multilinear schur multipliers and schatten properties of operator taylor remainders abstract we establish sharp conditions on scalar functions and perturbations that guarantee schatten summability of n th order operator taylor remainders in the special case of dimension one our estimates of these remainders deliver well known classical estimates of scalar taylor remainders we prove that if a scalar function f is in the set c n and a perturbation is in the p th schatten class s p p n then the respective n th order operator taylor remainder is an element of s p n and has an estimate like the one in 16 we construct examples of f c n and perturbations in s n such that the n th order taylor remainder of the respective operator function is not in s 1 our construction relies in particular on novel dimension dependent estimates for schatten norms of multilinear schur multipliers from below that are of interest in their own right our results apply to both self adjoint and unitary operators

trace formulas for relative schatten class perturbations abstract we derive trace formulas for a pair of self adjoint operators h v and h under the assumption that h i 1 v is in a schatten class this extends the trace formulas of 8 where v alone is assumed to be in a schatten class our trace formulas apply in particular in the setting of differential operators and are based on taylor like approximations of operator functions this significantly improves non taylor based trace formulas of 10

functions of noncommuting self adjoint operators under perturbation and estimates of triple operator integrals abstract we define functions of noncommuting self adjoint operators with the help of double operator integrals we are studying the problem to find conditions on a function f on r 2 for which the map a b f a b is lipschitz in the operator norm and in schatten von neumann norms s p it turns out that for functions f in the besov class b 1 1 r 2 the above map is lipschitz in the s p norm for p 1 2 however it is not lipschitz in the operator norm nor in the s p norm for p 2 the main tool is triple operator integrals to obtain the results we introduce new haagerup like tensor products of l spaces and obtain schatten von neumann norm estimates of triple operator integrals we also obtain similar results for functions of noncommuting unitary operators

10e9 solution to the elliptic painleve equation a function formalism for sakai s elliptic painleve equation is presented this establishes the equivalence between the two formulations by sakai and by ohta ramani grammaticos we also give a simple geometric description of the elliptic painleve equation as a non autonomous deformation of the addition formula on elliptic curves by using these formulations we construct a particular solution of the elliptic painleve equation expressed in terms of the elliptic hypergeometric function 10e9

functions of almost commuting operators and an extension of the helton howe trace formula abstract let a and b be almost commuting i e the commutator ab ba belongs to trace class self adjoint operators we construct a functional calculus a b for functions in the besov class b 1 1 r 2 this functional calculus is linear the operators a b and a b almost commute for b 1 1 r 2 and a b u a v b whenever s t u s v t we extend the helton howe trace formula for arbitrary functions in b 1 1 r 2 the main tool is triple operator integrals with integrands in haagerup like tensor products of l spaces

linear and nonlinear associative memories for parameter estimation this paper proposes the use of associative memories for obtaining preliminary parameter estimates for nonlinear systems for each parameter vector r in a selected training set the system equations are used to determine a vector s of system outputs an associative memory matrix m is then constructed which optimally in the least squares sense associates each system output vector s with its corresponding parameter vector i given any observed system output vector s an estimate i for the system parameters is obtained by setting i ms numerical experiments are reported which indicate the effectiveness of this approach especially for the nonlinear associative memory case in which the training vectors s include not only the system output levels but also products of these levels training with noisy output vectors is shown to improve the accuracy of the parameter estimates when the

multidimensional operator multipliers we introduce multidimensional schur multipliers and characterise them generalising well known results by grothendieck and peller we define a multidimensional version of the two dimensional operator multipliers studied recently by kissin and shulman the multidimensional operator multipliers are defined as elements of the minimal tensor product of several c algebras satisfying certain boundedness conditions in the case of commutative c algebras the multidimensional operator multipliers reduce to continuous multidimensional schur multipliers we show that the multipliers with respect to some given representations of the corresponding c algebras do not change if the representations are replaced by approximately equivalent ones we establish a non commutative and multidimensional version of the characterisations by grothendieck and peller which shows that universal operator multipliers can be obtained as certain weak limits of elements of the algebraic tensor product of the corresponding c algebras

resolution of peller s problem concerning koplienko neidhardt trace formulae a formula for the norm of a bilinear schur multiplier acting from the cartesian product s 2 s 2 of two copies of the hilbert schmidt classes into the trace class s 1 is established in terms of linear schur multipliers acting on the space s 1 of all compact operators using this formula we resolve peller s problem on koplienko neidhardt trace formulae namely we prove that there exist a twice continuously differentiable function f with a bounded second derivative a self adjoint unbounded operator a and a self adjoint operator b 2 s 2 such that f a b f a d dt f a tb t 0 2 s 1

higher order spectral shift abstract we construct higher order spectral shift functions extending the perturbation theory results of m g krein m g krein on a trace formula in perturbation theory mat sb 33 1953 597 626 in russian and l s koplienko l s koplienko trace formula for perturbations of nonnuclear type sibirsk mat zh 25 1984 62 71 in russian translation in trace formula for nontrace class perturbations siberian math j 25 1984 735 743 on representations for the remainders of the first and second order taylor type approximations of operator functions the higher order spectral shift functions represent the remainders of higher order taylor type approximations they can be expressed recursively via the lower order in particular krein s and koplienko s ones we also obtain higher order spectral averaging formulas generalizing the birman solomyak spectral averaging formula the results are obtained in the semi finite von neumann algebra setting with the perturbation taken in the hilbert schmidt class of the algebra

estimates and trace formulas for unitary and resolvent comparable perturbations abstract we prove estimates for schatten norms of multiple operator integrals with symbols arising from fourier series and as a consequence derive higher order trace formulas for functions given by absolutely convergent fourier series of unitary operators and for rational functions bounded on the real line of resolvent comparable self adjoint operators these results substantially extend the main results of 12 14 13 where the estimates were proved for multiple operator integrals with symbols arising from polynomials the trace formulas in the unitary case for analytic functions on the unit disc and the trace formulas in the resolvent comparable self adjoint case for rational functions bounded and analytic in the lower half plane our results rely crucially on an algebraic approach to symbols of multiple operator integrals developed in this paper to replace the prior restrictive analytic approach of 12

sigma function solution of the initial value problem for somos 5 sequences the somos 5 sequences are a family of sequences defined by a fifth order bilinear recurrence relation with constant coefficients for particular choices of coefficients and initial data integer sequences arise by making the connection with a second order nonlinear mapping with a first integral we prove that the two subsequences of odd even index terms each satisfy a somos 4 fourth order recurrence this leads directly to the explicit solution of the initial value problem for the somos 5 sequences in terms of the weierstrass sigma function for an associated elliptic curve

higher order extension of lowner s theory operator k tone functions the new notion of operator matrix k tone functions is introduced which is a higher order extension of operator matrix monotone and convex functions differential properties of matrix k tone functions are shown characterizations properties and examples of operator k tone functions are presented in particular integral representations of operator k tone functions are given generalizing familiar representations of operator monotone and convex functions

characterizing singular curves in parametrized families of biquadratics we consider families of biquadratic curves b 0 on defined with respect to arbitrarily many complex parameters due to the fact that these families include curve intersections across different parameter combinations they represent a generalization of the non intersecting foliations of one parameter invariant curves associated with the qrt mapping we use algebraic methods involving discriminants to provide a complete classification of the singular curves in these families in developing this classification we exploit the special symmetric nature of b namely that it is a quadratic in x and y whose reflection in the line y x is given by a simple change of parameters we also define a range of conditions in the biquadratic s parameters and demonstrate the manner in which they correspond to different geometric realizations of the singular curves

integrable mappings and soliton equations ii abstract some simple solutions of discrete soliton equations are shown to satisfy 2d mappings we show that these belong to a recently introduced 18 parameter family of integrable reversible mappings of the plane thus lending weight to a previous conjecture we also give an example of an integrable mapping occuring in an exactly solvable model in statistical mechanics finally we discuss the notion of generalized reversibility

operator holder zygmund functions abstract it is well known that a lipschitz function on the real line does not have to be operator lipschitz we show that the situation changes dramatically if we pass to holder classes namely we prove that if f belongs to the holder class r with 0 1 then f a f b const a b for arbitrary self adjoint operators a and b we prove a similar result for functions f in the zygmund class 1 r for arbitrary self adjoint operators a and k we have f a k 2 f a f a k const k we also obtain analogs of this result for all holder zygmund classes r 0 then we find a sharp estimate for f a f b for functions f of class def f f const for an arbitrary modulus of continuity in particular we study moduli of continuity for which f a f b const a b for self adjoint a and b and for an arbitrary function f in we obtain similar estimates for commutators f a q q f a and quasicommutators f a q q f b finally we estimate the norms of finite differences j 0 m 1 m j m j f a j k for f in the class m that is defined in terms of finite differences and a modulus continuity of order m we also obtain similar results for unitary operators and for contractions

spectral shift function of higher order this paper resolves affirmatively koplienko s sib mat zh 25 62 71 1984 conjecture on existence of higher order spectral shift measures moreover the paper establishes absolute continuity of these measures and thus existence of the higher order spectral shift functions a spectral shift function of order n is the function n n h v such that operatorname tr biggl f h v sum k 0 n 1 frac 1 k frac d k dt k bigl f h tv bigr bigg t 0 biggr int mathbb r f n t eta n t dt for every sufficiently smooth function f where h is a self adjoint operator defined in a separable hilbert space and v is a self adjoint operator in the n th schatten von neumann ideal sn existence and summability of 1 and 2 were established by krein mat sb 33 597 626 1953 and koplienko sib mat zh 25 62 71 1984 respectively whereas for n 2 the problem was unresolved we show that n h v exists integrable and vert eta n vert l 1 mathbb r leq c n vert v vert s n n for some constant cn depending only on n our results for n rely on estimates for multiple operator integrals obtained in this paper our method also applies to the general semi finite von neumann algebra setting of the perturbation theory

functions of operators under perturbations of class sp this is a continuation of our paper 2 we prove that for functions f in the holder class r and 1 p the operator f a f b belongs to sp whenever a and b are self adjoint operators with a b sp we also obtain sharp estimates for the schatten von neumann norms f a f b sp in terms of a b sp and establish similar results for other operator ideals we also estimate schatten von neumann norms of higher order differences j 0m 1 m j mj f a jk we prove that analogous results hold for functions on the unit circle and unitary operators and for analytic functions in the unit disk and contractions then we find necessary conditions on f for f a f b to belong to sq under the assumption that a b sp we also obtain schatten von neumann estimates for quasicommutators f a r rf b and introduce a spectral shift function and find a trace formula for operators of the form f a k 2f a f a k

integrable mappings of the plane preserving biquadratic invariant curves we provide a general framework to construct integrable mappings of the plane that preserve a one parameter family b x y k of biquadratic invariant curves where parametrization by k is very general these mappings are reversible by construction i e they are the composition of two involutions and can be shown to be measure preserving they generalize integrable maps previously given by mcmillan and quispel roberts and thompson by considering a transformation of the case of the symmetric biquadratic to a canonical form we provide a normal form for the symmetric integrable map acting on each invariant curve we give a lax pair for a large subclass of our symmetric integrable maps including at least a 10 parameter subfamily of the 12 parameter symmetric quispel roberts thompson maps

zero entropy for some birational maps of c2 abstract in this study we consider a special case of the family of birational maps f c 2 c 2 which were dynamically classified by 13 we identify the zero entropy subfamilies of f and explicitly give the associated invariant fibrations in particular we highlight all of the integrable and periodic mappings

a classification of four state spin edge potts models we classify four state spin models with interactions along the edges according to their behaviour under a specific group of symmetry transformations this analysis uses the measure of complexity of the action of the symmetries in the spirit of the study of discrete dynamical systems on the space of parameters of the models and aims at uncovering solvable ones we find that the action of these symmetries has low complexity polynomial growth zero entropy we obtain natural parametrizations of various models among which an unexpected elliptic parametrization of the four state chiral potts model which we use to localize possible integrability conditions associated with high genus curves

integrable mappings via rational elliptic surfaces we present a geometric description of the qrt map which is an integrable mapping introduced by quispel roberts and thompson in terms of the addition formula of a rational elliptic surface by this formulation we classify all the cases when the qrt map is periodic and show that its period is 2 3 4 5 or 6 a generalization of the qrt map which acts birationally on a pencil of k3 surfaces or calabi yau manifolds is also presented

the correlation memory matrix for parameter estimation dynamic systems described by nonlinear differential or difference equations are abound in the literature until recently estimation of system parameters has been mainly carried out using iterative search techniques such as conjugate gradient steepest descent and evolutionary algorithms these methods vary in terms of their noise immunity and robustness but they all tend to require a considerable number of iterations in order for the solution to converge especially in the presence of noise this is a disadvantage for systems requiring real time control in continuation of a series of papers utilizing linear associative memories lam for parameter estimation this paper proposes a non iterative single pass technique for estimating these parameters using the well known correlation memory matrix cmm previously used in pattern recognition the cmm method is used to estimate the parameters of the lotka volterra predator prey model and the van der pol oscillator it is shown that this class of linear associators is superior to other types of lam in that it provides equally good parameter estimates while requiring neither matrix inversion nor extensive training as nonlinear quadratic associations are incorporated into the cmm the method becomes more robust than the linear version

birational maps that send biquadratic curves to biquadratic curves recently many papers have begun to consider so called non quispel roberts thompson qrt birational maps of the plane compared to the qrt family of maps which preserve each biquadratic curve in a fibration of the plane non qrt maps send a biquadratic curve to another biquadratic curve belonging to the same fibration or to a biquadratic curve from a different fibration of the plane in this communication we give the general form of a birational map derived from a difference equation that sends a biquadratic curve to another the necessary and sufficient condition for such a map to exist is that the discriminants of the two biquadratic curves are the same and hence so are the j invariants the result allows existing examples in the literature to be better understood and allows some statements to be made concerning their generality

an algebraic geometric approach to integrable maps of the plane we show that the dynamics of a birational map on an elliptic curve over a field is typically conjugate to addition by a point under the associated group law when the field is taken to be the function field of rational complex functions of one variable this amounts to an algebraic geometric version of the arnold liouville integrability theorem for planar integrable maps by products of this approach are that birational maps preserving foliations are necessarily the composition of two involutions and that relationships between birational maps preserving the same foliation can be described in terms of the respective points they add on the corresponding weierstrass curves when the result is applied to finite fields it helps explain some universal features of the periodic orbit distribution function for the reductions of integrable maps

multiple operator integrals and higher operator derivatives abstract in this paper we consider the problem of the existence of higher derivatives of the function t a tk where is a function on the real line a is a self adjoint operator and k is a bounded self adjoint operator we improve earlier results by sten kin in order to do this we give a new approach to multiple operator integrals this approach improves the earlier approach given by sten kin we also consider a similar problem for unitary operators

lie symmetries of birational maps preserving genus 0 fibrations we prove that any planar birational integrable map which preserves a fibration given by genus 0 curves has a lie symmetry and some associated invariant measures the obtained results allow to study in a systematic way the global dynamics of these maps using this approach the dynamics of several maps is described in particular we are able to give for particular examples the explicit expression of the rotation number function and the set of periods of the considered maps

a new class of integrable discrete systems we derive a new class of integrable autonomous mappings which possess invariants biquartic in the dependent variables we link these mappings to discrete painleve equations we show how mappings with biquartic invariants can be naturally built up from the requirement that their solutions be parametrized in terms of elliptic functions

higher order s2 differentiability and application to koplienko trace formula abstract let a be a selfadjoint operator in a separable hilbert space k a selfadjoint hilbert schmidt operator and f c n r we establish that t f a t k f a is n times continuously differentiable on r in the hilbert schmidt norm provided either a is bounded or the derivatives f i i 1 n are bounded this substantially extends the results of 3 on higher order differentiability of in the hilbert schmidt norm for f in a certain wiener class as an application of the second order s 2 differentiability we extend the koplienko trace formula from the besov class b 1 2 r 20 to functions f for which the divided difference f 2 admits a certain hilbert space factorization

an extension of the koplienko neidhardt trace formulae abstract koplienko sibirsk mat zh 25 5 1984 62 found a trace formula for perturbations of self adjoint operators by operators of hilbert schmidt class s 2 a similar formula in the case of unitary operators was obtained by neidhardt math nachr 138 1988 7 in this paper we improve their results and obtain sharp conditions under which the koplienko neidhardt trace formulae hold

frechet differentiability of sp norms abstract one of the long standing questions in the theory of schatten von neumann ideals of compact operators is whether their norms have the same differentiability properties as the norms of their commutative counterparts we answer this question in the affirmative a key technical observation underlying our proof is a discovery of connection between this question and recent affirmative resolution of l s koplienko s conjecture concerning existence of higher order spectral shift functions

time reversal symmetry in dynamical systems a survey abstract in this paper we survey the topic of time reversal symmetry in dynamical systems we begin with a brief discussion of the position of time reversal symmetry in physics after defining time reversal symmetry as it applies to dynamical systems we then introduce a major theme of our survey namely the relation of time reversible dynamical sytems to equivariant and hamiltonian dynamical systems we follow with a survey of the state of the art on the theory of reversible dynamical systems including results on symmetric periodic orbits local bifurcation theory homoclinic orbits and renormalization and scaling some areas of physics and mathematics in which reversible dynamical systems arise are discussed in an appendix we provide an extensive bibliography on the topic of time reversal symmetry in dynamical systems

functions of unitary operators derivatives and trace formulas abstract let a be a bounded self adjoint operator on a separable hilbert space h and u 0 an arbitrary unitary operator we derive formulas for higher order derivatives of the function s e i s a u 0 for sufficiently smooth functions we compare our formulas with earlier results in this area and apply them to establish higher order analogs of trace formulas of krein koplienko neidhardt

linear algebra approach to neural associative memories and noise performance of neural classifiers the authors present an analytic evaluation of saturation and noise performance for a large class of associative memories based on matrix operations the importance of using standard linear algebra techniques for evaluating noise performance of associative memories is emphasized the authors present a detailed comparative analysis of the correlation matrix memory and the generalized inverse memory construction rules for auto associative memory and neural classifiers analytic results for the noise performance of neural classifiers that can store several prototypes in one class are presented the analysis indicates that for neural classifiers the simple correlation matrix memory provides better noise performance than the more complex generalized inverse memory

noncommutative riesz transforms a probabilistic approach for 2 p 0 in the classical setting this is certainly satisfied for the maps tt f f 1 t where t is a semigroup of measure preserving maps on a probability space in that case n l then the maps tt act on all noncommutative lp spaces lp n and in particular on the hilbert space l2 l2 n let a be the negative generator i e tt e ta for technical reasons we have to assume that there is a weakly dense algebra a n which lies in the domain of the unbounded operator a i e a a a the algebra a plays the role of smooth functions in the classical setting this assumption might be too restrictive for some classical applications but it is easily verified in the new noncommutative examples for elements x y a we may define the gradient form 2 x y a x y x a y a x y the correct definition of is the one given by sauvageot as a bilinear form with values in l1 again our arguments require stronger assumptions in fact in all our examples we will assume that the tt s are selfadjoint i e ttxy xtty and unital i e tt 1 1 then a is indeed a positive unbounded operator under these circumstances we can formulate p a meyer s problem is it true that x x 2 p c p a 1 2 x p 0 1 holds for all selfadjoint elements x a let us illustrate this question by considering the laplace operator a f f where i 2 x2 i then it is easily verified that

on the arazy conjecture concerning schur multipliers on schatten ideals abstract we confirm in the affirmative the conjecture made by arazy in 2 suppose that f is a continuously differentiable function on 1 1 such that f t c t t 1 1 for some constant c 0 and 0 let j j 1 l r for some 0 r if 0 p and 1 q satisfy 1 p r 1 q then the matrix f j f k j k j k 1 is a schur multiplier from the schatten class s q into s p we also consider a general version of this conjecture for arbitrary quasi banach ideals of compact operators

rational surfaces associated with affine root systems and geometry of the painleve equations we present a geometric approach to the theory of painleve equations based on rational surfaces our starting point is a compact smooth rational surface x which has a unique anti canonical divisor d of canonical type we classify all such surfaces x to each x there corresponds a root subsystem of e 1 8 inside the picard lattice of x we realize the action of the corresponding affine weyl group as the cremona action on a family of these surfaces we show that the translation part of the affine weyl group gives rise to discrete painleve equations and that the above action constitutes their group of symmetries by backlund transformations the six painleve differential equations appear as degenerate cases of this construction in the latter context x is okamoto s space of initial conditions and d is the pole divisor of the symplectic form defining the hamiltonian structure