existence of nonexpansive retractions for amenable semigroups of nonexpansive mappings and nonlinear ergodic theorems in banach spaces in this paper we study nonlinear ergodic properties for an amenable semigroup of nonexpansive mappings in a banach space we prove that ifsis an amenable semigroup and s tt t s is a nonexpansive semigroup on a closed convex subsetcin a uniformly convex banach spaceesuch that the setf s of common fixed points of s is nonempty then there exists a nonexpansive retractionpfromcontof s such thatptt ttp pfor eacht sandpx co ttx t s for eachx c in this case there exists a net a of finite averages of s such that for eacht sand for each bounded subsetbofc lim a ttx a x 0 and lim tta x a x 0 uniformly forx b also if the norm ofeis frechet differentiable then for eachx c pxis the unique common fixed point in s s co ttsx t s furthermore if is an asymptotically invariant net of means then for eachx c t x converges weakly topx

nonlinear ergodic theorems of nonexpansive type mappings abstract let s be a semitopological semigroup let c be a closed convex subset of a uniformly convex banach space e whose norm is frechet differentiable and t t t s be a continuous representation of s as almost asymptotically nonexpansive type mapping of c into c such that the common fixed point set f of in c is nonempty in this paper we prove that if s is right reversible then for each x c the closed convex set s s c o t t x t s f consists of at most one point we also prove that if s is reversible then the intersection s s c o t t x t s f is nonempty for each x c if and only if there exists a nonexpansive retraction p of c onto f such that p t t t t p p for all t s and px is in the closed convex hull of t t x t s for each x c

ergodic retractions for amenable semigroups in banach spaces with normal structure abstract in this paper we discuss the existence of nonexpansive retraction onto the set of common fixed points assume that t s s s is an amenable semigroup of nonexpansive mappings on a closed convex subset c in a reflexive banach space e such that the set f of common fixed points of is nonempty among other things it is shown that if either c has normal structure or the t s s are affine then there exists a nonexpansive retraction p from c onto f such that p t t t t p p for each t s and every closed convex invariant subset of c is also p invariant in the case that the mappings are affine p is also affine and p x c o t t x t s for each x c and it is unique regarding the latter property our results extend corresponding results of t suzuki some remarks on the set of common fixed points of one parameter semigroups of nonexpansive mappings in banach spaces with the opial property nonlinear anal 58 2004 441 458 and r e bruck a common fixed point theorem for a commuting family of nonexpansive mappings pacific j math 53 1974 59 71

knotted fields and explicit fibrations for lemniscate knots we give an explicit construction of complex maps whose nodal lines have the form of lemniscate knots we review the properties of lemniscate knots defined as closures of braids where all strands follow the same transverse 1 l lissajous figure and are therefore a subfamily of spiral knots generalizing the torus knots we then prove that such maps exist and are in fact fibrations with appropriate choices of parameters we describe how this may be useful in physics for creating knotted fields in quantum mechanics optics and generalizing to rational maps with application to the skyrme faddeev model we also prove how this construction extends to maps with weakly isolated singularities

helicity and the calugareanu invariant the helicity of a localized solenoidal vector field i e the integrated scalar product of the field and its vector potential is known to be a conserved quantity under frozen field distortion of the ambient medium in this paper we present a number of results concerning the helicity of linked and knotted flux tubes particularly as regards the topological interpretation of helicity in terms of the gauss linking number and its limiting form the c lug reanu invariant the helicity of a single knotted flux tube is shown to be intimately related to the c lug reanu invariant and a new and direct derivation of this topological invariant from the invariance of helicity is given helicity is decomposed into writhe and twist contributions the writhe contribution involving the gauss integral for definition see equation 4 8 which admits interpretation in terms of the sum of signed crossings of the knot averaged over all projections part of the twist contribution is shown to be associated with the torsion of the knot and part with what may be described as intrinsic twist of the field lines in the flux tube around the knot see equations 5 13 and 5 15 the generic behaviour associated with the deformation of the knot through a configuration with points of inflexion points at which the curvature vanishes is analysed and the role of the twist parameter is discussed the derivation of the c lug reanu invariant from first principles of fluid mechanics provides a good demonstration of the relevance of fluid dynamical techniques to topological problems

primitive vassiliev invariants and factorization in chern simons perturbation theory the general structure of the perturbative expansion of the vacuum expectation value of a wilson line operator in chern simons gauge field theory is analyzed the expansion is organized according to the independent group structures that appear at each order it is shown that the analysis is greatly simplified if the group factors are chosen in a certain way that we call canonical this enables us to show that the logarithm of a polynomial knot invariant can be written in terms of primitive vassiliev invariants only

a d step predictor in lattice and ladder form we use the orthogonalizing property of the two multiplier linear prediction lattice filter to construct a d step ahead predictor in lattice form the predictor generates d step forward and backward residuals in a recursive way and possesses most of the interesting properties of the basic one step prediction lattice filter an exact solution is presented first assuming a stationary observation process using orthogonal projections in hilbert space two adaptive implementations are also proposed for the case where the statistics of the signal process are unknown or time varying a gradient method and a recursive least squares scheme finally we show how to construct an adaptive d step ahead predictor by adding a ladder part to the d step lattice structure

the wilson loop observables of chern simons theory on ℝ3 in axial gauge we define the wilson loop observables wlos for pure chern simons models with base manifold m 3 rigorously as infinite dimensional oscillatory integrals by exploiting an axial gauge fixing and applying certain regularization techniques like loop smearing and framing the values of the wlos can be computed explicitly if the structure group g of the model considered is abelian one obtains well known linking number expressions for the wlos if g is non abelian one obtains expressions which are similar but not identical to the state sum representations for the homfly and kauffman polynomials by jones and turaev

enhancement of sinusoids in colored noise and the whitening performance of exact least squares predictors the extraction of sinusoids in white noise using least squares predictors has attracted a lot of attention in the past mainly in the context of the adaptive line enhancer ale however very few results exist for the practical colored noise case or for the whitening performance of the predictors we use a matrix formulation to derive the optimal least squares coefficients and frequency response of the d step predictor or ale for sinusoids real or complex in additive colored noise several cases are considered and in particular new formulas for the amplitude gain are obtained in low pass background noise the amplitude gain of the sinusoids becomes essentially a monotonically increasing function of their frequency and a decreasing function for high pass noise for the whitening application signal to noise ratio snr bounds of the output are derived when the input is a white signal plus a sinusoidal interference we also give a state space model and stochastic interpretations of our analysis of the d step predictor providing connections to other related areas to enable filtering of nonstationary complex inputs as well as multichannel and multiexperiment data a complex vector version of the ladder algorithm is presented that can be used to implement the ale noise cancelling and noise inversion for narrow band interference rejection

existence of ergodic retractions for semigroups in banach spaces abstract in this work we prove among other results that if s is a right amenable semigroup and t s s s is a quasi nonexpansive semigroup on a closed convex subset c in a strictly convex reflexive banach space e such that the set f of common fixed points of is nonempty then there exists a quasi nonexpansive retraction p from c onto f such that p t t t t p p for each t s and every closed convex invariant subset of c is also p invariant moreover if the mappings are also affine then t g rode an ergodic theorem for semigroups of nonexpansive mappings in a hilbert space j math anal appl 85 1982 172 178 12 is a quasi contractive affine retraction from c onto f such that t t t t t t t for each t s and t x c o t t x t s for each x c and if r is an arbitrary retraction from c onto f such that r x c o t t x t s for each x c then r t it is shown that if the t t s are f quasi contractive then the results hold without the strict convexity condition on e

an ergodic theorem for semigroups of nonexpansive mappings in a hilbert space abstract j b baillon c r acad sci paris ser a 280 1975 1511 1514 proved an ergodic theorem for a single nonexpansive mapping in a hilbert space which is a nonlinear version of von neumann s mean ergodic theorem in this paper we study the ergodic behavior of a semigroup of nonexpansive mappings we try to find a sequence of means on the semigroup generalizing the cesaro means on n such that the corresponding sequence of nonexpansive mappings converges to a projection onto the set of common fixed points our method of proof is an appropriate modification of a pazy s proof israel j math 26 1977 197 204 of baillon s theorem

integration of singular braid invariants and graph cohomology we prove necessary and sufficient conditions for an arbitrary in variant of braids with m double points to be the mth derivative of a braid invariant we show that the primary obstruction to integration i5 the only obstruction this gives a slight generalization of the existence theorem for vassiliev invariants of braids we give a direct proof by induction on m which works for invariants with values in any abelian group we snd that to prove our theorem we must show that every relation among four term relations satisses a certain geometric condition to snd the relations among relations we show that h1 of a variant of kontsevich s graph complex vanishes we discuss related open questions for invariants of links and other things

the jones polynomial for fluid knots from helicity in this paper we prove that under ideal conditions the helicity of fluid knots such as vortex filaments or magnetic flux tubes provides a fundamentally new topological means by which we may associate a topological invariant the jones polynomial that is much stronger than prior interpretations in terms of gauss linking numbers our proof is based on an extension of the kauffman bracket polynomial for unoriented knots explicit calculations of the jones polynomial for the left and right handed trefoil knots and for the whitehead link via the figure of eight knot are presented for illustration this novel approach establishes a topological foundation of classical field theory in general and of mathematical fluid dynamics in particular by opening up new directions of work both in theory and applications

nonexpansive retractions and fixed point properties abstract among other results we prove that if a dual banach space e has the weak fixed point property then e has the weak fixed point property for commuting separable semitopological semigroups of nonexpansive mappings and study the existence of ergodic retractions for right amenable semigroups and commuting families of nonexpansive mappings this answers an open problem of lau and mah 1976 12 for commuting separable semi topological semigroups

an exact recursion for the composite nearest neighbor degeneracy for a 2 n lattice space a set theoretic argument is utilized to develop a recursion relation that yields exactly the composite nearest neighbor degeneracy for simple indistinguishable particles distributed on a 2 n lattice space the associated generating functions as well as the expectation of the resulting statistics are also treated

from simplicial chern simons theory to the shadow invariant ii this is the second of a series of papers in which we introduce and study a rigorous simplicial realization of the non abelian chern simons path integral for manifolds m of the form m s1 and arbitrary simply connected compact structure groups g more precisely we introduce for general links l in m a rigorous simplicial version wlorig l of the corresponding wilson loop observable wlo l in the so called torus gauge by blau and thompson nucl phys b 408 2 345 390 1993 for a simple class of links l we then evaluate wlorig l explicitly in a non perturbative way finding agreement with turaev s shadow invariant l

on the melvin morton rozansky conjecture we prove a conjecture stated by melvin and morton and elucidated further by rozansky saying that the alexander conway polynomial of a knot can be read from some of the coefficients of the jones polynomials of cables of that knot i e coefficients of the colored jones polynomial we first reduce the problem to the level of weight systems using a general principle which may be of some independent interest and which sometimes allows to deduce equality of vassiliev invariants from the equality of their weight systems we then prove the conjecture combinatorially on the level of weight systems finally we prove a generalization of the melvin morton rozansky mmr conjecture to knot invariants coming from arbitrary semi simple lie algebras as side benefits we discuss a relation between the conway polynomial and immanants and a curious formula for the weight system of the colored jones polynomial

d step lattice ladder predictor in terms of one step lattice ladder predictors in this note it is shown that a d step lattice ladder predictor may be expressed in terms of one step lattice ladder predictors

nonlinear ergodic theorems for general curves defined on general semigroups let g be a algebra semigroup and let h be a real hilbert space let f g be a set of all mappings from g into h in this paper we first try to provide a nonlinear ergodic retraction theorem for the mappings from g to h moreover we introduce the notion of asymptotically nonexpansive type curves and give the ergodic convergence theorems for this kind of curves using the main results we can also prove the ergodic convergence theorem for the almost orbits of semitopological semigroups of non lipschitzian mappings without convexity

a mean ergodic theorem for nonlinear semigroups which are asymptotically nonexpansive in the intermediate sense abstract let x be a uniformly convex banach space such that its dual x has the kadec klee property c a bounded closed convex subset of x and t t t t 0 a one parameter nonlinear semigroup which is asymptotically nonexpansive in the intermediate sense we show that every continuous almost orbit of t is weakly almost convergent to a common fixed point of t a discrete version of our main result is also included

integral invariants of 3 manifolds ii this note is a sequel to our earlier paper of the same title 4 and describes invariants of rational homology 3 spheres associated to acyclic orthogonal local systems our work is in the spirit of the axelrod singer papers 1 generalizes some of their results and furnishes a new setting for the purely topological implications of their work

new perspectives on self linking abstract we initiate the study of classical knots through the homotopy class of the nth evaluation map of the knot which is the induced map on the compactified n point configuration space sending a knot to its nth evaluation map realizes the space of knots as a subspace of what we call the nth mapping space model for knots we compute the homotopy types of the first three mapping space models showing that the third model gives rise to an integer valued invariant we realize this invariant in two ways in terms of collinearities of three or four points on the knot and give some explicit computations we show this invariant coincides with the second coefficient of the conway polynomial thus giving a new geometric definition of the simplest finite type invariant finally using this geometric definition we give some new applications of this invariant relating to quadrisecants in the knot and to complexity of polygonal and polynomial realizations of a knot

on the existence of fixed points and ergodic retractions for lipschitzian semigroups in hilbert spaces and especially nonexpansive in the case of k 1 let s be a semitopological semigroup and let s c s e s be a continuous representation of s as lipschitzian mappings of c into itself which is said to be a lipschitzian semigroup on c a lipschitzian semigroup s t s e s with lipschitz constants k s e s is called a uniformly k lipschitzian semigroup and a nonexpansive semigroup when k k and k 1 for all s e s respectively goebel and kirk 7 first called a mapping t of c into itself a uniformly k lipschitzian mapping on c in the case when s t n e n is a lipschitzian semigroup with k k for all n e n lifschitz 12 obtained a fixed point theorem for uniformly k lipschitzian mappings on c with k 5 and this result was extended to uniformly k lipschitzian semigroups which are left reversible by downing and ray 6 lau l l showed that a nonexpansive semigroup has a common fixed point if there exists a left invariant mean on the space rug s of bounded right uniformly continuous functions on the semigroup furthermore ishihara and takahashi 9 extended lau s result to uniformly k lipschitzian semigroups and also gave a simpler proof of downing and ray s theorem on the other hand the first nonlinear ergodic theorem for nonexpansive mappings was established by baillon 11 let t be a nonexpansive mapping of c into itself and suppose that the set f t of fixed points of t is nonempty the the cesaro means

how tangled is a tangle abstract new measures of algebraic geometric and topological complexity are introduced and tested to quantify morphological aspects of a generic tangle of filaments the tangle is produced by standard numerical simulation of superfluid helium turbulence which we use as a benchmark for numerical investigation of complex systems we find that the measures used based on crossing number information are good indicators of generic behaviour and detect accurately a tangle s complexity direct measurements of kinetic helicity are found to be in agreement with the other complexity based measures proving that helicity is also a good indicator of structural complexity we find that complexity based measure growth rates are consistently similar to one another the growth rate of kinetic helicity is found to be twice that of energy

vassiliev knot invariants and chern simons perturbation theory to all orders at any order the perturbative expansion of the expectation values of wilson lines in chern simons theory gives certain integral expressions we show that they all lead to knot invariants moreover these are finite type invariants whose order coincides with the order in the perturbative expansion together they combine to give a universal vassiliev invariant

surface holonomy for non abelian 2 bundles via double groupoids in the context of non abelian gerbes we define a cubical version of categorical group 2 bundles with connection over a smooth manifold we address their two dimensional parallel transport study its properties and construct non abelian wilson surface functionals

semigroup of nonexpansive mappings on a hilbert space soit s un semi groupe semi topologique soit h un espace de hilbert et s t s s s une representation continue de s comme applications non dilatation sur un sous ensemble convexe clos c de h dans c soit x c on etudie la convergence de t s x s s vers un point fixe commun z de s dans c

chern simons perturbation theory ii we study the perturbation theory for three dimensional chern simons quantum field theory on a general compact three manifold without boundary we show that after a simple change of variables the action obtained by brs gauge fixing in the lorentz gauge has a superspace formulation the basic properties of the propagator and the feynman rules are written in a precise manner in the language of differential forms using the explicit description of the propagator singularities we prove that the theory is finite finally the anomalous metric dependence of the 2 loop partition function on the riemannian metric which was introduced to define the gauge fixing can be cancelled by a local counterterm as in the 1 loop case in fact the counterterm is equal to the chern simons action of the metric connection normalized precisely as one would expect based on the framing dependence of witten s exact solution invited talk at the xxth conference on differential geometric methods in physics new york june 1991

knot polynomials and vassiliev s invariants a fundamental relationship is established between jones knot invariants and vassiliev s knot invariants since vassiliev s knot invariants have a firm grounding in classical topology one obtains as a result a first step in understanding the jones polynomial by topological methods

kontsevich integral for vassiliev invariants from chern simons perturbation theory in the light cone gauge we analyze the structure of the perturbative series expansion of chern simons gauge theory in the light cone gauge after introducing a regularization prescription that entails the consideration of framed knots we present the general form of the vacuum expectation value of a wilson loop the resulting expression turns out to give the same framing dependence as the one obtained using nonperturbative methods and perturbative methods in covariant gauges it also contains the kontsevich integral for vassiliev invariants of framed knots

inequalities in banach spaces with applications well known that many problems in a hilbert space h are solved by applying the following identity ilax 1 a yi12 a412 1 llyl12 a 1 mx yl12 1 1 for all x y in h and 0 5 i 5 1 therefore one natural method to solve problems in a banach space xis to establish equalities and usually inequalities in xanalogous to 1 1 this is in fact shown by some authors recent work see for example reich l poffald and reich 2 lim 3 lim et af 4 prus and smarzewski 5 smarzewski 6 and reich 21 the purpose of this paper is to continue carrying out investigations in this line more precisely in section 3 we establish some inequalities analogous to 1 l in either uniformly convex or uniformly smooth banach spaces as corollaries we obtain new characteristics of p uniform convexity and q uniform smoothness of a banach space x via the functionals i lip and ii llq respectively we give in section 4 applications of the inequalities obtained in section 3 to fixed points of uniformly lipschitzian mappings and to best approximation theory

asymptotic behavior of uniformly asymptotically almost nonexpansive curves in a hilbert space abstract we extend the notion of uniformly asymptotically almost nonexpansive curves which was first introduced in rouhani and kim j nonlinear convex anal 4 2003 175 for a commutative semigroup to a general semitopological semigroup study the asymptotic behavior of such curves in a hilbert space and apply our results to study the asymptotic behavior for almost orbits of semigroups of asymptotically nonexpansive type mappings over a general semitopological semigroup in a hilbert space since in this case the semigroup is not directed the proofs in rouhani and kim j nonlinear convex anal 4 2003 175 do not extend to this case and new methods have to be used for the proofs in addition to rouhani j math anal appl 147 1990 465 j math anal appl 151 1990 226 nonlinear anal 44 2001 627 nonlinear anal 51 2002 735 and rouhani and kim j nonlinear convex anal 4 2003 175 our results extend and unify many previously known results j math anal appl 127 1987 206 proc amer math soc 104 1988 431 nonlinear anal 29 1997 539 topol methods nonlinear anal 5 1995 305 j math anal appl 105 1985 514 nonlinear anal 26 1996 1411 pacific j math 126 1987 277 j math anal appl 206 1997 451 chinese ann math 6 1996 729 nonlinear anal 14 1990 69 j math anal appl 85 1982 172 proc amer math soc 81 1981 253 proc amer math soc 97 1986 55 canad j math 44 1992 880 j math anal appl 142 1989 242 proc amer math soc 117 1993 385

on the self linking of knots this note describes a subcomplex f of the de rham complex of parametrized knot space which is combinatorial over a number of universal anomaly integrals the self linking integrals of guadaguini martellini and mintchev perturbative aspects of chern simons field theory phys lett b 227 111 1989 and bar natan perturbative aspects of the chern simons topological quantum field theory ph d thesis princeton university 1991 also on the vassiliev knot invariants to appear in topology are seen to represent the first nontrivial element in h0 f occurring at level 4 and are anomaly free however already at the next level an anomalous term is possible

chern simons theory on r3 in axial gauge a rigorous approach we study pure chern simons models on m r3 using a functional integral quantization approach which is based on axial gauge fixing it is well known see e g comm math phys 126 1989 167 comm math phys 186 1997 563 that in axial gauge the chern simons action function is quadratic and that the faddeev popov determinant of this gauge fixing procedure is a constant function this means that the wilson loop observables wlos of the model considered can be obtained heuristically by integrating certain quantities against a functional measure of gaussian type we demonstrate that although these heuristic integral expressions look rather singular it is possible to give a rigorous meaning to them by combining constructions from white noise analysis with certain regularization techniques like loop smearing and framing for the special case g u 1 g being the group of the model we carry out the details of this approach which is also applicable to the case of non abelian g and find well known linking number expressions for the wlos

the retractions onto the common fixed points of some families and semigroups of mappings abstract assume that c is a closed convex subset of a reflexive banach space e and t i i i is a family of self mappings on c of type such that f the common fixed point set of is nonempty from our results in this paper it can be derived that a if i i f t i is contained in a 3 dimensional subspace of e then f is a nonexpansive retract of c b if is commutative there exists a retraction r of type from c onto f such that r t i t i r r i and every closed convex invariant subset of c is r invariant the same result holds for a non commutative right amenable semigroup under some additional assumptions moreover the existence of a t i ergodic retraction r of type from c x i l e x i c i i onto f in l e for the family is discussed we also apply some of our results to find ergodic retractions for nonexpansive affine mappings