a polynomial time approximation algorithm for the permanent of a matrix with nonnegative entries we present a polynomial time randomized algorithm for estimating the permanent of an arbitrary n n matrix with nonnegative entries this algorithm technically a fully polynomial randomized approximation scheme computes an approximation that is with high probability within arbitrarily small specified relative error of the true value of the permanent

hit and run algorithms for the identification of nonredundant linear inequalities two probabilistic hit and run algorithms are presented to detect nonredundant constraints in a full dimensional system of linear inequalities the algorithms proceed by generating a random sequence of interior points whose limiting distribution is uniform and by searching for a nonredundant constraint in the direction of a random vector from each point in the sequence in the hypersphere directions algorithm the direction vector is drawn from a uniform distribution on a hypersphere in the computationally superior coordinate directions algorithm a search is carried out along one of the coordinate vectors the algorithms are terminated through the use of a bayesian stopping rule computational experience with the algorithms and the stopping rule will be reported

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

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

the ternary commutator obstruction for internal crossed modules abstract in finitely cocomplete homological categories co smash products give rise to possibly higher order commutators of subobjects we use binary and ternary co smash products and the associated commutators to give characterisations of internal crossed modules and internal categories respectively the ternary terms are redundant if the category has the smith is huq property which means that two equivalence relations on a given object commute precisely when their normalisations do in fact we show that the difference between the smith commutator of such relations and the huq commutator of their normalisations is measured by a ternary commutator so that the smith is huq property itself can be characterised by the relation between the latter two commutators this allows us to show that the category of loops does not have the smith is huq property which also implies that ternary commutators are generally not decomposable into nested binary ones thus in contexts where smith is huq need not hold we obtain a new description of internal categories beck modules and double central extensions as well as a decomposition formula for the smith commutator the ternary commutator now also appears in the hopf formula for the third homology with coefficients in the abelianisation functor

nonconvex variational problems with anisotropic perturbations we obtain the l 1 limit for a class of variational problems which arise as an anisotropic singular perturbation of a nonconvex functional any limit of minimizers of the perturbed problems must minimize the new variational problem which arises in the limit and the primary goal is to understand the dependence of this new variational problem on the type of perturbation chosen

gelfand numbers of operators with values in a hilbert space in the paper we prove two inequalities involving gelfand numbers of operators with values in a hilbert space the first inequality is a rademacher version of the main result in pa to 1 which relates the gelfand numbers of an operator from a banach spacex intol 2 n with a certain rademacher average for the dual operator the second inequality states that the gelfand numbers of an operatoru froml 1 n into a hilbert space satisfy the inequality k 1 2 c k u leqq c parallel u parallel log 1 n k 1 2 wherec is a universal constant several applications of these inequalities in the geometry of banach spaces are given

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

volume of the steady state space of financial flows in a monetary stock flow consistent model we show that a steady state stock flow consistent macro economic model can be represented as a constraint satisfaction problem csp the set of solutions is a polytope which volume depends on the constraints applied and reveals the potential fragility of the economic circuit with no need to study the dynamics several methods to compute the volume are compared inspired by operations research methods and the analysis of metabolic networks both exact and approximate we also introduce a random transaction matrix and study the particular case of linear flows with respect to money stocks

calculating the free energy of nearly jammed hard particle packings using molecular dynamics we present a new event driven molecular dynamics md algorithm for measuring the free energy of nearly jammed packings of spherical and non spherical hard particles this bounding cell molecular dynamics bcmd algorithm exactly calculates the free energy of a single occupancy cell soc model in which each particle is restricted to a neighborhood of its initial position using a hard wall bounding cell our md algorithm generalizes previous ones in the literature by enabling us to study non spherical particles as well as to measure the free energy change during continuous irreversible transformations moreover we make connections to the well studied problem of computing the volume of convex bodies in high dimensions using random walks we test and verify the numerical accuracy of the method by comparing against rigorous asymptotic results for the free energy of jammed and isostatic disordered packings of both hard spheres and ellipsoids for which the free energy can be calculated directly as the volume of a high dimensional simplex we also compare our results to previously published monte carlo results for hard sphere crystals near melting and jamming and find excellent agreement we have successfully used the bcmd algorithm to determine the configurational and free volume contributions to the free energy of glassy states of binary hard disks a donev f h stillinger s torquato do binary hard disks exhibit an ideal glass transition phys rev lett 96 22 2006 225502 the algorithm can also be used to determine phases with locally or globally minimal free energy to calculate the free energy cost of point and extended crystal defects or to calculate the elastic moduli of glassy or crystalline solids among other potential applications

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

ambiguous joint chance constraints under mean and dispersion information we study joint chance constraints where the distribution of the uncertain parameters is only known to belong to an ambiguity set characterized by the mean and support of the uncertainties and by an upper bound on their dispersion this setting gives rise to pessimistic optimistic ambiguous chance constraints which require the corresponding classical chance constraints to be satisfied for every for at least one distribution in the ambiguity set we demonstrate that the pessimistic joint chance constraints are conic representable if i the constraint coefficients of the decisions are deterministic ii the support set of the uncertain parameters is a cone and iii the dispersion function is of first order that is it is positively homogeneous we also show that pessimistic joint chance constrained programs become intractable as soon as any of the conditions i ii or iii is relaxed in the mildest possible way we further prove that the optimistic joint chance constraints are conic representab

on linear characterizations of combinatorial optimization problems we show that there can be no computationally tractable description by linear inequalities of the polyhedron associated with any np complete combinatorial optimization problem unless np co np a very unlikely event we also apply the ellipsoid method for linear programming to show that a combinatorial optimization problem is solvable in polynomial time if and only if it admits a small generator of violated inequalities

approximate counting uniform generation and rapidly mixing markov chains the paper studies effective approximate solutions to combinatorial counting and uniform generation problems using a technique based on the simulation of ergodic markov chains it is shown that for self reducible structures almost uniform generation is possible in polynomial time provided only that randomised approximate counting to within some arbitrary polynomial factor is possible in polynomial time it follows that for self reducible structures polynomial time randomised algorithms for counting to within factors of the form 1 n are available either for all fl e r or for no fi e r a substantial part of the paper is devoted to investigating the rate of convergence of finite ergodic markov chains and a simple but powerful characterisation of rapid convergence for a broad class of chains based on a structural property of the underlying graph is established finally the general techniques of the paper are used to derive an almost uniform generation procedure for labelled graphs with a given degree sequence which is valid over a much wider range of degrees than previous methods this in turn leads to randomised approximate counting algorithms for these graphs with very good

odd minimum cut sets and b matchings we show that the determination of a minimum cut set of odd cardinality in a graph with even and odd vertices can be dealt with by a minor modification of the polynomially bounded algorithm of gomory and hu for multi terminal networks we connect this problem to the problem of identifying a matching or blossom constraint that chops off a point which is not contained in the convex hull of matchings or proving that no such inequality exists both the b matching problems without and with upper bounds are considered we discuss how the results of this paper can be used in conjunction with commercial lp packages lo solve b matching problems

hit and run mixes fast it is shown that the hit and run algorithm for sampling from a convex body k introduced by r l smith mixes in time o n2r2 r2 where r and r are the radii of the inscribed and circumscribed balls of k thus after appropriate preprocessing hit and run produces an approximately uniformly distributed sample point in time o n3 which matches the best known bound for other sampling algorithms we show that the bound is best possible in terms of r r and n

an fptas for the volume of some v polytopes it is hard to compute the volume of the intersection of two cross polytopes abstract given an n dimensional convex body by a membership oracle in general it is known that any polynomial time deterministic algorithm cannot approximate its volume within ratio n log n n there is a substantial progress on randomized approximation such as markov chain monte carlo for a high dimensional volume and for many p hard problems while only a few p hard problems are known to yield deterministic approximation motivated by the problem of deterministically approximating the volume of a v polytope that is a polytope with a small number of vertices and possibly exponentially many facets this paper investigates the problem of computing the volume of a knapsack dual polytope which is known to be p hard due to khachiyan 1989 16 we reduce an approximate volume of a knapsack dual polytope to that of the intersection of two cross polytopes in a short distance and give fptass for those volume computations interestingly computing the volume of the intersection of two cross polytopes i e l 1 balls is p hard unlike the cases of l balls or l 2 balls

the quasistationary phase field equations with neumann boundary conditions abstract we prove that the quasistationary phase field equations t u u f 2e 1 e w u where w t t 2 1 2 is a double well potential admit a solution when the space dimension n 3 and that the solutions converge for e 0 to solutions of the stefan problem with gibbs thomson law

an analysis of a phase field model of a free boundary a mathematical analysis of a new approach to solidification problems is presented a free boundary arising from a phase transition is assumed to have finite thickness the physics leads to a system of nonlinear parabolic differential equations existence and regularity of solutions are proved invariant regions of the solution space lead to physical interpretations of the interface a rigorous asymptotic analysis leads to the gibbs thompson condition which relates the temperature at the interface to the surface tension and curvature

the gibbs thompson relation within the gradient theory of phase transitions this paper discusses the asymptotic behavior as 0 of the chemical potentials associated with solutions of variational problems within the van der waals cahn hilliard theory of phase transitions in a fluid with free energy per unit volume given by 2 2 w where is the density the main result is that is asymptotically equal to e d o with e the interfacial energy per unit surface area of the interface between phases the constant sum of principal curvatures of the interface and d the density jump across the interface this result is in agreement with a formula conjectured by m gurtin and corresponds to the gibbs thompson relation for surface tension proved by g caginalp within the context of the phase field model of free boundaries arising from phase transitions

projective re normalization for improving the behavior of a homogeneous conic linear system in this paper we study the homogeneous conic system f ax 0 x in c setminus 0 we choose a point bar s in rm int c that serves as a normalizer and consider computational properties of the normalized system f bar s ax 0 bar s t x 1 x in c we show that the computational complexity of solving f via an interior point method depends only on the complexity value vartheta of the barrier for c and on the symmetry of the origin in the image set h bar s ax bar s tx 1 x in c where the symmetry of 0 in h bar s is rm sym 0 h bar s rm max alpha y in h bar s rightarrow alpha y in h bar s interior point iterations in order to improve the theoretical and practical computation of a solution of f we next present a general theory for projective re normalization of the feasible region f bar s and the image set h bar s and prove the existence of a normalizer bar s such that rm sym 0 h bar s ge 1 m provided that f has an interior solution we develop a methodology for constructing a normalizer bar s such that rm sym 0 h bar s ge 1 m with high probability based on sampling on a geometric random walk with associated probabilistic complexity analysis while such a normalizer is not itself computable in strongly polynomial time the normalizer will yield a conic system that is solvable in o sqrt vartheta rm ln m vartheta iterations which is strongly polynomial time finally we implement this methodology on randomly generated homogeneous linear programming feasibility problems constructed to be poorly behaved our computational results indicate that the projective re normalization methodology holds the promise to markedly reduce the overall computation time for conic feasibility problems for instance we observe a 46 decrease in average ipm iterations for 100 randomly generated poorly behaved problem instances of dimension 1 000 5 000

mean utility in the assurance region model abstract this paper addresses the problem of ranking a finite set of alternatives through a linear utility with coefficients randomly distributed in order to model relative importance of the different criteria assurance regions on weights are used a closed formula for the expected utility of each alternative is obtained under uniformity assumptions and as a consequence different ranking methods are proposed

approximating probabilistic inference in bayesian belief networks is np hard abstract it is known that exact computation of conditional probabilities in belief networks is np hard many investigators in the ai community have tacitly assumed that algorithms for performing approximate inference with belief networks are of polynomial complexity indeed special cases of approximate inference can be performed in time polynomial in the input size however we have discovered that the general problem of approximating conditional probabilities with belief networks like exact inference resides in the np hard complexity class we develop a complexity analysis to elucidate the difficulty of approximate probabilistic inference more specifically we show that the existence of a polynomial time relative approximation algorithm for major classes of problem instances implies that np p we present our proof and explore the implications of the result

baer sums in homological categories abstract we give a unified treatment of the baer sums in the context of efficiently homological categories which on the one hand contains any category of groups with multiple operators and more generally any semi abelian variety and on the other hand the category of hausdorff groups and more generally any category of semi abelian hausdorff algebras this gives rise to a generalized euclide s postulate and a five terms exact sequence

discrete quantum walks hit exponentially faster this paper addresses the question what processes take polynomial time on a quantum computer that require exponential time classically we show that the hitting time of the discrete time quantum walk on the n bit hypercube from one corner to its opposite is polynomial in n this gives the first exponential quantum classical gap in the hitting time of discrete quantum walks we provide the basic framework for quantum hitting time and give two alternative definitions to set the ground for its study on general graphs we outline a possible application to sequential packet routing

easy and optimal queries to reduce set uncertainty abstract in this paper we address the problem of optimally querying a single expert to reduce set interval uncertainty we propose optimal querying strategies for two particular query formats local bound and pairwise comparisons based on two main selection criteria the minimax and the bayesian rules we study the computational aspects of the optimal solution in the general case and for the specific functions of practical interest monotonic and multi linear the use of the proposed approach is illustrated through numerical simulations on a common estimation problem in reliability analysis

on the complete instability of interval polynomials abstract in this paper we study complete instability of interval polynomials which is the counterpart of classical robust stability that is the objective is to check if all polynomials in the family are unstable if not a subsequent goal is to find a stable polynomial to this end we first propose a randomized algorithm which is based on a recursive necessary condition for hurwitz stability the second contribution of this paper is to provide a probability one estimate of the volume of stable polynomials these results are based on a combination of deterministic and randomized methods finally we present two numerical examples and simulations showing the efficiency of the proposed methodology for small and medium size problems

note on the computational complexity of j radii of polytopes in r n we show that for fixed dimensionn the approximation of inner and outer j radii of polytopes in n endowed with the euclidean norm is in our method is based on the standard polynomial time algorithms for solving a system of polynomial inequalities over the reals in fixed dimension

approximately counting hamilton paths and cycles in dense graphs we describe fully polynomial randomized approximation schemes for the problems of determining the number of hamilton paths and cycles in an n vertex graph with minimum degree frac 1 2 a n for any fixed a 0 we show that the exact counting problems are p complete we also describe fully polynomial randomized approximation schemes for counting paths and cycles of all sizes in such graphs

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

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

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

integer programming with a fixed number of variables it is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable the proof depends on methods from geometry of numbers

the effect of a singular perturbation on nonconvex variational problems we study the effect of a singular perturbation on certain nonconvex variational problems the goal is to characterize the limit of minimizers as some perturbation parameter 0 the technique utilizes the notion of convergence of variational problems developed by de giorgi the essential idea is to identify the first nontrivial term in an asymptotic expansion for the energy of the perturbed problem in so doing one characterizes the limit of minimizers as the solution of a new variational problem for the cases considered here these new problems have a simple geometric nature involving minimal surfaces and geodesics

complexity of approximating the vertex centroid of a polyhedron let p be an h polytope in r d with vertex set v the vertex centroid is defined as the average of the vertices in v we first prove that computing the vertex centroid of anh polytope or even just checking whether it lies in a given halfspace is p hard we also consider the problem of approximating the vertex centroid by finding a point within an e distance from it and prove this problem to be p easy in the sense that it can be solved efficiently using an oracle for some p complete problem in particular we show that given an oracle for counting the number of vertices of an h polytope one can approximate the vertex centroid in polynomial time counting the number of vertices of a polytope defined as the intersection of halfspaces is known to be p complete we also show that any algorithm approximating the vertex centroid to any sufficiently non trivial for example constant distance can be used to construct a fully polynomial time approximation scheme for approximating the centroid and also an output sensitive polynomial algorithm for the vertex enumeration problem finally we show that for unbounded polyhedra the vertex centroid cannot be approximated to a distance of d 1 2 d for any fixed constant d 0 unless p np

local minimisers and singular perturbations we construct local minimisers to certain variational problems the method is quite general and relies on the theory of convergence the approach is demonstrated through the model problem it is shown that in certain nonconvex domains n and for e small there exist nonconstant local minimisers u e satisfying u e 1 except in a thin transition layer the location of the layer is determined through the requirement that in the limit u e u 0 the hypersurface separating the states u 0 1 and u 0 1 locally minimises surface area generalisations are discussed with for example vector valued u and anisotropic perturbations replacing u 2

on the complexity of approximating the maximal inscribed ellipsoid for a polytope we give a new polynomial bound on the complexity of approximating the maximal inscribed ellipsoid for a polytope

random generation of combinatorial structures from a uniform distribution abstract the class of problems involving the random generation of combinatorial structures from a uniform distribution is considered uniform generation problems are in computational difficulty intermediate between classical existence and counting problems it is shown that exactly uniform generation of efficiently verifiable combinatorial structures is reducible to approximate counting and hence is within the third level of the polynomial hierarchy natural combinatorial problems are presented which exhibit complexity gaps between their existence and generation and between their generation and counting versions it is further shown that for self reducible problems almost uniform generation and randomized approximate counting are inter reducible and hence of similar complexity

a random polynomial time algorithm for approximating the volume of convex bodies a randomized polynomial time algorithm for approximating the volume of a convex body k in n dimensional euclidean space is presented the proof of correctness of the algorithm relies on recent theory of rapidly mixing markov chains and isoperimetric inequalities to show that a certain random walk can be used to sample nearly uniformly from within k

efficient monte carlo procedures for generating points uniformly distributed over bounded regions we consider the monte carlo problem of generating points uniformly distributed within an arbitrary bounded measurable region the class of markovian methods considered generate points asymptotically uniformly distributed within the region computational experience suggests the methods are potentially superior to conventional rejection techniques for large dimensional regions

weighted commutators in semi abelian categories abstract we introduce new notions of weighted centrality and weighted commutators corresponding to each other in the same way as centrality of congruences and commutators do in the smith commutator theory both the huq commutator of subobjects and pedicchio s categorical generalization of smith commutator are special cases of our weighted commutators in fact we obtain them by taking the smallest and the largest weight respectively at the end of the paper we briefly consider the universal algebraic context in connection with an older work of the third author on the ideal theory version of the commutator theory

random walks totally unimodular matrices and a randomised dual simplex algorithm we discuss the application of random walks to generating a random basis of a totally unimodular matrix and to solving a linear program with such a constraint matrix we also derive polynomial upper bounds on the combinatorial diameter of an associated polyhedron

stochastic billiards for sampling from the boundary of a convex set stochastic billiards can be used for approximate sampling from the boundary of a bounded convex set through the markov chain monte carlo paradigm this paper studies how many steps of the underlying markov chain are required to get samples approximately from the uniform distribution on the boundary of the set for sets with an upper bound on the curvature of the boundary our main theorem implies a polynomial time algorithm for sampling from the boundary of such sets

dispersion of mass and the complexity of randomized geometric algorithms abstract how much can randomness help computation motivated by this general question and by volume computation one of the few instances where randomness provably helps we analyze a notion of dispersion and connect it to asymptotic convex geometry we obtain a nearly quadratic lower bound on the complexity of randomized volume algorithms for convex bodies in r n the current best algorithm has complexity roughly n 4 conjectured to be n 3 our main tools dispersion of random determinants and dispersion of the length of a random point from a convex body are of independent interest and applicable more generally in particular the latter is closely related to the variance hypothesis from convex geometry this geometric dispersion also leads to lower bounds for matrix problems and property testing

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

random vectors in the isotropic position abstract letybe a random vector in r n satisfying e y y id letmbe a natural number and lety1 ymbe independent copies ofy we study the question of approximation of the identity operator by finite sums of the tensorsyi yi we prove that for some absolute constantc e 1 m i 1 m y i y i id c log n m e y log m 1 log m provided that the last expression is smaller than 1 we apply this estimate to improve a result of bourgain concerning the number of random points needed to bring a convex body into a nearly isotropic position

computing and estimating the volume of the solution space of smt la constraints abstract the satisfiability modulo theories smt problem is a decision problem i e deciding the satisfiability of logical formulas with respect to combinations of background theories like reals integers arrays bit vectors in this paper we study the counting version of smt with respect to linear arithmetic smt la which generalizes both model counting and volume computation of convex polytopes we describe a method for estimating the volume of convex polytopes based on the multiphase monte carlo method it employs a new technique to reutilize random points so that the number of random points can be significantly reduced we prove that the reutilization technique has no side effect on the error we also investigate a simplified version of hit and run method the coordinate directions method based on volume estimation method for polytopes we present an approach to estimating the volume of the solution space of smt la formulas it employs a heuristic strategy to accelerate the volume estimation procedure in addition we devise some specific techniques for instances that arise from program analysis

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

finiteness and positivity results for global minimizers of a semilinear elliptic problem abstract we consider the landau ginzburg free energy functional f e e 2 1 e w u wherew t t2 1 2is a double well potential and prove that it has only finitely many global minimizers we investigate the stability properties of these minimizers and show that the second variationf eis positive definite in one dimension for nondegenerateu

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

hit and run algorithms for generating multivariate distributions we introduce a general class of hit and run algorithms for generating essentially arbitrary absolutely continuous distributions on rd they include the hypersphere directions algorithm and the coordinate directions algorithm that have been proposed for identifying nonredundant linear constraints and for generating uniform distributions over subsets of rd given a bounded open set s in rd an absolutely continuous probability distribution on s the target distribution and an arbitrary probability distribution i on the boundary of the d dimensional unit sphere centered at the origin the direction distribution the i hit and run algorithm produces a sequence of iteration points as follows given the nth iteration point x choose a direction i according to the distribution i and then choose the n 1st iteration point according to the conditionalization of the distribution along the line defined by x and x i under mild conditions we show that this sequence of points is a harris recurrent reversible markov chain converging in total variation to the target distribution

computational results of an o n4 volume algorithm abstract recently an o n4 volume algorithm has been presented for convex bodies by lovasz and vempala where n is the number of dimensions of the convex body essentially the algorithm is a series of monte carlo integrations in this paper we describe a computer implementation of the volume algorithm where we improved the computational aspects of the original algorithm by adding variance decreasing modifications a stratified sampling strategy double point integration and orthonormalised estimators formulas and methodology were developed so that the errors in each phase of the algorithm can be controlled some computational results for convex bodies in dimensions ranging from 2 to 10 are presented as well

convergence analysis for lasserre s measure based hierarchy of upper bounds for polynomial optimization we consider the problem of minimizing a continuous function f over a compact set mathbf k k we analyze a hierarchy of upper bounds proposed by lasserre siam j optim 21 3 864 885 2011 obtained by searching for an optimal probability density function h on mathbf k k which is a sum of squares of polynomials so that the expectation int mathbf k f x h x dx kf x h x dx is minimized we show that the rate of convergence is no worse than o 1 sqrt r o 1 r where 2r is the degree bound on the density function this analysis applies to the case when f is lipschitz continuous and mathbf k k is a full dimensional compact set satisfying some boundary condition which is satisfied e g for convex bodies the rth upper bound in the hierarchy may be computed using semidefinite programming if f is a polynomial of degree d and if all moments of order up to 2r d 2r d of the lebesgue measure on mathbf k k are known which holds for example if mathbf k k is a simplex hypercube or a euclidean ball

generalized haar dwt and transformations between decision trees and neural networks the core contribution of this paper is a three fold improvement of the haar discrete wavelet transform dwt it is modified to efficiently transform a multiclass rather than numerical valued function over a multidimensional rather than low dimensional domain or transform a multiclass valued decision tree into another useful representation we prove that this multidimensional multiclass dwt uses dynamic programming to minimize within its framework the number of nontrivial wavelet coefficients needed to summarize a training set or decision tree it is a spatially localized algorithm that takes linear time in the number of training samples after a sort convergence of the dwt to benchmark training sets seems to degrade with rising dimension in this test of high dimensional wavelets which have been seen as difficult to implement this multiclass multidimensional dwt has tightly coupled applications from learning dyadic decision trees directly from training data rebalancing or converting preexisting decision trees to fixed depth boolean or threshold neural networks in effect parallelizing the evaluation of the trees or learning rule exception sets represented as a new form of tree called an e tree which could greatly help interpretation visualization of a dataset

intrinsic centrality and associated classifying properties abstract the notion of central morphism is introduced in the very soft algebraic context of unital categories it reveals inside any such category an additive core which has a strong classifying potential