some non self similar solutions to a nonlinear diffusion equation the author generates exact solutions to the nonlinear diffusion equation ut del u 1 2 del u which are not similarity solutions some applications and generalizations are indicated

group classification and symmetry reduction of variable coefficient nonlinear diffusion convection equation based on classical lie group method the group for a general class of variable coefficient nonlinear diffusion convection equation in 1 1 dimensions is obtained new symmetries are found seven models have been studied

motives and siegel modular forms for a motive m over q we define the fundamental periods of m using invariant theory our definition generalizes deligne s periods we show that if a motive m is constructed from motives m 1 m 2 m n by a standard algebraic operation then the fundamental periods of m can be expressed as monomials of the fundamental periods of m 1 m 2 m n applying this theory we discuss two hypothetical motives attached to a siegel modular form we show that a siegel modular form of degree m has at most m 1 period invariants

analytic calculation of phase and group velocities of p waves in orthorhombic media abstractwe have developed an approximate method to calculate the p wave phase and group velocities for orthorhombic media two forms of analytic approximations for p wave velocities in orthorhombic media were built by analogy with the five parameter moveout approximation and the four parameter velocity approximation for transversely isotropic media respectively they are called the generalized moveout approximation gma type approximation and the fomel approximation respectively we have developed approximations for elastic and acoustic orthorhombic media we have characterized the elastic orthorhombic media in voigt notation and we can describe the acoustic orthorhombic media by introducing the modified alkhalifah s notation our numerical evaluations indicate that the gma type and fomel approximations are accurate for elastic and acoustic orthorhombic media with strong anisotropy and the gma type approximation is comparable with the approximation recently proposed by sripanich and fomel potential

exact seismic velocities for transversely isotropic media and extended thomsen formulas for stronger anisotropies a different type of approximation to the exact anisotropic wave velocities as a function of incidence angle in transversely isotropic ti media is explored this formulation extends thomsen s weak anisotropy approach to stronger deviations from isotropy without significantly affecting the equations simplicity one easily recognized improvement is that the extreme value of the quasi sv wave speed vsv is located at the correct incidence angle ex rather than always being at the position 45 this holds universally for thomsen s approximation although ex 45 actually is never correct for any ti anisotropic medium wave speed magnitudes are more closely approximated for most values of the incidence angle although there may be some exceptions depending on actual angular location of the extreme value furthermore a special angle m close to theextreme point of the sv wave speed and also needed by the new formulas can be deduced from the same data normally used in weak anisotropy data analy

group classification of heat conductivity equations with a nonlinear source we suggest a systematic procedure for classifying partial differential equations pdes invariant with respect to low dimensional lie algebras this procedure is a proper synthesis of the infinitesimal lie method the technique of equivalence transformations and the theory of classification of abstract low dimensional lie algebras as an application we consider the problem of classifying heat conductivity equations in one variable with nonlinear convection and source terms we have derived a complete classification of nonlinear equations of this type admitting nontrivial symmetry it is shown that there are 3 7 28 and 12 inequivalent classes of pdes of the type considered that are invariant under the one two three and four dimensional lie algebras correspondingly furthermore we prove that any pde belonging to the class under study and admitting the symmetry group of a dimension higher than four is locally equivalent to a linear equation this classification is compared with existing group classifications of nonlinear heat conductivity equations and one of the conclusions is that all of them can be obtained within the framework of our approach furthermore a number of new invariant equations are constructed which have rich symmetry properties and therefore may be used for mathematical modelling of say nonlinear heat transfer processes

on special l values attached to metaplectic modular forms in this paper we establish some algebraic properties of special l values attached to siegel modular forms of half integral weight often called metaplectic modular forms these results are motivated by some exercises left by shimura to the reader in his marvellous book arithmeticity in the theory of automorphic forms

wide angle phase slowness approximations in vti media an anisotropic medium with vertical symmetry axis vti often presents a good model for describing real rocks propagation of quasi p and quasi sv waves in such media requires an expression of the vertical phase slowness a complicated function of the horizontal phase slowness and the medium parameters for converted wave phase shift migration methods it is desired to have slowness expressions that are simple and accurate at wide angles of propagation taylor series representations of the squared vertical slowness for quasi p and quasi sv waves result in new wide angle phase slowness approximations based on truncated series and continued fraction representations slowness approximations that are exact for both vertical propagation and at a horizontal slowness corresponding to horizontally traveling qp waves are derived the approximation for quasi sv waves can be used in phase shift migration in media where the quasi sv wavefront contains triplications these approximations are tested on several models and compared to previously published approximations the numerical tests suggest that the new continued fraction approximations are more accurate they can be used in phase shift migration algorithms which are more efficient for large angles than the existing approximations

potential symmetries of nonlinear diffusion convection equations in this paper potential symmetries are sought for the nonlinear diffusion convection equations the functional forms of f u and k u that admit such symmetries are completely classified

a simple technique for finding effective elastic constants of cracked solids for arbitrary crack orientation statistics abstract the effective elastic properties of cracked solids are anisotropic if the cracks have preferred orientations in this paper a simple scheme for evaluating the elastic stiffness tensor for an arbitrary orientation distribution of cracks at finite crack densities is presented the approach is based on a tensorial transformation of the effective elastic constants for isotropic orientation statistics through the use of a second order crack density tensor

on the transformation formulas of theta series here s is a nondegenerate rational symmetric matrix of signature q s s 3 x iy is a variable in the siegel half space hn of degree n u is a variable in the space cn of all complex n x q matrices runs over the set qq of all rational q x n matrices a is a c valued function on qn that is a finite linear combination of characteristic functions of cosets of qq modulo a z lattice a is a real matrix such that s ta diag lq 1 a and e c exp 2iric for c e c if q 1 s a 1 and a is suitably chosen the series g becomes a classical theta function of jacobi riemann if s is positive definite n 1 u 0 and q is even g is an elliptic modular form of weight q 2 studied by hecke also siegel introduced the series with u 0 in his theory of quadratic forms in si now g satisfies a transformation formula of the following type for every

on form preserving point transformations of partial differential equations new identities are presented relating arbitrary order partial derivatives of and for the general point transformation these identities are used to study the nature of those point transformations which preserve the general form of a wide class of 1 1 partial differential equations these results facilitate the search for point symmetries both discrete and continuous lie and assist the search for point transformations which reduce equations to canonical but similar form a simple test for the existence of hodograph type transformations between equations of similar form is given

new results on group classification of nonlinear diffusion convection equations using a new method and additional conditional and partial equivalence transformations we performed group classification in a class of variable coefficient 1 1 dimensional nonlinear diffusion convection equations of the general form f x ut d u ux x k u ux we obtain new interesting cases of such equations with the density f localized in space which have non trivial invariance algebra exact solutions of these equations are constructed we also consider the problem of investigation of the possible local transformations for an arbitrary pair of equations from the class under consideration i e of describing all the possible partial equivalence transformations in this class

zeta functions and eisenstein series on metaplectic groups the main objectives of this paper are 1 to construct an euler product from a hecke eigenform of half integral weight 2 to prove its meromorphic continuation to the whole plane 3 to examine its relationship with the fourier coefficients of the form and 4 to apply these results to the investigations of eisenstein series of half integral weight the automorphic forms are holomorphic ones considered on the metaplectic cover m of g where g n sp n f with a totally real algebraic number field f the first three are generalizations of the corresponding results in the elliptic and hilbert modular cases obtained in 2 and 6 at the same time all four closely parallel similar results in the case of integral weight in 9 and s10 to describe our results let us take for simplicity f to be q in which case the weight k is half an odd positive integer we consider a holomorphic form f of weight k with respect to a principal congruence subgroup of sp n z of level n which is an eigenform in the sense that f it m 2 m f with a complex number 2 m where n is a multiple of 4 and t m is a hecke operator defined for each positive integer m in a natural way as our first main result we shall prove theorem 4 4 that

on point transformations of evolution equations two evolution equations are considered of the form ut hi x t u ux uxx i 1 2 which are related by the general point transformation x p x t u t q x t u u r x t u it is shown that the time transformation must necessarily be of the form t q t and that if hi are polynomials in the spatial derivatives of u then x p x t

characterization of siegel cusp forms by the growth of their fourier coefficients we characterize all siegel cusp forms of degree n and large weight k by the growth of their fourier coefficients more precisely we prove among other related results that if the fourier coefficients of a modular form on the congruence subgroup gamma 0 n n of square free level n satisfy the hecke bound at the cusp infty then it must be a cusp form provided k 2n 1

differential constraints and exact solutions of nonlinear diffusion equations the differential constraints are applied to obtain explicit solutions of nonlinear diffusion equations certain linear determining equations with parameters are used to find such differential constraints they generalize the determining equations used in the search for classical lie symmetries