the matrix exponential there are many different methods to calculate the exponential of a matrix series methods differential equations methods polynomial methods matrix decomposition methods and splitting methods none of which is entirely satisfactory from either a theoretical or a computational point of view how then should the matrix exponential be introduced in an elementary differential equations course for engineering students for example with a minimum of mathematical prerequisites in this note a method is given that uses the students knowledge of homogeneous linear differential equations with constant coefficients and the cayley hamilton theorem the method is not new what is new is the approach

the algebra of complex 2 2 matrices and a general closed baker campbell hausdorff formula we derive a closed formula for the baker campbell hausdorff series expansion in the case of complex matrices for arbitrary matrices a and b and a matrix z such that our result expresses z as a linear combination of a and b their commutator and the identity matrix i the coefficients in this linear combination are functions of the traces and determinants of a and b and the trace of their product the derivation proceeds purely via algebraic manipulations of the given matrices and their products making use of relations developed here based on the cayley hamilton theorem as well as a characterization of the consequences of and or its determinant being zero or otherwise as a corollary of our main result we also derive a closed formula for the zassenhaus expansion we apply our results to several special cases most notably the parametrization of the product of two matrices and a verification of the recent result of van brunt and visser 2015 j phys a math theor 48 225207 for complex matrices in this latter case deriving also the related zassenhaus formula which turns out to be quite simple we then show that this simple formula should be valid for all matrices and operators

phase separation and coarsening in one dimensional driven diffusive systems local dynamics leading to long range hamiltonians a driven system of three species of particles diffusing on a ring is studied in detail the dynamics is local and conserves the three densities a simple argument suggesting that the model should phase separate and break the translational symmetry is given we show that for the special case where the three densities are equal the model obeys detailed balance and the steady state distribution is governed by a hamiltonian with asymmetric long range interactions this provides an explicit demonstration of a simple mechanism for breaking of ergodicity in one dimension the steady state of finite size systems is studied using a generalized matrix product ansatz the coarsening process leading to phase separation is studied numerically and in a mean field model the system exhibits slow dynamics due to trapping in metastable states whose number is exponentially large in the system size the typical domain size is shown to grow logarithmically in time generalizations to a larger number of species are discussed

exact results for one dimensional cellular automata with different types of updates we study two common types of time noncontinuous updates for one dimensional stochastic cellular automata with arbitrary nearest neighbor interactions and arbitrary open boundary conditions we first construct the stationary states using the matrix product formalism this construction then allows to prove a general connection between the stationary states which are produced by the two different types of updates using this connection we derive explicit relations between the densities and correlation functions for these different stationary states

practical computation of matrix functions abstract several new representations for an analytic function f a of a complex matrix a and in particular for e at and a t are derived which also are numerically useful in that they avoid the computation of eigenvalues of a

on exponential of split quaternionic matrices abstract the exponential of a matrix plays an important role in the theory of lie groups the main purpose of this paper is to examine matrix groups over the split quaternions and the exponential map from their lie algebras into the groups since the set of split quaternions is a noncommutative algebra the way of computing the exponential of a matrix over the split quaternions is more difficult than calculating the exponential of a matrix over the real or complex numbers therefore we give a method of finding exponential of a split quaternion matrix by its complex adjoint matrix

conjugacy of finite biprefix codes two languages x and y are called conjugates if they satisfy the conjugacy equation xz zy for some non empty language z we will compare solutions of this equation with those of the corresponding equation of words and study the case of finite biprefix codes x and y we show that the maximal z in this case is rational we will also characterize x and y in the case where they are both finite biprefix codes this yields the decidability of the conjugacy of two finite biprefix codes

some explicit formulas for the polynomial decomposition of the matrix exponential and applications abstract this paper is devoted to the study of some formulas for polynomial decomposition of the exponential of a square matrix a more precisely we suppose that the minimal polynomial ma x of a is known and has degree m therefore eta is given in terms of p0 a pm 1 a where the pj a are polynomials in a of degree less than m and some explicit analytic functions examples and applications are given in particular the two cases m 5 and m 6 are considered

phase diagram of a generalized abc model on the interval we study the equilibrium phase diagram of a generalized abc model on an interval of the one dimensional lattice each site i 1 n is occupied by a particle of type a b c with the average density of each particle species n n r fixed these particles interact via a mean field nonreflection symmetric pair interaction the interaction need not be invariant under cyclic permutation of the particle species as in the standard abc model studied earlier we prove in some cases and conjecture in others that the scaled infinite system n i n x 0 1 has a unique density profile x except for some special values of the r for which the system undergoes a second order phase transition from a uniform to a nonuniform periodic profile at a critical temperature t c 3 sqrt r a r b r c 2 pi

explicit factorization of the vandermonde matrix the lu factorization of the vandermonde matrix is obtained using complete symmetric functions and the lower and upper triangular matrices are in turn factorized into 1 banded matrices thus expressing the vandermonde matrix as a product of 1 banded matrices

solutions of linear recurrence equations solutions are given to general homogeneous and non homogeneous recurrence equations defined on the set of integers solutions to homogeneous recurrence equations are given as i an infinite series of terms involving the partial ordinary bell polynomial ii an infinite series of terms involving the complete ordinary bell polynomial iii a weighted finite sum of terms involving powers solutions to non homogeneous recurrence equations are given as i a finite series of terms involving the complete ordinary bell polynomial ii a weighted infinite sum of terms involving powers computational issues of these solutions are also discussed

exact solution of a cellular automaton for traffic we present an exact solution of a probabilistic cellular automaton for traffic with open boundary conditions e g cars can enter and leave a part of a highway with certain probabilities the model studied is the asymmetric exclusion process asep with simultaneous updating of all sites it is equivalent to a special case vmax 1 of the nagel schreckenberg model for highway traffic which has found many applications in real time traffic simulations the simultaneous updating induces additional strong short range correlations compared to other updating schemes the stationary state is written in terms of a matrix product solution the corresponding algebra which expresses a system size recursion relation for the weights of the configurations is quartic in contrast to previous cases in which the algebra is quadratic we derive the phase diagram and compute various properties such as density profiles two point functions and the fluctuations in the number of particles cars in the system the current and the density profiles can be mapped onto the asep with other time discrete updating procedures through use of this mapping our results also give new results for these models

pole assignment of linear uncertain systems in a sector via a lyapunov type approach the problem of designing robust control laws in performance and in stability for uncertain linear systems is considered performances are taken into account using root clustering of the closed loop dynamic matrix in a sector of the complex plane a synthesis procedure based on a sufficient condition for quadratic stabilization and root clustering such as stabilizability is given using an auxiliary convex problem the results are illustrated by a significant example from the literature

linear recurrence relations in the algebra of matrices and applications abstract we study here some linear recurrence relations in the algebra of square matrices with the aid of the cayley hamilton theorem we derive some explicit formulas for a n n r and e ta for every r r matrix a in terms of the coefficients of its characteristic polynomial and matrices a j where 0 j r 1

local inhomogeneity in asymmetric simple exclusion processes with extended objects totally asymmetric simple exclusion processes tasep with particles which occupy more than one lattice site and with a local inhomogeneity far away from the boundaries are investigated these non equilibrium processes are relevant for the understanding of many biological and chemical phenomena the steady state phase diagrams currents and bulk densities are calculated using a simple approximate theory and extensive monte carlo computer simulations it is found that the phase diagram for tasep with a local inhomogeneity is qualitatively similar to homogeneous models although the phase boundaries are significantly shifted the complex dynamics is discussed in terms of domain wall theory for driven lattice systems

two way traffic flow exactly solvable model of traffic jam we study completely asymmetric two channel exclusion processes in one dimension it describes a two way traffic flow with cars moving in opposite directions the interchannel interaction makes cars slow down in the vicinity of approaching cars in the other lane particularly we consider in detail the system with a finite density of cars on one lane and a single car on the other when the interchannel interaction reaches a critical value a traffic jam occurs which turns out to be of first order phase transition we derive exact expressions for the average velocities the current the density profile and the k point density correlation functions we also obtain the exact probability of two cars being in one lane of distance r apart provided there is a finite density of cars on the other lane and show that the two cars form a weakly bound state in the jammed phase

on the matrix powers and exponential by the r generalized fibonacci sequences methods the companion matrix case properties of some r generalized fibonacci sequences in the algebra of square matrices gl r c allow us to derive some explicit formulas for the matrix powers an n 0 and exponential eta for every a gl r c connection with the verde star method is displayed for a companion matrix a we present some explicit combinatorial formulas for an n 0 and eta furthermore the chen louck s theorem is derived

exact large deviation functional of a stationary open driven diffusive system the asymmetric exclusion process we consider the asymmetric exclusion process asep in one dimension on sites i 1 n in contact at sites i 1 and i n with infinite particle reservoirs at densities a and b as a and b are varied the typical macroscopic steady state density profile x x a b obtained in the limit n l b a exhibits shocks and phase transitions here we derive an exact asymptotic expression for the probability of observing an arbitrary macroscopic profile rho x text p n rho x sim exp l mathcal f a b rho x rho a rho b so that mathcal f is the large deviation functional a quantity similar to the free energy of equilibrium systems we find as in the symmetric purely diffusive case q 1 treated in an earlier work that mathcal f is in general a non local functional of x unlike the symmetric case however the asymmetric case exhibits ranges of the parameters for which mathcal f rho x is not convex and others for which mathcal f rho x has discontinuities in its second derivatives at x x in the latter ranges the fluctuations of order 1 sqrt n in the density profile near x are then non gaussian and cannot be calculated from the large deviation function

yang lee zeros for a nonequilibrium phase transition equilibrium systems which exhibit a phase transition can be studied by investigating the complex zeros of the partition function this method pioneered by yang and lee has been widely used in equilibrium statistical physics we show that an analogous treatment is possible for a nonequilibrium phase transition into an absorbing state by investigating the complex zeros of the survival probability of directed percolation processes we demonstrate that the zeros provide information about universal properties moreover we identify certain nontrivial points where the survival probability for bond percolation can be computed exactly

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

on the k generalized fibonacci numbers and high order linear recurrence relations in this paper using elementary symmetric function and complete homogeneous symmetric function we obtain a determinant formula for the k generalized fibonacci sequence the relationship between the k generalized fibonacci sequence and the order k linear homogeneous recurrence relation has been investigated and a general solution for the latter is also derived furthermore we obtain an explicit expression for the elements in the nth power of the companion matrix in terms of k generalized fibonacci numbers

more explicit formulas for the matrix exponential abstract the matrix exponential plays an important role in the study of dynamical systems and linear systems an elementary method is discussed for computing eta for general a c n n it is shown that this method can be applied to generate explicit formulas for eta in various forms using this method some well known formulas are rederived and some new formulas are also derived as applications explicit formulas for n 3 or n 4 are given in each case a characterization of eta is also made based on the entries of a alone

matrix exponentials another approach the exponential of a matrix and the spectral decomposition of a matrix can be computed knowing nothing more than the eigenvalues of the matrix and the cayley hamilton theorem the arrangement of the ideas in this paper is simple enough to be taught to beginning students of odes

disordered totally asymmetric simple exclusion process exact results we study the effect of quenched spatial disorder on the current carrying steady states of a totally asymmetric simple exclusion process with spatially disordered jump rates expressions for the steady state weights and the current are derived for this model in one dimension these solutions are exploited to study analytically the exact symmetries of the system in particular we confirm the recent numerical observation of tripathy and barma phys rev e 58 1911 1998 that the magnitude of the steady state current is left invariant when the direction of all the allowed particle jumps are reversed

richardson lucy deblurring for scenes under a projective motion path this paper addresses how to model and correct image blur that arises when a camera undergoes ego motion while observing a distant scene in particular we discuss how the blurred image can be modeled as an integration of the clear scene under a sequence of planar projective transformations i e homographies that describe the camera s path this projective motion path blur model is more effective at modeling the spatially varying motion blur exhibited by ego motion than conventional methods based on space invariant blur kernels to correct the blurred image we describe how to modify the richardson lucy rl algorithm to incorporate this new blur model in addition we show that our projective motion rl algorithm can incorporate state of the art regularization priors to improve the deblurred results the projective motion path blur model along with the modified rl algorithm is detailed together with experimental results demonstrating its overall effectiveness statistical analysis on the algorithm s convergence properties and robustness to noise is also provided

some explicit formulas for the matrix exponential formulas are derived for the exponential of an arbitrary 2 2 matrix in terms of either its eigenvalues or entries these results are then applied to the second order mechanical vibration equation with weak or strong damping some formulas for the exponential of n n matrices are given for matrices that satisfy an arbitrary quadratic polynomial besides the above mentioned 2 2 matrices these results encompass involutory rank 1 and idempotent matrices consideration is then given to n n matrices that satisfy a special cubic polynomial these results are applied to the case of a 3 3 skew symmetric matrix whose exponential represents the constant rotation of a rigid body about a fixed axis

the combinatorial power of the companion matrix abstract using combinatorial methods we obtain the explicit polynomials for all elements in an arbitrary power of the companion matrix depending on n variables and provide some interesting applications and relationships to waring s formula on symmetric functions the general solution to homogeneous linear recurrence relations the multiplicative inverse of formal power series the generating function of compositions of numbers a unified approach to chebyshev polynomials including two recently discovered classes that satisfy analogous smallest norm and orthogonality properties subject to different weight functions dickson polynomials of various kinds arising from the theory of finite fields combinatorial expansions of toeplitz matrices and the recent notion of cycle dissections involving a bijective study of waring s formula

computations with infinite toeplitz matrices and polynomials we relate polynomial computations with operations involving infinite band toeplitz matrices and show applications to the numerical solution of markov chains of nonlinear matrix equations to spectral factorizations and to the solution of finite toeplitz systems in particular two matrix versions of graeffe s iteration are introduced and their convergence properties are analyzed correlations between graeffe s iteration for matrix polynomials and cyclic reduction for block toeplitz matrices are pointed out the paper contains a systematic treatment of known topics and presentation of new results improvements and extensions

pushdown processes a pushdown game is a two player perfect information infinite game on a transition graph of a pushdown automaton a winning condition in such a game is defined in terms of states appearing infinitely often in the play it is shown that if there is a winning strategy in a pushdown game then there is a winning strategy realized by a pushdown automaton an exptime procedure for finding a winner in a pushdown game is presented the procedure is then used to solve the model checking problem for the pushdown processes and the propositional calculus the problem is shown to be dexptime complete

conway s problem for three word sets we prove two results on commutation of languages first we show that the maximal language commuting with a three element language i e its centralizer is rational thus giving an affirmative answer to a special case of a problem proposed by conway in 1971 second we characterize all languages commuting with a three element code the characterization is similar to the one proved by bergman for polynomials over noncommuting variables see trans am math soc 137 1969 327 and algebraic combinatorics on words cambridge university press cambridge 2000 a language commutes with a three element code x if and only if it is a union of powers of x

phase diagram of the abc model on an interval the three species asymmetric abc model was initially defined on a ring by evans kafri koduvely and mukamel and the weakly asymmetric version was later studied by clincy derrida and evans here the latter model is studied on a one dimensional lattice of n sites with closed zero flux boundaries in this geometry the local particle conserving dynamics satisfies detailed balance with respect to a canonical gibbs measure with long range asymmetric pair interactions this generalizes results for the ring case where detailed balance holds and in fact the steady state measure is known only for the case of equal densities of the different species in the latter case the stationary states of the system on a ring and on an interval are the same we prove that in the limit n the scaled density profiles are given by pieces of the periodic trajectory of a particle moving in a quartic confining potential we further prove uniqueness of the profiles i e the existence of a single phase in all regions of the parameter space of average densities and temperature except at low temperature with all densities equal in this case a continuum of phases differing by translation coexist the results for the equal density case apply also to the system on the ring and there extend results of clincy et al

functions and jordan canonical forms of riordan matrices abstract this paper collects results about riordan arrays in the framework of matrix functions actually the following methodology applies to any square matrix m m with exactly one eigenvalue of algebraic multiplicity m n generalized lagrange bases are used to construct hermite polynomials that interpolate a family of functions moreover we show a parallel application of such functions via jordan canonical forms and case studies are given

on explicit formulas of the principal matrix pth root by polynomial decompositions abstract we present some explicit formulas for calculating the principal p th root of a square matrix the main tools are based on various polynomial decompositions of the principal matrix p th root and well known properties of the linear recursive sequences

the asymmetric exclusion process comparison of update procedures the asymmetric exclusion process asep has attracted a lot of interest not only because of its many applications e g in the context of the kinetics of biopolymerization and traffic flow theory but also because it is a paradigmatic model for nonequilibrium systems here we study the asep for different types of updates namely random sequential sequential sublattice parallel and parallel in order to compare the effects of the different update procedures on the properties of the stationary state we use large scale monte carlo simulations and analytical methods especially the so called matrix product ansatz mpa we present in detail the exact solution for the model with sublattice parallel and sequential updates using the mpa for the case of parallel update which is important for applications like traffic flow theory we determine the phase diagram the current and density profiles based on monte carlo simulations we furthermore suggest an mpa for that case and derive the corresponding matrix algebra

exact solution of an exclusion process with three classes of particles and vacancies we present an exact solution for an asymmetric exclusion process on a ring with three classes of particles and vacancies using a matrix product ansatz we find exact expressions for the weights of the configurations in the stationary state the solution involves tensor products of quadratic algebras

on explicit formulas for the principal matrix logarithm we describe a method for evaluating both the fibonacci horner and the polynomial decomposition of the principal matrix logarithm with a view to solve the lifting problem of its explicit computation the binet formula for linear recursive sequences serves as a triggering factor for giving the exact formula we supply some illustrative examples

improving formulas for the eigenvalues of finite block toeplitz tridiagonal matrices abstract after a short overview improvements based on the kronecker product are proposed for the eigenvalues of n n block toeplitz tridiagonal block tt matrices with k k matrix entries common in applications some extensions of the spectral properties of the toeplitz tridiagonal matrices are pointed out the eigenvalues of diagonalizable symmetric and skew symmetric block tt matrices are studied besides if certain matrix square root is well defined it is proved that every block tt matrix with commuting matrix entries is isospectral to a related symmetric block tt one further insight about the eigenvalues of hierarchical hermitian block tt matrices of use in the solution of pdes is also achieved

large deviation functional of the weakly asymmetric exclusion process we obtain the large deviation functional of a density profile for the asymmetric exclusion process of l sites with open boundary conditions when the asymmetry scales like l 1 we recover as limiting cases the expressions derived recently for the symmetric ssep and the asymmetric asep cases in the asep limit the nonlinear differential equation one needs to solve can be analysed by a method which resembles the wkb method

some open problems in matrix theory arising in linear systems and control control theory has long provided a rich source of motivation for developments in matrix theory accordingly we discuss some open problems in matrix theory arising from theoretical and practical issues in linear systems theory and feedback control the problems discussed include robust stability matrix exponent induced norms stabilizability and pole assignability and nonstandard matrix equations a substantial number of references are included to acquaint matrix theorists with problems and trends in this application area feedback control theory has long provided a rich source of motivation for developments in matrix theory the purpose of this paper is to discuss several open problems in matrix theory that arise from theoretical and practical issues in feedback control theory and the associated area of linear systems theory many of these problems are remarkably simple to state are of intense interest in control theory and applications and yet remain unsolved besides leading to the resolution of these problems it is hoped that this paper will help to stimulate increased interaction between matrix and control theorists accord

companion factorization in the general linear group gl n c and applications abstract the companion factorization for nonsingular matrices belonging to the general linear group gl n c is studied here the entries in the last row of the companion matrices are explicitly represented in terms of determinants of proper sub matrices of the matrix being factorized the computation of the inverse of a nonsingular matrix is given as an application of this factorization

one dimensional partially asymmetric simple exclusion process on a ring with a defect particle the effect of a moving defect particle for the one dimensional partially asymmetric simple exclusion process on a ring is considered the current of the ordinary particles the speed of the defect particle and the density profile of the ordinary particles are calculated exactly the phase diagram for the correlation length is identified as a by product the average and the variance of the particle density of the one dimensional partially asymmetric simple exclusion process with open boundaries are also computed

algebraic solution of the lindblad equation for a collection of multilevel systems coupled to independent environments we consider the lindblad equation for a collection of multilevel systems coupled to independent environments the equation is symmetric under the exchange of the labels associated with each system and thus the open system dynamics takes place in the permutation symmetric subspace of the operator space the dimension of this space grows polynomially with the number of systems we construct a basis of this space and a set of superoperators whose action on this basis is easily specified for a given number of levels m these superoperators are written in terms of a bosonic realization of the generators of the lie algebra in some cases these results enable finding an analytic solution of the master equation using known lie algebraic methods to demonstrate this we obtain an analytic expression for the state operator of a collection of three level atoms coupled to independent radiation baths when analytic solutions are difficult to find the basis and the superoperators can be used to considerably reduce the computational resources required for simulations

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

commutation with codes the centralizer of a set of words x is the largest set of words c x commuting with x xc x c x x it has been a long standing open question due to j h conway regular algebra and finite machines chapman hall london 1971 whether the centralizer of any rational set is rational while the answer turned out to be negative in general see m kunc proc of icalp 2004 lecture notes in computer science vol 3142 springer berlin 2004 pp 870 881 we prove here that the situation is different for codes the centralizer of any rational code is rational and if the code is finite then the centralizer is finitely generated this result has been previously proved only for binary and ternary sets of words in a series of papers by the authors and for prefix codes in an ingenious paper by b ratoandromanana rairo inform theor 23 4 1989 425 444 many of the techniques we use in this paper follow her ideas we also give in this paper an elementary proof for the prefix case

nonequilibrium steady states of matrix product form a solver s guide we consider the general problem of determining the steady state of stochastic nonequilibrium systems such as those that have been used to model among other things biological transport and traffic flow we begin with a broad overview of this class of driven diffusive systems which includes exclusion processes focusing on interesting physical properties such as shocks and phase transitions we then turn our attention specifically to those models for which the exact distribution of microstates in the steady state can be expressed in a matrix product form in addition to a gentle introduction to this matrix product approach how it works and how it relates to similar constructions that arise in other physical contexts we present a unified pedagogical account of the various means by which the statistical mechanical calculations of macroscopic physical quantities are actually performed we also review a number of more advanced topics including nonequilibrium free energy functionals the classification of exclusion processes involving multiple particle species existence proofs of a matrix product state for a given model and more complicated variants of the matrix product state that allow various types of parallel dynamics to be handled we conclude with a brief discussion of open problems for future research

finite dimensional representations of the quadratic algebra applications to the exclusion process we study the one dimensional partially asymmetric simple exclusion process asep with open boundaries that describes a system of hard core particles hopping stochastically on a chain coupled to reservoirs at both ends derrida and coworkers showed in 1993 that the stationary probability distribution of this model can be represented as a trace on a quadratic algebra closely related to the deformed oscillator algebra we construct all finite dimensional irreducible representations of this algebra this enables us to compute the stationary bulk density as well as all correlation lengths for the asep on a set of special curves of the phase diagram

an efficient algorithm for computing the baker campbell hausdorff series and some of its applications we provide a new algorithm for generating the baker campbell hausdorff bch series z log exey in an arbitrary generalized hall basis of the free lie algebra l x y generated by x and y it is based on the close relationship of l x y with a lie algebraic structure of labeled rooted trees with this algorithm the computation of the bch series up to degree of 20 111 013 independent elements in l x y takes less than 15min on a personal computer and requires 1 5gbytes of memory we also address the issue of the convergence of the series providing an optimal convergence domain when x and y are real or complex matrices

stochastic dynamics of discrete curves and multi type exclusion processes this study deals with continuous limits of interacting one dimensional diffusive systems arising from stochastic distortions of discrete curves with various kinds of coding representations these systems are essentially of a reaction diffusion nature in the non reversible case the invariant measure has in general a non gibbs form the corresponding steady state regime is analyzed in detail by using a tagged particle together with a state graph cycle expansion of the probability currents as a consequence the constants appearing in lotka volterra equations which describe the fluid limits of stationary states can be traced back directly at the discrete level to tagged particle cycles coefficients current fluctuations are also studied and the lagrangian is obtained via an iterative scheme the related hamilton jacobi equation which leads to the large deviation functional is investigated and solved in the reversible case just for the sake of checking

special cases of a matrix exponential formula abstract some low dimension or low rank cases of a formula for a product of exponentials of matrices are studied using eigenvalue considerations counterexamples to certain natural conjectures involving a product of matrix exponentials are also given and comments relating to certain recently discovered exponential identities are included

determination of a matrix function using the divided difference method of newton and the interpolation technique of hermite computing a function f a of an n by n matrix a is a frequently occurring problem in control theory and other applications in this paper we introduce an effective approach for the determination of matrix function f a we propose a new technique which is based on the extension of newton divided difference and the interpolation technique of hermite and using the eigenvalues of the given matrix a the new algorithm is tested on several problems to show the efficiency of the presented method finally the application of this method in control theory is highlighted

on the inversion of the vandermonde matrix abstract the inversion of the vandermonde matrix has received much attention for its role in the solution of some problems of numerical analysis and control theory this work deals with the problem of getting an explicit formula for the generic element of the inverse we derive two algorithms in o n2 and o n3 and compare them with the parker traub and the bjorck pereyra algorithms

spectral degeneracies in the totally asymmetric exclusion process we study the spectrum of the markov matrix of the totally asymmetric exclusion process tasep on a one dimensional periodic lattice at arbitrary filling although the system does not possess obvious symmetries except translation invariance the spectrum presents many multiplets with degeneracies of high order when the size of the lattice and the number of particles obey some simple arithmetic rules this behaviour is explained by a hidden symmetry property of the bethe ansatz assuming a one to one correspondence between the solutions of the bethe equations and the eigenmodes of the markov matrix we derive combinatorial formulae for the orders of degeneracy and the number of multiplets these results are confirmed by exact diagonalisations of small size systems this unexpected structure of the tasep spectrum suggests the existence of an underlying large invariance group

synchronous asymmetric exclusion processes for a fully synchronous asymmetric exclusion process with open or closed boundaries only partial analytic results are known owing to the appearance of strong short range correlations which invalidate simple mean field approximations here we present a new method for calculating basic properties of nonequilibrium steady states and calculate densities fluxes travel times spatial and temporal correlation functions phase diagrams profiles and widths of boundary layers and interfaces between phases in coexistence as well as their microstructures this paper is based on two new elements i a microscopic characterization of order parameters and local configurations in the relevant phases based on the microdynamics of the model and ii an improved mean field approximation which neglects certain four point and higher order correlation functions it is conjectured that the density profiles obtained here are exact up to terms that are exponentially small in the system size

functions of matrices abstract we use the cayley hamilton theorem and the sequence of horner polynomials associated with a polynomial w z to obtain explicit formulas for functions of the form f ta where f is defined by a convergent power series and a is a square matrix we use a well known explicit formula for the resolvent of a and show that f ta is the hadamard product of f t and the function i ta 1 which is easily obtained from the resolvent

solving some generalized vandermonde systems and inverse of their associate matrices via new approaches for the binet formula the aim of this paper is to exhibit new expressions for the binet formula without solving any type of vandermonde systems this allows us to establish a tractable formula for generalized fibonacci sequences as a consequence new solutions of the vandermonde systems and some special generalized vandermonde systems are provided moreover the inverses of their associated matrices are explicated some new results are formulated also others are recovered and generalized finally comparisons with other methods are broached and some numerical examples are supplied

phase transition in the abc model recent studies have shown that one dimensional driven systems can exhibit phase separation even if the dynamics is governed by local rules the mathrm abc model which comprises three particle species that diffuse asymmetrically around a ring shows anomalous coarsening into a phase separated steady state in the limiting case in which the dynamics is symmetric and the parameter q describing the asymmetry tends to one no phase separation occurs and the steady state of the system is disordered in the present work we consider the weak asymmetry regime q mathrm exp ensuremath ensuremath beta n where n is the system size and study how the disordered state is approached in the case of equal densities we find that the system exhibits a second order phase transition at some nonzero ensuremath beta c the value of ensuremath beta c 2 ensuremath pi sqrt 3 and the optimal profiles can be obtained by writing the exact large deviation functional for nonequal densities we write down mean field equations and analyze some of their predictions

the commutation of finite sets a challenging problem we prove that given a set x of two nonempty words a set y of nonempty words commutes with x if and only if either y is a union of powers of x or x y subseteq t for some primitive word t we also show that the same holds for certain special types of codes but does not hold in general for sets of cardinality at least four

exact solution of the master equation for the asymmetric exclusion process using the bethe ansatz we obtain the exact solution of the master equation for the totally asymmetric exclusion process on an infinite one dimensional lattice we derive explicit expressions for the conditional probabilitiesp x1 xn t y1 yn 0 of findingn particles on lattices sitesx1 xn at timet with initial occupationy1 yn at timet 0

the grand canonical abc model a reflection asymmetric mean field potts model we investigate the phase diagram of a three component system of particles on a one dimensional filled lattice or equivalently of a one dimensional three state potts model with reflection asymmetric mean field interactions the three types of particles are designated as a b and c the system is described by a grand canonical ensemble with temperature t and chemical potentials t a t b and t c we find that for a b c the system undergoes a phase transition from a uniform density to a continuum of phases at a critical temperature for other values of the chemical potentials the system has a unique equilibrium state as is the case for the canonical ensemble for this abc model the grand canonical ensemble is the stationary measure satisfying detailed balance for a natural dynamics we note that where tc is the critical temperature for a similar transition in the canonical ensemble at fixed equal densities ra rb rc 1 3

traffic of molecular motors through tube like compartments the traffic of molecular motors through open tube like compartments is studied using lattice models these models exhibit boundary induced phase transitions related to those of the asymmetric simple exclusion process asep in one dimension the location of the transition lines depends on the boundary conditions at the two ends of the tubes three types of boundary conditions are studied a periodic boundary conditions which correspond to a closed torus like tube b fixed motor densities at the two tube ends where radial equilibrium holds locally and c diffusive motor injection at one end and diffusive motor extraction at the other end in addition to the phase diagrams we also determine the profiles for the bound and unbound motor densities using mean field approximations and monte carlo simulations our theoretical predictions are accessible to experiments

truncated lucas sequence and its period abstract we prove some theorems concerning the truncated lucas sequences we also state some conjectures concerning the period mod m of truncated lucas sequences where the integer m is at least 2 in addition we present some computer generated results that tend to support these conjectures

classroom note a note on the matrix exponential this note is devoted to simplifying one method to calculate the exponential of a matrix presented by i e leonard in siam rev 38 1996 pp 507 512

geometric direct search algorithms for image registration a widely used approach to image registration involves finding the general linear transformation that maximizes the mutual information between two images with the transformation being rigid body i e belonging to se 3 or volume preserving i e belonging to sl 3 in this paper we present coordinate invariant geometric versions of the nelder mead optimization algorithm on the groups sl 3 se 3 and their various subgroups that are applicable to a wide class of image registration problems because the algorithms respect the geometric structure of the underlying groups they are numerically more stable and exhibit better convergence properties than existing local coordinate based algorithms experimental results demonstrate the improved convergence properties of our geometric algorithms

matrix product ground states for exclusion processes with parallel dynamics we show in the example of a one dimensional asymmetric exclusion process that stationary states of models with parallel dynamics may be written in a matrix product form the corresponding algebra is quadratic and involves three different matrices using this formalism we prove previous conjectures for the equal time correlation functions of the model

large deviation of the density profile in the steady state of the open symmetric simple exclusion process we consider an open one dimensional lattice gas on sites i 1 n with particles jumping independently with rate 1 to neighboring interior empty sites the simple symmetric exclusion process the particle fluxes at the left and right boundaries corresponding to exchanges with reservoirs at different chemical potentials create a stationary nonequilibrium state sns with a steady flux of particles through the system the mean density profile in this state which is linear describes the typical behavior of a macroscopic system i e this profile occurs with probability 1 when n the probability of microscopic configurations corresponding to some other profile x x i n has the asymptotic form exp n f f is the large deviation functional in contrast to equilibrium systems for which f eq is just the integral of the appropriately normalized local free energy density the f we find here for the nonequilibrium system is a nonlocal function of this gives rise to the long range correlations in the sns predicted by fluctuating hydrodynamics and suggests similar non local behavior of f in general sns where the long range correlations have been observed experimentally

on effective construction of the greatest solution of language inequality xa bx in this paper we consider effective constructions of the greatest solution of the language inequality xa bx it has been proved by kunc in 2005 that the greatest solution of xa bx is regular provided that b is regular no matter what a is however this proof is based on kruskal s tree theorem and does not provide any effective way to construct the greatest solution we focus on this gap in this paper we give an effective construction of the greatest solution for the following two cases i a b are regular and there exists k 1 such that pref b a k b kpref b a where pref b is the set of prefixes of words in b ii a b are regular and b is a code with finite decoding delay our construction takes the point of view of games as shown by kunc in his regularity proof the construction of the greatest solution can be reduced to the construction of the winning region of a two player game our contribution is to show that the winning regions of the two player game for the two cases can be constructed effectively the main ingredient of the construction for the first case is a shrinking lemma for the words on which one of the players has a winning strategy while the construction for the second case is based on the observation that the two player game can be reduced to a two player reachability game played on the transition graph of a one counter machine