on the complexity of computing the volume of a polyhedron we show that computing the volume of a polyhedron given either as a list of facets or as a list of vertices is as hard as computing the permanent of a matrix

on the number of quintic fields we show that the number of quintic number fields whose discriminant does not exceed x in absolute value is bounded by a constant times x1 e for any e 0 this may be compared with the conjecture that the number of such fields is asymptotic to a constant times x as x tends to infinity

on the poles of p adic complex powers and the b functions of prehomogeneous vector spaces introduction in the real case it is well known that there exists an intimate relation between poles of complex powers of polynomial functions and the roots of the b functions cf b many examples suggest that a similar relation between them would exist also in the p adic case recently the case of polynomials over a p adic field in two variables has been closely examined cf f loeser l w veys v in this paper we shall deal with the case of relative invariants of prehomogeneous vector spaces over a p adic field let us explain our problem more precisely let k be a p adic field i e a finite algebraic extension of qp we denote by k the maximal compact subring of k by rrt k the ideal of nonunits of k and put q dki xc k we denote by ikthe absolute value on k normalized as itwk q let fq kx be the set of quasi characters of kx then the identity component fl kx o of fl kx consists of w t itls for all s in c and fl kx o cx under ws q s let g p v be a prehomogeneous vector space abbrev p v defined over k where we assume that

polytope volume computation a combinatorial form of gram s relation for convex polytopes can be adapted for use in computing polytope volume we present an algorithm for volume computation based on this observation this algorithm is useful in finding the volume of a polytope given as the solution set of a system of linear inequalities p x e r ax b as an illustration we compute a formula for the volume of a projective image of the n cube from this formula we deduce that when a and b have rational entries so that the volume of p is also a rational number the number of binary digits in the denominator of the volume cannot be bounded by a polynomial in the total number of digits in the numerators and denominators of entries of a and b this settles a question posed by dyer and frieze

computing motivic zeta functions on log smooth models we give an explicit formula for the motivic zeta function in terms of a log smooth model it generalizes the classical formulas for snc models but it gives rise to much fewer candidate poles in general this formula plays an essential role in recent work on motivic zeta functions of degenerating calabi yau varieties by the second named author and his collaborators as a further illustration we explain how the formula for newton non degenerate polynomials can be viewed as a special case of our results

fast mcmc sampling algorithms on polytopes we propose and analyze two new mcmc sampling algorithms the vaidya walk and the john walk for generating samples from the uniform distribution over a polytope both random walks are sampling algorithms derived from interior point methods the former is based on volumetric logarithmic barrier introduced by vaidya whereas the latter uses john s ellipsoids we show that the vaidya walk mixes in significantly fewer steps than the logarithmic barrier based dikin walk studied in past work for a polytope in d defined by n d linear constraints we show that the mixing time from a warm start is bounded as o n0 5d1 5 compared to the o nd mixing time bound for the dikin walk the cost of each step of the vaidya walk is of the same order as the dikin walk and at most twice as large in terms of constant pre factors for the john walk we prove an o d2 5 log4 n d bound on its mixing time and conjecture that an improved variant of it could achieve a mixing time of o d2 poly log n d additionally we propose variants of the vaidya and john walks that mix in polynomial time from a deterministic starting point the speed up of the vaidya walk over the dikin walk are illustrated in numerical examples

vanishing of principal value integrals on surfaces principal value integrals are associated to multi valued rational differential forms with normal crossings support on a non singular algebraic variety we prove their vanishing on rational surfaces in the context of a conjecture of denef jacobs as an appli cation we obtain a strong vanishing result for candidate poles of p adic and motivic igusa zeta functions

computing volumes of polyhedra in this note we give two simple methods for calculating the volume of any closed bounded polyhedron in r having an orientable boundary which is triangulated into a set of n 1 dimensional simplices the formulas given require only coordinates of the vertices of the polyhedron 1 introduction the purpose of this note is to give two simple methods for calculating the volume of any closed bounded polyhedron p in r having an orientable boundary ap which is triangulated into a set t of n 1 dimensional simplices following hadwiger 2 we define a polyhedron to be the union of pairwise disjoint convex polyhedra each of which is the convex hull of a finite number of points in 1 we have described an algorithm for obtaining a piecewise linear manifold which closely approximates an implicitly defined manifold if p has been given in such a way then the affine pieces of ap are in general easy to triangulate with an inherited orientation for polyhedra p which are determined by a given system of inequalities methods and programs for triangulating p have been given in 5 6 for such polyhedra a triangulation of the boundary is not easily available so our method is inappropriate of course our approach would also be unnecessary for computing the volume of a parallelotope practical applications of the methods given here may be made to the approxima tion of an area bounded by an implicitly defined curve or to approximation of the volume of a solid which is bounded by an implicitly defined surface although some formulas for the volume of convex polyhedra in rn appear in the literature e g 3 these formulas generally require the computation of the n 1 volume of the facets and additionally they involve extra computations of certain distances the volume formulas we give here involve only the coordinates of the vertices of p

poles of maximal order of motivic zeta functions we prove a 1999 conjecture of veys which says that the opposite of the log canonical threshold is the only possible pole of maximal order of denef and loeser s motivic zeta function associated with a germ of a regular function on a smooth variety over a field of characteristic zero we apply similar methods to study the weight function on the berkovich skeleton associated with a degeneration of calabi yau varieties our results suggest that the weight function induces a flow on the non archimedean analytification of the degeneration towards the kontsevich soibelman skeleton

algorithmic and modeling insights via volumetric comparison of polyhedral relaxations this is mostly a survey on some mathematical results concerning volumes of polytopes of interest in non convex optimization our motivation is in geometrically comparing relaxations in the context of mixed integer linear and nonlinear optimization with the goal of gaining algorithmic and modeling insights we consider relaxations of fixed charge formulations vertex packing polytopes boolean quadric polytopes and relaxations of graphs of monomials on box domains besides surveying the area we do give a few new results and we provide many directions for further work

on the spectral side of arthur s trace formula combinatorial setup in arthur s trace formula a ubiquitous role is played by certain limiting expressions arising from piecewise smooth functions with respect to projections of the coxeter fan g m families these include terms resulting from intertwining operators on the spectral side and volumes of polytopes on the geometric side we introduce the combinatorial concept of a compatible family with respect to an arbitrary polyhedral fan and obtain new formulas for the corresponding limiting expressions in this general framework our formulas can be regarded as algebraic generalizations of certain volume formulas for convex polytopes in a companion paper the results are used to study the spectral side of the trace formula

on the smallest poles of igusa s p adic zeta functions let k be a p adic field we explore igusa s p adic zeta function which is associated to a k analytic function on an open and compact subset of k n first we deduce a formula for an important coefficient in the laurent series of this meromorphic function at a candidate pole afterwards we use this formula to determine all values less than 1 2 for n 2 and less than 1 for n 3 which occur as the real part of a pole

bayesian analysis of binary sequences this manuscript details bayesian methodology for learning by example with binary n sequences encoding the objects under consideration priors prove influential conformable priors are described laplace approximation of bayes integrals yields posterior likelihoods for all n sequences this involves the optimization of a definite function over a convex domain efficiently effectuated by the sequential application of the quadratic program

a sweep plane algorithm for generating random tuples in simple polytopes a sweep plane algorithm by lawrence for convex polytope computation is adapted to generate random tuples on simple polytopes in our method an affine hyperplane is swept through the given polytope until a random fraction sampled from a proper univariate distribution of the volume of the polytope is covered then the intersection of the plane with the polytope is a simple polytope with smaller dimension in the second part we apply this method to construct a black box algorithm for log concave and t concave multivariate distributions by means of transformed density rejection author s abstract

on the discretization of single input single output bilinear systems prompted by the observation that the parameters of a discrete bilinear model structure corresponding to a time invariant continuous bilinear model structure are input dependent quantities this paper deduces a correction factor that relates the continuous and discrete time models the correction factor that is obtained analytically is shown to provide an exact discretization an approximation to the correction factor is also included which produces a satisfactory discretization provided realistic bounds are placed on the choice of the sampling interval building on the developed approach for discretizing a first order bilinear system the procedure is extended to the case of higher order systems the new approach differs from existing schemes in that a discrete time bilinear model structure is retained with consideration given to a first order model of a high temperature furnace application it is shown that the resulting parameter variation is significantly reduced when a bilinear model structure is re

an efficient algorithm for obtaining the volume of a special kind of pyramid and application to convex polyhedra an efficient method is given for obtaining the volume of a pyramid of species n 2 whose base is a convex polygon the pyramid is transformed into a simplex whose volume is computed directly a refinement is provided to the cohen hickey method for determining volumes of convex polyhedra

a classification of 2 simple prehomogeneous vector spaces of type i let p g gl v be a rational representation of a connected linear algebraic group g on a finite dimensional vector space v all defined over an algebraically closed field k of characteristic zero if v has a zariskidense g orbit we call a triplet g p v a prehomogeneous oector space abbrev p v when p is irreducible such p v s have been classified in 11 since then it has turned out gradually that the complete classification of reductive p v s i e p v s with reductive groups g is an extremely laborious task therefore it is natural to classify some restricted class of p v s e g 2 to get some insight into the general situation a p v g p v is called a 2 simple p v when 1 g gl 1 x g x g with simple algebraic groups g and gz 2 p is the composition of a rational representation p of g x g of the form p p1 0 p pkol a a l l z t with k s t l where pi tv resp pi t are nontrivial irreducible representations of g resp g and the scalar multiplications gz 1 on each irreducible component v for i 1 f where v i v we say that a 2 simple p v g p v is of type zif k l and at least one of gl l xg xg pi pi i 1 k is a nontrivial p v see definition 5 p 43 in 11 on the other hand if k l and all gl l xg xg p op i l k are trivial p v s it is called a 2 simple p v of type ii in 3 all 2 simple p v s of type ii has been already classified in this paper we shall classify all 2simple p v s of type i thus together with 3 we complete a classification of all 2 simple p v s for example the fact that all irreducible p v s are castling equivalent to 2 simple p v s or to x m x y m x g 2 1 1 i i n 1 with m 2 3 see 1 indicates the importance of 2 simple p v s for simplicity we write g p v or g p instead of g p v

the classification of spinors under over finite fields the spinors of a 14 dimensional vector space v are studied with respect to the group gspin 14 of the 14 dimensional vector space v over finite fields f q results are given as follows 1 the decomposition of the space of spinors into gspin 14 equivalence classes or orbits over f q 2 the structure of the fixer of gspin 14 for each orbit as an f q group

arithmetic calculus of fourier transforms by igusa local zeta functions we show the possibility of explicit calculation of the fourier transforms of complex powers of relative invariants of some prehomogeneous vector spaces over jr by using the explicit form of p adic igusa local zeta functions let g p v be a regular prehomogeneous vector space defined over r and fi f the basic r relative invariants of g p v cf f sato 2 p 444 for explicit calculation of fourier transforms of the complex power r if x is 1 if1 x isi i 1 with s si sr e cr two general methods are known when g p v has finitely many g orbits one can use a microlocal calculus established by m kashiwara according to m sato s idea see p 404 in m k k m when r 1 and the relative invariant is linear for each variable one can use igusa s method see p 8 in igusa 2 in this paper we shall show the possibility of calculation for some cases by using the explicit form of the igusa s local zeta function based on the idea of iwasawa tate theory iwasawa tate as an example we shall calculate the fourier transform of if x is for gl4 x sl2m l a2 d al d a d a alt2m 1 d aff2m l d aff2m i 1 d aff2m which has infinitely many orbits and r 4 so that it has not been calculated by other methods the igusa local zeta function of this prehomogeneous vector space has been explicitly calculated by hosokawa our method is applicable for a prehomogeneous vector space with za rzm for some constant r 0 for example irreducible resp simple 2 simple of type i regular prehomogeneous vector spaces with finitely many adelic open orbits satisfy za rzm see igusa 4 and k k this paper has four sections 1 preliminaries 2 functional equation 3 unramified f factors and igusa local zeta functions 4 arithmetic calculus example the last section of this paper was completed when the author was received by the editors july 27 1993 and in revised form march 30 1994 1991 mathematics subject classification primary 32m1 2 20g20 secondary 1 1s40

generalised tangential interpolation for model reduction of discrete time mimo bilinear systems in this article we discuss a model order reduction method for multiple input and multiple output discrete time bilinear control systems similar to the continuous time case we will show that a system can be characterised by a series of generalised transfer functions this will be achieved by a multivariate z transform of kernels corresponding to an explicit solution formula for discrete time systems we will further address the problem of generalised tangential interpolation which naturally comes along with this approach we will introduce a reasonable generalisation of the linear 2 norm based on this concept we discuss the choice of interpolation points furthermore an efficient discretisation of continuous time systems is provided the performance of the proposed method is evaluated in some numerical examples and compared with the method of balanced truncation for bilinear systems

the adelic zeta function associated with the space of binary cubic forms with coefficients in a function field in this paper we study the adelic zeta function associated with the prehomogeneous vector space of binary cubic forms defined over a function field we establish its rationality find its poles and residues and a simple functional equation that this zeta function satisfies introduction the subject of integral equivalence of binary cubic forms although not as popular or well developed as the theory of integral equivalence of binary quadratic forms has received a fair amount of attention the first paper on this topic 9 written by eisenstein appeared in 1844 later on several other first rate mathematicians tried their hand at the subject among others the list includes arndt cayley hermite and davenport the classical method of attacking this problem is to draw upon a one to one discriminant preserving relation between binary cubic and binary quadratic forms which assigns to a cubic form its hessian and then use the reduction theory for binary quadratic forms to obtain results on class numbers of binary cubic forms the best of the classical results are due to davenport 5 who showed that e h ir n 2 36x o x15 16 nax e h ir n v2 12x o x 5116 nax where h ir n is the class number of irreducible binary cubic forms of discriminant n the credit for a new surge of interest in the study of integral equivalence of binary cubic forms must indisputably go to shintani for it was his 1972 paper 25 that put this study on modern footing in that paper shintani introduced a certain zeta function which as it turns out decomposes into a linear combination of products of dirichlet series whose coefficients are weighted class numbers of binary cubic forms and certain mellin transforms by studying analytic properties of the zeta function shintani was able to improve the error term in davenport s result to o x576 it is of interest to note that the zeta function and the dirichlet series it received by the editors september 3 1985 1980 mathematics subject classification 1985 revision primary 12a70 secondary iocio 12a90 1987 american mathematical society 0002 9947 87 1 00 25 per page

on the arithmetic of a singular invariant introduction we know by schouten 13 and gurevich 1 that the ring of invariants under sl7 acting on the space x of trivectors in aff7 the affine 7 space is generated by an irreducible homogeneous polynomial f x of degree 7 with rational coefficients we normalize f x up to sign by the condition that its coefficients are relatively prime integers this is the singular invariant in the title if k is a number field the image g of gl7 in gl x is a connected irreducible k subgroup of gl x acting transitively on y x f f1 0 and f gx v g f x for everyg in g with a rational character v of g we take a hecke quasicharacter x of k and introduce a zeta distribution z w in the adelization xa of x as

new polytope decompositions and euler maclaurin formulas for simple integral polytopes abstract we give new weighted decompositions for simple polytopes generalizing previous results of lawrence varchenko and brianchon gram we start with witten s non abelian localization principle in equivariant cohomology for the norm square of the moment map in the context of toric varieties to obtain a decomposition for delzant polytopes then by a purely combinatorial argument we show that this formula holds for any simple polytope as an application we study euler maclaurin formulas

robust multi criteria ranking with additive value models and holistic pair wise preference statements we consider a problem of ranking alternatives based on their deterministic performance evaluations on multiple criteria we apply additive value theory and assume the decision maker s dm preferences to be representable with general additive monotone value functions the dm provides indirect preference information in form of pair wise comparisons of reference alternatives and we use this to derive the set of compatible value functions then this set is analyzed to describe 1 the possible and necessary preference relations 2 probabilities of the possible relations 3 ranges of ranks the alternatives may obtain and 4 the distributions of these ranks our work combines previous results from robust ordinal regression extreme ranking analysis and stochastic multicriteria acceptability analysis under a unified decision support framework we show how the four different results complement each other discuss extensions of the main proposal and demonstrate practical use of the approach by considering a problem of ranking 20 european countries in terms of 4 criteria reflecting the quality of their universities

how to integrate a polynomial over a simplex this paper starts by settling the computational complexity of the problem of integrating a polynomial function f over a rational simplex we prove that the problem is np hard for arbitrary polynomials via a generalization of a theorem of motzkin and straus on the other hand if the polynomial depends only on a fixed number of variables while its degree and the dimension of the simplex are allowed to vary we prove that integration can be done in polynomial time as a consequence for polynomials of fixed total degree there is a polynomial time algorithm as well we explore our algorithms with some experiments we conclude the article with extensions to other polytopes and discussion of other available methods 1

optimality of the boundary control for n n parabolic lag system abstract optimal boundary control problem for n n coupled system of second order parabolic lag partial differential equation with infinitely many variables is considered by lions scheme j l lions optimal control of systems governed by partial differential equations grundlehren math wiss vol 170 springer verlag 1971 necessary and sufficient condition of optimality for the neumann problem with quadratic performance functional and constrained control is derived finally several mathematical examples for derived optimality conditions are presented

approximate volume and integration for basic semialgebraic sets given a basic compact semialgebraic set mathbf k subset mathbb r n we introduce a methodology that generates a sequence converging to the volume of mathbf k this sequence is obtained from optimal values of a hierarchy of either semidefinite or linear programs not only the volume but also every finite vector of moments of the probability measure that is uniformly distributed on mathbf k can be approximated as closely as desired which permits the approximation of the integral on mathbf k of any given polynomial the extension to integration against some weight functions is also provided finally some numerical issues associated with the algorithms involved are briefly discussed

euler maclaurin with remainder for a simple integral polytope we give an euler maclaurin formula with remainder for the sum of the values of a smooth function on the integral points in a simple integral polytope we prove this formula by elementary methods

the complexity of vertex enumeration methods the computational complexity of problems relating to the enumeration of all the vertices of a convex polyhedron defined by linear inequalities is examined several published approaches are evaluated in this light a new algorithm is described and shown to have a better complexity estimate than existing methods empirical evidence supporting the theoretical superiority is presented finally vertex enumeration is discussed when the space containing the polyhedra is of fixed dimension and only the size of the inequality system is permitted to vary

twists of the iwasawa tate zeta function the intent of this paper is to develop the theory of a simple variation of the adelic zeta function first described by tate 14 and iwasawa 9 in independent efforts to simplify hecke s analysis of his l functions attached to gr6bencharakters our twists of the iwasawa tate zeta function arise in the framework of a more general theory of zeta functions associated to representations of algebraic groups many of the basic problems of this general theory have already been solved by yukie and the author see 20 and 21 1 but still await appearance in final written form thus it seems worthwhile to give here a general statement of these problems before proceeding to the very special case considered in this paper let k be a global field i e algebraic number field or function field of a curve defined over a finite field let n and a denote the ring of adeles and the group of ideles of k respectively let co be a quasicharacter of the idele group trivial on the embedded group k of nonzero elements of k the group f2 f2 k of all such quasicharacters co has a natural riemann surface structure let l i denote the usual idele norm on a and for s tl2 let co be the principal quasicharacter co t ltl let re co be the real part of co f2 defined as the unique real number a such that l o i co for any reductive linear algebraic group g defined over k let g denote the adelizat ion of g over k as defined in 16 and let gk be the group of k rational points of g embedded as a discrete group in ga for simplicity we shall fix g as g1 now although the theory to be described below can also be effected for any reductive algebraic group with at least one nontrivial k rational character let p g g1 v be an irreducible k rational representation of g in the finite dimensional vector space v of dimension m let i be the identity matrix in g by schur s lemma there is an integer d such that p ti tdp i for all scalars t we shall always assume that d 0 let dg be any convenient left invariant measure on the quotient ga gk let v be the schwartz bruhat space of functions on v for the time being let vk be any gk invariant subset of vk the zeta function associated to the representation p may now be defined

monodromy eigenvalues and zeta functions with differential forms abstract for a complex polynomial or analytic function f there is a strong correspondence between poles of the so called local zeta functions or complex powers f 2 s where the are c differential forms with compact support and eigenvalues of the local monodromy of f in particular barlet showed that each monodromy eigenvalue of f is of the form exp 2 1 s 0 where s 0 is such a pole we prove an analogous result for similar p adic complex powers called igusa local zeta functions but mainly for the related algebro geometric topological and motivic zeta functions

motivic zeta functions of abelian varieties and the monodromy conjecture we prove for abelian varieties a global form of denef and loeser s motivic monodromy conjecture in arbitrary characteristic more precisely we prove that for every tamely ramified abelian variety a over a complete discretely valued field with algebraically closed residue field its motivic zeta function has a unique pole at chai s base change conductor c a of a and that the order of this pole equals one plus the potential toric rank of a moreover we show that for every embedding of ql in c the value exp 2 ic a is an l adic tame monodromy eigenvalue of a the main tool in the paper is edixhoven s filtration on the special fiber of the neron model of a which measures the behavior of the neron model under tame base change

monodromy jordan blocks b functions and poles of zeta functions for germs of plane curves abstract we study the poles of several local zeta functions the igusa topological and motivic zeta function associated to a germ of a holomorphic function in two variables it was known that there is at most one double pole for any of these zeta functions which is then given by the log canonical threshold of the function at the singular point if the germ is reduced loeser showed that such a double pole always induces a monodromy eigenvalue with a jordan block of size 2 here we settle the non reduced situation describing precisely in which case such a jordan block of maximal size 2 occurs we also provide detailed information about the bernstein sato polynomial in the relevant non reduced situation confirming a conjecture of igusa denef and loeser

the euler maclaurin formula for simple integral polytopes we give a euler maclaurin formula with remainder for the sum of a smooth function on the integral points in a simple integral lattice polytope our proof uses elementary methods

monodromy conjecture for nondegenerate surface singularities we prove the monodromy conjecture for the topological zeta function for all non degenerate surface singularities fundamental in our work is a detailed study of the formula for the zeta function of monodromy by varchenko and the study of the candidate poles of the topological zeta function yielded by what we call b1 facets in particular new cases among the nondegenerate surface singularities for which the monodromy conjecture is proven now are the non isolated singularities the singularities giving rise to a topological zeta function with multiple candidate pole and the ones for which the newton polyhedron contains a b1 facet

lower bound for the poles of igusa s p adic zeta functions let k be a p adic field r the valuation ring of k p the maximal ideal of r and q the cardinality of the residue field r p let f be a polynomial over r in n 1 variables and let be a character of r times let mi u be the number of solutions of f u in r pi n for i in mathbb z geq 0 and u in r p i these numbers are related with igusa s p adic zeta function zf s of f we explain the connection between the mi u and the smallest real part of a pole of zf s we also prove that mi u is divisible by q ulcorner n 2 i 1 urcorner where the corners indicate that we have to round up this will imply our main result zf s has no poles with real part less than n 2 we will also consider arbitrary k analytic functions f

some p v equivalences and a classification of 2 simple prehomogeneous vector spaces of type a classification of 2 simple prehomogeneous vector spaces is completed by using some p v equivalences together with 3 some part is very different from the previous classification of the irreducible or simple cases 1 2 and some new method is necessary this result shows the difficult point of a classification problem of reductive prehomogeneous vector spaces introduction in m sato and t kimura 1 all irreducible prehomogeneous vector spaces abbreviated irreducible p v s are classified up to castling equivalence and it is proved that any irreducible p v is castling equivalent to a 2 simple p v or to sl m x sl m x gl 2 a1 x a1 x a1 with m 2 3 therefore as a step to a classification of the general case it is natural to classify all nonirreducible 2 simple p v s in this paper we shall classify all 2 simple p v s of type ii and give the complete list of them up to strong equivalence in the sense of definition 4 p 36 in 1 however we need various p v equivalences cf propositions 1 7 1 12 1 34 1 36 1 37 theorem 1 16 and its corollary corollary 1 26 definition 1 31 remark 1 32 to carry out the classification and hence first we shall prove them we also investigate the regularity in some cases since all 2 simple p v s of type i are already classified in 3 this completes the classification of all 2 simple p v s here we say that g p v or simply g p is a 2 simple p v if it is a prehomogeneous vector space of the following form g gl 1 k s t xg x g2 where g1 and g2 are simple algebraic groups p the composition of the scalar multiplications gl l k s on each irreducible component and the representation p p1 p 1 pk p k a1 s 1 1 t1 t ofthegroupg1 x g2 where pi aj resp p tj are nontrivial irreducible representations of g1 resp g2 we write g p v instead of g p v for simplicity to avoid confusion we write instead of clearly each gl 1 x g1 x g2 pi p i 1 k is an irreducible 2 simple p v if at least one of them is a nontrivial p v see definition 1 1 we call it a 2 simple p v of type i on the other hand if all of them are trivial p v s we call it a 2 simple p v of type ii in this case we have g2 sl n received by the editors march 3 1986 and in revised form july 30 1986 1980 mathematics subject classification 1985 revision primary 11r20 11r29 secondary 11n25 11r18 01988 american mathematical society 0002 9947 88 1 00 25 per page

valuative invariants for polymatroids abstract many important invariants for matroids and polymatroids such as the tutte polynomial the billera jia reiner quasi symmetric function and the invariant g introduced by the first author are valuative in this paper we construct the z modules of all z valued valuative functions for labeled matroids and polymatroids on a fixed ground set and their unlabeled counterparts the z modules of valuative invariants we give explicit bases for these modules and for their dual modules generated by indicator functions of polytopes and explicit formulas for their ranks our results confirm a conjecture of the first author that g is universal for valuative invariants

how to compute the volume in high dimension in some areas of theoretical computer science we feel that randomized algorithms are better and in some others we can prove that they are more efficient than the deterministic ones

local zeta functions of certain prehomogeneous vector spaces in which k is a p adic field k is an absolute value on k ok is the ring of integers of k x is an affine space with an ok structure and f x is a polynomial on x we take as f x the discriminant f x1 x2 of the binary cubic form n ux1 vx2 in u v where x1 x2 are in a simple k split jordan algebra a of degree 3 with n as its generic norm and we shall determine z s in general if f x is a polynomial on an affine space v equipped with a nondegenerate quadratic form q x and if f x y is the coefficient of a variable t in f x ty then a morphism v v is defined as q x y f x y for every x y in v in this notation the above n x can be characterized up to a normalization as a cubic form satisfying x n x x cf springer 18 in general if gi is a subgroup of gl vi for i 1 2 then we shall denote by g1 0 g2 0 the image of g1 x g2 x in gl v1 0 v20 in this and other standard notations the above f x1 x2 becomes a generator of the ring of relative invariants of

a statistical approach to the analytic hierarchy process with interval judgements i distributions on feasible regions abstract this paper addresses the problem of extracting preferences for alternatives from interval judgement matrices in the analytic hierarchy process ahp the method of arbel for extracting such preferences which is based on the assumption that the interval judgements specify a feasible region in the weight space of the alternatives is critically appraised from a statistical perspective and some new ideas emanating from this approach are developed and discussed in particular it is proposed that a distribution for the weights on the feasible region which is both tractable and meaningful be adopted the mean of the distribution can then be used as an assessment of the overall ranking of the alternatives and quantities of interest such as probabilities and marginal distributions can immediately be quantified two specific distributions on the feasible region the uniform distribution and the distribution of random convex combinations with coefficients which are uniform spacings are examined in some detail and the ideas which emerge are illustrated by means of selected examples

finding simplicial subdivisions of polytopes subdivisions of polytopes are described which place any chosen vertex in a near minimal number of simplices ancillary procedures to find volumes and centroids of polytopes are indicated

on the complexity of computing mixed volumes this paper gives various positive and negative results on the complexity of the problem of computing and approximating mixed volumes of polytopes and more general convex bodies in arbitrary dimension on the negative side we present several p hardness results that focus on the difference of computing mixed volumes versus computing the volume of polytopes we show that computing the volume of zonotopes is p hard while each corresponding mixed volume can be computed easily but also give examples showing that computing mixed volumes is hard even when computing the volume is easy on the positive side we derive a randomized algorithm for computing the mixed volumes v overbrace k 1 ld k 1 m 1 overbrace k 2 dots k 2 m 2 dots overbrace k s dots k s m s of well presented convex bodies k 1 dots k s where m 1 dots m s in n 0 and m 1 geq n psi n with psi n o frac log n log log n the algorithm is an interpolation method based on polynomial time randomized algorithms for computing the volume of convex bodies this paper concludes with applications of our results to various problems in discrete mathematics combinatorics computational convexity algebraic geometry geometry of numbers and operations research

a laplace transform algorithm for the volume of a convex polytope we provide two algorithms for computing the volume of the convex polytope x n ax b for a m n b n the computational complexity of both algorithms is essentially described by nm which makes them especially attractive for large n and relatively small m when the other methods with o mn complexity fail the methodology which differs from previous existing methods uses a laplace transform technique that is well suited to the half space representation of

survey paper direction dependent systems a survey in this paper a survey on systems which exhibit gains and or dynamics depending on the input or output direction is presented several examples of such systems covering a wide range of application areas and the causes of their direction dependent responses are given direction dependent systems are mathematically classified into three classes depending on their state space representation and switching function the related linear dynamic transient and steady state characteristics of these systems are investigated approaches to model direction dependent systems using wiener models piecewise linear models bilinear models and recurrent neural networks are described three application examples are used to illustrate the usefulness of the direction dependent description the advantages and disadvantages of direction dependent models as well as possible directions for future work are pointed out

zeta functions and oscillatory integrals for meromorphic functions abstract in the 70 s igusa developed a uniform theory for local zeta functions and oscillatory integrals attached to polynomials with coefficients in a local field of characteristic zero in the present article this theory is extended to the case of rational functions or more generally meromorphic functions f g with coefficients in a local field of characteristic zero this generalization is far from being straightforward due to the fact that several new geometric phenomena appear also the oscillatory integrals have two different asymptotic expansions the usual one when the norm of the parameter tends to infinity and another one when the norm of the parameter tends to zero the first asymptotic expansion is controlled by the poles with negative real parts of all the twisted local zeta functions associated to the meromorphic functions f g c for certain special values c the second expansion is controlled by the poles with positive real parts of all the twisted local zeta functions associated to f g

poles of the topological zeta function associated to an ideal in dimension two to an ideal in mathbb c x y one can associate a topological zeta function this is an extension of the topological zeta function associated to one polynomial but in this case we use a principalization of the ideal instead of an embedded resolution of the curve in this paper we will study two questions about the poles of this zeta function first we will give a criterion to determine whether or not a candidate pole is a pole it turns out that we can know this immediately by looking at the intersection diagram of the principalization together with the numerical data of the exceptional curves afterwards we will completely describe the set of rational numbers that can occur as poles of a topological zeta function associated to an ideal in dimension two the same results are valid for related zeta functions as for instance the motivic zeta function

relations between numerical data of an embedded resolution introduction let k be an algebraically closed field of characteristic zero and f x y e k x y let x h be an embedded resolution of f 0 in the affine plane a2 constructed by successive blowing ups and denote by ei i e i the irreducible components of h f 0 we associate to each ei i e i a pair of numerical data n vi where ni and vi 1 are the multiplicities of ei in the divisor of respectively f o h and h dx a dy on x fix one exceptional curve e with numerical data n v and say e intersects k times another irreducible component denote these components by e1 ek then we have

linear programming system identification we define a version of the inverse linear programming problem that we call linear programming system identification this version of the problem seeks to identify both the objective function coefficient vector and the constraint matrix of a linear programming problem that best fits a set of observed vector pairs one vector is that of actual decisions that we call outputs these are regarded as approximations of optimal decision vectors the other vector consists of the inputs or resources actually used to produce the corresponding outputs we propose an algorithm for approximating the maximum likelihood solution the major limitation of the method is the computation of exact volumes of convex polytopes a numerical illustration is given for simulated data

wiener hammerstein modeling of nonlinear effects in bilinear systems the modeling of nonlinear effects in bilinear systems using wiener hammerstein models is considered such models are chosen based on the block oriented structure of bilinear systems and the shapes of their volterra kernels theoretical analysis is given for first order bilinear systems and simulation results are presented for first and second order systems while the nonlinearity is only approximately wiener hammerstein the models are able to capture a significant part of the nonlinear dynamics in order to reduce the complexity of optimizing the model parameters the linear subsystems are estimated using the technique of linear interpolation in the frequency domain