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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 (mathematics), 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. 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 the more ge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The usual notation for the square of a number is not the product , but the equivalent exponentiation , usually pronounced as " squared". The name ''square'' number comes from the name of the shape. The unit of area is defined as the area of a unit square (). Hence, a square with side length has area . If a square number is represented by ''n'' points, the points can be arranged in rows as a square each side of which has the same number of points as the square root of ''n''; thus, square numbers are a type of Figurate number, figurate numbers (other examples being Cube (algebra), cube numbers and triangular numbers). In the Real number, real number system, square numbers are non-negative. A non-negative integer ... [...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 directional coupler which is designed to be configured as a circuit having four ports that are conjugate in pairs, implemented using one or more 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 primary use of a voiceband hybrid transformer is to convert between 2-wire and 4-wire operatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conference Matrix 2-port
A conference is a meeting, often lasting a few days, which is organized on a particular subject, or to bring together people who have a common interest. Conferences can be used as a form of group decision-making, although discussion, not always decisions, is the primary purpose of conferences. The term derives from the word ''confer''. 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 t ... [...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. Similarly, the columns are pairwise orthogonal. * Each row and each column of W has exactly w non-zero elements. * W^\mathsfW = w I, since the d ... [...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 totally ordered set of any given ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Problem
In science and mathematics, an open problem or an open question is a known problem which can be accurately stated, and which is assumed to have an objective and verifiable solution, but which has not yet been solved (i.e., no solution for it is known). In the history of science, some of these supposed open problems were "solved" by means of showing that they were not well-defined. In mathematics, many open problems are concerned with the question of whether a certain definition is or is not consistent. Two notable examples in mathematics that have been solved and ''closed'' by researchers in the late twentieth century are Fermat's Last Theorem and the four-color theorem.K. Appel and W. Haken (1977), "Every planar map is four colorable. Part I. Discharging", ''Illinois J. Math'' 21: 429–490. K. Appel, W. Haken, and J. Koch (1977), "Every planar map is four colorable. Part II. Reducibility", ''Illinois J. Math'' 21: 491–567. An important open mathematics problem solved ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paley Graph
In mathematics, Paley graphs are 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 matrix, 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 matrix, Hadamard matrices from quadratic residues. They were introduced as graphs independently by and . Horst Sachs, Sachs was interested in them for their self-complementarity properties, while Paul Erdős, Erdős and Alfréd Rényi, Rényi studied their symmetries. Paley digraphs are directed graph, directed analogs of Paley graphs that yield antisymmetric conf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conference Graph
In the 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 (modulo 4) and a 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 graph In mathematics, Paley graphs are 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 yiel ...s) 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 '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strongly Regular Graph
In graph theory, a strongly regular graph (SRG) is a regular graph with vertices and degree such that for some given integers \lambda, \mu \ge 0 * every two adjacent vertices have common neighbours, and * every two non-adjacent vertices have common neighbours. Such a strongly regular graph is denoted by . Its complement graph is also strongly regular: it is an . 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 as 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 bu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph (discrete Mathematics)
In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a Set (mathematics), set of objects where some pairs of the objects are in some sense "related". The objects are represented by abstractions called ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') and each of the related pairs of vertices is called an ''edge'' (also called ''link'' or ''line''). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person ''A'' can shake hands with a person ''B'' only if ''B'' also shakes hands with ''A''. In contrast, if an edge from a person ''A'' to a person ''B'' means that ''A'' owes money to ''B'', then this graph is directed, because owing mon ... [...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 ''G'' a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
