algebraic self similar renormalization in the theory of critical phenomena we consider the method of self similar renormalization for calculating critical temperatures and critical indices a new optimized variant of the method for an effective summation of asymptotic series is suggested and illustrated by several different examples the advantage of the method is in combining simplicity with high accuracy

arithmetic properties of power series solutions of algebraic differential equations in some non trivial disk we considered such a problem at a regular singular point and also at an irregular singular point the problem at a regular singular point was more difficult to deal with this difficulty we utilized a newton iteration procedure this procedure was originally used by j moser and v i arnol d to deal with small divisors in celestial mechanics cf s sternberg 1141 in the present work we improve our previous results so that they imply the following main theorem cf chapter 4 let q be the algebraic closure of the rational numbers let y

self semilar approximations for strongly interacting systems abstract a new type of approximations for strongly interacting systems is suggested the basis of the method is the formulation of self similar transformations for a continuous iteractive procedure the method is illustrated by a number of examples by reconstructing some simple analytical functions summing the wilson expansion for the critical exponents considering the gell mann low function in scalar field theory calculating the ground state energies for the optic polaron model the one dimensional bose system and for the quartic anharmonic oscillator notwithstanding an extreme simplicity of the method it is shown to give quite good results in the whole range of coupling parameters varying from zero up to infinity

fokker planck equation for bistable potential in the optimized expansion the optimized expansion is used to formulate a systematic approximation scheme to the probability distribution of a stochastic system the first order approximation for the one dimensional system driven by noise in an anharmonic potential is shown to agree well with the exact solution of the fokker planck equation even for a bistable system the whole period of evolution to equilibrium is correctly described at various noise intensities

gpu accelerated bernstein bezier discontinuous galerkin methods for wave problems we evaluate the computational performance of the bernstein bezier basis for discontinuous galerkin dg discretizations and show how to exploit properties of derivative and lift operators specific to bernstein polynomials for an optimal complexity quadrature free evaluation of the dg formulation issues of efficiency and numerical stability are discussed in the context of a model wave propagation problem we compare the performance of bernstein bezier kernels to both a straightforward and a block partitioned implementation of nodal dg kernels in a time explicit gpu accelerated dg solver computational experiments confirm the advantage of bernstein bezier dg kernels over both straightforward and block partitioned nodal dg kernels at high orders of approximation

bernstein bezier weight adjusted discontinuous galerkin methods for wave propagation in heterogeneous media abstract this paper presents an efficient discontinuous galerkin method to simulate wave propagation in heterogeneous media with sub cell variations this method is based on a weight adjusted discontinuous galerkin method wadg which achieves high order accuracy for arbitrary heterogeneous media 1 however the computational cost of wadg grows rapidly with the order of approximation in this work we propose a bernstein bezier weight adjusted discontinuous galerkin method bbwadg to address this cost by approximating sub cell heterogeneities by a fixed degree polynomial the main steps of wadg can be expressed as polynomial multiplication and l 2 projection which we carry out using fast bernstein algorithms the proposed approach reduces the overall computational complexity from o n 2 d to o n d 1 in d dimensions numerical experiments illustrate the accuracy of the proposed approach and computational experiments for a gpu implementation of bbwadg verify that this theoretical complexity is achieved in practice

self similar exponential approximants an approach is suggested for defining effective sums of divergent series in the form of self similar exponential approximants the procedure of constructing these approximants from divergent series with arbitrary noninteger powers is developed the basis of this construction is the self similar approximation theory control functions governing the convergence of exponentially renormalized series are defined from stability and fixed point conditions and from additional asymptotic conditions when the latter are available the stability of the calculational procedure is checked by analyzing cascade multipliers a number of physical examples for different statistical systems illustrate the generality and high accuracy of the approach

stability conditions for method of self similar approximations a generalization for the method of self similar approximations suggested recently by the author is given stability conditions for the method are formulated first the mapping multipliers for the self similar recurrence relations have to be less than unity second the lyapunov exponent for the differential form of the self similar relations has to be negative the fixed point conditions defining the governing functions are analyzed from the point of view of stability it is shown that the fixed point condition expressed in the form of the principle of minimal sensitivity provides a contracting mapping contrary to the condition in the form of the principle of minimal difference this explains why in general the former fixed point condition yields more accurate results than the latter

a hermite pade perspective on the renormalization group with an application to the correlation function of lieb liniger gas renormalization group rg methods are based on symmetry principles that are peculiar to quantum field theory and certain related settings however in cases when the function depends only on the coupling the rg technology does include a part which is related to the equation for the running coupling and which we will call the single variable rg svrg that turns out to be a purely formal method for improving the convergence properties of series expansions in fact we show that svrg is an integral hermite pade approximation preceded by a series reversion a complementary viewpoint on what svrg does to an expansion is that it performs a minimal upgrade to a self similar form the minimality being crucial in applications svrg is especially useful in problems in which one knows not only the first few terms of an expansion valid for small values of a variable but also the exponent but not the prefactor of the scaling law for large values of the variable as a practical example we extract the scaling law prefactor for the one body density matrix of the lieb liniger gas using a new result for the fourth order term in the short distance expansion we find remarkable agreement with known quantum diffusion monte carlo numerical results

higher orders of self similar approximations for thermodynamic potentials abstract the problem of calculating thermodynamic potentials in statistical mechanics by using the method of self similar approximations developed by the authors is considered an emphasis is made on the search for a regular procedure of defining higher order terms providing a good accuracy and stability of the method it is shown how the renormalized perturbation theory can be reformulated to the language of dynamical theory then instead of sequences of approximants one can study trajectories of cascades whose fixed point attractors play the role of the limits for the corresponding sequences of perturbative terms as an illustration of the procedure the free energy of a zero dimensional 4 model is calculated up to the third order of the self similar approximation whose precision is found to be greater than 0 1 for all coupling parameters ranging from zero to infinity

method of self similar approximations a new method is suggested to approximately find out the limit for a sequence of functions when solely several first terms of a sequence are known the method seems to be very useful for those complicated problems of mathematical physics statistical mechanics and field theory in which because of technical difficulties one is able to calculate using some iteration procedure or perturbation scheme only several first approximations for quantities of interest as an illustration the quartic oscillator is considered and it is shown that invoking only the terms of a perturbation theory up to the second order the ground state energy is defined with an accuracy of 0 1 for all values of the anharmonicity parameter from zero up to infinity

unified approach to crossover phenomena a general analytical method is developed for describing crossover phenomena of arbitrary nature the method is based on the algebraic self similar renormalization of asymptotic series with control functions defined by crossover conditions the method can be employed for such difficult problems for which only a few terms of asymptotic expansions are available and no other techniques are applicable as an illustration analytical solutions for several important physical problems are presented

critical exponents of the n vector model recently the series for two renormalization group functions corresponding to the anomalous dimensions of the fields and of the three dimensional field theory have been extended to next order seven loops by murray and nickel we examine the influence of these additional terms on the estimates of critical exponents of the n vector model using some new ideas in the context of the borel summation techniques the estimates have slightly changed but remain within the errors of the previous evaluation exponents such as related to the field anomalous dimension which were poorly determined in the previous evaluation of le guillou zinn justin have seen their apparent errors significantly decrease more importantly perhaps summation errors are better determined the change in exponents affects the recently determined ratios of amplitudes and we report the corresponding new values finally because an error has been discovered in the last order of the published expansions order we have also re analysed the determination of exponents from the expansion the conclusion is that the general agreement between expansion and three dimensional series has improved with respect to le guillou zinn justin

self similar bootstrap of divergent series a method is developed for calculating effective sums of divergent series this approach is a variant of the self similar approximation theory the interesting component here is using an algebraic transformation with a power providing maximal stability of the self similar renormalization procedure the latter is to be repeated as many times as necessary in order to convert into closed self similar expressions all sums from the series considered this multiple and complete renormalization is called a self similar bootstrap the method is illustrated by several examples from statistical physics

self similar approximations for thermodynamic potentials abstract the method of self similar approximation is extended here to the problem of calculating thermodynamic potentials in statistical mechanics to formulate an optimal scheme of the method we compare its different variants connected with different definitions of governing functions and damping parameters stability conditions are analyzed all concions are made apparent by calculating the free energy of a zero dimensional 4 model the optimal variant of the method is shown to be stable providing a good accuracy in the whole range of a coupling parameter from zero up to infinity

self similar crossover in statistical physics abstract an analytical method is advanced for constructing interpolation formulae for complicated problems of statistical mechanics in which just a few terms of asymptotic expansions are available the method is based on the self similar approximation theory being its variant where control functions are defined from asymptotic crossover conditions several examples from statistical physics demonstrate that the suggested method results in rather simple and surprisingly accurate formulae

summation of power series by self similar factor approximants abstract a novel method of summation for power series is developed the method is based on the self similar approximation theory the trick employed is in transforming first a series expansion into a product expansion and in applying the self similar renormalization to the latter rather to the former this results in self similar factor approximants extrapolating the sought functions from the region of asymptotically small variables to their whole domains the method of constructing crossover formulas interpolating between small and large values of variables is also analysed the techniques are illustrated on different series which are typical of problems in statistical mechanics condensed matter physics and generally in many body theory

weight adjusted discontinuous galerkin methods matrix valued weights and elastic wave propagation in heterogeneous media summary weight adjusted inner products 1 2 are easily invertible approximations to weighted l2 inner products these approximations can be paired with a discontinuous galerkin dg discretization to produce a time domain method for wave propagation which is low storage energy stable and high order accurate for arbitrary heterogeneous media and curvilinear meshes in this work we extend weight adjusted dg wadg methods to the case of matrix valued weights with the linear elastic wave equation as an application we present a dg formulation of the symmetric form of the linear elastic wave equation with upwind like dissipation incorporated through simple penalty fluxes a semi discrete convergence analysis is given and numerical results confirm the stability and high order accuracy of wadg for several problems in elastic wave propagation this article is protected by copyright all rights reserved

stochastic theory of log periodic patterns we introduce an analytical model based on birth death clustering processes to help in understanding the empirical log periodic corrections to power law scaling and the finite time singularity as reported in several domains including rupture earthquakes world population and financial systems in our stochastic theory log periodicities are a consequence of transient clusters induced by an entropy like term that may reflect the amount of co operative information carried by the state of a large system of different species the clustering completion rates for the system are assumed to be given by a simple linear death process the singularity at t0 is derived in terms of birth death clustering coefficients

temporal dynamics in perturbation theory abstract perturbation theory can be reformulated as dynamical theory then a sequence of perturbative approximations is bijective to a trajectory of dynamical system with discrete time called the approximation cascade here we concentrate our attention on the stability conditions permitting to control the convergence of approximation sequences we show that several types of mapping multipliers and lyapunov exponents can be introduced and respectively several types of conditions controlling local stability can be formulated the ideas are illustrated by calculating the energy levels of an anharmonic oscillator

on kummer s twenty four solutions of the hypergeometric differential equation the p adic analyticity of the boyarsky matrix associated with the hypergeometric function f a b c x has been investigated in an earlier article the transformation of this matrix under translation of a b c by z3 was determined at that time this article gives the transformation of this matrix under the extended kummer group the p adic implications of quadratic and higher degree transformations remain open the object of this article is to explain the relation between kummer s twenty four solutions of this equation 1 ax i a 12 c a b ab y 0 and the p adic theory of this equation for simplicity of exposition let us assume that a b c satisfy the condition 2 a b c a c b all lie outside z the unique solution of 1 analytic at a 0 is given by the classical formula 3 f a b c e a s b s ms s o c s where a s f a s f a s e n s 0 the euler integral formula f a b c n 17 c jb f 4 bs c a f b r c b l a b c wa b c x b x 1 1 x cb1 l x a d valid for re c re b 0 a restriction which may be removed 7 p 60 equation 13 by changing the path of integration shows that we are concerned with the periods of a certain differential as is well known 5 6 11 p 107 12 the kummer list may be obtained by transformation of this integral representation indeed there received by the editors september 1 1982 and in revised form september 28 1983 1980 mathematics subject classification primary 12b40 34a30 1984 american mathematical society 0002 9947 84 1 00 25 per page