random polynomials having few or no real zeros consider a polynomial of large degree n whose coefficients are independent identically distributed nondegenerate random variables having zero mean and finite moments of all orders we show that such a polynomial has exactly k real zeros with probability n b o 1 as n through integers of the same parity as the fixed integer k 0 in particular the probability that a random polynomial of large even degree n has no real zeros is n b o 1 the finite positive constant b is characterized via the centered stationary gaussian process of correlation function sech t 2 the value of b depends neither on k nor upon the specific law of the coefficients under an extra smoothness assumption about the law of the coefficients with probability n b o 1 one may specify also the approximate locations of the k zeros on the real line the constant b is replaced by b 2 in case the i i d coefficients have a nonzero mean

approximate fundamental solutions and wave fronts for general anisotropic materials abstract the time dependent differential equations of elastodynamics for homogeneous solids with a general structure of anisotropy are considered in the paper a new method of computation of the fundamental solution for these equations is proposed this method consists of the following applying the fourier transformation with respect to space variables to these equations we obtain a system of second order ordinary differential equations whose coefficients depend on fourier parameters using the matrix transformations and properties of the coefficients the fourier image of the fundamental solution is computed finally the fundamental solution is calculated by the inverse fourier transformation to the obtained fourier transform the implementation and justification of the suggested method have been made by computational experiments in matlab these experiments confirm the robustness of the suggested method the visualization of the displacement components in general homogeneous anisotropic solids by modern computer tools allows us to see and evaluate the dependence between the structure of solids and the behavior of the displacement field our method allows users to observe the elastic wave propagation arising from pulse point forces of the form em x t in monoclinic triclinic and other anisotropic solids the visualization of displacement components gives knowledge about the form of fronts of elastic wave propagation in sodium thiosulfate with monoclinic and copper sulphate pentahydrate with triclinic structures of anisotropy

static fundamental solutions for a bi material full space in this paper the complete static response of two joined dissimilar half spaces due to an arbitrary interior point load is derived by means of a method of displacement potentials and integral transforms a dual format of the solution in the form of a hankel integral representation and algebraic closed form expressions is presented as illustrations its analytical degeneration to benchmark solutions for a homogeneous medium as well as its variation under general geometric and material conditions are shown the importance of the dual format of the bi material solution in connection with the method of asymptotic decomposition to the development of a rigorous treatment of its dynamic counterpart is also demonstrated

self normalized large deviations let x x n n geq 1 be a sequence of independent and identically distributed random variables the classical cramer chernoff large deviation states that lim n to infty n 1 ln p sum i 1 n x i n geq x ln rho x if and only if the moment generating function of x is finite in a right neighborhood of zero this paper uses n p 1 p v n p n p 1 p sum i 1 n x i p 1 p p 1 as the normalizing constant to establish a self normalized large deviation without any moment conditions a self normalized moderate deviation that is the asymptotic probability of p s n v n p geq x n for x n o n p 1 p is also found for any x in the domain of attraction of a normal or stable law as a consequence a precise constant in the self normalized law of the iterated logarithm of griffin and kuelbs is obtained applications to the limit distribution of self normalized sums the asymptotic probability of the t statistic as well as to the erdos renyi shepp law of large numbers are also discussed

brownian motion in cones we study the asymptotic behavior of brownian motion and its conditioned process in cones using an infinite series representation of its transition density a concise probabilistic interpretation of this series in terms of the skew product decomposition of brownian motion is derived and used to show properties of the transition density

almost sure convergence for the maxima of strongly dependent stationary gaussian vector sequences in this paper we prove the almost sure limit theorem of the maxima for a kind of strongly dependent stationary gaussian vector sequences

limit results on finite dimensional gaussian random fields in this paper we establish some limit results on csorgo revesz type increments combined with moduli of continuity for finite dimensional gaussian random fields under explicit conditions via estimating upper bounds of large deviation probabilities on suprema of the finite dimensional gaussian random fields

almost sure limit theorems for the maxima of some strongly dependent gaussian sequences we proved some almost sure limit theorems for standard strongly dependent gaussian sequences in nonstationary cases under some mild conditions

computation of p units in ray class fields of real quadratic number fields let k be a real quadratic field let p be a prime number which is inert in k and let k p be the completion of k at p as part of a ph d thesis we constructed a certain p adic invariant u e kp and conjectured that u is in fact a p unit in a suitable narrow ray class field of k in this paper we give numerical evidence in support of that conjecture our method of computation is similar to the one developed by dasgupta and relies on partial modular symbols attached to eisenstein series

a note on the continuity of solutions of parametric linear programs this note shows that if a parametric linear program min cx ax b1 b2 x 0 has optimal solutions in an interval left underset raise0 3em hbox smash scriptscriptstyle theta bar theta right for then depending on degeneracy the solutions is a continuous vector function or a continuous point to set mapping in the latter case an algorithm is introduced to solve the problem and generate a continuous vector solution in left underset raise0 3em hbox smash scriptscriptstyle theta bar theta right

three dimensional green s functions in anisotropic trimaterials in this paper three dimensional green s functions for anisotropic bimaterials are studied based on stroh formalism and two dimensional fourier transforms although the green s functions can be expressed exactly in the fourier transform domain it is diacult to obtain the explicit expressions of the green s functions in the physical domain due to the general anisotropy of the material and a geometry plane involved utilizing fourier inverse transform in the polar coordinate and combining with mindlin s superposition method the physical domain bimaterial green s functions are derived as a sum of a full space green s function and a complementary part while the full space green s function is in an explicit form the complementary part is expressed in terms of simple regular line integrals over 0 2pa that are suitable for standard numerical integration furthermore the present bimaterial green s functions can be reduced to the special cases such as half space surface interfacial and full space green s functions numerical examples are given for both half space and bimaterial cases with isotropic transversely isotropic and anisotropic material properties to verify the applicability of the technique for the half space case with isotropic or transversely isotropic material properties the green s function solutions are in excellent agreement with the existing analytical solutions for anisotropic half space and bimaterial cases numerical results show the strong dependence of the green s functions on the material properties 7 2000 elsevier science ltd all rights reserved

considerations of the existence of interfacial stoneley waves in bonded anisotropic elastic half spaces the questions of uniqueness and existence of subsonic stoneley waves in bonded anisotropic linear elastic half spaces are settled by using the notion of the interface impedance tensor which is a simple linear combination of the hermitian surface impedance tensors of the separate half spaces a definite existence criterion is presented in a form that proves most useful in numerical searches for stoneley waves in the sense that such searches need not be conducted when a stoneley wave mode

three dimensional dynamic green s functions for a multilayered transversely isotropic half space abstract with the aid of a method of displacement potentials an efficient and accurate analytical derivation of the three dimensional dynamic green s functions for a transversely isotropic multilayered half space is presented constituted by proper algebraic factorizations a set of generalized transmission reflection matrices and internal source fields that are free of any numerically unstable exponential terms are proposed for effective computations of the potential solution three dimensional point load green s functions for stresses and displacements are given for the first time in the complex plane line integral representations the present formulations and solutions are analytically in exact agreement with the existing solutions given by pak and guzina 2002 for the isotropic case for the numerical computation of the integrals a robust and effective methodology which gives the necessary account of the presence of singularities including branch points and poles on the path of integration is laid out a comparison with the existing numerical solutions for multilayered isotropic half space is made to confirm the accuracy of the numerical solutions

three dimensional dynamic green s functions in transversely isotropic bi materials by virtue of a complete representation using two displacement potentials an analytical derivation of the elastodynamic green s functions for a linear elastic transversely isotropic bi material full space is presented three dimensional point load green s functions for stresses and displacements are given in complex plane line integral representations the formulation includes a complete set of transformed stress potential and displacement potential relations within the framework of fourier expansions and hankel integral transforms that is useful in a variety of elastodynamic as well as elastostatic problems for numerical computation of the integrals a robust and effective methodology is laid out which gives the necessary account of the presence of singularities including branch points and pole on the path of integration as illustrations the present green s functions are analytically degenerated to the special cases such as half space surface and full space green s functions some typical numerical examples are also given to show the general features of the bi material green s functions

a normal comparison inequality and its applications let xi xi i 1 leq i leq n and eta eta i 1 leq i leq n be standard normal random variables with covariance matrices r 1 r ij 1 and r 0 r ij 0 respectively slepian s lemma says that if r ij 1 geq r ij 0 for 1 leq i j leq n the lower bound p xi i leq u mb for 1 leq i leq n p eta i leq u mb for 1 leq i leq n is at least 1 in this paper an upper bound is given the usefulness of the upper bound is justified with three concrete applications i the new law of the iterated logarithm of erd s and revesz ii the probability that a random polynomial does not have a real zero and iii the random pursuit problem for fractional brownian particles in particular a conjecture of kesten 1992 on the random pursuit problem for brownian particles is confirmed which leads to estimates of principal eigenvalues

vertical vibration of a rigid circular disc at the interface of a transversely isotropic bi material a theoretical formulation is presented for the determination of the dynamic interaction of a vertically loaded rigid disc embedded at the interface of a transversely isotropic bi material full space with the aid of hankel integral transforms a relaxed treatment of the mixed boundary value problem is formulated as dual integral equations which can be reduced to a fredholm integral equation of the second kind the dynamic contact pressure under the disc and the impedance function are analytically evaluated in the general dynamic case it is shown that the impedance functions are in exact agreement with the existing solutions for a an elastic half space with isotropic material properties to confirm the accuracy of the numerical evaluation of the integrals involved numerical results are included for cases of different degrees of the material anisotropy and compared with previously published solutions

equations of anisotropic elastodynamics as a symmetric hyperbolic system deriving the time dependent fundamental solution the dynamic system of anisotropic elasticity from three second order partial differential equations is written in the form of the time dependent first order symmetric hyperbolic system with respect to displacement velocity and stress components a new method of deriving the time dependent fundamental solution of the obtained system is suggested in this paper this method consists of the following the fourier transform image of the fundamental solution with respect to a space variable is presented as a power series expansion relative to the fourier parameters then explicit formulae for the coefficients of these power series are derived successively using these formulae the computer calculation of fundamental solution components displacement velocity and stress components arising from pulse point forces has been made for general anisotropic media orthorhombic and monoclinic and the simulation of elastic waves has been obtained these computational examples confirm the robustness of the suggested method

ordering of random walks the leader and the laggard we investigate two complementary problems related to maintaining the relative positions of n random walks on the line i the leader problem that is the probability n t that the leftmost particle remains the leftmost as a function of time and ii the laggard problem the probability n t that the rightmost particle never becomes the leftmost we map these ordering problems onto an equivalent n 1 dimensional electrostatic problem from this construction we obtain a very accurate estimate for n t for n 4 the first case that is not exactly solvable 4 t t 4 with 4 0 91342 8 the probability of being the laggard also decays algebraically n t t n we derive 2 1 2 3 3 8 and argue that n n 1 ln n as n

lower tail probabilities for gaussian processes let x xt t s be a real valued gaussian random process indexed by s with mean zero general upper and lower estimates are given for the lower tail probability p supt s xt xt0 x as x 0 with t0 s fixed in particular sharp rates are given for fractional brownian sheet furthermore connections between lower tail probabilities for gaussian processes with stationary increments and level crossing probabilities for stationary gaussian processes are studied our methods also provide useful information on a random pursuit problem for fractional brownian particles

elliptic units for real quadratic fields 1 a review of the classical setting 2 elliptic units for real quadratic fields 2 1 p adic measures 2 2 double integrals 2 3 splitting a two cocycle 2 4 the main conjecture 2 5 modular symbols and dedekind sums 2 6 measures and the bruhat tits tree 2 7 indefinite integrals 2 8 the action of complex conjugation and of up 3 special values of zeta functions 3 1 the zeta function 3 2 values at negative integers 3 3 the p adic valuation 3 4 the brumer stark conjecture 3 5 connection with the gross stark conjecture 4 a kronecker limit formula 4 1 measures associated to eisenstein series 4 2 construction of the p adic l function 4 3 an explicit splitting of a two cocycle 4 4 generalized dedekind sums 4 5 measures on zp zp 4 6 a partial modular symbol of measures on zp zp 4 7 from zp zp to x 4 8 the measures and invariance

shintani zeta functions and gross stark units for totally real fields let f be a totally real number field and let p be a finite prime of f such that p splits completely in the finite abelian extension h of f tate has proposed a conjecture 22 conjecture 5 4 stating the existence of a p unit u in h with absolute values at the places above p specified in terms of the values at zero of the partial zeta functions associated to h f this conjecture is an analogue of stark s conjecture which tate called the brumer stark conjecture gross 12 conjecture 7 6 proposed a refinement of the brumer stark conjecture that gives a conjectural formula for the image of u in fp e where fp denotes the completion of f at p and e denotes the topological closure of the group of totally positive units e of f we present a further refinement of gross s conjecture by proposing a conjectural formula for the exact value of u in fp

small values of gaussian processes and functional laws of the iterated logarithm we estimate small ball probabilities for locally nondeterministic gaussian processes with stationary increments a class of processes that includes the fractional brownian motions these estimates are used to prove chung type laws of the iterated logarithm