In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, these are the groups that obey the axiom of commutativity. Abelian groups generalize the arithmetic of addition of integers. They are named after early 19th century mathematician Niels Henrik Abel.
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is automorphism. For example, an endomorphism of a vector space V is a linear map f: V → V, and an endomorphism of a group G is a group homomorphism f: G → G. In general, we can talk about endomorphisms in any category. In the category of sets, endomorphisms are functions from a set S to itself.
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in an effort to unify several cohomology theories by Alexander Grothendieck and independently in the slightly earlier work of David Buchsbaum. Abelian categories are very stable categories; for example they are regular and they satisfy the snake lemma. The class of Abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an Abelian category, or the category of functors from a small category to an Abelian category are Abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometry, cohomology and pure category theory. Abelian categories are named after Niels Henrik Abel.
In abstract algebra, an abelian group (G, +) is called finitely generated if there exist finitely many elements x1, ..., xs in G such that every x in G can be written in the form
A polygraph, popularly referred to as a lie detector test, is a device or procedure that measures and records several physiological indicators such as blood pressure, pulse, respiration, and skin conductivity while a person is asked and answers a series of questions. The belief underpinning the use of the polygraph is that deceptive answers will produce physiological responses that can be differentiated from those associated with non-deceptive answers. There are, however, no specific physiological reactions associated with lying, making it difficult to identify factors that separate liars from truth tellers. Polygraph examiners also prefer to use their own individual scoring method, as opposed to computerized techniques, as they may more easily defend their own evaluations.
Categorization is something that humans and other organisms do: "doing the right thing with the right kind of thing." The doing can be nonverbal or verbal. For humans, both concrete objects and abstract ideas are recognized, differentiated, and understood through categorization. Objects are usually categorized for some adaptive or pragmatic purpose. Categorization is grounded in the features that distinguish the category's members from nonmembers. Categorization is important in learning, prediction, inference, decision making, language, and many forms of organisms' interaction with their environments.
In the theory of abelian groups, the torsion subgroupAT of an abelian group A is the subgroup of A consisting of all elements that have finite order. An abelian group A is called a torsion group if every element of A has finite order and is called torsion-free if every element of A except the identity is of infinite order.
In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra. A module over a ring is a generalization of the notion of vector space over a field, wherein the corresponding scalars are the elements of an arbitrary given ring and a multiplication is defined between elements of the ring and elements of the module.
In category theory, the coproduct, or categorical sum, is a category-theoretic construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism. It is the category-theoretic dual notion to the categorical product, which means the definition is the same as the product but with all arrows reversed. Despite this seemingly innocuous change in the name and notation, coproducts can be and typically are dramatically different from products.
In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated R-module also may be called a finite R-module, finite over R, or a module of finite type.
In mathematics, a monoidal category is a category C equipped with a bifunctor
In mathematics, the Euclidean group E(n), also known as ISO(n) or similar, is the symmetry group of n-dimensional Euclidean space. Its elements are the isometries associated with the Euclidean distance, and are called Euclidean isometries, Euclidean transformations or rigid transformations.
In topology, a compactly generated space is a topological space whose topology is coherent with the family of all compact subspaces. Specifically, a topological space X is compactly generated if it satisfies the following condition:
In the mathematical subject of universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satisfying a given set of identities. For example, the groups form a variety of algebras, as do the abelian groups, the rings, the monoids etc. According to Birkhoff's theorem, a class of algebraic structures of the same signature is a variety if and only if it is closed under the taking of homomorphic images, subalgebras and (direct) products. In the context of category theory, a variety of algebras, together with its homomorphisms, forms a category; these are usually called finitary algebraic categories.
In mathematics, a simplicial set is an object made up of "simplices" in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined as a contravariant functor from the simplex category to the category of sets. Simplicial sets were introduced in 1950 by Samuel Eilenberg and J. A. Zilber.
In category theory, a (strict) n-monoid is an n-category with only one 0-cell. In particular, a 1-monoid is a monoid and a 2-monoid is a strict monoidal category.
Lie detection is an assessment of a verbal statement with the goal to reveal a possible intentional deceit. Lie detection may refer to a cognitive process of detecting deception by evaluating message content as well as non-verbal cues. It also may refer to questioning techniques used along with technology that record physiological functions to ascertain truth and falsehood in response. The latter is commonly used by law enforcement in the United States, but rarely in other countries because it is based on pseudoscience. There are a wide variety of technologies available for this purpose. The most common and long used measure is the polygraph, which the U.S. National Academy of Sciences states, in populations untrained in countermeasures, can discriminate lying from truth telling at rates above chance, though below perfection. They added that the results apply only to specific events and not to screening, where it is assumed that the polygraph works less well.
Brain-reading uses the responses of multiple voxels in the brain evoked by stimulus then detected by fMRI in order to decode the original stimulus. Brain reading studies differ in the type of decoding employed, the target, and the decoding algorithms employed.
In mathematics, a Grothendieck category is a certain kind of abelian category, introduced in Alexander Grothendieck's Tôhoku paper of 1957 in order to develop the machinery of homological algebra for modules and for sheaves in a unified manner. The theory of these categories was further developed in Peter Gabriel's seminal thèse in 1962.
In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of simple objects, and simple objects are those that do not contain non-trivial proper sub-objects. The precise definitions of these words depends on the context.
