artin hasse exponentials of derivations abstract we show how artin hasse exponentials of nilpotent derivations can be used to produce new cyclic gradings of non associative algebras of prime characteristic from a given one we discuss the connection with the technique of switching tori and consequently the associated root space decompositions in restricted modular lie algebras however differently from toral switching our method can produce gradings over cyclic p groups of order higher than p in particular we apply it to construct cyclic gradings of the zassenhaus lie algebra of dimension p n for p 3

transient analysis of viscoelastic helical rods subject to time dependent loads abstract the transient analysis of viscoelastic helical rods subject to time dependent loads are examined in the laplace domain the governing equations for naturally twisted and curved spatial rods obtained using the timoshenko beam theory are rewritten for cylindrical helical rods the curvature of the rod axis effect of rotary inertia and shear and axial deformations are considered in the formulation the material of the rod is assumed to be homogeneous isotropic and linear viscoelastic the viscoelastic constitutive equations are written in the boltzmann volterra form ordinary differential equations in canonical form obtained in the laplace domain are solved numerically using the complementary functions method to calculate the dynamic stiffness matrix of the problem the solutions obtained are transformed to the real space using an appropriate numerical inverse laplace transform method numerical results for quasi static and dynamic response of viscoelastic models are presented in the form of graphics

an optimal filtering method for stable analytic continuation in this paper we consider the problem of numerical analytic continuation of an analytic function f z f x iy on a strip domain w email protected c email protected r 0

deformations of w1 n a and modular semisimple lie algebras with a solvable maximal subalgebra abstract in one of his last papers boris weisfeiler proved that if modular semisimple lie algebra possesses a solvable maximal subalgebra which defines in it a long filtration then associated graded algebra is isomorphic to one constructed from the zassenhaus algebra tensored with the divided powers algebra we completely determine such class of algebras calculating in process low dimensional cohomology groups of zassenhaus algebra tensored with any associative commutative algebra

the restricted simple lie algebras are of classical or cartan type abstract the classification of the finite dimensional restricted simple lie algebras over an algebraically closed field f of prime characteristic p 7 is announced such an algebra is either of classical type an analogue over f of a finite dimensional simple lie algebra over the complex numbers or of cartan type an analogue over f of one of the infinite lie algebras of cartan over the complex numbers

a solution to certain polynomial equations with applications to nonlinear fitting we present a combinatorial method for solving a certain system of polynomial equations of vandermonde type in 2n variables by reducing it to the problem of solving two special linear systems of size n and rooting a single univariate polynomial of degree n over c all solutions can be found with fixed precision using up to polylogarithmic factors o n 2 bitwise operations in the worst case furthermore if the data is well conditioned then this can be reduced to o n bit operations up to polylogarithmic factors as an application we show how this can be used to fit data to a complex exponential sum with n terms in the same nearly optimal time

correction to cartan subalgebras of simple lie algebras trans amer math soc 234 1977 no 2 435 446 mr0480650 58 806

infinitesimal deformations of restricted simple lie algebras i abstract we compute the infinitesimal deformations of two families of restricted simple modular lie algebras of cartan type the witt jacobson and the special lie algebras

on spline regularized inversion of noisy laplace transforms abstract in this paper we have converted the laplace transform to an integral equation of the first kind of convolution type which is an ill posed problem and used the spline regularization method to solve it inversion of perturbed laplace transforms also plays an important role in system theory the method is applied to several test examples taken from 1 3 8 11 22 it gives a good approximation to the true solution and compares well with the methods discussed in 1 3 8 11 22 the results are shown in table 1 and respective diagrams

simple lie algebras of small characteristic ii exceptional roots abstract letlbe a finite dimensional simple lie algebra of absolute toral rank 2 over an algebraically closed field of characteristicp 3 andta 2 dimensional torus in the semisimplep envelope ofl suppose thatlis not isomorphic to a melikian algebra it is proved in this paper that for every root l t the subalgebrak generated by i f pki whereki x li x l i 0 acts triangulably onl in particular this implies that in the terminology of r e block and r l wilson 1988 j algebra114 115 259 all roots of l t are nonexceptional

improvement of accuracy in numerical methods for inverting laplace transforms based on the post widder formula abstract the paper proposes a new computational version for numerical inversion of laplace transforms at a point and on an interval based on the post widder formula in this version the original sequence of post widder approximants is calculated using operations on series and the approximate value of the sought function is constructed as a limit of this sequence using polynomial and rational extrapolation to the limit some procedures are proposed for improvement of accuracy in the problem of inversion at equidistant points of an interval without computing additional approximants an extension of the method to the two dimensional case is discussed the one dimensional version is illustrated with some probabilistic examples involving both explicit and implicit functions the accuracy achieved in these examples with using the standard double precision arithmetic evaluated in terms of relative errors is about 10 10 10 14 even in case of out of scale parameters computational aspects are discussed in comparison with previously known realizations of the post widder method

simple lie algebras of small characteristic vi completion of the classification let l be a finite dimensional simple lie algebra over an algebraically closed field of characteristic p 3 it is proved in this paper that if the p envelope of adl in derl contains a torus of maximal dimension whose centralizer in adl acts nontriangulably on l then p 5 and l is isomorphic to one of the melikian algebras m m n in conjunction with a premet h strade simple lie algebras of small characteristic v the non melikian case j algebra 314 2007 664 692 theorem 1 2 this implies that up to isomorphism any finite dimensional simple lie algebra over an algebraically closed field of characteristic p 3 is either classical or a filtered lie algebra of cartan type or a melikian algebra of characteristic 5 this result finally settles the classification problem for finite dimensional simple lie algebras over algebraically closed fields of characteristic 2 3

quasi static and dynamic response of viscoelastic helical rods in this study the dynamic behaviour of cylindrical helical rods made of linear viscoelastic materials are investigated in the laplace domain the governing equations for naturally twisted and curved spatial rods obtained using the timoshenko beam theory are rewritten for cylindrical helical rods the curvature of the rod axis effect of rotary inertia and shear and axial deformations are considered in the formulation the material of the rod is assumed to be homogeneous isotropic and linear viscoelastic in the viscoelastic material case according to the correspondence principle the material constants are replaced with their complex counterparts in the laplace domain ordinary differential equations in scalar form obtained in the laplace domain are solved numerically using the complementary functions method to calculate the dynamic stiffness matrix of the problem in the solutions the kelvin model is employed the solutions obtained are transformed to the real space using the durbin s numerical inverse laplace transform method numerical results for quasi static and dynamic response of viscoelastic models are presented in the form of graphics

an iteration method for stable analytic continuation abstract in the present paper we consider the problem of numerical analytic continuation for an analytic function f z f x iy on a strip domain z x iy c x r 0 y y 0 the key difference with the known methods is that a novel iteration regularization method with a priori and a posteriori parameter choice rule is presented the comparison in numerical aspect with other methods is also discussed

simple lie algebras of small characteristic iv solvable and classical roots abstract let l be a finite dimensional simple lie algebra over an algebraically closed field f of characteristic p 3 and t a torus of maximal dimension in the p envelope of l in der l in this paper we describe the t semisimple quotients of the 2 sections of l relative to t and prove that if all 1 sections of l relative to t are compositionally classical or solvable then l is either classical or a block algebra or a filtered lie algebra of type s

an analysis of bilinear transform polynomial methods of inversion of laplace transforms methods for the numerical inversion of a laplace transform f s which use a special bilinear transformation of s are particularly effective in many cases and are widely used the main purpose of this paper is to analyze the convergence and conditioning properties of a special class of such methods characterized by the use of lagrange interpolation the results derived apply both to complex and real inversion and show that some known inversion methods are in fact in this class

rank one lie algebras let l denote a finite dimensional lie algebra over an algebraically closed field such an algebra is called a rank one lie algebra provided l has a one dimensional cartan subalgebra when d is of characteristic zero a fundamental result in the theory of simple lie algebras over d is that the only rank one algebra is the 3 dimensional lie algebra b t 2 of traceless 2 x 2 matrices the representation theory of t t 2 plays a critical role in the structure and classification of simple algebras over as well as in the study of their representations when the field d is of prime characteristic p 3 in addition to b 2 the albert zassenhaus lie algebras 1 see also 3 are known to be rank one algebras by an albert zassenhaus lie algebra is meant an algebra over d with basis gala e g where g is a finite additive subgroup of d and with multiplication defined by

a method for the numerical inversion of laplace transforms abstract in this paper a numerical inversion method for laplace transforms based on a fourier series expansion developed by durbin 5 is presented the disadvantage of the inversion methods of that type the encountered dependence of discretization and truncation error on the free parameters is removed by the simultaneous application of a procedure for the reduction of the discretization error a method for accelerating the convergence of the fourier series and a procedure that computes approximately the best choice of the free parameters suitable for a given problem the inversion method allows the adequate application of these procedures therefore in a big range of applications a high accuracy can be achieved with only a few function evaluations of the laplace transform the inversion method is implemented as a fortran subroutine

methods of interconversion between linear viscoelastic material functions part ii an approximate analytical method abstract a new very simple approximate interconversion method is proposed and verified by examples this technique employing the slope of the source function on logarithmic scales is found to substantially enhance the accuracy compared to existing approximate methods the new method is based on the characteristic mathematical properties of the narrow band weight functions involved in the interrelationships between broad band material functions with the material functions represented locally by a power law they are interrelated in terms of adjustment factors expressed through the local log log slope of the given source function a number of existing approximate interconversion methods are also tested and compared with the new method in part i park and schapery 1998 an efficient numerical interconversion method based on a prony exponential series representation of both the source and target functions was presented such a series representation is not needed here the new method when applied to the prediction of broad band time dependent functions from laplace or fourier transforms is an approximate method of transform inversion that is applicable to functions which are not necessarily viscoelastic material functions

a regularization method for the numerical inversion of the laplace transform a method for numerical inversion of the laplace transform based on tikhonov regularization is described the method requires values of u s only for real s and appears relatively reliable in practice since the ill posedness of this problem has been considered the precondition to use the algorithm is that the exact solution of the inversion exists and it is lipschitz continuous with finite and known support

simple lie algebras of small characteristic v the non melikian case abstract let l be a finite dimensional simple lie algebra over an algebraically closed field f of characteristic p 3 we prove in this paper that if for every torus t of maximal dimension in the p envelope of adl in derl the centralizer of t in adl acts triangulably on l then l is either classical or of cartan type as a consequence we obtain that any finite dimensional simple lie algebra over an algebraically closed field of characteristic p 5 is either classical or of cartan type this settles the last remaining case of the generalized kostrikin shafarevich conjecture the case where p 7

a modular analogue of morozov s theorem on maximal subalgebras of simple lie algebras abstract let g be a simple algebraic group over an algebraically closed field of characteristic p 0 and suppose that p is a very good prime for g in this paper we prove that any maximal lie subalgebra m of g lie g with rad m 0 has the form m lie p for some maximal parabolic subgroup p of g this means that morozov s theorem on maximal subalgebras is valid under mild assumptions on g we show that such assumptions are necessary by providing a counterexample to morozov s theorem for groups of type e 8 over fields of characteristic 5 our proof relies on the main results and methods of the classification theory of finite dimensional simple lie algebras over fields of prime characteristic

simple lie algebras of small characteristic iii the toral rank 2 case abstract it is proved that every finite dimensional simple lie algebra of absolute toral rank 2 over an algebraically closed field of characteristic p 3 is of classical or cartan type or a melikian algebra

an iterative method for the numerical inversion of laplace transforms we present an algorithm for the numerical inversion of laplace transforms that is a particular case of the iterated regularization method proposed by vainikko in 1982 to construct the finite dimensional space we use laguerre polynomials error bounds for the approximations are derived

a modified tikhonov regularization for stable analytic continuation this paper is devoted to a new regularization method for solving the numerical analytic continuation of an analytic function f z f x iy on a strip domain omega z x iy in mathbb c mid x in mathbb r 0 y y 0 where the data is given approximately only on the line y 0 this problem is severely ill posed and has important practical applications the theoretical optimal error bound for the problem is proved which is independent of the selected regularization methods a modified tikhonov regularization method with asymptotic order optimal error estimates is proposed this method can be numerically implemented easily by the fast fourier transform some numerical examples are provided and a comparison with a fourier regularization method is given which show the modified tikhonov method works very well

on the structure of the graded lie algebra associated with a noncontractible filtration the largest r 0 such that g 0 is called the depth of g any such yr g is semisimple and contains a unique minimal homogeneous ideal which will be denoted as g9 the structure of g9 can be described in terms of derivation simple commutative associative graded algebras and simple w x graded lie algebras 23 there are two cases nondegenerate and degenerate in the terminology of weisfeiler which expose different features these two are distinguished by whether g9 has a nonzero positive part or

a combined approach of the laplace transform and pade approximation solving viscoelasticity problems in this paper the laplace transform method is combined with pade approximations to solve linear viscoelastic problems this approach allows to avoid the usual difficulties of original function determination an algorithm is given to find solution with arbitrary precision as an example the solution for problem of viscoelastic orthotropic half plane stress state under concentrated normal force is given

root filtration spaces from lie algebras and abstract root groups both timmesfeld s abstract root subgroups and simple lie algebras generated by extremal elements lead to root filtration spaces synthetically defined geometries on points and lines which can be characterized as root shadow spaces of buildings here we show how to obtain the root filtration space axioms from root subgroups and classical lie algebras

classification of the restricted simple lie algebras soit l une algebre de lie simple restreinte de dimension finie sur un corps algebriquement clos f de caracteristique p 7 alors l est de type classique ou de cartan

simple lie algebras of small characteristic i sandwich elements abstract let l be a finite dimensional simple lie algebra over an algebraically closed field of characteristic p 3 of absolute toral rank 2 and t a 2 dimensional torus in the semisimple p envelope of l it is proved that l is either classical or a block algebra or contains sandwich elements which are homogeneous with respect to t or a conjugate to t in addition several dimension estimates are given

numerical inversion of the laplace transform from the real axis a numerical method for inversion of the laplace transform f p given for p 0 only is proposed recommendations for the choice of the abscissa of convergence and parameters of numerical integration are given the results of the numerical tests are discussed

tori in the melikian algebra this paper is concerned with the tori in the restricted melikian algebra g g 1 1 according to the latest classification result of premet and strade 16 g is the unique simple lie algebra of toral rank 2 over an algebraically closed field of characteristic p 5 distinct from the classical algebras and algebras of cartan type here we determine the one dimensional subtori in g up to conjugation by automorphisms and compute their centralizers in g one consequence of this investigation is that all maximal tori in g are two dimensional for each two dimensional torus we find the number of its roots of any possible type classical witt hamiltonian proper the answer depends only on the position of the torus relative to the invariant filtration of g the conjugacy classes of two dimensional tori are not determined in this paper it can be shown that their number is infinite the classification of simple lie algebras requires detailed information about the toral structure of low rank lie algebras see e g 3 19 one of the key points is the possibility of finding a torus of maximal dimension in the p envelope of a simple lie algebra l having all roots proper the classical argument is based on checking that when switching an improper root to a proper one the number of proper roots in every 2 section of l increases in the case of the melikian algebra we encounter a situation when the number of proper roots remains unchanged when this does occur we show that the number of hamiltonian roots becomes strictly less

contributions to the classification of simple modular lie algebras we develop results directed towards the problem of classifying the finite dimensional simple lie algebras over an algebraically closed field of characteristic p 7 a 1 section of such a lie algebra relative to a torus t of maximal absolute toral rank possesses a unique subalgebra maximal with respect to having a composition series with factors which are abelian or classical simple in this paper we show that the sum q of those compositionally classical subalgebras is a subalgebra this extends to the general case a crucial step in the classification by block and wilson of the restricted simple lie algebras we derive properties of the filtration which can be constructed using q and obtain structural information about the 1 sections and 2 sections of q relative to t we further classify all those algebras in which q is solvable

c sections of lie algebras let m be a maximal subalgebra of a lie algebra l and a b a chief factor of l such that b m and a 6 m we call the factor algebra m a b a c section of m all such c sections are isomorphic and this concept is related those of c ideals and ideal index previously introduced by the author properties of c sections are studied and some new characterizations of solvable lie algebras are obtained mathematics subject classification 2000 17b05 17b20 17b30 17b50

the classification of the simple modular lie algebras ii the toral structure abstract let l be a simple lie algebra over an algebraically closed field of characteristic p 7 and t an optimal torus in some p envelope l p we determine the action of t on the two sections of l which have been described in st4 we also give some new and noncomputational proofs to determine the conjugacy classes of the tori in w n 1 and of the cartan subalgebras of w 1 n

a modified lavrentiev iterative regularization method for analytic continuation abstract we consider the problem of numerical analytic continuation of an analytic function f z f x i y on a strip domain z x i y c x r 0 y y 0 where the data is given approximately only on the line y 0 this is a severely ill posed problem motivated by the advantage of iterative methods for solving ill posed problems we propose a new modified iterative method to solve this problem under both a priori and a posteriori parameter choice rules moreover some sharp error estimates between the exact solution and its approximation are proved some interesting numerical examples are conducted for showing that the newly developed method works well

an inverse model and mathematical solution for inferring viscoelastic properties and dynamic deformations of heterogeneous structures abstract we proposed an adjoint based inverse method and mathematical solutions using lagrange multiplier theorem for inferring dynamic viscoelastic properties of heterogeneous structures with displacement observations to improve computation reliability and efficiency existing methods such as the one using finite difference approximated gradient may not be efficient and accurate enough for considering complex situations of the coupled effects of dynamic loading material deformation memory and structural heterogeneity the proposed method derives both gradient and hessian mathematically to satisfy the first order necessary and second order sufficient optimal conditions which has great advantages compared to other approaches we also proposed robust numerical algorithms for accurate and fast computations including a regularization to control reasonable parameter ranges a newton s method with hessian function to determine search direction for fast convenience and a modified armijo rule to find a stable step length we developed a galerkin time domain finite element method for numerical solutions of resulting partial differential equations and validated the model for a layered structure under dynamic loading tests results indicate that the proposed method has greatly improved computation accuracy and speed as compared to existing approaches it may be applied to a broad range of heterogeneous materials and structures at different length and time scales

determination of mechanical spectra from experimental responses abstract a recursive computer algorithm was developed which generates line spectra from experimental response functions the method allows storing information on the mechanical properties of polymeric materials in a convenient way the algorithm also interconverts between relaxation and retardation spectra from the spectra any desired response function can then be recovered the algorithm essentially utilizes the fact that the kernel functions resemble step functions slightly different codes are used for each kernel function the appearance of negative relaxation or retardation lines is obviated mathematically such lines would be acceptable and they do not seriously affect reconstruction of responses within relaxation or retardation behavior however they would seriously interfere with interconversion between the two types of behavior and they would also pose problems in the interpretation of the spectra

methods of interconversion between linear viscoelastic material functions part i a numerical method based on prony series abstract an efficient and accurate numerical method of interconversion between linear viscoelastic material functions based on a prony series representation is presented and tested using experimental data from selected polymeric materials the method is straightforward and applicable to interconversion between modulus and compliance functions in time frequency and laplace transform domains good agreement is shown between solutions obtained from the method in different domains a detailed computational procedure and selection of values for parameters involved in the method are presented and illustrated in particular the effects of different choices of relaxation and retardation times on the accuracy of the method are discussed the mathematical efficiency associated with prony series representations of both source and target transient material functions is fully utilized the method is general enough to cover both viscoelastic solids and liquids the tensile relaxation data from polymethyl methacrylate pmma and the shear storage compliance data from polyisobutylene are used in illustrating the method in a companion paper schapery and park 1998 an extended approximate analytical interconversion method is presented

analytical modeling of viscoelastic dampers for structural and vibration control different approaches to the mathematical modeling of viscoelastic dampers are addressed and their theoretical basis and performance are compared the standard mechanical model smm comprising linear springs and dashpots is shown to accurately describe the broad band rheological behavior of common viscoelastic dampers and be more efficient than other models such as the fractional derivative model and the modified power law the smm renders a prony series expression for the modulus and compliance functions in the time domain and the remarkable mathematical efficiency associated with the exponential basis functions of a prony series greatly facilitates model calibration and interconversion while cumbersome nonlinear regression is usually required for other models a simple collocation or least squares method can be used to fit the smm to available experimental data the model allows viscoelastic material functions to be readily determined either directly from the experimental data or through interconversion from a function established in another domain numerical examples on two common viscoelastic dampers demonstrate the advantages of the smm over fractional derivative and power law models detailed computational procedures for fitting and interconversion are discussed and illustrated published experimental data from a viscoelastic liquid damper and a viscoelastic solid damper are used in the examples

the classification of the simple modular lie algebras iii solution of the classical case we investigate the structure of simple modular lie algebras over an algebraically closed field of characteristic p 7 if the distinguished subalgebra q l t with respect to an optimal torus t in some p envelope lp of l equals l then either l has a two section of type h 2 1 z or l is of classical type