some properties of a random walk on a comb structure abstract we analyze transport properties of a random walk on a comb structure which serves as a model for a random walk on the backbone of a percolation cluster it is shown that the random walk along the x axis which is the analog of the backbone exhibits anomalous diffusion in that x 2 n n 1 2 and the expected number of x sites visited is proportional to n 1 4 for large n the distribution function is found to be a two dimensional gaussian if a field in the x direction so that diffusion is asymmetric the expected displacement is found to be asymptotically proportional to n 1 2

exponential energy decay of solutions for an integro differential equation with strong damping abstract the initial boundary value problem for an integro differential equation with strong damping in 0 u t t m u 2 2 u 0 t g t s u s d s u t f u is considered where is a bounded domain with a smooth boundary the asymptotic behavior of solutions is discussed under some conditions on g decay estimates of the energy function of solutions are also given via an integral inequality introduced by komornik 1994 11

existence and nonexistence of global solutions of some system of semilinear wave equations abstract an initial boundary value problem for systems of semilinear wave equations in a bounded domain is considered we prove the global existence uniqueness and blow up of solutions by energy methods and give some estimates for the lifespan of solutions

excursions for polymers in elongational flows the study of the configurational statistical mechanics of polymers in elongational flows requires the evaluation of quadratic path integrals here we present a new technique for the evaluation of such path integrals the method relies on ito excursion theory and the ray knight theorem from probability theory in addition to providing a powerful computational method in standard cases the method generalizes to deal with quadratic path integrals for branching processes

energy decay rates for the semilinear wave equation with memory boundary condition and acoustic boundary conditions in this paper we consider the energy decay rates for the semilinear wave equation with memory boundary condition and acoustic boundary conditions motivated by results of gerbi and said houari 2011 2008 li and zhao 2011 liu and chen 2016 wu etal 2010 and lu etal 2011 we intend to study the energy decay rates for problem 1 1 1 6 by using the perturbed energy method and riemannian geometry method we obtained general energy decay rates

the structure preserving doubling algorithm and convergence analysis for a nonlinear matrix equation abstract in this paper applying special properties of doubling transformation a structure preserving doubling algorithm is developed for computing the positive definite solutions for a nonlinear matrix equation further by mathematical induction we establish the convergence theory of the structure preserving doubling algorithm finally we offer corresponding numerical examples to illustrate the effectiveness of the derived algorithm

semiflexible polymers in straining flows we introduce a model for semiflexible polymer chains based on the integral of an appropriate gaussian process the stiffness is characterized physically by adding a bending energy the degree of stiffness in the polymer chain is quantified by means of a parameter and as this parameter tends to infinity the limiting case reduces to the brownian model of completely flexible chains studied in earlier work the calculation of the partition function for the configuration statistical mechanics i e the distribution of shapes of such polymers in elongational flow or quadratic potentials is equivalent to the probabilistic problem of finding the law of a quadratic functional of the associated gaussian process an exact formula for the partition function is presented however in practice this formula is too complicated for most computations we therefore develop an asymptotic expansion for the partition function in terms of the stiffness parameter and obtain the first order term which gives the first order deviation from the completely flexible case in addition to the partition function the method presented here can also deal with other quadratic functionals such as the stochastic area associated with two polymer chains

brownian excursions on combs in this paper we use techniques from ito excursion theory to analyze brownian motion on generalized combs ito excursion theory is a little known area of probability theory and we therefore present a brief introduction for the uninitiated a general method for analyzing transport along the backbone of the comb is demonstrated and the specific case of a comb whose teeth are scaling branching trees is examined we then present a recursive method for evaluating the distribution of the first passage times on hierarchical combs

on the distribution of estimators of diffusion constants for brownian motion we discuss the distribution of various estimators for extracting the diffusion constant of single brownian trajectories obtained by fitting the squared displacement of the trajectory the analysis of the problem can be framed in terms of quadratic functionals of brownian motion that correspond to the euclidean path integral for simple harmonic oscillators with time dependent frequencies explicit analytical results are given for the distribution of the diffusion constant estimator in a number of cases and our results are confirmed by numerical simulations

snakes and spiders brownian motion on ℝ trees we consider diffusion processes on a class of trees the processes are defined in a manner similar to that of le gall s brownian snake each point in the tree has a real valued height or generation and the height of the diffusion process evolves as a brownian motion when the height process decreases the diffusion retreats back along a lineage whereas when the height process increases the diffusion chooses among branching lineages according to relative weights given by a possibly infinite measure on the family of lineages the class of trees we consider can have branch points with countably infinite branching and lineages along which the branch points have points of accumulation

optimal estimates of the diffusion coefficient of a single brownian trajectory modern developments in microscopy and image processing are revolutionizing areas of physics chemistry and biology as nanoscale objects can be tracked with unprecedented accuracy the goal of single particle tracking is to determine the interaction between the particle and its environment the price paid for having a direct visualization of a single particle is a consequent lack of statistics here we address the optimal way of extracting diffusion constants from single trajectories for pure brownian motion it is shown that the maximum likelihood estimator is much more efficient than the commonly used least squares estimate furthermore we investigate the effect of disorder on the distribution of estimated diffusion constants and show that it increases the probability of observing estimates much smaller than the true average value

general decay and blowup of solutions for a degenerate viscoelastic equation of kirchhoff type with source term abstract in this work we consider a nonlinear degenerate viscoelastic equation of kirchhoff type with source term in a bounded domain under suitable assumptions on the relaxation functions g and m we obtain global existence of solutions then we prove the general decay result finally we prove the finite time blow up result of solutions with negative initial energy this work generalizes and improves earlier results in the literature

on the existence of a positive definite solution of the matrix equation x atx 1a i the question is raised under which conditions on the real square matrix a the matrix equation x atx 1a i has a real symmetric positive definite solution x both necessary and sufficient solvability conditions on a are derived moreover we give an algorithm to calculate the solution for a number of special cases we also present an analytic solution

thompson metric method for solving a class of nonlinear matrix equation based on the elegant properties of the thompson metric we prove that the general nonlinear matrix equation x q a f x a q q 1 always has a unique positive definite solution an iterative method is proposed to compute the unique positive definite solution we show that the iterative method is more effective as q increases a perturbation bound for the unique positive definite solution is derived in the end

efficient computation of the extreme solutions of x a x 1 a q and x a x 1 a q we propose a new quadratically convergent algorithm having a low computational cost per step and good numerical stability properties which allows the simultaneous approximation of the extreme solutions of the matrix equations x a x 1 a q and x a x 1 a q the algorithm is based on the cyclic reduction method

low memory and low complexity iterative schemes for a nonsymmetric algebraic riccati equation arising from transport theory abstract we reconsider newton s method and two fixed point methods for finding the minimal positive solution of a nonsymmetric algebraic riccati equation arising from transport theory we rewrite the subproblem of the newton and fixed point iterative schemes into an equivalent form with some special structure by the use of the particular structure of the subproblem we then present low memory and low complexity versions of these iterative methods with a factored alternating direction implicit iteration some properties of eigenvalues for iterative coefficient matrices in solving the subproblem are derived and the convergence of the proposed methods is established numerical experiments show that the new iterative schemes are highly efficient to obtain the minimal positive solution the proposed low memory and low complexity newton s method is particularly efficient for solving large scale riccati equation arising from transport theory

decay estimates for some semilinear damped hyperbolic problems let be a bounded open domain in rn g r r a non decreasing continuous function such that g 0 0 and h e lloc1 r l2 under suitable assumptions on g and h the rate of decay of the difference of two solutions is studied for some abstract evolution equations of the general form u lu g u h t x as t the results obtained by use of differential inequalities can be applied to the case of the semilinear wave equation u u delta u g text u t text h text in r times omega text u text 0 on r times partial omega in r u 0 onr for instance if g s c left s right p 1 s d left s right q 1 s with c d 0 and 1 p q n 2 q n 2 then if h in l infty r l 2 omega all solutions are bounded in the energy space for t 0 and if u v are two such solutions the energy norm of u t v t decays like t 1 p 1 as t

scaling of the first passage time and the survival probability on exact and quasi exact self similar structures the authors calculate the asymptotic behaviour of the moments of the first passage time and survival probability for random walks on an exactly self similar tree and on a quasi self similar comb by applying an exact decimation approach to the master equations for the hierarchical comb a transition from ordinary to anomalous diffusion occurs at r 2 where r is the ratio of teeth length in successive iterations of the structure in the anomalous regime r 2 the positive integer moments of the first passage time tq scale as ltau q with tau q 1 2q 1 ln r ln 2 where l is the linear distance from input to output the asymptotic behaviour of the survival probability is studied using both scaling theory and by a direct solution of the master equations they find that the characteristic time t in the asymptotic exponential decay of the survival probability exp t t scales as t approximately ltau with tau ln r2 ln 2 i e tau is distinct from tau 1 however substantial corrections to this asymptotic form for tau exist and these are needed to account for the recent simulation data of havlin and matan 1988

some brownian functionals and their laws we develop some topics about brownian motion with a particular emphasis on the study of principal values of brownian local times we show some links between principal values and doob s h transforms of brownian motion for nonpositive harmonic functions h we also give a survey and complement some martingale approaches to ray knight theorems for local times

an analytic treatment of the first order correction to the poisson boltzmann interaction free energy in the case of counterion only coulomb fluid the author has computed the first order correlation correction to the poisson boltzmann result for the interaction energy of two charged surfaces immersed into a counterion only coulomb fluid the cases of small and large coupling were amenable to analytic treatment in the saddle point approximation

computing the extremal positive definite solutions of a matrix equation an efficient and numerically stable implementation of a known algorithm is suggested for finding the extremal positive definite solutions of the matrix equation x a x 1 a i if such solutions exist the convergence rate is analyzed a new algorithm that avoids matrix inversion is presented numerical examples are given to illustrate the effectiveness of the algorithms

hermitian solutions of the equation x q nx 1n we consider the matrix equation x q nx 1n its hermitian solutions are parametrized in terms of the generalized lagrangian eigenspaces of a certain matrix pencil we show that the equation admits both a largest and a smallest solution the largest solution corresponds to the unique positive definite solution the smallest solution is the unique negative definite solution if and only if n is nonsingular if n is singular no negative definite solution exists an interesting relation between the given equation and a standard algebraic riccati equation of kalman filtering theory is also obtained finally we present an algorithm which converges to the positive definite solution for a wide range of initial conditions

exponential energy decay of solutions of viscoelastic wave equations abstract in this paper we consider the nonlinear viscoelastic equation u t t u 0 t g t u d u t u p 1 u in 0 with initial conditions and dirichlet boundary condition under some appropriate assumption on g by introducing a potential well we establish the global existence result for the equation above furthermore we investigate explicit exponential energy decay of the viscoelastic wave equation under the potential well

efficient numerical solution of stochastic differential equations using exponential timestepping we present an exact timestepping method for brownian motion that does not require gaussian random variables to be generated time is incremented in steps that are exponentially distributed random variables boundaries can be explicitly accounted for at each timestep the method is illustrated by numerical solution of a system of diffusing particles

some properties of the nonlinear matrix equation xs a x ta q abstract in this paper the nonlinear matrix equation x s a x t a q is investigated where q is a hermitian positive definite matrix we consider three cases of this equation the case s t 0 the case s 1 0 t 1 and the case 0 s 1 t 1 in the case s t 0 we derive necessary conditions and sufficient condition for the existence of hermitian positive definite solutions for the matrix equation and obtain some properties of the solutions as compared to earlier works on these topics the results we present here are more general and the analysis here is much simpler in the cases s 1 0 t 1 and 0 s 1 t 1 necessary conditions for the existence of a hermitian positive definite solution is given which is sharper than that of cai and chen 2010 9

a structure preserving doubling algorithm for continuous time algebraic riccati equations abstract continuous time algebraic riccati equations cares can be transformed a la cayley to discrete time algebraic riccati equations dares the efficient structure preserving doubling algorithm sda for dares from e k w chu h y fan w w lin a structure preserving doubling algorithm for periodic discrete time algebraic riccati equations preprint 2002 28 ncts national tsing hua university hsinchu 300 taiwan 2003 e k w chu h y fan w w lin c s wang a structure preserving doubling algorithm for periodic discrete time algebraic riccati equations preprint 2002 18 ncts national tsing hua university hsinchu 300 taiwan 2003 can then be applied in this paper we develop the structure preserving doubling algorithm from a new point of view and show its quadratic convergence under assumptions which are weaker than stabilizability and detectability as well as practical issues involved in the application of the sda to cares a modified version of the sda developed for dares with a doubly symmetric structure is also presented extensive numerical results show that our approach is efficient and competitive

a note on the fixed point iteration for the matrix equations x a x 1a i the fixed point iteration is a simple method for finding the maximal hermitian positive definite solutions of the matrix equations x a x 1 a i the plus minus equations the convergence of this method may be very slow if the initial matrix is not chosen carefully a strategy for choosing better initial matrices has been recently proposed by ivanov hasanov and uhlig they proved that this strategy can improve the convergence in general and observed from numerical experiments that dramatic improvement happens for the plus equation with some matrices a it turns out that the matrices a are normal for those examples in this note we prove a result that explains the dramatic improvement in convergence for normal and thus nearly normal matrices for the plus equation a similar result is also proved for the minus equation ams classification 15a24 65f30 65h10

stochastic calculus application to dynamic bifurcations and threshold crossings for the dynamic pitchfork bifurcation in the presence of white noise the statistics of the last time at zero are calculated as a function of the noise level and the rate of change of the parameter the threshold crossing problem used for example to model the firing of a single cortical neuron is considered concentrating on quantities that may be experimentally measurable but have so far received little attention expressions for the statistics of pre threshold excursions occupation density and last crossing time of zero are compared with results from numerical generation of paths

structured doubling algorithms for weakly stabilizing hermitian solutions of algebraic riccati equations in this paper we propose structured doubling algorithms for the computation of the weakly stabilizing hermitian solutions of the continuous and discrete time algebraic riccati equations respectively assume that the partial multiplicities of purely imaginary and unimodular eigenvalues if any of the associated hamiltonian and symplectic pencil respectively are all even and the c dare and the dual c dare have weakly stabilizing hermitian solutions with property p under these assumptions we prove that if these structured doubling algorithms do not break down then they converge to the desired hermitian solutions globally and linearly numerical experiments show that the structured doubling algorithms perform efficiently and reliably

iterative solution of two matrix equations we study iterative methods for finding the maximal hermitian positive definite solutions of the matrix equations x a x 1 a q and x a x 1 a q where q is hermitian positive definite general convergence results are given for the basic fixed point iteration for both equations newton s method and inversion free variants of the basic fixed point iteration are discussed in some detail for the first equation numerical results are reported to illustrate the convergence behaviour of various algorithms

improved methods and starting values to solve the matrix equations x a x 1a i iteratively the two matrix iterations x k 1 i a x 1 k a are known to converge linearly to a positive definite solution of the matrix equations x a x 1 a i respectively for known choices of x 0 and under certain restrictions on a the convergence for previously suggested starting matrices x 0 is generally very slow this paper explores different initial choices of x 0 in both iterations that depend on the extreme singular values of a and lead to much more rapid convergence further the paper offers a new algorithm for solving the minus sign equation and explores mixed algorithms that use newton s method in part

on stochastic areas and averages of planar brownian motion duplantier 1989 has computed the characteristic function of the stochastic area of planar brownian motion taken with respect to its centre of gravity in this paper it is shown that this result can be deduced from levy s stochastic area formula moreover duplantier s stochastic area is shown to have the same law as the stochastic area of the restriction to the time interval 0 1 of the brownian ring defined on the time interval 0 4 3 the same method also applies to the stochastic area formulae obtained by biane and yor 1986 which are relative to certain planar processes associated with the orthogonal decomposition of brownian motion along the l2 0 1 basis of legendre polynomials

blow up of solutions for some non linear wave equations of kirchhoff type with some dissipation abstract the initial boundary value problem for non linear wave equations of kirchhoff type with dissipation in a bounded domain is considered we prove the blow up of solutions for the strong dissipative term u t and the linear dissipative term u t by the energy method and give some estimates for the life span of solutions we also show the nonexistence of global solutions with positive initial energy for non linear dissipative term by vitillaro s argument

on decay and blow up of solutions for a system of nonlinear wave equations abstract the initial boundary value problem for a system of viscoelastic wave equations of kirchhoff type with the nonlinear damping and the source terms in a bounded domain is considered we prove that under suitable conditions on the nonlinearity of the damping and the source terms and certain initial data in the stable set and for a wider class of relaxation functions the decay estimates of the energy function is exponential or polynomial depending on the exponents of the damping terms in both equations by using nakao s method conversely for certain initial data in the unstable set we obtain the blow up of solutions in finite time when the initial energy is nonnegative this improves earlier results in the literature

on the tripling algorithm for large scale nonlinear matrix equations with low rank structure for the large scale nonlinear matrix equations with low rank structure the well developed doubling algorithm in low rank form da lr is known as an efficient method to compute the stabilizing solution by further analyzing the global efficiency index constructed in this paper we propose a tripling algorithm in low rank form ta lr from two points of view the cyclic reduction and the symplectic structure preservation the new presented algorithm shares the same pre processing complexity with that of da lr but can attain the prescribed normalized residual level within less iterations by only consuming some negligible iteration time as an offset under the solvability condition the proposed algorithm is demonstrated to inherit a cubic convergence and is capable of delivering errors from the current iteration to the next with the same order numerical experiments including some from nano research show that the ta lr is highly efficient to compute the stabilizing solution of large scale nonlinear matrix equations with low rank structure

positive solutions to x a bx 1 b abstract we study the positive semidefinite solutions to the matrix equation x a bx 1 b under the assumption that a 0 we show that a positive solution exists if and only if a certain block tridiagonal operator is positive in which case the solution is given by the shorted operator or generalized schur complement of that operator

brownian motions on metric graphs brownian motions on a metric graph are defined their generators are characterized as laplace operators subject to wentzell boundary at every vertex conversely given a set of wentzell boundary conditions at the vertices of a metric graph a brownian motion is constructed pathwise on this graph so that its generator satisfies the given boundary conditions

areas of planar brownian curves addresses the problem of the algebraic area enclosed by a brownian curve in two dimensions recently reconsidered by khandekar and wiegel 1988 the author recalls the principal results actually first obtained by paul levy in 1950 another derivation by functional integrals is given for several probability distributions that of the area of a brownian ring the area between an open brownian arc and its chord and finally the area swept with respect to the mean position of the brownian path as a result which seems to be new

a structure preserving doubling algorithm for nonsymmetric algebraic riccati equation in this paper we propose a structure preserving doubling algorithm sda for the computation of the minimal nonnegative solution to the nonsymmetric algebraic riccati equation nare based on the techniques developed for the symmetric cases this method allows the simultaneous approximation to the minimal nonnegative solutions of the nare and its dual equation requiring only the solutions to two linear systems and several matrix multiplications per iteration similar to newton s method and the fixed point iteration methods for solving nares we also establish global convergence for sda under suitable conditions using only elementary matrix theory we show that sequences of matrices generated by sda are monotonically increasing and quadratically convergent to the minimal nonnegative solutions of the nare and its dual equation numerical experiments show that the sda algorithm is feasible and effective and outperforms newton s iteration and the fixed point iteration methods

necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation x a x 1a q abstract we consider the problem of when the matrix equation x a x 1 a q has a positive definite solution here q is positive definite we study both the real and the complex case this equation plays a crucial role in solving a special case of the discrete time riccati equation we present both necessary and sufficient conditions for its solvability this result is obtained by using an analytic factorization approach moreover we present algebraic recursive algorithms to compute the largest and smallest the solution of the equation respectively finally we discuss the number of solutions

highly accurate doubling algorithm for quadratic matrix equation from quasi birth and death process abstract a highly accurate doubling algorithm to solve the most fundamental quadratic matrix equation in the quasi birth and death qbd process is developed it follows from the general framework of the doubling algorithm for the first standard form sf1 but can be implemented to compute the minimal nonnegative solution with high entrywise relative accuracy for all entries large or tiny the algorithm is globally and quadratically convergent except for qbd equations in the critical case where convergence is linear with the linear rate 1 2 numerical examples are presented to demonstrate and confirm our claims the development here parallels the recent work of xue and li 2017 3 on the m matrix algebraic riccati equation

on polymer conformations in elongational flows we consider various models of polymer conformations using paths of gaussian processes such as brownian motion in each case the calculation of the law of the moment of inertia of a random polymer structure which is equivalent to the calculation of the partition function is reduced to the problem of finding the law of a certain quadratic functional of a gaussian process we present a new method for computing the laplace transforms of these quadratic functionals which exploit their special form via the ray knight theorem and which does not involve the classical method of eigenvalue expansions we apply the method to several simple examples then show how the same techniques can be applied to more complicated cases with the aid of a little excursion theory

on a quasilinear wave equation with memory equation 1 l with a 0 and n 1 has its roots in a model for the small amplitude vibrations of a string in which the dependence of the tension on the deformation cannot be ignored see for example bernstein 2 carrier 3 41 dickey s and nishida 13 when a 0 1 1 may be used to describe the dynamics of an extensible string with fading memory this equation states that the dynamic equilibrium of a body depends not only on the present state of deformation but also on the previous history of the deformation gradient since the publication of the aforementioned papers equation 1 1 with a 0 has been the subject of intense scrutiny see for example arosio and spagnolo 11 lions 9 medeiros lo menzala ll nishihara 14 151 pohoiaev 16 and yamada 17 181 to name but a few among the facts thus far proved pohoiaev s results and their extension to the degenerate case of arosio and spagnolo are of particular significance they show that if the data satisfy an analytic type condition then solutions are global in time in spite of the abundant literature on the memoryless form of equation l l to our best knowledge equation 1 1 with a f 0 appears to be untouched our main results theorem 5 1 theorem 5 2 and theorem 5 4 will show that under appropriate assumptions on the kernel

properties of positive definite solutions of the equation x a x 2a i abstract in this paper we discuss some properties of a positive definite solution of the matrix equation x a x 2 a i two effective iterative methods for computing a positive definite solution of this equation are proposed necessary and sufficient conditions for existence of a positive definite solution are derived numerical experiments are executed with these methods