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Ulam–Warburton Automaton
The Ulam–Warburton cellular automaton (UWCA) is a 2-dimensional fractal pattern that grows on a regular grid of cells consisting of squares. Starting with one square initially ON and all others OFF, successive iterations are generated by turning ON all squares that share precisely one edge with an ON square. This is the von Neumann neighborhood. The automaton is named after the Polish-American mathematician and scientist Stanislaw Ulam and the Scottish engineer, inventor and List of amateur mathematicians, amateur mathematician Mike Warburton. Properties and relations The UWCA is a 2D 5-neighbor outer totalistic cellular automaton using rule 686. The number of cells turned ON in each iteration is denoted u(n), with an explicit formula: u(0)=0, u(1)=1, and for n \ge 2 u(n) = 4\cdot 3^ where wt(n) is the Hamming weight function which counts the number of 1's in the binary expansion of n wt(n)=n-\sum_^ \left\lfloor\frac\right\rfloor The minimum upper bound of summation for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cellular Automaton
A cellular automaton (pl. cellular automata, abbrev. CA) is a discrete model of computation studied in automata theory. Cellular automata are also called cellular spaces, tessellation automata, homogeneous structures, cellular structures, tessellation structures, and iterative arrays. Cellular automata have found application in various areas, including physics, theoretical biology and microstructure modeling. A cellular automaton consists of a regular grid of ''cells'', each in one of a finite number of '' states'', such as ''on'' and ''off'' (in contrast to a coupled map lattice). The grid can be in any finite number of dimensions. For each cell, a set of cells called its ''neighborhood'' is defined relative to the specified cell. An initial state (time ''t'' = 0) is selected by assigning a state for each cell. A new ''generation'' is created (advancing ''t'' by 1), according to some fixed ''rule'' (generally, a mathematical function) that determines the new state ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Upper And Lower Bounds
In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of . Dually, a lower bound or minorant of is defined to be an element of that is less than or equal to every element of . A set with an upper (respectively, lower) bound is said to be bounded from above or majorized (respectively bounded from below or minorized) by that bound. The terms bounded above (bounded below) are also used in the mathematical literature for sets that have upper (respectively lower) bounds. Examples For example, is a lower bound for the set (as a subset of the integers or of the real numbers, etc.), and so is . On the other hand, is not a lower bound for since it is not smaller than every element in . The set has as both an upper bound and a lower bound; all other numbers are either an upper bound or a lower bound for that . Every subset of the natural numbers has a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Certificate Of Secondary Education
The General Certificate of Secondary Education (GCSE) is an academic qualification in a particular subject, taken in England, Wales, and Northern Ireland. State schools in Scotland use the Scottish Qualifications Certificate instead. Private schools in Scotland may choose to use GCSEs from England. Each GCSE qualification is offered in a specific school subject (English literature, English language, mathematics, science, history, geography, art and design, design and technology, business studies, classical civilisation, drama, music, foreign languages, etc). The Department for Education has drawn up a list of preferred subjects known as the English Baccalaureate for England on the results in eight GCSEs including English, mathematics, the sciences (physics, chemistry, biology, computer science), history, geography, and an ancient or modern foreign language. Studies for GCSE examinations take place over a period of two or three academic years (depending upon the subject, scho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George Heriot's School
George Heriot's School is a Scottish independent primary and secondary day school on Lauriston Place in the Old Town of Edinburgh, Scotland. In the early 21st century, it has more than 1600 pupils, 155 teaching staff, and 80 non-teaching staff. It was established in 1628 as George Heriot's Hospital, by bequest of the royal goldsmith George Heriot, and opened in 1659. It is governed by George Heriot's Trust, a Scottish charity. Architecture The main building of the school is notable for its renaissance architecture, the work of William Wallace, until his death in 1631. He was succeeded as master mason by William Aytoun, who was succeeded in turn by John Mylne. In 1676, Sir William Bruce drew up plans for the completion of Heriot's Hospital. His design, for the central tower of the north façade, was eventually executed in 1693. The school is a turreted building surrounding a large quadrangle, and built out of sandstone. The foundation stone is inscribed with the date 1628 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Von Neumann
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest coverage of any mathematician of his time and was said to have been "the last representative of the great mathematicians who were equally at home in both pure and applied mathematics". He integrated pure and applied sciences. Von Neumann made major contributions to many fields, including mathematics ( foundations of mathematics, measure theory, functional analysis, ergodic theory, group theory, lattice theory, representation theory, operator algebras, matrix theory, geometry, and numerical analysis), physics (quantum mechanics, hydrodynamics, ballistics, nuclear physics and quantum statistical mechanics), economics ( game theory and general equilibrium theory), computing ( Von Neumann architecture, linear programming, numerical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subtraction Game
In combinatorial game theory, a subtraction game is an abstract strategy game whose state can be represented by a natural number or vector of numbers (for instance, the numbers of game tokens in piles of tokens, or the positions of pieces on board) and in which the allowed moves reduce these numbers., "Subtraction games", pp. 83–86. Often, the moves of the game allow any number to be reduced by subtracting a value from a specified ''subtraction set'', and different subtraction games vary in their subtraction sets. These games also vary in whether the last player to move wins (the normal play convention) or loses ( misère play convention). Another winning convention that has also been used is that a player who moves to a position with all numbers zero wins, but that any other position with no moves possible is a draw. Examples Examples of notable subtraction games include the following: *Nim is a game whose state consists of multiple piles of tokens, such as coins or matchsticks, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree (graph Theory)
In graph theory, a tree is an undirected graph in which any two vertices are connected by ''exactly one'' path, or equivalently a connected acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by ''at most one'' path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees. A polytreeSee . (or directed tree or oriented treeSee .See . or singly connected networkSee .) is a directed acyclic graph (DAG) whose underlying undirected graph is a tree. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest. The various kinds of data structures referred to as trees in computer science have underlying graphs that are trees in graph theory, although such data structures are generally rooted trees. A rooted tree may be directed, called a directed rooted tree, either making all its edges point away from the root—in which case it is call ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph (discrete Mathematics)
In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' 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. Graphs are one of the objects of study in discrete mathematics. 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'', th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toothpick Sequence - Generation 13
A toothpick is a small thin stick of wood, plastic, bamboo, metal, bone or other substance with at least one and sometimes two pointed ends to insert between teeth to remove detritus, usually after a meal. Toothpicks are also used for festive occasions to hold or spear small appetizers (like cheese cubes or olives) or as a cocktail stick, and can be decorated with plastic frills or small paper umbrellas or flags. History Known in all cultures, the toothpick is the oldest instrument for dental cleaning. Hominin remains from Dmanisi, Georgia, dated to about 1.8 million years ago, bear lesions indicating the repeated use of a “toothpick”. A Neanderthal man's jawbone found in the Cova Foradà in Spain evidenced use of a toothpick to alleviate pain in his teeth caused by periodontal disease and dental wear. Toothpicks made of bronze have been found as burial objects in prehistoric graves in Northern Italy and in the East Alps. In 1986, researchers in Florida dis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wacław Sierpiński
Wacław Franciszek Sierpiński (; 14 March 1882 – 21 October 1969) was a Polish mathematician. He was known for contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of functions, and topology. He published over 700 papers and 50 books. Three well-known fractals are named after him (the Sierpiński triangle, the Sierpiński carpet, and the Sierpiński curve), as are Sierpiński numbers and the associated Sierpiński problem. Educational background Sierpiński enrolled in the Department of Mathematics and Physics at the University of Warsaw in 1899 and graduated four years later. In 1903, while still at the University of Warsaw, the Department of Mathematics and Physics offered a prize for the best essay from a student on Voronoy's contribution to number theory. Sierpiński was awarded a gold medal for his essay, thus laying the foundation for his first major mathematical contribution. Unwilling for his work to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
