automorphisms of the nottingham group abstract let k be a finite field of characteristic p 5 the nottingham group n over k is the group of normalised automorphisms of the local field k t in this paper we determine the automorphism group of n in particular we show that every automorphism of the nottingham group is standard we also give a complete description of the conjugacy classes of elements of order p in n

hausdorff dimension pro p groups and kac moody algebras every finitely generated profinite group can be given the structure of a metric space and as such it has a well defined hausdorff dimension function in this paper we study hausdorff dimension of closed subgroups of finitely generated pro p groups g we prove that if g is p adic analytic and h c g is a closed subgroup then the hausdorff dimension of h is dimh dimg where the dimensions are of h and g as lie groups letting the spectrum spec g of g denote the set of hausdorff dimensions of closed subgroups of g it follows that the spectrum of p adic analytic groups is finite and consists of rational numbers we then consider some non p adic analytic groups g and study their spectrum in particular we investigate the maximal hausdorff dimension of nonopen subgroups of g and show that it is equal to 1 1 d 1 in the case of g sld fp t where p 2 and to 1 2 if g is the so called nottingham group where p 5 we also determine the spectrum of sl2 fp t p 2 completely showing that it is equal to 0 2 3 1 some of the proofs rely on the description of maximal graded subalgebras of kac moody algebras recently obtained by the authors in joint work with e i zelmanov

analytic treatment for variable coefficient fourth order parabolic partial differential equations the variable coefficient fourth order parabolic partial differential equation in one space variable that arise in the study of the transverse vibrations of a uniform flexible beam is examined the analytic solution is obtained by using adomian decomposition method the nonhomogeneous problem is solved by using the noise terms phenomenon the method is tested on three problems which have appeared in physical applications

some physical structures for the 2 1 dimensional boussinesq water equation with positive and negative exponents in this paper a mathematical method is constructed to study two variants of the two dimensional boussinesq water equation with positive and negative exponents in terms of travelling wave solutions the partial differential equations are transformed to nonlinear ordinary differential equations exact solutions are then derived for various cases to describe the different physical structures such as compactons solitons solitary patterns and periodic solutions the exponent of the wave function u and the ratio of the two coefficients a and b in the boussinesq equation are shown to qualitatively determine the physical structures of the solutions

resonance and web structure in discrete soliton systems the two dimensional toda lattice and its fully discrete and ultra discrete analogues we present a class of solutions of the two dimensional toda lattice equation its fully discrete analogue and its ultra discrete limit these solutions demonstrate the existence of soliton resonance and web like structure in discrete integrable systems such as differential difference equations difference equations and cellular automata ultra discrete equations

an implicit high accuracy variable mesh scheme for 1 d non linear singular parabolic partial differential equations abstract in this paper we propose a new two level implicit difference scheme of o k 2 h l 1 kh l h l 3 for the solution of non linear parabolic equation eu xx x t u u x u t 0 x t 0 subject to appropriate initial and dirichlet boundary conditions prescribed where k 0 h l 0 are mesh sizes in t and x coordinates respectively and e 0 is a small parameter in addition we also discuss a new explicit variable mesh difference scheme of o kh l h l 3 for the estimates of u x in all cases we require only three spatial variable grid points the proposed schemes require less algebra and three evaluations of function a special technique is required to solve singular parabolic equations the proposed variable mesh scheme when applied to a linear diffusion equation is shown to be stable for all h l 0 and k 0 computational results are provided to support our derived schemes and analysis

the remarkable relations among pdes integrable by the inverse spectral transform method by the method of characteristics and by the hopf cole transformation we establish deep and remarkable connections among partial differential equations pdes integrable by different methods the inverse spectral transform method the method of characteristics and the hopf cole transformation more concretely 1 we show that the integrability properties lax pair infinitely many commuting symmetries large classes of analytic solutions of 2 1 dimensional pdes integrable by the inverse scattering transform method s integrable can be generated by the integrability prop erties of the 1 1 dimensional matrix burgers hierarchy integrable by the matrix hopf cole transformation c integrable 2 we show that the integrability properties i of s integrable pdes in 1 1 dimensions ii of the multidimensional generalizations of the gl m c self dual yang mills equations and iii of the multidimensional calogero equations can be generated by the integrability properties of a recently introduced mul tidimensional matrix equation solvable by the method of characteristics to establish the above links we consider a block frobenius matrix reduction of the relevant matrix fields leading to integrable chains of matrix equations for the blocks of such a frobenius matrix followed by a systematic elimination procedure of some of these blocks the con struction of large classes of solutions of the soliton equations from solutions of the matrix burgers hierarchy turns out to be intimately related to the construction of solutions in sato theory 3 we finally show that suitable generalizations of the block frobenius ma trix reduction of the matrix burgers hierarchy generates pdes exhibiting integrability properties in common with both s and c integrable equations

embedding properties of hereditarily just infinite profinite wreath products abstract we study infinitely iterated wreath products of finite permutation groups w r t product actions in particular we prove that for every non empty class of finite simple groups x there exists a finitely generated hereditarily just infinite profinite group w with composition factors in x such that any countably based profinite group with composition factors in x can be embedded into w additionally we investigate when infinitely iterated wreath products of finite simple groups w r t product actions are co hopfian or non co hopfian

pro p groups with few normal subgroups abstract motivated by the study of pro p groups with finite coclass we consider the class of pro p groups with few normal subgroups this is not a well defined class and we offer several different definitions and study the connections between them furthermore we propose a definition of periodicity for pro p groups thus providing a general framework for some periodic patterns that have already been observed in the existing literature we then focus on examples and show that strikingly all the interesting examples not only have few normal subgroups but in addition have periodicity in the lattice of normal subgroups

he s variational iteration method for fourth order parabolic equations he s variational iteration method is applied to fourth order parabolic partial differential equations with variable coefficients to illustrate the ability and reliability of the method some examples are given revealing its effectiveness and simplicity

abstract commensurators of profinite groups in this paper we initiate a systematic study of the abstract com mensurators of profinite groups the abstract commensurator of a profinite group g is a group comm g which depends only on the commensurability class of g we study various properties of comm g in particular we find two natural ways to turn it into a topological group we also use comm g to study topological groups which contain g as an open subgroup all such groups are totally disconnected and locally compact for instance we con struct a topologically simple group which contains the pro 2 completion of the grigorchuk group as an open subgroup on the other hand we show that some profinite groups cannot be embedded as open subgroups of compactly generated topologically simple groups several celebrated rigidity theorems like pink s analogue of mostow s strong rigidity theorem for simple algebraic groups defined over local fields and the neukirch uchida theorem can be re formulated as structure theorems for the commensurators of certain profinite groups

variable mesh methods for the numerical solution of two point singular perturbation problems abstract in this paper variable mesh difference methods of third order are derived for the solution of the two point second order singular perturbation problems y f x y y e these methods are applied to a number of examples and their superiority over the existing methods is shown

the nottingham group for p 2 abstract in this paper i consider the nottingham group over a finite field of characteristic 2 i obtain a number of results previously known only for nottingham groups in odd characteristic for example i obtain information on the derived series and i prove that the nottingham group is hereditarily just infinite as a pro 2 group as a consequence the nottingham group over any finite field of characteristic 2 is not analytic over any pro 2 ring

a class of two and three level implicit methods of order two in time and four in space based on half step discretization for two dimensional fourth order quasi linear parabolic equations abstract novel compact schemes based on half step discretization are developed to solve two dimensional fourth order quasi linear parabolic equations these discretizations use a single computational cell and the boundary conditions are incorporated in a natural way without the use of any fictitious points or discretization the schemes have second order accuracy in time and fourth order accuracy in space the proposed methods are directly applicable to problems constituting singular terms we also present operator splitting method for the solution of linear parabolic equations which allows the use of the one dimensional tri diagonal solver multiple times it is shown that the operator splitting method is unconditionally stable the numerical schemes have been successfully applied to the two dimensional extended fisher kolmogorov equation sivashinsky equation boussinesq equation and vibrations problem the illustrative results confirm the theoretical order of magnitude and accuracy of the proposed methods

new just infinite pro p groups of finite width and subgroups of the nottingham group we use the action of the nottingham group on the completion of its lie algebra to construct new examples of just infinite pro p groups of finite width whose graded lie algebras with respect to the lower central series are isomorphic to sl2 fp tfp t the groups constructed are non linear and thus answer the question posed in comm algebra 30 2002 1293 the proof of nonlinearity uses a new description of the centralizers of elements of order p in the nottingham group and the concept of hausdorff dimension

unconditionally stable compact theta schemes for solving the linear and semi linear fourth order diffusion equations abstract this paper is concerned with numerical methods for solving a class of fourth order diffusion equations combining the compact difference operator in space discretization and the linear method in time the compact theta scheme for the linear problem is first proposed by virtue of the fourier method the suggested scheme is shown to be unconditionally stable and convergent in the discrete l2 norm for any 1 2 1 and then this idea is generalized to the semi linear case the corresponding compact theta scheme is constructed and analyzed in detail numerical experiments corresponding to the linear and semi linear situations are carried out to support our theoretical statements

fifth degree b spline solution for a fourth order parabolic partial differential equations in this paper we find the numerical solution of a fourth order parabolic partial differential equation a family of b spline methods has been considered for the numerical solution of the problems the results show that the present method is an applicable technique and approximates the exact solution very well

on the commensurator of the nottingham group let p 5 be a prime number we prove that the abstract commensurator of the nottingham group n f p coincides with its automorphism group which is known to be a finite extension of n f p as a corollary we deduce that the nottingham group cannot be embedded as an open subgroup of a topologically simple group

index subgroups of the nottingham group abstract it is known that every countably based pro p group embeds as a closed subgroup into the nottingham group j j f p at the same time the subgroup structure of j remains rather mysterious in this paper we introduce the class of so called index subgroups of j and study their fundamental properties two main applications are given i we prove that a certain infinite subclass of index subgroups provides new examples of hereditarily just infinite pro p groups of finite width ii we determine which hausdorff dimensions occur for index subgroups and subsequently show that the hausdorff spectrum of j begins with a proper interval

high accuracy implicit variable mesh methods for numerical study of special types of fourth order non linear parabolic equations in this paper we propose a new implicit high order two level finite difference method of o k 2 k h l h l 3 for the solution of special type of 1d fourth order non linear parabolic partial differential equation of the form u x x x x 2 u x x t u t t f x t u u x u x x u t u x x x u x t a 0 and hl 0 are the mesh sizes in time and space coordinates respectively the third order variable mesh methods proposed are based on only three spatial grid points xl xl 1 meaning that no fictitious points are required for incorporating the boundary conditions we discuss how our formulation is able to handle linear singular problem the proposed two level finite difference method is shown to be unconditionally stable for a model linear problem we applied the proposed methods to find the numerical solution of more complex fourth order non linear equations like second order benjamin ono equation and the good boussinesq equation it is evident from the numerical experiments that the numerical results agree well with the exact solutions hence demonstrating the efficiency of the methods derived

a difference scheme with high accuracy in time for fourth order parabolic equations abstract a novel finite difference method is developed for the numerical solution of fourth order parabolic partial differential equations in one and two space variables the method is seen to evolve from a multiderivative method for second order ordinary differential equations the method is tested on three model problems with constant coefficients and variable coefficients which have appeared in the literature

some hereditarily just infinite subgroups of j abstract this work examines the commutator structure of some closed subgroups of the wild group of automorphisms of a local field with perfect residue field a group we call j in particular we establish a new approach to evaluating commutators in j and using this method investigate the normal subgroup structure of some classes of index subgroups of j as introduced by klopsch we provide new proofs of fesenko s results that lead to a proof that the torsion free group t t k 1 a k t qk 1 a k f p is hereditarily just infinite and by extending his work we also demonstrate the existence of a new class of hereditarily just infinite subgroups of j which have non trivial torsion

an accurate three spatial grid point discretization of o k2 h4 for the numerical solution of one space dimensional unsteady quasi linear biharmonic problem of second kind in this article using three spatial grid points we propose two new two level implicit finite difference approximations of o k 2 h 2 and o k 2 h 4 in a coupled manner to the one space dimensional unsteady quasi linear biharmonic equation a x t u u x x u x x x x u t f x t u u x u x x u x x x 00 subject to the initial and boundary conditions u x 0 email protected x u 0 t p 0 t u x x 0 t q 0 t u 1 t p 1 t u x x 1 t q 1 t are prescribed where h 0 and k 0 are mesh sizes in x and t directions respectively the numerical solution of u x x is obtained as a by product of the method and we do not require to discretize the boundary conditions the methods are successfully tested on the problems having singularities numerical results are provided to demonstrate the convergence of new methods

exact solutions for variable coefficients fourth order parabolic partial differential equations in higher dimensional spaces in this paper analytic solution of variable coefficient fourth order parabolic partial differential equation in two and three space variables are developed the calculations are accelerated by using the noise terms phenomenon for nonhomogeneous problems numerical examples are investigated to illustrate the pertinent features of the proposed algorithm