using regression match graphs to control search in planning abstract classical planning is the problem of finding a sequence of actions to achieve a goal given an exact characterization of a domain an algorithm to solve this problem is presented which searches a space of plan prefixes trying to extend one of them to a complete sequence of actions it is guided by a heuristic estimator based on regression match graphs which attempt to characterize the entire subgoal structure of the remaining part of the problem these graphs simplify the structure by neglecting goal interactions and by assuming that variables in goal conjunctions should be bound in such a way as to make as many conjuncts as possible true without further work in some domains these approximations work very well and experiments show that many classical planning problems can be solved with very little search

high probability analysis of the condition number of sparse polynomial systems let f f1 fn be a random polynomial system with fixed n tuple of supports our main result is an upper bound on the probability that the condition number of f in a region u is larger than 1 e the bound depends on an integral of a differential form on a toric manifold and admits a simple explicit upper bound when the newton polytopes and underlying variances are all identical we also consider polynomials with real coefficients and give bounds for the expected number of real roots and restricted condition number using a kahler geometric framework throughout we also express the expected number of roots of f inside a region u as the integral over u of a certain mixed volume form thus recovering the classical mixed volume when u c n

multiobjective heuristic state space planning modern domain independent heuristic planners evaluate their plans on the single basis of their length however in real world problems there are other criteria that also play an important role e g resource consumption profit safety etc this paper enhances the grt planner an efficient domain independent heuristic state space planner with the ability to consider multiple criteria the grt heuristic is based on the estimation of the distances between each fact of a problem and the goals the new planner called mo grt uses a weighted a strategy and a multiobjective heuristic function computed over a weighted hierarchy of user defined criteria its computation is based on sets of non dominated cost vectors assigned to the problem facts which estimate the total cost of achieving the facts from the goals using alternative paths experiments show that a change in the criteria weights or scales affects both the quality of the resulting plan and the planning time the proposed approach can easily be adapted to other modern heuristic state space planners

regenerative cascade homotopies for solving polynomial systems abstract a key step in the numerical computation of the irreducible decomposition of a polynomial system is the computation of a witness superset of the solution set in many problems involving a solution set of a polynomial system the witness superset contains all the needed information sommese and wampler gave the first numerical method to compute witness supersets based on dimension by dimension slicing of the solution set by generic linear spaces followed later by the cascade homotopy of sommese and verschelde recently the authors of this article introduced a new method regeneration to compute solution sets of polynomial systems tests showed that combining regeneration with the dimension by dimension algorithm was significantly faster than naively combining it with the cascade homotopy however in this article we combine an appropriate randomization of the polynomial system with the regeneration technique to construct a new cascade of homotopies for computing witness supersets computational tests give strong evidence that regenerative cascade is superior in practice to previous methods

lattice polytopes cut out by root systems and the koszul property we show that lattice polytopes cut out by root systems of classical type are normal and koszul generalizing a well known result of bruns gubeladze and trung in type a we prove similar results for cayley sums of collections of polytopes whose minkowski sums are cut out by root systems the proofs are based on a combinatorial characterization of diagonally split toric varieties

impulsive boundary value problems with nonlinear boundary conditions abstract in this paper the upper and lower solution method and schauder s fixed point theorem are employed in the study of boundary value problems for a class of second order impulsive ordinary differential equations with nonlinear boundary conditions we prove the existence of solutions to the problem under the assumption that there exist lower and upper solutions associated with the problem

an approach to efficient planning with numerical fluents and multi criteria plan quality dealing with numerical information is practically important in many real world planning domains where the executability of an action can depend on certain numerical conditions and the action effects can consume or renew some critical continuous resources which in pddl can be represented by numerical fluents when a planning problem involves numerical fluents the quality of the solutions can be expressed by an objective function that can take different plan quality criteria into account we propose an incremental approach to automated planning with numerical fluents and multi criteria objective functions for pddl numerical planning problems the techniques in this paper significantly extend the framework of planning with action graphs and local search implemented in the lpg planner we define the numerical action graph na graph representation for numerical plans and we propose some new local search techniques using this representation including a heuristic search neighborhood for na graphs a heuristic evaluation function based on relaxed numerical plans and an incremental method for plan quality optimization based on particular search restarts moreover we analyze our approach through an extensive experimental study aimed at evaluating the importance of some specific techniques for the performance of the approach and at analyzing its effectiveness in terms of fast computation of a valid plan and quality of the best plan that can be generated within a given cpu time limit overall the results show that our planner performs quite well compared to other state of the art planners handling numerical fluents

minimizing multi homogeneous bezout numbers by a local search method consider the multi homogeneous homotopy continuation method for solving a system of polynomial equations for any partition of variables the multi homogeneous bezout number bounds the number of isolated solution curves one has to follow in the method this paper presents a local search method for finding a partition of variables with minimal multi homogeneous bezout number as with any other local search method it may give a local minimum rather than the minimum over all possible homogenizations numerical examples show the efficiency of this local search method

a numerical algorithm to find all feedback nash equilibria in scalar affine quadratic differential games this note deals with solving scalar coupled algebraic riccati equations these equations arise in finding linear feedback nash equilibria of the scalar n player affine quadratic differential game a numerical procedure is provided to compute all the stabilizing solutions the main idea is to reformulate the riccati equations into an extended eigenvalue eigenvector problem for a specific parametrized matrix u in i r 2 n times 2 n since the size of u increases exponentially on n the algorithm only applies for games where the number of players is not too large

hyperdiscriminant polytopes chow polytopes and mabuchi energy asymptotics letx n p n be a smooth linearly normal algebraic variety it is shown that the mabuchi energy of x fsjx restricted to the bergman metrics is completely determined by the x hyperdiscriminant of format n 1 and the chow form of x as a corollary it is shown that the mabuchi energy is bounded from below for all degenerations in g if and only if the hyperdiscriminant polytope dominates the chow polytope for all maximal algebraic tori h of g

a hybrid polynomial system solving method for mixed trigonometric polynomial systems mixed trigonometric polynomial systems arise in many fields of science and engineering commonly this class of systems is transformed into polynomial systems by variable substituting and adding some quadratic equations and then solved by some polynomial system solving method in this paper by exploiting the special structure of the additional quadratic equations an efficient hybrid method for solving polynomial systems coming from mixed trigonometric polynomial systems is presented it combines the homotopy method in which the homotopy is a combination of coefficient parameter homotopy and the random product homotopy with decomposition variable substitution and reduction techniques numerical tests are given to show its effectiveness and it is applied to solve a practical problem

searching game trees under a partial order abstract the problem of partial order game tree search arises from game playing situations where multiple conflicting and non commensurate criteria dictate the merit of a position of the game in partial order game trees the outcomes evaluated at the tip nodes are vectors where each dimension of the vector represents a distinct criterion of merit this leads to an interesting variant of the game tree searching problem where corresponding to every game playing strategy of a player several outcomes are possible depending on the individual priorities of the opponent in this paper we identify the necessary and sufficient conditions for a set of outcomes to be inferior to another set of outcomes for every strategy using an algebra called dominance algebra on sets of outcomes we describe a bottom up approach to find the non inferior sets of outcomes at the root node we also identify shallow and deep pruning conditions for partial order game trees and present a partial order search algorithm on lines similar to the pruning algorithm for conventional game trees

a new existence theory for single and multiple positive periodic solutions to volterra integro differential equations with impulse effects this paper presents a new existence theory for single and multiple positive periodic solutions to a kind of nonautonomous volterra integro differential equations with impulse effects existence is established by using krasnoselskii fixed point theorem in cones this paper extends some previous results and reports some new results about impulsive differential equation

a homotopy for solving general polynomial systems that respects m homogeneous structures methods for finding all solutions to a polynomial system f z 0 of n equations in n unknowns using a generic homotopy have been presented by several researchers over the last ten years in this paper we describe a new method for defining a homotophy to find all solutions to a polynomial system a major feature of this new approach is that the homotopy need not to be generic or random the practical consequences are that the homotopy may be chosen to more closely resemble the system being solved with a consequent potential for reducing the arc length of paths and exploiting special structures in the original system in particular if the system is naturally m homogeneous then the number of homotopy paths that need to be tracked is reduced since many systems that arise in applications have such a structure this represents a significant advance over previous homotopies we also provide a proof that the homotopy parameter is strictly increasing on homotopy paths as a function of arc length a new result which has implications for the choice of numerical path tracking algorithms

periodic boundary value problems for second order impulsive differential systems on etudie le probleme aux valeurs limites periodiques pour un systeme d equations differentielles d ordre 2 faiblement couplees avec impulsion

quantitative equidistribution for the solutions of systems of sparse polynomial equations for a system of laurent polynomials f 1 ldots f n in bbb c x 1 pm1 ldots x n pm1 whose coefficients are not too big with respect to its directional resultants we show that the solutions in the algebraic torus bbb c times n of the system of equations f 1 cdots f n 0 are approximately equidistributed near the unit polycircle this generalizes to the multivariate case a classical result due to erd os and tur an on the distribution of the arguments of the roots of a univariate polynomial we apply this result to bound the number of real roots of a system of laurent polynomials and to study the asymptotic distribution of the roots of systems of laurent polynomials over bbb z and of random systems of laurent polynomials over bbb c

nonlinear second order equations with functional implicit impulses and nonlinear functional boundary conditions abstract in this paper we present existence results for solutions of nonlinear second order boundary value problems with impulses our impulses are applied at p points in the interval and given implicitly by nonlinear functions of the solution moreover we allow functional dependence on the solution our existence results follow from the existence of a pair of well ordered lower and upper solutions we generalize earlier results of cabada and tomec ek allowing more general compatible boundary conditions impulses and laplacian equations

heuristic methods for computing the minimal multi homogeneous bezout number the multi homogeneous bezout number of a polynomial system is the number of paths that one has to follow in computing all its isolated solutions with continuation method each partition of variables corresponds to a multi homogeneous bezout number it is a challenging problem to find a partition with minimal multi homogeneous bezout number two heuristic numerical methods for computing the minimal multi homogeneous bezout number are presented in this paper some analysis of computational complexity are given numerical examples show the efficiency of these two methods

a subdivision based algorithm for the sparse resultant multivariate resultants generalize the sylvester resultant of two polynomials and characterize the solvability of a polynomial system they also reduce the computation of all common roots to a problem in linear algebra we propose a determinantal formula for the sparse resultant of an arbitrary system of n 1 polynomials in n variables this resultant generalizes the classical one and has significantly lower degree for polynomials that are sparse in the sense that their mixed volume is lower than their bezout number our algorithm uses a mixed polyhedral subdivision of the minkowski sum of the newton polytopes in order to construct a newton matrix its determinant is a nonzero multiple of the sparse resultant and the latter equals the gcd of at most n 1 such determinants this construction implies a restricted version of an effective sparse nullstellensatz for an arbitrary specialization of the coefficients there are two methods that use one extra variable and yield the sparse resultant this is the first algorithm to handle the general case with complexity polynomial in the resultant degree and simply exponential in n we conjecture its extension to producing an exact rational expression for the sparse resultant

mixed multiplicities of ideals versus mixed volumes of polytopes the main results of this paper interpret mixed volumes of lattice polytopes as mixed multiplicities of ideals and mixed multiplicities of ideals as samuel s multiplicities in particular we can give a purely algebraic proof of bernstein s theorem which asserts that the number of common zeros of a system of laurent polynomial equations in the torus is bounded above by the mixed volume of their newton polytopes

macaulay style formulas for sparse resultants we present formulas for computing the resultant of sparse polynomials as a quotient of two determinants the denominator being a minor of the numerator these formulas extend the original formulation given by macaulay for homogeneous polynomials

existence result for impulsive third order periodic boundary value problems this paper is devoted to the study of periodic boundary value problems for nonlinear third order differential equations subjected to impulsive effects we provide sufficient conditions on the nonlinearity and the impulse functions that guarantee the existence of at least one periodic solution our approach is based on a priori estimates the method of upper and lower solutions combined with an iterative technique which need not be monotone

existence results for impulsive second order periodic problems abstract this paper provides existence results for the nonlinear impulsive periodic boundary value problem u f t u u u t i j i u t i u t i m i u t i i 1 2 m u 0 u t u 0 u t where f car 0 t r 2 and j i m i c r the basic assumption is the existence of lower upper functions 1 2 associated with the problem here we generalize and extend the existence results of our previous papers

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

homotopies for solving polynomial systems within a bounded domain abstract the problem considered in this paper is the computation of all solutions of a given polynomial system in a bounded domain proving rouche s theorem by homotopy continuation concepts yields a new class of homotopy methods the so called regional homotopy methods these methods rely on isolating a part of the system to be solved which dominates the rest of the system on the border of the domain as the dominant part has a sparser structure it is easier to solve it will be used as start system in the regional homotopy the paper further describes practical homotopy construction methods by presenting estimators to obtain bounds for polynomials over a bounded domain applications illustrate the usefulness of the approach

balancing the lifting values to improve the numerical stability of polyhedral homotopy continuation methods polyhedral homotopy continuation methods exploit the sparsity of polynomial systems so that the number of solution curves to reach all isolated solutions is optimal for generic systems the numerical stability of tracing solution curves of polyhedral homotopies is mainly determined by the height of the powers of the continuation parameter to reduce this height we propose a procedure that operates as an intermediate stage between the mixed volume computation and the tracing of solution curves this procedure computes new lifting values of the support of a polynomial system these values preserve the structure of the mixed cell configuration obtained from the mixed volume computation and produce better balanced powers of the continuation parameter in the polyhedral homotopies

the height of the mixed sparse resultant we present an upper bound for the height of the mixed sparse resultant defined as the logarithm of the maximum modulus of its coefficients we obtain a similar estimate for its mahler measure

homotopies exploiting newton polytopes for solving sparse polynomial systems this paper is concerned with the problem of finding all isolated solutions of a polynomial system the bkk bound defined as the mixed volume of the newton polytopes of the polynomials in the system is a sharp upper bound for the number of isolated solutions in mathbb c 0 n mathbb c 0 mathbb c backslash 0 of a polynomial system with a sparse monomial structure first an algorithm is described for computing the bkk bound following the lines of bernshtei n s proof the algorithmic construction of the cheater s homotopy or the coefficient homotopy is obtained the mixed homotopy methods can be combined with the random product start systems based on a generalized bezout number applications illustrate the effectiveness of the new approach

on a result of gelfand kapranov and zelevinsky abstract in this paper we give new elementary proofs of basic results due to gelfand kapranov and zelevinsky which express discriminants and results in terms of determinants of direct images of stably twisted cayley koszul complexes of sheaves

existence results for boundary value problems of second order impulsive differential equations abstract with the help of lower and upper solutions the extremal solutions of a second order impulsive differential equation are obtained as limits of monotone iterates several existence results for the second order impulsive differential equation subject to nonlinear boundary conditions which include the usual periodic and linear separated boundary conditions are also established

extremal solutions for nonlinear functional ϕ laplacian impulsive equations abstract in this paper we deal with some nonlinear laplacian second order boundary value problems with impulses subject to a general type of boundary value conditions which cover dirichlet and periodic boundary data as particular cases our approach is that of upper and lower solutions together with growth restrictions of nagumo s type the impulsive functions depend implicitly on the different considered variables and the boundary value conditions are nonlinear in both cases functional dependence on the solution is allowed

imperialist competitive algorithm for solving systems of nonlinear equations solving systems of nonlinear equations is a relatively complicated problem in which arise a diverse range of sciences there are a number of different approaches that have been proposed in this paper we employ the imperialist competitive algorithm ica for solving systems of nonlinear equations some well known problems are presented to demonstrate the efficiency of this new robust optimization method in comparison to other known methods

study of singular boundary value problems for second order impulsive differential equations this paper studies the existence of extremal solutions for a class of singular boundary value problems of second order impulsive differential equations by using the method of upper and lower solutions and the monotone iterative technique criteria of the existence of extremal solutions are established

chaotic quantum behaved particle swarm optimization algorithm for solving nonlinear system of equations abstract this study proposes a novel chaotic quantum behaved particle swarm optimization algorithm for solving nonlinear system of equations different chaotic maps are introduced to enhance the effectiveness and robustness of the algorithm several benchmark studies are carried out logistic map gives the best results and is utilized in solving nonlinear equation sets nine well known problems are solved with our algorithm and results are compared with quantum behaved particle swarm optimization intelligent tuned harmony search gravitational search algorithm and literature studies comparison results reveal that the proposed algorithm can cope with the highly non linear problems and outperforms many algorithms which exist in the literature

fixed points of belief propagation an analysis via polynomial homotopy continuation belief propagation bp is an iterative method to perform approximate inference on arbitrary graphical models whether bp converges and if the solution is a unique fixed point depends on both the structure and the parametrization of the model to understand this dependence it is interesting to find all fixed points in this work we formulate a set of polynomial equations the solutions of which correspond to bp fixed points to solve such a nonlinear system we present the numerical polynomial homotopy continuation nphc method experiments on binary ising models and on error correcting codes show how our method is capable of obtaining all bp fixed points on ising models with fixed parameters we show how the structure influences both the number of fixed points and the convergence properties we further asses the accuracy of the marginals and weighted combinations thereof weighting marginals with their respective partition function increases the accuracy in all experiments contrary to the conjecture that uniqueness of bp fixed points implies convergence we find graphs for which bp fails to converge even though a unique fixed point exists moreover we show that this fixed point gives a good approximation and the nphc method is able to obtain this fixed point

numerical solution of bivariate and polyanalytic polynomial systems finding the real solutions of a bivariate polynomial system is a central problem in robotics computer modeling and graphics computational geometry and numerical optimization we propose an efficient and numerically robust algorithm for solving bivariate and polyanalytic polynomial systems using a single generalized eigenvalue decomposition in contrast to existing eigen based solvers the proposed algorithm does not depend on grobner bases or normal sets nor does it require computing eigenvectors or solving additional eigenproblems to recover the solution the method transforms bivariate systems into polyanalytic systems and then uses resultants in a novel way to project the variables onto the real plane associated with the two variables solutions are returned counting multiplicity and their accuracy is maximized by means of numerical balancing and newton raphson refinement numerical experiments show that the proposed algorithm consistently recovers a higher percentage of solutions and is at the sa

nonlinear second order equations with functional implicit impulses and nonlinear functional boundary conditions in this paper we deal with nonlinear second order boundary value problems with impulses the impulsive functions depend implicitly on the different considered variables and the boundary value conditions are nonlinear in both cases functional dependence on the solution is allowed the existence results follow from the existence of a pair of well ordered lower and upper solutions

finiteness of relative equilibria of the four body problem we show that the number of relative equilibria of the newtonian four body problem is finite up to symmetry in fact we show that this number is always between 32 and 8472 the proof is based on symbolic and exact integer computations which are carried out by computer

on the toric ideal of a matroid abstract describing minimal generating set of a toric ideal is a well studied and difficult problem in 1980 white conjectured that the toric ideal associated to a matroid is equal to the ideal generated by quadratic binomials corresponding to symmetric exchanges we prove white s conjecture up to saturation that is that the saturations of both ideals are equal in the language of algebraic geometry this means that both ideals define the same projective scheme additionally we prove the full conjecture for strongly base orderable matroids

nonlinear boundary value problem of first order impulsive functional differential equations this paper discusses nonlinear boundary value problem for first order impulsive functional differential equations we establish several existence results by using the lower and upper solutions and monotone iterative techniques two examples are discussed to illustrate the efficiency of the obtained results

regeneration homotopies for solving systems of polynomials we present a new technique based on polynomial continuation for solving systems of n polynomials in n complex variables the method allows equations to be introduced one by one or in groups obtaining at each stage a representation of the solution set that can be extended to the next stage until finally obtaining the solution set for the entire system at any stage where positive dimensional solution components must be found they are sliced down to isolated points by the introduction of hyperplanes by moving these hyperplanes one may build up the solution set to an intermediate system in which a union of hyperplanes regenerates the intersection of the component with the variety of the polynomial or system of polynomials brought in at the next stage the theory underlying the approach guarantees that homotopy paths lead to all isolated solutions and this capability can be used to generate witness supersets for solution components at any dimension the first step in computing an irreducible decomposition of the solution set of a system of polynomial equations the method is illustrated on several challenging problems where it proves advantageous over both the polyhedral homotopy method and the diagonal equation by equation method formerly the two leading approaches to solving sparse polynomial systems by numerical continuation

polynomial equation solving by lifting procedures for ramified fibers let be given a parametric polynomial equation system which represents a generically unramified family of zero dimensional algebraic varieties we exhibit an efficient algorithm which computes a complete description of the solution set of an arbitrary parameter instance from a complete description of the infinitesimal structure of a particular ramified parameter instance of our family this generalizes in the case of space curves previous methods of heintz et al and schost which require the given parameter instance to be unramified we illustrate our method solving particular polynomial equation systems by deformation techniques

grobner bases and invariant theory abstract in this paper we study the relationship between buchberger s grobner basis method and the straightening algorithm in the bracket algebra these methods will be introduced in a self contained overview on the relevant areas from computational algebraic geometry and invariant theory we prove that a certain class of van der waerden syzygies forms a grobner basis for the syzygy ideal in the bracket ring we also give a description of a reduced grobner basis in terms of standard and non standard tableaux some possible applications of straightening for symbolic computations in projective geometry are indicated

a parametric representation of totally mixed nash equilibria we present an algorithm to compute a parametric description of the totally mixed nash equilibria of a generic game in normal form with a fixed structure using this representation we also show an algorithm to compute polynomial inequality conditions under which a game has the maximum possible number of this kind of equilibria then we present symbolic procedures to describe the set of isolated totally mixed nash equilibria of an arbitrary game and to compute under certain general assumptions the exact number of these equilibria the complexity of all these algorithms is polynomial in the number of players the number of each player s strategies and the generic number of totally mixed nash equilibria of a game with the considered structure

perturbed homotopies for finding all isolated solutions of polynomial systems abstract given a polynomial system f c n c n the methods of numerical algebraic geometry produce numerical approximations of the isolated solutions of f z 0 as well as points on any positive dimensional components of the solution set v f some of these methods are guaranteed to find all isolated solutions nonsingular and singular alike while others may not find singular solutions one of the most recent advances in this field is regeneration an equation by equation solver that is often more efficient than other methods however the basic form of regeneration will not necessarily find all isolated singular solutions of a polynomial system without the additional cost of using deflation the aim of this article is two fold first more generally we consider the use of perturbed homotopies for solving polynomial systems in particular we propose first solving a perturbed version of the polynomial system followed by a parameter homotopy to remove the perturbation such perturbed homotopies are sometimes more efficient than regular homotopies though they can also be less efficient second a useful consequence is that the application of this perturbation to regeneration will yield all isolated solutions including all singular isolated solutions this version of regeneration perturbed regeneration can decrease the efficiency of regeneration but increases its applicability

the analysis of elliptic families i metrics and connections on determinant bundles in this paper we construct the quillen metric on the determinant bundle associated with a family of elliptic first order differential operators we also introduce a unitary connection on and calculate its curvature our results will be applied to the case of dirac operators in a forthcoming paper

a poisson formula for the sparse resultant we present a poisson formula for sparse resultants and a formula for the product of the roots of a family of laurent polynomials which are valid for arbitrary families of supports to obtain these formulae we show that the sparse resultant associated to a family of supports can be identified with the resultant of a suitable multiprojective toric cycle in the sense of remond this connection allows to study sparse resultants using multiprojective elimination theory and intersection theory of toric varieties

intersection theory in differential algebraic geometry generic intersections and the differential chow form in this paper an intersection theory for generic differential polynomials is presented the intersection of an irreducible differential variety of dimension d and order h with a generic differential hypersurface of order s is shown to be an irreducible variety of dimension d 1 and order h s as a consequence the dimension conjecture for generic differential polynomials is proved based on intersection theory the chow form for an irreducible differential variety is defined and most of the properties of the chow form in the algebraic case are established for its differential counterpart furthermore the generalized differential chow form is defined and its properties are proved as an application of the generalized differential chow form the differential resultant of n 1 generic differential polynomials in n variables is defined and properties similar to that of the macaulay resultant for multivariate polynomials are proved

algebraic c actions and the inverse kinematics of a general 6r manipulator abstract let x be a smooth quadric of dimension 2 m in p c 2 m 1 and let y z x be subvarieties both of dimension m which intersect transversely in this paper we give an algorithm for computing the intersection points of y z based on a homotopy method the homotopy is constructed using a c action on x whose fixed points are isolated which induces bialynicki birula decompositions of x into locally closed invariant subsets as an application we present a new solution to the inverse kinematics problem of a general six revolute serial link manipulator

periodic boundary value problems of second order impulsive differential equations abstract this paper discusses periodic boundary value problems of second order impulsive differential equations by using schaeffer s fixed point theorem and monotone iterative technique some existence results are obtained

a polyhedral method for solving sparse polynomial systems a continuation method is presented for computing all isolated roots of a semimixed sparse system of polynomial equations we introduce mixed subdivisions of newton polytopes and we apply them to give a new proof and algorithm for bernstein s theorem on the expected number of roots this results in a numerical homotopy with the optimal number of paths to be followed in this homotopy there is one starting system for each cell of the mixed subdivision and the roots of these starting systems are obtained by an easy combinatorial construction

multiobjective a search with consistent heuristics the article describes and analyzes namoa an algorithm for multiobjective heuristic graph search problems the algorithm is presented as an extension of a an admissible scalar shortest path algorithm under consistent heuristics a is known to improve its efficiency with more informed heuristics and to be optimal over the class of admissible algorithms in terms of the set of expanded nodes and the number of node expansions equivalent beneficial properties are shown to prevail in the new algorithm

chow form for projective differential variety abstract in this paper a generic intersection theorem in projective differential algebraic geometry is given then the chow form for an irreducible projective differential variety is defined and basic properties of the differential chow form in affine differential case are shown to be valid for its projective differential counterpart finally we apply the differential chow form to a result of linear dependence over projective varieties given by kolchin

unification and extension of intersection algorithms in numerical algebraic geometry the solution set of a system of polynomial equations called an algebraic set can be decomposed into finitely many irreducible components in numerical algebraic geometry irreducible algebraic sets are represented by witness sets whereas general algebraic sets allow a numerical irreducible decomposition comprising a collection of witness sets one for each irreducible component we denote the solution set of any system of polynomials f c n c n as v f c n given a witness set for some algebraic set z c n and a system of polynomials f c n c n the algorithms of this paper compute a numerical irreducible decomposition of the set z v f while extending the types of intersection problems that can be solved via numerical algebraic geometry this approach is also a unification of two existing algorithms the diagonal intersection algorithm and the homotopy membership test the new approach includes as a special case the extension problem where one wishes to intersect an irreducible component a of v g x with v f x y where f introduces new variables y for example this problem arises in computing the singularities of a when the singularity conditions are expressed in terms of new variables associated to the tangent space of a several examples are included to demonstrate the effectiveness of our approach applied in a variety of scenarios

sharp estimates for the arithmetic nullstellensatz we present sharp estimates for the degree and the height of the polynomials in the nullstellensatz over the integer ring zz the result improves previous work of philippon berenstein yger and krick pardo we also present degree and height estimates of intrinsic type which depend mainly on the degree and the height of the input polynomial system as an application we derive an effective arithmetic nullstellensatz for sparse polynomial systems the proof of these results relies heavily on the notion of local height of an affine variety defined over a number field we introduce this notion and study its basic properties

the bkk root count in c n the root count developed by bernshtein kushnirenko and khovanskii only counts the number of isolated zeros of a polynomial system in the algebraic torus c n in this paper we modify this bound slightly so that it counts the number of isolated zeros in c n our bound is apparently significantly sharper than the recent root counts found by rojas and in many cases easier to compute as a consequence of our result the huber sturmfels homotopy for finding all the isolated zeros of a polynomial system in c n can be slightly modified to obtain all the isolated zeros in c n

a convex geometric approach to counting the roots of a polynomial system consider a polynomial system f f1 fn in n variables with complex coefficients a standard way to bound the number of isolated roots of f in cn is to multiply and add the degrees of the fi in a straightfoward computation described explicitly by some variant of bezout s theorem unfortunately even if one knows a priori which monomial terms will appear this upper bound can be far from exact in practice one frequently knows which monomial terms will appear in a polynomial system before one actually decides to count or approximate its roots so a fast accurate root count which takes this additional information into account is vital we propose a much tighter upper bound on the number of isolated roots of f in cn and give an explicit combinatorial geometric criterion for when it is exact as a corollary we obtain a geometric characterization of the average case computational complexity of root counting our root count is a formula derived from the theory of toric varieties and its computation involves finding the mixed volume of n shadowed polytopes in rn our formula generalizes the bkk bound bernshtein 1975 kushnirenko 1976 khovanskii 1978 canny and rojas 1991 in four ways 1 for any r 0 n our formula can be used to count the roots whose first r coordinates are nonzero the bkk bound only handles the r n case as a corollary we obtain that for fixed n all of these root counts for generic f can be done in time polynomial in the numbers of monomial terms of the fi 2 we refine bernshtein s 1975 algebraic criterion for exactness of the bkk bound into a complete classification of the subsets of the coefficients of f whose genericity guarantees exactness in our generalized bound 3 our results are stated in terms of arbitrary algebraically closed fields of any characteristic the bkk bound was originally stated only over the complex numbers 4 we generalize all of the above to results on the number of n k dimensional components of the zero set of f1 fk when k n

computing all nonsingular solutions of cyclic n polynomial using polyhedral homotopy continuation methods all isolated solutions of the cyclic n polynomial equations are not known for larger dimensions than 11 we exploit two types of symmetric structures in the cyclic n polynomial to compute all isolated nonsingular solutions of the equations efficiently by the polyhedral homotopy continuation method and to verify the correctness of the generated approximate solutions numerical results on the cyclic 8 to the cyclic 12 polynomial equations including their solution information are given

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

applying genetic algorithms for solving nonlinear algebraic equations solving of nonlinear algebraic equations is a prominent problem in science and engineering in this study the systems of these equations were solved using genetic algorithms three distinctive groups of the equations were studied a single nonlinear equation a simple set of nonlinear equations and a complex set of nonlinear equations in the two first cases the problem was converted into the unconstrained optimization problem with some simple mathematical relations then the genetic algorithm was used to solve the problem in the complex nonlinear equations the problem was changed into the constrained optimization problem and genetic algorithm in conjunction with augmented lagrangian function has been used to find the solution the proposed scheme has been evaluated with five benchmarks with different applications moreover the sensitivity analysis of the method to initial interval and size of population has been studied the results have a high precision and show good agreement in comparison with conventional methods

computing isolated roots of sparse polynomial systems in affine space we present a symbolic probabilistic algorithm to compute the isolated roots in c n of sparse polynomial equation systems as some already known numerical algorithms solving this task our procedure is based on polyhedral deformations and homotopies but it amounts to solving a smaller number of square systems of equations and in fewer variables the output of the algorithm is a geometric resolution of a finite set of points including the isolated roots of the system the complexity is polynomial in the size of the combinatorial structure of the system supports up to a pre processing yielding the mixed cells in a subdivision of the family of these supports

exact algorithms for linear matrix inequalities let a x a 0 x 1a 1 cdots x na n be a linear matrix or pencil generated by given symmetric matrices a 0 a 1 ldots a n of size m with rational entries the set of real vectors x such that the pencil is positive semidefinite is a convex semialgebraic set called spectrahedron described by a linear matrix inequality we design an exact algorithm that up to genericity assumptions on the input matrices computes an exact algebraic representation of at least one point in the spectrahedron or decides that it is empty the algorithm does not assume the existence of an interior point and the computed point minimizes the rank of the pencil on the spectrahedron the degree d of the algebraic representation of the point coincides experimentally with the algebraic degree of a generic semidefinite program associated to the pencil we provide explicit bounds for the complexity of our algorithm proving that the maximum number of arithmetic operations that are performed is essentially quadratic in a mu

stable reductive varieties ii projective case abstract we construct a moduli space of stable projective pairs with a nontrivial action of a connected reductive group these stable reductive pairs are higher dimensional analogs of stable n pointed curves and generalize to the non commutative case a functorial compactification of moduli of abelian varieties and of toric pairs we prove that confirming a prediction of log minimal program stable reductive pairs have semi log canonical singularities

the expected number of nash equilibria of a normal form game fix finite pure strategy sets s1 s and let s s x x s n in our model of a random game the agents payoffs are statistically independent with each agent s payoff uniformly distributed on the unit sphere in r a for given nonempty t 1 subset of s 1 t n subset of s n we give a computationally implementable formula for the mean number of nash equilibria in which each agent i s mixed strategy has support t i the formula is the product of two expressions the first is the expected number of totally mixed equilibria for the truncated game obtained by eliminating pure strategies outside the sets ti the second may be construed as the probability that such an equilibrium remains an equilibrium when the strategies in the sets s i t i become available

on the matroid stratification of grassmann varieties specialization of coordinates and a problem of n white abstract we resolve a problem of n white j combin theory ser b29 1980 168 175 by constructing representable matroids f and g with f g in the weak map order such that no coordinatization of g specializes to a coordinatization of f our approach is based upon the interplay of combinatorics and topology in the matroid stratification of the complex grassmann variety gn d introduced by i m gelfand r m goresky r d macpherson and v serganova adv in math 63 1987 301 316 related aspects and open problems in the algebraic geometry of matroids are discussed