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David Singmaster
DAVID BREYER SINGMASTER (born 1939, USA ) is a retired professor of mathematics at London South Bank University , England, UK . A self-described metagrobologist , he has a huge personal collection of mechanical puzzles and books of brain teasers . He is most famous for being an early adopter and enthusiastic promoter of the Rubik\'s Cube . His Notes on Rubik's "Magic Cube" which he began compiling in 1979 provided the first mathematical analysis of the Cube as well as providing one of the first published solutions. The book contained his cube notation which allowed the recording of Rubik\'s Cube moves, and which quickly became the standard. He is both a puzzle historian and a composer of puzzles, and many of his puzzles have been published in newspapers and magazines. In combinatorial number theory , Singmaster\'s conjecture states that there is an upper bound on the number of times a number other than 1 can appear in Pascal\'s triangle
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Group Theory
In mathematics and abstract algebra , GROUP THEORY studies the algebraic structures known as groups . The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings , fields , and vector spaces , can all be seen as groups endowed with additional operations and axioms . Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom , may be modelled by symmetry groups . Thus group theory and the closely related representation theory have many important applications in physics , chemistry , and materials science . Group theory
Group theory
is also central to public key cryptography
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Cayley Graph
In mathematics , a CAYLEY GRAPH, also known as a CAYLEY COLOUR GRAPH, CAYLEY DIAGRAM, GROUP DIAGRAM, or COLOUR GROUP is a graph that encodes the abstract structure of a group . Its definition is suggested by Cayley\'s theorem (named after Arthur Cayley
Arthur Cayley
) and uses a specified, usually finite, set of generators for the group. It is a central tool in combinatorial and geometric group theory . CONTENTS * 1 Definition * 2 Examples * 3 Characterization * 4 Elementary properties * 5 Schreier coset graph * 6 Connection to group theory * 6.1 Geometric group theory * 7 History * 8 Bethe lattice * 9 See also * 10 Notes * 11 External links DEFINITIONSuppose that G {displaystyle G} is a group and S {displaystyle S} is a generating set
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Wreath Product
In mathematics , the WREATH PRODUCT of group theory is a specialized product of two groups, based on a semidirect product . Wreath products are used in the classification of permutation groups and also provide a way of constructing interesting examples of groups. Given two groups A and H, there exist two variations of the wreath product: the UNRESTRICTED WREATH PRODUCT A Wr H (also written A≀H) and the RESTRICTED WREATH PRODUCT A wr H. Given a set Ω with an H-action there exists a generalisation of the wreath product which is denoted by A WrΩ H or A wrΩ H respectively. The notion generalizes to semigroups and is a central construction in the Krohn-Rhodes structure theory of finite semigroups. CONTENTS * 1 Definition * 2 Notation and conventions * 3 Properties * 4 Canonical actions of wreath products * 5 Examples * 6 References * 7 External links DEFINITIONLet A and H be groups and Ω a set with H acting on it
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Layer By Layer
LAYER-BY-LAYER (LBL) deposition is a thin film fabrication technique. The films are formed by depositing alternating layers of oppositely charged materials with wash steps in between. This can be accomplished by using various techniques such as immersion, spin, spray, electromagnetism, or fluidics. CONTENTS * 1 Development * 2 Implementation * 3 Applications * 4 See also * 5 References DEVELOPMENTThe first implementation of this technique is attributed to J. J. Kirkland and R. K. Iler of DuPont , who carried it out using microparticles in 1966. The method was later revitalized by the discovery of its applicability to a wide range of polyelectrolytes by Prof. Gero Decher at Johannes Gutenberg-Universität Mainz . IMPLEMENTATIONA simple representation can be made by defining two oppositely charged polyions as + and -, and defining the wash step as W
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The Observer
THE OBSERVER is a British newspaper, published on Sundays . In the same place on the political spectrum as its sister papers The Guardian and The Guardian
The Guardian
Weekly , whose parent company Guardian Media Group Limited acquired it in 1993, it takes a social liberal or social democratic line on most issues. First published in 1791, it is the world's oldest Sunday newspaper. CONTENTS* 1 History * 1.1 Origins * 1.2 Nineteenth century * 1.3 Twentieth century * 1.4 Twenty-first century * 2 Supplements and features * 3 The Newsroom * 4 Bans * 5 Editors * 6 Awards * 7 Conventions sponsored * 8 Bibliography * 9 See also * 10 References * 11 External links HISTORYORIGINSThe first issue, published on 4 December 1791 by W.S. Bourne , was the world's first Sunday newspaper . Believing that the paper would be a means of wealth, Bourne instead soon found himself facing debts of nearly £1,600
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M. C. Escher
MAURITS CORNELIS ESCHER (Dutch pronunciation: ; 17 June 1898 – 27 March 1972), or commonly M. C. ESCHER, was a Dutch graphic artist who made mathematically inspired woodcuts , lithographs , and mezzotints . His work features mathematical objects and operations including impossible objects , explorations of infinity , reflection , symmetry , perspective , truncated and stellated polyhedra , hyperbolic geometry , and tessellations . Although Escher believed he had no mathematical ability, he interacted with the mathematicians George Pólya , Roger Penrose
Roger Penrose
, Harold Coxeter and crystallographer Friedrich Haag , and conducted his own research into tessellation. Early in his career, he drew inspiration from nature , making studies of insects, landscapes , and plants such as lichens , all of which he used as details in his artworks
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Ferguson, Missouri
FERGUSON is a city in St. Louis County, Missouri
St. Louis County, Missouri
, United States. It is part of the Greater St. Louis
Greater St. Louis
metropolitan area. The population was 21,203 at the 2010 census . CONTENTS * 1 History * 2 Geography * 2.1 Climate * 3 Demographics * 3.1 2010 census * 3.2 Religion * 4 Economy * 5 Government * 5.1 DOJ Investigation into Ferguson PD * 6 Michael Brown shooting * 7 Education * 8 Notable people * 9 See also * 10 References * 11 External links HISTORYWhat is now the city of Ferguson was founded in 1855 when William B. Ferguson deeded 10 acres (4.0 ha) of land to the Wabash Railroad in exchange for a new depot and naming rights. The settlement that sprang up around the depot was called Ferguson Station. Ferguson was the first railroad station connected directly to St. Louis
St. Louis

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Helsinki
HELSINKI (/hɛlˈsɪŋki/ ; Finnish pronunciation: ( listen ); Swedish : Helsingfors; Swedish pronunciation: ( listen )) is the capital and largest city of Finland
Finland
. It is in the region of Uusimaa , in southern Finland, on the shore of the Gulf of Finland
Finland
. Helsinki has a population of 629,512, an urban population of 1,231,595, and a metropolitan population of over 1.4 million, making it the most populous municipality and urban area in Finland. Helsinki
Helsinki
is located some 80 kilometres (50 mi) north of Tallinn
Tallinn
, Estonia
Estonia
, 400 km (250 mi) east of Stockholm
Stockholm
, Sweden
Sweden
, and 390 km (240 mi) west of Saint Petersburg , Russia
Russia
. Helsinki
Helsinki
has close historical connections with these three cities
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John Horton Conway
JOHN HORTON CONWAY FRS (/ˈkɒnweɪ/ ; born 26 December 1937) is an English mathematician active in the theory of finite groups , knot theory , number theory , combinatorial game theory and coding theory . He has also contributed to many branches of recreational mathematics , notably the invention of the cellular automaton called the Game of Life . Conway is currently Professor Emeritus of Mathematics at Princeton University in New Jersey
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Roger Penrose
SIR ROGER PENROSE OM FRS (born 8 August 1931) is an English mathematical physicist , mathematician and philosopher of science . He is the Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute of the University of Oxford , as well as an Emeritus Fellow of Wadham College . Penrose is known for his work in mathematical physics, in particular for his contributions to general relativity and cosmology . He has received several prizes and awards, including the 1988 Wolf Prize for physics, which he shared with Stephen Hawking for their contribution to our understanding of the universe
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Penguin Books
Peter Field, CEO Madeline McIntosh, President USA PUBLICATION TYPES Books IMPRINTS Penguin Classics
Penguin Classics
, Viking Press OWNER(S) Bertelsmann
Bertelsmann
, Pearson PLC
Pearson PLC
OFFICIAL WEBSITE www.penguin.com Penguin Crime (details ) PENGUIN BOOKS is a British publishing house . It was founded in 1935 by Sir Allen Lane
Allen Lane
, his brothers Richard and John , as a line of the publishers The Bodley Head , only becoming a separate company the following year . Penguin revolutionised publishing in the 1930s through its inexpensive paperbacks , sold through Woolworths and other high street stores for sixpence , bringing high-quality paperback fiction and non-fiction to the mass market. Penguin's success demonstrated that large audiences existed for serious books
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Enslow Publishers
ENSLOW PUBLISHING is an American publisher of books and eBooks founded by Ridley M. Enslow, Jr. in 1976. Enslow publishes educational nonfiction, fiction, historical fiction, and trade books for children and young adults. Their books are intended to be sold to school and public libraries. Its current imprints include Enslow Elementary, Speeding Star and Chasing Roses. MyReportLinks.com Books and Bailey Books are currently out-of-print imprints. MyReportLinks.com Books is the properly formatted name. Enslow uses 3rd party authors to write the manuscripts, and uses in-house editorial and production staff to create their final products. Marketing, warehousing, and shipping operations are conducted at their headquarters in Berkeley Heights location. Enslow was acquired by Roger Rosen of Rosen Publishing in 2014
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Upper Bound
In mathematics , especially in order theory , an UPPER BOUND of a subset S of some partially ordered set (K, ≤) is an element of K which is greater than or equal to every element of S. The term LOWER BOUND is defined dually as an element of K which is less than or equal to every element of S. A set with an upper bound is said to be BOUNDED FROM ABOVE by that bound, a set with a lower bound is said to be BOUNDED FROM BELOW by that bound. The terms BOUNDED ABOVE (BOUNDED BELOW) are also used in the mathematical literature for sets that have upper (respectively lower) bounds. CONTENTS * 1 Examples * 2 Bounds of functions * 3 Tight bounds * 4 References EXAMPLESFor example, 5 is a lower bound for the set { 5, 8, 42, 34, 13934 }; so is 4; but 6 is not. Another example: for the set { 42 }, the number 42 is both an upper bound and a lower bound; all other real numbers are either an upper bound or a lower bound for that set
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Paul Erdős
PAUL ERDőS (Hungarian : Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician . He was one of the most prolific mathematicians of the 20th century. He was known both for his social practice of mathematics (he engaged more than 500 collaborators) and for his eccentric lifestyle (Time magazine called him The Oddball's Oddball). He devoted his waking hours to mathematics, even into his later years—indeed, his death came only hours after he solved a geometry problem in a conference in Warsaw
Warsaw
. Erdős pursued and proposed problems in discrete mathematics , graph theory , number theory , mathematical analysis , approximation theory , set theory , and probability theory . Much of his work centered around discrete mathematics , cracking many previously unsolved problems in the field
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Empirical Evidence
EMPIRICAL EVIDENCE, also known as SENSORY EXPERIENCE, is the knowledge received by means of the senses , particularly by observation and experimentation . The term comes from the Greek word for experience, ἐμπειρία (empeiría). After Immanuel Kant
Immanuel Kant
, in philosophy, it is common to call the knowledge gained a posteriori knowledge (in contrast to a priori knowledge). CONTENTS * 1 Meaning * 2 See also * 3 Footnotes * 4 References * 5 External links MEANING Empirical evidence is information that justifies the truth or falsity of a claim. In the empiricist view, one can claim to have knowledge only when based on empirical evidence. This stands in contrast to the rationalist view under which reason or reflection alone is considered evidence for the truth or falsity of some propositions . Empirical evidence is information acquired by observation or experimentation. This data is recorded and analyzed by scientists
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