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Equiangular Lines
In geometry, a set of lines is called equiangular if all the lines intersect at a single point, and every pair of lines makes the same angle. Equiangular lines in Euclidean space Computing the maximum number of equiangular lines in ''n''-dimensional Euclidean space is a difficult problem, and unsolved in general, though bounds are known. The maximal number of equiangular lines in 2-dimensional Euclidean space is 3: we can take the lines through opposite vertices of a regular hexagon, each at an angle 120 degrees from the other two. The maximum in 3 dimensions is 6: we can take lines through opposite vertices of an icosahedron. It is known that the maximum number in any dimension n is less than or equal to \binom. This upper bound is tight up to a constant factor to a construction by de Caen. The maximum in dimensions 1 through 16 is listed in the ''On-Line Encyclopedia of Integer Sequences'' as follows: :1, 3, 6, 6, 10, 16, 28, 28, 28, 28, 28, 28, 28, 28, 36, 40, ... In particu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, \begin1 & 9 & -13 \\20 & 5 & -6 \end is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a "-matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra. Therefore, the study of matrices is a large part of linear algebra, and most properties and operations of abstract linear algebra can be expressed in terms of matrices. For example, matrix multiplication represents composition of linear maps. Not all matrices are related to linear algebra. This is, in particular, the case in graph theory, of incidence matrices, and adjacency matrices. ''This article focuses on matrices related to linear algebra, and, unle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andries Brouwer
Andries Evert Brouwer (born 1951) is a Dutch mathematician and computer programmer, Professor Emeritus at Eindhoven University of Technology (TU/e). He is known as the creator of the greatly expanded 1984 to 1985 versions of the roguelike computer game ''Hack'' that formed the basis for ''NetHack''. He is also a Linux kernel hacker. He is sometimes referred to by the handle ''aeb''. Biography Born in Amsterdam, Brouwer attended the gymnasium, and obtained his MSc in mathematics at the University of Amsterdam in 1971. In 1976 he received his Ph.D. in mathematics from Vrije Universiteit with a thesis entitled "Treelike Spaces and Related Topological Spaces", under the supervision of Maarten Maurice and Pieter Baayen, both of whom were in turn students of Johannes de Groot. In 2004 he received an honorary doctorate from Aalborg University. After graduation Brouwer started his academic career at the Mathematisch Centrum, later Centrum Wiskunde & Informatica. From 1986 to 2012 he w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quanta Magazine
''Quanta Magazine'' is an editorially independent online publication of the Simons Foundation covering developments in physics, mathematics, biology and computer science. ''Undark Magazine'' described ''Quanta Magazine'' as "highly regarded for its masterful coverage of complex topics in science and math." The science news aggregator ''RealClearScience'' ranked ''Quanta Magazine'' first on its list of "The Top 10 Websites for Science in 2018." In 2020, the magazine received a National Magazine Award for General Excellence from the American Society of Magazine Editors for its "willingness to tackle some of the toughest and most difficult topics in science and math in a language that is accessible to the lay reader without condescension or oversimplification." The articles in the magazine are freely available to read online. ''Scientific American'', ''Wired'', ''The Atlantic'', and ''The Washington Post'', as well as international science publications like ''Spektrum der Wissensch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SIC-POVM
A symmetric, informationally complete, positive operator-valued measure (SIC-POVM) is a special case of a generalized measurement on a Hilbert space, used in the field of quantum mechanics. A measurement of the prescribed form satisfies certain defining qualities that makes it an interesting candidate for a "standard quantum measurement", utilized in the study of foundational quantum mechanics, most notably in QBism. Furthermore, it has been shown that applications exist in quantum state tomography and quantum cryptography, and a possible connection has been discovered with Hilbert's twelfth problem. Definition Due to the use of SIC-POVMs primarily in quantum mechanics, Dirac notation will be used throughout this article to represent elements in a Hilbert space. A POVM over a d-dimensional Hilbert space \mathcal is a set of m positive-semidefinite operators \left\_^m that sum to the identity: \sum_^m F_i = I. If a POVM consists of at least d^2 operators which span the space of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inner Product Space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often denoted with angle brackets such as in \langle a, b \rangle. Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or ''scalar product'' of Cartesian coordinates. Inner product spaces of infinite dimension are widely used in functional analysis. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898. An inner product naturally induces an associated norm, (denoted , x, and , y, in the picture); so, ... [...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] |
Symmetric Matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with respect to the main diagonal. So if a_ denotes the entry in the ith row and jth column then for all indices i and j. Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative. In linear algebra, a real symmetric matrix represents a self-adjoint operator represented in an orthonormal basis over a real inner product space. The corresponding object for a complex inner product space is a Hermitian matrix with complex-valued entries, which is equal to its conjugate transpose. Therefore, in linear algebra over the complex numbers, it is often assumed that a symmetric matrix refe ... [...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] |
Inner Product
In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two vectors in the space is a Scalar (mathematics), scalar, often denoted with angle brackets such as in \langle a, b \rangle. Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or ''scalar product'' of Cartesian coordinates. Inner product spaces of infinite Dimension (vector space), dimension are widely used in functional analysis. Inner product spaces over the Field (mathematics), field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Origin (mathematics)
In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter ''O'', used as a fixed point of reference for the geometry of the surrounding space. In physical problems, the choice of origin is often arbitrary, meaning any choice of origin will ultimately give the same answer. This allows one to pick an origin point that makes the mathematics as simple as possible, often by taking advantage of some kind of geometric symmetry. Cartesian coordinates In a Cartesian coordinate system, the origin is the point where the axes of the system intersect.. The origin divides each of these axes into two halves, a positive and a negative semiaxis. Points can then be located with reference to the origin by giving their numerical coordinates—that is, the positions of their projections along each axis, either in the positive or negative direction. The coordinates of the origin are always all zero, for example (0,0) in two dimensions and (0,0,0) in three. Ot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line (geometry)
In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segment in everyday life, which has two points to denote its ends. Lines can be referred by two points that lay on it (e.g., \overleftrightarrow) or by a single letter (e.g., \ell). Euclid described a line as "breadthless length" which "lies evenly with respect to the points on itself"; he introduced several postulates as basic unprovable properties from which he constructed all of geometry, which is now called Euclidean geometry to avoid confusion with other geometries which have been introduced since the end of the 19th century (such as non-Euclidean, projective and affine geometry). In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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