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

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

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

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

passive advection of a vector field anisotropy finite correlation time exact solution and logarithmic corrections to ordinary scaling in this work we study the generalization of the problem considered in phys rev e 91 013002 2015 to the case of finite correlation time of the environment velocity field the model describes a vector e g magnetic field passively advected by a strongly anisotropic turbulent flow inertial range asymptotic behavior is studied by means of the field theoretic renormalization group and the operator product expansion the advecting velocity field is gaussian with finite correlation time and preassigned pair correlation function due to the presence of distinguished direction n all the multiloop diagrams in this model vanish so that the results obtained are exact the inertial range behavior of the model is described by two regimes the limits of vanishing or infinite correlation time that correspond to the two nontrivial fixed points of the rg equations their stability depends on the relation between the exponents in the energy spectrum e k 1 and the dispersion law k 2 in contrast to the well known isotropic kraichnan s model where various correlation functions exhibit anomalous scaling behavior with infinite sets of anomalous exponents here the corrections to ordinary scaling are polynomials of logarithms of the integral turbulence scale l

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

renormalization group operator product expansion and anomalous scaling in models of turbulent advection recent progress on the anomalous scaling in models of turbulent heat and mass transport is reviewed with the emphasis on the approach based on the field theoretic renormalization group rg and operator product expansion ope in that approach the anomalous scaling is established as a consequence of the existence in the corresponding field theoretic models of an infinite number of dangerous composite fields operators with negative critical dimensions which are identified with the anomalous exponents this allows one to calculate the exponents in a systematic perturbation expansion similar to the e expansion in the theory of critical phenomena the rg and ope approach is presented in a self contained way for the example of a passive scalar field temperature concentration of an impurity etc advected by a self similar gaussian velocity ensemble with vanishing correlation time the so called kraichnan s rapid change model where the anomalous exponents are known up to order o e3 effects of anisotropy compressibility and the correlation time of the velocity field are discussed passive advection by non gaussian velocity field governed by the stochastic navier stokes equation and passively advected vector e g magnetic fields are considered

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

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

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

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

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

anomalous scaling of passive scalar fields advected by the navier stokes velocity ensemble effects of strong compressibility and large scale anisotropy the field theoretic renormalization group and the operator product expansion are applied to two models of passive scalar quantities the density and the tracer fields advected by a random turbulent velocity field the latter is governed by the navier stokes equation for compressible fluid subject to external random force with the covariance t t k 4 d y where d is the dimension of space and y is an arbitrary exponent the original stochastic problems are reformulated as multiplicatively renormalizable field theoretic models the corresponding renormalization group equations possess infrared attractive fixed points it is shown that various correlation functions of the scalar field its powers and gradients demonstrate anomalous scaling behavior in the inertial convective range already for small values of y the corresponding anomalous exponents identified with scaling critical dimensions of certain composite fields operators in the quantum field terminology can be systematically calculated as series in y the practical calculation is performed in the leading one loop approximation including exponents in anisotropic contributions it should be emphasized that in contrast to gaussian ensembles with finite correlation time the model and the perturbation theory presented here are manifestly galilean covariant the validity of the one loop approximation and comparison with gaussian models are briefly discussed

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

anomalous scaling in two and three dimensions for a passive vector field advected by a turbulent flow a model of the passive vector field advected by the uncorrelated in time gaussian velocity with power like covariance is studied by means of the renormalization group and the operator product expansion the structure functions of the admixture demonstrate essential power like dependence on the external scale in the inertial range the case of an anomalous scaling the method of finding independent tensor invariants in the cases of two and three dimensions is proposed to eliminate linear dependencies between the operators entering into the operator product expansions of the structure functions the constructed operator bases which include the powers of the dissipation operator and the enstrophy operator provide the possibility of calculating the exponents of the anomalous scaling

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

probabilistic ω automata probabilistic automata are variants of nondeterministic automata over infinite words where all choices are resolved by probabilistic distributions acceptance of a run for an infinite input word can be defined using traditional acceptance criteria for automata such as buchi rabin or streett conditions the accepted language of a probabilistic automata is then defined by imposing a constraint on the probability measure of the accepting runs in this paper we study a series of fundamental properties of probabilistic automata with three different language semantics 1 the probable semantics that requires positive acceptance probability 2 the almost sure semantics that requires acceptance with probability 1 and 3 the threshold semantics that relies on an additional parameter i r0 1l that specifies a lower probability bound for the acceptance probability we provide a comparison of probabilistic automata under these three semantics and nondeterministic automata concerning expressiveness and efficiency furthermore we address closure properties under the boolean operators union intersection and complementation and algorithmic aspects such as checking emptiness or language containment

shortest paths without a map we study several versions of the shortest path problem when the map is not known in advanced but is specified dynamically we are seeking dynamic decision rules that optimize the worst case ratio of the distance covered to the length of the statically optimal path we describe optimal decision rules for two cases layered graphs of bounded width and two dimensional scenes with unit square obstacles the optimal rules turn out to be intuitive common sense heuristics for slightly more general graphs and scenes we show that no bounded ratio is possible we also show that the computational problem of devising a strategy that achieves a given worst case ratio to the optimum path in a graph is a universal two person game and thus pspace complete whereas optimizing the expected ratio is p hard

anomalous scaling of a passive scalar advected by the turbulent velocity field with finite correlation time two loop approximation the renormalization group and operator product expansion are applied to the model of a passive scalar quantity advected by the gaussian self similar velocity field with finite and not small correlation time the inertial range energy spectrum of the velocity is chosen in the form e k ensuremath propto k 1 ensuremath 2 ensuremath varepsilon and the correlation time at the wave number k scales as k ensuremath 2 ensuremath eta inertial range anomalous scaling for the structure functions and other correlation functions emerges as a consequence of the existence in the model of composite operators with negative scaling dimensions identified with anomalous exponents for ensuremath eta g ensuremath varepsilon these exponents are the same as in the rapid change limit of the model for ensuremath eta l ensuremath varepsilon they are the same as in the limit of a time independent quenched velocity field for ensuremath varepsilon ensuremath eta local turnover exponent the anomalous exponents are nonuniversal through the dependence on a dimensionless parameter the ratio of the velocity correlation time and the scalar turnover time the nonuniversality reveals itself however only in the second order of the ensuremath varepsilon expansion and the exponents are derived to order ensuremath varepsilon 2 including anisotropic contributions it is shown that for moderate order of the structure function n and the space dimensionality d finite correlation time enhances the intermittency in comparison with both the limits the rapid change and quenched ones the situation changes when n and or d become large enough the correction to the rapid change limit due to the finite correlation time is positive that is the anomalous scaling is suppressed it is maximal for the quenched limit and monotonically decreases as the correlation time tends to zero

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

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

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

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

anomalous scaling of a passive vector advected by the navier stokes velocity field using the field theoretic renormalization group and the operator product expansion the model of a passive vector field a weak magnetic field in the framework of the kinematic mhd advected by the velocity field which is governed by the stochastic navier stokes equation with the gaussian random stirring force correlated in time and with the correlator proportional to k4 d 2e is investigated to the first order in e one loop approximation it is shown that the single time correlation functions of the advected vector field have anomalous scaling behavior and the corresponding exponents are calculated in the isotropic case as well as in the case with the presence of large scale anisotropy the hierarchy of the anisotropic critical dimensions is briefly discussed and the persistence of the anisotropy inside the inertial range is demonstrated on the behavior of the skewness and hyperskewness dimensionless ratios of correlation functions as functions of the reynolds number re it is shown that even though the present model of a passive vector field advected by the realistic velocity field is mathematically more complicated than on one hand the corresponding models of a passive vector field advected by synthetic gaussian velocity fields and on the other hand than the corresponding model of a passive scalar quantity advected by the velocity field driven by the stochastic navier stokes equation the final one loop approximate asymptotic scaling behavior of the single time correlation or structure functions of the advected fields of all models are defined by the same anomalous dimensions up to normalization

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

complexity of finite horizon markov decision process problems controlled stochastic systems occur in science engineering manufacturing social sciences and many other cntexts if the systems is modeled as a markov decision process mdp and will run ad infinitum the optimal control policy can be computed in polynomial time using linear programming the problems considered here assume that the time that the process will run is finite and based on the size of the input there are mny factors that compound the complexity of computing the optimal policy for instance there are many factors that compound the complexity of this computation for instance if the controller does not have complete information about the state of the system or if the system is represented in some very succint manner the optimal policy is provably not computable in time polynomial in the size of the input we analyze the computational complexity of evaluating policies and of determining whether a sufficiently good policy exists for a mdp based on a number of confounding factors including the observability of the system state the succinctness of the representation the type of policy even the number of actions relative to the number of states in almost every case we show that the decision problem is complete for some known complexity class some of these results are familiar from work by papadimitriou and tsitsiklis and others but some such as our pl completeness proofs are surprising we include proofs of completeness for natural problems in the as yet little studied classes np pp

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

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

on the complexity of partially observed markov decision processes abstract in the paper we consider the complexity of constructing optimal policies strategies for some type of partially observed markov decision processes this particular case of the classical problem deals with finite stationary processes and can be represented as constructing optimal strategies to reach target vertices from a starting vertex in a graph with colored vertices and probabilistic deviations from an edge chosen to follow the colors of the visited vertices is the only information available to a strategy the complexity of markov decision in the case of perfect information bijective coloring of vertices is known and briefly surveyed at the beginning of the paper for the unobservable case all the colors are equal we give an improvement of the result of papadimitriou and tsitsiklis namely we show that the problem of constructing even a very weak approximation to an optimal strategy is np hard our main results concern the case of a fixed bound on the multiplicity of coloring that is a case of partially observed processes where some upper bound on the unobservability is supposed we show that the problem of finding an optimal strategy is still np hard but polytime approximations are possible some relations of our results to the max word problem are also indicated

optimal control of infinite horizon partially observable decision processes modelled as generators of probabilistic regular languages decision processes with incomplete state feedback have been traditionally modelled as partially observable markov decision processes in this article we present an alternative formulation based on probabilistic regular languages the proposed approach generalises the recently reported work on language measure theoretic optimal control for perfectly observable situations and shows that such a framework is far more computationally tractable to the classical alternative in particular we show that the infinite horizon decision problem under partial observation modelled in the proposed framework is approximable and in general is not harder to solve compared to the fully observable case the approach is illustrated via two simple examples

turbulent compressible fluid renormalization group analysis scaling regimes and anomalous scaling of advected scalar fields we study a model of fully developed turbulence of a compressible fluid based on the stochastic navier stokes equation by means of the field theoretic renormalization group in this approach scaling properties are related to the fixed points of the renormalization group equations previous analysis of this model near the real world space dimension 3 identified a scaling regime n v antonov et al theor math phys 110 305 1997 tmphah0040 577910 1007 bf02630456 the aim of the present paper is to explore the existence of additional regimes which could not be found using the direct perturbative approach of the previous work and to analyze the crossover between different regimes it seems possible to determine them near the special value of space dimension 4 in the framework of double y and expansion where y is the exponent associated with the random force and 4 d is the deviation from the space dimension 4 our calculations show that there exists an additional fixed point that governs scaling behavior turbulent advection of a passive scalar density field by this velocity ensemble is considered as well we demonstrate that various correlation functions of the scalar field exhibit anomalous scaling behavior in the inertial convective range the corresponding anomalous exponents identified as scaling dimensions of certain composite fields can be systematically calculated as a series in y and all calculations are performed in the leading one loop approximation

the complexity of markov decision processes we investigate the complexity of the classical problem of optimal policy computation in markov decision processes all three variants of the problem finite horizon infinite horizon discounted and infinite horizon average cost were known to be solvable in polynomial time by dynamic programming finite horizon problems linear programming or successive approximation techniques infinite horizon we show that they are complete for p and therefore most likely cannot be solved by highly parallel algorithms we also show that in contrast the deterministic cases of all three problems can be solved very fast in parallel the version with partially observed states is shown to be pspace complete and thus even less likely to be solved in polynomial time than the np complete problems in fact we show that most likely it is not possible to have an efficient on line implementation involving polynomial time on line computations and memory of an optimal policy even if an arbitrary amount of precomputation is allowed finally the variant of the problem in which there are no observations is shown to be np complete

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

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

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

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

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

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

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 undecidability of probabilistic planning and related stochastic optimization problems automated planning the problem of how an agent achieves a goal given a repertoire of actions is one of the foundational and most widely studied problems in the ai literature the original formulation of the problem makes strong assumptions regarding the agent s knowledge and control over the world namely that its information is complete and correct and that the results of its actions are deterministic and known recent research in planning under uncertainty has endeavored te relax these assumptions providing formal and computation models wherein the agent has incomplete or noisy information about the world and has noisy sensors and effectors this research has mainly taken one of two approaches extend the classical planning paradigm to a semantics that admits uncertainty or adopt another framework for approaching the problem most commonly the markov decision process mdp model this paper presents a complexity analysis of planning under uncertainty it begins with the probabilistic classical planning problem showing that problem to be formally undecidable this fundamental result is then applied to a broad class of stochastic optimization problems in brief any problem statement where the agent a operates over an infinite or indefinite time horizon and b has available only probabilistic information about the system s state undecidability is established for policy existence problems for partially observable infinite horizon markov decision processes under discounted and undiscounted total reward models average reward models and state avoidance models the results also apply to corresponding approximation problems with undiscounted objective functions the paper answers a significant open question raised by papadimitriou and tsitsiklis math oper res 12 3 1987 441 450 about the complexity of infinite horizon pomdps

anomalous scaling of a passive scalar advected by the navier stokes velocity field two loop approximation the field theoretic renormalization group and operator product expansion are applied to the model of a passive scalar quantity advected by a non gaussian velocity field with finite correlation time the velocity is governed by the navier stokes equation subject to an external random stirring force with the correlation function proportional to ensuremath delta t ensuremath t ensuremath k 4 ensuremath d ensuremath 2 ensuremath epsilon it is shown that the scalar field is intermittent already for small ensuremath epsilon its structure functions display anomalous scaling behavior and the corresponding exponents can be systematically calculated as series in ensuremath epsilon the practical calculation is accomplished to order ensuremath epsilon 2 two loop approximation including anisotropic sectors as for the well known kraichnan rapid change model the anomalous scaling results from the existence in the model of composite fields operators with negative scaling dimensions identified with the anomalous exponents thus the mechanism of the origin of anomalous scaling appears similar for the gaussian model with zero correlation time and the non gaussian model with finite correlation time it should be emphasized that in contrast to gaussian velocity ensembles with finite correlation time the model and the perturbation theory discussed here are manifestly galilean covariant the relevance of these results for real passive advection and comparison with the gaussian models and experiments are briefly discussed

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

optimal replacement under partial observations in this paper we present a framework for the condition based maintenance optimization a technical system which can be in one of n operational states or in a failure state is considered the system state is not observable except the failure state the information that is stochastically related to the system state is obtained through condition monitoring at equidistant inspection times the system can be replaced at any time a preventive replacement is less costly than failure replacement the objective is to find a replacement policy minimizing the long run expected average cost per unit time the replacement problem is formulated as an optimal stopping problem with partial information and transformed to a problem with complete information by applying the projection theorem to a smooth semimartingale process in the objective function the dynamic equation is derived and analyzed in the piecewise deterministic markov process stopping framework the contraction property is shown and an algorithm for the calculation of the value function is presented illustrated by an example

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