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Identity element. In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. [1] [2] For example, 0 is an identity element of the addition of real numbers. This concept is used in algebraic structures such as groups and rings.
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures.It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science.
Formally, a group is an ordered pair of a set and a binary operation on this set that satisfies the group axioms. The set is called the underlying set of the group, and the operation is called the group operation or the group law . A group and its underlying set are thus two different mathematical objects.
A proof by double counting. A combinatorial identity is proven by counting the number of elements of some carefully chosen set in two different ways to obtain the different expressions in the identity. Since those expressions count the same objects, they must be equal to each other and thus the identity is established. A bijective proof.
The term convolution refers to both the result function and to the process of computing it. It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The integral is evaluated for all values of shift, producing the convolution function.
A group in which the objects are matrices and the group operation is matrix multiplication is called a matrix group. Since a group of every element must be invertible, the most general matrix groups are the groups of all invertible matrices of a given size, called the general linear groups.
Quantum mechanics. Quantum entanglement is the phenomenon of a group of particles being generated, interacting, or sharing spatial proximity in such a way that the quantum state of each particle of the group cannot be described independently of the state of the others, including when the particles are separated by a large distance.
Synthesis involves building a structure or pattern from diverse elements; it also refers to the act of putting parts together to form a whole or bringing pieces of information together to form a new meaning. Its characteristics include: Production of a unique communication; Production of a plan, or proposed set of operations