maps completely preserving commutativity and maps completely preserving jordan zero product abstract let x y be real or complex banach spaces with infinite dimension and let a b be standard operator algebras on x and y respectively in this paper we show that every map completely preserving commutativity from a onto b is a scalar multiple of either an isomorphism or in the complex case a conjugate isomorphism every map completely preserving jordan zero product from a onto b is a scalar multiple of either an isomorphism or in the complex case a conjugate isomorphism

maps on idempotent matrices over division rings abstract let d be an arbitrary division ring and p n d the set of all n n idempotent matrices over d under some mild conditions we give a complete description of maps on p n d that preserve either commutativity or order or orthogonality we give examples showing that our assumptions cannot be relaxed much further as an application we will prove a quaternionic analogue of ovchinnikov s result that is important in quantum mechanics other applications of our theorems include results on automorphisms of operator and matrix semigroups local automorphisms linear preserver problems and geometry of matrices and grassmannians

automorphisms of b h with respect to minus partial order abstract let h be an infinite dimensional hilbert space and b h the algebra of all bounded linear operators on h we discuss possible extensions of the concept of the minus partial order from matrix to operator algebras in particular we show that mitra s unified theory of matrix partial orders based on generalized inverses can be modified in such a way that we get a unified approach to partial orders on b h or even more general algebras and rings then we choose the most natural among possible definitions of minus partial order on b h and describe the structure of corresponding automorphisms

approximating spectral densities of large matrices in physics it is sometimes desirable to compute the so called density of states dos also known as the spectral density of a real symmetric matrix a the spectral density can be viewed as a probability density distribution that measures the likelihood of finding eigenvalues near some point on the real line the most straightforward way to obtain this density is to compute all eigenvalues of a but this approach is generally costly and wasteful especially for matrices of large dimension there exist alternative methods that allow us to estimate the spectral density function at much lower cost the major computational cost of these methods is in multiplying a with a number of vectors which makes them appealing for large scale problems where products of the matrix a with arbitrary vectors are relatively inexpensive this article defines the problem of estimating the spectral density carefully and discusses how to measure the accuracy of an approximate spectral density it then surveys a few kno

on injective jordan semi triple maps of matrix algebras abstract we show that every injective jordan semi triple map on the algebra m n f of all n n matrices with entries in a field f i e a map m n f m n f satisfying aba a b a for every a and b in m n f is given by a map of the following form there exist f 1 an injective homomorphism of f and an invertible t m n f such that either a t 1 a t for all a m n f or a t 1 a t t for all a m n f here a is the image of a under applied entrywise

reconstruction of thermally symmetrized quantum autocorrelation functions from imaginary time data in this paper i propose a technique for recovering quantum dynamical information from imaginary time data via the resolution of a one dimensional hamburger moment problem it is shown that the quantum autocorrelation functions are uniquely determined by and can be reconstructed from their sequence of derivatives at origin a general class of reconstruction algorithms is then identified according to theorem 3 the technique is advocated as especially effective for a certain class of quantum problems in continuum space for which only a few moments are necessary for such problems it is argued that the derivatives at origin can be evaluated by monte carlo simulations via estimators of finite variances in the limit of an infinite number of path variables finally a maximum entropy inversion algorithm for the hamburger moment problem is utilized to compute the quantum rate of reaction for a one dimensional symmetric eckart barrier

function reconstruction as a classical moment problem a maximum entropy approach we present a systematic study of the reconstruction of non negative functions via maximum entropy approach utilizing the information contained in a finite number of moments of the functions for testing the efficacy of the approach we reconstruct a set of functions using an iterative entropy optimization scheme and study the convergence profile as the number of moments is increased a wide variety of functions are considered that include a distribution with a sharp discontinuity an oscillatory function a distribution with singularities and finally a distribution with several spikes and fine structure the last example is important in the context of the determination of the natural density of the logistic map the convergence of the method is studied by comparing the moments of the approximated functions with the exact ones furthermore by varying the number of moments and iterations we examine to what extent the features of the functions such as the divergence behavior at singular points within the interval is reproduced the proximity of the reconstructed maximum entropy solution to the exact solution is examined via kullback leibler divergence and variation measures for different number of moments

jordan triple maps abstract let a be a standard operator algebra on a banach space of dimension 1 and b be an algebra over the field of rational numbers let k be a non zero rational number in this note we prove that if a bijection a b satisfies kaba k a b a for all a b a or k abc cba k a b c c b a for all a b c a then is additive

polynomial scheme for time evolution of open and closed quantum systems based on the generating function of laguerre polynomials we propose a laguerre polynomial expansion scheme in the calculation of the evolution of the time dependent schr odinger equation theoretical analysis and numerical tests show that the method is equally as good as the chebyshev polynomial expansion method in efficiency and accuracy with the additional merits that no scaling to the hamiltonian is needed and it has wider suitability

automorphisms of the poset of skew projections abstract any automorphism of the poset b of all idempotents of the algebra b h of all bounded linear operators in a hilbert space h dim h 3 is shown to be of the form p apa 1 or p ap a 1 with a c where c is the set of all semi linear bijections a h h if h is finite dimensional and the set of all continuous invertible linear or conjugate linear operators in h if h is infinite dimensional further applications of the method are outlined

orthogonality preserving transformations on indefinite inner product spaces generalization of uhlhorn s version of wigner s theorem abstract we present an analogue of uhlhorn s version of wigner s theorem on symmetry transformations for the case of indefinite inner product spaces this significantly generalizes a result of van den broek the proof is based on our main theorem which describes the form of all bijective transformations on the set of all rank one idempotents of a banach space which preserve zero products in both directions

stability analysis of finite difference schemes for quantum mechanical equations of motion for a pdf involving both space and time variables stability criteria are presently shown to change drastically when the equation contains i as in the quantum mechanical equations of motion it is further noted that the stability of finite difference schemes for quantum mechanical equations of motion depends on both spatial and temporal zoning it is possible to compare a free particle green s function to the solution of a simple diffusion equation and the quantum mechanical motion of a free particle to fresnel diffraction in optics

optimal choice of grid points in multidimensional pseudospectral fourier methods abstract the optimal choice of grid points for the multi dimensional pseudospectral fourier method is investigated optimal sampling is obtained by looking for the most isotropic sampling grid in momentum space the points of this grid are found to be positioned at the center of densely packed hard spheres forming an oblique grid it is found that by using this oblique grid the sampling efficiency can be enhanced compared to a rectangular grid by a factor ranging from 1 4 for three dimensions to 16 for eight dimensions the method is checked in five dimensions by calculating the laplacian in an oblique and in a cartesian grid the oblique grid was found to have superior accuracy using a factor of 2 5 less grid points the method is also illustrated in six dimensions by a calculation of the ground and first excited states of two interacting triplet hydrogen atoms

solving the maxwell equations by the chebyshev method a one step finite difference time domain algorithm we present a one step algorithm that solves the maxwell equations for systems with spatially varying permittivity and permeability by the chebyshev method we demonstrate that this algorithm may be orders of magnitude more efficient than current finite difference time domain fdtd algorithms

an averaging technique for highly oscillatory hamiltonian problems in this paper we are concerned with the numerical solution of highly oscillatory hamiltonian systems with a stiff linear part we construct an averaged system whose solution remains close to the exact one over bounded time intervals possesses the same adiabatic and hamiltonian invariants as the original system and is nonstiff we then investigate its numerical approximation through a method which combines a symplectic integration scheme and an acceleration technique for the evaluation of time averages developed in e cances et al numer math 100 2005 pp 211 232 eventually we demonstrate the efficiency of our approach on two test problems with one or several frequencies

a gautschi type method for oscillatory second order differential equations we study a numerical method for second order differential equations in which high frequency oscillations are generated by a linear part for example semilinear wave equations are of this type the numerical scheme is based on the requirement that it solves linear problems with constant inhomogeneity exactly we prove that the method admits second order error bounds which are independent of the product of the step size with the frequencies our analysis also provides new insight into the m ollified impulse method of garcia archilla sanz serna and skeel we include results of numerical experiments with the sine gordon equation

applying projective geometry to transformations on rank one idempotents abstract we present a short proof of molnar s characterization of bijective transformations on the set of all rank one idempotent operators on a banach space which preserve zero products in both directions an improvement in the finite dimensional case is given we apply these results to describe automorphisms of standard operator semigroups and to improve uhlhorn s version of wigner s theorem

maps on matrix algebras preserving idempotents abstract let m n be the algebra of all n n complex matrices and p n the set of all idempotents in m n suppose m n m n is a surjective map satisfying a b p n if and only if a b p n a b m n c then either is of the form a tat 1 a m n or is of the form a ta t t 1 a m n where t m n is a nonsingular matrix

a fourier method solution for the time dependent schrodinger equation as a tool in molecular dynamics a new method is presented for the solution of the time dependent schrbdinger equation in its application to physical and chemical molecular phenomena the method is based on discretizing space and time on a grid and using the fourier method to produce both spatial derivatives and second order differencing for time derivatives the method conserves norm and energy and preserves quantum mechanical commutation relations one and twodimensional examples where a comparison to analytic results is possible are investigated many phenomena of physics and chemistry have a common dynamical evolution pattern this pattern begins with an initial state which under the influence of the potentials evolves through time to produce a final asymptotic state among such phenomena are chemical reactions photodissociation unimolecular breakdown surface scattering and desorption the present study presents a numerical solution based on quantum mechanics for the common pattern or for the time dependent evolution of the state of the system the importance of exact numerical solutions is twofold first these solutions give a quantitative description of physical problems and also are of use as a bench mark to check approximation methods and to define their range of validity

random walk statistics in moment based o n tight binding and applications in carbon nanotubes a computational framework for a moment based o n tight binding atomistic method is presented analyzed and applied to the problem of electronic properties of deformed carbon nanotubes where n is the number of atoms in the system the moment based approach is based on the maximum entropy and kernel polynomial methods for constructing the electronic density of states from local statistical information about the environment around individual atoms random walk statistics are formally presented as the basis for several methods to collect the moments of the density of states in a computationally efficient manner the computational complexity and accuracy of these methods are systematically analyzed using these methods for the problem of deformed carbon nanotubes it is shown that the computational cost for some cases per atom scales as efficiently as o m phantom rule 0 2em 0ex mathrm log phantom rule 0 2em 0ex m where m is the desired number of moments in the expansion of the density of states these methods are compared to other methods such as direct diagonalization and a green s function approach

backward error analysis for numerical integrators backward error analysis has become an important tool for understanding the long time behavior of numerical integration methods this is true in particular for the integration of hamiltonian systems where backward error analysis can be used to show that a symplectic method will conserve energy over exponentially long periods of time such results are typically based on two aspects of backward error analysis i it can be shown that the modified vector fields have some qualitative properties which they share with the given problem and ii an estimate is given for the difference between the best interpolating vector field and the numerical method these aspects have been investigated recently for example by benettin and giorgilli in j statist phys 74 1994 pp 1117 1143 by hairer in ann numer math 1 1994 pp 107 132 and by hairer and lubich in numer math 76 1997 pp 441 462 in this paper we aim at providing a unifying framework and a simplification of the existing results and corresponding proofs our approach to backward error analysis is based on a simple recursive definition of the modified vector fields that does not require explicit taylor series expansion of the numerical method and the corresponding flow maps as in the above cited works as an application we discuss the long time integration of chaotic hamiltonian systems and the approximation of time averages along numerically computed trajectories

maps on matrix spaces it is well known that every automorphism of the full matrix algebra is inner we give a short proof of this statement and discuss several extensions of this theorem including structural results for multiplicative maps on matrix algebras characterizations of monotone and orthogonality preserving maps on idempotent matrices some nonlinear preserver results and some recent theorems concerning geometry of matrices we show that all these topics are closely related and point out the connections with physics and geometry several open problems are posed

endomorphisms of the poset of idempotent matrices abstract let d be a division ring n 3 an integer and p n d the poset of all n n idempotent matrices over d with the partial order defined by p q if p q q p p let t m n d be an invertible matrix and d d an endomorphism d d an anti endomorphism for any p p n d we denote by p p the idempotent matrix obtained from p by applying entrywise the map p n d p n d defined by p t p t 1 p p n d p t t p t 1 p p n d is an injective map preserving order in both directions every such map is called a standard map it has been known before that if d is an eas division ring then every injective order preserving map p n d p n d is either standard or of a very special degenerate form in this paper we use some ideas from geometry of algebraic homogeneous spaces and elementary field theory to give examples showing that the eas assumption is indispensable then we define generalized standard maps and using them we describe the general form of injective order preservers on p n d for an arbitrary division ring d our proof is shorter than the original one for the special case of eas division rings under somewhat stronger assumptions we get two characterizations of standard maps all the results are optimal as shown by counterexamples

chebyshev polynomial expansion of the localization length of hermitian and non hermitian random chains we study chebyshev polynomial expansion of the inverse localization length of hermitian and non hermitian random chains as a function of energy for hermitian models the expansion produces this energy dependent function numerically in one run of the algorithm this is in strong contrast to the standard transfer matrix method which produces the inverse localization length for a fixed energy in each run for non hermitian models as in the transfer matrix method our algorithm computes the inverse localization length for a fixed complex energy we also find a formula of the chebyshev polynomial expansion of the density of states of non hermitian models as explained in detail our algorithm for non hermitian models may be the only available efficient algorithm for finding the density of states of models with interactions

long time energy conservation of numerical methods for oscillatory differential equations we consider second order differential systems where high frequency oscillations are generated by a linear part we present a frequency expansion of the solution and we discuss two invariants of the system that determine the coefficients of the frequency expansion these invariants are related to the total energy and the oscillatory harmonic energy of the original system for the numerical solution we study a class of symmetric methods that discretize the linear part without error we are interested in the case where the product of the step size with the highest frequency can be large in the sense of backward error analysis we represent the numerical solution by a frequency expansion where the coefficients are the solution of a modified system this allows us to prove the near conservation of the total and the oscillatory energy over very long time intervals

generalized symmetry transformations on quaternionic indefinite inner product spaces an extension of quaternionic version of wigner s theorem uhlhorn s extension of wigner s unitary antiunitary theorem has been recently generalized by molnar to indefinite inner product spaces we present the quaternionic version of this result

automorphisms of mn partially ordered by rank subtractivity ordering let mn be the space of all n n matrices with coefficients in r or c where n 3 the rank subtractivity or minus partial order on mn is defined by a rsb iff rankb ranka rank b a we characterize bijective mappings on mn such that a rsb iff a rs b a b mn

error analysis of exponential integrators for oscillatory second order differential equations in this paper we analyse a family of exponential integrators for second order differential equations in which high frequency oscillations in the solution are generated by a linear part conditions are given which guarantee that the integrators allow second order error bounds independent of the product of the step size with the frequencies our convergence analysis generalizes

maximum entropy and the problem of moments a stable algorithm we present a technique for entropy optimization to calculate a distribution from its moments the technique is based upon maximizing a discretized form of the shannon entropy functional by mapping the problem onto a dual space where an optimal solution can be constructed iteratively we demonstrate the performance and stability of our algorithm with several tests on numerically difficult functions we then consider an electronic structure application the electronic density of states of amorphous silica and study the convergence of the fermi level with increasing number of moments

realization of holonomic constraints and freezing of high frequency degrees of freedom in the light of classical perturbation theory part ii the so called problem of the realization of the holonomic constraints of classical mechanics is here revisited in the light of nekhoroshev like classical perturbation theory precisely if constraints are physically represented by very steep potential wells with associated high frequency transversal vibrations then one shows that within suitable assumptions the vibrational energy and the energy associated to the constrained motion are separately almost constant for a very long time scale growing exponentially with the frequency i e with the rigidity of the constraint one aims to realize this result can also be applied to microscopic physics providing a possible entirely classical mechanism for the freezing of the high frequency degrees of freedom in terms of non equilibrium statistical mechanics according to some ideas expressed by boltzmann and jeans at the turn of the century in this part i we introduce the problem and prove a first theorem concerning the realization of a single constraint within a system of any number of degrees of freedom the problem of the realization of many constraints will be considered in a forthcoming part ii

analysis of a leap frog pseudospectral scheme for the schrodinger equation the numerical properties of a leap frog pseudospectral scheme for the schrodinger equation are analyzed stability second order accuracy in time and spectral accuracy in space are discussed considering the linear schrodinger equation with potential in a periodic setting further issues regarding phase error gauge invariance conservation properties and commutation relations are addressed results of numerical experiments are reported to demonstrate the validity and limitations of the theoretical findings and for comparison with the well known crank nicholson finite difference scheme

maps on essentially infinite idempotent operators abstract let h be an infinite dimensional real or complex hilbert space and i h the set of all bounded linear idempotent operators on h with infinite dimensional image and infinite dimensional kernel we characterize three types of maps on i h namely poset automorphisms bijective maps preserving orthogonality in both directions and bijective maps preserving commutativity in both directions

order preserving maps on the poset of upper triangular idempotent matrices abstract we obtain the general form of bijective order and orthogonality preserving maps on the poset of all n n upper triangular idempotent matrices over an arbitrary field f we also show that the assumption of surjectivity can be removed if every nonzero homomorphism g f f is surjective

modeling of the acoustic wave equation with transform methods numerical methods are described for the simulation of wave phenomena with application to the modeling of seismic data two separate topics are studied the first deals with the solution of the acoustic wave equation the second topic treats wave phenomena whose direction of propagation is restricted within 90 degrees from a given axis in the numerical methods developed here the wave field is advanced in time by using standard time differencing schemes on the other hand expressions including space derivative terms are computed by fourier transform methods this approach to computing derivatives minimizes truncation errors another benefit of transform methods becomes evident when attempting to restrict propagation to upward moving waves e g to avoid multiple reflections constraints imposed on the direction of the wave propagation are accomplished most precisely in the wavenumber domain the error analysis of the algorithms shows that truncation errors are due mainly to time discretization such er

randomized algorithms for estimating the trace of an implicit symmetric positive semi definite matrix we analyze the convergence of randomized trace estimators starting at 1989 several algorithms have been proposed for estimating the trace of a matrix by 1 m i 1m zit azi where the zi are random vectors different estimators use different distributions for the zis all of which lead to e 1 m i 1m zit azi trace a these algorithms are useful in applications in which there is no explicit representation of a but rather an efficient method compute ztaz given z existing results only analyze the variance of the different estimators in contrast we analyze the number of samples m required to guarantee that with probability at least 1 the relative error in the estimate is at most we argue that such bounds are much more useful in applications than the variance we found that these bounds rank the estimators differently than the variance this suggests that minimum variance estimators may not be the best we also make two additional contributions to this area the first is a specialized bound for projection matrices whose trace rank needs to be computed in electronic structure calculations the second is a new estimator that uses less randomness than all the existing estimators

long time step methods for oscillatory differential equations considered are numerical integration schemes for nondissipative dynamical systems in which multiple time scales are present it is assumed that one can do an explicit separation of the rhs forces into fast forces and slow forces such that i the fast forces contain the high frequency part of the solution ii the fast forces are conservative and iii the reduced problem consisting only of the fast forces can be integrated much more cheaply than the full problem the fast forces are allowed to have low frequency components particular applications for which the schemes are intended include n body problems for which most of the forces are slow and nonlinear wave phenomena for which the fast forces can be propagated by spectral methods the assumption of cheap integration of fast forces implies that the overall cost of integration is primarily determined by the step size used to sample the slow forces a long time step method is one in which this step size exceeds half the period of the fastest normal mode present in the full system an existing method that comes close to qualifying is the impulse method also known as verlet i and r respa it is shown that it might fail though for a couple of reasons first it suffers a serious loss of accuracy if the step size is near a multiple of the period of a normal mode and second it is unstable if the step size is near a multiple of half the period of a normal mode proposed in this paper is a mollified impulse method having an error bound that is independent of the frequency of the fast forces it is also shown to possess superior stability properties theoretical results are supplemented by numerical experiments the method is efficient and reasonably easy to implement

calculation of densities of states and spectral functions by chebyshev recursion and maximum entropy we present an efficient algorithm for calculating spectral properties of large sparse hamiltonian matrices such as densities of states and spectral functions the combination of chebyshev recursion and maximum entropy achieves high energy resolution without significant roundoff error machine precision or numerical instability limitations if controlled statistical or systematic errors are acceptable cpu and memory requirements scale linearly in the number of states the inference of spectral properties from moments is much better conditioned for chebyshev moments than for power moments we adapt concepts from the kernel polynomial method a linear chebyshev approximation with optimized gibbs damping to control the accuracy of fourier integrals of positive nonanalytic functions we compare the performance of kernel polynomial and maximum entropy algorithms for an electronic structure example

hua s fundamental theorems of the geometry of matrices and related results abstract we briefly survey some recent improvements of hua s fundamental theorem of the geometry of rectangular matrices then we discuss possible further generalizations as well as some related open problems in the theory of preservers we solve one such open problem using ovchinnikov s characterization of automorphisms of the poset of idempotent matrices using ovchinnikov s result we obtain a short proof of the fundamental theorem of the geometry of square matrices

maps on b h preserving involution abstract let b h be the algebra of all bounded linear operators on a complex hilbert space h we use the form of idempotence preserving maps to claracterize the form of involutivity preserving maps

hua s fundamental theorem of the geometry of matrices abstract let m n f be the space of all n n matrices over the field f n 2 two matrices a b m n f are adjacent if rank a b 1 hua s fundamental theorem of the geometry of square matrices characterizes bijective maps on m n f that preserve adjacency in both directions in this paper we treat a long standing open problem whether the result of hua holds true under the weaker assumption of preserving the adjacency in one direction only we answer this question in the affirmative in the case that every nonzero homomorphism f f f is surjective for example the field of real numbers has this property in order to prove this result we have to improve ovchinnikov s characterization of automorphisms of the poset of idempotent matrices

idempotence preserving maps without the linearity and surjectivity assumptions abstract let m n f be the space of all n n matrices over a field f of characteristic not 2 and let p n f be the subset of m n f consisting of all n n idempotent matrices we denote by n f the set of all maps from m n f to itself satisfying a b p n f if and only if a b p n f for every a b m n f and f it was shown that n f if and only if there exists an invertible matrix p m n f such that either a pap 1 for every a m n f or a patp 1 for every a m n f this improved dolinar s result by omitting the surjectivity assumption and extending the complex field to any field of characteristic not 2