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Domino Tiling
In geometry, a domino tiling of a region in the Euclidean plane is a tessellation of the region by domino (mathematics), dominoes, shapes formed by the union of two unit squares meeting edge-to-edge. Equivalently, it is a matching (graph theory), perfect matching in the grid graph formed by placing a vertex at the center of each square of the region and connecting two vertices when they correspond to adjacent squares. Height functions For some classes of tilings on a regular grid in two dimensions, it is possible to define a height function associating an integer to the vertex (graph theory), vertices of the grid. For instance, draw a chessboard, fix a node A_0 with height 0, then for any node there is a path from A_0 to it. On this path define the height of each node A_ (i.e. corners of the squares) to be the height of the previous node A_n plus one if the square on the right of the path from A_n to A_ is black, and minus one otherwise. More details can be found in . Thurston's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pavage Domino
Pavage was a medieval toll for the maintenance or improvement of a road or street in medieval Kingdom of England, England, Medieval Wales, Wales and Lordship of Ireland, Ireland. The king by letters patent granted the right to collect it to an individual, or the corporation of a town, or to the "bailiffs and good men" of a neighbouring village. Pavage grants can be divided into two classes: *Urban grants to enable the streets of a town (or its marketplace) to be paved. These represent the majority of grants. *Rural grants to enable a particular road to be repaired. These grants were mostly made in the 14th century, and largely for the great roads radiating from London, which were presumably those carrying the heaviest traffic. The first grant was in 1249 for the Yorkshire town of Beverley, where the pavage was associated with the cult of St John of Beverley, and was ultimately made permanent. Another early one was for Shrewsbury in 1266 for paving the new marketplace, remove ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chessboard
A chessboard is a game board used to play chess. It consists of 64 squares, 8 rows by 8 columns, on which the chess pieces are placed. It is square in shape and uses two colours of squares, one light and one dark, in a chequered pattern. During play, the board is oriented such that each player's near-right corner square is a light square. The columns of a chessboard are known as ', the rows are known as ', and the lines of adjoining same-coloured squares (each running from one edge of the board to an adjacent edge) are known as '. Each square of the board is named using algebraic, descriptive, or numeric chess notation; algebraic notation is the FIDE standard. In algebraic notation, using White's perspective, files are labeled ''a'' through ''h'' from left to right, and ranks are labeled ''1'' through ''8'' from bottom to top; each square is identified by the file and rank which it occupies. The a- through d-files constitute the , and the e- through h-files constitute the ; the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mutilated Chessboard Problem
The mutilated chessboard problem is a tiling puzzle posed by Max Black in 1946 that asks: Suppose a standard 8×8 chessboard (or checkerboard) has two diagonally opposite corners removed, leaving 62 squares. Is it possible to place 31 dominoes of size 2×1 so as to cover all of these squares? It is an impossible puzzle: there is no domino tiling meeting these conditions. One proof of its impossibility uses the fact that, with the corners removed, the chessboard has 32 squares of one color and 30 of the other, but each domino must cover equally many squares of each color. More generally, if any two squares are removed from the chessboard, the rest can be tiled by dominoes if and only if the removed squares are of different colors. This problem has been used as a test case for automated reasoning, creativity, and the philosophy of mathematics. History The mutilated chessboard problem is an instance of domino tiling of grids and polyominoes, also known as "dimer models", a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Free Field
In probability theory and statistical mechanics, the Gaussian free field (GFF) is a Gaussian random field, a central model of random surfaces (random height functions). The discrete version can be defined on any graph, usually a lattice in ''d''-dimensional Euclidean space. The continuum version is defined on R''d'' or on a bounded subdomain of R''d''. It can be thought of as a natural generalization of one-dimensional Brownian motion to ''d'' time (but still one space) dimensions: it is a random (generalized) function from R''d'' to R. In particular, the one-dimensional continuum GFF is just the standard one-dimensional Brownian motion or Brownian bridge on an interval. In the theory of random surfaces, it is also called the harmonic crystal. It is also the starting point for many constructions in quantum field theory, where it is called the Euclidean bosonic massless free field. A key property of the 2-dimensional GFF is conformal invariance, which relates it in several ways ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Lattice
In the theory of lattices, the dual lattice is a construction analogous to that of a dual vector space. In certain respects, the geometry of the dual lattice of a lattice L is the reciprocal of the geometry of L , a perspective which underlies many of its uses. Dual lattices have many applications inside of lattice theory, theoretical computer science, cryptography and mathematics more broadly. For instance, it is used in the statement of the Poisson summation formula, transference theorems provide connections between the geometry of a lattice and that of its dual, and many lattice algorithms exploit the dual lattice. For an article with emphasis on the physics / chemistry applications, see Reciprocal lattice. This article focuses on the mathematical notion of a dual lattice. Definition Let L \subseteq \mathbb^n be a lattice. That is, L = B \mathbb^n for some matrix B . The dual lattice is the set of linear functionals on L which take integer values on each poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometrical Frustration
In condensed matter physics, geometrical frustration (or in short, frustration) is a phenomenon where the combination of conflicting inter-atomic forces leads to complex structures. Frustration can imply a plenitude of distinct ground states at absolute zero, zero temperature, and usual thermal ordering may be suppressed at higher temperatures. Much-studied examples include amorphous materials, glasses, and dilute magnets. The term ''frustration'', in the context of magnetism, magnetic systems, was introduced by Gerard Toulouse in 1977. Frustrated magnetism, magnetic systems had been studied even before. Early work includes a study of the Ising model on a triangular lattice with nearest-neighbor Spin (physics), spins coupled Antiferromagnetism, antiferromagnetically, by Gregory Wannier, G. H. Wannier, published in 1950. Related features occur in magnets with ''competing interactions'', where both ferromagnetic as well as antiferromagnetic couplings between pairs of Spin (physics), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ising Model
The Ising model (or Lenz–Ising model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical models in physics, mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent Nuclear magnetic moment, magnetic dipole moments of atomic "spins" that can be in one of two states (+1 or −1). The spins are arranged in a Graph (abstract data type), graph, usually a lattice (group), lattice (where the local structure repeats periodically in all directions), allowing each spin to interact with its neighbors. Neighboring spins that agree have a lower energy than those that disagree; the system tends to the lowest energy but heat disturbs this tendency, thus creating the possibility of different structural phases.The two-dimensional square-lattice Ising model is one of the simplest statistical models to show a phase transition. Though it is a highly simplified model of a magnetic material, the Ising model can sti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bijection
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equivalently, a bijection is a relation between two sets such that each element of either set is paired with exactly one element of the other set. A function is bijective if it is invertible; that is, a function f:X\to Y is bijective if and only if there is a function g:Y\to X, the ''inverse'' of , such that each of the two ways for composing the two functions produces an identity function: g(f(x)) = x for each x in X and f(g(y)) = y for each y in Y. For example, the ''multiplication by two'' defines a bijection from the integers to the even numbers, which has the ''division by two'' as its inverse function. A function is bijective if and only if it is both injective (or ''one-to-one'')—meaning that each element in the codomain is mappe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-complete
In computational complexity theory, NP-complete problems are the hardest of the problems to which ''solutions'' can be verified ''quickly''. Somewhat more precisely, a problem is NP-complete when: # It is a decision problem, meaning that for any input to the problem, the output is either "yes" or "no". # When the answer is "yes", this can be demonstrated through the existence of a short (polynomial length) ''solution''. # The correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying all possible solutions. # The problem can be used to simulate every other problem for which we can verify quickly that a solution is correct. Hence, if we could find solutions of some NP-complete problem quickly, we could quickly find the solutions of every other problem to which a given solution can be easily verified. The name "NP-complete" is short for "nondeterministic polynomial-time complete". In this name, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tatami
are soft mats used as flooring material in traditional Japanese-style rooms. They are made in standard sizes, twice as long as wide, about , depending on the region. In martial arts, tatami are used for training in a dojo and for competition. Tatami are covered with a weft-faced weave of on a warp of hemp or weaker cotton. There are four warps per weft shed, two at each end (or sometimes two per shed, one at each end, to reduce cost). The (core) is traditionally made from sewn-together rice straw, but contemporary tatami sometimes have compressed wood chip boards or extruded polystyrene foam in their cores instead or as well. The long sides are usually with brocade or plain cloth, although some tatami have no edging. File:Modern tatami.JPG, Machine-sewing of tatami File:Tatami sectional view.jpg, Cross-section of a modern tatami with an extruded polystyrene foam core File:Men Making Tatami Mats, 1860 - ca. 1900.jpg, Making tatami mats, late 19th century. File:Tatami.jpg, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Super-exponential Growth
In mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though Knuth's up arrow notation \uparrow \uparrow and the left-exponent ^b are common. Under the definition as repeated exponentiation, means , where ' copies of ' are iterated via exponentiation, right-to-left, i.e. the application of exponentiation n-1 times. ' is called the "height" of the function, while ' is called the "base," analogous to exponentiation. It would be read as "the th tetration of ". For example, 2 tetrated to 4 (or the fourth tetration of 2) is =2^=2^=2^=65536. It is the next hyperoperation after exponentiation, but before pentation. The word was coined by Reuben Louis Goodstein from tetra- (four) and iteration. Tetration is also defined recursively as : := \begin 1 &\textn=0, \\ a^ &\textn>0, \end allowing for the holomorphic extension of tetration to non-natural numbers such as real, complex, and ord ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
