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Conference Matrix
In mathematics, a conference matrix (also called a C-matrix) is a square matrix ''C'' with 0 on the diagonal and +1 and −1 off the diagonal, such that ''C''T''C'' is a multiple of the identity matrix ''I''. Thus, if the matrix has order ''n'', ''C''T''C'' = (''n''−1)''I''. Some authors use a more general definition, which requires there to be a single 0 in each row and column but not necessarily on the diagonal. Conference matrices first arose in connection with a problem in telephony.Belevitch, pp. 231-244. They were first described by Vitold Belevitch, who also gave them their name. Belevitch was interested in constructing ideal telephone conference networks from ideal transformers and discovered that such networks were represented by conference matrices, hence the name. Other applications are in statistics, and another is in elliptic geometry. For ''n'' > 1, there are two kinds of conference matrix. Let us normalize ''C'' by, first (if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Seidel Adjacency Matrix
In mathematics, in graph theory, the Seidel adjacency matrix of a simple undirected graph ''G'' is a symmetric matrix with a row and column for each vertex, having 0 on the diagonal, −1 for positions whose rows and columns correspond to adjacent vertices, and +1 for positions corresponding to non-adjacent vertices. It is also called the Seidel matrix or—its original name—the (−1,1,0)-adjacency matrix. It can be interpreted as the result of subtracting the adjacency matrix of ''G'' from the adjacency matrix of the complement of ''G''. The multiset of eigenvalues of this matrix is called the Seidel spectrum. The Seidel matrix was introduced by J. H. van Lint and in 1966 and extensively exploited by Seidel and coauthors. The Seidel matrix of ''G'' is also the adjacency matrix of a signed complete graph ''KG'' in which the edges of ''G'' are negative and the edges not in ''G'' are positive. It is also the adjacency matrix of the two-graph associated with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derek Corneil
Derek Gordon Corneil is a Canadian mathematician and computer scientist, a professor ''emeritus'' of computer science at the University of Toronto, and an expert in graph algorithms and graph theory. Life When he was leaving high school, Corneil was told by his English teacher that doing a degree in mathematics and physics was a bad idea, and that the best he could hope for was to go to a technical college. His interest in computer science began when, as an undergraduate student at Queens College, he heard that a computer was purchased by the London Life insurance company in London, Ontario, where his father worked. As a freshman, he took a summer job operating the UNIVAC Mark II at the company. One of his main responsibilities was to operate a printer. An opportunity for a programming job with the company sponsoring his college scholarship appeared soon after. It was a chance that Corneil jumped at after being denied a similar position at London Life. There was an initial mix-up a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hybrid Transformer
A hybrid transformer (also known as a bridge transformer, hybrid coil, or just hybrid) is a type of Power dividers and directional couplers, directional coupler which is designed to be configured as a electrical network, circuit having four port (circuit theory), ports that are conjugate in pairs, implemented using one or more Transformer, transformers. It is a particular case of the more general concept of a hybrid coupler. A signal arriving at one port is divided equally between the two adjacent ports but does not appear at the opposite port. In the schematic diagram, the signal into W splits between X and Z, and no signal passes to Y. Similarly, signals into X split to W and Y with none to Z, etc. Correct operation requires matched characteristic impedance at all four ports. Forms of hybrid other than transformer coils are possible; any format of directional coupler can be designed to be a hybrid. These formats include transmission lines and waveguides. Motivation The p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conference Matrix 2-port
A conference is a meeting of two or more experts to discuss and exchange opinions or new information about a particular topic. Conferences can be used as a form of group decision-making, although discussion, not always decisions, are the main purpose of conferences. History The first known use of "conference" appears in 1527, meaning "a meeting of two or more persons for discussing matters of common concern". It came from the word "confer", which means "to compare views or take counsel". However the idea of a conference far predates the word. Arguably, as long as there have been people, there have been meetings and discussions between people. Evidence of ancient forms of conference can be seen in archaeological ruins of common areas where people would gather to discuss shared interests such as "hunting plans, wartime activities, negotiations for peace or the organisation of tribal celebrations". Since the 1960s, conferences have become a lucrative sector of the tourism ind ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Matrix
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are often used to represent simple linear transformations, such as shearing or rotation. For example, if R is a square matrix representing a rotation (rotation matrix) and \mathbf is a column vector describing the position of a point in space, the product R\mathbf yields another column vector describing the position of that point after that rotation. If \mathbf is a row vector, the same transformation can be obtained using where R^ is the transpose of Main diagonal The entries a_ (''i'' = 1, …, ''n'') form the main diagonal of a square matrix. They lie on the imaginary line which runs from the top left corner to the bottom right corner of the matrix. For instance, the main diagonal of the 4×4 matrix above contains the elements , , , . The d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weighing Matrix
In mathematics, a weighing matrix of order n and weight w is a matrix W with entries from the set \ such that: :WW^\mathsf = wI_n Where W^\mathsf is the transpose of W and I_n is the identity matrix of order n. The weight w is also called the ''degree'' of the matrix. For convenience, a weighing matrix of order n and weight w is often denoted by W(n,w). Weighing matrices are so called because of their use in optimally measuring the individual weights of multiple objects. When the weighing device is a balance scale, the statistical variance of the measurement can be minimized by weighing multiple objects at once, including some objects in the opposite pan of the scale where they subtract from the measurement. Properties Some properties are immediate from the definition. If W is a W(n,w), then: * The rows of W are pairwise orthogonal (that is, every pair of rows you pick from W will be orthogonal). Similarly, the columns are pairwise orthogonal. * Each row and each column o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antisymmetric Matrices
In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition In terms of the entries of the matrix, if a_ denotes the entry in the i-th row and j-th column, then the skew-symmetric condition is equivalent to Example The matrix :A = \begin 0 & 2 & -45 \\ -2 & 0 & -4 \\ 45 & 4 & 0 \end is skew-symmetric because : -A = \begin 0 & -2 & 45 \\ 2 & 0 & 4 \\ -45 & -4 & 0 \end = A^\textsf . Properties Throughout, we assume that all matrix entries belong to a field \mathbb whose characteristic is not equal to 2. That is, we assume that , where 1 denotes the multiplicative identity and 0 the additive identity of the given field. If the characteristic of the field is 2, then a skew-symmetric matrix is the same thing as a symmetric matrix. * The sum of two skew-symmetric matrices is skew-symmetric. * A sca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Essentially Unique
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 \ or \. In this case, the non-uniqueness of the isomorphism (e.g., match 1 to a or 1 to ''c'') is reflected in the symmetric group. On the other hand, there is an essentially unique ''ordered'' set of any given f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paley Graph
In mathematics, Paley graphs are dense undirected graphs constructed from the members of a suitable finite field by connecting pairs of elements that differ by a quadratic residue. The Paley graphs form an infinite family of conference graphs, which yield an infinite family of symmetric conference matrices. Paley graphs allow graph-theoretic tools to be applied to the number theory of quadratic residues, and have interesting properties that make them useful in graph theory more generally. Paley graphs are named after Raymond Paley. They are closely related to the Paley construction for constructing Hadamard matrices from quadratic residues . They were introduced as graphs independently by and . Sachs was interested in them for their self-complementarity properties, while Erdős and Rényi studied their symmetries. Paley digraphs are directed analogs of Paley graphs that yield antisymmetric conference matrices. They were introduced by (independently of Sachs, Erdős, and Rén ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conference Graph
In the mathematics, mathematical area of graph theory, a conference graph is a strongly regular graph with parameters ''v'', and It is the graph associated with a symmetric conference matrix, and consequently its order ''v'' must be 1 (modular arithmetic, modulo 4) and a Sum of two squares theorem, sum of two squares. Conference graphs are known to exist for all small values of ''v'' allowed by the restrictions, e.g., ''v'' = 5, 9, 13, 17, 25, 29, and (the Paley graphs) for all prime powers congruent to 1 (modulo 4). However, there are many values of ''v'' that are allowed, for which the existence of a conference graph is unknown. The eigenvalues of a conference graph need not be integers, unlike those of other strongly regular graphs. If the graph is connected, the eigenvalues are ''k'' with multiplicity 1, and two other eigenvalues, :\frac , each with multiplicity References Andries Brouwer, Brouwer, A.E., Cohen, A.M., and Neumaier, A. (1989), Distance Regular Graphs [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strongly Regular Graph
In graph theory, a strongly regular graph (SRG) is defined as follows. Let be a regular graph with vertices and degree . is said to be strongly regular if there are also integers and such that: * Every two adjacent vertices have common neighbours. * Every two non-adjacent vertices have common neighbours. The complement of an is also strongly regular. It is a . A strongly regular graph is a distance-regular graph with diameter 2 whenever μ is non-zero. It is a locally linear graph whenever . Etymology A strongly regular graph is denoted an srg(''v'', ''k'', λ, μ) in the literature. By convention, graphs which satisfy the definition trivially are excluded from detailed studies and lists of strongly regular graphs. These include the disjoint union of one or more equal-sized complete graphs, and their complements, the complete multipartite graphs with equal-sized independent sets. Andries Brouwer and Hendrik van Maldeghem (see #References) use an alternate but fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
