quasi sectorial contractions we revise the notion of the quasi sectorial contractions our main theorem establishes a relation between semigroups of quasi sectorial contractions and a class of m sectorial generators we discuss a relevance of this kind of contractions to the theory of operator norm approximations of strongly continuous semigroups

the norm convergence of the trotter kato product formula with error bound the norm convergence of the trotter kato product formula with error bound is shown for the semigroup generated by that operator sum of two nonnegative selfadjoint operators a and b which is selfadjoint

a generalization of chernoff s product formula for time dependent operators in this article we provide a set of sufficient conditions that allow a natural extension of chernoff s product formula to the case of certain one parameter family of functions taking values in the algebra l b of all bounded linear operators defined on a complex banach space b those functions need not be contraction valued and are intimately related to certain evolution operators u t s 0 s t t on b the most direct consequences of our main result are new formulae of the trotter kato type which involve either semigroups with time dependent generators or the resolvent operators associated with these generators in the general case we can apply such formulae to evolution problems of parabolic type as well as to schrodinger evolution equations albeit in some very special cases the formulae we prove may also be relevant to the numerical analysis of non autonomous ordinary and partial differential equations

some remarks on operator norm convergence of the trotter product formula on banach spaces abstract we extend the trotter kato chernoff theory of strong approximation of c 0 semigroups on banach spaces to operator norm approximation of analytic semigroups with error estimate as application we obtain a criterion for the operator norm convergence of the trotter product formula on banach spaces with error estimate n 1 log n provided one of the generators has a bounded h functional calculus for both results we present versions for approximation in operator ideal norms such as the trace norm or the hilbert schmidt norm finally we give some remarks on the operator norm convergence of the trotter product formula for semigroups acting on a scale of l p spaces

ionization for three dimensional time dependent point interactions we study the time evolution of a three dimensional quantum particle under the action of a time dependent point interaction fixed at the origin we assume that the strength of the interaction t is a periodic function with an arbitrary mean under very weak conditions on the fourier coefficients of t we prove that there is complete ionization as t starting from a bound state at time t 0 moreover we prove also that under the same conditions all the states of the system are scattering states

trotter kato product formula and operator norm convergence we find necessary and sufficient conditions for the operator norm convergence of the trotter kato product formula using them we prove that this convergence takes place i if the resolvent of one of the involved operators is compact either ii if one operator is relatively compact with respect to another one or iii if the product of resolvents of the involved operators is compact

general purpose computation with neural networks a survey of complexity theoretic results we survey and summarize the literature on the computational aspects of neural network models by presenting a detailed taxonomy of the various models according to their complexity theoretic characteristics the criteria of classification include the architecture of the network feedforward versus recurrent time model discrete versus continuous state type binary versus analog weight constraints symmetric versus asymmetric network size finite nets versus infinite families and computation type deterministic versus probabilistic among others the underlying results concerning the computational power and complexity issues of perceptron radial basis function winner take all and spiking neural networks are briefly surveyed with pointers to the relevant literature in our survey we focus mainly on the digital computation whose inputs and outputs are binary in nature although their values are quite often encoded as analog neuron states we omit the important learning issues

operator splitting for non autonomous evolution equations abstract we establish general product formulas for the solutions of non autonomous abstract cauchy problems the main technical tools are evolution semigroups allowing the direct application of existing results on autonomous problems the results obtained are illustrated by the example of an autonomous diffusion equation perturbed with time dependent potential we also prove convergence rates for the sequential splitting applied to this problem

simple local search problems that are hard to solve many algorithms for np hard optimization problems find solutions that are locally optimal in the sense that the solutions cannot be improved by a polynomially computable perturbation very little is known about the complexity of finding locally optimal solutions either by local search algorithms or using other indirect methods johnson papadimitriou and yannakakis j comput system sci 37 1988 pp 79 100 studied this question by defining a complexity class pls that captures local search problems it was proved that finding a partition of a graph that is locally optimal into equal parts with respect to the acclaimed kernighan lin algorithm is pls complete it is shown here that several natural simple local search problems are pls complete and thus just as hard two examples are finding a partition that cannot be improved by a single swap of two vertices and finding a stable configuration for an undirected connectionist network when edges or other objects are unweighted then a local optimum

on characterizations of the class pspace poly abstract we present several characterizations of the class pspace poly these characterizations are given in terms of pspace complete problems certain algebraic closure operations vectorial straight line programs and kolmogorov complexity to end we characterize the problems having an exponential lower bound on their nonuniform complexity measured by vectorial straight line programs

notes on the chernoff product formula abstract we revise the strong convergent chernoff product formula and extend it in a hilbert space to convergence in the operator norm topology main results deal with the self adjoint chernoff product formula the non self adjoint case concerns the quasi sectorial contractions

trotter kato product formula and symmetrically normed ideals abstract it is proven that the trotter product formula converges in the norm of a symmetrically normed ideal of compact operators away from t0 0 if the kac operator the transfer matrix f t e tb 2e tae tb 2 belongs to this ideal for t t0 the result is generalized to the trotter kato product formula moreover if the perturbation b is small relative to a then error bounds for convergence are obtained the results apply to the dixmier trace

on positivity shape and norm bound preservation of time stepping methods for semigroups abstract we use functional calculus methods to investigate qualitative properties of c 0 semigroups that are preserved by time discretization methods preservation of positivity concavity and other qualitative shape properties which can be described via positivity are treated in a banach lattice framework preservation of contractivity or norm bound of the semigroup is investigated in the banach space setting the use of the hille phillips h p functional calculus instead of the dunford taylor functional calculus allows us to extend fundamental qualitative results concerning time discretization methods and simplify their proofs including results on multi step schemes and variable step sizes since the h p functional calculus is used throughout the paper we present an elementary introduction to it based on the riemann stieltjes integral

a trotter kato product formula for a class of non autonomous evolution equations abstract in this article dedicated to professor v lakshmikantham on the occasion of the celebration of his 84th birthday we announce new results concerning the existence and various properties of an evolution system u a b t s 0 s t t generated by the sum a t b t of two linear time dependent and generally unbounded operators defined on time dependent domains in a complex and separable banach space b in particular writing l b for the algebra of all linear bounded operators on b we can express u a b t s 0 s t t as the strong limit in l b of a product of the holomorphic contraction semigroups generated by a t and b t thereby getting a product formula of the trotter kato type under very general conditions which allow the domain d a t b t to evolve with time provided there exists a fixed set d t 0 t d a t b t everywhere dense in b we then mention several possible applications of our product formula to various classes of non autonomous parabolic initial boundary value problems as well as to evolution problems of schrodinger type related to the theory of time dependent singular perturbations of self adjoint operators in quantum mechanics we defer all the proofs and all the details of the applications to a separate publication

dimension splitting for evolution equations in this paper we are concerned with splitting methods for the time integration of abstract evolution equations we introduce an analytic framework which allows us to prove optimal convergence orders for various splitting methods including the lie and peaceman rachford splittings our setting is applicable for a wide variety of linear equations and their dimension splittings in particular we analyze parabolic problems with dirichlet boundary conditions as well as degenerate equations on bounded domains we further illustrate our theoretical results with a set of numerical experiments

on the joint subword complexity of automatic sequences let the subword complexity of a sequence u u n n 0 over a finite set s be the function m p u m where p u m denotes the number of distinct blocks u n u n m 1 of size m in u in this paper we study the complexity of u n u 1 n u r n n 0 when each u i u i n n 0 i 1 r is a q i automatic sequence over a finite set s i and q 1 q r 2 are pairwise coprime integers as an application we answer a question of allouche and shallit regarding morphic real numbers

infinite order differential operators in spaces of entire functions abstract differential operators where is an exponential type entire function of a single complex variable and z d zd2 d z z c 0 r acting in the spaces of exponential type entire function are studied it is shown that for 0 such operators preserve the set of laguerre entire functions provided the function also belongs to this set the latter consists of the polynomials possessing real nonpositive zeros only and of their uniform limits on compact subsets of the complex plane c the operator exp a a 0 is studied in more details in particular it is shown that it preserves the set of laguerre entire functions for all r an integral representation of exp a a 0 is obtained these results are used to obtain the solutions to certain cauchy problems employing

a general trotter kato formula for a class of evolution operators in this article we prove new results concerning the existence and various properties of an evolution system ua b t s 0 s t t generated by the sum a t b t of two linear time dependent and generally unbounded operators de ned on time dependent domains in a complex and separable banach space b in particular writing l b for the algebra of all linear bounded operators on b we can express ua b t s 0 s t t as the strong limit in l b of a product of the holomorphic contraction semigroups generated by a t and b t respectively thereby proving a product formula of the trotter kato type under very general conditions which allow the domain d a t b t to evolve with time provided there exists a fi xed set d incl t in 0 t d a t b t everywhere dense in b we obtain a special case of our formula when b t 0 which in effect allows us to reconstruct ua t s 0 s t t very simply in terms of the semigroup generated by a t we then illustrate our results by considering various examples of non autonomous parabolic initial boundary value problems including one related to the theory of time dependent singular perturba tions of self adjoint operators we nally mention what we think remains an open problem for the corresponding equations of schrodinger type in quantum mechanics

on operator norm approximation of some semigroups by quasi sectorial operators abstract we use some results and methods of the probability theory to improve bounds for the convergence rates in some approximation formulas for operators

a maximum principle applied to quasi geostrophic equations we study the initial value problem for dissipative 2d quasi geostrophic equations proving local existence global results for small initial data in the super critical case decay of lp norms and asymptotic behavior of viscosity solution in the critical case our proofs are based on a maximum principle valid for more general flows

efficient simulation of finite automata by neural nets let k m denote the smallest number with the property that every m state finite automaton can be built as a neural net using k m or fewer neurons a counting argument shows that k m is at least o m log m 1 3 and a construction shows that k m is at most o m 3 4 the counting argument and the construction allow neural nets with arbitrarily complex local structure and thus may require neurons that themselves amount to complicated networks mild and in practical situations almost necessary constraints on the local structure of the network give again by a counting argument and a construction lower and upper bounds for k m that are both linear in m

the convergence of symmetric threshold automata we study the length of the transient in a threshold automata with values on the set 0 1 the original conjecture was that in the symmetric case this length was a linear function of the number of cells in the automata goles e 1981 comportement oscillatoire d une famille d automates cellulaires non uniformes these imag grenoble france and olivos j 1979 contribution a l algorithmique these orsay france we shall show that this length can be exponential

the computational power of discrete hopfield nets with hidden units we prove that polynomial size discrete hopfield networks with hidden units compute exactly the class of boolean functions pspace poly i e the same functions as are computed by polynomial space bounded nonuniform turing machines as a corollary to the construction we observe also that networks with polynomially bounded interconnection weights compute exactly the class of functions p poly i e the class computed by polynomial time bounded nonuniform turing machines

separators in infinite words generated by morphisms resume nous appelons separateur d un mot x a l indice n e n le plus petit facteur de x dont l indice du debut de la premiere occurrence est n notons sx n la longueur de ce separateur lorsqu il existe et posons sx n 0 sinon nous montrons que l application sx ainsi definie est q reguli e re au sens d allouche et shallit lorsque x est un mot infini engendre par iteration d un morphisme q uniforme et que x est circulaire de maniere equivalente cela revient a dire que la serie formelle dont les coefficients sont les valeurs sx n en base q est n rationnelle salomaa soittola

on the computational complexity of binary and analog symmetric hopfield nets we investigate the computational properties of finite binary and analog state discrete time symmetric hopfield nets for binary networks we obtain a simulation of convergent asymmetric networks by symmetric networks with only a linear increase in network size and computation time then we analyze the convergence time of hopfield nets in terms of the length of their bit representations here we construct an analog symmetric network whose convergence time exceeds the convergence time of any binary hopfield net with the same representation length further we prove that the min energy problem for analog hopfield nets is np hard and provide a polynomial time approximation algorithm for this problem in the case of binary nets finally we show that symmetric analog nets with an external clock are computationally turing universal

an approximation to wiener measure and quantization of the hamiltonian on manifolds with non positive sectional curvature this paper gives a rigorous interpretation of a feynman path integral on a riemannian manifold m with non positive sectional curvature an l2 riemannian metric gp is given on the space of piecewise geodesic paths hp m adapted to the partition p of 0 1 whence a finite dimensional approximation of wiener measure is developed it is proved that as mesh p 0 the approximate wiener measure converges in an l1 sense to the measure exp 2 3203 01scal s ds d on the wiener space w m with wiener measure this gives a possible prescription for the path integral representation of the quantized hamiltonian as well as yielding such a result for the natural geometric approximation schemes originating in andersson and driver 1999 3 and followed by lim 2007 34

operator splitting for delay equations operator splitting methods are widely used for partial differential equations up until now they have not been used for delay differential equations in this paper we introduce splitting methods for delay equations in an abstract setting we then prove the convergence of the method and discuss the results of some numerical experiments

a generalized convergence theorem for neural networks a neural network model is presented in which each neuron performs a threshold logic function the model always converges to a stable state when operating in a serial mode and to a cycle of length at most 2 when operating in a fully parallel mode this property is the basis for the potential applications of the model such as associative memory devices and combinatorial optimization the two convergence theorems for serial and fully parallel modes of operation are reviewed and a general convergence theorem is presented that unifies the two known cases new relations between the neural network model and the problem of finding a minimum cut in a graph are obtained

properties of words and recognizability of fixed points of a substitution resume soit u un mot infini non periodique point fixe d une substitution primitive nous montrons que pour n assez grand u ne contient aucune puissance n eme de mot nous donnons ensuite une condition necessaire et suffisante pour que soit reconnaissable au sens de host et queffelec et nous montrons qu ill suffit de bilateraliser cette definition pour obtenir une propriete verifiee cette fois par toutes les substitutions primitives a point fixe non periodique

the ring of k regular sequences ii in this paper we continue our study of k regular sequences begun in 1992 we prove some new results give many new examples from the literature and state some open problems

operator splitting for nonautonomous delay equations we provide a general product formula for the solution of nonautonomous abstract delay equations after having shown the convergence we obtain estimates on the order of convergence for differentiable history functions finally the theoretical results are demonstrated by some typical numerical examples

balancedness and coboundaries in symbolic systems abstract this paper studies balancedness for infinite words and subshifts both for letters and factors balancedness is a measure of disorder that amounts to strong convergence properties for frequencies it measures the difference between the numbers of occurrences of a given word in factors of the same length we focus on two families of words namely dendric words and words generated by substitutions the family of dendric words includes sturmian and arnoux rauzy words as well as codings of regular interval exchanges we prove that dendric words are balanced on letters if and only if they are balanced on words in the substitutive case we stress the role played by the existence of coboundaries taking rational values and show simple criteria when frequencies take rational values for exhibiting imbalancedness

trotter kato product formula and fractional powers of self adjoint generators let a and b be non negative self adjoint operators in a hilbert space such that their densely defined form sum h a b obeys dom h dom a dom b for some 1 2 1 it is proved that if in addition a and b satisfy dom a1 2 dom b1 2 then the symmetric and non symmetric trotter kato product formula converges in the operator norm e tb 2ne ta ne tb 2n n e th o n 2 1 e ta ne tb n n e th o n 2 1 uniformly in t 0 t 0 t as n both with the same optimal error bound the same is valid if one replaces the exponential function in the product by functions of the kato class that is by real valued borel measurable functions f defined on the non negative real axis obeying 0 f x 1 f 0 1 and f 0 1 with some additional smoothness property at zero the present result improves previous ones relaxing the smallness of b with respect to a to the milder assumption dom a1 2 dom b1 2 and extending essentially the admissible class of kato functions

operator norm approximation of semigroups by quasi sectorial contractions abstract we extend the chernoff theory of approximation of contraction semigroups a la trotter we show that the trotter neveu kato convergence theorem holds in operator norm for a family of uniformly m sectorial generators in a hilbert space then we obtain a chernoff type approximation theorem for quasi sectorial contractions on a hilbert space in the operator norm we give necessary and sufficient conditions for the operator norm convergence of trotter type product formulae

continuous time symmetric hopfield nets are computationally universal we establish a fundamental result in the theory of computation by continuous time dynamical systems by showing that systems corresponding to so called continuous time symmetric hopfield nets are capable of general computation as is well known such networks have very constrained lyapunov function controlled dynamics nevertheless we show that they are universal and efficient computational devices in the sense that any convergent synchronous fully parallel computation by a recurrent network of n discrete time binary neurons with in general asymmetric coupling weights can be simulated by a symmetric continuous time hopfield net containing only 18n 7 units employing the saturated linear activation function moreover if the asymmetric network has maximum integer weight size wmax and converges in discrete time t then the corresponding hopfield net can be designed to operate in continuous time t e for any e 0 such that wmax212n e21 e in terms of standard discrete computation models our result implies that any polynomially space bounded turing machine can be simulated by a family of polynomial size continuous time symmetric hopfield nets

a limit evolution problem for time dependent point interactions abstract we study the diffusion in r 3 of a particle interacting with n fixed points through point interactions whose strength varies in time under mild assumptions on the time dependence of the strengths we prove existence for all times and uniqueness of the solution for which we provide a rather explicit expression we also prove that under a suitable rescaling of the interaction strengths the solution converges when n to the solution of a diffusion equation with a regular killing term potential we use properties of the local self adjoint extensions of the laplacian and results from the theory of fractional integrals and derivatives

approximation of the semigroup generated by the hamiltonian of reggeon field theory in bargmann space abstract the reggeon field theory is governed by a non self adjoint operator constructed as a polynomial in a a the standard bose annihilation and creation operators in zero transverse dimension this hamiltonian acting in bargmann space is defined by h a 2 a 2 a a i a a a a where i 2 1 and are real numbers and the operators a a satisfy the commutation relation a a i as the quantum mechanical system described by h has a velocity dependent potential containing powers of momentum up to the fourth the problem of existence of hamiltonian path integral for the evolution operator e t h of this theory is of interest on its own in particular can we express e t h as a limit of integral operators in this article one considerably reduces the difficulty by studying the trotter product formula of h to reach two objectives the first objective is to prove a very specific error estimate for the error in a trotter product formula in trace norm for h viewed as the sum of the operators a 2 a 2 and a a i a a a a the second objective of this work is to give a approximation of the semigroup generated by h when h is split in the sum of a 2 a 2 a a and i a a a a we note that this case is entirely different in fact the usual trotter product formula is not defined because the interaction operator a a a a is not the infinitesimal generator of a semigroup on bargmann space for 0 and 0 we choose an approximation operator i i a a a a e a 2 a 2 a a and we give a connection between and e h this choice allows us to give in a intissar note on the path integral formulation of reggeon field theory preprint a generalized trotter product formula for t a a i a a a a i e for limit case as 0 and answers to the above question

lp bounds for quasi geostrophic equations via functional analysis we give a proof of lp bounds for the quasi geostrophic equation and other non local equations the proof uses mainly tools from functional analysis notably the product formulas also known as operator splitting methods and the bochner pollard subordination identities hence it could be applicable to other equations

palindrome complexity we study the palindrome complexity of infinite sequences on finite alphabets i e the number of palindromic factors blocks of given length occurring in a given sequence we survey the known results and obtain new results for some sequences in particular for rote sequences and for fixed points of primitive morphisms of constant length belonging to class p of hof knill simon we also give an upper bound for the palindrome complexity of a sequence in terms of its block complexity

exponential splitting for unbounded operators we present a convergence analysis for exponential splitting methods applied to linear evolution equations our main result states that the classical order of the splitting method is retained in a setting of unbounded operators without requiring any additional order condition this is achieved by basing the analysis on the abstract framework of semi groups the convergence analysis also includes generalizations to splittings consisting of more then two operators and to variable time steps we conclude by illustrating that the abstract results are applicable in the context of the schrodinger equation with an external magnetic field or with an unbounded potential

conjugacies for tiling dynamical systems we consider tiling dynamical systems and topological conjugacies between them we prove that the criterion of being of finite type is invariant under topological conjugacy for substitution tiling systems under rather general conditions including the penrose and pinwheel systems we show that substitutions are invertible and that conjugacies are generalized sliding block codes

on critical exponents in fixed points of non erasing morphisms let s be an alphabet of size t let f s s be a non erasing morphism let w be an infinite fixed point of f and let e w be the critical exponent of w we prove that if e w is finite then for a uniform f it is rational and for a non uniform f it lies in the field extension q r l 1 l where r l 1 l are the eigenvalues of the incidence matrix of f in particular e w is algebraic of degree at most t under certain conditions our proof implies an algorithm for computing e w

a note on the limiting procedures for path integrals limiting procedures for the feynman type path integral are considered the evolution operator is approximated with operators corresponding to the exponential of the hamiltonian s symbol the proof of convergence uses chernoff s theory and its extension on the class of stable operators the approach is especially adequate for the generalized coherent states path integral where decomposition of a hamiltonian into the sums used in the approach based on trotter s formula may be unnatural