semi smooth newton methods for state constrained optimal control problems abstract a regularized optimality system for state constrained optimal control problems is introduced and semi smooth newton methods for its solution are analyzed convergence of the regularized problems is proved numerical tests confirm the theoretical results and demonstrate the efficiency of the proposed methodology

counting lattice points of rational polyhedra abstract the generating function f p p zn x for a rational polytope p carries all essential information of p in this paper we show that for any positive integer n the generating function f p n of np nx x p can be written as f p n a p n x n where a is the set of all vertices of p and each p n is a certain periodic function of n the ehrhart reciprocity law follows automatically from the above formula we also present a formula for the coefficients of ehrhart polynomials in terms of elementary symmetric functions

positive solutions of a class of second order semilinear elliptic equations in the plane on considere l equation elliptique a 2 dimensions u f x u u x r 2 ou x x 1 x 2 et f x u p est une fonction continue non negative sur r 2 r xr 2 on considere l existence de solutions positives definies dans tout r 2

lattice points in simple polytopes in terms of fp h q x dx where the polytope p h is obtained from p by independent parallel motions of all facets this extends to simple lattice polytopes the eulermaclaurin summation formula of khovanskii and pukhlikov 8 valid for lattice polytopes such that the primitive vectors on edges through each vertex of p form a basis of the lattice as a corollary we recover results of pommersheim 9 and kantor khovanskii 6 on the coefficients of the ehrhart polynomial of p our proof is elementary in a subsequent article we will show how to adapt it to compute the equivariant todd class of any complete toric variety with quotient singularities the euler maclaurin summation formula for simple lattice polytopes has been obtained independently by ginzburg guillemin karshon 4 they used the dictionary between convex polytopes and projective toric varieties with an ample divisor class in combination with the riemann roch kawasaki formula 1 7 for complex manifolds with quotient singularities a counting formula for lattice points in lattice simplices has been announced by cappell and shaneson 2 as a consequence of their computation of the todd class of toric varieties with quotient singularities

existence theory and the maximum principle for relaxed infinite dimensional optimal control problems existence theorems are considered for relaxed optimal control problems described by semilinear systems in banach spaces relaxed controls are used whose values are finitely additive probability measures this class of relaxed controls does not require special assumptions such as compactness on the control set under suitable conditions relaxed trajectories coincide with those obtained from differential inclusions existence theorems for relaxed controls are obtained that apply to distributed parameter systems described by semilinear parabolic and wave equations as well as a version of pontryagin s maximum principle for relaxed optimal control problems

values of zeta functions at negative integers dedekind sums and toric geometry in the present paper we study relations among special values of zeta functions of real quadratic fields properties of generalized dedekind sums and todd classes of toric varieties the main theme of the paper is the use of toric geometry to explain in a conceptual way properties of the values of zeta functions and dedekind sums as well as to provide explicit computations both toric varieties and zeta functions associate numerical invariants to cones in lattices with different motivations and applications though we will focus on the case of two dimensional cones in the present paper we introduce notation and definitions that are valid for cones of arbitrary dimension the reason for this added generality is clarity as well as preparation for the results of a subsequent publication

generalized solutions of constrained optimization problems a constrained optimization problem on a uniform space x is considered under certain assumptions the problem may be extended onto the completion of x with respect to the precompact modification of the original uniformity on x which yields the generalized problem it is shown that the solution of the perturbed classical problem or of the penalized classical problem converges to the solution of the generalized problem which makes the generalized problem more convenient than the classical one finally the general framework is demonstrated in special cases in normed linear spaces

residue formulae vector partition functions and lattice points in rational polytopes we obtain residue formulae for certain functions of several vari ables as an application we obtain closed formulae for vector partition func tions and for their continuous analogs they imply an euler maclaurin sum mation formula for vector partition functions and for rational convex poly topes as well we express the sum of values of a polynomial function at all lattice points of a rational convex polytope in terms of the variation of the integral of the function over the deformed polytope institut fourier b p 74 38402 saint martin d heres cedex france e mail address mbrion fourier ujf grenoble fr ecole normale superieure 45 rue d ulm 75005 paris cedex 05 france e mail address vergne dmi ens fr

random polynomials with prescribed newton polytope the newton polytope pf of a polynomial f is well known to have a strong impact on its zeros as in the kouchnirenko bernstein theorem on the number of simultaneous zeros of m polynomials with given newton polytopes in this article we show that pf also has a strong impact on the distribution of zeros of one or several polynomials we equip the space of holomorphic polynomials of degreen in m complex variables with its usual su m 1 invariant gaussian measure and then consider the conditional measures np induced on the subspace of polynomials whose newton polytope pfnp when p the unit simplex then it is obvious and well known that the expected distribution of zeros zf1 fk regarded as a current of k polynomials f1 fk of degree n is uniform relative to the fubini study form our main results concern the conditional expectation e np zf1 fk of zeros of k polynomials with newton polytope npnp where p deg p these results are asymptotic as the scaling factor n 1 we show that e np zf1 fk is asymptotically uniform on the inverse image ap of the open scaled polytope 1 p via the moment map cp m for projective space however the zeros have an exotic distribution outside of ap and when k m the case of the kouchnirenko bernstein theorem the percentage of zeros outside ap approaches 0 as n 1

optimal processes governed by integral equation equations with unilateral constraint abstract an optimal process with unilateral constraint is considered using relaxed controls existence of optimal control is automatic further problems with linear and certain quadratic cost functions with linear integral and state constraints are such that the controls are actually ordinary i e bounded measurable functions in addition if the control set is a convex polyhedron the optimal controls take their values in the vertices of the polyhedron control processes governed by integral equations have been considered by v r vinokurov siam 7 1969 324 355 however there existence is not shown for the general problem and optimality conditions are not available for the problem when state constraints are present furthermore the above work has some errors as pointed out by l w neustadt and j warga siam 8 1970 572 finally we point out that the basic idea in our approach here is the same as in a forthcoming paper j optim theory appl in press

algebraic birkhoff factorization and the euler maclaurin formula on cones we equip the space of lattice cones with a coproduct which makes it a cograded coaugmented connnected coalgebra the exponential generating sum and exponential generating integral on lattice cones can be viewed as linear maps on this space with values in the space of meromorphic germs with linear poles at zero we investigate the subdivision properties reminiscent of the inclusion exclusion principle for the cardinal on finite sets of such linear maps and show that these properties are compatible with the convolution quotient of maps on the coalgebra implementing the algebraic birkhoff factorization procedure on the linear maps under consideration we factorize the exponential generating sum as a convolution quotient of two maps with each of the maps in the factorization satisfying a subdivision property a direct computation shows that the polar decomposition of the exponential generating sum on a smooth lattice cone yields an euler maclaurin formula the compatibility with subdivisions of the convolution quotient arising in the algebraic birkhoff factorization then yields the euler maclaurin formula for any lattice cone this provides a simple formula for the interpolating factor by means of a projection formula

existence of entire explosive positive solutions of quasi linear elliptic equations in this paper our main purpose is to establish the existence of entire explosive positive radial solutions for quasi linear elliptic equations the main results of the paper extend previously known results both for the super linear and sub linear cases

special values of partial zeta functions of real quadratic fields at nonpositive integers and the euler maclaurin formula we compute the special values at nonpositive integers of the partial zeta function of an ideal of a real quadratic field applying an asymptotic version of euler maclaurin formula to the lattice cone associated to the ideal considered the euler maclaurin formula in volved is obtained by applying the todd series of differential opera tors to an integral of a small perturbation of the cone the additive property of todd series w r t the cone decomposition enables us to express the partial zeta values in terms of the continued fraction of the reduced element of the ideal the expression obtained uses the positive continued fraction which yields a virtual decomposition of the cone we apply the expression to some indexed families of real quadratic fields satisfying certain condition on the shape of the continued frac tions the families considered include those appeared in 20 and 21 as well as the richaud degert types we show that the partial zeta values at a given nonpositive integerk in the family indexed by n is a polynomial of n finally we compute explicitly the polynomials producing the par tial zeta values at s k for small k of some chosen families and compare these with some previously known results

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

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

asymptotic euler maclaurin formula over lattice polytopes abstract formulas for the riemann sums over lattice polytopes determined by the lattice points in the polytopes are often called euler maclaurin formulas an asymptotic euler maclaurin formula by which we mean an asymptotic expansion formula for riemann sums over lattice polytopes was first obtained by guillemin and sternberg 2007 11 then the problem is to find a concrete formula for each term of the expansion in this paper an asymptotic euler maclaurin formula of the riemann sums over general lattice polytopes is given the formula given here is an asymptotic form of the so called local euler maclaurin formula of berline and vergne 2007 3 for delzant polytopes our proof given here is independent of the local euler maclaurin formula furthermore a concrete description of differential operators which appear in each term of the asymptotic expansion for delzant lattice polytopes is given by using this description when the polytopes are delzant lattice a concrete formula for each term of the expansion in two dimension and a formula for the third term of the expansion in arbitrary dimension are given

some new results on the existence of bounded positive entire solutions for quasilinear elliptic equations in this paper our main purpose is to establish the existence of positive bounded entire solutions of second order quasilinear elliptic equations under new conditions the main results of the present paper are new and extend the previously known results

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

an extension of pontryagin s principle for state constrained optimal control of semilinear elliptic equations and variational inequalities this paper deals with state constrained optimal control problems governed by semilinear elliptic equations or variational inequalities by using ekeland s principle a minimum principle of pontryagin s type under some stability conditions of the optimal cost with respect to the state constraints is derived

state constrained relaxed problems for semilinear elliptic equations in this paper we study optimal control problems governed by semilinear elliptic equations in the presence of pointwise state constraints since no convexity condition is assumed on data of the problem we define a relaxed control problem prove the existence of relaxed solutions and give some relaxation results by adapting the penalty method of berkovitz we prove a pontryagin s minimum principle for relaxed solutions in nonqualified form and in qualified form under a stability condition

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

the ehrhart polynomial of a lattice polytope the problem of counting the number of lattice points inside a convex lattice polytope in r a polytope whose vertices have integer coordinates has been studied from a variety of perspectives here we use the poisson summation formula and related techniques in fourier analysis to obtain a formula for the number of lattice points inside all integral dilates of a lattice polytope we show that an associated generating function has an explicit representation in terms of sums of cotangents which are higher dimensional generalizations of dedekind sums let en denote the n dimensional integer lattice in r and let p be an n dimensional lattice polytope in r which is a compact simplicial complex of pure dimension n whose vertices lie on the lattice consider the function of an integer valued variable t that describes the number of lattice points that lie inside the dilated polytope tp

characteristic classes of singular toric varieties in this paper we compute the motivic chern classes and homology hirzebruch characteristic classes of possibly singular toric varieties which in the context of complete toric varieties fit nicely with a generalized hirzebruch riemann roch theorem as important special cases we obtain new or recover well known formulae for the baum fulton macpherson todd or macpherson s chern classes of toric varieties as well as for the thom milnor l classes of simplicial projective toric varieties we present two different perspectives for the computation of these characteristic classes of toric varieties first we take advantage of the torus orbit decomposition and the motivic properties of the motivic chern and respectively homology hirzebruch classes to express the latter in terms of dualizing sheaves and respectively the dual todd classes of closures of orbits this method even applies to torus invariant subspaces of a given toric variety the obtained formula is then applied to weighted lattice point counting in lattice polytopes and their subcomplexes yielding generalized pick type formulae second in the case of simplicial toric varieties we compute our characteristic classes by using the lefschetz riemann roch theorem of edidin graham in the context of the geometric quotient description of such varieties in this setting we define mock hirzebruch classes of simplicial toric varieties which specialize to the mock chern mock todd and mock l classes of such varieties and investigate the difference between the actual homology hirzebruch class and the mock hirzebruch class we show that this difference is localized on the singular locus and we obtain a formula for it in which the contribution of each singular cone is identified explicitly finally the two methods of computing characteristic classes are combined for proving several characteristic class formulae originally obtained by cappell and shaneson in the early 1990s 2015 wiley periodicals inc

wall crossings for double hurwitz numbers abstract double hurwitz numbers count covers of p 1 by genus g curves with assigned ramification profiles over 0 and and simple ramification over a fixed branch divisor goulden jackson and vakil have shown double hurwitz numbers are piecewise polynomial in the orders of ramification goulden et al 2005 10 and shadrin shapiro and vainshtein have determined the chamber structure and wall crossing formulas for g 0 shadrin et al 2008 15 this paper gives a unified approach to these results and strengthens them in several ways the most important being the extension of the results of shadrin et al 2008 15 to arbitrary genus the main tool is the authors previous work cavalieri et al 2010 6 expressing double hurwitz number as a sum over certain labeled graphs we identify the labels of the graphs with lattice points in the chambers of certain hyperplane arrangements which give rise to piecewise polynomial functions our understanding of the wall crossing for these functions builds on the work of varchenko 1987 17 and could have broader applications

on the optimal control of strongly nonlinear evolution equations abstract in this paper we study optimal control problems governed by nonlinear evolution equations with nonmonotone nonlinearities in the state first we prove an existence theorem then using the penalty method we obtain necessary conditions for optimality from them we derive a bang bang characterization of optimal controls finally an example of a nonlinear parabolic optimal control system is worked out in detail

boundary control of semilinear elliptic equations with pointwise state constraints this paper is concerned with state constrained optimal control problems of semilinear elliptic equations the control being on the boundary optimality conditions are derived and regularity of the optimal solution is investigated