properties of some weighted sobolev spaces and application to spectral approximations the weighted sobolev spaces on a square whose weight is a given power of the product of the distances to the edges are introduced trace theorems related to these spaces are proved then a regularity result for the dirichlet problem for the laplace operator is proved next several projection operators with polynomial values are considered for which approximation results in weighted norms are stated finally a collocation spectral method for the dirichlet problem for the laplace operator with inhomogeneous boundary conditions is analyzed this work includes new results on chebyshev approximation

generalized quantum onsager coefficients from a symmetrized campbell hausdorff expansion the analogue of the classical onsager theory of entropy production is systematically derived for weakly irreversible processes in open quantum systems with finite dimensional hilbert space the dynamics is assumed to be given by a quantum dynamical semigroup with infinitesimal generator of gorini kossakowski sudarshan type the basic spohn formula for entropy production is used to obtain an expansion in terms of powers of the deviation of the initial state relative to the final stationary state of irreversible dynamics to this end an appropriate lie series is constructed from a particular symmetrization procedure applied to the ordinary campbell hausdorff expansion in this way only hermitian contributions by higher order commutators are generated which allow an identification with so called generalized onsager coefficients the explicit derivations concentrate on second third and fourth order coefficients whereas complete detailed expressions are worked out for second and third order in a suitable coherence vector representation of density matrices the results can be given in terms of the dynamical parameters fixing the infinitesimal semigroup generator and in terms of symmetric and antisymmetric structure constants of the lie algebra of su n as an illustration an application to generalized bloch equations for two level systems is studied where the onsager like expansion can be compared with exact results for entropy production we find that convergence is good even for rather large deviations between initial and final state if the calculation includes second and third order coefficients only the formalism presented in this paper generalizes restrictions on admitted final states adopted in much simpler earlier treatments to the most general case of arbitrary unique final states of irreversible processes

on the rapid computation of various polylogarithmic constants we give algorithms for the computation of the d th digit of certain transcendental numbers in various bases these algorithms can be easily implemented multiple precision arithmetic is not needed require virtually no memory and feature run times that scale nearly linearly with the order of the digit desired they make it feasible to compute for example the billionth binary digit of log 2 or on a modest work station in a few hours run time

quantum theory of dissipative processes the markov approximation revisited adopting the standard mathematical framework for describing reduced dynamics we derive two formal identities for the density operator of an open quantum system each of these is equivalent to the old nakajima zwanzig equation the first identity is local in time it contains the inverse of the dynamical map which govern the evolution of the density operator we indicate a time interval on which this inverse exists the second identity constitutes a suitable starting point for going beyond the markov approximation in a controlled way on the basis of the bloch equations we argue once more that in studying quantum dissipation one has to pay attention to the von neumann conditions in the nakajima zwanzig equation we make the first born approximation the ensuing master equation possesses the correct weak coupling limit while proving this rather obvious but at the same time important statement we elucidate the mathematical methods which underlie the weak coupling limit moving to a two dimensional hilbert space we find that both for short and for long times our approximate master equation respects the von neumann conditions assuming exponential decay for correlation functions we propose a physical limit in which the solutions for the density operator become markovian in character we confirm the well known statement that as seen from a macroscopic standpoint the system starts from an effective initial condition the approach to equilibrium is exponential the accessory relaxation constants can differ from the usual bloch parameters and by more than 50

higher order quantum onsager coefficients from dynamical invariants functionals representing dynamical invariants under unitary quantum dynamics of open systems are used to derive onsager coefficients for entropy production in irreversible processes if the nonunitary time evolution is determined by quantum dynamical semigroups the procedure allows a derivation from first principles of the quantum analogue to the classical case

evolution matrix in a coherence vector formulation for quantum markovian master equations of n level systems quantum markovian master equation in n dimensions are systematically transformed into vector form by using the lie algebra of the special unitary group su n density operators are represented through coherence vectors and the infinitesimal kossakowski generator of completely positive quantum dynamical semigroups appears as a real evolution matrix that is not completely diagonalisable in general a complete classification of the spectrum of the latter and its jordan canonical form together with all types of associated stationary states is given in detail the results of this analysis are particularly suited for computations in practical applications

regularization of quantum relative entropy in finite dimensions and application to entropy production the fundamental concept of relative entropy is extended to a functional that is regular valued also on arbitrary pairs of nonfaithful states of open quantum systems this regularized version preserves almost all important properties of ordinary relative entropy such as joint convexity and contractivity under completely positive quantum dynamical semigroup time evolution on this basis a generalized formula for entropy production is proposed the applicability of which is tested in models of irreversible processes the dynamics of the latter is determined by either markovian or non markovian master equations and involves all types of states

the exponential accuracy of fourier and chebyshev differencing methods it is shown that when differencing analytic functions using the pseudospectral fourier or chebyshev methods the error committed decays to zero at an exponential rate

an enhanced pseudospectral chebyshev method for dissipative partial differential equations we present an enhanced pseudospectral chebyshev method based on a postprocess previously introduced for the galerkin method de frutos et al applied mathematics and computation reports july 1996 when the regularity of the solution is limited the new method is shown to possess a superior rate of convergence and improved efficiency than the classical pseudospectral scheme in the particular case of smooth solutions the reduction of the error obtained by postprocessing is not so clear however the use of a filter in the numerical solution before applying the postprocess improves again the efficiency of the method the numerical analysis of the new method and some numerical experiments showing its superior efficiency are presented

unfolding a quantum master equation into a system of real valued equations computationally effective expansion over the basis of su n generators dynamics of an open n state quantum system is often modeled with a markovian master equation describing the evolution of the system density operator by using generators of su n group as a basis the density operator can be transformed into a real valued coherence vector a generator of the dissipative evolution so called lindbladian can be expanded over the same basis and recast in the form of a real matrix together these expansions result is a nonhomogeneous system of n 2 1 real valued linear ordinary differential equations now one can e g implement standard high performance algorithms to integrate the system of equations forward in time while being sure in exact preservation of the trace norm and hermiticity of the density operator however when performed in a straightforward way the expansion turns to be an operation of the time complexity o n 10 the complexity can be reduced when the number of dissipative operators is independent of n which is often the case for physically meaningful models here we present an algorithm to transform quantum master equation into a system of real valued differential equations and propagate it forward in time by using a specific scalable model we evaluate computational efficiency of the algorithm and demonstrate that it is possible to handle the model system with n 10 3 states on a single node of a computer cluster

efficient methods using high accuracy approximate inertial manifolds summary we extend the idea of the post processing galerkin method in the context of dissipative evolution equations to the nonlinear galerkin the filtered galerkin and the filtered nonlinear galerkin methods in general the post processing algorithm takes advantage of the fact that the error committed in the lower modes of the nonlinear galerkin method and galerkin method for approximating smooth bounded solutions is much smaller than the total error of the method in each case an improvement in accuracy is obtained by post processing these more accurate lower modes with an appropriately chosen highly accurate approximate inertial manifold aim we present numerical experiments that support the theoretical improvements in accuracy both the theory and computations are presented in the framework of a two dimensional reaction diffusion system with polynomial nonlinearity however the algorithm is very general and can be implemented for other dissipative evolution systems the computations clearly show the post processed filtered galerkin method to be the most efficient method

analysis of pslq an integer relation finding algorithm let k be either the real complex or quaternion number system and let o k be the corresponding integers let x x 1 x n be a vector in k n the vector x has an integer relation if there exists a vector m m 1 m n o k n m 0 such that m 1 x 1 m 2 x 2 m n x n 0 in this paper we define the parameterized integer relation construction algorithm pslq where the parameter can be freely chosen in a certain interval beginning with an arbitrary vector x x 1 x n k n iterations of pslq will produce lower bounds on the norm of any possible relation for x thus pslq can be used to prove that there are no relations for x of norm less than a given size let m x be the smallest norm of any relation for x for the real and complex case and each fixed parameter in a certain interval we prove that pslq constructs a relation in less than o n 3 n 2 logm x iterations

measure of irreversibility and entropy production in open quantum systems a new concept of a measure of irreversibility for quantum dynamics in open systems is introduced as a suitably regularized substitute for the common notion of entropy production which unfortunately yields infinite values for so many irreversible processes of physical relevance

nonlinear galerkin method using chebyshev and legendre polynomials i the one dimensional case a new strategy stemming from the nonlinear galerkin method m marion and r temam siam j numer anal 26 1989 pp 1139 1157 for solving linear elliptic and nonlinear dissipative evolution equations by using chebyshev and legendre polynomials is presented the essential idea is to decompose the solution into a low mode part and a high mode part and to treat them separately the robustness of the method over the classical galerkin method is substantiated by rigorous error estimates and preliminary numerical experiments

a postprocessed galerkin method with chebyshev or legendre polynomials we present an approximate inertial manifold based postprocess to enhance chebyshev or legendre spectral galerkin methods we prove that the postprocess improves the order of convergence of the galerkin solution yielding the same accuracy as the nonlinear galerkin method numerical experiments show that the new method is computationally more efficient than galerkin and nonlinear galerkin methods new approximation results for chebyshev polynomials are presented

a variation on karmarkar s algorithm for solving linear programming problems the algorithm described here is a variation on karmarkar s algorithm for linear programming it has several advantages over karmarkar s original algorithm in the first place it applies to the standard form of a linear programming problem and produces a monotone decreasing sequence of values of the objective function the minimum value of the objective function does not have to be known in advance secondly in the absence of degeneracy the algorithm converges to an optimal basic feasible solution with the nonbasic variables converging monotonically to zero this makes it possible to identify an optimal basis before the algorithm converges

postprocessing the galerkin method a novel approach to approximate inertial manifolds a postprocess of the standard galerkin method for the discretization of dissipative equations is presented the postprocessed galerkin method uses the same approximate inertial manifold phi app to approximate the high wave number modes of the solution as in the nonlinear galerkin nlg method however in this postprocessed galerkin method the value of phi app is calculated only once after the time integration of the standard galerkin method is completed contrary to the nlg in which phi app evolves with time and affects the time evolution of the lower wave number modes the postprocessed galerkin method which is much cheaper to implement computationally than the nlg is shown in the case of fourier modes to possess the same rate of convergence accuracy as the simplest version of the nlg which is based on either the foias manley temam approximate inertial manifold or the euler galerkin approximate inertial manifold this is proved for some problems in one and two spatial dimensions including the navier stokes equations under periodic boundary conditions the advantages of postprocessing that we present here apply not only to the standard galerkin method but also to the computationally more efficient pseudospectral method

best k digit rational approximation of irrational numbers pre computer versus computer era abstract the best k digit rational approximation for a given irrational number where the numerator has k digits is presented such an approximation allows better error free computation in the real world scenario than that permitted by the k digit decimal approximation of the irrational number it is pointed out that such a best k digit rational approximation though exponential has been made possible due to current high speed computation performed by a digital computer to appreciate the importance of such an approximation we have also focused on the rational approximation not always the best one of some of the famous irrational numbers such as the and the exponential function e by the ingenuity of super mathematicians in several parts of the world during the pre computer era spanning over centuries

an approximate inertial manifolds approach to postprocessing the galerkin method for the navier stokes equation in a recent paper we have introduced a postprocessing procedure for the galerkin method for dissipative evolution partial differential equations with periodic boundary conditions the postprocessing technique uses approximate inertial manifolds to approximate the high modes the small scale components in the exact solutions in terms of the galerkin approximations which in this case play the role of the lower modes large scale components this procedure can be seen as a defect correction technique but contrary to standard procedures the correction is computed only when the time evolution is completed here we extend these results to more realistic boundary conditions specifically we study in detail the two dimensional navier stokes equations subject to homogeneous nonslip dirichlet boundary conditions we also discuss other equations such as reaction diffusion systems and the cahn hilliard equations

efficient spectral galerkin method ii direct solvers of second and fourth order equations using chebyshev polynomials efficient direct solvers based on the chebyshev galerkin methods for second and fourth order equations are presented they are based on appropriate base functions for the galerkin formulation that lead to discrete systems with special structured matrices that can be efficiently inverted numerical results indicate that the direct solvers presented in this paper are significantly more accurate and efficient than those based on the chebyshev tau method

the computation of π to 29 360 000 decimal digits using borweins quartically convergent algorithm in a recent work 6 borwein and borwein derived a class of algorithms based on the theory of elliptic integrals that yield very rapidly convergent approximations to elementary constants the author has implemented borweins quartically convergent algorithm for 1 using a prime modulus transform multi precision technique to compute over 29 360 000 digits of the decimal expansion of the result was checked by using a different algorithm also due to the borweins that converges quadratically to these computations were performed as a system test of the cray 2 operated by the numerical aerodynamical simulation nas program at nasa ames research center the calculations were made possible by the very large memory of the cray 2

conditions for bounded solutions of non markovian quantum master equations for the most general quantum master equations also called nakajima zwanzig or non markovian equations we define suitable boundedness conditions on integral kernels and inhomogeneity terms in order to derive with mathematical rigor an upper bound on solutions as required by the von neumann conditions such equations are of importance for quantum dynamics of open systems with arbitrary couplings to environment and arbitrary entangled initial states the derivation is based on an equivalent coherence vector representation in finite dimension n leading to coupled volterra integro differential equations of second kind and convolution type in an n2 1 dimensional real vector space as examples analytical and numerical model solutions are worked out for 2 level systems in order to test suitable trial functions for input quantities all this is motivated by the fact that exact solutions can hardly be found but appropriate trial functions may provide a reasonable semiphenomenological description of complicated quantum dynamics

quantum skew divergence in this paper we study the quantum generalisation of the skew divergence which is a dissimilarity measure between distributions introduced by lee in the context of natural language processing we provide an in depth study of the quantum skew divergence including its relation to other state distinguishability measures finally we present a number of important applications new continuity inequalities for the quantum jensen shannon divergence and the holevo information and a new and short proof of bravyi s small incremental mixing conjecture

virtues and limitations of markovian master equations with a time dependent generator a markovian master equation with time dependent generator is constructed that respects basic constraints of quantum mechanics in particular the von neumann conditions for the case of a two level system bloch equations with time dependent parameters are obtained necessary conditions on the latter are formulated by employing a time local counterpart of the nakajima zwanzig equation we establish a relation with unitary dynamics we also discuss the relation with the weak coupling limit on the basis of a uniqueness theorem a standard form for the generator of time local master equations is proposed the jaynes cummings model with atomic damping is solved the solution explicitly demonstrates that reduced dynamics can be described by time local master equations only on a finite time interval this limitation is caused by divergencies in the generator a limit of maximum entropy is presented that corroborates the foregoing statements a second limiting case demonstrates that divergencies may even occur for small perturbations of the weak coupling regime

efficient spectral galerkin method i direct solvers of second and fourth order equations using legendre polynomials this paper presents some efficient algorithms based on the legendre galerkin approximations for the direct solution of the second and fourth order elliptic equations the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with sparse matrices for the discrete variational formulations the complexities of the algorithms are a small multiple of n d 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions in addition the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix matrix multiplications