a chevalley theorem for difference equations by a theorem of chevalley the image of a morphism of varieties is a constructible set the algebraic version of this fact is usually stated as a result on extension of specializations or lifting of prime ideals we present a difference analog of this theorem the approach is based on the philosophy that occasionally one needs to pass to higher powers of where is the endomorphism defining the difference structure in other words we consider difference pseudo fields which are finite direct products of fields rather than difference fields we also prove a result on compatibility of pseudo fields and present some applications of the main theorem e g constrained extension and uniqueness of differential picard vessiot rings with a difference parameter

analytic and algorithmic aspects of generalized harmonic sums and polylogarithms in recent three loop calculations of massive feynman integrals within quantum chromodynamics qcd and e g in recent combinatorial problems the so called generalized harmonic sums in short s sums arise they are characterized by rational or real numerator weights also different from 1 in this article we explore the algorithmic and analytic properties of these sums systematically we work out the mellin and inverse mellin transform which connects the sums under consideration with the associated poincare iterated integrals also called generalized harmonic polylogarithms in this regard we obtain explicit analytic continuations by means of asymptotic expansions of the s sums which started to occur frequently in current qcd calculations in addition we derive algebraic and structural relations like differentiation with respect to the external summation index and different multi argument relations for the compactification of s sum expressions finally we calculate algebraic relations for infinite s sums or equivalently for generalized harmonic polylogarithms evaluated at special values the corresponding algorithms and relations are encoded in the computer algebra package harmonicsums

non integrability by discrete quadratures we give a necessary condition for integrability by discrete quadratures of systems of difference equations the discrete variational equations along algebraic solutions must have virtually solvable galois groups this necessary condition a la morales and ramis is used in order to prove that q analogues of painleve i and painleve iii equations are not integrable by discrete quadratures

multiple eigenvalue sensitivity analysis abstract this paper surveys the development of the sensitivity analysis of multiple eigenvalues and the associated invariant subspaces of two kinds of matrix eigenvalue problems in which the matrices are analytically dependent on several parameters

differential galois theory of linear difference equations we present a galois theory of difference equations designed to measure the differential dependencies among solutions of linear difference equations with this we are able to reprove holder s theorem that the gamma function satisfies no polynomial differential equation and are able to give general results that imply for example that no differential relationship holds among solutions of certain classes of q hypergeometric equations

on the nature of the generating series of walks in the quarter plane in the present paper we introduce a new approach relying on the galois theory of difference equations to study the nature of the generating series of walks in the quarter plane using this approach we are not only able to recover many of the recent results about these series but also to go beyond them for instance we give for the first time hypertranscendency results i e we prove that certain of these generating series do not satisfy any nontrivial nonlinear algebraic differential equation with rational function coefficients

fuzzy human reliability analysis on the chernobyl accident abstract this paper compares the result of fuzzy reliability analysis using error possibility with that of probabilistic analysis on the accident at the chernobyl nuclear power plant the case study shows that fuzzy reliability analysis gives information from more points of view than probabilistic analysis

differential tannakian categories abstract we define a differential tannakian category and show that under a natural assumption it has a fibre functor if in addition this category is neutral that is the target category for the fibre functor are finite dimensional vector spaces over the base field then it is equivalent to the category of representations of a pro linear differential algebraic group our treatment of the problem is via differential hopf algebras and deligne s fibre functor construction p deligne categories tannakiennes in the grothendieck festschrift vol ii in progr math vol 87 birkhauser boston boston ma 1990 pp 111 195

zariski closures of reductive linear differential algebraic groups linear differential algebraic groups ldags appear as galois groups of systems of linear differential and difference equations with parameters these groups measure differential algebraic dependencies among solutions of the equations ldags are now also used in factoring partial differential operators in this paper we study zariski closures of ldags in particular we give a tannakian characterization of algebraic groups that are zariski closures of a given ldag moreover we show that the zariski closures that correspond to representations of minimal dimension of a reductive ldag are all isomorphic in addition we give a tannakian description of simple ldags this substantially extends the classical results of p cassidy and we hope will have an impact on developing algorithms that compute differential galois groups of the above equations and factoring partial differential operators

fuzzy logic in the nuclear research world abstract a glance through this special issue entitled fuzzy logic applications in nuclear science and technology reveals a significant influence of fuzzy logic applications this paper outlines briefly current results of fuzzy logic applications in the field of nuclear science and technology and emphasizes that fuzzy logic and novel intelligent systems are needed in today s nuclear world

generalized basic hypergeometric equations this paper deals with regular singular generalized q hypergeometric equations with either large or small galois groups in particular we consider the fundamental problem of finding appropriate galoisian substitutes for the usual notion of local monodromy

on multiple eigenvalues of matrices depending on several parameters the eigenvalues and eigenvectors as well as their derivatives of matrices depending on one or several parameters have been studied by sun j comput math 3 1985 pp 351 364 using the implicit function theorems and others but not for the more general multiple eigenvalues in this paper the mean of the multiple eigenvalues in terms of the trace operator tr the corresponding invariant subspaces and their derivatives are studied using a generalization of sun s approach the numerical aspects concerning the computation of such derivatives by direct and iterative methods e g simultaneous and inverse simultaneous iterations will be discussed briefly the main results in this paper apply to clusters of nonmultiple eigenvalues as well the implications e g on simultaneous iteration will be explored

existence of parameterized picard vessiot extensions over fields with algebraically closed constants abstract the purpose of this short note is to establish the existence of parameterized picard vessiot extensions for systems of linear difference differential equations over difference differential fields with an algebraically closed field of constants

galois groups for integrable and projectively integrable linear difference equations we consider first order linear difference systems over c x c x with respect to a difference operator that is either a shift x x 1 x x 1 q dilation x qx x qx with q c q c not a root of unity or mahler operator x xq x xq with q z 2q z 2 such a system is integrable if its solutions also satisfy a linear differential system it is projectively integrable if it becomes integrable after moding out by scalars we apply recent results of schafke and singer to characterize which groups can occur as galois groups of integrable or projectively integrable linear difference systems in particular such groups must be solvable finally we give hypertranscendence criteria

the condition number of equivalence transformations that block diagonalize matrix pencils how ill conditioned must a matrix s be if its columns are constrained to span certain subspaces we answer this question in order to find nearly best conditioned matrices sr and sl that block diagonalize a given matrix pencil t a b i e s l 1 tsr is bloc diagonal we show that the best conditioned sr has a condition number approximately equal to the cosecant of the smallest angle between right subspaces belonging to different diagonal blocks of thus the more nearly the right subspaces overlap the more ill conditioned sr must be the same is true of sl and the left subspaces for the standard eigenproblem t a i sl sr and the cosecant of the angle between subspaces turns out equal to an earlier estimate of the smallest condition number namely the norm of the projection matrix associated with one of the subspaces we apply this result to bound the error in an algorithm to compute analytic functions of matrices for instance exp t

calculating differential galois groups of parametrized differential equations with applications to hypertranscendence the main motivation of our work is to create an efficient algorithm that decides hypertranscendence of solutions of linear differential equations via the parameterized and differential galois theories to achieve this we expand the representation theory of linear differential algebraic groups and develop new algorithms that calculate unipotent radicals of parameterized differential galois groups for differential equations whose coefficients are rational functions berman and singer presented an algorithm calculating the differential galois group for differential equations without parameters whose differential operator is a composition of two completely reducible differential operators we use their algorithm as a part of our algorithm as a result we find an effective criterion for the algebraic independence of the solutions of parameterized differential equations and all of their derivatives with respect to the parameter

difference galois theory of linear differential equations abstract we develop a galois theory for linear differential equations equipped with the action of an endomorphism this theory is aimed at studying the difference algebraic relations among the solutions of a linear differential equation the galois groups here are linear difference algebraic groups i e matrix groups defined by algebraic difference equations

linear algebraic groups as parameterized picard vessiot galois groups we show that a linear algebraic group is the galois group of a parameterized picard vessiot extension of k x x 1 for certain differential fields k if and only if its identity component has no one dimensional quotient as a linear algebraic group

a galois theoretic proof of the differential transcendence of the incomplete gamma function abstract we give simple necessary and sufficient conditions for the t transcendence of the solutions to a parameterized second order linear differential equation of the form 2 y x 2 p y x 0 where p f x is a rational function in x with coefficients in a t field f our criteria imply in particular the t transcendence of the incomplete gamma function t x generalizing a result of johnson reinhart and rubel this result is also an important part of an efficient algorithm to compute the parameterized picard vessiot group of an arbitrary parameterized second order linear differential equation over f x

parameterized picard vessiot extensions and atiyah extensions abstract generalizing atiyah extensions we introduce and study differential abelian tensor categories over differential rings by a differential ring we mean a commutative ring with an action of a lie ring by derivations in particular these derivations act on a differential category a differential tannakian theory is developed the main application is to the galois theory of linear differential equations with parameters namely we show the existence of a parameterized picard vessiot extension and therefore the galois correspondence for many differential fields with possibly non differentially closed fields of constants that is fields of functions of parameters other applications include a substantially simplified test for a system of linear differential equations with parameters to be isomonodromic which will appear in a separate paper this application is based on differential categories developed in the present paper and not just differential algebraic groups and their representations

fundamental solutions for meromorphic linear difference equations in the complex plane and related problems soit a m z y z m a 0 z y z 0 une equation aux differences lineaires a coefficients entiers alors il existe un systeme de m solutions meromorphes y 1 y m sur c lineairement independantes sur les fonctions periodiques on demontre ce theoreme en considerant des solutions de l equation comme des sections d un faisceau de modules sur un faisceau de fonctions periodiques meromorphes

differential transcendence of solutions of difference riccati equations and tietze s treatment abstract there is the paper by h tietze published in 1905 on differential transcendence of solutions of difference riccati equations in this paper we clarify the essence of tietze s treatment and make it purely algebraic as an application the q airy equation is studied

an approach to human reliability on man machine systems using error possibility abstract this paper is concerned with proposing the concept error possibility instead of the error rate and presenting a method of fault tree analysis of human errors based on this concept the dependence between consecutive human tasks is considered as a fuzzy causal relation human reliability is defined as the evaluation of the error possibility from several points of view 1 the likelihood of error which gives the maximal grade of membership 2 potentiality for error 3 fuzziness of error possibility and 4 importance of a basic event as an illustrative example the emergency tasks in blackout at a pwr plant are shown and the fault tree concerning the error of these tasks is analyzed it is shown that the proposed method gives us information as well