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fixed point





Encyclopedia results for fixed point

  1. Fixed point

    Fixed point has many meanings in science, most of them mathematical. Fixed point mathematics Fixed point combinator Fixed point arithmetic , a manner of doing arithmetic on computers Benchmark surveying , fixed points used by geodesists For fixed points in physics, see Renormalization group Fixed points are necessary for Mooring watercraft a watercraft to be moored to a quay. Archimedes said , which is sometimes translated as Give me a fixed point and I will move the world. disambiguation Category Mathematical disambiguation es Punto fijo desambiguaci n ...   more details



  1. Gaussian fixed point

    Unreferenced date December 2009 A Gaussian fixed point is a Fixed point mathematics fixed point of the renormalization group flow which is noninteracting in the sense that it is described by a free field theory . The word Gaussian comes from the fact that the probability distribution is Gaussian at the Gaussian fixed point. This means that Gaussian fixed points are exactly solvable Quantum triviality trivially solvable in fact . Slight deviations from the Gaussian fixed point can be described by perturbation theory. See also UV fixed point IR fixed point Quantum triviality DEFAULTSORT Gaussian Fixed Point Category Quantum field theory Category Statistical mechanics Category Renormalization group ...   more details



  1. Fixed-point theorem

    In mathematics , a fixed point theorem is a result saying that a function mathematics function F will have at least one fixed point mathematics fixed point a point x for which F x x , under some conditions ... in mathematics. Well known fixed point theorems include Atiyah Bott fixed point theorem Banach fixed point theorem , for contraction mapping s Borel fixed point theorem Brouwer fixed point theorem ... space to itself Caristi fixed point theorem Fixed point lemma for normal functions , for continuous strictly increasing functions from ordinal number ordinals to ordinals Fixed point theorems in infinite dimensional spaces Kakutani fixed point theorem Kleene fixpoint theorem Lefschetz fixed point theorem Nielsen theory Nielsen fixed point theorem Knaster&ndash Tarski theorem , which states that any monotonic order preserving function on a complete lattice has a smallest fixed point Tychonoff fixed point theorem Woods Hole fixed point theorem See also Fixed point property Fixed point combinator Collage theorem Diagonal lemma , also known as the fixed point lemma, for producing self referential ..., Maria O Regan, Donal title Fixed Point Theory and Applications year 2001 publisher Cambridge University ... Methods in fixed point theory year 1990 publisher Springer Verlag isbn 0 387 97364 8 cite book author Border, Kim C. title Fixed Point Theorems with Applications to Economics and Game Theory year 1989 ... Fixed Point Theory and Its Applications year 1988 publisher American Mathematical Society isbn 0 8218 5080 6 cite book author Dugundji, James Granas, Andrzej title Fixed Point Theory year 2003 publisher ... in Metric Fixed Point Theory year 1990 publisher Cambridge University Press isbn 0 521 38289 0 cite book author Kirk, William A. Khamsi, Mohamed A. title An Introduction to Metric Spaces and Fixed Point Theory year 2001 publisher John Wiley, New York. isbn 978 0 471 41825 2 cite book author Kirk, William A. Sims, Brailey title Handbook of Metric Fixed Point Theory year 2001 publisher Springer Verlag ...   more details



  1. Fixed-point space

    In mathematics , a Hausdorff space X is called a fixed point space if every continuous function math f X rightarrow X math has a fixed point mathematics fixed point . For example, any closed interval a,b in math mathbb R math is a fixed point space, and it can be proved from the intermediate value property of real continuous function. The open interval a ,  b , however, is not a fixed point space. To see it, consider the function math f x a frac 1 b a cdot x a 2 math , for example. Any linearly ordered space that is connected and has a top and a bottom element is a fixed point space. Note that, in the definition, we could easily have disposed of the condition that the space is Hausdorff. References Vasile I. Istratescu, Fixed Point Theory, An Introduction , D. Reidel, the Netherlands 1981 . ISBN 90 277 1224 7 Andrzej Granas and James Dugundji, Fixed Point Theory 2003 Springer Verlag, New York, ISBN 0 387 00173 5 William A. Kirk and Brailey Sims, Handbook of Metric Fixed Point Theory 2001 , Kluwer Academic, London ISBN 0 7923 7073 2 mathanalysis stub Category Fixed points Category Topology ...   more details



  1. Fixed point (mathematics)

    Not to be confused with a stationary point where f x 0 . Image Fixed Point Graph.png thumb right A function with three fixed points In mathematics , a fixed point sometimes shortened to fixpoint , also ... s by math f x x 2 3 x 4, math then 2 is a fixed point of f , because f 2 2. Not all functions have ... points, since x is never equal to x 1 for any real number. In graphical terms, a fixed point means ... of iterated function iterations of the function are known as periodic point s a fixed point is a periodic point with period equal to one. In projective geometry , a fixed point of a collineation ... Press ref Attractive fixed points Image Cosine fixed point.svg 250px thumb The fixed point ... fixed point of a function f is a fixed point x sub 0 sub of f such that for any value of x in the domain ... of the existence of such solution is given by Banach fixed point theorem . The natural cosine function natural means in radians , not degrees or other units has exactly one fixed point, which is attractive ... is in radians mode . It eventually converges to about 0.739085133, which is a fixed point ... points are attractive for example, x 0 is a fixed point of the function f x 2 x , but iteration of this function ... differentiable in an open neighbourhood of a fixed point x sub 0 sub , and math f , x 0 1 math , attraction ... of attractor s. An attractive fixed point is said to be a stable fixed point if it is also Lyapunov stable . A fixed point is said to be a neutrally stable fixed point if it is Lyapunov stable but not attracting ... fixed point. Theorems guaranteeing fixed points There are numerous theorems in different parts ... fixed point. These are amongst the most basic qualitative results available such fixed point theorem ..., in economics , a Nash equilibrium of a game theory game is a fixed point of the game s best ... theory of Phase Transitions , linearisation near an unstable fixed point has led to Kenneth G. Wilson ... explanation of the term critical phenomenon . In compilers , fixed point computations are used for whole ...   more details



  1. Least fixed point

    In order theory , a branch of mathematics , the least fixed point lfp or LFP of a function mathematics function is the fixed point mathematics fixed point which is less than or equal to all other fixed points, according to some partial order . For example, the least fixed point of the real function f x x sup 2 sup is x 0 with the usual order on the real numbers since the only other fixpoint is 1 and 0 < 1 . Many fixed point theorem s yield algorithms for locating the least fixed point. Least fixed points often have desirable properties that arbitrary fixed points do not. In mathematical logic and computer science , the least fixed point is related to making Recursion recursive definitions see domain theory and or denotational semantics for details . Immerman ref N. Immerman, Relational queries computable in polynomial time, Information and Control 68 1 3 1986 86 104. ref and Moshe Y. Vardi Vardi ref M. Y. Vardi, The complexity of relational query languages, in Proc. 14th ACM Symp. on Theory of Computing, 1982, pp. 137 146. ref independently showed the descriptive complexity result that the polynomial time computable properties of linearly ordered structures are de nable in LFP. However, LFP is too weak to express all polynomial time properties of unordered structures for instance that a structure has even size . Greatest fixed points Greatest fixed points can also be determined, but they are less commonly used than least fixed points. Notes reflist See also Fixed point mathematics Fixed point Kleene fixpoint theorem Knaster Tarski theorem References Immerman, Neil. Descriptive Complexity , 1999, Springer Verlag. Libkin, Leonid. Elements of Finite Model Theory , 2004, Springer. Category Order theory Category Fixed points ja zh mathlogic stub ...   more details



  1. Fixed-point index

    Refimprove date November 2010 In mathematics , the fixed point index is a concept in topological fixed point mathematics fixed point theory, and in particular Nielsen theory . The fixed point index can be thought of as a Multiplicity mathematics 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 sub 0 sub be a fixed point of f . Then the function f z     z is holomorphic, and has an isolated zero at z sub 0 sub . We define the fixed point index of f at z sub 0 sub , denoted i f , z sub 0 sub , to be the multiplicity of the zero of the function f z     z at the point z sub 0 sub . In real Euclidean space, the fixed point index is defined as follows If x sub 0 sub is an isolated fixed point of f , then let g be the function defined by math g x frac x f x x f x . , math Then g has an isolated singularity at x sub 0 sub , and maps the boundary of some deleted neighborhood of x sub 0 sub to the unit sphere. We define i f ,  x sub 0 sub to be the L. E. J. Brouwer Brouwer Degree of a continuous mapping degree of the mapping induced by g on some suitably chosen small sphere around x sub 0 sub . ref A. Katok and B. Hasselblatt 1995 , Introduction to the modern theory of dynamical systems, Cambridge University Press, Chapter 8. ref The Lefschetz Hopf theorem The importance of the fixed point index is largely due to its role in the Solomon Lefschetz ... where Fix f is the set of fixed points of f , and sub f sub is the Lefschetz number of f . Since the quantity on the left hand side of the above is clearly zero when f has no fixed points, the Lefschetz Hopf theorem trivially implies the Lefschetz fixed point theorem . References reflist Robert F. Brown Fixed Point Theory , in I. M. James, History of Topology , Amsterdam 1999, ISBN 0 444 82375 1, 271 299. DEFAULTSORT Fixed Point Index Category Fixed points Category Topology ...   more details



  1. Fixed-point property

    A mathematics mathematical object X has the fixed point property if every suitably well behaved mapping mathematics mapping from X to itself has a fixed point mathematics fixed point . It is a special ... every continuous mathematics continuous mapping has a fixed point. But another use is in order theory , where a partially ordered set P is said to have the fixed point property if every increasing function on P has a fixed point. Definition Let A be an object in the concrete category C . Then A has the fixed point property if every morphism i.e., every Function mathematics function math f A to A math has a fixed point. The most common usage is when C Top is the category of topological spaces. Then a topological space X has the fixed point property if every continuous map math f X to X math has a fixed point. Examples The closed interval The closed interval 0,1 has the fixed point property Let f 0,1 0,1 be a continuous mapping. If f 0 0 or f 1 1, then our mapping has a fixed point ... x sub 0 sub is a fixed point. The open interval does not have the fixed point property. The mapping f x x sup 2 sup has no fixed point on the interval 0,1 . The closed disc The closed interval is a special case of the closed disc , which in any finite dimension has the fixed point property by the Brouwer fixed point theorem . Topology A Deformation retract retract A of a space X with the fixed point property also has the fixed point property. This is because if math r X to A math is a retraction ... math where math i A to X math is inclusion has a fixed point. That is, there is math x in A math ... math f x x. math A topological space has the fixed point property if and only if its identity map is universal map universal . A product topology product of spaces with the fixed point property in general fails to have the fixed point property even if one of the spaces is the closed real interval. The FPP ... by any retraction . According to Brouwer fixed point theorem every compact space compact and convex ...   more details



  1. Fixed-point iteration

    refimprove date May 2010 In numerical analysis , fixed point iteration is a method of computing fixed point mathematics fixed points of iterated function s. More specifically, given a function math f math ... domain of math f math , the fixed point iteration is math x n 1 f x n , , n 0, 1, 2, dots math ... math x math is a fixed point of math f math , i.e., math f x x math . More generally, the function ... case of Newton s method quoted below. Image Sine fixed point.svg 250px thumb The fixed point iteration ... does not satisfy the hypotheses of the Banach fixed point theorem and so its speed of convergence is very slow. The fixed point iteration math x n 1 cos x n , math converges to the unique fixed ... satisfy the hypotheses of the Banach fixed point theorem . Hence, the error after n steps satisfies ... say that we have linear convergence . The Banach fixed point theorem allows one to obtain fixed point iterations with linear convergence. The fixed point iteration math x n 1 2x n , math will diverge unless math x 0 0 math . We say that the fixed point of math f x 2x , math is repelling. The requirement ... x 0 math . However, 0 is not a fixed point of the function math f x begin cases frac x 2 , & x ne 0 ... iteration as the fixed point iteration math x n 1 g x n math . If this iteration converges to a fixed ... of the Banach fixed point theorem , the Newton iteration, framed as the fixed point method, demonstrates ... s and numerical Ordinary Differential Equation solvers in general can be viewed as fixed point ... to the fixed point math y 0 math whenever the real part of a is negative. ref One may also consider ... have solutions, is essentially an application of the Banach fixed point theorem to a special sequence of functions which forms a fixed point iteration. The goal seeking function in Excel can be used ... Bellman s functional equation are based on fixed point iterations in the space of the return function ... with Lipschitz constant math L 1 math , then this function has precisely one fixed point, and the fixed ...   more details



  1. UV fixed point

    they are not effective field theories. If the UV fixed point is trivial fixed point trivial aka Gaussian , we say that we have asymptotic freedom . If the UV fixed point is nontrivial, we say that we ... Asymptotic Safety gravity References Reflist DEFAULTSORT Uv Fixed Point Category Quantum field ... Category Fixed points Quantum stub ...   more details



  1. Infrared fixed point

    Unreferenced date December 2009 In physics , an infrared fixed point is a set of coupling constants, or other parameters that evolve from initial values at very high energies short distance , to fixed ... fixed values, then we have ultraviolet fixed point s. The fixed points are generally independent ... system approaches an infrared fixed point that is independent of the initial short distance ... constituents. Particle Physics In particle physics the best known fixed point is that the strong ... fixed point, associated with the phenomenon known as asymptotic freedom . This causes quark s and gluon ... fixed point of the renormalization group equation for the Yukawa coupling. No matter what the initial starting value of the coupling is, if it is sufficiently large it will reach this fixed point value, and the corresponding quark mass is predicted. The value of the fixed point is fairly precisely ... to a fixed point where the top mass is smaller, 170 200  GeV. Some theorists believe this is supporting evidence for the MSSM. The quasi infrared fixed point was proposed in 1981 by C. T. Hill ... lie in a range of 15 to 26 GeV. The quasi infrared fixed point has formed the basis of top quark condensation ... DEFAULTSORT Infrared Fixed Point Category Quantum field theory Category Statistical mechanics Category Conformal field theory Category Renormalization group Category Fixed points es Punto fijo infrarrojo ... temperature , or critical point thermodynamics critical point . Observables, such as critical .... It was critical to the development of quantum chromodynamics . There is a remarkable infrared fixed point of the coupling constants that determine the masses of very heavy quarks. In the Standard Model ... other aspects of infrared fixed points to understand the anticipated spectrum of Higgs bosons in multi Higgs models. Another example of an infrared fixed point is the Banks Zaks fixed point in which the coupling constant of a Yang Mills theory evolves to a fixed large value. The beta function vanishes ...   more details



  1. Schauder fixed point theorem

    The Schauder fixed point theorem is an extension of the Brouwer fixed point theorem to topological vector space s, which may be of infinite dimension. It asserts that if math K math is a convex set convex subset of a topological vector space math V math and math T math is a continuous mapping of math K math into itself so that math T K math is contained in a compact set compact subset of math K math , then math T math has a fixed point mathematics fixed point . A consequence, called Schaefer s fixed point theorem , is particularly useful for proving existence of solutions to nonlinear partial differential equations . Schaefer s theorem is in fact a special case of the far reaching Leray Schauder theorem which was discovered earlier by Juliusz Schauder and Jean Leray . The statement is as follows. Let math T math be a continuous and compact mapping of a Banach space math X math into itself, such that the set math x in X x lambda T x mbox for some 0 leq lambda leq 1 math is bounded. Then math T math has a fixed point. History The theorem was conjectured and proven for special cases, such as Banach spaces, by Juliusz Schauder in 1930. His conjecture for the general case was published in the Scottish ... convex space. This version is known as the Schauder Tychonoff fixed point theorem . B. V. Singbal ... was finally proven by Robert Cauty in 2001. See also Banach fixed point theorem Kakutani fixed point theorem References J. Schauder, Der Fixpunktsatz in Funktionalr umen , Studia Math. 2 1930 , 171 ... on some fixed point theorems of functional analysis , Bombay 1962 Robert Cauty, Solution du probl me de point fixe de Schauder , Fund. Math. 170 2001 , 231 246 D. Gilbarg, N. Trudinger , Elliptic ... Analysis and its Applications, I Fixed Point Theorems External links planetmath reference title Schauder fixed point theorem id 4455 with attached proof for the Banach space case . Category Fixed ... du point fixe de Schauder it Teorema di punto fisso di Schaefer pl Twierdzenie Schaudera o punkcie ...   more details



  1. Borel fixed-point theorem

    In mathematics, the Borel fixed point theorem is a fixed point theorem in algebraic geometry generalizing the Lie Kolchin theorem . The result was proved by harvs txt authorlink Armand Borel first Armand last Borel year 1956 . Statement of the theorem Let G be a connected space connected , solvable group solvable algebraic group Group action Types of actions acting regularly on a non empty , complete algebraic variety complete algebraic variety V over an algebraically closed field k . Then G has a fixed point in V . References cite journal last Borel first Armand title Groupes lin aires alg briques journal Ann. Of Math. 2 year 1956 pages 20&ndash 82 volume 64 doi 10.2307 1969949 issue 1 publisher Annals of Mathematics jstor 1969949 mr 0093006 External links springer id b b017070 title Borel fixed point theorem author V.P. Platonov Category Algebraic geometry Category Fixed points Category Group actions Category Theorems in algebraic geometry ...   more details



  1. Fixed-point arithmetic

    Fixed point mathematics In computing , a fixed point number representation is a real data type for a number that has a fixed number of digits after and sometimes also before the radix point after the decimal point . in English decimal notation . Fixed point number representation can be compared to the more complicated and more computationally demanding floating point number representation. Fixed ... the executing central processing unit processor has no floating point unit FPU or if fixed point provides .... For example, the value 1.23 can be represented as 1230 in a fixed point data type with scaling factor ... may be used occasionally, e.g. a time value in hours may be represented as a fixed point type ... by the scaling factor and similarly for the minimum value. For example, consider a fixed point ... a number from a fixed point type with scaling factor R to another type with scaling factor S , the underlying ... fixed point type, it is sufficient to add or subtract the underlying integers, and keep their common ... integer type. If the numbers have different fixed point types, with different scaling factors, then one of them must be converted to the other before the sum. To multiply two fixed point numbers, it suffices ... method used, to result in a final scale factor of 1 100. To divide two fixed point numbers, one ... . If both operands and the desired result are represented in the same fixed point type, then the quotient ... November 2011 Binary vs. decimal The two most common fixed point types are decimal and binary. Decimal fixed point types have a scaling factor that is a power of ten, for binary fixed point types it is a power of two. Binary fixed point types are most commonly used, because the rescaling operations can be implemented as fast bit shift s. Binary fixed point numbers can represent fractional powers ..., one tenth 0.1 and one hundredth 0.01 can be represented only approximately by binary fixed point or binary floating point representations, while they can be represented exactly in decimal fixed ...   more details



  1. Banks?Zaks fixed point

    Refimprove date November 2010 Orphan date December 2009 In quantum chromodynamics and also N     1 superquantum chromodynamics with massless flavors, if the number of flavors, N sub f sub , is sufficiently small that is small enough to guarantee asymptotic freedom , the theory can flow to an interacting conformal Fixed point mathematics fixed point of the renormalization group . If the value of the coupling at that point is less than one, then the fixed point is called a Banks Zaks fixed point . More specifically, suppose that we find that the beta function of a theory up to two loops has the form math beta g b 0 g 3 b 1 g 5 , math where math b 0 math and math b 1 math are positive constants. Then, there exists a value math g g ast math such that math beta g ast 0 math math g ast 2 frac b 0 b 1 . math If we can arrange math b 0 math to be smaller than math b 1 math , then we have math g 2 ast 1 math . It follows that the theory in the IR is a conformal, weakly coupled theory with coupling math g ast math . For the case of QCD the number of flavors, math N f math , should lie just below math tfrac 11 2 N c math , where math N c math is the number of colors, in order for the Banks Zaks fixed point to appear. References T. Banks and A. Zaks, Nucl.Phys. B196, 189 1982 . DEFAULTSORT Banks Zaks Fixed Point Category Quantum field theory Category Quantum chromodynamics Category Fixed points Category Renormalization group Category Conformal field theory ...   more details



  1. Lefschetz fixed-point theorem

    In mathematics , the Lefschetz fixed point theorem is a formula that counts the fixed point mathematics fixed point s of a continuous function topology continuous mapping from a compact space compact topological ... 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 .... A simple version of the Lefschetz fixed point theorem states if math Lambda f neq 0 , math then f has at least one fixed point, i.e. there exists at least one x in X such that f x x . In fact, since ... that any map homotopic to f has a fixed point as well. Note however that the converse is not true in general sub f sub may be zero even if f has fixed points. A stronger form of the theorem, also known as the Lefschetz Hopf theorem , states that, if f has only finitely many fixed points, then math sum x in mathrm Fix f i f,x Lambda f, math where Fix f is the set of fixed points of f , and i f , x denotes the fixed point index index of the fixed point x . ref Cite book last1 Dold first1 Albrecht ... . Thus we have math Lambda mathrm id chi X . math Relation to the Brouwer fixed point theorem The Lefschetz fixed point theorem generalizes the Brouwer fixed point theorem , which states that every continuous ... at least one fixed point. This can be seen as follows D sup n sup is compact and triangulable, all ... context Lefschetz presented his fixed point theorem in Lefschetz 1926 . Lefschetz s focus was not on fixed points of mappings, but rather on what are now called coincidence point s of mappings ... is nonzero, then f and g have a coincidence point. He notes in his paper that letting X Y and letting g be the identity map gives a simpler result, which we now know as the fixed point theorem. Frobenius ... s over finite fields. See also Fixed point theorem s Lefschetz zeta function Notes references References ... maps a point with coordinates math x 1, ldots,x n math to the point with coordinates math x 1 q, ldots ...   more details



  1. Kleene fixed-point theorem

    Unreferenced date December 2011 In the mathematics mathematical areas of order theory order and lattice theory , the Kleene fixed point theorem , named after American mathematician Stephen Cole Kleene , states the following Let L be a complete partial order , and let f    L    L be a Scott continuity continuous and therefore monotone function monotone function mathematics function . Then the least fixed point of f is the supremum of the ascending Kleene chain of f. It is often attributed to Alfred Tarski , but the original statement of Tarski s fixed point theorem is about monotone functions on complete lattices. The ascending Kleene chain of f is the chain order theory chain math bot le f bot le f left f bot right le dots le f n bot le dots math obtained by iterated function iterating f on the least element of L . Expressed in a formula, the theorem states that math textrm lfp f sup left left f n bot mid n in mathbb N right right math where math textrm lfp math denotes the least fixed point. See also Knaster Tarski theorem Other fixed point theorem s Category Order theory Category Fixed points Category Theorems in discrete mathematics mathlogic stub fr Th or me du point fixe de Kleene zh ...   more details



  1. Caristi fixed-point theorem

    In mathematics , the Caristi fixed point theorem also known as the Caristi Kirk fixed point theorem generalizes the Banach fixed point theorem for maps of a complete space complete metric space into itself. Caristi s fixed point theorem is a variation of the Ekeland s variational principle variational principle of Ivar Ekeland Ekeland 1974, 1979 . Moreover, the conclusion of Caristi s theorem is equivalent to metric completeness, as proved by Weston 1977 . The original result is due to the mathematicians James Caristi and William Arthur Kirk . Statement of the theorem Let X ,  d be a complete metric space. Let T     X     X and f     X     0,  be a lower semicontinuous function from X into the non negative real numbers . Suppose that, for all points x in X , math d big x, T x big leq f x f big T x big . math Then T has a fixed point in X , i.e. a point x sub 0 sub such that T x sub 0 sub     x sub 0 sub . References cite journal last Caristi first James title Fixed point theorems for mappings satisfying inwardness conditions journal Transactions of the American Mathematical Society Trans. Amer. Math. Soc. volume 215 year 1976 pages 241&ndash 251 issn 0002 9947 doi 10.2307 1999724 jstor 1999724 cite journal doi 10.1016 0022 247X 74 90025 0 last Ekeland first Ivar title On the variational principle journal J. Math. Anal. Appl. volume 47 year 1974 pages 324&ndash 353 issn 0022 247x issue 2 cite journal last Ekeland first Ivar title Nonconvex minimization problems journal Bulletin of the American Mathematical Society Bull. Amer. Math. Soc. N.S. volume 1 year 1979 issue 3 pages 443&ndash 474 issn 0002 9904 doi 10.1090 S0273 0979 1979 14595 6 cite journal last Weston first J. D. title A characterization of metric completeness journal Proceedings of the American Mathematical Society Proc. Amer. Math. Soc. volume 64 year 1977 issue 1 pages 186&ndash 188 issn 0002 9939 doi 10.2307 2041008 jstor 2041008 Category Fixed points Category ...   more details



  1. Kakutani fixed-point theorem

    In mathematical analysis , the Kakutani fixed point theorem is a fixed point theorem for set valued function ... , compact set compact subset of a Euclidean space to have a fixed point mathematics fixed point , i.e. a point which is map mathematics map ped to a set containing it. The Kakutani fixed point theorem is a generalization of Brouwer fixed point theorem . The Brouwer fixed point theorem is a fundamental ... cite book last Border first Kim C. title Fixed Point Theorems with Applications to Economics and Game ... x  &isin   S. Then &phi has a fixed point mathematics fixed point . When we say that the graph ... set closed subset of X × Y in the product topology . Fixed point Let   X 2 sup X sup be a set valued function. Then a     X is a fixed point of if a     a . Example ... x satisfies all the assumptions of the theorem and must have fixed points. In the diagram, any point ... point, so in fact there is an infinity of fixed points in this particular case. For example, x     0.72 dashed line in blue is a fixed point since 0.72    1  &minus   0.72 ... no fixed point. Though it satisfies all other requirements of Kakutani s theorem, its value fails to be convex ... has a fixed point mathematics fixed point . This statement of Kakutani s theorem is completely equivalent ... Forbes Nash John Nash used the Kakutani fixed point theorem to prove a major result in game theory ... equilibrium of the game is defined as a fixed point of , i.e. a tuple of strategies where ... ensures that this fixed point exists. General equilibrium See also General equilibrium In general ... s theorem. If this can be done then has a fixed point according to the theorem. Given the way it was constructed, this fixed point must correspond to a price tuple which equates supply with demand ... on the closed interval nowiki 0,1 nowiki which satisfies the conditions of Kakutani s fixed point ... that q &isin &phi x &le x &le p &isin &phi x . Show that the limiting point is a fixed point. If p ...   more details



  1. Brouwer fixed-point theorem

    proof in 1912. Brouwer s fixed point theorem is a fixed point theorem in topology , named after Luitzen ... K of Euclidean space to itself. Among hundreds of fixed point theorems, ref E.g. F & V Bayart http ... fields such as game theory . In economics, Brouwer s fixed point theorem and its extension, the Kakutani fixed point theorem , play a central role in the proof of existence of general equilibrium in market ... from a Closed set closed Disk mathematics disk to itself has at least one fixed point. ref D. Violette ... has a fixed point. ref Page 15 of D. Leborgne Calcul diff rentiel et g om trie Puf 1982 ISBN ... to K itself has a fixed point. ref V. & F. Bayart http www.bibmath.net dico index.php3?action affiche&quoi ... form is better known under a different name Schauder fixed point theorem Every continuous function from a convex compact subset K of a Banach space to K itself has a fixed point. ref C. Minazzo ..., being a fixed point are invariant under homeomorphism s, the theorem is equivalent to forms in which ... to the right, it cannot have a fixed point. But it does have a fixed point for the closed interval 1,1 ... the consequence of the Brouwer fixed point theorem is that no matter how much you stir a cocktail ... archimed 19990921 ftext sujet5.html Archim de , Arte , 21 septembre 1999 ref The fixed point is not necessarily ... bit. The result is not intuitive, since the original fixed point may become mobile when another fixed ... interval. Saying that this function has a fixed point amounts to saying that its graph black ... is a fixed point. Brouwer is said to have expressed this as follows Instead of examining a surface ..., Brouwer s fixed point theorem is equivalent to the intermediate value theorem . History The Brouwer fixed point theorem was one of the early achievements of algebraic topology , and is the basis of more general fixed point theorem s which are important in functional analysis . The case n 3 ... disk shaped area, where it guarantees the existence of a fixed point. To understand the prehistory ...   more details



  1. Banach fixed-point theorem

    In mathematics, the Banach fixed point theorem also known as the contraction mapping theorem or contraction ... and uniqueness of fixed point mathematics fixed points of certain self maps of metric spaces, and provides ... d x,y math for all x , y in X . Then the map T admits one and only one fixed point x sup sup in X this means T x sup sup x sup sup . Furthermore, this fixed point can be found as follows start with an arbitrary ... to ensure the existence of a fixed point, as is shown by the map T nowiki 1,&infin &rarr 1,&infin nowiki with T x     x     1 x , which lacks a fixed point. However, if the metric space ... of a fixed point, that can be easily found as a minimizer of d x ,  T x indeed, a minimizer exists by compactness, and has to be a fixed point of  T . It then easily follows that the fixed point ... x n math . We make two claims 1 math x , math is a Fixed point mathematics fixed point of math T , math . That is, math T x x , math 2 math x , math is the only fixed point of math T , math in math ... equation is expressed as a fixed point of a suitable integral operator which transforms continuous functions into continuous functions. The Banach fixed point theorem is then used to show that this integral operator has a unique fixed point. One consequence of the Banach fixed point ... iterated function iterate &fnof sup n sup has a unique fixed point. Let q be a real number, 0 ... a unique fixed point. Assume that for all math x math and math y math in math X math , math sum n d T n x ,T n y infty. math Then T has a unique fixed point. However, in most applications the existence and unicity of a fixed point can be shown directly with the standard Banach fixed point theorem ... by Bessaga strongly suggests to look for such a metric. See also the article on fixed point theorems ... 1922 , 133 181. http matwbn.icm.edu.pl ksiazki or or2 or215.pdf Vasile I. Istratescu, Fixed Point ... Granas and James Dugundji, Fixed Point Theory 2003 Springer Verlag, New York, ISBN 0 387 00173 5. cite ...   more details



  1. Fixed-point theorems in infinite-dimensional spaces

    In mathematics , a number of fixed point mathematics fixed point theorems in infinite dimensional spaces generalise the Brouwer fixed point theorem . They have applications, for example, to the proof of existence theorem s for partial differential equation s. The first result in the field was the Schauder fixed point theorem , proved in 1930 by Juliusz Schauder . Quite a number of further results followed. One way in which fixed point theorems of this kind have had a larger influence on mathematics as a whole has been that one approach is to try to carry over methods of algebraic topology , first proved for finite simplicial complex es, to spaces of infinite dimension. For example, the research of Jean Leray who founded sheaf theory came out of efforts to extend Schauder s work. The Schauder fixed point theorem states, in one version, that if C is a nonempty Closed set closed Convex set convex subset of a Banach space V and f is a continuous function continuous map from C to C whose image is compact set compact , then f has a fixed point. The Tikhonov Tychonoff fixed point theorem is applied ... convex set X in V , any continuous function &fnof X X , has a fixed point. Other results are the Shizuo Kakutani Kakutani and Markov fixed point theorems, as well as the Ryll Nardzewski fixed point theorem 1967 . Kakutani fixed point theorem Kakutani s fixed point theorem states that Every correspondence ... nonempty images has a fixed point. See also Topological degree theory References Vasile I. Istratescu, Fixed Point Theory, An Introduction , D.Reidel, Holland 1981 . ISBN 90 277 1224 7. Andrzej Granas and James Dugundji, Fixed Point Theory 2003 Springer Verlag, New York, ISBN 0 387 00173 5. William A. Kirk and Brailey Sims, Handbook of Metric Fixed Point Theory 2001 , Kluwer Academic, London ... PlanetMath article on the Tychonoff Fixed Point Theorem Category Fixed points Category Functional ... Th or me du point fixe de Schauder ...   more details



  1. Ryll-Nardzewski fixed-point theorem

    In functional analysis , a branch of mathematics, the Ryll Nardzewski fixed point theorem states that if math E math is a normed vector space and math K math is a nonempty Convex set convex subset of math E math which is compact space compact under the weak topology , then every group mathematics group or equivalently every semigroup of affine map affine isometry isometries of math K math has at least one fixed point. Here, a fixed point of a set of maps is a point that is Fixed point mathematics fixed by each map in the set. This theorem was announced by Czes aw Ryll Nardzewski . ref cite journal first C. last Ryll Nardzewski title Generalized random ergodic theorems and weakly almost periodic functions journal Bull. Acad. Polon. Sci. S r. Sci. Math. Astronom. Phys. volume 10 year 1962 pages 271 275 ref Later Namioka and Asplund ref cite journal doi 10.1090 S0002 9904 1967 11779 8 first I. last Namioka coauthors Asplund, E. title A geometric proof of Ryll Nardzewski s fixed point theorem journal Bull. Amer. Math. Soc. volume 73 issue 3 year 1967 pages 443 445 ref gave a proof based on a different approach. Ryll Nardzewski himself gave a complete proof in the original spirit. ref cite journal first C. last Ryll Nardzewski title On fixed points of semi groups of endomorphisms of linear spaces journal Proc. 5 th Berkeley Symp. Probab. Math. Stat volume 2 1 publisher Univ. California Press year 1967 pages 55 61 ref Applications The Ryll Nardzewski theorem yields the existence of a Haar measure on compact groups. ref cite book first N. last Bourbaki title Espaces vectoriels topologiques ... in Functional Analysis , about the treatment of weak vs. weak topology See also Fixed point theorem s Fixed point theorems in infinite dimensional spaces References references Andrzej Granas and James Dugundji, Fixed Point Theory 2003 Springer Verlag, New York, ISBN 0 387 00173 5. http www.math.harvard.edu lurie 261ynotes lecture26.pdf A proof written by J. Lurie Category Fixed points Category ...   more details



  1. Atiyah?Bott fixed-point theorem

    In mathematics , the Atiyah Bott fixed point theorem , proven by Michael Atiyah and Raoul Bott in the 1960s, is a general form of the Lefschetz fixed point theorem for smooth manifold s M , which uses an elliptic complex on M . This is a system of elliptic differential operator s on vector bundle s, generalizing the de Rham complex constructed from smooth differential form s which appears in the original Lefschetz fixed point theorem. Formulation The idea is to find the correct replacement for the Lefschetz number , which in the classical result is an integer counting the correct contribution of a Fixed point mathematics fixed point of a smooth mapping f M M . Intuitively, the fixed points are the points of intersection of the graph of a function graph of f with the diagonal graph of the identity ... at fixed points of f . Counting codimension s in M × M , a transversality assumption for the graph of f and the diagonal should ensure that the fixed point set is zero dimensional. Assuming ... sub j , sub at a fixed point x of f , and x is the determinant of the endomorphism I &minus Df at x ... yields the original Lefschetz fixed point formula. A famous application of the Atiyah Bott theorem ..., as is suggested by the alternate name Woods Hole fixed point theorem http www.whoi.edu mpcweb meetings atiyah bott 35.html that was used in the past referring properly to the case of isolated fixed ... between fixed point theorems and automorphic form s. Goro Shimura Shimura played an important ... references References M. F. Atiyah R. Bott A Lefschetz Fixed Point Formula for Elliptic Differential ... number of an endomorphism of an elliptic complex. M. F. Atiyah R. Bott A Lefschetz Fixed Point Formula ... 3E2.0.CO 3B2 N A Lefschetz Fixed Point Formula for Elliptic Complexes I http links.jstor.org sici ... Bott Fixed Point Theorem Category Differential topology Category Fixed points Category Theorems ... summation is over the fixed points x , and the inner summation over the index j in the elliptic ...   more details



  1. Fixed-point lemma for normal functions

    The fixed point lemma for normal functions is a basic result in axiomatic set theory stating that any normal function has arbitrarily large fixed point mathematics fixed point s Levy 1979 p. 117 . It was first proved by Oswald Veblen in 1908. Background and formal statement A normal function is a proper class class function f from the class Ord of ordinal numbers to itself such that f is strictly increasing f &alpha f &beta whenever &alpha &beta . f is continuous for every limit ordinal &lambda i.e. &lambda is neither zero nor a successor , f &lambda sup f &alpha &alpha &lambda . It can be shown that if f is normal then f commutes with supremum suprema for any nonempty set A of ordinals, f sup A sup f &alpha &alpha A . Indeed, if sup A is a successor ordinal then sup A is an element of A and the equality follows from the increasing property of f . If sup A is a limit ordinal then the equality follows from the continuous property of f . A fixed point of a normal function is an ordinal &beta such that f &beta &beta . The fixed point lemma states that the class of fixed points of any normal function is nonempty and in fact is unbounded given any ordinal , there exists an ordinal such that and f . The continuity of the normal function implies the class of fixed points is closed the supremum of any subset of the class of fixed points is again a fixed point . Thus the fixed point lemma is equivalent to the statement that the fixed points of a normal function form a club set closed and unbounded class. Proof The first step of the proof is to verify that f for all ordinals and that f commutes with suprema. Given these results, inductively define an increasing sequence < sub n sub > n < by setting sub 0 sub , and sub n 1 sub f sub n sub for n . Let sup sub n sub n &omega , so . Moreover, because f commutes with suprema, f f sup sub n ... Mathematical Society jstor 1988605 issn 0002 9947 Category Ordinal numbers Category Fixed points ...   more details




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