a pearson random walk with steps of uniform orientation and dirichlet distributed lengths a constrained diffusive random walk of n steps in d and a random flight in d which are equivalent were investigated independently in recent papers j stat phys 127 813 2007 j theor probab 20 769 2007 and j stat phys 131 1039 2008 the n steps of the walk are independent and identically distributed random vectors of exponential length and uniform orientation conditioned on the sum of their lengths being equal to a given value l closed form expressions for the distribution of the endpoint of the walk were obtained altogether for any n for d 1 2 4 uniform distributions of the endpoint inside a ball of radius l were evidenced for a walk of three steps in 2d and of two steps in 4d

isotropic random motion at finite speed with k erlang distributed direction alternations we study uniformly distributed direction of motion at finite speed where the direction alternations occur according to the renewal epochs of a k erlang pdf at first sight our generalizations of previous markovian results appears to be a small step however it must be seen as an important non markovian case where we have found closed form expressions for the pdf and the conditional characteristic function of this semi markov transport process we present detailed calculations of a three dimensional example for the 2 erlang case which is important not only from physical applications point of view but also to understand more general models for instance in principle the example of the 2 erlang case can be extended to a k erlang case k 3 4 but some of the mathematical expressions may be cumbersome

when a random walk of fixed length can lead uniformly anywhere inside a hypersphere a variation of the pearson rayleigh random walk in which the steps are i i d random vectors of exponential length and uniform orientation is considered conditioned on the total path length the probability density function of the position of the walker after n steps is determined analytically in one and two dimensions it is shown that in two dimensions n 3 marks a critical transition point in the behavior of the random walk by taking less than three steps and walking a total length l one is more likely to end the walk near the boundary of the disc of radius l while by taking more than three steps one is more likely to end near the origin somehow surprisingly by taking exactly three steps one can end uniformly anywhere inside the disc of radius l this means that conditioned on l the sum of three vectors of exponential length and uniform direction has a uniform probability density

a 4 node finite shell element for the implementation of general hyperelastic 3d elasticity at finite strains abstract in this paper a finite shell element for large deformations is presented based on extensible director kinematics the essential feature is an interface to arbitrary three dimensional material laws the non linear lagrangian formulation is based on the three field variational principle parametrized with the displacement vector enhanced green lagrangian strain tensor and second piola kirchhoff stress tensor the developed quadrilateral shell element is characterized by a course mesh accuracy and distortion insensitivity compared with bilinear displacement approaches furthermore plane stress response is approximately recovered in the asymptotic case of vanishing thickness a number of example problems investigating large deformation as well as finite strain applications are presented compressible and incompressible hyperelastic materials of the st venant kirchhoff neo hookean and mooney rivlin type are particularly used

eigenvalues invariant factors highest weights and schubert calculus we describe recent work of klyachko totaro knutson and tao that characterizes eigenvalues of sums of hermitian matrices and decomposition of tensor products of representations of gln c we explain related applications to invariant factors of products of matrices intersections in grassmann varieties and singular values of sums and products of arbitrary matrices recent breakthroughs primarily by a klyachko b totaro a knutson and t tao with contributions by p belkale and c woodward have led to complete solutions of several old problems involving the various notions in the title our aim here is to describe this work and especially to show how these solutions are derived from it along the way we will see that these problems are also related to other areas of mathematics including geometric invariant theory symplectic geometry and combinatorics in addition we present some related applications to singular values of arbitrary matrices although many of the theorems we state here have not appeared elsewhere their proofs are mostly soft algebra based on the hard geometric or combinatorial work of others indeed this paper emphasizes concrete elementary arguments we do give some new examples and counterexamples and raise some new open questions we have attempted to point to the sources and to some of the key partial results that had been conjectured or proved before however there is a very large literature particularly for linear algebra problems about eigenvalues singular values and invariant factors we have listed only a few of these articles from whose bibliographies we hope an interested reader can trace the history we apologize to the many whose work is not cited directly we begin in the first five sections by describing each of the problems together with some of their early histories and we state as theorems the new solutions to these problems in section 6 we describe the steps toward these solutions that were carried out before the recent breakthroughs then we discuss the recent solutions and explain how these theorems follow from the work of the above mathematicians sections 7 8 9 and 10 also contain variations and generalizations of some of the theorems stated in the first five sections as well as attributions of the theorems to their authors one of our fascinations with this subject even now that we have proofs of the theorems is the challenge to understand in a deeper way why all these subjects are received by the editors in july 1999 and in revised form january 3 2000 2000 mathematics subject classification primary 15a42 22e46 14m15 secondary 05e15 13f10 14c17 15a18 47b07 the author was partly supported by nsf grant dms9970435 c 2000 american mathematical society

convex functions on symmetric spaces side lengths of polygons and the stability inequalities for weighted configurations at infinity in a symmetric space of noncompact type x g k oriented geodesic segments correspond modulo isometries to vectors in the euclidean weyl chamber we can hence assign vector valued lengths to segments our main result is a system of homogeneous linear inequalities which we call the generalized triangle inequalities or stability inequalities describing the restrictions on the vector valued side lengths of oriented polygons it is based on the mod 2 schubert calculus in the real grassmannians g p for maximal parabolic subgroups p the side lengths of polygons in euclidean buildings are studied in the related paper klm2 applications of the geometric results in both papers to algebraic group theory are given in klm3

a holonomic systems approach to special functions identities abstract we observe that many special functions are solutions of so called holonomic systems bernstein s deep theory of holonomic systems is then invoked to show that any identity involving sums and integrals of products of these special functions can be verified in a finite number of steps this is partially substantiated by an algorithm that proves terminating hypergeometric series identities and that is given both in english and in maple

a new family of solvable pearson dirichlet random walks an n step pearson gamma random walk in d starts at the origin and consists of n independent steps with gamma distributed lengths and uniform orientations the gamma distribution of each step length has a shape parameter q 0 constrained random walks of n steps in d are obtained from the latter walks by imposing that the sum of the step lengths is equal to a fixed value simple closed form expressions were obtained in particular for the distribution of the endpoint of such constrained walks for any d d0 and any n 2 when q is either q frac d 2 1 d0 3 or q d 1 d0 2 le caer in j stat phys 140 728 751 2010 when the total walk length is chosen without loss of generality to be equal to 1 then the constrained step lengths have a dirichlet distribution whose parameters are all equal to q and the associated walk is thus named a pearson dirichlet random walk the density of the endpoint position of a n step planar walk of this type n 2 with q d 2 was shown recently to be a weighted mixture of 1 floor n 2 endpoint densities of planar pearson dirichlet walks with q 1 beghin and orsingher in stochastics 82 201 229 2010 the previous result is generalized to any walk space dimension and any number of steps n 2 when the parameter of the pearson dirichlet random walk is q d 1 we rely on the connection between an unconstrained random walk and a constrained one which have both the same n and the same q d to obtain a closed form expression of the endpoint density the latter is a weighted mixture of 1 floor n 2 densities with simple forms equivalently expressed as a product of a power and a gauss hypergeometric function the weights are products of factors which depends both on d and n and bessel numbers independent of d

multiplicative lidskii s inequalities and optimal perturbations of frames abstract in this paper we study two design problems in frame theory on the one hand given a fixed finite frame f f j j i n for c d we compute those dual frames g of f that are optimal perturbations of the canonical dual frame for f under certain restrictions on the norms of the elements of g on the other hand we compute those v f v f j j i n for invertible operators v which are close to the identity that are optimal perturbations of f that is we compute the optimal perturbations of f among frames g g j j i n that have the same linear relations as f in both cases optimality is measured with respect to submajorization of the eigenvalues of the frame operators hence our optimal designs are minimizers of a family of convex potentials that include the frame potential and the mean squared error the key tool for these results is a multiplicative analogue of lidskii s inequality in terms of log majorization and a characterization of the case of equality

the high road to an exponential formula a conjectured exponential formula is proved using a recent result of klyachko linear algebra appl 319 2000 37 then the exponential formula is applied to confirm the equality of two types of spectral triples used to describe all homogeneous eigenvalue inequalities for sum of hermitian matrices and product of positive definite matrices

random motions at finite speed in higher dimensions we present a general method of studying the transport process bold x t t 0 in the euclidean space m m 2 based on the analysis of the integral transforms of its distributions we show that the joint characteristic functions of bold x t are connected with each other by a convolution type recurrent relation this enables us to prove that the characteristic function fourier transform of bold x t in any dimension m 2 satisfies a convolution type volterra integral equation of second kind we give its solution and obtain the characteristic function of bold x t in terms of the multiple convolutions of the kernel of the equation with itself an explicit form of the laplace transform of the characteristic function in any dimension is given the complete solution of the problem of finding the initial conditions for the governing partial differential equations is given

pivoting to normalize a basic matrix given a matrix in basic form a sequence of pivots is described which brings the matrix into a basic form where the maximum absolute component is unity

random walks on symmetric spaces and inequalities for matrix spectra abstract using harmonic analysis on symmetric spaces we reduce the singular spectral problem for products of matrices to the recently solved spectral problem for sums of hermitian matrices this proves r c thompson s conjecture matrix spectral inequalities johns hopkins university press baltimore md 1988

application of transversely isotropic materials to multi layer shell elements undergoing finite rotations and large strains abstract the contribution deals with an extension of a classical neo hookean model for compressible isotropic materials to transverse isotropy with this enhancement for one preferred material direction there is a possibility to simulate large strains in volume changes of the isotropic basic continuum and supplementary in fiber direction the integrity basis of polynomial invariants in case of transversely isotropic hyperelasticity consists of three principal invariants of the isotropic basic continuum and additionally of two principal invariants for the preferred material direction the proposed stored energy function for transverse isotropy contains the classical theory near to the natural state and fulfills the restriction on polyconvexity and coerciveness by numerical enforcement of the material model into shell kinematics without rotational variables a four node isoparametric finite element is developed using special concepts to avoid locking the capability of the algorithms proposed is demonstrated by a numerical example involving large strains as well as finite rotations

probabilistic methods in crystal structure analysis abstract one of the main goals of modern crystallography is the determination of the detailed internal structure of crystalline matter at the atomic level statistical analyses and in particular random walk models play a central role in inferring structural information from crystallographic data such methods are routinely employed by crystallographers in the determination of crystal symmetry from the experimental data and in the solution of the outstandingly important problem for this discipline the phase problem three classes of approaches are discussed a methods based entirely on the central limit theorem b approximate expansions in terms of orthogonal polynomials that have the central limit theorem pdf as their weight function that is gram charlier and edgeworth expansions and c pdf s that are exactly formulated and reduced to computable forms represented as fourier and fourier bessel series both univariate and multivariate pdf s of crystallographic interest are derived and discussed

a family of random walks with generalized dirichlet steps we analyze a class of continuous time random walks in rd d 2 with uniformly distributed directions the steps performed by these processes are distributed according to a generalized dirichlet law given the number of changes of orientation we provide the analytic form of the probability density function of the position xd t t 0 reached at time t 0 by the random motion in particular we analyze the case of random walks with two steps in general it is a hard task to obtain the explicit probability distributions for the process xd t t 0 nevertheless for suitable values for the basic parameters of the generalized dirichlet probability distribution we are able to derive the explicit conditional density functions of xd t t 0 furthermore in some cases by exploiting the fractional poisson process the unconditional probability distributions of the random walk are obtained this paper extends in a more general setting the random walks with dirichlet displacements introduced in some prev

exact solutions for isotropic random flights in odd dimensions we give a method to solve the isotropic random flight in arbitrary dimension and show that when the dimension is odd it leads to a piecewise polynomial expression for the probability density the method is based on the fact that the projection of an isotropic function has an inverse and on the fact that the convolution commutes with the projection and the inverse projection in order to develop this method the abel transform and its inverse are extended by comparing with the expression for the isotropic random flight obtained using the convolution theorem some integrals of bessel functions can be expressed as piecewise polynomials the expressions for the five dimensional case for 2 3 and 4 steps are found explicitly although the main result derived from this method concerns odd dimensions it also yields an analytical result in the four dimensional isotropic random flight which is derived here the results of this article can also be considered as results on the convolution with itself of an isotropic probability density whose support is the surface of a sphere

random motion with uniformly distributed directions and random velocity in this paper we deal with uniformly distributed direction of motion or isotropic motion at random speed or velocity where the direction alternations occur according to the renewal epochs of a general distribution we derive the renewal equation for the characteristic function of the transition density of the multidimensional motion then by using the renewal equation we study the behavior of the transition density near the sphere of its singularity for one two three and four dimensional cases to illustrate our solution methodology we present detailed calculations of many solvable examples

boundedness of iterated spherical average on modulation spaces abstract the spherical average a 1 f and its iteration a 1 n are important operators in harmonic analysis and probability theory also a 1 n is used to study the k functional in approximation theory where is the laplace operator in this paper we obtain the sufficient and necessary conditions to ensure the boundedness of a 1 n from the modulation space m p 1 q 1 s 1 to the modulation space m p 2 q 2 s 2 for 1 p 1 p 2 q 1 q 2 and s 1 s 2 r

some applications of persistent random walks and the telegrapher s equation abstract a persistent random walk can be regarded as a multidimensional markov process the bias free telegraphers equation is 2 p t 2 1 t p t v 2 2 p it can be regarded as interpolating between the wave equation t and the diffusion equation t 0 previously it has found application in thermodynamics cf the review in rev mod phys 61 1989 41 62 1990 375 more recent applications are reviewed in the present article

quadratic and cubic transformations and zeros of hypergeometric polynomials in this paper we consider the location of the zeros of the hypergeometric polynomials that lie in either the quadratic or the cubic class where each of these classes is determined by a necessary and sufficient condition due to kummer we show that the zeros of most polynomials in these classes can be specified by simple applications of the results proved in recent papers of driver and duren

the poisson geometry of su 1 1 we study the natural poisson structure on the group su 1 1 and related questions in particular we give an explicit description of the ginzburg weinstein isomorphism for the sets of admissible elements we also establish an analog of thompson s conjecture for this group

the sum of unitary similarity orbits containing only special operators abstract let b h be the algebra of bounded linear operator acting on a hilbert space h over the complex or real field characterization is given to a 1 a k b h such that for any unitary operators u 1 u k j 1 k u j a j u j is always in a special class s of operators such as normal operators self adjoint operators unitary operators as corollaries characterizations are given to a b h such that complex real or nonnegative linear combinations of operators in its unitary orbit u a u au u unitary always lie in s

asymptotic relation for the transition density of the three dimensional markov random flight on small time intervals we consider the markov random flight varvec x t t 0 in the three dimensional euclidean space mathbb r 3 with constant finite speed c 0 and the uniform choice of the initial and each new direction at random time instants that form a homogeneous poisson flow of rate lambda 0 series representations for the conditional characteristic functions of varvec x t corresponding to two and three changes of direction are obtained based on these results an asymptotic formula as t rightarrow 0 for the unconditional characteristic function of varvec x t is derived by inverting it we obtain an asymptotic relation for the transition density of the process we show that the error in this formula has the order o t 3 and therefore it gives a good approximation on small time intervals whose lengths depend on lambda an asymptotic formula as t rightarrow 0 for the probability of being in a three dimensional ball of radius r ct is also derived estimate of the accuracy of the approximation is analysed

decomposable representations and lagrangian submanifolds of moduli spaces associated to surface groups the importance of explicit examples of lagrangian submanifolds of moduli spaces is revealed by papers such as dostoglou and salamon ann of math 2 139 3 581 640 1994 and salamon proceedings of the international congress of mathematicians vol 1 2 zurich 1994 pp 526 536 birkhauser basel 1995 given a 3 manifold m with boundary m dostoglou and salamon use such examples to obtain a proof of the atiyah floer conjecture relating the symplectic floer homology of the representation space hom 1 m u u associated to an explicit pair of lagrangian submanifolds of this representation space and the instanton homology of the 3 manifold m in the present paper we construct a lagrangian submanifold of the space of representations mathcal m g l hom mathcal c pi g l u u of the fundamental group g lof a punctured riemann surface g l into an arbitrary compact connected lie group u this lagrangian submanifold is obtained as the fixed point set of an anti symplectic involution hat beta defined on mathcal m g l we show that the involution hat beta is induced by a form reversing involution defined on the quasi hamiltonian space u times u g times mathcal c 1 times cdots times mathcal c l the fact that hat beta has a non empty fixed point set is a consequence of the real convexity theorem for group valued momentum maps proved in schaffhauser a real convexity theorem for quasi hamiltonian actions submitted 25 p 2007 http arxiv org abs math 0705 0858 the notion of decomposable representation provides a geometric interpretation of the lagrangian submanifold thus obtained

computing zeros of sections of vector bundles using homotopies and relocalization an algorithm is described for computing fixed points on a grassmannian manifold the method can be applied in the more general setting of solving equations on abstract smooth manifolds into vector bundles this development is part of a project to compute economic equilibria in the presence of incomplete markets

refined shear deformation models for composite laminates with finite rotations abstract for arbitrary multilayered shell structures made particularly of composite material layers a refined finite rotation theory with seven independent displacement variables is developed approximating the displacement field by a cubic series expansion of thickness coordinates this model allows a quadratic shear deformation distribution across the thickness procedures are given permitting a unique determination of the first order displacement term in the case of finite rotations kinematic relations are formulated in two alternative forms suitable for both classical and isoparametric finite element formulations the constitutive relations presented model orthotropic material properties varying arbitrarily across the thickness this third order single layer theory is then transformed by introducing further constraints into three simplified models a third order theory with five independent displacement variables a mindlin reissner type theory and a kirchhoff love type theory these four models differ however from each other essentially in the constraints imposed on the first and third order displacement variables a significant advantage for a unified finite element development finally the mindlin reissner type theory is generalized to a layer wise model being the most predictive one in dealing with local interlaminar effects the theoretical models are transformed into adequate finite shell elements and then compared by means of appropriate examples concerning their prediction capability also examples are given demonstrating their applicability to finite rotation phenomena

on products of sln characters and support containment abstract let and be dominant weights of sl n satisfying let v denote the highest weight module corresponding to lam postnikov pylyavskyy conjectured a sufficient condition for v v to be contained in v v as sl n modules in this note we prove a weaker version of the conjecture namely we prove that under the conjectured conditions every irreducible sl n module which appears in the decomposition of v v does appear in the decomposition of v v

tail asymptotics under beta random scaling abstract let x y b be three independent random variables such that x has the same distribution function as yb assume that b is a beta random variable with positive parameters and y has distribution function h with h 0 0 in this paper we derive a recursive formula for calculation of h if the distribution function h of x is known furthermore we investigate the relation between the tail asymptotic behaviour of x and y which is closely related to asymptotics of weyl fractional order integral operators we present three applications of our asymptotic results concerning the extremes of two random samples with underlying distribution functions h and h respectively and the conditional limiting distribution of bivariate elliptical distributions

a note on iterated spherical average on lebesgue spaces abstract for the spherical average a and the laplace operator we study the iterated operator a n and find the minimal positive integer n to ensure the l 1 r n boundedness of a n using a constructive proof we completely solve a question posted by belinsky et al 2003

probability law for the euclidean distance between two planar random flights we consider two independent symmetric markov random flights z1 t and z2 t performed by the particles that simultaneously start from the origin of the euclidean plane mathbb r 2 in random directions distributed uniformly on the unit circumference s1 and move with constant finite velocities c1 0 c2 0 respectively the new random directions are taking uniformly on s1 at random time instants that form independent homogeneous poisson flows of rates 1 0 2 0

eigenvalues singular values and littlewood richardson coefficients we characterize the relationship between the singular values of a hermitian resp real symmetric complex symmetric matrix and the singular values of its off diagonal block we also characterize the eigenvalues of a hermitian or real symmetric matrix c a b in terms of the combined list of eigenvalues of a and b the answers are given by horn type linear inequalities the proofs depend on a new inequality among littlewood richardson coefficients

a cellation of the grassmann manifold motivated by the computation of equilibria in economic models with incomplete asset markets a cellation of the grassmann manifold is constructed by restricting a common atlas the grassmann manifold ofm planes inn dimensional space is shown to be a union ofn choosem congruentm n m dimensional topological disks whose interiors are disjoint

two step dirichlet random walks abstract random walks of n steps taken into independent uniformly random directions in a d dimensional euclidean space d 2 which are characterized by a sum of step lengths which is fixed and taken to be 1 without loss of generality are named dirichlet when this constraint is realized via a dirichlet law of step lengths the latter continuous multivariate distribution which depends on n positive parameters generalizes the beta distribution n 2 it is simply obtained from n independent gamma random variables with identical scale factors previous literature studies of these random walks dealt with symmetric dirichlet distributions whose parameters are all equal to a value q which takes half integer or integer values in the present work the probability density function of the distance from the endpoint to the origin is first made explicit for a symmetric dirichlet random walk of two steps it is valid for any positive value of q and for all d 2 the latter pdf is used in turn to express the related density of a random walk of two steps whose step length is distributed according to an asymmetric beta distribution which depends on two parameters namely q and q s where s is a positive integer

joint densities for random walks in the plane abstract we point out the existence of computationally convenient techniques for calculating the joint probability density for the position of a pearson random walk after n steps a new fourier bessel function expansion for p n r is developed for this purpose which does not require radial symmetry but does require that p n r 0 when r exceeds some maximum radius r

stein s method and characters of compact lie groups stein s method is used to study the trace of a random element from a compact lie group or symmetric space central limit theorems are proved using very little information character values on a single element and the decomposition of the square of the trace into irreducible components this is illustrated for lie groups of classical type and dyson s circular ensembles the approach in this paper will be useful for the study of higher dimensional characters where normal approximations need not hold

collision densities and mean residence times for d dimensional exponential flights in this paper we analyze some aspects of exponential flights a stochastic process that governs the evolution of many random transport phenomena such as neutron propagation chemical or biological species migration and electron motion we introduce a general framework for d dimensional setups and emphasize that exponential flights represent a deceivingly simple system where in most cases closed form formulas can hardly be obtained we derive a number of exact where possible or asymptotic results among which are the stationary probability density for two dimensional systems a long standing issue in physics and the mean residence time in a given volume bounded or unbounded domains as well as scattering or absorbing domains are examined and monte carlo simulations are performed so as to support our findings

random flights through spaces of different dimensions we shall study random flights that start in a space of one given dimension and after performing a definite number of steps continue to develop in a space of higher dimension we show that if the difference of the dimension of spaces is even then the probability density describing the composite flight can be expressed as marginalizations of the probability density associated to a random flight in the space of less dimensions this dimensional reduction is a consequence of gegenbauer addition theorem

some two and three dimensional persistent random walks abstract we formulate non markovian versions of the persistent random walk in two and three dimensions and in continuous time these models can be regarded as being generalizations of the original pearson random walk and of the freely jointed chain which has been of some importance in polymer physics solutions for the probability density of the displacement of the random walker can be furnished for a restricted essentially markovian set of these models it is shown that in two dimensions the solution to one of these models is equivalent to the solution to an inhomogeneous telegrapher s equation it does not appear to be possible starting from a similar model in three dumensions to find any form of the telegrapher s equation that follows from our solution of the equations in the fourier laplace transform domains

the canonical correlations of a 2 2 block matrix with given eigenvalues abstract let 1 k n 2 and a a 11 a 12 1pt a 21 a 22 be an n n positive definite matrix so that a11 is k k suppose that a has given eigenvalues 1 n 0 the singular values j a11 1 2a12a22 1 2 j 1 k are known as the canonical correlations of the partitioned matrix a and have been extensively studied with regard to the inefficiency of the ordinary least squares method in statistics the object of this paper is to provide proofs of some new inequalities for the canonical correlations in terms of 1 n

a general high order finite element formulation for shells at large strains and finite rotations for hyperelastic shells with finite rotations and large strains a p finite element formulation is presented accommodating general kinematic assumptions interpolation polynomials and particularly general three dimensional hyperelastic constitutive laws this goal is achieved by hierarchical high order shell models the tangent stiffness matrices for the hierarchical shell models are derived by computer algebra both non hierarchical nodal as well as hierarchical element shape functions are admissible numerical experiments show the high order formulation to be less prone to locking effects copyright 2003 john wiley sons ltd

short hyperuniform random walks random walks of two steps with fixed sums of lengths of 1 taken into uniformly random directions in d dimensional euclidean spaces d ge 2 are investigated to construct continuous step length distributions which make them hyperuniform the endpoint positions of hyperuniform walks are spread out in the unit ball as the projections in the walk space of points uniformly distributed on the surface of the unit hypersphere of some k dimensional euclidean space k d unique symmetric continuous step length distributions exist for given d and k provided that d k 2d the walk becomes uniform on the unit ball when k d 2 the symmetric densities reduce to simple polynomials for uniform random walks and are mixtures of two pairs of asymmetric beta distributions

joint distribution function for position and rotation angle in plane random walks abstract the angle by which the tangent to the trajectory rotates in the course of a plane random walk is studied the joint probability distribution function for position and rotation angle in a random walk of n steps is obtained in closed form this result is a generalization of kluyver s integral representation for the distribution function of the position alone explicit expressions are presented for n 1 2 and n in the latter case both by application of the central limit theorem and the steepest descent method the distribution of the rotation angle modulo 2 is also discussed and turns out to be particularly simple

the exact probability distribution of a two dimensional random walk a calculation is made of the exact probability distribution of the two dimensional displacement of a particle at timet that starts at the origin moves in straight line paths at constant speed and changes its direction after exponentially distributed time intervals where the lengths of the straight line paths and the turn angles are independent the angles being uniformly distributed this random walk is the simplest model for the locomotion of microorganisms on surfaces its weak convergence to a wiener process is also shown

the influence of elasticity on analysis modern developments on examine plusieurs domaines de l analyse auxquels l elasticite a apporte une contribution importante questions de connectivite de la theorie globale des bifurcations particularites de l analyse non lineaire calcul des variations convergence faible inegalites variationnelles autres contributions

densities of short uniform random walks in higher dimensions we study arithmetic properties of short uniform random walks in arbitrary dimensions with a focus on explicit hypergeometric evaluations of the moment functions and probability densities in the case of up to five steps somewhat to our surprise we are able to provide complete extensions to arbitrary dimensions for most of the central results known in the two dimensional case

the pearson random walk with unequal step sizes there have been many analyses of pearson random walks with equal step sizes several applications suggest the importance of these walks with unequal step sizes numerical comparisons are made of approximations obtained using the central limit theorem and the steepest descents method these suggest that the latter generally but not always leads to more accurate results the authors also derive an approximation that is extremely accurate at near maximal extensions

a note on enhanced strain methods for large deformations abstract enhanced strain elements have been applied to simulate geometrically and materially non linear problems their main advantage is due to the fact that they perform very good in the incompressible limit as well as in bending situations however in compressive deformation states these elements depict stability modes which are associated with hour glass forms and thus denote a rank deficiency for such deformations in this paper the appearance of these modes will be investigated analytically by means of a simple representative example