Common Fixed Point Problem

	Common Fixed Point Problem In mathematics, the common fixed point problem is the conjecture that, for any two continuous functions that map the unit interval into itself and commute under functional composition, there must be a point that is a fixed point of both functions. In other words, if the functions f and g are continuous, and f(g(x)) = g(f(x)) for all x in the unit interval, then there must be some x in the unit interval for which f(x) = x = g(x). First posed in 1954, the problem remained unsolved for more than a decade, during which several mathematicians made incremental progress toward an affirmative answer. In 1967, William M. Boyce and John P. Huneke independently proved the conjecture to be false by providing examples of commuting functions on a closed interval that do not have a common fixed point. History A 1951 paper by H. D. Block and H. P. Thielman sparked interest in the subject of fixed points of commuting functions. Building on earlier work by J. F. Ritt and A. G. Walker, Block a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Continuous Function In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Until the 19th century, mathematicians largely relied on intuitive notions of continuity and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the most general continuous functions, and their d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Glen E A glen is a valley, typically one that is long and bounded by gently sloped concave sides, unlike a ravine, which is deep and bounded by steep slopes. The word is Goidelic languages, Goidelic in origin: ''gleann'' in Irish language, Irish and Scottish Gaelic, ''glion'' in Manx language, Manx. The designation "glen" also occurs often in place names. Glens are appreciated by tourists for their tranquility and scenery. Etymology The word is Goidelic languages, Goidelic in origin: ''gleann'' in Irish language, Irish and Scottish Gaelic, ''glion'' in Manx language, Manx. In Manx, ''glan'' is also to be found meaning glen. It is cognate with Welsh language, Welsh ''glyn''. Whittow defines it as a "Scottish term for a deep valley in the Highlands" that is "narrower than a strath". Examples in Northern England, such as Glenridding, Westmorland, or Glendue, near Haltwhistle, Northumberland, are thought to derive from the aforementioned Cumbric cognate, or another Brittonic languages, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Fixed Points (mathematics) Fixed point may refer to: * Fixed point (mathematics), a value that does not change under a given transformation * Fixed-point arithmetic, a manner of doing arithmetic on computers * Fixed point, a benchmark (surveying) used by geodesists * Fixed point join, also called a recursive join * Fixed point, in quantum field theory, a coupling where the beta function vanishes – see * Temperature reference point, usually defined by a phase change or triple point In thermodynamics, the triple point of a substance is the temperature and pressure at which the three Phase (matter), phases (gas, liquid, and solid) of that substance coexist in thermodynamic equilibrium.. It is that temperature and pressure at ... . {{disambiguation, math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Journal Of Combinatorial Theory The ''Journal of Combinatorial Theory'', Series A and Series B, are mathematical journals specializing in combinatorics and related areas. They are published by Elsevier. ''Series A'' is concerned primarily with structures, designs, and applications of combinatorics. ''Series B'' is concerned primarily with graph and matroid theory. The two series are two of the leading journals in the field and are widely known as ''JCTA'' and ''JCTB''. The journal was founded in 1966 by Frank Harary and Gian-Carlo Rota.They are acknowledged on the journals' title pages and Web sites. SeEditorial board of JCTAEditorial board of JCTB Originally there was only one journal, which was split into two parts in 1971 as the field grew rapidly. In 2020, most of the editorial board of ''JCTA'' resigned to form a new, [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Mountain Climbing Problem In mathematics, the mountain climbing problem is a mathematical problem that considers a two-dimensional mountain range (represented as a continuous function), and asks whether it is possible for two mountaineering, mountain climbers starting at sea level on the left and right sides of the mountain to meet at the summit, while maintaining equal altitudes at all times. It has been shown that when the mountain range has only a finite number of peaks and valleys, it is always possible to coordinate the climbers' movements, but this does not necessarily hold when it has an infinite number of peaks and valleys. This problem was named and posed in this form by , but its history goes back to , who solved a version of it. The problem has been repeatedly rediscovered and solved independently in different contexts by a number of people (see references below). Since the 1990s, the problem was shown to be connected to the weak Fréchet distance of curves in the plane, various planar motion ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Wesleyan University Wesleyan University ( ) is a Private university, private liberal arts college, liberal arts university in Middletown, Connecticut, United States. It was founded in 1831 as a Men's colleges in the United States, men's college under the Methodist Episcopal Church and with the support of prominent residents of Middletown. It is now a secular, coeducational institution. The college accepted female applicants from 1872 to 1909, but did not become fully coeducational until 1970. Before full coeducation, Wesleyan alumni and other supporters of Women's colleges in the United States, women's education established Connecticut College in 1912. Wesleyan, along with Amherst College, Amherst and Williams College, Williams colleges, is part of "The Little Three". Its teams compete athletically as a member of the NESCAC in NCAA Division III. History Before Wesleyan was founded, a military academy established by Alden Partridge existed, consisting of the campus's North and South Colleges. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Source Code In computing, source code, or simply code or source, is a plain text computer program written in a programming language. A programmer writes the human readable source code to control the behavior of a computer. Since a computer, at base, only understands machine code, source code must be Translator (computing), translated before a computer can Execution (computing), execute it. The translation process can be implemented three ways. Source code can be converted into machine code by a compiler or an assembler (computing), assembler. The resulting executable is machine code ready for the computer. Alternatively, source code can be executed without conversion via an interpreter (computing), interpreter. An interpreter loads the source code into memory. It simultaneously translates and executes each statement (computer science), statement. A method that combines compilation and interpretation is to first produce bytecode. Bytecode is an intermediate representation of source code tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	MIT Lincoln Laboratory The MIT Lincoln Laboratory, located in Lexington, Massachusetts, is a United States Department of Defense federally funded research and development center chartered to apply advanced technology to problems of national security. Research and development activities focus on long-term technology development as well as rapid system prototyping and demonstration. Its core competencies are in sensors, integrated sensing, signal processing for information extraction, decision-making support, and communications. These efforts are aligned within ten mission areas. The laboratory also maintains several field sites around the world. The laboratory transfers much of its advanced technology to government agencies, industry, and academia, and has launched more than 100 start-ups. History Origins At the urging of the United States Air Force, the Lincoln Laboratory was created in 1951 at the Massachusetts Institute of Technology (MIT) as part of an effort to improve the U.S. air defense syste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Computer-assisted Proof Automation describes a wide range of technologies that reduce human intervention in processes, mainly by predetermining decision criteria, subprocess relationships, and related actions, as well as embodying those predeterminations in machines. Automation has been achieved by various means including mechanical, hydraulic, pneumatic, electrical, electronic devices, and computers, usually in combination. Complicated systems, such as modern factories, airplanes, and ships typically use combinations of all of these techniques. The benefit of automation includes labor savings, reducing waste, savings in electricity costs, savings in material costs, and improvements to quality, accuracy, and precision. Automation includes the use of various equipment and control systems such as machinery, processes in factories, boilers, and heat-treating ovens, switching on telephone networks, steering, stabilization of ships, aircraft and other applications and vehicles with reduced human ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Baxter Permutation In combinatorial mathematics, a Baxter permutation is a permutation \sigma \in S_n which satisfies the following generalized pattern avoidance property: * There are no indices i such that $\sigma(j+1)<\sigma(i)<\sigma(k)<\sigma(j)$ or $\sigma(j)<\sigma(k)<\sigma(i)<\sigma(j+1)$ . Equivalently, using the notation for vincular pattern s, a Baxter permutation is one that avoids the two dashed patterns $2-41-3$ and $3-14-2$ . For example, the permutation $\sigma=2413$ in $S_4$ (written in one-line notation ) is ''not'' a Baxter permutation because, taking $i= 1$ , $j=2$ and $k = 4$ , this permutation violates the first condit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Jon Folkman Jon Hal Folkman (December 8, 1938 – January 23, 1969) was an American mathematician, a student of John Milnor, and a researcher at the RAND Corporation. Schooling Folkman was a William Lowell Putnam Mathematical Competition, Putnam Fellow in 1960. He received his Ph.D. in 1964 from Princeton University, under the supervision of Milnor, with a thesis entitled ''Equivariant Maps of Spheres into the Classical Groups''. Research Jon Folkman contributed important theorems in many areas of combinatorics. In geometric combinatorics, Folkman is known for his pioneering and posthumously-published studies of oriented matroids; in particular, the Folkman–Lawrence topological representation theorem is "one of the cornerstones of the theory of oriented matroids". In lattice (order), lattice theory, Folkman solved an open problem on the foundations of enumerative combinatorics, combinatorics by proving a conjecture of Gian-Carlo Rota, Gian–Carlo Rota; in proving Rota's conjecture ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]