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Monge Array
In mathematics applied to computer science, Monge arrays, or Monge matrices, are mathematical objects named for their discoverer, the French mathematician Gaspard Monge. An ''m''-by-''n'' matrix is said to be a ''Monge array'' if, for all \scriptstyle i,\, j,\, k,\, \ell such that :1\le i < k\le m\text1\le j < \ell\le n one obtains : So for any two rows and two columns of a Monge array (a 2 × 2 sub-matrix) the four elements at the intersection points have the property that the sum of the upper-left and lower right elements (across the |
Computer Science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical disciplines (including the design and implementation of Computer architecture, hardware and Computer programming, software). Computer science is generally considered an area of research, academic research and distinct from computer programming. Algorithms and data structures are central to computer science. The theory of computation concerns abstract models of computation and general classes of computational problem, problems that can be solved using them. The fields of cryptography and computer security involve studying the means for secure communication and for preventing Vulnerability (computing), security vulnerabilities. Computer graphics (computer science), Computer graphics and computational geometry address the generation of images. Progr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaspard Monge
Gaspard Monge, Comte de Péluse (9 May 1746 – 28 July 1818) was a French mathematician, commonly presented as the inventor of descriptive geometry, (the mathematical basis of) technical drawing, and the father of differential geometry. During the French Revolution he served as the Minister of the Marine, and was involved in the reform of the French educational system, helping to found the École Polytechnique. Biography Early life Monge was born at Beaune, Côte-d'Or, the son of a merchant. He was educated at the college of the Oratorians at Beaune. In 1762 he went to the Collège de la Trinité at Lyon, where, one year after he had begun studying, he was made a teacher of physics at the age of just seventeen. After finishing his education in 1764 he returned to Beaune, where he made a large-scale plan of the town, inventing the methods of observation and constructing the necessary instruments; the plan was presented to the town, and is still preserved in their library. ... [...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] |
Main Diagonal
In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix A is the list of entries a_ where i = j. All off-diagonal elements are zero in a diagonal matrix. The following four matrices have their main diagonals indicated by red ones: :\begin \color & 0 & 0\\ 0 & \color & 0\\ 0 & 0 & \color\end \qquad \begin \color & 0 & 0 & 0 \\ 0 & \color & 0 & 0 \\ 0 & 0 & \color & 0 \end \qquad \begin \color & 0 & 0 \\ 0 & \color & 0 \\ 0 & 0 & \color \\ 0 & 0 & 0 \end \qquad \begin \color & 0 & 0 & 0 \\ 0 & \color & 0 & 0 \\ 0 & 0 &\color & 0 \\ 0 & 0 & 0 & \color \end \qquad Antidiagonal The antidiagonal (sometimes counter diagonal, secondary diagonal, trailing diagonal, minor diagonal, off diagonal, or bad diagonal) of an order N square matrix B is the collection of entries b_ such that i + j = N+1 for all 1 \leq i, j \leq N. That is, it runs from the top right corner to the bottom left corner. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antidiagonal
In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix A is the list of entries a_ where i = j. All off-diagonal elements are zero in a diagonal matrix. The following four matrices have their main diagonals indicated by red ones: :\begin \color & 0 & 0\\ 0 & \color & 0\\ 0 & 0 & \color\end \qquad \begin \color & 0 & 0 & 0 \\ 0 & \color & 0 & 0 \\ 0 & 0 & \color & 0 \end \qquad \begin \color & 0 & 0 \\ 0 & \color & 0 \\ 0 & 0 & \color \\ 0 & 0 & 0 \end \qquad \begin \color & 0 & 0 & 0 \\ 0 & \color & 0 & 0 \\ 0 & 0 &\color & 0 \\ 0 & 0 & 0 & \color \end \qquad Antidiagonal The antidiagonal (sometimes counter diagonal, secondary diagonal, trailing diagonal, minor diagonal, off diagonal, or bad diagonal) of an order N square matrix B is the collection of entries b_ such that i + j = N+1 for all 1 \leq i, j \leq N. That is, it runs from the top right corner to the bottom left corner. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q'', there could be other scenarios where ''P'' is true and ''Q'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SMAWK Algorithm
The SMAWK algorithm is an algorithm for finding the minimum value in each row of an implicitly-defined totally monotone Matrix (mathematics), matrix. It is named after the initials of its five inventors, Peter Shor, Shlomo Moran, Alok Aggarwal, Robert Wilber, and Maria Klawe.. Input For the purposes of this algorithm, a matrix is defined to be monotone if each row's minimum value occurs in a column which is equal to or greater than the column of the previous row's minimum. It is totally monotone if the same property is true for every submatrix (defined by an arbitrary subset of the rows and columns of the given matrix). Equivalently, a matrix is totally monotone if there does not exist a 2×2 submatrix whose row minima are in the top right and bottom left corners. Every Monge array is totally monotone, but not necessarily vice versa. For the SMAWK algorithm, the matrix to be searched should be defined as a function, and this function is given as input to the algorithm (together with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supermodular Function
In mathematics, a function :f\colon \mathbb^k \to \mathbb is supermodular if : f(x \uparrow y) + f(x \downarrow y) \geq f(x) + f(y) for all x, y \isin \mathbb^, where x \uparrow y denotes the componentwise maximum and x \downarrow y the componentwise minimum of x and y. If −''f'' is supermodular then ''f'' is called submodular, and if the inequality is changed to an equality the function is modular. If ''f'' is twice continuously differentiable, then supermodularity is equivalent to the condition : \frac \geq 0 \mbox i \neq j. Supermodularity in economics and game theory The concept of supermodularity is used in the social sciences to analyze how one Agent (economics), agent's decision affects the incentives of others. Consider a symmetric game with a smooth payoff function \,f defined over actions \,z_i of two or more players i \in . Suppose the action space is continuous; for simplicity, suppose each action is chosen from an interval: z_i \in [a,b]. In this context, sup ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supnick Matrix
A Supnick matrix or Supnick array – named after Fred Supnick of the City College of New York, who introduced the notion in 1957 – is a Monge array which is also a symmetric matrix. Mathematical definition A Supnick matrix is a square Monge array that is symmetric around the main diagonal. An ''n''-by-''n'' matrix is a Supnick matrix if, for all ''i'', ''j'', ''k'', ''l'' such that if :1\le i < k\le n and then : and also : A logically equivalent definition is given by Rudolf & Woeginger who in 1995 proved that :''A matrix is a Supnick matrix it can be written as the sum of a sum matrix ''S'' and a non-negative linear combination of LL-UR block matrices.'' The ''sum matrix'' is defined in terms of a sequence of ''n'' real numbers : : |
Fred Supnick
Fred may refer to: People * Fred (name), including a list of people and characters with the name Mononym * Fred (cartoonist) (1931–2013), pen name of Fred Othon Aristidès, French * Fred (footballer, born 1949) (1949–2022), Frederico Rodrigues de Oliveira, Brazilian * Fred (footballer, born 1979), Helbert Frederico Carreiro da Silva, Brazilian * Fred (footballer, born 1983), Frederico Chaves Guedes, Brazilian * Fred (footballer, born 1986), Frederico Burgel Xavier, Brazilian * Fred (footballer, born 1993), Frederico Rodrigues de Paula Santos, Brazilian * Fred Again (born 1993), British songwriter known as FRED Television and movies * ''Fred Claus'', a 2007 Christmas film * Fred (2014 film), ''Fred'' (2014 film), a 2014 documentary film * Fred Figglehorn, a YouTube character created by Lucas Cruikshank ** Fred (franchise), ''Fred'' (franchise), a Nickelodeon media franchise ** ''Fred: The Movie'', a 2010 independent comedy film * ''Fred the Caveman'', French Teletoon prod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Traveling Salesman Problem
The travelling salesman problem (also called the travelling salesperson problem or TSP) asks the following question: "Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city?" It is an NP-hard problem in combinatorial optimization, important in theoretical computer science and operations research. The travelling purchaser problem and the vehicle routing problem are both generalizations of TSP. In the theory of computational complexity, the decision version of the TSP (where given a length ''L'', the task is to decide whether the graph has a tour of at most ''L'') belongs to the class of NP-complete problems. Thus, it is possible that the worst-case running time for any algorithm for the TSP increases superpolynomially (but no more than exponentially) with the number of cities. The problem was first formulated in 1930 and is one of the most intensively studied p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
