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Sharkovskii's Theorem
In mathematics, Sharkovskii's theorem, named after Oleksandr Mykolaiovych Sharkovskii, who published it in 1964, is a result about discrete dynamical systems. One of the implications of the theorem is that if a discrete dynamical system on the real line has a periodic point of period 3, then it must have periodic points of every other period. Statement For some interval I\subset \mathbb, suppose that f : I \to I is a continuous function. The number x is called a ''periodic point of period m'' if f^(x)=x, where f^ denotes the iterated function obtained by composition of m copies of f. The number x is said to have ''least period m'' if, in addition, f^(x)\ne x for all 0 |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Upper Set
In mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in ''X'') of a partially ordered set (X, \leq) is a subset S \subseteq X with the following property: if ''s'' is in ''S'' and if ''x'' in ''X'' is larger than ''s'' (that is, if s \leq x), then ''x'' is in ''S''. In words, this means that any ''x'' element of ''X'' that is \,\geq\, to some element of ''S'' is necessarily also an element of ''S''. The term lower set (also called a downward closed set, down set, decreasing set, initial segment, or semi-ideal) is defined similarly as being a subset ''S'' of ''X'' with the property that any element ''x'' of ''X'' that is \,\leq\, to some element of ''S'' is necessarily also an element of ''S''. Definition Let (X, \leq) be a preordered set. An in X (also called an , an , or an set) is a subset U \subseteq X that is "closed under going up", in the sense that :for all u \in U and all x \in X, if u \leq x then x \in U. The dual notion is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The American Mathematical Monthly
''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content. In this the ''American Mathematical Monthly'' fulfills a different role from that of typical mathematical research journals. The ''American Mathematical Monthly'' is the most widely read mathematics journal in the world according to records on JSTOR. Tables of contents with article abstracts from 1997–2010 are availablonline The MAA gives the Lester R. Ford Awards annually to "authors of articles of expository excellence" published in the ''American Mathematical Monthly''. Editors *2022� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Providence, Rhode Island
Providence is the capital and most populous city of the U.S. state of Rhode Island. One of the oldest cities in New England, it was founded in 1636 by Roger Williams, a Reformed Baptist theologian and religious exile from the Massachusetts Bay Colony. He named the area in honor of "God's merciful Providence" which he believed was responsible for revealing such a haven for him and his followers. The city developed as a busy port as it is situated at the mouth of the Providence River in Providence County, at the head of Narragansett Bay. Providence was one of the first cities in the country to industrialize and became noted for its textile manufacturing and subsequent machine tool, jewelry, and silverware industries. Today, the city of Providence is home to eight hospitals and eight institutions of higher learning which have shifted the city's economy into service industries, though it still retains some manufacturing activity. At the 2020 census, Providence had a populati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the '' Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Kloeden
Peter may refer to: People * List of people named Peter, a list of people and fictional characters with the given name * Peter (given name) ** Saint Peter (died 60s), apostle of Jesus, leader of the early Christian Church * Peter (surname), a surname (including a list of people with the name) Culture * Peter (actor) (born 1952), stage name Shinnosuke Ikehata, Japanese dancer and actor * ''Peter'' (album), a 1993 EP by Canadian band Eric's Trip * ''Peter'' (1934 film), a 1934 film directed by Henry Koster * ''Peter'' (2021 film), Marathi language film * "Peter" (''Fringe'' episode), an episode of the television series ''Fringe'' * ''Peter'' (novel), a 1908 book by Francis Hopkinson Smith * "Peter" (short story), an 1892 short story by Willa Cather Animals * Peter, the Lord's cat, cat at Lord's Cricket Ground in London * Peter (chief mouser), Chief Mouser between 1929 and 1946 * Peter II (cat), Chief Mouser between 1946 and 1947 * Peter III (cat), Chief Mouser between 1947 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle Map
In mathematics, particularly in dynamical systems, Arnold tongues (named after Vladimir Arnold) Section 12 in page 78 has a figure showing Arnold tongues. are a pictorial phenomenon that occur when visualizing how the rotation number of a dynamical system, or other related invariant property thereof, changes according to two or more of its parameters. The regions of constant rotation number have been observed, for some dynamical systems, to form geometric shapes that resemble tongues, in which case they are called Arnold tongues. Arnold tongues are observed in a large variety of natural phenomena that involve oscillating quantities, such as concentration of enzymes and substrates in biological processes and cardiac electric waves. Sometimes the frequency of oscillation depends on, or is constrained (i.e., ''phase-locked'' or ''mode-locked'', in some contexts) based on some quantity, and it is often of interest to study this relation. For instance, the outset of a tumor triggers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Period Three Implies Chaos
James A. Yorke (born August 3, 1941) is a Distinguished University Research Professor of Mathematics and Physics and former chair of the Mathematics Department at the University of Maryland, College Park. Born in Plainfield, New Jersey, United States, Yorke attended The Pingry School, then located in Hillside, New Jersey. Yorke is now a Distinguished University Research Professor of Mathematics and Physics with the Institute for Physical Science and Technology at the University of Maryland. In June 2013, Dr. Yorke retired as chair of the University of Maryland's Math department. He devotes his university efforts to collaborative research in chaos theory and genomics. He and Benoit Mandelbrot were the recipients of the 2003 Japan Prize in Science and Technology: Yorke was selected for his work in chaotic systems. In 2003 He was elected a Fellow of the American Physical Society. and in 2012 became a fellow of the American Mathematical Society. He received the Doctor Honoris ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chaos (mathematics)
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have completely random states of disorder and irregularities. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals, and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning that there is sensitive dependence on initial conditions). A metaphor for this behavior is that a butterfly flapping its wings in Brazil can cause a tornado in Texas. Small differences in initial conditions, such as those due to errors in measurements or due to rounding error ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James A
James is a common English language surname and given name: * James (name), the typically masculine first name James * James (surname), various people with the last name James James or James City may also refer to: People * King James (other), various kings named James * Saint James (other) * James (musician) * James, brother of Jesus Places Canada * James Bay, a large body of water * James, Ontario United Kingdom * James College, a college of the University of York United States * James, Georgia, an unincorporated community * James, Iowa, an unincorporated community * James City, North Carolina * James City County, Virginia ** James City (Virginia Company) ** James City Shire * James City, Pennsylvania * St. James City, Florida Arts, entertainment, and media * ''James'' (2005 film), a Bollywood film * ''James'' (2008 film), an Irish short film * ''James'' (2022 film), an Indian Kannada-language film * James the Red Engine, a character in ''Thomas the Tan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tien-Yien Li
Tien-Yien Li (李天岩) (June 28, 1945 – June 25, 2020) was a University Distinguished Professor of Mathematics and University Distinguished Professor Emeritus at Michigan State University. There, he spent 42 years and supervised 26 Ph.D. dissertations. Early life and education Li was born on June 28, 1945 in Sha County, Fujian Province, China. At age three, he was brought to Taiwan by his parents. He earned his B.S. in Mathematics at the National Tsinghua University in 1968. Li received his doctorate in 1974 from University of Maryland under the guidance of James Yorke. Academic career Li joined the faculty of the Department of Mathematics at Michigan State University in 1976 and was promoted to the rank of full professor in 1983. He retired as a University Distinguished Professor Emeritus in 2018 after spending 42 years at the university. Li and his supervisor James Yorke published a paper in 1975 entitled '' Period three implies chaos,'' in which the mathemat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
