set: the set containing no or null members Trivial group: the mathematical group containing only the identity element Trivial ring: a ring defined on a singleton set "Trivial" can also be used to


id: triviality-mathematics-20260505-1
file: data/en.wikipedia.org/wiki/Triviality_(mathematics)-1.md
category: reference
title: Triviality (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Triviality_(mathematics)
date_saved: 2026-05-05T07:25:21.450294+00:00
instance: kb-cron
parent: triviality-mathematics-20260505-0
summary:
A common joke in the mathematical community is to say that "trivial" is synonymous with "proved"—that is, any theorem can be considered "trivial" once it is known to be proved as true. Two mathematicians who are discussing a theorem: the first mathematician says that the theorem is "trivial". In response to the other's request for an explanation, he then proceeds with twenty minutes of exposition. At the end of the explanation, the second mathematician agrees that the theorem is trivial. But can we say that this theorem is trivial even if it takes a lot of time and effort to prove it? When a mathematician says that a theorem is trivial, but he is unable to prove it by himself at the moment that he pronounces it as trivial, is the theorem trivial? Often, as a joke, a problem is referred to as "intuitively obvious". For example, someone experienced in calculus would consider the following statement trivial: ∫ 0 1 x 2 d x = 1 3 . {\displaystyle \int _{0}^{1}x^{2}\,dx={\frac {1}{3}}.} However, to someone with no knowledge of integral calculus, this is not obvious, so it is not trivial. == Examples == In number theory, it is often important


id: undefined-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Undefined_(mathematics)-0.md
category: reference
title: Undefined (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Undefined_(mathematics)
date_saved: 2026-05-05T07:25:22.620521+00:00
instance: kb-cron
parent: undefined-mathematics-20260505-0
summary:
In mathematics, the term undefined refers to a value, function, or other expression that cannot be assigned a meaning within a specific formal system. Attempting to assign or use an undefined value within a particular formal system, may produce contradictory or meaningless results within that system. In practice, mathematicians may use the term undefined to warn that a particular calculation or property can produce mathematically inconsistent results, and therefore, it should be avoided. Caution must be taken to avoid the use of such undefined values in a deduction or proof. Whether a particular function or value is undefined, depends on the rules of the formal system in which it is used. For example, the imaginary number − 1 {\displaystyle {\sqrt {-1}}} is undefined within the set of real numbers. So it is meaningless to reason about the value, solely within the discourse of real numbers. However, defining the imaginary number i {\displaystyle i} to be equal to − 1 {\displaystyle {\sqrt {-1}}} , allows there to be a consistent set of mathematics referred to as the complex number plane. Therefore, within the discourse of complex numbers, − 1 {\displaystyle {\sqrt {-1}}} is in fact defined. Many new fields of mathematics


id: undefined-mathematics-20260505-1
file: data/en.wikipedia.org/wiki/Undefined_(mathematics)-1.md
category: reference
title: Undefined (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Undefined_(mathematics)
date_saved: 2026-05-05T07:25:22.620521+00:00
instance: kb-cron
parent: undefined-mathematics-20260505-0
summary:
=== Indeterminate === The term undefined should be contrasted with the term indeterminate. In the first case, undefined generally indicates that a value or property can have no meaningful definition. In the second case, indeterminate generally indicates that a value or property can have many meaningful definitions. Additionally, it seems to be generally accepted that undefined values may not be safely used within a particular formal system, whereas indeterminate values might be, depending on the relevant rules of the particular formal system. == See also == Analytic function - a function locally given by a convergent power series, which may be useful for dealing with otherwise undefined values L'Hôpital's rule - a method in calculus for evaluating indeterminate forms Indeterminate form - a mathematical expression for which many assignments exist NaN - the IEEE-754 expression indicating that the result of a calculation is not a number Primitive notion - a concept that is not defined in terms of previously defined concepts Singularity - a point at which a mathematical function ceases to be well-behaved == References == == Further reading == Smart, James R. (1988). Modern Geometries (3rd ed.). Brooks/Cole. ISBN 0-534-08310-2. Lo Bello, Anthony (2013). Origins of Mathematical Words.


id: uniqueness-quantification-20260505-0
file: data/en.wikipedia.org/wiki/Uniqueness_quantification-0.md
category: reference
title: Uniqueness quantification
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Uniqueness_quantification
date_saved: 2026-05-05T07:25:23.930015+00:00
instance: kb-cron
parent: uniqueness-quantification-20260505-0
summary:
In mathematics and logic, the term "uniqueness" refers to the property of being the one and only object satisfying a certain condition. This sort of quantification is known as uniqueness quantification or unique existential quantification, and is often denoted with the symbols "∃!" or "∃=1". It is defined to mean there exists an object with the given property, and all objects with this property are equal. For example, the formal statement ∃ ! n ∈ N ( n − 2 = 4 ) {\displaystyle \exists !n\in \mathbb {N} \,(n-2=4)} may be read as "there is exactly one natural number n {\displaystyle n} such that ⁠ n − 2 = 4 {\displaystyle n-2=4} ⁠". == Proving uniqueness == The most common technique to prove the unique existence of an object is to first prove the existence of the entity with the desired condition, and then to prove that any two such entities (say, a {\displaystyle a} and ⁠ b {\displaystyle b} ⁠) must be equal to each other (i.e. ⁠ a = b {\displaystyle a=b} ⁠). For example, to show that the equation x + 2 = 5 {\displaystyle x+2=5} has exactly one solution, one would first start by establishing that


id: uniqueness-theorem-20260505-0
file: data/en.wikipedia.org/wiki/Uniqueness_theorem-0.md
category: reference
title: Uniqueness theorem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Uniqueness_theorem
date_saved: 2026-05-05T07:25:25.125766+00:00
instance: kb-cron
parent: uniqueness-theorem-20260505-0
summary:
In mathematics, a uniqueness theorem, also called a unicity theorem, is a theorem asserting the uniqueness of an object satisfying certain conditions, or the equivalence of all objects satisfying the said conditions. Examples of uniqueness theorems include: Cauchy's rigidity theorem and Alexandrov's uniqueness theorem for three-dimensional polyhedra. Black hole uniqueness theorem Cauchy–Kowalevski theorem is the main local existence and uniqueness theorem for analytic partial differential equations associated with Cauchy initial value problems. Cauchy–Kowalevski–Kashiwara theorem is a wide generalization of the Cauchy–Kowalevski theorem for systems of linear partial differential equations with analytic coefficients. Division theorem, the uniqueness of quotient and remainder under Euclidean division. Fundamental theorem of arithmetic, the uniqueness of prime factorization. Holmgren's uniqueness theorem for linear partial differential equations with real analytic coefficients. Picard–Lindelöf theorem, the uniqueness of solutions to first-order differential equations. Thompson uniqueness theorem in finite group theory. Uniqueness theorem for Poisson's equation. Electromagnetism uniqueness theorem for the solution of Maxwell's equation. Uniqueness case in finite group theory. The word unique is sometimes replaced by essentially unique, whenever one wants to stress that the uniqueness is only referred to the underlying structure, whereas the form may vary in all ways that do not affect the mathematical content.


id: univariate-20260505-0
file: data/en.wikipedia.org/wiki/Univariate-0.md
category: reference
title: Univariate
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Univariate
date_saved: 2026-05-05T07:25:26.386149+00:00
instance: kb-cron
parent: univariate-20260505-0
summary:
In mathematics, a univariate object is an expression, equation, function or polynomial involving only one variable. Objects involving more than one variable are multivariate. In some cases the distinction between the univariate and multivariate cases is fundamental; for example, the fundamental theorem of algebra and Euclid's algorithm for polynomials are fundamental properties of univariate polynomials that cannot be generalized to multivariate polynomials. In statistics, a univariate distribution characterizes one variable, although it can be applied in other ways as well. For example, univariate data are composed of a single scalar component. In time series analysis, the whole time series is the "variable": a univariate time series is the series of values over time of a single quantity. Correspondingly, a "multivariate time series" characterizes the changing values over time of several quantities. In some cases, the terminology is ambiguous, since the values within a univariate time series may be treated using certain types of multivariate statistical analyses and may be represented using multivariate distributions. In addition to the question of scaling, a criterion (variable) in univariate statistics can be described by two important measures (also key figures or parameters): Location & Variation. Measures of Location Scales (e.g. mode, median, arithmetic mean)


id: univariate-statistics-20260505-0
file: data/en.wikipedia.org/wiki/Univariate_(statistics)-0.md
category: reference
title: Univariate (statistics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Univariate_(statistics)
date_saved: 2026-05-05T07:25:27.600546+00:00
instance: kb-cron
parent: univariate-statistics-20260505-0
summary:
Univariate is a term commonly used in statistics to describe a type of data which consists of observations on only a single characteristic or attribute. A simple example of univariate data would be the salaries of workers in industry. Similar to other data, univariate data can be visualized using graphs, images, or other analysis tools after the data are measured, collected, reported, and analyzed. == Data types == Some univariate data consist of numbers (such as the height of 1.65 m, or the mass of 70 kg), whilst others are non-numerical (such as eye colors of brown or blue). Generally, the terms categorical univariate data and numerical univariate data are used to distinguish between these types. === Categorical univariate data === Categorical univariate data consists of non-numerical observations that may be placed in categories. It includes labels or names used to identify an attribute of each element. Categorical univariate data usually use either nominal or ordinal scale of measurement. === Numerical univariate data === Numerical univariate data consists of observations that are numbers. They are obtained using either interval or ratio scale of measurement. This type of univariate data can be classified even further into two subcategories: discrete and continuous.


id: univariate-statistics-20260505-1
file: data/en.wikipedia.org/wiki/Univariate_(statistics)-1.md
category: reference
title: Univariate (statistics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Univariate_(statistics)
date_saved: 2026-05-05T07:25:27.600546+00:00
instance: kb-cron
parent: univariate-statistics-20260505-0
summary:
The most frequently used graphical illustrations for univariate data are: === Frequency distribution tables === Frequency is how many times a number occurs. The frequency of an observation in statistics indicates the number of times the observation occurs in the data. For example, in the following list of numbers {1, 2, 3, 4, 6, 9, 9, 8, 5, 1, 1, 9, 9, 0, 6, 9}, the frequency of the number 9 is 5 (because it occurs 5 times in this data set). === Bar charts === A bar chart is a graph consisting of rectangular bars. These bars actually represents number or percentage of observations of existing categories in a variable. The length or height of bars gives a visual representation of the proportional differences among categories. === Histograms === Histograms are used to estimate the distribution of the data, with the frequency of values assigned to a value range called a bin. === Pie charts === A pie chart is a circle divided into portions that represent the relative frequencies or percentages of a population or a sample belonging to different categories. == Distributions == Univariate distribution is a dispersal type for a single random variable described either with


id: universal-space-20260505-0
file: data/en.wikipedia.org/wiki/Universal_space-0.md
category: reference
title: Universal space
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Universal_space
date_saved: 2026-05-05T07:25:28.849483+00:00
instance: kb-cron
parent: universal-space-20260505-0
summary:
In mathematics, a universal space is a certain metric space that contains all metric spaces whose dimension is bounded by some fixed constant. A similar definition exists in topological dynamics. == Definition == Given a class C {\displaystyle \textstyle {\mathcal {C}}} of topological spaces, U ∈ C {\displaystyle \textstyle \mathbb {U} \in {\mathcal {C}}} is universal for C {\displaystyle \textstyle {\mathcal {C}}} if each member of C {\displaystyle \textstyle {\mathcal {C}}} embeds in U {\displaystyle \textstyle \mathbb {U} } . Menger stated and proved the case d = 1 {\displaystyle \textstyle d=1} of the following theorem. The theorem in full generality was proven by Nöbeling. Theorem: The ( 2 d + 1 ) {\displaystyle \textstyle (2d+1)} -dimensional cube [ 0 , 1 ] 2 d + 1 {\displaystyle \textstyle [0,1]^{2d+1}} is universal for the class of compact metric spaces whose Lebesgue covering dimension is less than d {\displaystyle \textstyle d} . Nöbeling went further and proved: Theorem: The subspace of [ 0 , 1 ] 2 d + 1 {\displaystyle \textstyle [0,1]^{2d+1}} consisting of set of points, at most d {\displaystyle \textstyle d} of whose coordinates are rational, is universal for the class of separable metric spaces whose Lebesgue covering


id: up-to-20260505-0
file: data/en.wikipedia.org/wiki/Up_to-0.md
category: reference
title: Up to
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Up_to
date_saved: 2026-05-05T07:25:29.989667+00:00
instance: kb-cron
parent: up-to-20260505-0
summary:
Two mathematical objects a and b are called "equal up to an equivalence relation R" if a and b are related by R, that is, if aRb holds, that is, if the equivalence classes of a and b with respect to R are equal. This figure of speech is mostly used in connection with expressions derived from equality, such as uniqueness or count. For example, "x is unique up to R" means that all objects x under consideration are in the same equivalence class with respect to the relation R. Moreover, the equivalence relation R is often designated rather implicitly by a generating condition or transformation. For example, the statement "an integer's prime factorization is unique up to ordering" is a concise way to say that any two lists of prime factors of a given integer are equivalent with respect to the relation R that relates two lists if one can be obtained by reordering (permuting) the other. As another example, the statement "the solution to an indefinite integral is sin(x), up to addition of a constant" tacitly employs the equivalence relation R between functions, defined by fRg if the pointwise difference f−g is a constant function, and means that


id: upper-and-lower-bounds-20260505-0
file: data/en.wikipedia.org/wiki/Upper_and_lower_bounds-0.md
category: reference
title: Upper and lower bounds
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Upper_and_lower_bounds
date_saved: 2026-05-05T07:25:31.217787+00:00
instance: kb-cron
parent: upper-and-lower-bounds-20260505-0
summary:
In mathematics, particularly in order theory, an upper bound or majorant of a subset S of some preordered set (K, ≤) is an element of K that is greater than or equal to every element of S. Dually, a lower bound or minorant of S is defined to be an element of K that is less than or equal to every element of S. A set with an upper (respectively, lower) bound is said to be bounded from above or majorized (respectively bounded from below or minorized) by that bound. The terms bounded above (bounded below) are also used in the mathematical literature for sets that have upper (respectively lower) bounds. == Examples == For example, 5 is a lower bound for the set S = {5, 8, 42, 34, 13934} (as a subset of the integers or of the real numbers, etc.), and so is 4. On the other hand, 6 is not a lower bound for S since it is not smaller than every element in S. 13934 and other numbers x such that x ≥ 13934 would be an upper bound for S. The set S = {42} has 42 as both an upper bound and a


id: well-defined-expression-20260505-0
file: data/en.wikipedia.org/wiki/Well-defined_expression-0.md
category: reference
title: Well-defined expression
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Well-defined_expression
date_saved: 2026-05-05T07:25:32.423393+00:00
instance: kb-cron
parent: well-defined-expression-20260505-0
summary:
In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be not well defined, ill-defined or ambiguous. A function is well defined if it gives the same result when the representation of the input is changed without changing the value of the input. For instance, if f {\displaystyle f} takes real numbers as input, and if f ( 0.5 ) {\displaystyle f(0.5)} does not equal f ( 1 / 2 ) {\displaystyle f(1/2)} then f {\displaystyle f} is not well defined (and thus not a function). The term well-defined can also be used to indicate that a logical expression is unambiguous or uncontradictory. A function that is not well defined is not the same as a function that is undefined. For example, if f ( x ) = 1 x {\displaystyle f(x)={\frac {1}{x}}} , then even though f ( 0 ) {\displaystyle f(0)} is undefined, this does not mean that the function is not well defined; rather, 0 is not in the domain of f {\displaystyle f} . == Example == Let A 0 , A 1 {\displaystyle A_{0},A_{1}} be sets, let A =


id: well-defined-expression-20260505-1
file: data/en.wikipedia.org/wiki/Well-defined_expression-1.md
category: reference
title: Well-defined expression
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Well-defined_expression
date_saved: 2026-05-05T07:25:32.423393+00:00
instance: kb-cron
parent: well-defined-expression-20260505-0
summary:
g : Z / 4 Z → Z / 8 Z n ¯ 4 ↦ n ¯ 8 , {\displaystyle {\begin{matrix}g:&\mathbb {Z} /4\mathbb {Z} &\to &\mathbb {Z} /8\mathbb {Z} \\&{\overline {n}}_{4}&\mapsto &{\overline {n}}_{8},\end{matrix}}} does not lead to a well-defined function, since e.g. 1 ¯ 4 {\displaystyle {\overline {1}}_{4}} equals 5 ¯ 4 {\displaystyle {\overline {5}}_{4}} in Z / 4 Z {\displaystyle \mathbb {Z} /4\mathbb {Z} } , but the first would be mapped by g {\displaystyle g} to 1 ¯ 8 {\displaystyle {\overline {1}}_{8}} , while the second would be mapped to 5 ¯ 8 {\displaystyle {\overline {5}}_{8}} , and 1 ¯ 8 {\displaystyle {\overline {1}}_{8}} and 5 ¯ 8 {\displaystyle {\overline {5}}_{8}} are unequal in Z / 8 Z {\displaystyle \mathbb {Z} /8\mathbb {Z} } . === Operations === In particular, the term well-defined is used with respect to (binary) operations on cosets. In this case, one can view the operation as a function of two variables, and the property of being well-defined is the same as that for a function. For example, addition on the integers modulo some n can be defined naturally in terms of integer addition. [ a ] ⊕ [ b ] = [ a


id: without-loss-of-generality-20260505-0
file: data/en.wikipedia.org/wiki/Without_loss_of_generality-0.md
category: reference
title: Without loss of generality
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Without_loss_of_generality
date_saved: 2026-05-05T07:25:33.661377+00:00
instance: kb-cron
parent: without-loss-of-generality-20260505-0
summary:
Without loss of generality (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as without any loss of generality or with no loss of generality) is a frequently used expression in mathematics. The term is used to indicate the assumption that what follows is chosen arbitrarily, narrowing the premise to a particular case, but does not affect the validity of the proof in general. The other cases are sufficiently similar to the one presented that proving them follows by essentially the same logic. As a result, once a proof is given for the particular case, it is trivial to adapt it to prove the conclusion in all other cases. In many scenarios, the use of "without loss of generality" is made possible by the presence of symmetry. For example, if some property P(x,y) of real numbers is known to be symmetric in x and y, namely that P(x,y) is equivalent to P(y,x), then in proving that P(x,y) holds for every x and y, one may assume "without loss of generality" that x ≤ y. There is no loss of generality in this assumption, since once the case x ≤ y ⇒ P(x,y) has been proved, the other case follows


id: medical-terminology-20260505-0
file: data/en.wikipedia.org/wiki/Medical_terminology-0.md
category: reference
title: Medical terminology
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Medical_terminology
date_saved: 2026-05-05T07:25:35.931317+00:00
instance: kb-cron
parent: medical-terminology-20260505-0
summary:
Medical terminology is language used to describe the components, processes, conditions, medical procedures and treatments of the human body. In the English language, medical terminology generally has a regular morphology; the same prefixes and suffixes are used to add meanings to different roots. The root of a term often refers to an organ, tissue, or condition, and medical roots and affixes are often derived from Ancient Greek or Latin (particularly Neo-Latin). Many medical terms are examples of neoclassical compounds. Historically, all European universities used Latin as the dominant language of instruction and research, as Neo-Latin was the lingua franca of science, medicine, and education in Europe during the early modern period. Medical terminology includes a large part of anatomical terminology, which includes the anatomical terms of location, motion, muscle, bone, and histology. It also includes language from biology, chemistry, physics, and physiology, as well as vocabulary unique to the field of medicine, such as medical abbreviations. Each branch of medicine has its own clinical and scientific terminology. Medical dictionaries are specialised dictionaries for medical terminology and may be organised alphabetically or according to medical classification systems such as the Systematized Nomenclature of Medicine, International Classification of Diseases, or Unified Medical


id: medical-terminology-20260505-1
file: data/en.wikipedia.org/wiki/Medical_terminology-1.md
category: reference
title: Medical terminology
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Medical_terminology
date_saved: 2026-05-05T07:25:35.931317+00:00
instance: kb-cron
parent: medical-terminology-20260505-0
summary:
The parasagittal (Greek πᾰρᾰ́ (pará), beside) or paramedian or sagittal planes (Latin sagitta, arrow), which include the median or midsagittal plane and divide the body into left and right (in reference to the subject and not the observer). The frontal or coronal plane (Latin corona, crown), which divide the body into front and back. The axial or horizontal or transverse plane (Latin trans, across; vetere, to turn), which is perpendicular to the other two planes. The transpyloric plane, the subcostal plane, and the transumbilical plane planes are also considered in the division of the torso into the quadrants and regions of the abdomen. The three main axes of a human are the left-right (or horizontal or frontal), the craniocaudal (or rostrocaudal, longitudinal, or cephalocaudal), and the anteroposterior (or dorsoventral or sagittal) axes. Other anatomical lines include the axillary lines, parasternal line, and scapular line. ==== Location ==== Many anatomical terms can be combined to indicate a position in two axes simultaneously or the direction of a movement relative to the body: Anterolateral indicates a position that is both anterior and lateral to the standard anatomical position (such as the bulk of the pectoralis major muscle) or a named organ such


id: medical-terminology-20260505-2
file: data/en.wikipedia.org/wiki/Medical_terminology-2.md
category: reference
title: Medical terminology
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Medical_terminology
date_saved: 2026-05-05T07:25:35.931317+00:00
instance: kb-cron
parent: medical-terminology-20260505-0
summary:
Bones are surrounded by a membrane known as the periosteum (Greek περῐ́ (perí), around; Greek ὀστέον (ostéon), bone), and comprise multiple layers, depending on the type of bone, including yellow and red bone marrow, spongy bone, and the endosteum. There are five general classifications of bones: Long bones are characterized by a shaft known as the diaphysis, which is much longer than its width, and an epiphysis (Greek ἐπι- (epi-) upon, outside, over), a rounded head at each end of the shaft, connected to the diaphysis by the epiphyseal plate. They are made up mostly of compact bone, with lesser amounts of marrow, located within the medullary cavity, and areas of spongy, cancellous bone at the ends of the bones. Short bones are roughly cube-shaped, and have only a thin layer of compact bone surrounding a spongy interior. Short bones provide stability and support as well as some limited motion. Flat bones are thin and generally curved, with two parallel layers of compact bone sandwiching a layer of spongy bone. Sesamoid bones are bones embedded in tendons. Since they act to hold the tendon further away from the joint, the angle of the tendon is increased and thus the leverage


id: medical-terminology-20260505-3
file: data/en.wikipedia.org/wiki/Medical_terminology-3.md
category: reference
title: Medical terminology
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Medical_terminology
date_saved: 2026-05-05T07:25:35.931317+00:00
instance: kb-cron
parent: medical-terminology-20260505-0
summary:
==== Circulatory system ==== The network of blood vessels include the great vessels (comprising large elastic arteries and large veins), other arteries (which carry blood away from the heart) and veins (which carry blood to the heart), smaller arterioles, and capillaries, which join with venules. Blood is a fluid consisting of plasma (comprising serum and clotting factors), red blood cells, white blood cells, and platelets. Components of the blood include nutrients (such as proteins and minerals), hemoglobin, hormones, and gases such as oxygen and carbon dioxide. These substances provide nourishment, help the immune system to fight diseases, and help maintain homeostasis through mechanisms such as thermoregulation, osmoregulation, and acid-base regulation. The circulatory system has two divisions, a systemic circuit (i.e. the left heart pumping oxygenated blood to the rest of the body (via the aorta) and into the right heart (via the venae cava)) and a pulmonary circuit (i.e. the right heart pumping deoxygenated blood to the lungs (via the pulmonary artery) and into the left heart (via the pulmonary vein)). In the human heart: There is one atrium (which receives blood) and one ventricle (which expels blood) for each circulation, with a total of four chambers: left atrium, left


id: medical-terminology-20260505-4
file: data/en.wikipedia.org/wiki/Medical_terminology-4.md
category: reference
title: Medical terminology
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Medical_terminology
date_saved: 2026-05-05T07:25:35.931317+00:00
instance: kb-cron
parent: medical-terminology-20260505-0
summary:
The thoracic cavity is protected by the thoracic wall (comprising the rib cage and associated skin, muscle, and fascia), comprising the mediastinum, heart and great vessels, and the bulk of the respiratory tract. The abdominal cavity, lined by the peritoneum, contains the bulk of the gastrointestinal tract, the spleen and the kidneys. Abdominal organs may be classified as solid organs or hollow organs. The solid organs are the liver, pancreas, spleen, kidneys, and adrenal glands. The hollow organs of the abdomen are the stomach, intestines, gallbladder, bladder, and rectum. The pelvic cavity contains the urinary bladder, internal reproductive organs, and rectum. Viscera are the internal organs of the ventral cavity. The term "visceral" is contrasted with the term "parietal", meaning "of or relating to the wall of a body part, organ or cavity". The two terms are often used in describing a membrane or piece of connective tissue, referring to the opposing sides. ==== Respiratory system ==== The respiratory system allows for gas exchange, particularly of carbon dioxide and oxygen, in human beings. In the process of breathing or ventilation, the muscles of respiration pump air into the lungs, bringing it into close contact with the blood via millions of


id: medical-terminology-20260505-5
file: data/en.wikipedia.org/wiki/Medical_terminology-5.md
category: reference
title: Medical terminology
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Medical_terminology
date_saved: 2026-05-05T07:25:35.931317+00:00
instance: kb-cron
parent: medical-terminology-20260505-0
summary:
Muscle tissue comprises cardiac, skeletal, and smooth muscle. Smooth muscles control the flow of substances within the lumens of hollow organs, and are not consciously controlled. In the small intestine, smooth muscle contraction is characterised by peristalsis. Skeletal muscles are attached to bones and arranged in opposing groups around joints. Skeletal muscles and cardiac muscles have striations, unlike smooth muscle. Only skeletal and smooth muscles are part of the musculoskeletal system and can move the body. Cardiac muscles are found in the heart and are used only to circulate blood; like the smooth muscles, these muscles are not under conscious control. Nervous tissue comprises neuroglia, which provide structural and metabolic support, and neurons, which send signals in the form of electrochemical impulses traveling along axons. Bundles of axons are known as nerve tracts and neural pathways. Impulses can be directly transmitted to neighbouring cells through electrical synapses or cause chemicals called neurotransmitters to be released at chemical synapses. The CNS includes astrocytes (-cyte, 'cell'; Greek κύτος (kútos), a hollow, vessel), oligodendrocytes, ependyma, and radial glia; the PNS includes Schwann cells, satellite glia, and enteric glia. Astrocytes are star-shaped glial cells that have been observed to turn into neurons by virtue


id: medical-terminology-20260505-6
file: data/en.wikipedia.org/wiki/Medical_terminology-6.md
category: reference
title: Medical terminology
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Medical_terminology
date_saved: 2026-05-05T07:25:35.931317+00:00
instance: kb-cron
parent: medical-terminology-20260505-0
summary:
Fibrocyte Mesenchyme Adipocyte Fibroblast (Greek βλαστός (blastós), germinate or bud) ==== Epithelial ==== The basal surface of epithelial tissue rests on a basement membrane and the free, apical, or apex surface faces body fluid or the outside. The basement membrane acts as a scaffolding on which epithelium can grow and regenerate after injuries, and comprises the basal lamina and reticular lamina; although, some older sources use basement membrane and basal lamina synonymously. The basement membrane acts as a selectively permeable membrane that determines which substances will be able to enter the epithelium, as epithelial tissue has a nerve supply though no blood supply. There are three principal shapes of epithelial cell: squamous (scaly), columnar, and cuboidal. Transitional epithelium has cells that can change from squamous to cuboidal, depending on the amount of tension on the epithelium. Epithelial tissue can be further categorised as having a singular layer of cells as simple epithelium; or as layers of two or more cells deep as stratified epithelium—stratified squamous epithelium, stratified cuboidal epithelium, and stratified columnar epithelium. When taller simple columnar epithelial cells are viewed in cross section showing several nuclei appearing at different heights, they can be confused with stratified epithelia, and are


id: medical-terminology-20260505-7
file: data/en.wikipedia.org/wiki/Medical_terminology-7.md
category: reference
title: Medical terminology
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Medical_terminology
date_saved: 2026-05-05T07:25:35.931317+00:00
instance: kb-cron
parent: medical-terminology-20260505-0
summary:
The professional phagocytes, which generally patrol the body searching for pathogens, but can be called to specific locations by cytokines. Once a pathogen has been engulfed by a phagocyte, it becomes trapped in an intracellular vesicle called a phagosome, which subsequently fuses with a lysosome vesicle to form a phagolysosome. The pathogen is then killed by the activity of digestive enzymes or following a respiratory burst that releases free radicals into the phagolysosome. Macrophages and neutrophils, which travel around the body in pursuit of invading pathogens. Macrophages are cells that reside within tissues and produce an array of chemicals including enzymes, complement proteins, and cytokines. They also rid the body of worn-out cells and other debris and act as antigen-presenting cells that activate the adaptive immune system. Neutrophils are normally found in the bloodstream and are the most abundant type of phagocyte, representing 50% to 60% of total circulating leukocytes. During the acute phase of inflammation, neutrophils migrate toward the site of inflammation in a process called chemotaxis and are usually the first cells to arrive at the scene of infection. Dendritic cells are phagocytes in tissues that are in contact with the external environment, located mainly in the skin,


id: medical-terminology-20260505-8
file: data/en.wikipedia.org/wiki/Medical_terminology-8.md
category: reference
title: Medical terminology
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Medical_terminology
date_saved: 2026-05-05T07:25:35.931317+00:00
instance: kb-cron
parent: medical-terminology-20260505-0
summary:
Killer T cells kill cells that are infected with pathogens or otherwise damaged or dysfunctional, which contain a complex of a specific antigen coupled to a Class I MHC receptor. When the receptor of a cytotoxic or "killer" T-cell contacts such cells, it releases cytotoxins, such as perforin, which form pores in the target cell's plasma membrane, allowing ions, water and toxins to enter. The entry of another toxin called granulysin induces the target cell to undergo apoptosis. Helper T cells and regulatory T cells only recognize antigens coupled to Class II MHC molecules. Helper T cells regulate both the innate and adaptive immune responses and help determine which immune responses the body makes to a particular pathogen. These cells have no cytotoxic activity and do not kill infected cells or clear pathogens directly. They instead control the immune response by directing other cells to perform these tasks. A third, minor subtype are the γδ T cells, which recognise intact antigens that are not bound to MHC receptors. A B cell identifies pathogens when antibodies on its surface bind to a specific foreign antigen. This antigen/antibody complex is taken up by the B cell and processed by proteolysis into peptides.


id: medical-terminology-20260505-9
file: data/en.wikipedia.org/wiki/Medical_terminology-9.md
category: reference
title: Medical terminology
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Medical_terminology
date_saved: 2026-05-05T07:25:35.931317+00:00
instance: kb-cron
parent: medical-terminology-20260505-0
summary:
The human brain contains multiple regions, as does the spinal cord. There are multiple ways to divide the brain, including by the embryological development of the brain; into three parts: the cerebellum, brainstem, and cerebrum; the evolution of the brain; and cytoarchitecture, as in the case of Brodmann areas. A popular model of the 'triune brain' — comprising the reptilian complex (basal ganglia), the paleomammalian complex (limbic system), and the neomammalian complex (neocortex) — was formerly popular during the 1960s, though is now regarded as a myth. 'Limbic system' and associated terms, however, remain in neuroanatomical use, although some neuroscientists have argued against such use. Various other neuroanatomical systems have been developed according to functions, connections, and systems of the brain. Neuroendocrine axes Hypothalamic–pituitary–adrenal axis Hypothalamic–neurohypophyseal system Hypothalamic–pituitary–gonadal axis Hypothalamic–pituitary–thyroid axis Limbic system, corresponding to Cortical areas: Limbic lobe Orbitofrontal cortex Piriform cortex part of the olfactory system Entorhinal cortex Hippocampus and associated structures Fornix and septal nuclei Subcortical areas: Septal nuclei Amygdala Nucleus accumbens Diencephalic structures: Hypothalamus Mammillary bodies Anterior nuclei of thalamus ==== Neuropathology ==== In the peripheral nervous system, the most common problem is the failure of nerve conduction, which can be due to different causes including


id: medical-terminology-20260505-10
file: data/en.wikipedia.org/wiki/Medical_terminology-10.md
category: reference
title: Medical terminology
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Medical_terminology
date_saved: 2026-05-05T07:25:35.931317+00:00
instance: kb-cron
parent: medical-terminology-20260505-0
summary:
== History == The earliest known glossaries of medical terms were discovered on Egyptian papyrus authored around 1600 B.C. Other precursors to modern medical dictionaries include lists of terms compiled from the Hippocratic Corpus in the first century AD. The Synonyma Simonis Genuensis (the Synonyms of Simon of Genoa), attributed to the physician to Pope Nicholas IV in the year 1288, was printed by Antonius Zarotus at Milan in 1473. Referring to a copy held in the library of the College of Physicians of Philadelphia, Henry wrote in 1905 that "It is the first edition of the first medical dictionary." However, this claim is disputed as the composition only included lists of herbs and drugs. At the beginning of the Renaissance, Italian universities began teaching a broader range of courses, such as law and medicine, whilst universities in northern Europe were still dominated by theology and related topics. All European universities required Latin proficiency for admission. Latin was the dominant language of university education, where rules were enforced against the use of vernacular languages. Lectures and debates took place in Latin, and writing was in Latin, across the curriculum. Latin dominated topics of international academic and scientific interest, especially at


id: medical-terminology-20260505-11
file: data/en.wikipedia.org/wiki/Medical_terminology-11.md
category: reference
title: Medical terminology
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Medical_terminology
date_saved: 2026-05-05T07:25:35.931317+00:00
instance: kb-cron
parent: medical-terminology-20260505-0
summary:
== Works cited == Bertok L, Chow D (2005). Bertok L, Chow D (eds.). Natural Immunity. Vol. 5 (1st ed.). Elsevier Science. ISBN 978-0-44451-755-5. Breinstrup, Thomas (2006). "Un revolution in le mundo scientific" [A revolution in the scientific world]. Historia de interlingua (in Interlingua). Archived from the original on 2019-04-27. Retrieved 2025-01-14. Chung, Kyung Won (2005). Gross Anatomy (Board Review). Lippincott Williams & Wilkins. ISBN 0-7817-5309-0. Cook, Chad E. (2012). Orthopedic Manual Therapy: An Evidence Based Approach (2nd ed.). Upper Saddle River, NJ: Pearson Education. ISBN 978-0-13-802173-3. Breinstrup, Th.; Stenström, Ingvar; Olsson, Jesper, eds. (2007) [2001]. "Biographias: Ingvar Stenström". Interlingua (in Interlingua). Archived from the original on 2019-04-26. Retrieved 2025-01-15. Dinour, Lauren M. (2019). "Speaking Out on 'Breastfeeding' Terminology: Recommendations for Gender-Inclusive Language in Research and Reporting". Breastfeeding Medicine. 14 (8). Mary Ann Liebert: 523–532. doi:10.1089/bfm.2019.0110. PMID 31364867. Fielding, Lucie (2021). Trans Sex: Clinical Approaches to Trans Sexualities and Erotic Embodiments. New York: Routledge. ISBN 978-0-429-31829-0. Edwards, Heather. "Anatomy Talk". pp. 165–168. Gopsill, Frank Peter (1990). International languages: a matter for Interlingua. Sheffield, England: British Interlingua Society. ISBN 0-9511695-6-4. OCLC 27813762. Gopsill, Frank Peter (1994). Interlingua today: A course for beginners. Sheffield, England: British Interlingua Society. ISBN 978-1-898017-01-1. Gopsill, Frank


id: list-of-medical-roots-and-affixes-20260505-0
file: data/en.wikipedia.org/wiki/List_of_medical_roots_and_affixes-0.md
category: reference
title: List of medical roots and affixes
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_medical_roots_and_affixes
date_saved: 2026-05-05T07:25:37.200007+00:00
instance: kb-cron
parent: list-of-medical-roots-and-affixes-20260505-0
summary:
This is a list of roots, suffixes, and prefixes used in medical terminology, their meanings, and their etymologies. Most of them are combining forms in Neo-Latin and hence international scientific vocabulary. There are a few general rules about how they combine. First, prefixes and suffixes, most of which are derived from ancient Greek or classical Latin, have a droppable vowel, usually -o-. As a general rule, this vowel almost always acts as a joint-stem to connect two consonantal roots (e.g. arthr- + -o- + -logy = arthrology), but generally, the -o- is dropped when connecting to a vowel-stem (e.g. arthr- + -itis = arthritis, instead of arthr-o-itis). Second, medical roots generally go together according to language, i.e., Greek prefixes occur with Greek suffixes and Latin prefixes with Latin suffixes. Although international scientific vocabulary is not stringent about segregating combining forms of different languages, it is advisable when coining new words not to mix different lingual roots. == Prefixes and suffixes == The following is an alphabetical list of medical prefixes and suffixes, along with their meanings, origins, and English examples. === A === === B === === C === === D === === E === === F === === G


id: abrasion-collar-20260505-0
file: data/en.wikipedia.org/wiki/Abrasion_collar-0.md
category: reference
title: Abrasion collar
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Abrasion_collar
date_saved: 2026-05-05T07:25:38.336536+00:00
instance: kb-cron
parent: abrasion-collar-20260505-0
summary:
An abrasion collar, also known as an abrasion ring or abrasion rim, is a narrow ring of stretched, abraded skin immediately surrounding projectile wounds, such as gunshot wounds. It is most commonly associated with entrance wounds and is a mechanical defect due to a projectile's penetration through the skin. It is caused by a temporary over-stretching of the skin surrounding the projectile's point of penetration. Like all skin abrasions, the abrasion collar tends to dry out due to scraping away of the skin's outer layers and the collapse and dehydration of the underlying cells; it therefore becomes easier to discern with time. This defect is most often seen around rifled firearm entrance wounds due to the striations or grooves in the bullet's surface caused by the rifling on the inside of the weapon's barrel; however, certain other high-velocity projectile wounds can also have the same effect. An abrasion collar is usually found in association with a contusion collar of bruising caused by damaged blood vessels in the skin by hydrostatic forces from the bullet's entry. An abrasion rim defect is also possible in firearm exit wounds under certain circumstances, such as if the skin at the exit was crushed between


id: absolute-risk-20260505-0
file: data/en.wikipedia.org/wiki/Absolute_risk-0.md
category: reference
title: Absolute risk
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Absolute_risk
date_saved: 2026-05-05T07:25:39.613272+00:00
instance: kb-cron
parent: absolute-risk-20260505-0
summary:
Absolute risk (or AR) is the probability or chance of an event. It is usually used for the number of events (such as a disease) that occurred in a group, divided by the number of people in that group. Absolute risk is one of the most understandable ways of communicating health risks to the general public. In difference to absolute risk, the relative risk (RR) is the ratio of the probability of an outcome in an exposed group to the probability of the outcome in an unexposed group. == See also == Absolute risk reduction Relative risk reduction == References == == Further reading == Woloshin, Steven; Schwartz, Lisa M.; Welch, H. Gilbert (2008). Know Your Chances: Understanding Health Statistics. Berkeley (CA): University of California Press. PMID 23469386.


id: acholia-20260505-0
file: data/en.wikipedia.org/wiki/Acholia-0.md
category: reference
title: Acholia
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Acholia
date_saved: 2026-05-05T07:25:40.769428+00:00
instance: kb-cron
parent: acholia-20260505-0
summary:
Acholia is pale feces, due to lack of bile which results in the normal brown colour. It is a sign of reduced conjugated bilirubin into the bowel, as a result of a problem in the liver itself or in the biliary tree. == Signs and symptoms == Acholia results in pale feces. == Cause == A condition in which little or no bile is secreted or the flow of bile into the digestive tract is obstructed. The acholia is a sign of many diseases, such as hepatitis. == Etymology == Ancient Greek: a + chole (without bile). == See also == Choluria == References == == External links ==


id: active-placebo-20260505-0
file: data/en.wikipedia.org/wiki/Active_placebo-0.md
category: reference
title: Active placebo
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Active_placebo
date_saved: 2026-05-05T07:25:41.989643+00:00
instance: kb-cron
parent: active-placebo-20260505-0
summary:
An active placebo is a placebo that produces noticeable side effects that may convince the person being treated that they are receiving a legitimate treatment, rather than an inert placebo. == Nomenclature == According to a 1965 paper, the term "concealed placebo" (German: Kaschiertes Placebo) was suggested in a 1959 paper published in German. == Example == An example of an active placebo is the 1964 work of Shader and colleagues who used a combination of low-dose phenobarbital plus atropine to mimic the sedation and dry mouth produced by phenothiazines. Morphine and gabapentin are painkillers with the common side effects of sleepiness and dizziness. In a 2005 study assessing the effects of these painkillers on neuropathic pain, lorazepam was chosen as an active placebo because it is not a painkiller but it does cause sleepiness and can cause dizziness. Testing from the late 1950s onwards on narcotic analgesics like morphine also has used dicyclomine as an active placebo, and on some occasions it was reported to cause the Straub mouse tail reaction, as do most narcotics. Clonidine is now finding more use as an active placebo for narcotics. == References == Shader, R. I.; Cohler, J.; Elashoff, R.; Grinspoon, L.


id: acute-medicine-20260505-0
file: data/en.wikipedia.org/wiki/Acute_(medicine)-0.md
category: reference
title: Acute (medicine)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Acute_(medicine)
date_saved: 2026-05-05T07:25:43.224820+00:00
instance: kb-cron
parent: acute-medicine-20260505-0
summary:
In medicine, describing a disease as acute denotes that it is of recent onset; it occasionally denotes a short duration. The quantification of how much time constitutes "short" and "recent" varies by disease and by context, but the core denotation of "acute" is always qualitatively in contrast with "chronic", which denotes long-lasting disease (for example, in acute leukaemia and chronic leukaemia). In the context of the mass noun "acute disease", it refers to the acute phase (that is, a short course) of any disease entity. For example, in an article on ulcerative enteritis in poultry, the author says, "in acute disease there may be increased mortality without any obvious signs", referring to the acute form or phase of ulcerative enteritis. == Meaning variations == A mild stubbed toe is an acute injury. Similarly, many acute upper respiratory infections and acute gastroenteritis cases in adults are mild and usually resolve within a few days or weeks. The term "acute" is also included in the definition of several diseases, such as severe acute respiratory syndrome, acute leukaemia, acute myocardial infarction, and acute hepatitis. This is often to distinguish diseases from their chronic forms, such as chronic leukaemia, or to highlight the sudden


id: acute-abdomen-20260505-0
file: data/en.wikipedia.org/wiki/Acute_abdomen-0.md
category: reference
title: Acute abdomen
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Acute_abdomen
date_saved: 2026-05-05T07:25:44.430078+00:00
instance: kb-cron
parent: acute-abdomen-20260505-0
summary:
An acute abdomen refers to a sudden, severe abdominal pain. It is in many cases a medical emergency, requiring urgent and specific diagnosis. Several causes need immediate surgical treatment. == Differential diagnosis == Common causes of an acute abdomen include a gastrointestinal perforation, peptic ulcer disease, mesenteric ischemia, acute cholecystitis, appendicitis, diverticulitis, pancreatitis, and an abdominal hemorrhage. However, this is a non-exhaustive list and other less common causes may also lead to an acute abdomen. In pregnant patient, a tubo-ovarian abscess, ruptured ovarian cyst or a ruptured ectopic pregnancy are common causes of an acute abdomen. === Ischemic acute abdomen === Vascular disorders are more likely to affect the small bowel than the large bowel. Arterial supply to the intestines is provided by the superior and inferior mesenteric arteries (SMA and IMA respectively), both of which are direct branches of the aorta. Clinically, patients present with diffuse abdominal pain, bowel distention, and bloody diarrhea. On physical exam, bowel sounds will be absent. Laboratory tests reveal a neutrophilic leukocytosis, sometimes with a left shift, and increased serum amylase. Abdominal radiography will show many air-fluid levels, as well as widespread edema. Acute ischemic abdomen is a surgical emergency. Typically, treatment involves removal


id: acute-pericarditis-20260505-0
file: data/en.wikipedia.org/wiki/Acute_pericarditis-0.md
category: reference
title: Acute pericarditis
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Acute_pericarditis
date_saved: 2026-05-05T07:25:45.765948+00:00
instance: kb-cron
parent: acute-pericarditis-20260505-0
summary:
Acute pericarditis is a type of pericarditis (inflammation of the sac surrounding the heart, the pericardium) usually lasting less than 4 to 6 weeks. It is the most common condition affecting the pericardium. == Signs and symptoms == Chest pain is one of the common symptoms of acute pericarditis. It is usually of sudden onset, occurring in the anterior chest and often has a sharp quality that worsens with breathing in or coughing, due to inflammation of the pleural surface at the same time. The pain may be reduced with sitting up and leaning forward while worsened with lying down, and also may radiate to the back, to one or both trapezius ridges. However, the pain can also be dull and steady, resembling the chest pain in an acute myocardial infarction. As with any chest pain, other causes must also be ruled out, such as GERD, pulmonary embolism, muscular pain, etc. A pericardial friction rub is a very specific sign of acute pericarditis, meaning the presence of this sign invariably indicates presence of disease. However, absence of this sign does not rule out disease. This rub can be best heard by the diaphragm of the stethoscope at the left sternal


id: acute-pericarditis-20260505-1
file: data/en.wikipedia.org/wiki/Acute_pericarditis-1.md
category: reference
title: Acute pericarditis
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Acute_pericarditis
date_saved: 2026-05-05T07:25:45.765948+00:00
instance: kb-cron
parent: acute-pericarditis-20260505-0
summary:
stage 1 -- diffuse, positive, ST elevations with reciprocal ST depression in aVR and V1. Elevation of PR segment in aVR and depression of PR in other leads especially left heart V5, V6 leads indicates atrial injury. stage 2 -- normalization of ST and PR deviations stage 3 -- diffuse T wave inversions (may not be present in all patients) stage 4 -- EKG becomes normal OR T waves may be indefinitely inverted The two most common clinical conditions where ECG findings may mimic pericarditis are acute myocardial infarction (AMI) and generalized early repolarization. As opposed to pericarditis, AMI usually causes localized convex ST-elevation usually associated with reciprocal ST-depression which may also be frequently accompanied by Q-waves, T-wave inversions (while ST is still elevated unlike pericarditis), arrhythmias and conduction abnormalities. In AMI, PR-depressions are rarely present. Early repolarization usually occurs in young males (age <40 years) and ECG changes are characterized by terminal R-S slurring, temporal stability of ST-deviations and J-height/ T-amplitude ratio in V5 and V6 of <25% as opposed to pericarditis where terminal R-S slurring is very uncommon and J-height/ T-amplitude ratio is ≥ 25%. Very rarely, ECG changes in hypothermia may mimic pericarditis, however differentiation can be


id: acute-pericarditis-20260505-2
file: data/en.wikipedia.org/wiki/Acute_pericarditis-2.md
category: reference
title: Acute pericarditis
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Acute_pericarditis
date_saved: 2026-05-05T07:25:45.765948+00:00
instance: kb-cron
parent: acute-pericarditis-20260505-0
summary:
presence of moderate or severe cardiac tamponade diagnostic purpose for suspected purulent, tuberculosis, or neoplastic pericarditis persistent symptomatic pericardial effusion NSAIDs in viral or idiopathic pericarditis. In patients with underlying causes other than viral, the specific etiology should be treated. With idiopathic or viral pericarditis, NSAID is the mainstay treatment. Goal of therapy is to reduce pain and inflammation. The course of the disease may not be affected. The preferred NSAID is ibuprofen because of rare side effects, better effect on coronary flow, and larger dose range. Depending on severity, dosing is between 300 and 800 mg every 6–8 hours for days or weeks as needed. An alternative protocol is aspirin 800 mg every 6–8 hours. Dose tapering of NSAIDs may be needed. In pericarditis following acute myocardial infarction, NSAIDs other than aspirin should be avoided since they can impair scar formation. As with all NSAID use, GI protection should be engaged. Failure to respond to NSAIDs within one week (indicated by persistence of fever, worsening of condition, new pericardial effusion, or continuing chest pain) likely indicates that a cause other than viral or idiopathic is in process. Colchicine, which has been essential to treat recurrent pericarditis, has been supported


id: purulent-pericarditis-20260505-0
file: data/en.wikipedia.org/wiki/Purulent_pericarditis-0.md
category: reference
title: Purulent pericarditis
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Purulent_pericarditis
date_saved: 2026-05-05T07:25:47.030592+00:00
instance: kb-cron
parent: purulent-pericarditis-20260505-0
summary:
Purulent pericarditis refers to localized inflammation in the setting of infection of the pericardial sac surrounding the heart. In contrast to other causes of pericarditis which may have a viral etiology, purulent pericarditis refers specifically to bacterial or fungal infection of the pericardial sac. Clinical etiologies of purulent pericarditis may include recent surgery, adjacent infection, trauma, or even primary infection. The onset of purulent pericarditis is usually acute, with most individuals presenting to a medical facility approximately 3 days following the onset of symptoms. As a subtype of pericarditis, purulent pericarditis often presents with substernal chest pain that is exacerbated by deep breathing and lying in the supine position. Other presenting features that may be more specific for purulent pericarditis include fever, rigors/chills, and cardiorespiratory signs (i.e., tachycardia, friction rub, pulsus paradoxus, pericardial effusion, cardiac tamponade, pleural effusion). The incidence of cardiac tamponade varies from 42-77% and is associated with rapid-onset mortality, especially without prompt intervention. Chest radiography may reveal cardiomegaly, pneumonia, pleural effusion, and/or mediastinal widening. Electrocardiogram (ECG) is a component of the diagnostic work-up which may suggest pericarditis as the underlying cause of symptoms. The ECG findings for purulent pericarditis are similar to those for other etiologies of


id: adherence-medicine-20260505-0
file: data/en.wikipedia.org/wiki/Adherence_(medicine)-0.md
category: reference
title: Adherence (medicine)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Adherence_(medicine)
date_saved: 2026-05-05T07:25:48.247796+00:00
instance: kb-cron
parent: adherence-medicine-20260505-0
summary:
In medicine, patient adherence describes the degree to which a person correctly follows medical advice. Most commonly, it refers to medication or drug compliance, but it can also apply to other situations such as medical device use, self care, self-directed exercises, therapy sessions, or medical follow-up visits. Patient compliance and patient adherence may be used interchangeably but adherence emphasizes the role of the patient in decision-making and factors that influence the ability to follow instructions. Concordance is the process by which a patient and clinician make decisions together about treatment. Differences in terminology reflect regional variation, deliberate distinctions, and the preferences of various groups and organizations. Both patient and health-care provider affect adherence, and a positive physician-patient relationship is the most important factor in improving compliance. Access to care plays a role in patient adherence, whereby greater wait times to access care contributing to greater absenteeism. The cost of prescription medication and potential side effects also play a role. Worldwide, non-compliance is a major obstacle to the effective delivery of health care. 2003 estimates from the World Health Organization indicated that only about 50% of patients with chronic diseases living in developed countries follow treatment recommendations with particularly low rates


id: adherence-medicine-20260505-1
file: data/en.wikipedia.org/wiki/Adherence_(medicine)-1.md
category: reference
title: Adherence (medicine)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Adherence_(medicine)
date_saved: 2026-05-05T07:25:48.247796+00:00
instance: kb-cron
parent: adherence-medicine-20260505-0
summary:
=== Literacy === In 1999 one fifth of UK adults, nearly seven million people, had problems with basic skills, especially functional literacy and functional numeracy, described as: "The ability to read, write and speak in English, and to use mathematics at a level necessary to function at work and in society in general." This made it impossible for them to effectively take medication, read labels, follow drug regimes, and find out more. In 2003, 20% of adults in the UK had a long-standing illness or disability and a national study for the UK Department of Health, found more than one-third of people with poor or very poor health had literary skills of Entry Level 3 or below. A study of the relationship of literacy to asthma knowledge revealed that only 31% of asthma patients with a reading level of a ten-year-old knew they needed to see the doctors, even when they were not having an asthma attack, compared to 90% with a high school graduate reading level. === Treatment cost === In 2013 the US National Community Pharmacists Association sampled for one month 1,020 Americans above age 40 for with an ongoing prescription to take medication for a chronic condition


id: adherence-medicine-20260505-2
file: data/en.wikipedia.org/wiki/Adherence_(medicine)-2.md
category: reference
title: Adherence (medicine)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Adherence_(medicine)
date_saved: 2026-05-05T07:25:48.247796+00:00
instance: kb-cron
parent: adherence-medicine-20260505-0
summary:
==== Medication Possession Ratio ==== Prescription medical claims records can be used to estimate medication adherence based on fill rate. Patients can be routinely defined as being 'Adherent Patients' if the amount of medication furnished is at least 80% based on days' supply of medication divided by the number of days patient should be consuming the medication. This percentage is called the medication possession ratio (MPR). 2013 work has suggested that a medication possession ratio of 90% or above may be a better threshold for deeming consumption as 'Adherent'. Two forms of MPR can be calculated, fixed and variable. Calculating either is relatively straightforward, for Variable MPR (VMPR) it is calculated as the number of days' supply divided by the number of elapsed days including the last prescription. V M P R = All days' supply Elapsed days (inclusive of last prescription) {\displaystyle VMPR={\dfrac {\text{All days' supply}}{\text{Elapsed days (inclusive of last prescription)}}}} For the Fixed MPR (FMPR) the calculation is similar but the denominator is the number of days in a year whilst the numerator is constrained to be the number of days' supply within the year that the patient has been prescribed. F M P R = All days'


id: adherence-medicine-20260505-3
file: data/en.wikipedia.org/wiki/Adherence_(medicine)-3.md
category: reference
title: Adherence (medicine)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Adherence_(medicine)
date_saved: 2026-05-05T07:25:48.247796+00:00
instance: kb-cron
parent: adherence-medicine-20260505-0
summary:
=== Technology === In 2012 it was predicted that as telemedicine technology improves, physicians will have better capabilities to remotely monitor patients in real-time and to communicate recommendations and medication adjustments using personal mobile devices, such as smartphones, rather than waiting until the next office visit. Medication Event Monitoring Systems (MEMS), as in the form of smart medicine bottle tops, smart pharmacy vials or smart blister packages as used in clinical trials and other applications where exact compliance data are required, work without any patient input, and record the time and date the bottle or vial was accessed, or the medication removed from a blister package. The data can be read via proprietary readers, or NFC enabled devices, such as smartphones or tablets. A 2009 study stated that such devices can help improve adherence. More recently a 2016 scoping review suggested that in comparison to MEMS, median mediction adherence was grossly overestimated by 17% using self-report, by 8% using pill count and by 6% using rating as alternative methods for measuring medication adherence. The effectiveness of two-way email communication between health care professionals and their patients has not been adequately assessed. ==== Mobile phones ==== As of 2019, 5.15 billion


id: adherence-medicine-20260505-4
file: data/en.wikipedia.org/wiki/Adherence_(medicine)-4.md
category: reference
title: Adherence (medicine)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Adherence_(medicine)
date_saved: 2026-05-05T07:25:48.247796+00:00
instance: kb-cron
parent: adherence-medicine-20260505-0
summary:
Diabetes non-compliance is 98% in US and the principal cause of complications related to diabetes including nerve damage and kidney failure. Among patients with Type 2 Diabetes, adherence was found in less than one third of those prescribed sulphonylureas and/or metformin. Patients taking both drugs achieve only 13% adherence. Other aspects that drive medicine adherence rates is the idea of perceived self-efficacy and risk assessment in managing diabetes symptoms and decision making surrounding rigorous medication regiments. Perceived control and self-efficacy not only significantly correlate with each other, but also with diabetes distress psychological symptoms and have been directly related to better medication adherence outcomes. Various external factors also impact diabetic patients' self-management behaviors including health-related knowledge/beliefs, problem-solving skills, and self-regulatory skills, which all impact perceived control over diabetic symptoms. Additionally, it is crucial to understand the decision-making processes that drive diabetics in their choices surrounding risks of not adhering to their medication. While patient decision aids (PtDAs), sets of tools used to help individuals engage with their clinicians in making decisions about their healthcare options, have been useful in decreasing decisional conflict, improving transfer of diabetes treatment knowledge, and achieving greater risk perception for disease complications, their efficacy in medication


id: adhesion-medicine-20260505-0
file: data/en.wikipedia.org/wiki/Adhesion_(medicine)-0.md
category: reference
title: Adhesion (medicine)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Adhesion_(medicine)
date_saved: 2026-05-05T07:25:49.521012+00:00
instance: kb-cron
parent: adhesion-medicine-20260505-0
summary:
Adhesions are fibrous bands that form between tissues and organs, often as a result of irritation of internal surfaces during surgery, infections or trauma. They may be thought of as internal scar tissue that connects tissues not normally connected. == Pathophysiology == Adhesions form as a natural part of the body's healing process after surgery in a similar way that a scar forms. The term "adhesion" is applied when the scar extends from within one tissue across to another, usually across a virtual space such as the peritoneal cavity. Adhesion formation post-surgery typically occurs when two injured surfaces are close to one another. According to the "classical paradigm" of adhesion formation, the pathogenesis starts with inflammation and activation of the coagulation system which causes fibrin deposits onto the damaged tissues. The fibrin then connects the two adjacent structures where damage of the tissues occurred. The fibrin acts like a glue to seal the injury and builds the fledgling adhesion, said at this point to be "fibrinous." In body cavities such as the peritoneal, pericardial, and synovial cavities, a family of fibrinolytic enzymes may act to limit the extent of the initial fibrinous adhesion, and may even dissolve it. In many


id: adhesion-medicine-20260505-1
file: data/en.wikipedia.org/wiki/Adhesion_(medicine)-1.md
category: reference
title: Adhesion (medicine)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Adhesion_(medicine)
date_saved: 2026-05-05T07:25:49.521012+00:00
instance: kb-cron
parent: adhesion-medicine-20260505-0
summary:
=== Abdominal surgery === A study showed that more than 90% of people develop adhesions following open abdominal surgery and that 55–100% of women develop adhesions following pelvic surgery. Adhesions from prior abdominal or pelvic surgery may obscure visibility and access at subsequent abdominal or pelvic surgery. In a very large study (29,790 participants) published in British medical journal The Lancet, 35% of patients who underwent open abdominal or pelvic surgery were readmitted to the hospital an average of two times after their surgery, due to adhesion-related or adhesion-suspected complications. Over 22% of all readmissions occurred in the first year after the initial surgery. Adhesion-related complexity at reoperation adds significant risk to subsequent surgical procedures. Certain organs and structures in the body are more prone to adhesion formation than others. The omentum is particularly susceptible to adhesion formation; one study found that 92% of post-operative adhesions were found in the omentum. It appears that the omentum is the chief organ responsible for "spontaneous" adhesion formation (i.e. no prior history of surgery). In another study, 100% of spontaneous adhesion formations were associated with the omentum. One method to reduce the formation of adhesions following abdominal surgery is hydroflotation, in which the


id: admission-note-20260505-0
file: data/en.wikipedia.org/wiki/Admission_note-0.md
category: reference
title: Admission note
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Admission_note
date_saved: 2026-05-05T07:25:50.831652+00:00
instance: kb-cron
parent: admission-note-20260505-0
summary:
An admission note is part of a medical record that documents the patient's status (including history and physical examination findings), reasons why the patient is being admitted for inpatient care to a hospital or other facility, and the initial instructions for that patient's care. == Purpose == Admission notes document the reasons why a patient is being admitted for inpatient care to a hospital or other facility, the patient's baseline status, and the initial instructions for that patient's care. Health care professionals use them to record a patient's baseline status and may write additional on-service notes, progress notes (SOAP notes), preoperative notes, operative notes, postoperative notes, procedure notes, delivery notes, postpartum notes, and discharge notes. These notes constitute a large part of the medical record. Medical students often develop their clinical reasoning skills by writing admission notes. The traditional, rational definition of being admitted usually involves spending an overnight in the hospital. This definition is sometimes stretched in the U.S. medical billing industry, where hospital corporations may blur the definitions of "admission" and "observation" because of reimbursement rules under which healthcare payors pay less for the care if an "admission" was involved. == Components == An admission note may sometimes


id: adrenergic-storm-20260505-0
file: data/en.wikipedia.org/wiki/Adrenergic_storm-0.md
category: reference
title: Adrenergic storm
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Adrenergic_storm
date_saved: 2026-05-05T07:25:52.151132+00:00
instance: kb-cron
parent: adrenergic-storm-20260505-0
summary:
An adrenergic storm is a sudden and dramatic increase in serum levels of the catecholamines adrenaline and noradrenaline (also known as epinephrine and norepinephrine respectively), with a less significant increase in dopamine transmission. It is a life-threatening condition because of extreme tachycardia and hypertension, and is especially dire for those with prior heart problems. If treatment is prompt, prognosis is good; typically large amounts of diazepam or other benzodiazepines are administered alongside beta blockers. Beta blockers are contraindicated in some patients, so other antihypertensive medication such as clonidine may be used. Antipsychotics are also used to treat the most severe psychiatric reactions such as psychosis, paranoia or terror, after their use was formerly discouraged because of their potential to prolong the QT interval; however, more recent research performed since 2019 has revealed that this and other severe side effects are rare and their occurrence does not warrant banning antipsychotics from the treatment of adrenergic crises for which they can be extremely useful. Adrenergic storms are usually caused by overdoses of stimulants, especially cocaine or methamphetamine, or eating foods high in tyramine while taking monoamine oxidase inhibitors. A subarachnoid hemorrhage can also cause an adrenergic storm. A catecholamine storm is part


id: adrenergic-storm-20260505-1
file: data/en.wikipedia.org/wiki/Adrenergic_storm-1.md
category: reference
title: Adrenergic storm
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Adrenergic_storm
date_saved: 2026-05-05T07:25:52.151132+00:00
instance: kb-cron
parent: adrenergic-storm-20260505-0
summary:
=== Differential diagnosis === Because the adrenergic storm overlaps with so many other similar conditions, such as hypertensive crises, stimulant intoxication or overdose, or even panic attack, and because the treatments for these overlapping conditions are largely alike, it is not necessary to obtain a differential and definitive diagnosis before initiating treatment. However, analysis of the patient's medical history, checked against the possible causes of the adrenergic storm such as those above, should be done, because some adrenergic storms can be caused by serious underlying conditions. If a patient has an adrenergic storm and all or most of the other factors are ruled out, the adrenergic storm could lead to the discovery of a pheochromocytoma, which can become malignant. However, not all cases of adrenergic storm have an identifiable cause. Serotonin syndrome, in which an excess of serotonin in the synapses causes a similar crisis of hypertension and mental confusion, could be confused with an adrenergic storm. Serotonin, being a tryptamine (non-catecholamine) involved in higher brain functions, can cause dangerous hypertension and tachycardia from its effects on the sympathetic nervous system. Symptoms caused by excessive adrenergic signalling can occur alongside those of serotonergic signalling. Abnormal echocardiograms or chest pain are


id: adverse-effect-20260505-0
file: data/en.wikipedia.org/wiki/Adverse_effect-0.md
category: reference
title: Adverse effect
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Adverse_effect
date_saved: 2026-05-05T07:25:53.424672+00:00
instance: kb-cron
parent: adverse-effect-20260505-0
summary:
An adverse effect is an undesired harmful effect resulting from a medication or other intervention, such as surgery. An adverse effect may be termed a "side effect", when judged to be secondary to a main or therapeutic effect. The term complication is similar to adverse effect, but the latter is typically used in pharmacological contexts, or when the negative effect is expected or common. If the negative effect results from an unsuitable or incorrect dosage or procedure, this is called a medical error and not an adverse effect. Adverse effects are sometimes referred to as "iatrogenic" because they are generated by a physician/treatment. Some adverse effects occur only when starting, increasing or discontinuing a treatment. Using a drug or other medical intervention which is contraindicated may increase the risk of adverse effects. Adverse effects may cause complications of a disease or procedure and negatively affect its prognosis. They may also lead to non-compliance with a treatment regimen. Adverse effects of medical treatment resulted in 142,000 deaths in 2013 up from 94,000 deaths in 1990 globally. The harmful outcome is usually indicated by some result such as morbidity, mortality, alteration in body weight, levels of enzymes, loss of function, or as


id: adverse-effect-20260505-1
file: data/en.wikipedia.org/wiki/Adverse_effect-1.md
category: reference
title: Adverse effect
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Adverse_effect
date_saved: 2026-05-05T07:25:53.424672+00:00
instance: kb-cron
parent: adverse-effect-20260505-0
summary:
=== Limitations === In principle, medical professionals are required to report all adverse effects related to a specific form of therapy. In practice, it is at the discretion of the professional to determine whether a medical event is at all related to the therapy. As a result, routine adverse effects reporting often may not include long-term and subtle effects that may ultimately be attributed to a therapy. Part of the difficulty is identifying the source of a complaint. A headache in a patient taking medication for influenza may be caused by the underlying disease or may be an adverse effect of the treatment. In patients with end-stage cancer, death is a very likely outcome and whether the drug is the cause or a bystander is often difficult to discern. == By situation == === Medical procedures === Surgery may have a number of undesirable or harmful effects, such as infection, hemorrhage, inflammation, scarring, loss of function, or changes in local blood flow. They can be reversible or irreversible, and a compromise must be found by the physician and the patient between the beneficial or life-saving consequences of surgery versus its adverse effects. For example, a limb may be lost to


id: adverse-effect-20260505-2
file: data/en.wikipedia.org/wiki/Adverse_effect-2.md
category: reference
title: Adverse effect
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Adverse_effect
date_saved: 2026-05-05T07:25:53.424672+00:00
instance: kb-cron
parent: adverse-effect-20260505-0
summary:
==== Examples with specific medications ==== Abortion, miscarriage or uterine hemorrhage associated with misoprostol (Cytotec), a labor-inducing drug (this is a case where the adverse effect has been used legally and illegally for performing abortions) Addiction to many sedatives and analgesics, such as diazepam, morphine, etc. Birth defects associated with thalidomide Bleeding of the intestine associated with aspirin therapy Cardiovascular disease associated with COX-2 inhibitors (i.e. Vioxx) Deafness and kidney failure associated with gentamicin (an antibiotic) Death, following sedation, in children using propofol (Diprivan) Depression or hepatic injury caused by interferon Diabetes caused by atypical antipsychotic medications (neuroleptic psychiatric drugs) Diarrhea caused by the use of orlistat (Xenical) Erectile dysfunction associated with many drugs, such as antidepressants Fever associated with vaccination Glaucoma associated with corticosteroid-based eye drops Hair loss and anemia may be caused by chemotherapy against cancer, leukemia, etc. Headache following spinal anaesthesia Hypertension in ephedrine users, which prompted FDA to remove the dietary supplement status of ephedra extracts Insomnia caused by stimulants, methylphenidate (Ritalin), Adderall, etc. Lactic acidosis associated with the use of stavudine (Zerit, for HIV therapy) or metformin (for diabetes) Mania caused by corticosteroids Liver damage from paracetamol Melasma and thrombosis associated with use of estrogen-containing


id: adverse-outcome-pathway-20260505-0
file: data/en.wikipedia.org/wiki/Adverse_outcome_pathway-0.md
category: reference
title: Adverse outcome pathway
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Adverse_outcome_pathway
date_saved: 2026-05-05T07:25:55.006766+00:00
instance: kb-cron
parent: adverse-outcome-pathway-20260505-0
summary:
An adverse outcome pathway (AOP) is structured representation of biological events leading to adverse effects and is considered relevant to risk assessment. The AOP links in a linear way existing knowledge along one or more series of causally connected key events (KE) between two points — a molecular initiating event (MIE) and an adverse outcome (AO) that occur at a level of biological organization relevant to risk assessment. The linkage between the events is described by key event relationships (KER) that describe the causal relationships between the key events. AOPs are important for expanding the use of mechanistic toxicological data for risk assessment and regulatory applications with recent applications in further disciplines such as climate science. == Background == In 2012, the Organisation for Economic Co-operation and Development (OECD) launched a new programme on the development of adverse outcome pathways. A guidance document describes in detail how AOPs are to be developed, reviewed, agreed and published at the OECD level. The AOP development and reviewing workflow is intended to take place via a web-based IT management tool: the Adverse Outcome Pathway Knowledge Base (AOP-KB) which is currently still under development. It is the wiki-based, user-friendly tool providing open-source interface for


id: age-of-onset-20260505-0
file: data/en.wikipedia.org/wiki/Age_of_onset-0.md
category: reference
title: Age of onset
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Age_of_onset
date_saved: 2026-05-05T07:25:56.198547+00:00
instance: kb-cron
parent: age-of-onset-20260505-0
summary:
The age of onset is the age at which an individual acquires, develops, or first experiences a condition or symptoms of a disease or disorder. For instance, the general age of onset for the spinal disease scoliosis is "10-15 years old," meaning that most people develop scoliosis when they are of age between ten and fifteen years. Diseases are often categorized by their age of onset as congenital, infantile, juvenile, or adult. Missed or delayed diagnosis often occurs if a disease that is typically diagnosed in juveniles (such as asthma) is present in adults, and vice versa (such as arthritis). Depending on the disease, ages of onset may impact features such as phenotype, as is the case in Parkinson's and Huntington's diseases. For example, the phenotype for juvenile Huntington's disease clearly differs from adult-onset Huntington's disease and late-onset Parkinson's exhibits more severe motor and non-motor phenotypes. == Causes == Germ-line mutations are often at least in part the cause of disease onset at an earlier age. Though many germ-line mutations are deleterious, the genetic lens through which they may be viewed may provide insights to treatment, possibly through genetic counseling. In some cases, the age of onset may be the


id: agenesis-20260505-0
file: data/en.wikipedia.org/wiki/Agenesis-0.md
category: reference
title: Agenesis
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Agenesis
date_saved: 2026-05-05T07:25:57.418271+00:00
instance: kb-cron
parent: agenesis-20260505-0
summary:
In medicine, agenesis () refers to the failure of an organ to develop during embryonic growth and development due to the absence of primordial tissue. Many forms of agenesis are referred to by individual names, depending on the organ affected: Agenesis of the corpus callosum - failure in the development of the band of fibers connecting the cerebral hemispheres Renal agenesis - failure of one or both of the kidneys to develop Amelia - failure of the arms or legs to develop Penile agenesis - failure of penis to develop Müllerian agenesis - failure of the uterus and part of the vagina to develop Agenesis of the gallbladder - failure of the Gallbladder to develop. A person may not realize they have this condition unless they undergo surgery or medical imaging, since the gallbladder is neither externally visible nor essential. == Eye agenesis == Eye agenesis is a medical condition in which people are born with no eyes. == Dental & oral agenesis == Anodontia, absence of all primary or permanent teeth. Aglossia, absence of the tongue. Agnathia, absence of the jaw. Wisdom tooth agenesis - most adult humans have three molars (on each upper/lower left/right side), with the third


id: agnosia-20260505-0
file: data/en.wikipedia.org/wiki/Agnosia-0.md
category: reference
title: Agnosia
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Agnosia
date_saved: 2026-05-05T07:25:58.633985+00:00
instance: kb-cron
parent: agnosia-20260505-0
summary:
Agnosia is a neurological disorder characterized by an inability to process sensory information. Often there is a loss of ability to recognize objects, persons, sounds, shapes, or smells while the specific sense is neither defective nor is there any significant memory loss. It is usually associated with brain injury or neurological illness, particularly after damage to the occipitotemporal border, which is part of the ventral stream. Agnosia affects only a single modality, such as vision or hearing. More recently, a top-down interruption is considered to cause the disturbance of handling perceptual information. == Types == === Visual agnosia === Visual agnosia is a broad category that refers to a deficiency in the ability to recognize visual objects. Visual agnosia can be further subdivided into two different subtypes: apperceptive visual agnosia and associative visual agnosia. Individuals with apperceptive visual agnosia display the ability to see contours and outlines when shown an object, but they experience difficulty if asked to categorize objects. Apperceptive visual agnosia is associated with damage to one hemisphere, specifically damage to the posterior sections of the right hemisphere. In contrast, individuals with associative visual agnosia experience difficulty when asked to name objects. Associative agnosia is associated with damage


id: agnosia-20260505-1
file: data/en.wikipedia.org/wiki/Agnosia-1.md
category: reference
title: Agnosia
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Agnosia
date_saved: 2026-05-05T07:25:58.633985+00:00
instance: kb-cron
parent: agnosia-20260505-0
summary:
== Treatment == For all practical purposes, there is no direct cure. Patients may improve if information is presented in other modalities than the damaged one. Different types of therapies can help to reverse the effects of agnosia. In some cases, occupational therapy or speech therapy can improve agnosia, depending on its cause. Initially many individuals with a form of agnosia are unaware of the extent to which they have either a perceptual or recognition deficit. This may be caused by anosognosia, which is the lack of awareness of a deficit. This lack of awareness usually leads to a form of denial and resistance to any form of help or treatment. There are various methods that can be used which can help the individual recognize the impairment in perception or recognition that they may have. A patient can be presented with a stimulus to the impaired modality only to help increase their awareness of their deficit. Alternatively, a task can be broken down into its component parts so that the individual can see each part of the problem caused by the deficit. Once the individual acknowledges their perceptual or recognition deficit, a form of treatment may be recommended. There are


id: agonist-20260505-0
file: data/en.wikipedia.org/wiki/Agonist-0.md
category: reference
title: Agonist
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Agonist
date_saved: 2026-05-05T07:25:59.888594+00:00
instance: kb-cron
parent: agonist-20260505-0
summary:
An agonist is a chemical that activates a receptor to produce a biological response. Receptors are cellular proteins whose activation causes the cell to modify what it is currently doing. In contrast, an antagonist blocks the action of the agonist, while an inverse agonist causes an action opposite to that of the agonist. == Etymology == The word originates from the Greek word ἀγωνιστής (agōnistēs), "contestant; champion; rival" < ἀγών (agōn), "contest, combat; exertion, struggle" < ἄγω (agō), "I lead, lead towards, conduct; drive." == Types of agonists == Receptors can be activated by either endogenous agonists (such as hormones and neurotransmitters) or exogenous agonists (such as drugs), resulting in a biological response. A physiological agonist is a substance that creates the same bodily responses but does not bind to the same receptor. An endogenous agonist for a particular receptor is a compound naturally produced by the body that binds to and activates that receptor. For example, the endogenous agonist for serotonin receptors is serotonin, and the endogenous agonist for dopamine receptors is dopamine. Full agonists bind to and activate a receptor with the maximum response that an agonist can elicit at the receptor. One example of a drug that


id: allopathic-medicine-20260505-0
file: data/en.wikipedia.org/wiki/Allopathic_medicine-0.md
category: reference
title: Allopathic medicine
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Allopathic_medicine
date_saved: 2026-05-05T07:26:01.122654+00:00
instance: kb-cron
parent: allopathic-medicine-20260505-0
summary:
Allopathic medicine, or allopathy, from Ancient Greek ἄλλος (állos), meaning "other", and πάθος (páthos), meaning "pain", is a label originally used derogatorily by 19th-century homeopaths to describe heroic medicine. In its current usage, the term generally refers to contemporary conventional medicine. However, there are regional variations in usage of the term. For example, in the United States the term is primarily used in contrast with osteopathic medicine, especially in the field of medical education; whereas in India the term is used to distinguish modern medicine from Siddha medicine, Ayurveda, homeopathy, Unani and other alternative and traditional medicine traditions, especially when comparing treatments and drugs. The terms were coined in 1810 by the creator of homeopathy, Samuel Hahnemann. Heroic medicine was the conventional European medicine of the time and did not rely on evidence of effectiveness. It was based on the belief that disease is caused by an imbalance of the four "humours" (blood, phlegm, yellow bile, and black bile) and sought to treat disease symptoms by correcting that imbalance, using "harsh and abusive" methods to induce symptoms seen as opposite to those of diseases rather than treating their underlying causes: disease was caused by an excess of one humour and


id: allopathic-medicine-20260505-1
file: data/en.wikipedia.org/wiki/Allopathic_medicine-1.md
category: reference
title: Allopathic medicine
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Allopathic_medicine
date_saved: 2026-05-05T07:26:01.122654+00:00
instance: kb-cron
parent: allopathic-medicine-20260505-0
summary:
James Whorton discusses this historical pejorative usage: One form of verbal warfare used in retaliation by irregulars was the word "allopathy". ..."Allopathy" and "allopathic" were liberally employed as pejoratives by all irregular physicians of the nineteenth century, and the terms were considered highly offensive by those at whom they were directed. The generally uncomplaining acceptance of [the term] "allopathic medicine" by today's physicians is an indication of both a lack of awareness of the term's historical use and the recent thawing of relations between irregulars and allopaths. The controversy surrounding the term can be traced to its original usage during a heated 19th-century debate between practitioners of homeopathy and those they derisively referred to as "allopaths." Hahnemann used "allopathy" to refer to what he saw as a system of medicine that combats disease by using remedies that produce effects in a healthy subject that are different (hence the Greek root allo- "different") from the effects produced by the disease to be treated. The distinction comes from the use in homeopathy of substances that are meant to cause similar effects as the symptoms of a disease to treat patients (homeo - meaning "similar"). As used by homeopaths, the term allopathy has


id: allow-natural-death-20260505-0
file: data/en.wikipedia.org/wiki/Allow_natural_death-0.md
category: reference
title: Allow natural death
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Allow_natural_death
date_saved: 2026-05-05T07:26:02.316918+00:00
instance: kb-cron
parent: allow-natural-death-20260505-0
summary:
Allow Natural Death (AND) is a medical term defining the use of life-extending measures such as cardiopulmonary resuscitation (CPR). These orders emphasize patient comfort and pain management instead of life extension. Currently, American medical communities utilize "do not resuscitate," (DNR) orders to define patients' medical wishes. Those who propose to replace DNR with AND posit that DNR orders are ambiguous and require complex understanding between several parties, while AND orders are clearer. Proponents of replacing DNR with AND believe that AND terminology is more ethically conscientious DNR terminology. Research has been conducted regarding participant preference for AND vs. DNR terminology. The ease with which the terminology change can be practically incorporated depends on many factors such as costs and staff reeducation. == DNR vs. AND == DNR orders range in the extent of life-saving measures to be avoided, from solely prohibiting the use of resuscitation to prohibiting any action seen as life extending. Because there are many parties involved in a patient's end of life care - significant others, family, personal doctors, specialists and nurses - DNR orders are not always completely clear, leaving open possible violation of the patient's wishes. DNR terminology was replaced in 2005 by the American


id: allow-natural-death-20260505-1
file: data/en.wikipedia.org/wiki/Allow_natural_death-1.md
category: reference
title: Allow natural death
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Allow_natural_death
date_saved: 2026-05-05T07:26:02.316918+00:00
instance: kb-cron
parent: allow-natural-death-20260505-0
summary:
== Bibliography == Fan, S.-Y., Wang, Y.-W., & Lin, I.-M. (2018). Allow natural death versus do-not-resuscitate: Titles, information contents, outcomes, and the considerations related to do-not-resuscitate decision. BMC Palliative Care, 17(1). https://doi.org/10.1186/s12904-018-0367-4. This is a peer-reviewed scientific journal, so it should be a reliable source. It covers the topic in some depth, so it's helpful in establishing notability. Miljković, M. D., Emuron, D., Rhodes, L., Abraham, J., & Miller, K. (2015). "allow natural death" versus "do not resuscitate": What do patients with advanced cancer choose? Journal of Palliative Medicine, 18(5), 457–460. https://doi.org/10.1089/jpm.2014.0369. This is a peer-reviewed scientific journal, so it should be a reliable source. It covers the topic in some depth, so it's helpful in establishing notability. Wittmann-Price, Ruth; Celia, Linda M. (Nov–Dec 2010). "Exploring perceptions of "do not resuscitate" and "allowing natural death" among physicians and nurses". Holistic Nursing Practice. 24 (6): 333–337. This is a peer-reviewed scientific journal, so it should be a reliable source. It is an observational study that was used to understand the differences in perception of the terms "allow natural death" and "do not resuscitate." Akdeniz, Melahat; Yardımcı, Bülent; Kavukcu, Ethem (2021). "Ethical considerations at the end-of-life care". SAGE open medicine. 9: 20503121211000918.


id: ambulatory-20260505-0
file: data/en.wikipedia.org/wiki/Ambulatory-0.md
category: reference
title: Ambulatory
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Ambulatory
date_saved: 2026-05-05T07:26:03.561509+00:00
instance: kb-cron
parent: ambulatory-20260505-0
summary:
The ambulatory (Latin: ambulatorium 'walking place') is the covered passage around a cloister or the processional way around the east end of a cathedral or large church and behind the high altar. The first ambulatory was in France in the 11th century but by the 13th century ambulatories had been introduced in England and many English cathedrals were extended to provide an ambulatory. The same feature is often found in Indian architecture and Buddhist architecture generally, especially in older periods. Ritual circumambulation or parikrama around a stupa or cult image is important in Buddhism and Hinduism. Often the whole building was circumambulated, often many times. The Buddhist chaitya hall always allowed a path for this, and the Durga temple, Aihole (7th or 8th century) is a famous Hindu example. The term is also used to describe a garden feature in the grounds of a country house. A typical example is the one shown, which stands in the grounds of Horton Court in Gloucestershire, England. == Medical term == Ambulatory is also an adjective used to describe patients who can walk despite their illness or injury. outpatients generally including those needing a wheelchair. medical staff providing outpatient care (see Ambulatory care


id: anatomical-terminology-20260505-0
file: data/en.wikipedia.org/wiki/Anatomical_terminology-0.md
category: reference
title: Anatomical terminology
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Anatomical_terminology
date_saved: 2026-05-05T07:26:04.798609+00:00
instance: kb-cron
parent: anatomical-terminology-20260505-0
summary:
Anatomical terminology is a specialized system of terms used by anatomists, zoologists, and health professionals, such as doctors, surgeons, and pharmacists, to describe the structures and functions of the body. This terminology incorporates a range of unique terms, prefixes, and suffixes derived primarily from Ancient Greek and Latin. While these terms can be challenging for those unfamiliar with them, they provide a level of precision that reduces ambiguity and minimizes the risk of errors. Because anatomical terminology is not commonly used in everyday language, its meanings are less likely to evolve or be misinterpreted. For example, everyday language can lead to confusion in descriptions: the phrase "a scar above the wrist" could refer to a location several inches away from the hand, possibly on the forearm, or it could be at the base of the hand, either on the palm or dorsal (back) side. By using precise anatomical terms, such as "proximal," "distal," "palmar," or "dorsal," this ambiguity is eliminated, ensuring clear communication. To standardize this system of terminology, Terminologia Anatomica was established as an international reference for anatomical terms. == Word formation == Anatomical terminology follows a regular morphology, with consistent prefixes and suffixes used to modify different roots.


id: anatomical-terminology-20260505-1
file: data/en.wikipedia.org/wiki/Anatomical_terminology-1.md
category: reference
title: Anatomical terminology
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Anatomical_terminology
date_saved: 2026-05-05T07:26:04.798609+00:00
instance: kb-cron
parent: anatomical-terminology-20260505-0
summary:
=== Regions === In terms of anatomy, anatomical lines are theoretical lines used to divide the body into regions, such as the nine regions of the abdomen. Axillary lines provide reference points for the underarm region In the front, the trunk is referred to as the "thorax" and "abdomen". The back as a general area is the dorsum or dorsal area, and the lower back is the lumbus or lumbar region. The shoulder blades are the scapular area and the breastbone is the sternal region. The abdominal region is the area between the chest and the pelvis. The breast is also called the mammary region, the armpit as the axilla and axillary, and the navel as the umbilicus and umbilical. The pelvis is the lower torso, between the abdomen and the thighs. The groin, where the thigh joins the trunk, are the inguen and inguinal area. The entire arm is referred to as the brachium and brachial, the front of the elbow as the antecubitis and antecubital, the back of the elbow as the olecranon or olecranal, the forearm as the antebrachium and antebrachial, the wrist as the carpus and carpal area, the hand as the manus and manual, the


id: anatomical-terminology-20260505-2
file: data/en.wikipedia.org/wiki/Anatomical_terminology-2.md
category: reference
title: Anatomical terminology
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Anatomical_terminology
date_saved: 2026-05-05T07:26:04.798609+00:00
instance: kb-cron
parent: anatomical-terminology-20260505-0
summary:
Anterior and posterior, which describe structures at the front (anterior) and back (posterior) of the body. For example, the toes are anterior to the heel, and the popliteus is posterior to the patella. Superior and inferior, which describe a position above (superior) or below (inferior) another part of the body. For example, the orbits are superior to the oris, and the pelvis is inferior to the abdomen. Proximal and distal, which describe a position that is closer to (proximal) or farther from (distal) the trunk of the body. For example, the shoulder is proximal to the arm, and the foot is distal to the knee. Superficial and deep, which describe structures that are closer to (superficial) or farther from (deep) the surface of the body. For example, the skin is superficial to the bones, and the brain is deep to the skull. Sometimes profound is used synonymously with deep. Medial and lateral, which describe a position that is closer to (medial) or farther from (lateral) the midline of the body. For example, the shoulders are lateral to the heart, and the umbilicus is medial to the hips. The medial side of the left knee is the side toward the opposite


id: anatomical-terminology-20260505-3
file: data/en.wikipedia.org/wiki/Anatomical_terminology-3.md
category: reference
title: Anatomical terminology
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Anatomical_terminology
date_saved: 2026-05-05T07:26:04.798609+00:00
instance: kb-cron
parent: anatomical-terminology-20260505-0
summary:
Flexion and extension, which refer to a movement that decreases (flexion) or increases (extension) the angle between body parts. For example, when standing up, the knees are extended. Abduction and adduction refers to a motion that pulls a structure away from (abduction) or towards (adduction) the midline of the body or limb. For example, a star jump requires the legs to be abducted. Internal rotation (or medial rotation) and external rotation (or lateral rotation) refers to rotation towards (internal) or away from (external) the center of the body. For example, the lotus position posture in yoga requires the legs to be externally rotated. Elevation and depression refer to movement in a superior (elevation) or inferior (depression) direction. Primarily refers to movements involving the scapula and mandible. === Special motions of the hands and feet === These terms refer to movements that are regarded as unique to the hands and feet: Dorsiflexion and plantarflexion refers to flexion (dorsiflexion) or extension (plantarflexion) of the foot at the ankle. For example, plantarflexion occurs when pressing the brake pedal of a car. Palmarflexion and dorsiflexion refer to movement of the flexion (palmarflexion) or extension (dorsiflexion) of the hand at the wrist. For example, prayer


id: anatomical-terms-of-location-20260505-0
file: data/en.wikipedia.org/wiki/Anatomical_terms_of_location-0.md
category: reference
title: Anatomical terms of location
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Anatomical_terms_of_location
date_saved: 2026-05-05T07:26:06.044970+00:00
instance: kb-cron
parent: anatomical-terms-of-location-20260505-0
summary:
Standard anatomical terms of location are used to describe unambiguously the anatomy of humans and other animals. The terms, typically derived from Latin or Greek roots, describe something in its standard anatomical position. This position provides a definition of what is at the front ("anterior"), behind ("posterior") and so on. As part of defining and describing terms, the body is described through the use of anatomical planes and axes. The meaning of terms that are used can change depending on whether a vertebrate is a biped or a quadruped, due to the difference in the neuraxis, or if an invertebrate is a non-bilaterian. A non-bilaterian has no anterior or posterior surface for example but can still have a descriptor used such as proximal or distal in relation to a body part that is nearest to, or furthest from its middle. International organisations have determined vocabularies that are often used as standards for subdisciplines of anatomy. For example, Terminologia Anatomica, Terminologia Neuroanatomica, and Terminologia Embryologica for humans and Nomina Anatomica Veterinaria for animals. These allow parties that use anatomical terms, such as anatomists, veterinarians, and medical doctors, to have a standard set of terms to communicate clearly the position of a


id: anatomical-terms-of-location-20260505-1
file: data/en.wikipedia.org/wiki/Anatomical_terms_of_location-1.md
category: reference
title: Anatomical terms of location
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Anatomical_terms_of_location
date_saved: 2026-05-05T07:26:06.044970+00:00
instance: kb-cron
parent: anatomical-terms-of-location-20260505-0
summary:
=== Dorsal and ventral === These two terms, used in veterinary anatomy, are also used in human anatomy mostly in neuroanatomy, and embryology, to describe something at the back (dorsal, posterior) or front (ventral, anterior) of an organ, or organism. The dorsal (from Latin dorsum 'back') surface, (also dorsum) of an organism or organ, refers to the back, or upper side, such as in the human, the dorsum of the tongue, the dorsum of the hand, and the dorsum of the foot. If talking about the skull, the dorsal side is the top. The ventral (from Latin venter 'belly') surface refers to the front, or lower side, of an organism, or organ such as the undersurface of the tongue. In a fish, the dorsal fin is on the upper surface and its ventral fins (pelvic fins) are on the belly or undersurface. The terms are used in other contexts, for example in dorsal and ventral gun turrets on a bomber aircraft. === Medial and lateral === These terms describe how close something is to the median plane. Lateral (from Latin lateralis 'to the side') describes something to the sides of an animal, as in "left lateral" and "right lateral". Medial


id: anatomical-terms-of-location-20260505-2
file: data/en.wikipedia.org/wiki/Anatomical_terms_of_location-2.md
category: reference
title: Anatomical terms of location
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Anatomical_terms_of_location
date_saved: 2026-05-05T07:26:06.044970+00:00
instance: kb-cron
parent: anatomical-terms-of-location-20260505-0
summary:
=== Superficial and deep === These terms refer to the distance of a structure from the surface. Deep (from Old English) describes something further away from the surface of the organism. For example, the external oblique muscle of the abdomen is deep to the skin. "Deep" is one of the few anatomical terms of location derived from Old English rather than Latin – the anglicised Latin term would have been "profound" (from Latin profundus 'due to depth'). Superficial (from Latin superficies 'surface') describes something near the outer surface of the organism. For example, in skin, the epidermis is superficial to the subcutis. === Combined terms === Many anatomical terms can be combined, either to indicate a position in two axes simultaneously or to indicate the direction of a movement relative to the body. For example, anterolateral indicates a position that is both anterior and lateral to the body axis (such as the bulk of the pectoralis major muscle), or to a named organ such as the anterolateral tibial tubercle. The term can also describe the direction and location of something that enters or courses through the body such as the anterolateral system in the spinal cord, and the anterolateral central


id: anatomical-terms-of-location-20260505-3
file: data/en.wikipedia.org/wiki/Anatomical_terms_of_location-3.md
category: reference
title: Anatomical terms of location
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Anatomical_terms_of_location
date_saved: 2026-05-05T07:26:06.044970+00:00
instance: kb-cron
parent: anatomical-terms-of-location-20260505-0
summary:
Special terms are used to describe the mouth and teeth. Fields such as osteology, paleontology and dentistry apply special terms of location to describe the mouth and teeth. This is because although teeth may be aligned with their main axes within the jaw, some different relationships require special terminology as well; for example, teeth also can be rotated, and in such contexts terms like "anterior" or "lateral" become ambiguous. For example, the terms "distal" and "proximal" (or "mesial") are used for surfaces of individual teeth relative to the midpoint of the dental arch, and "medial" and "lateral" are used in the standard sense relative to the median plane. Terms used to describe structures include "buccal" (from Latin bucca 'cheek') and "palatal" (from Latin palatum 'palate') referring to structures close to the cheek and hard palate respectively. === Hands and feet === Several anatomical terms are particular to the hands and feet. Additional terms may be used to avoid confusion when describing the surfaces of the hand and what is the "anterior" or "posterior" surface. The term "anterior", while anatomically correct, can be confusing when describing the palm of the hand; Similarly is "posterior", used to describe the back of the


id: anatomical-terms-of-location-20260505-4
file: data/en.wikipedia.org/wiki/Anatomical_terms_of_location-4.md
category: reference
title: Anatomical terms of location
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Anatomical_terms_of_location
date_saved: 2026-05-05T07:26:06.044970+00:00
instance: kb-cron
parent: anatomical-terms-of-location-20260505-0
summary:
== Other animals == Different terms are used because of different body plans in animals, whether animals stand on two or four legs, and whether an animal is symmetrical or asymmetrical. For example, as humans are bilaterally symmetrical, anatomical descriptions usually use the same terms as those for other vertebrates. However, the standard human anatomical position means that their anterior/posterior and ventral/dorsal directions are the same, so the inferior/superior directions are used due to longstanding tradition instead of cranial/caudal, which apply regardless of position, as in other species. The term "rostral" used to refer to the beak or nose in some animals is used less frequently in humans, with the exception of parts of the brain; while humans do not have a visible tail (the coccygeal vertebrae are present and commonly called the "tailbone") the term "caudal" that refers to the tail-end is also sometimes used in humans and animals without tails to refer to the hind part of the body. Flounder and other flatfish which lie on the seabed on their left or right side are asymmetric, with both eyes on the 'up' side, making anatomical nomenclature a challenge. Invertebrates have a large variety of body shapes that can


id: anginal-equivalent-20260505-0
file: data/en.wikipedia.org/wiki/Anginal_equivalent-0.md
category: reference
title: Anginal equivalent
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Anginal_equivalent
date_saved: 2026-05-05T07:26:07.251155+00:00
instance: kb-cron
parent: anginal-equivalent-20260505-0
summary:
An anginal equivalent is a symptom such as shortness of breath (dyspnea), diaphoresis (sweating), extreme fatigue, or pain at a site other than the chest, occurring in a patient at high cardiac risk. Anginal equivalents are considered to be symptoms of myocardial ischemia. Anginal equivalents are considered to have the same importance as angina pectoris in patients presenting with elevation of cardiac enzymes or certain EKG changes which are diagnostic of myocardial ischemia. == References ==


id: anisomastia-20260505-0
file: data/en.wikipedia.org/wiki/Anisomastia-0.md
category: reference
title: Anisomastia
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Anisomastia
date_saved: 2026-05-05T07:26:08.506878+00:00
instance: kb-cron
parent: anisomastia-20260505-0
summary:
Anisomastia is a medical condition in which there is a severe asymmetry or unequalness in the size of the breasts, generally related to a difference in volume. In other words, when one of the breasts is much larger than the other. In contrast to anisomastia, a slight asymmetry of the breasts is common. Anisomastia may be corrected by surgical breast augmentation or reduction. == See also == Micromastia Breast hypertrophy == References == == External links ==


id: anophthalmia-20260505-0
file: data/en.wikipedia.org/wiki/Anophthalmia-0.md
category: reference
title: Anophthalmia
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Anophthalmia
date_saved: 2026-05-05T07:26:09.740164+00:00
instance: kb-cron
parent: anophthalmia-20260505-0
summary:
Anophthalmia (Greek: ἀνόφθαλμος, "without eye") is the medical term for the absence of one or both eyes. Both the globe and the ocular tissue are missing from the orbit. The absence of the eye will cause a small bony orbit, a constricted mucosal socket, short eyelids, reduced palpebral fissure and malar prominence. Genetic mutations, chromosomal abnormalities, and prenatal environment can all cause anophthalmia. Anophthalmia is an extremely rare disease and is mostly rooted in genetic abnormalities. It can also be associated with other syndromes. == Classifications == There are three classifications for this condition: Primary anophthalmia is a complete absence of eye tissue due to a failure of the part of the brain that forms the eye. Secondary anophthalmia the eye starts to develop and for some reason stops, leaving the infant with only residual eye tissue or extremely small eyes which can only be seen under close examination. Degenerative anophthalmia the eye started to form and, for some reason, degenerated. One reason for this occurring could be a lack of blood supply to the eye. == Associations == There are a few conditions that are associated with anophthalmia. These include: Trisomy 13 Lenz syndrome Goldenhar-Gorlin syndrome Waardenburg syndrome Aside


id: anophthalmia-20260505-1
file: data/en.wikipedia.org/wiki/Anophthalmia-1.md
category: reference
title: Anophthalmia
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Anophthalmia
date_saved: 2026-05-05T07:26:09.740164+00:00
instance: kb-cron
parent: anophthalmia-20260505-0
summary:
=== Prosthetic eye === Currently, there is not a treatment option for regaining vision by developing a new eye. There are, however, cosmetic options so the absence of the eye is not as noticeable. Typically, the child will need to go to an ocularist to have conformers fitted into the eye. Conformers are made of clear plastic and are fitted into the socket to promote socket growth and expansion. As the child's face grows and develops, the conformer will need to be changed. An expander may also be needed in anophthalmia to expand the socket that is present. The conformer is changed every few weeks the first two years of life. After that, a painted prosthetic eye can be fitted for the child's socket. The prosthetic eye can be cleaned with mild baby soap and water. Rubbing alcohol should be avoided because it may damage the prosthetic eye. Children need to be checked regularly to ensure the fit and size is appropriate. A Cochrane Review published in 2016 asked whether the type of material used to make the prosthetic eye affects the success of the operation. Prosthetic eyes can be made from two types of material; porous or non-porous material.


id: aplasia-20260505-0
file: data/en.wikipedia.org/wiki/Aplasia-0.md
category: reference
title: Aplasia
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Aplasia
date_saved: 2026-05-05T07:26:11.073294+00:00
instance: kb-cron
parent: aplasia-20260505-0
summary:
Aplasia ( ; from Greek a, "not", "no" + plasis, "formation") is a birth defect where an organ or tissue is wholly or largely absent. It is caused by a defect in a developmental process. Aplastic anemia is the failure of the body to produce blood cells. It may occur at any time, and has multiple causes. == Types == === Pure red cell aplasia === Pure red cell aplasia (PRCA) is caused by the selective destruction or inhibition of erythroid progenitor or precursor cells. It is characterized by anemia and reticulocytopenia and can be chronic or acute. Diamond–Blackfan anemia is a type of PRCA that occurs at birth. PRCA can be acquired as a primary disorder or as a result of another disorder. Immunosuppressive drugs, particularly corticosteroids, will usually result in a temporary or permanent remission. The final outcome is primarily determined by the underlying disorder. === Aplasia cutis congenita === Aplasia cutis congenita is a condition in which some or large portions of the skin is missing at birth. The disorder is most commonly seen on the scalp, often as a solitary lesion without other abnormalities. The condition may be caused by epidermolysis bullosa, specific teratogens, or intrauterine


id: arrested-development-20260505-0
file: data/en.wikipedia.org/wiki/Arrested_development-0.md
category: reference
title: Arrested development
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Arrested_development
date_saved: 2026-05-05T07:26:12.406155+00:00
instance: kb-cron
parent: arrested-development-20260505-0
summary:
The term "arrested development" has had multiple meanings for over 200 years. In the field of medicine, the term "arrested development" was first used, circa 1835–1836, to mean a stoppage of physical development; the term continues to be used in the same way. In contrast, the UK's Mental Health Act 1983 used the term "arrested development" to characterize a form of mental disorder comprising severe mental impairment, resulting in a lack of intelligence. However, some researchers have objected to the notion that mental development can be "arrested" or stopped, preferring to consider mental status as continuing to develop in other ways. Consequently, the term "arrested development" is no longer used when referring to a developmental disorder in mental health. In anthropology and archaeology, the term "arrested development" means that a plateau of development in some sphere has been reached. Often it is a technological plateau such as the development of high temperature ceramics without glaze because of a lack of materials, or copper smelting without the development of bronze, because of a lack of tin. Arrested development is key in the insight of self-domestication in the evolution of hominidae where it involves being in an environment that favors reduction in


id: assessment-and-plan-20260505-0
file: data/en.wikipedia.org/wiki/Assessment_and_plan-0.md
category: reference
title: Assessment and plan
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Assessment_and_plan
date_saved: 2026-05-05T07:26:13.697908+00:00
instance: kb-cron
parent: assessment-and-plan-20260505-0
summary:
The assessment and plan (abbreviated A/P" or A&P) is a component of an admission note. Assessment includes a discussion of the differential diagnosis and supporting history and exam findings. The plan is typically broken out by problem or system. Each problem should include: brief summary of the problem, perhaps including what has been done thus far orders for medications, labs, studies, procedures and surgeries to address the problem. problems are commonly derived from chief complaint history of present illness review of systems (rarely; these should have been picked up and incorporated as new chief complaints during the interview) physical exam (rarely; these should have been picked up and incorporated as new chief complaints during the exam) social history, including counseling for smoking, alcohol, and illicit drug use family history medications may indicate problems that need to be addressed in the treatment of the other problems, such as dyslipidemia controlled with a statin. == References ==


id: asymptomatic-20260505-0
file: data/en.wikipedia.org/wiki/Asymptomatic-0.md
category: reference
title: Asymptomatic
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Asymptomatic
date_saved: 2026-05-05T07:26:14.855425+00:00
instance: kb-cron
parent: asymptomatic-20260505-0
summary:
Asymptomatic (or clinically silent) is an adjective categorising the medical conditions (i.e., injuries or diseases) that patients carry but without experiencing their symptoms, despite an explicit diagnosis (e.g., a positive medical test). Pre-symptomatic is the adjective categorising the time periods during which the medical conditions are asymptomatic. Subclinical and paucisymptomatic are other adjectives categorising either the asymptomatic infections (i.e., subclinical infections), or the psychosomatic illnesses and mental disorders expressing a subset of symptoms but not the entire set an explicit medical diagnosis requires. == Examples == An example of an asymptomatic disease is cytomegalovirus (CMV) which is a member of the herpes virus family. "It is estimated that 1% of all newborns are infected with CMV, but the majority of infections are asymptomatic." (Knox, 1983; Kumar et al. 1984) In some diseases, the proportion of asymptomatic cases can be important. For example, in multiple sclerosis it is estimated that around 25% of the cases are asymptomatic, with these cases detected postmortem or just by coincidence (as incidental findings) while treating other diseases. == Importance == Knowing that a condition is asymptomatic is important because: It may be contagious, and the contribution of asymptomatic and pre-symptomatic infections to the transmission level


id: asymptomatic-carrier-20260505-0
file: data/en.wikipedia.org/wiki/Asymptomatic_carrier-0.md
category: reference
title: Asymptomatic carrier
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Asymptomatic_carrier
date_saved: 2026-05-05T07:26:16.115969+00:00
instance: kb-cron
parent: asymptomatic-carrier-20260505-0
summary:
An asymptomatic carrier is a person or other organism that has become infected with a pathogen, but shows no signs or symptoms. Although unaffected by the pathogen, carriers can transmit it to others or develop symptoms in later stages of the disease. Asymptomatic carriers play a critical role in the transmission of common infectious diseases such as typhoid, HIV, C. difficile, influenzas, cholera, tuberculosis, and COVID-19, although the latter is often associated with "robust T-cell immunity" in more than a quarter of patients studied. While the mechanism of disease-carrying is still unknown, researchers have made progress towards understanding how certain pathogens can remain dormant in a human for a period of time. A better understanding of asymptomatic disease carriers is crucial to the fields of medicine and public health as they work towards mitigating the spread of common infectious diseases. == Types of asymptomatic carriers == Asymptomatic carriers can be categorized by their current disease state. When an individual transmits pathogens immediately following infection but prior to developing symptoms, they are known as an incubatory carrier. Humans are also capable of spreading disease following a period of illness. Typically thinking themselves cured of the disease, these individuals are known as


id: asymptomatic-carrier-20260505-1
file: data/en.wikipedia.org/wiki/Asymptomatic_carrier-1.md
category: reference
title: Asymptomatic carrier
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Asymptomatic_carrier
date_saved: 2026-05-05T07:26:16.115969+00:00
instance: kb-cron
parent: asymptomatic-carrier-20260505-0
summary:
Mary Mallon, known as "Typhoid Mary", was an asymptomatic carrier of Salmonella enterica serovar typhi, the causative agent of typhoid fever. She was a cook for several families and soldiers in New York City during the late 1800s, and several cases of typhoid fever were traced to her by the Health Department. At the time, there was no way of eradicating the disease, and it was spread primarily through fecal-oral transmission. Most of Mary Mallon's transmission risk was thought to arise from her continued involvement in occupations involving food preparation and handling. New York City's public health officials initially sought to merely restrict her from such employment rather than permanently quarantining her. When she continued to be non-compliant, the Health Commission ordered that she be quarantined on one of the islands surrounding Manhattan. She remained there until her death. Despite appearing perfectly healthy, it is estimated that Mallon infected about 50 people before she was quarantined on North Brother Island. Scientists calculate that between 1% and 6% of individuals infected with Salmonella typhi become chronic, asymptomatic carriers like Mary. === HIV === HIV infection has a long period during which the person is asymptomatic. Although the host may not be


id: atresia-20260505-0
file: data/en.wikipedia.org/wiki/Atresia-0.md
category: reference
title: Atresia
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Atresia
date_saved: 2026-05-05T07:26:18.542702+00:00
instance: kb-cron
parent: atresia-20260505-0
summary:
Atresia is a condition in which an orifice or passage in the body is (usually abnormally) closed or absent. == Types == === Anotia === Anotia is characterized by the complete absence of the ear and is extremely rare. This condition may affect one or both ears, though one missing ear is more common. Anotia is also linked to conductive hearing loss, a condition in which sound waves do not travel well through the ear and sound is not efficiently conducted from the outer ear canal to the eardrum. Anotia has no known cause. An associated syndrome, such as Treacher Collins or Goldenhar syndrome, may affect up to 40% of patients. Anotia is typically diagnosed through a physical examination at birth. Prenatal ultrasounds may help with early detection. Total ear reconstruction is the standard treatment for Anotia. === Biliary atresia === Biliary atresia (BA) is a rare disease marked by an unknown-origin biliary obstruction that manifests in the neonatal period. The classic clinical triad of Biliary atresia is acholic stools, and dark urine, jaundice, and hepatomegaly. The clinical manifestations are used to make the diagnosis, which is supported by liver ultrasonography, cholangiography, and a liver biopsy. The initial treatment is


id: atresia-20260505-1
file: data/en.wikipedia.org/wiki/Atresia-1.md
category: reference
title: Atresia
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Atresia
date_saved: 2026-05-05T07:26:18.542702+00:00
instance: kb-cron
parent: atresia-20260505-0
summary:
Renal agenesis occurs when the ureteric bud doesn't fuse with the metanephric blastema during embryogenesis, leading to the nephron and, in some cases, the ureter being absent. Unilateral renal agenesis occurs in 1 in 1000 live births, in contrast bilateral renal agenesis occurs in 1 in 3000 to 4000 pregnancies. Unilateral renal agenesis has a very good prognosis, whereas bilateral renal agenesis has a high rate of perinatal mortality and morbidity due to the lack of amniotic fluid, resulting in lethal pulmonary hypoplasia. The diagnosis of renal agenesis is usually made during a midgestation anatomy ultrasound examination. A genetic syndrome or other anomalies are linked to approximately 30% of cases of renal agenesis. === Tricuspid atresia === Tricuspid atresia is a form of congenital heart disease whereby there is a complete absence of the tricuspid valve. Therefore, there is an absence of right atrioventricular connection. This leads to a hypoplastic (undersized) or absent right ventricle. This defect is contracted during prenatal development, when the heart does not finish developing. It causes the systemic circulation to be filled with relatively deoxygenated blood. The causes of tricuspid atresia are unknown. === Vaginal atresia === Vaginal atresia is a birth defect that causes


id: atypia-20260505-0
file: data/en.wikipedia.org/wiki/Atypia-0.md
category: reference
title: Atypia
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Atypia
date_saved: 2026-05-05T07:26:19.731619+00:00
instance: kb-cron
parent: atypia-20260505-0
summary:
Atypia (from Greek, a + typos, without type; a condition of being irregular or nonstandard) is a histopathologic term for a structural abnormality in a cell, i.e. it is used to describe atypical cells. Atypia can be caused by infection or irritation. If, for example it were diagnosed in a Pap smear in the uterus it is more likely to be precancerous. The related concept of dysplasia refers to an abnormality of development, and includes abnormalities on larger, histopathologic scales. == Example features == Features that constitute atypia have different definitions for different diseases, but often include the following nucleus abnormalities: Enlargement Pleomorphism Nuclear polychromasia, which means variability in nuclear chromatin content. Polychromasia otherwise refers to a disease of immature red blood cells. Numerous mitotic figures === Examples for Barrett's esophagus === In Barrett's esophagus, features that are classified as atypia but not as dysplasia are mainly: Nuclear stratification, wherein cell nuclei, which are normally located nearly at the same level between adjacent cells, are instead located at different levels. Crowding Hyperchromatism Prominent nucleoli == Prognosis == It may or may not be a precancerous indication associated with later malignancy, but the level of appropriate concern is highly dependent on


id: auditory-agnosia-20260505-0
file: data/en.wikipedia.org/wiki/Auditory_agnosia-0.md
category: reference
title: Auditory agnosia
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Auditory_agnosia
date_saved: 2026-05-05T07:26:21.081452+00:00
instance: kb-cron
parent: auditory-agnosia-20260505-0
summary:
Auditory agnosia is a form of agnosia that manifests itself primarily in the inability to recognize or differentiate between sounds. It is not a defect of the ear or "hearing", but rather a neurological inability of the brain to process sound meaning. While auditory agnosia impairs the understanding of sounds, other abilities such as reading, writing, and speaking are not hindered. It is caused by bilateral damage to the anterior superior temporal gyrus, which is part of the auditory pathway responsible for sound recognition, the auditory "what" pathway. Persons with auditory agnosia can physically hear the sounds and describe them using unrelated terms, but are unable to recognize them. They might describe the sound of some environmental sounds, such as a motor starting, as resembling a lion roaring, but would not be able to associate the sound with "car" or "engine", nor would they say that it was a lion creating the noise. All auditory agnosia patients read lips in order to enhance the speech comprehension. It is yet unclear whether auditory agnosia (also called general auditory agnosia) is a combination of milder disorders, such auditory verbal agnosia (pure word deafness), non-verbal auditory agnosia, amusia and word-meaning deafness, or a


id: auditory-agnosia-20260505-1
file: data/en.wikipedia.org/wiki/Auditory_agnosia-1.md
category: reference
title: Auditory agnosia
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Auditory_agnosia
date_saved: 2026-05-05T07:26:21.081452+00:00
instance: kb-cron
parent: auditory-agnosia-20260505-0
summary:
== History == A relationship between hearing and the brain was first documented by Ambroise Paré, a 16th century battlefield doctor, who associated parietal lobe damage with acquired deafness (reported in Henschen, 1918). Systematic research into the manner in which the brain processes sounds, however, only began toward the end of the 19th century. In 1874, Wernicke was the first to ascribe to a brain region a role in auditory perception. Wernicke proposed that the impaired perception of language in his patients was due to losing the ability to register sound frequencies that are specific to spoken words (he also suggested that other aphasic symptoms, such as speaking, reading and writing errors occur because these speech specific frequencies are required for feedback). Wernicke localized the perception of spoken words to the posterior half of the left STG (superior temporal gyrus). Wernicke also distinguished between patients with auditory agnosia (which he labels as receptive aphasia) with patients who cannot detect sound at any frequency (which he labels as cortical deafness). In 1877, Kussmaul was the first to report auditory agnosia in a patient with intact hearing, speaking, and reading-writing abilities. This case-study led Kussmaul to propose of distinction between the word


id: auditory-agnosia-20260505-2
file: data/en.wikipedia.org/wiki/Auditory_agnosia-2.md
category: reference
title: Auditory agnosia
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Auditory_agnosia
date_saved: 2026-05-05T07:26:21.081452+00:00
instance: kb-cron
parent: auditory-agnosia-20260505-0
summary:
=== Cerebral deafness === Cerebral deafness (also known as cortical deafness or central deafness) is a disorder characterized by complete deafness that is the result of damage to the central nervous system. The primary distinction between auditory agnosia and cerebral deafness is the ability to detect pure tones, as measured with pure tone audiometry. Using this test, auditory agnosia patients were often reported capable of detecting pure tones almost as good as healthy individuals, whereas cerebral deafness patients found this task almost impossible or they required very loud presentations of sounds (above 100 dB). In all reported cases, cerebral deafness was associated with bilateral temporal lobe lesions. A study that compared the lesions of two cerebral deafness patients to an auditory agnosia patient concluded that cerebral deafness is the result of complete de-afferentation of the auditory cortices, whereas in auditory agnosia some thalamo-cortical fibers are spared. In most cases the disorder is transient and the symptoms mitigate into auditory agnosia (although chronic cases were reported). Similarly, a monkey study that ablated both auditory cortices of monkeys reported of deafness that lasted 1 week in all cases, and that was gradually mitigated into auditory agnosia in a period of 3–7 weeks.


id: auditory-agnosia-20260505-3
file: data/en.wikipedia.org/wiki/Auditory_agnosia-3.md
category: reference
title: Auditory agnosia
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Auditory_agnosia
date_saved: 2026-05-05T07:26:21.081452+00:00
instance: kb-cron
parent: auditory-agnosia-20260505-0
summary:
=== Non-verbal auditory agnosia === The term auditory agnosia was originally coined by Sigmund Freud in 1891, to describe patients with selective impairment of environmental sounds. In a review of the auditory agnosia literature, Ulrich re-named this disorder as non-verbal auditory agnosia (although sound auditory agnosia and environmental sound auditory agnosia are also commonly used). This disorder is very rare and only 18 cases have been documented. In contradiction to pure word deafness and general auditory agnosia, this disorder is likely under-diagnosed because patients are often not aware of their disorder, and thus don't seek medical intervention. Throughout the 20th century, all reported non-verbal auditory agnosia patients had bilateral or right temporal lobe damage. For this reason, the right hemisphere was traditionally attributed with the perception of environmental sounds. However, Tanaka et al reported 8 patients with non-verbal auditory agnosia, 4 with right hemisphere lesions and 4 with left hemisphere lesions. Saygin et al also reported a patient with damage to the left auditory cortex. The underlying deficit in non-verbal auditory agnosia appears to be varied. Several patients were characterized by impaired discrimination of pitch whereas others reported with impaired discrimination of timbre and rhythm (discrimination of pitch was relatively


id: auditory-agnosia-20260505-4
file: data/en.wikipedia.org/wiki/Auditory_agnosia-4.md
category: reference
title: Auditory agnosia
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Auditory_agnosia
date_saved: 2026-05-05T07:26:21.081452+00:00
instance: kb-cron
parent: auditory-agnosia-20260505-0
summary:
== Neurological mechanism == Auditory agnosia (with the exception of non-verbal auditory agnosia and amusia) is strongly dependent on damage to both hemispheres. The order of hemispheric damage is irrelevant to manifestation of symptoms, and years could take between the damage of the first hemisphere and the second hemisphere (after which the symptoms suddenly emerge). A study that compared lesion locations, reported that in all cases with bilateral hemispheric damage, at least in one side the lesion included Heschl's gyrus or its underlying white matter. A rare insight into the etiology of this disorder was reported in a study of an auditory agnosia patient with damage to the brainstem, instead of cortex. fMRI scanning of the patient revealed weak activation of the anterior Heschl's gyrus (area R) and anterior superior temporal gyrus. These brain areas are part of the auditory 'what' pathway, and are known from both human and monkey research to participate in the recognition of sounds. == See also == Amusia Aphasia Apraxia Auditory verbal agnosia == References == == Further reading == Polster MR, Rose SB (February 1998). "Disorders of auditory processing: evidence for modularity in audition" (PDF). Cortex; A Journal Devoted to the Study of the


id: aura-symptom-20260505-0
file: data/en.wikipedia.org/wiki/Aura_(symptom)-0.md
category: reference
title: Aura (symptom)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Aura_(symptom)
date_saved: 2026-05-05T07:26:22.252744+00:00
instance: kb-cron
parent: aura-symptom-20260505-0
summary:
An aura is a perceptual disturbance experienced by some with epilepsy or migraine. An epileptic aura is a form of minor seizure. Epileptic and migraine auras are due to the involvement of specific areas of the brain, which are those that determine the symptoms of the aura. Therefore, if the visual area is affected, the aura will consist of visual symptoms, while if a tactile sensory one, then tactile sensory symptoms will occur. Epileptic auras are subjective sensory or psychic phenomena due to a focal seizure, i.e. a seizure that originates from that area of the brain responsible for the function which then expresses itself with the symptoms of the aura. It is important because it makes it clear where the alteration causing the seizure is located. An epileptic aura is in most cases followed by other manifestations of a seizure, for example a convulsion, since the epileptic discharge spreads to other parts of the brain. Rarely it remains isolated. Auras, when they occur, allow some people who have epilepsy time to prevent injury to themselves and/or others when they lose consciousness. == Migraine == The aura of migraine is visual in the vast majority of cases, because dysfunction starts


id: autoamputation-20260505-0
file: data/en.wikipedia.org/wiki/Autoamputation-0.md
category: reference
title: Autoamputation
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Autoamputation
date_saved: 2026-05-05T07:26:23.447574+00:00
instance: kb-cron
parent: autoamputation-20260505-0
summary:
Autoamputation is the spontaneous detachment (amputation) of an appendage or organ from the body. This is not to be confused with self-amputation, which is performed at will. It is usually due to destruction of the blood vessels feeding an extremity such as the finger tips. Once the vessels are destroyed, the tissue is starved of oxygen and dies, which is often followed by gangrene. Autoamputation is a feature of ainhum, cryoglobulinemia and thromboangiitis obliterans. In 1881, Thornton made the case of autoamputation. Autoamputation could be the result of severe cases of certain chronic wounds, such as frostbite. These chronic wounds might be due to some vascular and pathogenic conditions like Buerger disease or Raynaud's phenomenon. Also, uncontrolled diabetes can predispose one to autoamputation. However, autoamputation has been described as spontaneous. Autoamputation has often been associated with fingers and toes but other parts of the body can suffer this condition as well. There have been reported cases of ovarian autoamputation in a newborn and also in a mature ovary of adults. Autoamputation has been reported to affect an infant of closely consanguineous parentage. Though autoamputation is often regarded as an acquired ailment, it could also be congenital. Chronic torsion or a


id: autonomic-dysreflexia-20260505-0
file: data/en.wikipedia.org/wiki/Autonomic_dysreflexia-0.md
category: reference
title: Autonomic dysreflexia
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Autonomic_dysreflexia
date_saved: 2026-05-05T07:26:24.727358+00:00
instance: kb-cron
parent: autonomic-dysreflexia-20260505-0
summary:
Autonomic dysreflexia (AD) is a life-threatening medical emergency characterized by hypertension and cardiac arrhythmias. This condition is sometimes referred to as autonomic hyperreflexia. Most cases of AD occur in individuals with spinal cord injuries. Lesions at or above the T6 spinal cord level are more frequently reported, although there are reports of AD in patients with lesions as low as T10. Guillain–Barré syndrome may also cause autonomic dysreflexia. Hypertension in AD may result in mild symptoms, such as sweating above the lesion level, goosebumps, blurred vision, or headache. Severe symptoms may result in life-threatening complications including seizure, intracranial bleeds (stroke), myocardial infarction, and retinal detachment. Both noxious and non-noxious stimuli can trigger AD. The result is stimulation and hyperactivity of the sympathetic nervous system. The noxious stimuli activate a sympathetic surge that travels through intact peripheral nerves, resulting in systemic vasoconstriction below the level of the spinal cord lesion. The peripheral arterial vasoconstriction and hypertension activates the baroreceptors, resulting in a parasympathetic surge. This surge originates in the central nervous system to inhibit the sympathetic outflow. However, the parasympathetic signal is unable to transmit below the level of the spinal cord lesion to reduce elevated blood pressure. This can result


id: autonomic-dysreflexia-20260505-1
file: data/en.wikipedia.org/wiki/Autonomic_dysreflexia-1.md
category: reference
title: Autonomic dysreflexia
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Autonomic_dysreflexia
date_saved: 2026-05-05T07:26:24.727358+00:00
instance: kb-cron
parent: autonomic-dysreflexia-20260505-0
summary:
The autonomic nervous system comprises the sympathetic, parasympathetic, and enteric nervous systems. The mechanism of autonomic dysreflexia has to do with the relationship of the sympathetic and parasympathetic systems. Supraspinal vasomotor neurons send projections to the intermediolateral cell column, which is composed of sympathetic preganglionic neurons (SPN) through the T1-L2 segments of the spinal cord. The supraspinal neurons act on the SPN and its tonic firing by modulating its action on the peripheral sympathetic chain ganglia and the adrenal medulla. The sympathetic ganglia act directly on the blood vessels they innervate throughout the body. This controls vessel diameter and resistance. The adrenal medulla indirectly controls the same action through the release of epinephrine and norepinephrine. In a patient with a spinal cord lesion, the descending autonomic pathways that are responsible for the supraspinal communication with the SPN are interrupted. This results in decreased sympathetic outflow below the level of the injury. In this circumstance, the SPN is controlled only by spinal influences. After a spinal injury, the decreased sympathetic outflow causes reduced blood pressure and sympathetic reflex. Eventually, synaptic reorganization and plasticity of the SPN develops into an overly sensitive state. Because of this, there is abnormal reflex activation of


id: autonomic-dysreflexia-20260505-2
file: data/en.wikipedia.org/wiki/Autonomic_dysreflexia-2.md
category: reference
title: Autonomic dysreflexia
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Autonomic_dysreflexia
date_saved: 2026-05-05T07:26:24.727358+00:00
instance: kb-cron
parent: autonomic-dysreflexia-20260505-0
summary:
== Research directions == Most future work on the topic of autonomic dysreflexia is directed at earlier detection and intervention. Overall, the goal of these research projects involves minimizing complications that result from late detection of autonomic dysreflexia. Some research is aimed at investigating the use of non-invasive sensors to track nerve activity to detect signs of AD. Other work has begun to look at the use of AI for this role, although it has been limited to rat models. Results from a study showed that AI can serve as another non-invasive tool in combination with sensors that track nerve activity. Future work of studies such as these includes using more sensors to track other variables for increasingly accurate results. Other work revolves around increasing the understanding of the mechanism behind AD. While it is understood that spinal cord injury results in inhibited parasympathetic surges and a heightened sympathetic response that can lead to AD, other details are yet to be defined. It is also understood that the renin-angiotensin system (RAS) plays a significant role in cardiovascular function in addition to the autonomic nervous system (ANS), which includes the sympathetic and parasympathetic systems. What remains to be studied is the


id: azygos-20260505-0
file: data/en.wikipedia.org/wiki/Azygos-0.md
category: reference
title: Azygos
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Azygos
date_saved: 2026-05-05T07:26:25.966903+00:00
instance: kb-cron
parent: azygos-20260505-0
summary:
Azygos (impar), from the Greek άζυξ, refers to an anatomical structure that is unpaired. This is relatively unusual, as most elements of anatomy reflect bilateral symmetry. Azygos may refer to: Azygos anterior cerebral artery Azygos artery of vagina Azygos lobe Azygos vein Ganglion impar


id: b-type-inclusion-20260505-0
file: data/en.wikipedia.org/wiki/B_type_inclusion-0.md
category: reference
title: B type inclusion
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/B_type_inclusion
date_saved: 2026-05-05T07:26:27.133905+00:00
instance: kb-cron
parent: b-type-inclusion-20260505-0
summary:
B-type inclusions, formerly known as Guarnieri bodies are cellular features found upon microscopic inspection of epithelial cells of individuals suspected of having poxvirus (e.g. smallpox or vaccinia). In cells stained with eosin, they appear as pink blobs in the cytoplasm of affected epithelial cells. The absence of Guarnieri bodies cannot be used as to rule out smallpox, however, as more sensitive test need to be performed. B-type inclusions are the sites of viral replication and are found in all poxvirus-infected cells, unlike A-type inclusions which are more strongly eosinophilic and only found in infections with certain poxviruses. They are named after the Italian physician Giuseppe Guarnieri. == References == == Further reading == GOODPASTURE EW (1959). "Cytoplasmic inclusions resembling Guarnieri bodies, and other phenomena induced by mutants of the virus of fowlpox". Am. J. Pathol. 35 (2): 213–31. PMC 1934859. PMID 13627123.


id: ballard-maturational-assessment-20260505-0
file: data/en.wikipedia.org/wiki/Ballard_Maturational_Assessment-0.md
category: reference
title: Ballard Maturational Assessment
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Ballard_Maturational_Assessment
date_saved: 2026-05-05T07:26:28.282771+00:00
instance: kb-cron
parent: ballard-maturational-assessment-20260505-0
summary:
The Ballard maturational assessment, Ballard score, or Ballard scale, is a gestational age assessment technique. It was devised by Dr. Jeanne L. Ballard, professor emeritus of the Department of Pediatrics, Obstetrics, and Gynecology at the University of Cincinnati College of Medicine. It was developed in 1979. The assessment scores various criteria, the sum of which is then extrapolated to the gestational age of the fetus. These criteria are divided into physical and neuromuscular criteria, which allows for the estimation of age in the range of 26 weeks to 44 weeks. The new Ballard score is an extension of the above to include extremely preterm babies, i.e., up to 20 weeks. The scoring relies on the intrauterine changes the fetus undergoes during maturation. Whereas the neuromuscular criteria depend mainly upon muscle tone, the physical scale relies on anatomical changes. Neonate fetuses (less than 37 weeks of age) are in a state of physiological hypotonia, and since muscle tone increases throughout the fetal growth period, it can be used to identify fetal maturation. == Neuromuscular criteria == Posture – score the infant's posture from flexed to extended Square window – assess the flexibility of the wrist Arm recoil – measure the angle


id: benign-tumor-20260505-0
file: data/en.wikipedia.org/wiki/Benign_tumor-0.md
category: reference
title: Benign tumor
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Benign_tumor
date_saved: 2026-05-05T07:26:29.480012+00:00
instance: kb-cron
parent: benign-tumor-20260505-0
summary:
A benign tumor is a mass of cells (tumor) that does not invade neighboring tissue or metastasize (spread throughout the body). Compared to malignant (cancerous) tumors, benign tumors generally have a slower growth rate. Benign tumors have relatively well differentiated cells. They are often surrounded by an outer surface (fibrous sheath of connective tissue) or stay contained within the epithelium. Common examples of benign tumors include moles and uterine fibroids. Some forms of benign tumors may be harmful to health. Benign tumor growth causes a mass effect that can compress neighboring tissues. This can lead to nerve damage, blood flow reduction (ischemia), tissue death (necrosis), or organ damage. The health effects of benign tumor growth may be more prominent if the tumor is contained within an enclosed space such as the cranium, respiratory tract, sinus, or bones. For example, unlike most benign tumors elsewhere in the body, benign brain tumors can be life-threatening. Tumors may exhibit behaviors characteristic of their cell type of origin; as an example, endocrine tumors such as thyroid adenomas and adrenocortical adenomas may overproduce certain hormones. The word benign means 'favourable, kind, fortunate, salutary, propitious'. However, a benign tumor is not benign in the usual sense;


id: benign-tumor-20260505-1
file: data/en.wikipedia.org/wiki/Benign_tumor-1.md
category: reference
title: Benign tumor
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Benign_tumor
date_saved: 2026-05-05T07:26:29.480012+00:00
instance: kb-cron
parent: benign-tumor-20260505-0
summary:
==== Giant cell tumors ==== Giant cell tumors of bone frequently occur in long bone epiphyses of the appendicular skeleton or the sacrum of the axial skeleton. Local growth can cause destruction of neighboring cortical bone and soft tissue, leading to pain and limiting range of motion. The characteristic radiologic finding of giant cell tumors of bone is a lytic lesion that does not have marginal sclerosis of bone. On histology, giant cells of fused osteoclasts are seen as a response to neoplastic mononucleated cells. Notably, giant cells are not unique among benign bone tumors to giant cell tumors of bone. Molecular characteristics of the neoplastic cells causing giant cell tumors of bone indicate an origin of pluripotent mesenchymal stem cells that adopt preosteoblastic markers. Cytogenetic causes of giant cell tumors of bone involve telomeres. Treatment involves surgical curettage with adjuvant bisphosphonates. ==== Osteochondroma ==== Osteochondromas form cartilage-capped projections of bone. Structures such as the marrow cavity and cortical bone of the osteochondroma are contiguous to those of the originating bone. Sites of origin often involve metaphyses of long bones. While many osteochondromas occur spontaneously, there are cases in which several osteochondromas can occur in the same individual; these may


id: benign-tumor-20260505-2
file: data/en.wikipedia.org/wiki/Benign_tumor-2.md
category: reference
title: Benign tumor
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Benign_tumor
date_saved: 2026-05-05T07:26:29.480012+00:00
instance: kb-cron
parent: benign-tumor-20260505-0
summary:
Benign neoplasms are typically, but not always, composed of cells which bear a strong resemblance to a normal cell type in their organ of origin. These tumors are named for the cell or tissue type from which they originate. The suffix "-oma" (but not -carcinoma, -sarcoma, or -blastoma, which are generally cancers) is applied to indicate a benign tumor. For example, a lipoma is a common benign tumor of fat cells (lipocytes), and a chondroma is a benign tumor of cartilage-forming cells (chondrocytes). Adenomas are benign tumors of gland-forming cells, and are usually specified further by their cell or organ of origin, as in hepatic adenoma (a benign tumor of hepatocytes, or liver cells). Teratomas contain many cell types such as skin, nerve, brain and thyroid, among others, because they are derived from germ cells. Hamartomas are a group of benign tumors that have relatively normal cellular differentiation but exhibit disorganized tissue organization. Exceptions to the nomenclature rules exist for historical reasons. Malignant examples include melanoma (a skin cancer of pigmented melanocytes) and seminoma (a cancer of male reproductive cells). Not all benign growths are tumors. Skin tags, vocal chord polyps, and hyperplastic polyps of the colon are often called


id: bethesda-system-20260505-0
file: data/en.wikipedia.org/wiki/Bethesda_system-0.md
category: reference
title: Bethesda system
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Bethesda_system
date_saved: 2026-05-05T07:26:30.688221+00:00
instance: kb-cron
parent: bethesda-system-20260505-0
summary:
The Bethesda system (TBS), officially called The Bethesda System for Reporting Cervical Cytology, is a system for reporting cervical or vaginal cytologic diagnoses, used for reporting Pap smear results. It was introduced in 1988 and revised in 1991, 2001, and 2014. The name comes from the location (Bethesda, Maryland) of the conference, sponsored by the National Institutes of Health, that established the system. Since 2010, the Bethesda system has been used for cytopathology of thyroid nodules, which is called The Bethesda System for Reporting Thyroid Cytopathology (TBSRTC or BSRTC). Like TBS, it was the result of a conference sponsored by the NIH and is published in book editions (currently by Springer). Mentions of "the Bethesda system" without further specification usually refer to the cervical system, unless the thyroid context of a discussion is implicit. == Cervix == Abnormal results include: Atypical squamous cells Atypical squamous cells of undetermined significance (ASC-US) Atypical squamous cells – cannot exclude HSIL (ASC-H) Low-grade squamous intraepithelial lesion (LGSIL or LSIL) High-grade squamous intraepithelial lesion (HGSIL or HSIL) Squamous cell carcinoma Atypical Glandular Cells not otherwise specified (AGC-NOS) Atypical Glandular Cells, suspicious for AIS or cancer (AGC-neoplastic) Adenocarcinoma in situ (AIS) The results are calculated differently


id: biliary-microlithiasis-20260505-0
file: data/en.wikipedia.org/wiki/Biliary_microlithiasis-0.md
category: reference
title: Biliary microlithiasis
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Biliary_microlithiasis
date_saved: 2026-05-05T07:26:31.871930+00:00
instance: kb-cron
parent: biliary-microlithiasis-20260505-0
summary:
Biliary microlithiasis refers to the creation of small gallstones less than 3 mm in diameter in the biliary duct or gallbladder. It has been suggested as a cause of postcholecystectomy syndrome, or PCS, the symptoms of which include: Upset stomach, nausea, and vomiting. Gas, bloating, and diarrhea. Persistent pain in the upper right abdomen. == Diagnostics == Biliary Microlithiasis may be detectable by ultrasound using a Rapid Patient Rotation Ultrasound Protocol Analysis of biliary sludge obtained through endoscopic retrograde cholangiopancreatography (ERCP) == Treatment == Oral ursodeoxycholic acid can be used to dissolve these crystals. == See also == Biliary sludge == References ==


id: biliary-pseudolithiasis-20260505-0
file: data/en.wikipedia.org/wiki/Biliary_pseudolithiasis-0.md
category: reference
title: Biliary pseudolithiasis
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Biliary_pseudolithiasis
date_saved: 2026-05-05T07:26:33.115723+00:00
instance: kb-cron
parent: biliary-pseudolithiasis-20260505-0
summary:
Biliary pseudolithiasis is an unusual complication of ceftriaxone where the drug complexes with calcium and mimics gallstones. It is reversed when ceftriaxone administration is stopped. It was first described in 1988 by Schaad et al. as "reversible ceftriaxone-associated biliary pseudolithiasis". Ceftriaxone has been frequently associated with biliary sludge or biliary pseudolithiasis in subsequent reports. Ceftriaxone is excreted primarily through the urine, but also through the bile, up to 40% of its excretion, with concentrations in the bile 20-150 times higher than in the serum. It forms a calcium salt in the gallbladder, which can exceed its solubility and create precipitates that resemble gallstones on ultrasonography. The incidence of pseudolithiasis in children treated with ceftriaxone is up to 25%, but most patients are asymptomatic. Risk factors for biliary pseudolithiasis include age greater than 24 months, gram-negative sepsis, high doses of ceftriaxone, hypercalcemia, surgery, and decreased bile flow/increased ceftriaxone excretion in bile. Conservative management with serial ultrasounds is recommended until the "stones" completely resolve. If associated with ceftriaxone, it resolves on average about 2 weeks after the ceftriaxone is stopped. == Ceftriaxone == Ceftriaxone sold under the brand name Rocephin, is a third-generation cephalosporin antibiotic used for the treatment of a number


id: biliary-sludge-20260505-0
file: data/en.wikipedia.org/wiki/Biliary_sludge-0.md
category: reference
title: Biliary sludge
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Biliary_sludge
date_saved: 2026-05-05T07:26:34.361744+00:00
instance: kb-cron
parent: biliary-sludge-20260505-0
summary:
Biliary sludge refers to a viscous mixture of small particles derived from bile. These sediments consist of cholesterol crystals, calcium salts, calcium bilirubinate, mucin, and other materials. == Signs and symptoms == === Complications === Biliary sludge may cause complications such as biliary colic, acute cholecystitis, acute cholangitis, and acute pancreatitis. == Cause == Biliary sludge has been associated with pregnancy, rapid weight loss, total parenteral nutrition, drugs such as ceftriaxone and octreotide, solid organ transplantation, and gastric surgery. In many of these conditions, it is thought that the impairment in the contractility of the gallbladder leads to the formation of the sludge. == Pathophysiology == The pathophysiology of biliary sludge formation is likely related to gallbladder dysmotility. It is presumed that because the gallbladder is unable to effectively empty, the biliary sludge can start to accumulate. == Diagnosis == Biliary sludge is typically diagnosed by CT scan or transabdominal ultrasonography. Endoscopic ultrasonography is another more sensitive option. However, the gold standard is considered to be direct microscopy of aspirated gallbladder bile. This method is much more sensitive, although it is less practical. == Treatment == For patients without symptoms, no treatment is recommended. If patients become symptomatic and/or develop


id: list-of-biological-development-disorders-20260505-0
file: data/en.wikipedia.org/wiki/List_of_biological_development_disorders-0.md
category: reference
title: List of biological development disorders
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_biological_development_disorders
date_saved: 2026-05-05T07:26:35.537754+00:00
instance: kb-cron
parent: list-of-biological-development-disorders-20260505-0
summary:
The following is a list of terms used to describe biological disorders of development, arranged by root word and shared prefix: == References == == Bibliography == Reece, JB; Urry, LA; Cain, ML; Wasserman, SA; Minorsky, PV; Jackson, RB (2014). Campbell Biology (10th ed.). Addison Wesley Longman. ISBN 978-0321834959.


id: biotrauma-20260505-0
file: data/en.wikipedia.org/wiki/Biotrauma-0.md
category: reference
title: Biotrauma
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Biotrauma
date_saved: 2026-05-05T07:26:36.815898+00:00
instance: kb-cron
parent: biotrauma-20260505-0
summary:
Although the term has occasionally been used in other ways, in medical literature biotrauma is usually defined as a severe inflammatory response produced in the lungs of patients who breathe by means of a mechanical ventilator for a long period of time. The term was coined in a 1998 paper by L. N. Tremblay and A. S. Slutsky, titled Ventilator-induced injury: from barotrauma to biotrauma. The message of that paper was that barotrauma caused by pressure differentials is only one of several types of lung damage that a ventilator can produce. == References ==


id: bleb-medicine-20260505-0
file: data/en.wikipedia.org/wiki/Bleb_(medicine)-0.md
category: reference
title: Bleb (medicine)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Bleb_(medicine)
date_saved: 2026-05-05T07:26:38.141600+00:00
instance: kb-cron
parent: bleb-medicine-20260505-0
summary:
In medicine, a bleb is a blister-like protrusion (often hemispherical) or vesicle filled with serous fluid. Blebs can form in a number of tissues by different pathologies, including frostbite and can "appear and disappear within a short time interval". In pathology, pulmonary blebs are small subpleural thin-walled air-containing spaces, not larger than 1-2 cm in diameter, found by the upper lobe of the lung, between the lung and the visceral pleura. Their walls are thin, being less than 1 mm thick. If they rupture, they allow air to escape into pleural space, resulting in a spontaneous pneumothorax and possibly a collapsed lung. Blebs can grow larger or join together to create a larger cyst, or bulla. There are usually no symptoms unless a pneumothorax occurs or the bulla grows very large. Blebs are usually associated with emphysema. In ophthalmology, blebs may be formed intentionally in the treatment of glaucoma. In such treatments, functional blebs facilitate the circulation of aqueous humor, the blockage of which will lead to increase in eye pressure. Use of collagen matrix wound modulation device such as ologen during glaucoma surgery is known to produce vascular and functional blebs, which are positively correlated with treatment success rate.


id: body-fluid-20260505-0
file: data/en.wikipedia.org/wiki/Body_fluid-0.md
category: reference
title: Body fluid
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Body_fluid
date_saved: 2026-05-05T07:26:39.339802+00:00
instance: kb-cron
parent: body-fluid-20260505-0
summary:
Body fluids, bodily fluids, or biofluids, sometimes body liquids, are liquids within the body of an organism. In lean healthy adult men, the total body water is about 60% (60–67%) of the total body weight; it is usually slightly lower in women (52–55%). The exact percentage of fluid relative to body weight is inversely proportional to the percentage of body fat. A lean 70 kg (150 lb) man, for example, has about 42 (42–47) liters of water in his body. The total body of water is divided into fluid compartments, between the intracellular fluid compartment (also called space, or volume) and the extracellular fluid (ECF) compartment (space, volume) in a two-to-one ratio: 28 (28–32) liters are inside cells and 14 (14–15) liters are outside cells. The ECF compartment is divided into the interstitial fluid volume – the fluid outside both the cells and the blood vessels – and the intravascular volume (also called the vascular volume and blood plasma volume) – the fluid inside the blood vessels – in a three-to-one ratio: the interstitial fluid volume is about 12 liters; the vascular volume is about 4 liters. The interstitial fluid compartment is divided into the lymphatic fluid compartment – about


id: body-image-neuroscience-20260505-0
file: data/en.wikipedia.org/wiki/Body_image_(neuroscience)-0.md
category: reference
title: Body image (neuroscience)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Body_image_(neuroscience)
date_saved: 2026-05-05T07:26:40.513246+00:00
instance: kb-cron
parent: body-image-neuroscience-20260505-0
summary:
Body image is a complex construct, often used in the clinical context of describing a patient's cognitive perception of their own body. The medical concept began with the work of the Austrian neuropsychiatrist and psychoanalyst Paul Schilder, described in his book The Image and Appearance of the Human Body first published in 1935. The term "body image" was officially introduced by Schilder himself and his widely used definition is: "body image is the picture of our own body we form in our mind, that is to say the way in which the body appears to ourselves". In research with the term "body image" we currently refer to a conscious mental representation of one's own body, which involves affects, attitudes, perceptual components and cognition. On the contrary, the term "body schema" was initially used to describe an unconscious body mental representation fundamental for action. Keizer and colleagues (2013) suggest the following definition: "[body schema is] an unconscious, sensorimotor, representation of the body that is invoked in action". In light of recent scientific developments regarding the multisensory integration of body sensations, the distinction between body image and body schema appears simplistic and probably no longer useful for scientific research and clinical purposes.


id: body-image-neuroscience-20260505-1
file: data/en.wikipedia.org/wiki/Body_image_(neuroscience)-1.md
category: reference
title: Body image (neuroscience)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Body_image_(neuroscience)
date_saved: 2026-05-05T07:26:40.513246+00:00
instance: kb-cron
parent: body-image-neuroscience-20260505-0
summary:
=== Body integrity disorder === Body integrity dysphoria is a mental disorder in which a person gets the feeling that a certain part of their body no longer belongs on their body. People with this disorder know that the part is useful and healthy, but don't believe that they should be able to use it. It often causes people to try and get amputations or attempt to amputate themselves. There's no known cause of body integrity disorder, but it is thought to have to do with issues regarding the structure of the brain because multiple parts of the brain are involved in body perception. === Measurements === Attempts by researchers to measure variances in body image include the FAI index, developed in a 2014 study (Zaccagni 2014). The FAI (feel-status minus actual-status inconsistency) index is used to assess someone's weight perception. FAI scores range from -3 to +3: Negative FAI values mean weight status underestimation, positive FAI values mean weight status overestimation and a FAI score of 0 means a realistic perception of one's weight status. The study found that women tend to have positive FAI values (overestimating their weight) while men had negative FAI values (underestimating their weight). Further


id: bogart-bacall-syndrome-20260505-0
file: data/en.wikipedia.org/wiki/Bogart–Bacall_syndrome-0.md
category: reference
title: Bogart–Bacall syndrome
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Bogart–Bacall_syndrome
date_saved: 2026-05-05T07:26:41.835054+00:00
instance: kb-cron
parent: bogart-bacall-syndrome-20260505-0
summary:
Bogart–Bacall syndrome (BBS) is a voice disorder that is caused by abuse or overuse of the vocal cords. People who speak or sing outside their normal vocal range can develop BBS; symptoms are chiefly an unusually deep or rough voice, or dysphonia, and vocal fatigue. The people most commonly affected are those who speak in a low-pitched voice, particularly if they have poor breath and vocal control. The syndrome can affect both men and women. In 1988, an article was published describing a discrete type of vocal dysfunction which results in men sounding like actor Humphrey Bogart and women sounding like actress Lauren Bacall; Bogart and Bacall were married to each other and made several films together. BBS is now the medical term for an ongoing hoarseness that often affects actors, singers or TV/radio voice workers who routinely speak in a very low pitch. Treatment usually involves voice therapy by a speech language pathologist. == Signs and symptoms == Signs and symptoms of Bogart–Bacall Syndrome can appear differently depending on the vocal use of the individual. In singers, symptoms may appear more subtly due to their extreme sensitivity to small changes in the laryngeal mechanism and being able to control


id: bogart-bacall-syndrome-20260505-1
file: data/en.wikipedia.org/wiki/Bogart–Bacall_syndrome-1.md
category: reference
title: Bogart–Bacall syndrome
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Bogart–Bacall_syndrome
date_saved: 2026-05-05T07:26:41.835054+00:00
instance: kb-cron
parent: bogart-bacall-syndrome-20260505-0
summary:
== Prevention and treatment == Prevention of Bogart–Bacall syndrome aims to target the two main symptoms of this disorder, dysphonia and vocal fatigue. Having effective posture allows a person to shift the tension between the muscles allowing free movement of the larynx without blockage leading to effective voice production. Improvement of breath control with a professional vocal coach is also important for individuals who may have work-related conditions that require continuous strain on the vocal cords and larynx. Speech therapy with a speech-language pathologist is a very common method of treatment to return speaking level to its normal pitch range. Consultation with a speech-language pathologist will determine what levels of rest, fluid, and performance corrections are needed for the vocal cords and larynx to sustain a healthy voice. If there is a particular underlying cause of the condition that can be treated with medication, this can be coupled with speech therapy as a form of treatment. Individuals with persistent symptoms after speech therapy may require more invasive treatment. Invasive treatment may require the removal of lesions, nodules, or masses on the vocal cords if visualized during diagnostic imaging. If the disorder is found to be nerve related through electromyography the


id: breast-atrophy-20260505-0
file: data/en.wikipedia.org/wiki/Breast_atrophy-0.md
category: reference
title: Breast atrophy
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Breast_atrophy
date_saved: 2026-05-05T07:26:43.043100+00:00
instance: kb-cron
parent: breast-atrophy-20260505-0
summary:
Breast atrophy is the normal or spontaneous atrophy or shrinkage of the breasts. Breast atrophy commonly occurs in women during menopause when estrogen levels decrease. It can also be caused by hypoestrogenism and/or hyperandrogenism in women in general, such as in antiestrogen treatment for breast cancer, in polycystic ovary syndrome (PCOS), and in malnutrition such as that associated with eating disorders like anorexia nervosa or with chronic disease. It can also be an effect of weight loss. In the treatment of gynecomastia in males and macromastia in women, and in hormone replacement therapy (HRT) for trans men, breast atrophy may be a desired effect. == See also == Mammoplasia Micromastia == References == == External links ==


id: brief-resolved-unexplained-event-20260505-0
file: data/en.wikipedia.org/wiki/Brief_resolved_unexplained_event-0.md
category: reference
title: Brief resolved unexplained event
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Brief_resolved_unexplained_event
date_saved: 2026-05-05T07:26:44.192552+00:00
instance: kb-cron
parent: brief-resolved-unexplained-event-20260505-0
summary:
Brief resolved unexplained event (BRUE), previously apparent life-threatening event (ALTE), is a medical term in pediatrics that describes an event that occurs during infancy. The event is noted by an observer, typically the infant's caregiver. It is characterized by one or more concerning symptoms such as change in skin color, lack of breathing, weakness, or poor responsiveness. By definition, by the time they are assessed in a healthcare environment they must be back to normal without obvious explanation after the clinician takes the appropriate clinical history and physical examination. The American Academy of Pediatrics (AAP) clarified the use of both terms in a 2016 consensus statement that recommended the term BRUE be used whenever possible as it is more specifically defined. Thus, it is more useful for assessing risk of further events. The cause for BRUEs is often unknown, although some of the more common causes include gastroesophageal reflux, seizure, and child maltreatment. Evaluation after an ALTE or BRUE is diagnostically important, as some events represent the first sign or symptom of an underlying medical condition. In most cases, assuming the infants are otherwise healthy and no underlying medical issue is found, the infants who have a BRUE are unlikely


id: brief-resolved-unexplained-event-20260505-1
file: data/en.wikipedia.org/wiki/Brief_resolved_unexplained_event-1.md
category: reference
title: Brief resolved unexplained event
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Brief_resolved_unexplained_event
date_saved: 2026-05-05T07:26:44.192552+00:00
instance: kb-cron
parent: brief-resolved-unexplained-event-20260505-0
summary:
== Prognosis == The risk of death of patients who have a BRUE has been studied by using the literature about ALTEs, since this data is more abundant. The studies concluded that there is no increased risk of death for these patients compared to the rest of the infant population. As for the prognosis of these infants into adulthood, research still needs to be conducted to assess for any long-term health effects. == History == In 1986, the National Institute of Health defined an apparent life-threatening event (ALTE) as an observed frightening event of an infant that includes at least one component of lack of breathing (apnea), skin color change (such as cyanosis), weakness, choking, or gagging. The term was invented to avoid previously used terms such as "near-miss SIDS" to dissociate the event from SIDS, a separate condition in infancy. There had been literature discussion in the past about the increased risk of SIDS in these infants, but more recently the research has concluded that there is no direct relationship between an ALTE and SIDS. It also was defined as part of an attempt to characterize the different forms of apnea, or sudden lack of breathing, in infants. In


id: calcinosis-20260505-0
file: data/en.wikipedia.org/wiki/Calcinosis-0.md
category: reference
title: Calcinosis
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Calcinosis
date_saved: 2026-05-05T07:26:45.452780+00:00
instance: kb-cron
parent: calcinosis-20260505-0
summary:
Calcinosis is the formation of calcium deposits in any soft tissue. It is a rare condition that has many different causes. These range from infection and injury to systemic diseases like kidney failure. == Types == === Dystrophic calcification === The most common type of calcinosis is dystrophic calcification. This type of calcification can occur as a response to any soft tissue damage, including that involved in implantation of medical devices. === Metastatic calcification === Metastatic calcification involves a systemic calcium excess imbalance, which can be caused by hypercalcemia, kidney failure, milk-alkali syndrome, lack or excess of other minerals, or other causes. === Tumoral calcinosis === The cause of the rare condition of tumoral calcinosis is not entirely understood. It is generally characterized by large, globular calcifications near joints. == See also == Calcification Calcinosis cutis Dermatomyositis Fahr's syndrome Hyperphosphatemia Primrose syndrome Scleroderma == References == == External links == Media related to Calcinosis at Wikimedia Commons


id: calculus-medicine-20260505-0
file: data/en.wikipedia.org/wiki/Calculus_(medicine)-0.md
category: reference
title: Calculus (medicine)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Calculus_(medicine)
date_saved: 2026-05-05T07:26:46.702674+00:00
instance: kb-cron
parent: calculus-medicine-20260505-0
summary:
A calculus (pl.: calculi), often called a stone, is a concretion of material, usually mineral salts, that forms in an organ or duct of the body. Formation of calculi is known as lithiasis (). Stones can cause a number of medical conditions. Some common principles (below) apply to stones at any location, but for specifics see the particular stone type in question. Calculi are not to be confused with gastroliths, which are ingested rather than grown endogenously. == Types == Calculi in the inner ear are called otoliths. Unlike other entries in this list, otoliths are not pathological but are a normal feature of the inner ear. Calculi in the urinary system are called urinary calculi and include kidney stones (also called renal calculi or nephroliths) and bladder stones (also called vesical calculi or cystoliths). They can have any of several compositions, including mixed. Principal compositions include oxalate and urate. Calculi in the prostate are called prostatic calculi. Calculi in the mammary gland are called breast microcalcifications or mammary microcalcifications. Calculi of the gallbladder and bile ducts are called gallstones and are primarily developed from bile salts and cholesterol derivatives. Calculi in the nasal passages (rhinoliths) are rare. Calculi in


id: cardiac-shunt-20260505-0
file: data/en.wikipedia.org/wiki/Cardiac_shunt-0.md
category: reference
title: Cardiac shunt
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Cardiac_shunt
date_saved: 2026-05-05T07:26:47.861966+00:00
instance: kb-cron
parent: cardiac-shunt-20260505-0
summary:
In cardiology, a cardiac shunt is a pattern of blood flow in the heart that deviates from the normal circuit of the circulatory system. It may be described as right-left, left-right or bidirectional, or as systemic-to-pulmonary or pulmonary-to-systemic. The direction may be controlled by left and/or right heart pressure, a biological or artificial heart valve or both. The presence of a shunt may also affect left and/or right heart pressure either beneficially or detrimentally. == Terminology == The left and right sides of the heart are named from a dorsal view, i.e., looking at the heart from the back or from the perspective of the person whose heart it is. There are four chambers in a heart: an atrium (upper) and a ventricle (lower) on both the left and right sides. In mammals and birds, blood from the body goes to the right side of the heart first. Blood enters the upper right atrium, is pumped down to the right ventricle and from there to the lungs via the pulmonary artery. Blood going to the lungs is called the pulmonary circulation. When the blood returns to the heart from the lungs via the pulmonary vein, it goes to the left


id: carnoy-s-solution-20260505-0
file: data/en.wikipedia.org/wiki/Carnoy's_solution-0.md
category: reference
title: Carnoy's solution
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Carnoy's_solution
date_saved: 2026-05-05T07:26:50.446534+00:00
instance: kb-cron
parent: carnoy-s-solution-20260505-0
summary:
Carnoy's solution is a fixative composed of 60% ethanol, 30% chloroform and 10% glacial acetic acid, 1 gram of ferric chloride. Carnoy's solution is also the name of a different fixation composed of ethanol and glacial acetic acid (3:1). The invention of Carnoy's solution is attributed to Jean-Baptiste Carnoy, a pioneering 19th century cytologist. == Uses == Some of the uses of Carnoy's solution are: Enhancing lymph node detection during dissection of cadavers. Immunohistochemical fixation and detection of NMDA receptors within the murine hippocampus. Applied directly following enucleation for the treatment of odontogenic keratocysts. Direct application following enucleation (Cuba) for certain kinds of unicystic ameloblastomas. This appears to decrease the likelihood of recurrence over enucleation alone. Protein coagulation is thought to limit uptake of these toxic materials by surrounding tissues, however it is this fact that limits its usefulness as a treatment agent in general. As a fixative for pap smear samples. As a fixative agent for both nuclear and mitochondrial DNA in various tissues. As a fixative agent to preserve mucus, useful for tissue preparation before staining with periodic acid-Schiff base. == References ==


id: case-presentation-20260505-0
file: data/en.wikipedia.org/wiki/Case_presentation-0.md
category: reference
title: Case presentation
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Case_presentation
date_saved: 2026-05-05T07:26:51.626584+00:00
instance: kb-cron
parent: case-presentation-20260505-0
summary:
A case presentation is a formal communication between health care professionals such as doctors and nurses regarding a patient's clinical information. Essential parts of a case presentation include: Identification Reason for consultation/admission Chief complaints (CC) - what made patients seek medical attention. History of present illness (HPI) - circumstances relating to chief complaints. Past medical history (PMHx) Past surgical history Current medications Allergies Family history (FHx) Social history (SocHx) Physical examination (PE) Laboratory results (Lab) Other investigations (imaging, biopsy etc.) Case summary and impression Management plans follow up in clinic or hospital Adherence of the patient to treatment success of the treatment or failure. causes of success or failure. == See also == Case report Case series == References ==


id: case-report-20260505-0
file: data/en.wikipedia.org/wiki/Case_report-0.md
category: reference
title: Case report
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Case_report
date_saved: 2026-05-05T07:26:52.887502+00:00
instance: kb-cron
parent: case-report-20260505-0
summary:
In medicine, a case report is a detailed report of the symptoms, signs, diagnosis, treatment, and follow-up of an individual patient. Case reports may contain a demographic profile of the patient, but usually describe an unusual or novel occurrence. Some case reports also contain a literature review of other reported cases. Case reports are professional narratives that provide feedback on clinical practice guidelines and offer a framework for early signals of effectiveness, adverse events, and cost. They can be shared for medical, scientific, or educational purposes. == Types == Most case reports are on one of six topics: An unexpected association between diseases or symptoms. An unexpected event in the course of observing or treating a patient. Findings that shed new light on the possible pathogenesis of a disease or an adverse effect. Unique or rare features of a disease. Unique therapeutic approaches. A positional or quantitative variation of the anatomical structures. == Roles in research and education == A case report is generally considered a type of anecdotal evidence. Given their intrinsic methodological limitations, including lack of statistical sampling, case reports are placed at the bottom of the hierarchy of clinical evidence, together with case series. Nevertheless, case reports


id: cause-of-death-20260505-0
file: data/en.wikipedia.org/wiki/Cause_of_death-0.md
category: reference
title: Cause of death
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Cause_of_death
date_saved: 2026-05-05T07:26:54.150733+00:00
instance: kb-cron
parent: cause-of-death-20260505-0
summary:
In law, medicine, and statistics, cause of death is an official determination of the conditions resulting in a human's death, which may be recorded on a death certificate. A cause of death is determined by a medical examiner. In rare cases, an autopsy needs to be performed by a pathologist. The cause of death is a specific disease or injury, in contrast to the manner of death, which is a small number of categories like "natural", "accident", "suicide", and "homicide", each with different legal implications. The standards for cause of death recording and reporting are set by WHO. All death certificates typically include a medical certificate of cause of death collecting information for disease prevention and health policy. The recommended international format for this certificate is established by the World Health Organization (WHO) and applies uniformly across member states. The World Health Organization's ICD-11 Reference Guide provides the authoritative specification for its design and completion. Extensive detail is also available in the format of a manual for medical certification. International Classification of Disease (ICD) codes are used to record manner and cause of death in a systematic way that makes it easy to compile statistics and more feasible to compare


id: central-venous-pressure-20260505-0
file: data/en.wikipedia.org/wiki/Central_venous_pressure-0.md
category: reference
title: Central venous pressure
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Central_venous_pressure
date_saved: 2026-05-05T07:26:55.347879+00:00
instance: kb-cron
parent: central-venous-pressure-20260505-0
summary:
Central venous pressure (CVP) is the blood pressure in the venae cavae, near the right atrium of the heart. CVP reflects the amount of blood returning to the heart and the ability of the heart to pump the blood back into the arterial system. CVP is often a good approximation of right atrial pressure (RAP), although the two terms are not identical, as a pressure differential can sometimes exist between the venae cavae and the right atrium. CVP and RAP can differ when arterial tone is altered. This can be graphically depicted as changes in the slope of the venous return (VR) plotted against right atrial pressure (where central venous pressure (CVP) increases, but right atrial pressure (RAP) stays the same; VR = CVP − RAP). CVP has been, and often still is, used as a surrogate for preload, and changes in CVP in response to infusions of intravenous fluid have been used to predict volume-responsiveness (i.e. whether more fluid will improve cardiac output). However, there is increasing evidence that CVP, whether as an absolute value or in terms of changes in response to fluid, does not correlate with ventricular volume (i.e. preload) or volume-responsiveness, and so should not be


id: cerebellar-degeneration-20260505-0
file: data/en.wikipedia.org/wiki/Cerebellar_degeneration-0.md
category: reference
title: Cerebellar degeneration
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Cerebellar_degeneration
date_saved: 2026-05-05T07:26:56.519839+00:00
instance: kb-cron
parent: cerebellar-degeneration-20260505-0
summary:
Cerebellar degeneration is a condition in which cerebellar cells, otherwise known as neurons, become damaged and progressively weaken in the cerebellum. There are two types of cerebellar degeneration; paraneoplastic cerebellar degeneration, and alcoholic or nutritional cerebellar degeneration. As the cerebellum contributes to the coordination and regulation of motor activities, as well as controlling equilibrium of the human body, any degeneration to this part of the organ can be life-threatening. Cerebellar degeneration can result in disorders in fine movement, posture, and motor learning in humans, due to a disturbance of the vestibular system. This condition may not only cause cerebellar damage on a temporary or permanent basis, but can also affect other tissues of the central nervous system, those including the cerebral cortex, spinal cord and the brainstem (made up of the medulla oblongata, midbrain, and pons). Cerebellar degeneration can be attributed to a plethora of hereditary and non-hereditary conditions. More commonly, cerebellar degeneration can also be classified according to conditions that an individual may acquire during their lifetime, including infectious, metabolic, autoimmune, paraneoplastic, nutritional or toxic triggers. == Signs and symptoms == Patients with cerebellar degeneration experience a progressive loss of nerve cells (Purkinje cells) throughout the cerebellum. As well


id: cerebellar-degeneration-20260505-1
file: data/en.wikipedia.org/wiki/Cerebellar_degeneration-1.md
category: reference
title: Cerebellar degeneration
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Cerebellar_degeneration
date_saved: 2026-05-05T07:26:56.519839+00:00
instance: kb-cron
parent: cerebellar-degeneration-20260505-0
summary:
To select an appropriate and accurate diagnostic test for cerebellar degeneration, it is crucial that a range of factors specific to each patient are taken into consideration. These include; the patient's age, acuity of their signs and symptoms, associated neurological conditions, and family history of hereditary forms of cerebellar degeneration. A diagnosis for cerebellar degeneration is regarded after any of the aforementioned signs and symptoms surface. For genetically classified forms of cerebellar degeneration, genetic testing can be carried out in order to confirm or deny the diagnosis, where this form of testing is only possible if the gene responsible for the cause of the condition is recognised. In saying this, for most conditions the genetic cause of cerebellar degeneration is unidentified, hence these patients cannot proceed with genetic testing. In cases where cerebellar degeneration is acquired, a diagnosis can be established using imaging methods such as computerised tomography (CT scans) and magnetic resonance imaging (MRI), necessary to detect brain abnormalities in patients with cerebellar degeneration. == Treatment == Like any other disease, treatment for cerebellar degeneration is contingent on the underlying cause, unique to each patient. As of present time, hereditary forms of cerebellar degeneration are incurable, though they can


id: cerebral-perfusion-pressure-20260505-0
file: data/en.wikipedia.org/wiki/Cerebral_perfusion_pressure-0.md
category: reference
title: Cerebral perfusion pressure
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Cerebral_perfusion_pressure
date_saved: 2026-05-05T07:26:57.755648+00:00
instance: kb-cron
parent: cerebral-perfusion-pressure-20260505-0
summary:
Cerebral perfusion pressure (CPP) is the net pressure gradient causing cerebral blood flow to the brain (brain perfusion). It must be maintained within narrow limits because too little pressure could cause brain tissue to become ischemic (having inadequate blood flow), and too much could raise intracranial pressure (ICP). == Definitions == The cranium encloses a fixed-volume space that holds three components: blood, cerebrospinal fluid (CSF), and very soft tissue (the brain). While both the blood and CSF have poor compression capacity, the brain is easily compressible. Every increase of ICP can cause a change in tissue perfusion and an increase in stroke events. === From resistance === CPP can be defined as the pressure gradient causing cerebral blood flow (CBF) such that C B F = C P P / C V R {\displaystyle CBF=CPP/CVR} where: CVR is cerebrovascular resistance === By intracranial pressure === An alternative definition of CPP is: C P P = M A P − I C P {\displaystyle CPP=MAP-ICP} where: MAP is mean arterial pressure ICP is intracranial pressure JVP is jugular venous pressure This definition may be more appropriate if considering the circulatory system in the brain as a Starling resistor, where an external


id: cervical-conization-20260505-0
file: data/en.wikipedia.org/wiki/Cervical_conization-0.md
category: reference
title: Cervical conization
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Cervical_conization
date_saved: 2026-05-05T07:26:59.084677+00:00
instance: kb-cron
parent: cervical-conization-20260505-0
summary:
Cervical conization refers to an excision of a cone-shaped portion of tissue from the mucous membrane of the cervix. Conization is used for diagnostic purposes as part of a biopsy and for therapeutic purposes to remove pre-cancerous cells (cervical intraepithelial neoplasia) or early stage cervical cancer. Ablative treatments are also available to treat abnormal cervical cells. The decision to perform a cervical conization procedure is made with consideration of a patient's pap smear, colposcopy, and HPV test results. The American College of Obstetricians and Gynecologists (ACOG) recommends that decisions regarding excision should be based on the risk of CIN3+. A conization can be performed in the office or the operating room, depending on the type of conization performed. This procedure carries few risks, with the most common one being bleeding after the procedure. == History == Before the introduction of the speculum, cervical cancer was only found once it was advanced. With the invention and use of a speculum, changes in the cervix could be appreciated. First, they were evaluated macroscopically and eventually were also assessed using a microscope. In 1927, H. Hinselmann discovered the transformation zone, where metaplastic squamous epithelium is found between the columnar epithelium of the endocervix


id: cervical-conization-20260505-1
file: data/en.wikipedia.org/wiki/Cervical_conization-1.md
category: reference
title: Cervical conization
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Cervical_conization
date_saved: 2026-05-05T07:26:59.084677+00:00
instance: kb-cron
parent: cervical-conization-20260505-0
summary:
The vagina is prepped using antimicrobial scrub or iodine. Draping is placed to maintain a sterile surgical field. Some physicians may choose to drain the bladder using a catheter. The speculum will be placed, and the cervix visualized. The tissue is then excised from the cervix. The tissue will include the transformation zone and will be shaped like a cone, as the procedure name suggests. The physician will ensure hemostasis has been achieved before removing the speculum and ending the procedure. Typically, the physician will place a suture at the 12 o'clock position of the excised tissue to serve as a reference point during histological examination. The main difference between cold knife conization and LEEP is the instrument used to excise the tissue. In a LEEP, a thin wire loop electrode is used to remove the cone-shaped surgical specimen. During a cold knife cone, a scalpel is used to excise the tissue. Both LEEP and cold knife cone have shown equal effectiveness, so the decision for which procedure is often based on the physician's comfort with each procedure or other clinical considerations. Cold knife cone is performed with a scalpel, and one advantage of this procedure is that the margins


id: cervical-dislocation-20260505-0
file: data/en.wikipedia.org/wiki/Cervical_dislocation-0.md
category: reference
title: Cervical dislocation
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Cervical_dislocation
date_saved: 2026-05-05T07:27:00.313128+00:00
instance: kb-cron
parent: cervical-dislocation-20260505-0
summary:
Cervical dislocation is a common method of animal euthanasia. It refers to a technique used in physical euthanasia of small animals by applying pressure to the neck and dislocating the spinal column from the skull or brain. The aim is to quickly separate the spinal cord from the brain so as to provide the animal with a fast, painless, and easy death. == Technique == Firm pressure is applied at the base of the skull, along with a sharp pinching and twisting of the thumb and forefinger. At the same time, the tail is pulled backward. This severs the spinal cord at the base of the brain or within the cervical spine area (the upper third of the neck). According to the Canadian Council on Animal Care (CCAC), cervical dislocation is normally only conducted on small animals. == Ethics == The University of Iowa and some veterinary associations consider the technique to be an ethically acceptable method for killing small rodents such as rats, mice, squirrels, etc. == See also == Cervical fracture Blunt trauma Slaughter (livestock) == References ==


id: chain-of-survival-20260505-0
file: data/en.wikipedia.org/wiki/Chain_of_survival-0.md
category: reference
title: Chain of survival
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Chain_of_survival
date_saved: 2026-05-05T07:27:01.490920+00:00
instance: kb-cron
parent: chain-of-survival-20260505-0
summary:
The chain of survival refers to a series of actions that, properly executed, reduce the mortality associated with sudden cardiac arrest. Like any chain, the chain of survival is only as strong as its weakest link. The six interdependent links in the chain of survival are early recognition of sudden cardiac arrest and access to emergency medical care, early CPR, early defibrillation, early advanced cardiac life support, and physical and emotional recovery. The first three links in the chain can be performed by lay bystanders, while the second three links are designated to medical professionals. Currently, between 70 and 90% of cardiac arrest patients die before they reach the hospital. However, a cardiac arrest does not have to be lethal if bystanders can take the right steps immediately. == Background == According to the American Heart Association, out-of-hospital cardiac arrest can affect more than 300,000 people in the United States each year. Three minutes after the onset of cardiac arrest, a lack of blood flow starts to damage the brain, and 10 minutes after, the chances of survival are low. Therefore, bystanders have only a few minutes to act to optimize a person's chances of survival and recovery. To improve


id: chain-of-survival-20260505-1
file: data/en.wikipedia.org/wiki/Chain_of_survival-1.md
category: reference
title: Chain of survival
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Chain_of_survival
date_saved: 2026-05-05T07:27:01.490920+00:00
instance: kb-cron
parent: chain-of-survival-20260505-0
summary:
== Early advanced care == Early advanced cardiac life support by paramedics is another critical link in the chain of survival. In communities with survival rates > 20%, a minimum of two of the rescuers are trained to the advanced level. Some ACLS ambulance providers will administer medications to manage pain, arrhythmias, shock, and pulmonary congestion; monitor the heart rhythm to identify any potentially lethal cardiac arrhythmias; or initiate transcutaneous pacing. ACLS ambulance providers use the mnemonic "MONA" (morphine, oxygen, nitroglycerin, and aspirin) to reflect the out-of-hospital therapies they will use for cardiac arrest. Often, ACLS ambulance providers will attach an electrocardiogram to the patient and transmit its findings to the receiving hospital or care facility, which leads to earlier diagnosis of a heart attack, and significantly reduces time to treatment at the hospital. This prearrival ECG and notification has been shown to improve patient outcomes. In the event of a complication at the scene of the event or on the way to the hospital, ACLS ambulance providers can administer life saving therapies, including CPR, rapid defibrillation, airway management, and intravenous medications. == Recovery == In October 2020, the American Heart Association added the recovery phase as the sixth link


id: cheesewiring-20260505-0
file: data/en.wikipedia.org/wiki/Cheesewiring-0.md
category: reference
title: Cheesewiring
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Cheesewiring
date_saved: 2026-05-05T07:27:02.729782+00:00
instance: kb-cron
parent: cheesewiring-20260505-0
summary:
In medicine, the term cheesewiring or cheesewire effect (used interchangeably) describes any process in which cells or intercellular matrix are dissected or extruded either by the material being pressed through a taut element, or by the tension of a taut element pulling through tissue. The procedure is typically conducted in a surgical setting. Cheesewiring or the cheesewire effect can also describe the process of suture material cutting or tearing through viscera at the time of suture anastomosis and tension. == As a complication == Cheesewiring can be a complication or part of a negative outcome of a procedure. Examples include tumor growth penetrating the openings in a bowel stent that was placed to open an obstructed bowel, sutures used to hold a transplanted cornea in place, and treatment of tendon rupture, when sutures pull through the tendon. Evidence that the cheesewire effect has occurred is most clearly seen after surgery has been completed. Postoperatively as wound margins begin to swell, a suture can drag through the thin portion of tissue present near the surface. This suture drag is most often the result of tension being reduced in tying or placing the suture. It is apparent that cheesewiring has occurred if


id: chemoprophylaxis-20260505-0
file: data/en.wikipedia.org/wiki/Chemoprophylaxis-0.md
category: reference
title: Chemoprophylaxis
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Chemoprophylaxis
date_saved: 2026-05-05T07:27:03.934272+00:00
instance: kb-cron
parent: chemoprophylaxis-20260505-0
summary:
Chemoprevention or chemoprophylaxis refers to the administration of a medication for the purpose of preventing disease or infection. Antibiotics, for example, may be administered to patients with disorders of immune system function to prevent bacterial infections (particularly opportunistic infection). Antibiotics may also be administered to healthy individuals to limit the spread of an epidemic, or to patients who have repeated infections (such as urinary tract infections) to prevent recurrence. It may also refer to the administration of heparin to prevent deep venous thrombosis in hospitalized patients. In some cases, chemoprophylaxis is initiated to prevent the spread of an existing infection in an individual to a new organ system, as when intrathecal chemotherapy is administered in patients with malignancy to prevent further infection. The use of chemoprophylaxis is limited primarily by two factors: risk and financial costs. All medications have the potential to cause side effects. In general, chemoprophylaxis should be initiated only when the benefits of treatment outweigh the risks. The cost associated with chemoprophylaxis may be prohibitive, particularly when the cost of treatment is high or the incidence of the target disease is low. Many forms of chemoprophylaxis are therefore not cost-effective. == Specific diseases == Using chemoprophylaxis as


id: chief-complaint-20260505-0
file: data/en.wikipedia.org/wiki/Chief_complaint-0.md
category: reference
title: Chief complaint
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Chief_complaint
date_saved: 2026-05-05T07:27:05.058536+00:00
instance: kb-cron
parent: chief-complaint-20260505-0
summary:
The chief complaint, formally known as CC in the medical field, or termed presenting complaint (PC) in Europe and Canada, forms the second step of medical history taking. It is sometimes also referred to as reason for encounter (RFE), presenting problem, problem on admission or reason for presenting. The chief complaint is a concise statement describing the symptom, problem, condition, diagnosis, physician-recommended return, or other reason for a medical encounter. In some instances, the nature of a patient's chief complaint may determine if services are covered by health insurance. When obtaining the chief complaint, medical students are advised to use open-ended questions. Once the presenting problem is elucidated, a history of present illness can be done using acronyms such as SOCRATES or OPQRST to further analyze the severity, onset and nature of the presenting problem. The patient's initial comments to a physician, nurse, or other health care professionals are important for formulating differential diagnoses. == Prevalence == The collection of chief complaint data may be useful in addressing public health issues. Certain complaints are more common in certain settings and among certain populations. Fatigue has been reported as one of the ten most common reasons for seeing a physician. In


id: children-with-special-healthcare-needs-in-the-united-states-20260505-0
file: data/en.wikipedia.org/wiki/Children_with_Special_Healthcare_Needs_in_the_United_States-0.md
category: reference
title: Children with Special Healthcare Needs in the United States
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Children_with_Special_Healthcare_Needs_in_the_United_States
date_saved: 2026-05-05T07:27:06.242840+00:00
instance: kb-cron
parent: children-with-special-healthcare-needs-in-the-united-states-20260505-0
summary:
Children with Special Healthcare Needs (CSHCN) are defined by the Maternal and Child Health Bureau as: "Those who have one or more chronic physical, developmental, behavioral, or emotional conditions and who also require health and related services of a type or amount beyond that required by children generally" == Types of special healthcare needs == There are a wide variety of physical, mental, and psychological health conditions considered to be special healthcare needs in the United States. They range from relatively mild to chronic and severe. The functional impairments of CSHCN include problems with one or more of the following criteria: breathing, swallowing/digestion/metabolism, blood circulation, chronic pain, hearing even with corrective devices, seeing even with corrective devices, taking care of self, coordination/moving around, learning/understanding/paying attention, speaking/communicating, making/keeping friends, and behavior. The list below states health conditions considered to be special healthcare needs. Attention deficit disorder (ADD) Attention deficit hyperactivity disorder (ADHD) Depression Anxiety problems Behavioral problems Autism spectrum disorders (ASD) Diabetes Developmental disability Intellectual disability (ID) Epilepsy Migraines Traumatic brain injury (TBI) Heart problems Blood problems Cystic fibrosis Cerebral palsy Muscular dystrophy Down syndrome Arthritis Joint problems Allergies == Population == As of 2009, 15.1% of all children in the


id: children-with-special-healthcare-needs-in-the-united-states-20260505-1
file: data/en.wikipedia.org/wiki/Children_with_Special_Healthcare_Needs_in_the_United_States-1.md
category: reference
title: Children with Special Healthcare Needs in the United States
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Children_with_Special_Healthcare_Needs_in_the_United_States
date_saved: 2026-05-05T07:27:06.242840+00:00
instance: kb-cron
parent: children-with-special-healthcare-needs-in-the-united-states-20260505-0
summary:
Inpatient/outpatient hospital resources Transportation for non-emergency medical care Home health services Laboratory procedures Physician services Family planning Nursing facilities Early Periodic Screening, Diagnosis, and Treatment (EPDST) program Tobacco cessation programs for pregnant women Medicaid requires no co-payment or deductibles. To be eligible for the program one must be in one of four categories. The first is income level, the second is disability criteria, the third is eligibility for institutional level of care, and the fourth is out of home placement. Many CSHCN qualify for Medicaid based on low income level, but some children qualify independently of income level due to their disabilities. Medicaid has a more restrictive definition of disabilities and special healthcare needs than the Maternal and Children's Health Bureau (MCHB), and defines special needs as needs that must impede daily functioning. Medicaid accepts children who need to receive Supplemental Security Income program money, and children who are defined as medically needy. Medically needy children are those whose families have above the maximum income to receive Medicaid, but due to health expenditures their income is lowered to the level required. 40 states currently offer this program. Medicaid programs in each state are administered differently, and federal dollars go to


id: children-with-special-healthcare-needs-in-the-united-states-20260505-2
file: data/en.wikipedia.org/wiki/Children_with_Special_Healthcare_Needs_in_the_United_States-2.md
category: reference
title: Children with Special Healthcare Needs in the United States
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Children_with_Special_Healthcare_Needs_in_the_United_States
date_saved: 2026-05-05T07:27:06.242840+00:00
instance: kb-cron
parent: children-with-special-healthcare-needs-in-the-united-states-20260505-0
summary:
=== Therapy for CSHCN === Therapies for children with special healthcare needs can be accessed via public schools or private therapists. The Individuals with Disabilities Education Act (IDEA) includes occupational and physical therapy as well as other therapies, as part of the special education that should be offered in all public schools to CSHCN. This act states that all children with disabilities should have access to education that suits their SHCN, including needed therapies. Out of school therapies can also be used be employed by children with SHCN but only 3.2% of CSHCN qualify for uses of special therapy under their insurance programs. One type of therapy for children with SHCN is occupational therapy. Occupational therapists work with CSHCN by supporting them and their families to learn how to participate in everyday routines and daily activities. They encourage children with physical, cognitive, communication and behavior challenges to develop ways to live, play, learn and make friends despite their special needs. Occupational therapists can work with these children and their schools to create more accommodating learning environments. Another skill these professionals teach to children with SHCN is how to use adaptive equipment such as wheelchairs, eating aids and braces in daily


id: chronic-condition-20260505-0
file: data/en.wikipedia.org/wiki/Chronic_condition-0.md
category: reference
title: Chronic condition
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Chronic_condition
date_saved: 2026-05-05T07:27:07.415511+00:00
instance: kb-cron
parent: chronic-condition-20260505-0
summary:
A chronic condition, also known as chronic disease or chronic illness, is a health condition or disease that is persistent or otherwise long-lasting in its effects. The term chronic is often applied when the course of the disease lasts for more than three months. Common chronic diseases include diabetes, functional gastrointestinal disorder, eczema, arthritis, asthma, chronic obstructive pulmonary disease, autoimmune diseases, genetic disorders and some viral diseases such as acquired immunodeficiency syndrome. Chronic illnesses are sometimes contrasted with terminal illnesses, which will end the affected person’s life. It is possible for an illness to change in definition from terminal to chronic as medicine progresses. For example, type 1 diabetes and HIV were once terminal, but are now considered chronic illnesses due to the availability of daily medications which allow these individuals to live while managing symptoms. In medicine, chronic conditions are distinguished from those that are acute. An acute condition typically affects a person for a few days to several weeks. On the other hand, a chronic condition persists for an extended period of time or is lifelong, even with treatment. Chronic conditions may have periods of remission, where a person has few or no symptoms, and relapse, where symptoms


id: chronic-condition-20260505-1
file: data/en.wikipedia.org/wiki/Chronic_condition-1.md
category: reference
title: Chronic condition
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Chronic_condition
date_saved: 2026-05-05T07:27:07.415511+00:00
instance: kb-cron
parent: chronic-condition-20260505-0
summary:
While risk factors vary, many common chronic diseases are caused by dietary, lifestyle, and metabolic risk factors. Therefore, these conditions might be prevented by behavioral changes, such as quitting smoking, adopting a healthy diet, and increasing physical activity. Social determinants are important risk factors for chronic diseases. Social factors, e.g., socioeconomic status, education level, and race/ethnicity, are a major cause for the disparities observed in the care of chronic disease. Lack of access and delay in receiving care result in worse outcomes for patients from minorities and underserved populations. Those barriers to medical care complicate patients monitoring and continuity in treatment. In the US, minorities and low-income populations are less likely to seek, access and receive preventive services necessary to detect conditions at an early stage. The majority of US health care and economic costs associated with medical conditions are incurred by chronic diseases and conditions and associated health risk behaviors. Eighty-four percent of all health care spending in 2006 was for the 50% of the population who have one or more common chronic medical conditions (CDC, 2014). There are several psychosocial risk and resistance factors among children with chronic illness and their family members. Adults with chronic illness were


