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Fixed-point Index
In mathematics, the fixed-point index is a concept in topological fixed-point theory, and in particular Nielsen theory. The fixed-point index can be thought of as a multiplicity measurement for fixed points. The index can be easily defined in the setting of complex analysis: Let ''f''(''z'') be a holomorphic mapping on the complex plane, and let ''z''0 be a fixed point of ''f''. Then the function ''f''(''z'') − ''z'' is holomorphic, and has an isolated zero at ''z''0. We define the fixed-point index of ''f'' at ''z''0, denoted ''i''(''f'', ''z''0), to be the multiplicity of the zero of the function ''f''(''z'') − ''z'' at the point ''z''0. In real Euclidean space, the fixed-point index is defined as follows: If ''x''0 is an isolated fixed point of ''f'', then let ''g'' be the function defined by :g(x) = \frac. Then ''g'' has an isolated singularity at ''x''0, and maps the boundary of some deleted neighborhood of ''x''0 to the unit sphere. We define ''i'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set endowed with a structure, called a ''topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. Basic examples of topological properties are: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connecte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fixed Point (mathematics)
A fixed point (sometimes shortened to fixpoint, also known as an invariant point) is a value that does not change under a given transformation. Specifically, in mathematics, a fixed point of a function is an element that is mapped to itself by the function. In physics, the term fixed point can refer to a temperature that can be used as a reproducible reference point, usually defined by a phase change or triple point. Fixed point of a function Formally, is a fixed point of a function if belongs to both the domain and the codomain of , and . For example, if is defined on the real numbers by f(x) = x^2 - 3 x + 4, then 2 is a fixed point of , because . Not all functions have fixed points: for example, , has no fixed points, since is never equal to for any real number. In graphical terms, a fixed point means the point is on the line , or in other words the graph of has a point in common with that line. Fixed-point iteration In numerical analysis, ''fixed-point iter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nielsen Theory
Nielsen theory is a branch of mathematical research with its origins in topological fixed-point theory. Its central ideas were developed by Danish mathematician Jakob Nielsen (mathematician), Jakob Nielsen, and bear his name. The theory developed in the study of the so-called ''minimal number'' of a map (mathematics), map ''f'' from a compact (topology), compact space to itself, denoted ''MF''[''f'']. This is defined as: :\mathit[f] = \min \, where ''~'' indicates homotopy of mappings, and #Fix(''g'') indicates the number of fixed points of ''g''. The minimal number was very difficult to compute in Nielsen's time, and remains so today. Nielsen's approach is to group the fixed-point set into classes, which are judged "essential" or "nonessential" according to whether or not they can be "removed" by a homotopy. Nielsen's original formulation is equivalent to the following: We define an equivalence relation on the set of fixed points of a self-map ''f'' on a space ''X''. We say that '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplicity (mathematics)
In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial has a root at a given point is the multiplicity of that root. The notion of multiplicity is important to be able to count correctly without specifying exceptions (for example, ''double roots'' counted twice). Hence the expression, "counted with multiplicity". If multiplicity is ignored, this may be emphasized by counting the number of ''distinct'' elements, as in "the number of distinct roots". However, whenever a set (as opposed to multiset) is formed, multiplicity is automatically ignored, without requiring use of the term "distinct". Multiplicity of a prime factor In prime factorization, the multiplicity of a prime factor is its p-adic valuation. For example, the prime factorization of the integer is : the multiplicity of the prime factor is , while the multiplicity of each of the prime factors and is . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, and particularly quantum mechanics. By extension, use of complex analysis also has applications in engineering fields such as nuclear engineering, nuclear, aerospace engineering, aerospace, mechanical engineering, mechanical and electrical engineering. As a differentiable function of a complex variable is equal to its Taylor series (that is, it is Analyticity of holomorphic functions, analytic), complex analysis is particularly concerned with analytic functions of a complex variable (that is, holomorphic functions). History Complex analysis is one of the classical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Holomorphic Map
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivative in a neighbourhood is a very strong condition: it implies that a holomorphic function is infinitely differentiable and locally equal to its own Taylor series (''analytic''). Holomorphic functions are the central objects of study in complex analysis. Though the term ''analytic function'' is often used interchangeably with "holomorphic function", the word "analytic" is defined in a broader sense to denote any function (real, complex, or of more general type) that can be written as a convergent power series in a neighbourhood of each point in its domain. That all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis. Holomorphic functions are also sometimes referred to as ''regular fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree Of A Continuous Mapping
In topology, the degree of a continuous mapping between two compact oriented manifolds of the same dimension is a number that represents the number of times that the domain manifold wraps around the range manifold under the mapping. The degree is always an integer, but may be positive or negative depending on the orientations. The degree of a map was first defined by Brouwer, who showed that the degree is homotopy invariant (invariant among homotopies), and used it to prove the Brouwer fixed point theorem. In modern mathematics, the degree of a map plays an important role in topology and geometry. In physics, the degree of a continuous map (for instance a map from space to some order parameter set) is one example of a topological quantum number. Definitions of the degree From ''S''''n'' to ''S''''n'' The simplest and most important case is the degree of a continuous map from the n-sphere S^n to itself (in the case n=1, this is called the winding number): Let f\colon S^n\t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solomon Lefschetz
Solomon Lefschetz (russian: Соломо́н Ле́фшец; 3 September 1884 – 5 October 1972) was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations. Life He was born in Moscow, the son of Alexander Lefschetz and his wife Sarah or Vera Lifschitz, Jewish traders who used to travel around Europe and the Middle East (they held Ottoman passports). Shortly thereafter, the family moved to Paris. He was educated there in engineering at the École Centrale Paris, but emigrated to the US in 1905. He was badly injured in an industrial accident in 1907, losing both hands. He moved towards mathematics, receiving a Ph.D. in algebraic geometry from Clark University in Worcester, Massachusetts in 1911. He then took positions in University of Nebraska and University of Kansas, moving to Princeton University in 1924, where he was soon given a permanent position. He rema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heinz Hopf
Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of topology and geometry. Early life and education Hopf was born in Gräbschen, Germany (now , part of Wrocław, Poland), the son of Elizabeth (née Kirchner) and Wilhelm Hopf. His father was born Jewish and converted to Protestantism a year after Heinz was born; his mother was from a Protestant family. Hopf attended Karl Mittelhaus higher boys' school from 1901 to 1904, and then entered the König-Wilhelm- Gymnasium in Breslau. He showed mathematical talent from an early age. In 1913 he entered the Silesian Friedrich Wilhelm University where he attended lectures by Ernst Steinitz, Adolf Kneser, Max Dehn, Erhard Schmidt, and Rudolf Sturm. When World War I broke out in 1914, Hopf eagerly enlisted. He was wounded twice and received the iron cross (first class) in 1918. After the war Hopf continued his mathematical education in Heidelberg (winter 1919/20 and summer 1920) and Berl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lefschetz Number
In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X. It is named after Solomon Lefschetz, who first stated it in 1926. The counting is subject to an imputed multiplicity at a fixed point called the fixed-point index. A weak version of the theorem is enough to show that a mapping without ''any'' fixed point must have rather special topological properties (like a rotation of a circle). Formal statement For a formal statement of the theorem, let :f\colon X \rightarrow X\, be a continuous map from a compact triangulable space X to itself. Define the Lefschetz number \Lambda_f of f by :\Lambda_f:=\sum_(-1)^k\mathrm(f_*, H_k(X,\Q)), the alternating (finite) sum of the matrix traces of the linear maps induced by f on H_k(X,\Q), the singular homology groups of X with rational coefficients. A simple versi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lefschetz Fixed-point Theorem
In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X. It is named after Solomon Lefschetz, who first stated it in 1926. The counting is subject to an imputed multiplicity at a fixed point called the fixed-point index. A weak version of the theorem is enough to show that a mapping without ''any'' fixed point must have rather special topological properties (like a rotation of a circle). Formal statement For a formal statement of the theorem, let :f\colon X \rightarrow X\, be a continuous map from a compact triangulable space X to itself. Define the Lefschetz number \Lambda_f of f by :\Lambda_f:=\sum_(-1)^k\mathrm(f_*, H_k(X,\Q)), the alternating (finite) sum of the matrix traces of the linear maps induced by f on H_k(X,\Q), the singular homology groups of X with rational coefficients. A simple vers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
