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Hadamard Gate
The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogonal, symmetric, involutive, linear operation on real numbers (or complex, or hypercomplex numbers, although the Hadamard matrices themselves are purely real). The Hadamard transform can be regarded as being built out of size-2 discrete Fourier transforms (DFTs), and is in fact equivalent to a multidimensional DFT of size . It decomposes an arbitrary input vector into a superposition of Walsh functions. The transform is named for the French mathematician Jacques Hadamard (), the German-American mathematician Hans Rademacher, and the American mathematician Joseph L. Walsh. Definition The Hadamard transform ''H''''m'' is a 2''m'' × 2''m'' matrix, the Hadamard matrix (scaled by a normalization factor), that transforms 2''m'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1010 0110 Walsh Spectrum (single Row)
1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment of length 1. In conventions of sign where zero is considered neither positive nor negative, 1 is the first and smallest positive integer. It is also sometimes considered the first of the infinite sequence of natural numbers, followed by 2, although by other definitions 1 is the second natural number, following 0. The fundamental mathematical property of 1 is to be a multiplicative identity, meaning that any number multiplied by 1 equals the same number. Most if not all properties of 1 can be deduced from this. In advanced mathematics, a multiplicative identity is often denoted 1, even if it is not a number. 1 is by convention not considered a prime number; this was not universally accepted until the mid-20th century. Additionally, 1 is the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joseph L
Joseph is a common male given name, derived from the Hebrew Yosef (יוֹסֵף). "Joseph" is used, along with "Josef", mostly in English, French and partially German languages. This spelling is also found as a variant in the languages of the modern-day Nordic countries. In Portuguese and Spanish, the name is "José". In Arabic, including in the Quran, the name is spelled '' Yūsuf''. In Persian, the name is "Yousef". The name has enjoyed significant popularity in its many forms in numerous countries, and ''Joseph'' was one of the two names, along with '' Robert'', to have remained in the top 10 boys' names list in the US from 1925 to 1972. It is especially common in contemporary Israel, as either "Yossi" or "Yossef", and in Italy, where the name "Giuseppe" was the most common male name in the 20th century. In the first century CE, Joseph was the second most popular male name for Palestine Jews. In the Book of Genesis Joseph is Jacob's eleventh son and Rachel's first s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fast Hadamard Transform
Fast or FAST may refer to: * Fast (noun), high speed or velocity * Fast (noun, verb), to practice fasting, abstaining from food and/or water for a certain period of time Acronyms and coded Computing and software * ''Faceted Application of Subject Terminology'', a thesaurus of subject headings * Facilitated Application Specification Techniques, a team-oriented approach for requirement gathering * FAST protocol, an adaptation of the FIX protocol, optimized for streaming * FAST TCP, a TCP congestion avoidance algorithm * FAST and later as Fast Search & Transfer, a Norwegian company focusing on data search technologies * Fatigue Avoidance Scheduling Tool, software to develop work schedules * Features from accelerated segment test, computer vision method for corner detection * Federation Against Software Theft, a UK organization that pursues those who illegally distribute software * Feedback arc set in Tournaments, a computational problem in graph theory * USENIX Conference on Fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element ''g'' such that every other element of the group may be obtained by repeatedly applying the group operation to ''g'' or its inverse. Each element can be written as an integer power of ''g'' in multiplicative notation, or as an integer multiple of ''g'' in additive notation. This element ''g'' is called a '' generator'' of the group. Every infinite cyclic group is isomorphic to the additive group of Z, the integers. Every finite cyclic group of order ''n'' is isomorphic to the additive group of Z/''n''Z, the integers modulo ''n''. Every cyclic group is an abelian group (meaning that its group operation is commutative), and every finitely generated abelian group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pontryagin Duality
In mathematics, Pontryagin duality is a duality (mathematics), duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numbers of modulus one), the finite abelian groups (with the discrete topology), and the additive group of the integers (also with the discrete topology), the real numbers, and every dimension (vector space), finite dimensional vector space over the reals or a p-adic field, -adic field. The Pontryagin dual of a locally compact abelian group is the locally compact abelian topological group formed by the continuous group homomorphisms from the group to the circle group with the operation of pointwise multiplication and the topology of uniform convergence on compact sets. The Pontryagin duality theorem establishes Pontryagin duality by stating that any locally compact abelian group is naturally isomorphic with its bidual (the dual of its dual). T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Character (mathematics)
In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ..., a character is (most commonly) a special kind of function from a group to a field (mathematics), field (such as the complex numbers). There are at least two distinct, but overlapping meanings. Other uses of the word "character" are almost always qualified. Multiplicative character A multiplicative character (or linear character, or simply character) on a group ''G'' is a group homomorphism from ''G'' to the unit group, multiplicative group of a field , usually the field of complex numbers. If ''G'' is any group, then the set Ch(''G'') of these morphisms forms an abelian group under pointwise multiplication. This group is referred to as the character group of ''G''. Sometimes only ''unitary'' characters are consid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Transform On Finite Groups
In mathematics, the Fourier transform on finite groups is a generalization of the discrete Fourier transform from cyclic to arbitrary finite groups. Definitions The Fourier transform of a function f : G \to \Complex at a representation \varrho : G \to \mathrm(d_\varrho, \Complex) of G is \widehat(\varrho) = \sum_ f(a) \varrho(a). For each representation \varrho of G, \widehat(\varrho) is a d_\varrho \times d_\varrho matrix, where d_\varrho is the degree of \varrho. The inverse Fourier transform at an element a of G is given by f(a) = \frac \sum_i d_ \text\left(\varrho_i(a^)\widehat(\varrho_i)\right). Properties Transform of a convolution The convolution of two functions f, g : G \to \mathbb is defined as (f \ast g)(a) = \sum_ f\!\left(ab^\right) g(b). The Fourier transform of a convolution at any representation \varrho of G is given by \widehat(\varrho) = \hat(\varrho)\hat(\varrho). Plancherel formula For functions f, g : G \to \mathbb, the Plancherel formula st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Group
In mathematics, specifically in group theory, an elementary abelian group (or elementary abelian ''p''-group) is an abelian group in which every nontrivial element has order ''p''. The number ''p'' must be prime, and the elementary abelian groups are a particular kind of ''p''-group. The case where ''p'' = 2, i.e., an elementary abelian 2-group, is sometimes called a Boolean group. Every elementary abelian ''p''-group is a vector space over the prime field with ''p'' elements, and conversely every such vector space is an elementary abelian group. By the classification of finitely generated abelian groups, or by the fact that every vector space has a basis, every finite elementary abelian group must be of the form (Z/''p''Z)''n'' for ''n'' a non-negative integer (sometimes called the group's ''rank''). Here, Z/''p''Z denotes the cyclic group of order ''p'' (or equivalently the integers mod ''p''), and the superscript notation means the ''n''-fold direct product of groups. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary Operator
In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Unitary operators are usually taken as operating ''on'' a Hilbert space, but the same notion serves to define the concept of isomorphism ''between'' Hilbert spaces. A unitary element is a generalization of a unitary operator. In a unital algebra, an element of the algebra is called a unitary element if , where is the identity element. Definition Definition 1. A ''unitary operator'' is a bounded linear operator on a Hilbert space that satisfies , where is the adjoint of , and is the identity operator. The weaker condition defines an ''isometry''. The other condition, , defines a ''coisometry''. Thus a unitary operator is a bounded linear operator which is both an isometry and a coisometry, or, equivalently, a surjective isometry. An equivalent definition is the following: Definition 2. A ''unitary operator'' is a bounded linear operator ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kronecker Product
In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer product (which is denoted by the same symbol) from vectors to matrices, and gives the matrix of the tensor product linear map with respect to a standard choice of basis. The Kronecker product is to be distinguished from the usual matrix multiplication, which is an entirely different operation. The Kronecker product is also sometimes called matrix direct product. The Kronecker product is named after the German mathematician Leopold Kronecker (1823–1891), even though there is little evidence that he was the first to define and use it. The Kronecker product has also been called the ''Zehfuss matrix'', and the ''Zehfuss product'', after , who in 1858 described this matrix operation, but Kronecker product is currently the most widely used. Definition If A is an matrix and B is a matrix, then the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or can be trivially determined by the context. I_1 = \begin 1 \end ,\ I_2 = \begin 1 & 0 \\ 0 & 1 \end ,\ I_3 = \begin 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end ,\ \dots ,\ I_n = \begin 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \end. The term unit matrix has also been widely used, but the term ''identity matrix'' is now standard. The term ''unit matrix'' is ambiguous, because it is also used for a matrix of ones and for any unit of the ring of all n\times n matrices. In some fields, such as group theory or quantum mechanics, the identity matrix is sometimes denoted by a boldface one, \mathbf, or called "id" (short for identity) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base (exponentiation)
In exponentiation, the base is the number b in an expression of the form bn. Related terms The number n is called the exponent and the expression is known formally as exponentiation of b by n or the exponential of n with base b. It is more commonly expressed as "the nth power of b", "b to the nth power" or "b to the power n". For example, the fourth power of 10 is 10,000 because . The term ''power'' strictly refers to the entire expression, but is sometimes used to refer to the exponent. Radix is the traditional term for ''base'', but usually refers then to one of the common bases: decimal (10), binary (2), hexadecimal (16), or sexagesimal (60). When the concepts of variable and constant came to be distinguished, the process of exponentiation was seen to transcend the algebraic functions. In his 1748 ''Introductio in analysin infinitorum'', Leonhard Euler referred to "base a = 10" in an example. He referred to ''a'' as a "constant number" in an extensive consideration of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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