discrete painleve equations coalescences limits and degeneracies abstract starting from the standard form of the five discrete painleve equations we show how one can obtain through appropriate limits a host of new equations which are also the discrete analogues of the continuous painleve equations a particularly interesting technique is the one based on the assumption that some simplification takes place in the autonomous form of the mapping following which the deautonomization leads to a new n dependence and introduces more new discrete painleve equations

on the linear complexity of nonuniformity decimated pn sequences a lower bound is derived on the probability that when a pn sequence of period n 2 sup n 1 is nonuniformly decimated by means of a sequence whose period divides m the decimated sequence will have maximum linear complexity nm it is shown that by choosing m and n appropriately this probability can be made arbitrarily close to one with nm arbitrarily large

the spectral gap for some spin chains with discrete symmetry breaking we prove that for any finite set of generalized valence bond solid gvbs states of a quantum spin chain there exists a translation invariant finite range hamiltonian for which this set is the set of ground states this result implies that there are gvbs models with arbitrary broken discrete symmetries that are described as combinations of lattice translations lattice reflections and local unitary or anti unitary transformations we also show that all gvbs models that satisfy some natural conditions have a spectral gap the existence of a spectral gap is obtained by applying a simple and quite general strategy for proving lower bounds on the spectral gap of the generator of a classical or quantum spin dynamics this general scheme is interesting in its own right and threfore although the basic idea is not new we present it in a system independent setting the results are illustrated with a number of examples

minimal linear equivalent analysis of a variable memory binary sequence generator starting from a memory based nonlinear generator due to p r geffe 1973 a binary sequence generator mem bsg consisting of three linear feedback shift registers and a variable memory is defined and analyzed the mem bsg is shown to be convenient for generating fast binary sequences of large period and linear complexity which may be useful in spread spectrum and cryptographic applications

on the linear complexity of nonlinearly filtered pn sequences binary sequences of period 2 sup n 1 generated by a linear feedback shift register lfsr whose stages are filtered by a nonlinear function f are studied new iterative formulas are derived for the calculation of the linear complexity of the output sequences it is shown that these tools provide an efficient mechanism for controlling the linear complexity of the nonlinearly filtered maximal length sequences

products of linear recurring sequences with maximum complexity conditions are derived which guarantee that products of linear recurring sequences attain maximum linear complexity it is shown that the product of any number of maximum length gf q sequences has maximum linear complexity provided only the degrees of the corresponding minimal polynomials are distinct and greater than two it is also shown that if the roots of any number of not necessarily irreducible minimal polynomials are simple and lie in extension fields of pairwise relatively prime degrees then the product of the corresponding gf q sequences attains maximum linear complexity provided only that no two roots of any minimal polynomial are linearly dependent over the groundfield gf q which is automatically satisfied when q 2 the results obtained for products are extended to arbitrary linear combinations of product sequences

meromorphic solutions of difference equations integrability and the discrete painleve equations the painlev e property is closely connected to differential equations that are integrable via related iso monodromy problems many apparently integrable discrete analogues of the painlev e equations have appeared in the literature the existence of sufficiently many finite order meromorphic solutions appears to be a good analogue of the painlev e property for discrete equations in which the independent variable is taken to be complex a general introduction to nevanlinna theory is presented together with an overview of recent applications to meromorphic solutions of difference equations and the difference and q difference operators new results are presented concerning equations of the form w z 1 w z 1 r z w where r is rational in w with meromorphic coefficients

existence of finite order meromorphic solutions as a detector of integrability in difference equations abstract the existence of sufficiently many finite order in the sense of nevanlinna meromorphic solutions of a difference equation appears to be a good indicator of integrability it is shown that out of a large class of second order difference equations the only equation that can admit a sufficiently general finite order meromorphic solution is the difference painleve ii equation the proof given relies on estimates obtained by arguments related to singularity confinement the existence of meromorphic solutions of a general class of first order difference equations is also proven by a simple method based on banach s fixed point theorem

spectral properties of the two photon absorption and emission process the quantum markov semigroup of the two photon absorption and emission process has two extremal normal invariant states starting from an arbitrary initial state it converges toward some convex combination of these states as time goes to infinity approach to equilibrium we compute the exact exponential rate of this convergence showing that it depends only on the emission rates moreover we show that off diagonal matrix elements of any initial state go to zero with an exponential rate which is smaller than the exponential rate of convergence of the diagonal part in other words quantum features of a state survive longer than the relaxation time of its classical part

dirichlet forms and markovian semigroups on standard forms of von neumann algebras we characterize w continuous markovian semigroups on a von neumann algebra m which are 0 symmetric w r t a faithful normal state 0in m in terms of quadratic forms on the hilbert space h of a standard form m h p j we characterize also symmetric strongly continuous contraction semigroups on a real hilbert space h which leave invariant a closed convex set in h in terms of a contraction property of the associated quadratic forms we apply the results to give criteria of essential selfadjointness for quadratic form sums and to give a characterization of w continuous markovian semigroups on m which commute with the modular automorphism group 0t

ergodic property of markovian semigroups on standard forms of von neumann algebras we give sufficient conditions for ergodicity of the markovian semigroups associated to dirichlet forms on standard forms of von neumann algebras constructed by the method proposed by park we apply our result to show that the diffusion type markovian semigroups for quantum spin systems are ergodic in the region of high temperatures where the uniqueness of the kms state holds

hypercontractivity for a quantum ornstein uhlenbeck semigroup we prove hypercontractivity for a quantum ornstein uhlenbeck semigroup on the entire algebra mathcal b h of bounded operators on a separable hilbert space h we exploit the particular structure of the spectrum together with hypercontractivity of the corresponding birth and death process and a proper decomposition of the domain then we deduce a logarithmic sobolev inequality for the semigroup and gain an elementary estimate of the best constant

on the linear complexity of functions of periodic gf q sequences it is proved that the product of arbitrary periodic gf q sequences attains maximum linear complexity if their periods are pairwise coprime the necessary and sufficient conditions are derived for maximum linear complexity of the product of two periodic gf q sequences with irreducible minimal characteristic polynomials for a linear combination of products of arbitrary periodic gf q sequences it is shown that maximum linear complexity is achieved if their periods are pairwise coprime and the polynomial x 1 does not divide any of their minimal characteristic polynomials assuming only that their periods are pairwise coprime the author establishes a lower bound on the linear complexity which is of the same order of magnitude as maximum linear complexity boolean functions are derived that are optimal with respect to the maximum linear complexity possible applications of the results in the design of sequence generators are discussed

finitely correlated states on quantum spin chains we study a construction that yields a class of translation invariant states on quantum spin chains characterized by the property that the correlations across any bond can be modeled on a finite dimensional vector space these states can be considered as generalized valence bond states and they are dense in the set of all translation invariant states we develop a complete theory of the ergodic decomposition of such states including the decomposition into periodic neel ordered states the ergodic components have exponential decay of correlations all states considered can be obtained as local functions of states of a special kind so called purely generated states which are shown to be ground states for suitably chosen finite range vbs interactions we show that all these generalized vbs models have a spectral gap our theory does not require symmetry of the state with respect to a local gauge group in particular we illustrate our results with a one parameter family of examples which are not isotropic except for one special case this isotropic model coincides with the one dimensional antiferromagnet recently studied by affleck kennedy lieb and tasaki

spectral analysis and feller property for quantum ornstein uhlenbeck semigroups a class of dynamical semigroups arising in quantum optics models of masers and lasers is investigated the semigroups are constructed by means of noncommutative dirichlet forms on the full algebra of bounded operators on a separable hilbert space the explicit action of their generators on a core in the domain is used to demonstrate the feller property of the semigroups with respect to the c subalgebra of compact operators the dirichlet forms are analysed and the c 2 spectrum together with eigenspaces are found when reduced to certain maximal abelian subalgebras the semigroups give rise to the markov semigroups of classical ornstein uhlenbeck processes on the one hand and of classical birth and death processes on the other

continuity and generators of dynamical semigroups for infinite bose systems abstract for a class of quasifree quantum dynamical semigroups on the algebra of the canonical commutation relations ccr we give sufficient conditions for these semigroups to extend to ultraweakly continuous semigroups of normal operators on the von neumann algebra associated with a representation of the ccr then the explicit form of the generators of the extended semigroups is calculated

dirichlet forms and symmetric markovian semigroups on ccr algebras with respect to quasi free states employing the construction method of dirichlet forms on standard forms of von neumann algebras developed in infinite dimensional analysis quantum probability and related topics 2000 vol 3 no 1 pp 1 14 ref 1 we construct dirichlet forms and associated symmetric markovian semigroups on ccr algebras with respect to quasi free states more precisely let a h0 be the ccr algebra over a complex separable pre hilbert space h0 and let be a quasi free state on a h0 for any normalized admissible function f and complete orthonormal system cons gn h0 we construct a dirichlet form and corresponding symmetric markovian semigroup on the natural standard form associated to the gns representation of a h0 it turns out that the form is independent of admissible function f and cons gn chosen by analyzing the spectrum of the generator dirichlet operator of the semigroup we show that the semigroup is ergodic and tends to the equilibrium exponentially fast

the road to the discrete analogue of the painleve property nevanlinna meets singularity confinement the question of integrability of discrete systems is analysed in the light of the recent findings of ablowitz et al who have conjectured that a fast growth of the solutions of a differ ence equation is an indication of nonintegrability the study of the behaviour of the solutions of a mapping is based on the theory of nevanlinna in this paper we show how this approach can be implemented in the csse of second order mappings which include the discrete painlevk equations since the nevanlinna approach does offer only a necessary condition which is not restrictive enough we complement it by the singularity confinement requirement first in an autonomous setting and then for deautonomisation we believe that this three tiered approach is the closest one can get to a discrete analogue of the painlevb property 2003 elsevier science ltd all rights reserved

from continuous to discrete painleve equations abstract discrete analogues of the painleve equations have recently been derived using either a method based on orthogonal polynomials or the singularity confinement method here we analyze additional methods for deriving integrable non autonomous discrete equations these methods are based on the discrete akns approach on the use of backlund and schlesinger transforms of the continuous painleve equations and on discrete analogues of miura transformations

a transformation from the fibonacci to the galois nlfsrs conventional nonlinear feedback shift registers nlfsrs use the fibonacci configuration in which the feedback is applied to the last bit only in this paper we show how to transform a fibonacci nlfsr into an equivalent nlfsr in the galois configuration in which the feedback can be applied to every bit such a transformation can potentially reduce the depth of the circuits implementing feedback functions thus decreasing the propagation time and increasing the throughput

construction of a family of quantum ornstein uhlenbeck semigroups for a given quasi free state on the ccr algebra over one dimensional hilbert space a family of markovian semigroups which leave the quasi free state invariant is constructed by means of noncommutative elliptic operators and dirichlet forms on von neumann algebras the generators dirichlet operators of the semigroups are analyzed and the spectrums together with eigenspaces are found when restricted to a maximal abelian subalgebra the semigroups are reduced to a unique markovian semigroup of classical ornstein uhlenbeck process

gradient flow and entropy inequalities for quantum markov semigroups with detailed balance abstract we study a class of ergodic quantum markov semigroups on finite dimensional unital c algebras these semigroups have a unique stationary state and we are concerned with those that satisfy a quantum detailed balance condition with respect to we show that the evolution on the set of states that is given by such a quantum markov semigroup is gradient flow for the relative entropy with respect to in a particular riemannian metric on the set of states this metric is a non commutative analog of the 2 wasserstein metric and in several interesting cases we are able to show in analogy with work of otto on gradient flows with respect to the classical 2 wasserstein metric that the relative entropy is strictly and uniformly convex with respect to the riemannian metric introduced here as a consequence we obtain a number of new inequalities for the decay of relative entropy for ergodic quantum markov semigroups with detailed balance

unitary matrix models as exactly solvable string theories models of unitary matrices are solved exactly in a double scaling limit using orthogonal polynomials on a circle exact differential equations are found for the scaling functions of these models for the simplest model k 1 the painlev e ii equation with constant 0 is obtained there are possible nonperturbative phase transitions in these models the scaling function is of the form mathit n mathrm ensuremath 1 2 mathit k 1 ifmmode times else texttimes fi f mathit n 2 mathit k 2 mathit k 1 ensuremath lambda mathit x ensuremath lambda for the kth multicritical point the specific heat is mathit f 2 and is therefore manifestly positive equations are given for k 2 and 3 with a discussion of asymptotic behavior

quantum ito s formula and stochastic evolutions using only the boson canonical commutation relations and the riemann lebesgue integral we construct a simple theory of stochastic integrals and differentials with respect to the basic field operator processes this leads to a noncommutative ito product formula a realisation of the classical poisson process in fock space which gives a noncommutative central limit theorem the construction of solutions of certain noncommutative stochastic differential equations and finally to the integration of certain irreversible equations of motion governed by semigroups of completely positive maps the classical ito product formula for stochastic differentials with respect to brownian motion and the poisson process is a special case

derivations as square roots of dirichlet forms abstract the aim of this work is to analyze the structure of a tracially symmetric dirichlet form on a c algebra in terms of a killing weight and a closable derivation taking values in a hilbert space with a bimodule structure it is shown that the generator of the associate markovian semigroup always appears in a natural way as the divergence of a closable derivation applications are shown to the decomposition of dirichlet forms and to the construction of differential calculus on metric spaces

remarks on the decomposition of dirichlet forms on standard forms of von neumann algebras for a bounded generator g of a weakly continuous completely positive kms symmetric markovian semigroup on a von neumann algebra m acting on a separable hilbert space h let h be the operator induced by g via the symmetric embedding of m into h we decompose the dirichlet form associated with h into a direct integral of forms whose associated generators are divergences of derivations moreover if the derivations are inner then the dirichlet form can be written as the form given by park infinite dimen anal quantum probab relat top 3 1 2000 8 179 2005

non symmetric dirichlet forms on semifinite von neumann algebras abstract the theory of non symmetric dirichlet forms is generalized to the non abelian setting also establishing the natural correspondences among dirichlet forms sub markovian semigroups and sub markovian resolvents within this context some results on the allowed functional calculus for closed derivations on hilbert algebras are obtained examples of non symmetric dirichlet forms given by derivations on hilbert algebras are studied

meromorphic solutions of difference painleve equations we show that under certain conditions some difference painleve equations have nontrivial meromorphic solutions in the whole complex plane these meromorphic solutions are obtained by analytic continuation of asymptotic solutions given in sectors of zero opening angle the existence of these asymptotic solutions is shown by using fundamental matrix solutions of the associated linear systems with double unit characteristic roots and with coefficients expressed by asymptotic series in fractional powers

finite order meromorphic solutions and the discrete painleve equations let w z be a finite order meromorphic solution of the second order difference equation w z 1 w z 1 r z w z where r z w z is rational in w z and meromorphic in z then either w z satisfies a difference linear or riccati equation or else equation can be transformed to one of a list of canonical difference equations this list consists of all known difference painleve equation of the form together with their autonomous versions this suggests that the existence of finite order meromorphic solutions is a good detector of integrable difference equations