In mathematics, 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, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel.
In mathematics, the factorial of a non-negative integer , denoted by , is the product of all positive integers less than or equal to . The factorial of also equals the product of with the next smaller factorial:
In abstract algebra, a group isomorphism is a function between two groups that sets up a bijection between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic. From the standpoint of group theory, isomorphic groups have the same properties and need not be distinguished.
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, making it an example of a coproduct. Contrast with the direct product, which is the dual notion.
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. As formulas are entirely constituted with symbols of various types, many symbols are needed for expressing all mathematics.
An exact sequence is a sequence of morphisms between objects such that the image of one morphism equals the kernel of the next.
In mathematics, a recurrence relation is an equation according to which the th term of a sequence of numbers is equal to some combination of the previous terms. Often, only previous terms of the sequence appear in the equation, for a parameter that is independent of ; this number is called the order of the relation. If the values of the first numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation.
In mathematics and computer science, a recursive definition, or inductive definition, is used to define the elements in a set in terms of other elements in the set. Some examples of recursively-definable objects include factorials, natural numbers, Fibonacci numbers, and the Cantor ternary set.
In computer science, corecursion is a type of operation that is dual to recursion. Whereas recursion works analytically, starting on data further from a base case and breaking it down into smaller data and repeating until one reaches a base case, corecursion works synthetically, starting from a base case and building it up, iteratively producing data further removed from a base case. Put simply, corecursive algorithms use the data that they themselves produce, bit by bit, as they become available, and needed, to produce further bits of data. A similar but distinct concept is generative recursion which may lack a definite "direction" inherent in corecursion and recursion.
In mathematics, the Grothendieck group, or group of differences, of a commutative monoid M is a certain abelian group. This abelian group is constructed from M in the most universal way, in the sense that any abelian group containing a homomorphic image of M will also contain a homomorphic image of the Grothendieck group of M. The Grothendieck group construction takes its name from a specific case in category theory, introduced by Alexander Grothendieck in his proof of the Grothendieck–Riemann–Roch theorem, which resulted in the development of K-theory. This specific case is the monoid of isomorphism classes of objects of an abelian category, with the direct sum as its operation.
In computer science, recursion is a method of solving a computational problem where the solution depends on solutions to smaller instances of the same problem. Recursion solves such recursive problems by using functions that call themselves from within their own code. The approach can be applied to many types of problems, and recursion is one of the central ideas of computer science.
The power of recursion evidently lies in the possibility of defining an infinite set of objects by a finite statement. In the same manner, an infinite number of computations can be described by a finite recursive program, even if this program contains no explicit repetitions.
In category theory, the concept of catamorphism denotes the unique homomorphism from an initial algebra into some other algebra.
In algebraic geometry, the Chow groups of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties in a similar way to how simplicial or cellular homology groups are formed out of subcomplexes. When the variety is smooth, the Chow groups can be interpreted as cohomology groups and have a multiplication called the intersection product. The Chow groups carry rich information about an algebraic variety, and they are correspondingly hard to compute in general.
The Bird–Meertens formalism (BMF) is a calculus for deriving programs from program specifications by a process of equational reasoning. It was devised by Richard Bird and Lambert Meertens as part of their work within IFIP Working Group 2.1.
In computer programming, an anamorphism is a function that generates a sequence by repeated application of the function to its previous result. You begin with some value A and apply a function f to it to get B. Then you apply f to B to get C, and so on until some terminating condition is reached. The anamorphism is the function that generates the list of A, B, C, etc. You can think of the anamorphism as unfolding the initial value into a sequence.
In formal methods of computer science, a paramorphism (from Greek παρά, meaning "close together") is an extension of the concept of catamorphism first introduced by Lambert Meertens to deal with a form which “eats its argument and keeps it too”, as exemplified by the factorial function. Its categorical dual is the apomorphism.
A formal grammar describes how to form strings from an alphabet of a formal language that are valid according to the language's syntax. A grammar does not describe the meaning of the strings or what can be done with them in whatever context—only their form. A formal grammar is defined as a set of production rules for such strings in a formal language.
This article describes the features in the programming language Haskell.
In algebra, the cofree coalgebra of a vector space or module is a coalgebra analog of the free algebra of a vector space. The cofree coalgebra of any vector space over a field exists, though it is more complicated than one might expect by analogy with the free algebra.
