Number theory is a branch of pure mathematics devoted to the study of the integers. German mathematician Carl Friedrich Gauss said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of objects made out of integers or defined as generalizations of the integers. Integers can be considered either as solutions to equations. Questions in number theory are best understood through the study of analytical objects that encode properties of the integers, primes or other number-theoretic objects in some fashion. One may study real numbers in relation to rational numbers, for example, as approximated by the latter; the older term for number theory is arithmetic. By the early twentieth century, it had been superseded by "number theory"; the use of the term arithmetic for number theory regained some ground in the second half of the 20th century, arguably in part due to French influence. In particular, arithmetical is preferred as an adjective to number-theoretic.
The first historical find of an arithmetical nature is a fragment of a table: the broken clay tablet Plimpton 322 contains a list of "Pythagorean triples", that is, integers such that a 2 + b 2 = c 2. The triples are too large to have been obtained by brute force; the heading over the first column reads: "The takiltum of the diagonal, subtracted such that the width..." The table's layout suggests that it was constructed by means of what amounts, in modern language, to the identity 2 + 1 = 2, implicit in routine Old Babylonian exercises. If some other method was used, the triples were first constructed and reordered by c / a for actual use as a "table", for example, with a view to applications, it is not known whether there could have been any. It has been suggested instead. While Babylonian number theory—or what survives of Babylonian mathematics that can be called thus—consists of this single, striking fragment, Babylonian algebra was exceptionally well developed. Late Neoplatonic sources state.
Much earlier sources state that Pythagoras traveled and studied in Egypt. Euclid IX 21–34 is probably Pythagorean. Pythagorean mystics gave great importance to the even; the discovery that 2 is irrational is credited to the early Pythagoreans. By revealing that numbers could be irrational, this discovery seems to have provoked the first foundational crisis in mathematical history; this forced a distinction between numbers, on the one hand, lengths and proportions, on the other hand. The Pythagorean tradition spoke of so-called polygonal or figurate numbers. While square numbers, cubic numbers, etc. are seen now as more natural than triangular numbers, pentagonal numbers, etc. the study of the sums of triangular and pentagonal numbers would prove fruitful in the early modern period. We know of no arithmetical material in ancient Egyptian or Vedic sources, though there is some algebra in both; the Chinese remainder theorem appears as an exercise in Sunzi Suanjing There is some numerical mysticism in Chinese mathematics, unlike that of the Pythagoreans, it seems to have led nowhere.
Like the Pythagoreans' perfect numbers, magic squares have passed from superstition into recreation. Aside from a few fragments, the mathematics of Classical Greece is known to us either through the reports of contemporary non-m
Fan-Rong King Chung Graham, known professionally as Fan Chung, is a Taiwanese-born American mathematician who works in the areas of spectral graph theory, extremal graph theory and random graphs, in particular in generalizing the Erdős–Rényi model for graphs with general degree distribution. Since 1998, Chung has been the Akamai Professor in Internet Mathematics at the University of California, San Diego, she received her doctorate from the University of Pennsylvania in 1974, under the direction of Herbert Wilf. After working at Bell Laboratories and Bellcore for nineteen years, she joined the faculty of the University of Pennsylvania as the first female tenured professor in mathematics, she serves on the editorial boards of more than a dozen international journals. Since 2003 she has been the editor-in-chief of Internet Mathematics, she has been invited to give lectures at many conferences, including the International Congress of Mathematicians in 1994 and a plenary lecture on the mathematics of PageRank at the 2008 Annual meeting of the American Mathematical Society.
She was selected to be a Noether Lecturer in 2009. Chung has two children. Since 1983 she has been married to the mathematician Ronald Graham, they were close friends of the mathematician Paul Erdős, have both published papers with him – 13 in her case. She has published more than 200 research papers and three books: Erdős on Graphs: His Legacy of Unsolved Problems, A K Peters, Ltd. 1998, ISBN 1-56881-079-2 Spectral Graph Theory, American Mathematical Society, 1997, ISBN 0-8218-0315-8 Complex Graphs and Networks (CBMS Regional Conference Series in Mathematics, No. 107 ", American Mathematical Society, 2006, ISBN 0-8218-3657-9In 2012, she became a fellow of the American Mathematical Society. Fan Chung was born on October 1949 in Kaohsiung, Taiwan. Under the influence of her father, an engineer, she became interested in mathematics in the area of combinatorics in high school in Kaohsiung. After high school, Chung entered the National Taiwan University to start her career in mathematics formally.
While Chung was an undergraduate, she was surrounded by many female mathematicians, this helped encourage her to pursue and study mathematics. After graduating from NTU with a B. S. in mathematics, Chung went on to the University of Pennsylvania to pursue a career in mathematics. There she obtained the highest score in the qualifying exam by a wide margin, catching the attention of Herbert Wilf, who would become her doctoral advisor. Wilf suggested Ramsey theory. During a single week studying material Chung had come up with new proofs for established results in the field. Wilf said: "My eyes were bulging. I was excited. I asked her to show me. What she wrote was incredible! In just one week, from a cold start, she had a major result in Ramsey theory. I told her she had just done two-thirds of a doctoral dissertation."Chung was awarded a M. S. in 1972 and a Ph. D. two years later. By this time, she was married and had given birth to her first child; the same year she received her Ph. D. and started working for the Mathematical Foundations of Computing Department at Bell Laboratories in Murray Hill, New Jersey.
The position at Bell Laboratories was an opportunity to work with other excellent mathematicians, but it contributed to her mathematical world powerfully. She published many impressive mathematical papers and published many joint papers with Ron Graham. After twenty years of work at Bell Laboratories and Bellcore, Chung decided to go back University of Pennsylvania to become a professor of mathematics. In 1998, she was named Distinguished Professor of Mathematics at University of California, San Diego. To date, she has over 200 publications to her name; the two best known books are Spectral Graph Erdős on Graphs. Spectral Graph Theory studies how the spectrum of the Laplacian of a graph is related to its combinatorial properties. Erdős on Graphs, jointly written by Fan Chung and Ron Graham, studies many of Paul Erdős problems and conjectures in graph theory. Beyond her contributions to graph theory, Chung has used her knowledge to connect different fields of science; as she wrote in "Graph Theory in the Information Age", In the past decade, graph theory has gone through a remarkable shift and a profound transformation.
The change is in large part due to the humongous amount of information. A main way to sort through massive data sets is to build and examine the network formed by interrelations. For example, Google’s successful Web search algorithms are based on the WWW graph, which contains all Web pages as vertices and hyperlinks as edges. There are all sorts of information networks, such as biological networks built from biological databases and social networks formed by email, phone calls, instant messaging, etc. as well as various types of physical networks. Of particular interest to mathematicians is the collaboration graph, based on the data from Mathematical Reviews. In the collaboration graph, every mathematician is a vertex, two mathematicians who wrote a joint paper are connected. Chung's life was profiled in the 2017 documentary film Girls. In 1974, Fan Chung graduated from the University of Pennsylvania and became a member of Technical Staff working for the Mathematical Foundations of Computing Department at Bell Laboratories in Murray Hill, New Jersey.
She worked under
A game is a structured form of play undertaken for enjoyment and sometimes used as an educational tool. Games are distinct from work, carried out for remuneration, from art, more an expression of aesthetic or ideological elements. However, the distinction is not clear-cut, many games are considered to be work or art. Games are sometimes played purely sometimes for achievement or reward as well, they can be played alone, in online. The players may have an audience of non-players, such as when people are entertained by watching a chess championship. On the other hand, players in a game may constitute their own audience as they take their turn to play. Part of the entertainment for children playing a game is deciding, part of their audience and, a player. Key components of games are goals, rules and interaction. Games involve mental or physical stimulation, both. Many games help develop practical skills, serve as a form of exercise, or otherwise perform an educational, simulational, or psychological role.
Attested as early as 2600 BC, games are a universal part of human experience and present in all cultures. The Royal Game of Ur, Mancala are some of the oldest known games. Ludwig Wittgenstein was the first academic philosopher to address the definition of the word game. In his Philosophical Investigations, Wittgenstein argued that the elements of games, such as play and competition, all fail to adequately define what games are. From this, Wittgenstein concluded that people apply the term game to a range of disparate human activities that bear to one another only what one might call family resemblances; as the following game definitions show, this conclusion was not a final one and today many philosophers, like Thomas Hurka, think that Wittgenstein was wrong and that Bernard Suits' definition is a good answer to the problem. French sociologist Roger Caillois, in his book Les jeux et les hommes, defined a game as an activity that must have the following characteristics: fun: the activity is chosen for its light-hearted character separate: it is circumscribed in time and place uncertain: the outcome of the activity is unforeseeable non-productive: participation does not accomplish anything useful governed by rules: the activity has rules that are different from everyday life fictitious: it is accompanied by the awareness of a different reality Computer game designer Chris Crawford, founder of The Journal of Computer Game Design, has attempted to define the term game using a series of dichotomies: Creative expression is art if made for its own beauty, entertainment if made for money.
A piece of entertainment is a plaything. Movies and books are cited as examples of non-interactive entertainment. If no goals are associated with a plaything, it is a toy. If it has goals, a plaything is a challenge. If a challenge has no "active agent against whom you compete", it is a puzzle. If the player can only outperform the opponent, but not attack them to interfere with their performance, the conflict is a competition. However, if attacks are allowed the conflict qualifies as a game. Crawford's definition may thus be rendered as: an interactive, goal-oriented activity made for money, with active agents to play against, in which players can interfere with each other. "A game is a system in which players engage in an artificial conflict, defined by rules, that results in a quantifiable outcome." "A game is a form of art in which participants, termed players, make decisions in order to manage resources through game tokens in the pursuit of a goal." According to this definition, some "games" that do not involve choices, such as Chutes and Ladders, Candy Land, War are not technically games any more than a slot machine is.
"A game is an activity among two or more independent decision-makers seeking to achieve their objectives in some limiting context." "At its most elementary level we can define game as an exercise of voluntary control systems in which there is an opposition between forces, confined by a procedure and rules in order to produce a disequilibrial outcome." "A game is a form of play with goals and structure." "to play a game is to engage in activity directed toward bringing about a specific state of affairs, using only means permitted by specific rules, where the means permitted by the rules are more limited in scope than they would be in the absence of the rules, where the sole reason for accepting such limitation is to make possible such activity." "When you strip away the genre differences and the technological complexities, all games share four defining traits: a goal, rules, a feedback system, voluntary participation." Games can be characterized by "what the player does". This is referred to as gameplay.
Major key elements identified in this context are tools and rules that define the overall context of game. Games are classified by the com
Lior Samuel Pachter is a computational biologist. He works at the California Institute of Technology, where he is the Bren Professor of Computational Biology, he has varied research interests including genomics, computational geometry, machine learning, scientific computing, statistics. Pachter grew up in South Africa, he earned a bachelor's degree in mathematics from the California Institute of Technology in 1994. He completed his doctorate in mathematics from the Massachusetts Institute of Technology in 1999, supervised by Bonnie Berger, with Eric Lander and Daniel Kleitman as co-advisors. Pachter joined the University of California, Berkeley faculty in 1999 and was given the Sackler Chair in 2012; as well as for his technical contributions, Pachter is known for using new media to promote open science and for a thought experiment he posted on his blog according to which'the nearest neighbor to the "perfect human"' is from Puerto Rico. This received considerable media attention, a response was published in Scientific American.
In 2017, Pachter was elected a Fellow of the International Society for Computational Biology
ArXiv is a repository of electronic preprints approved for posting after moderation, but not full peer review. It consists of scientific papers in the fields of mathematics, astronomy, electrical engineering, computer science, quantitative biology, mathematical finance and economics, which can be accessed online. In many fields of mathematics and physics all scientific papers are self-archived on the arXiv repository. Begun on August 14, 1991, arXiv.org passed the half-million-article milestone on October 3, 2008, had hit a million by the end of 2014. By October 2016 the submission rate had grown to more than 10,000 per month. ArXiv was made possible by the compact TeX file format, which allowed scientific papers to be transmitted over the Internet and rendered client-side. Around 1990, Joanne Cohn began emailing physics preprints to colleagues as TeX files, but the number of papers being sent soon filled mailboxes to capacity. Paul Ginsparg recognized the need for central storage, in August 1991 he created a central repository mailbox stored at the Los Alamos National Laboratory which could be accessed from any computer.
Additional modes of access were soon added: FTP in 1991, Gopher in 1992, the World Wide Web in 1993. The term e-print was adopted to describe the articles, it began as a physics archive, called the LANL preprint archive, but soon expanded to include astronomy, computer science, quantitative biology and, most statistics. Its original domain name was xxx.lanl.gov. Due to LANL's lack of interest in the expanding technology, in 2001 Ginsparg changed institutions to Cornell University and changed the name of the repository to arXiv.org. It is now hosted principally with eight mirrors around the world, its existence was one of the precipitating factors that led to the current movement in scientific publishing known as open access. Mathematicians and scientists upload their papers to arXiv.org for worldwide access and sometimes for reviews before they are published in peer-reviewed journals. Ginsparg was awarded a MacArthur Fellowship in 2002 for his establishment of arXiv; the annual budget for arXiv is $826,000 for 2013 to 2017, funded jointly by Cornell University Library, the Simons Foundation and annual fee income from member institutions.
This model arose in 2010, when Cornell sought to broaden the financial funding of the project by asking institutions to make annual voluntary contributions based on the amount of download usage by each institution. Each member institution pledges a five-year funding commitment to support arXiv. Based on institutional usage ranking, the annual fees are set in four tiers from $1,000 to $4,400. Cornell's goal is to raise at least $504,000 per year through membership fees generated by 220 institutions. In September 2011, Cornell University Library took overall administrative and financial responsibility for arXiv's operation and development. Ginsparg was quoted in the Chronicle of Higher Education as saying it "was supposed to be a three-hour tour, not a life sentence". However, Ginsparg remains on the arXiv Scientific Advisory Board and on the arXiv Physics Advisory Committee. Although arXiv is not peer reviewed, a collection of moderators for each area review the submissions; the lists of moderators for many sections of arXiv are publicly available, but moderators for most of the physics sections remain unlisted.
Additionally, an "endorsement" system was introduced in 2004 as part of an effort to ensure content is relevant and of interest to current research in the specified disciplines. Under the system, for categories that use it, an author must be endorsed by an established arXiv author before being allowed to submit papers to those categories. Endorsers are not asked to review the paper for errors, but to check whether the paper is appropriate for the intended subject area. New authors from recognized academic institutions receive automatic endorsement, which in practice means that they do not need to deal with the endorsement system at all. However, the endorsement system has attracted criticism for restricting scientific inquiry. A majority of the e-prints are submitted to journals for publication, but some work, including some influential papers, remain purely as e-prints and are never published in a peer-reviewed journal. A well-known example of the latter is an outline of a proof of Thurston's geometrization conjecture, including the Poincaré conjecture as a particular case, uploaded by Grigori Perelman in November 2002.
Perelman appears content to forgo the traditional peer-reviewed journal process, stating: "If anybody is interested in my way of solving the problem, it's all there – let them go and read about it". Despite this non-traditional method of publication, other mathematicians recognized this work by offering the Fields Medal and Clay Mathematics Millennium Prizes to Perelman, both of which he refused. Papers can be submitted in any of several formats, including LaTeX, PDF printed from a word processor other than TeX or LaTeX; the submission is rejected by the arXiv software if generating the final PDF file fails, if any image file is too large, or if the total size of the submission is too large. ArXiv now allows one to store and modify an incomplete submission, only finalize the submission when ready; the time stamp on the article is set. The standard access route is through one of several mirrors. Sev
In mathematics, the natural numbers are those used for counting and ordering. In common mathematical terminology, words colloquially used for counting are "cardinal numbers" and words connected to ordering represent "ordinal numbers"; the natural numbers can, at times, appear as a convenient set of codes. Some definitions, including the standard ISO 80000-2, begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, …, whereas others start with 1, corresponding to the positive integers 1, 2, 3, …. Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers; the natural numbers are a basis from which many other number sets may be built by extension: the integers, by including the neutral element 0 and an additive inverse for each nonzero natural number n. These chains of extensions make the natural numbers canonically embedded in the other number systems.
Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics. In common language, for example in primary school, natural numbers may be called counting numbers both to intuitively exclude the negative integers and zero, to contrast the discreteness of counting to the continuity of measurement, established by the real numbers; the most primitive method of representing a natural number is to put down a mark for each object. A set of objects could be tested for equality, excess or shortage, by striking out a mark and removing an object from the set; the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers; the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1, 10, all the powers of 10 up to over 1 million.
A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, 6 ones. The Babylonians had a place-value system based on the numerals for 1 and 10, using base sixty, so that the symbol for sixty was the same as the symbol for one, its value being determined from context. A much advance was the development of the idea that 0 can be considered as a number, with its own numeral; the use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, but they omitted such a digit when it would have been the last symbol in the number. The Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica; the use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628. However, 0 had been used as a number in the medieval computus, beginning with Dionysius Exiguus in 525, without being denoted by a numeral; the first systematic study of numbers as abstractions is credited to the Greek philosophers Pythagoras and Archimedes.
Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes not as a number at all. Independent studies occurred at around the same time in India and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. A school of Naturalism stated that the natural numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized "God made the integers, all else is the work of man". In opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not natural but a consequence of definitions. Two classes of such formal definitions were constructed. Set-theoretical definitions of natural numbers were initiated by Frege and he defined a natural number as the class of all sets that are in one-to-one correspondence with a particular set, but this definition turned out to lead to paradoxes including Russell's paradox.
Therefore, this formalism was modified so that a natural number is defined as a particular set, any set that can be put into one-to-one correspondence with that set is said to have that number of elements. The second class of definitions was introduced by Charles Sanders Peirce, refined by Richard Dedekind, further explored by Giuseppe Peano, it is based on an axiomatization of the properties of ordinal numbers: each natural number has a