las vegas algorithms for linear and integer programming when the dimension is small this paper gives an algorithm for solving linear programming problems for a problem with n constraints and d variables the algorithm requires an expected o d2n logn o d d 2 o 1 o d4 n logn arithmetic operations as n the constant factors do not depend on d also an algorithm is given for integer linear programming let 4 bound the number of bits required to specify the rational numbers defining an input constraint or the objective function vector let n and d be as before then the algorithm requires expected o 2ddn 8dd n l n n ln n do d 4 ln n operations on numbers with do 1 4 bits as n where the constant factors do not depend on d or 4 to other convex programming problems for example an algorithm for finding the smallest sphere enclosing a set of n points in ed has the same time bound

asymptotic expansions in the clt in free probability we prove edgeworth type expansions for distribution functions of sums of free random variables under minimal moment conditions the proofs are based on the analytic definition of free convolution

mutations of puzzles and equivariant cohomology of two step flag varieties we introduce a mutation algorithm for puzzles that is a three direction analogue of the classical jeu de taquin algorithm for semistandard tableaux we apply this algorithm to prove our conjectured puzzle formula for the equivariant schubert structure constants of two step flag varieties this formula gives an expression for the structure constants that is positive in the sense of graham thanks to the equivariant version of the quantum equals classical result our formula specializes to a littlewood richardson rule for the equivariant quantum cohomology of grassmannians

fuzzy versions of the covering circle problem abstract in this paper we consider two fuzzy versions of the well known problem of determining the smallest circle center and radius that would cover a given finite set of points in the plane when the locations of points are not precise but fuzzy the first is modeled as a possibility constrained mathematical program while the second is modeled as a necessity constrained one polynomial algorithms are presented for both the versions also a numerical example is included for one case these models and solutions are of interest in both theoretical and practical contexts

a randomized algorithm for fixed dimensional linear programming we give a las vegas randomized algorithm for linear programming in a fixed dimensiond for which the expected computation time is o d 3 varepsilon d d n where limd e d 0 this improves the corresponding worst case complexity o 3 d 2 n the method is based on a recent idea of clarkson two variations on the algorithm are examined briefly

addition of freely independent random variables abstract a direct proof is given of voiculescu s addition theorem for freely independent real valued random variables using resolvents of self adjoint operators in contrast to the original proof no assumption is made on the existence of moments above the second

stable laws and domains of attraction in free probability theory in this paper we determine the distributional behavior of sums of free in the sense of voiculescu identically distributed inflnitesimal random variables the theory is shown to parallel the classical theory of independent random variables though the limit laws are usually quite difierent our work subsumes all previously known instances of weak convergence of sums of free identically distributed random variables in particular we determine the domains of attraction of stable distributions in the free theory these freely stable distributions are studied in detail in the appendix where their unimodality and duality properties are demonstrated

superconvergence to the central limit and failure of the cramer theorem for free random variables we show that convergence of the semicircle law in the free central limit theorem for bounded random variables is much better than expected thus the distributions which tend to the semicircle become absolutely continuous in finite time and the densities converge in a very strong sense we also show that the semicircle law is the free convolution of laws which are not semicircular thus proving that cramer s classical result for the normal distribution does not have a free counterpart

parallel linear programming in fixed dimension almost surely in constant time for any fixed dimension d thelinear programming problem with n inequality constraints can be solved on a probabilistic crcw pram with o n processors almost surely in constant time the algorithm always findsthe correct solution with nd log 2 d processors the probability that the algorithm will not finish within o d 2 log 2 d time tends to zero exponentially with n authors abstract

application of decision analysis techniques to the weber facility location problem a location problem with future uncertainties about the data is considered several possible scenarios about the future values of the parameters are postulated however it is not clear which of these scenarios will actually happen we find the location that will best accommodate the possible scenarios four rules utilized in decision theory are examined the expected value rule the optimistic rule the pessimistic rule and the minimax regret rule the solution for the squared euclidean distance is explicitly found algorithms are suggested for general convex distance metrics an example problem is solved in detail to illustrate the findings and computational experiments with randomly generated problems are reported

on the supports of measures in free multiplicative convolution semigroups in this paper we study the supports of measures in free multiplicative semigroups on the positive real line and on the unit circle we provide formulas for the densities of the absolutely continuous parts of measures in these semigroups the descriptions of the densities rely on the characterizations of the images of the upper half plane and the unit disc under certain subordination functions these subordination functions are eta transforms of infinitely divisible measures with respect to free multiplicative convolution the characterizations also help us study the regularity properties of these measures one of the main results is that the number of components in the support of measures in the semigroups is a decreasing function of the semigroup parameter

the central limit theorem for free additive convolution abstract let xn n 1be a sequence of free identically distributed random variables with common distribution then there exist sequences bn n 1and an n 1of positive and real numbers respectively such that sequence of random variables formula converges in distribution to the semicircle law if and only if the function formula is slowly varying in karamata s sense in other words the free domain of attraction of the semicircle law coincides with the classical domain of attraction of the gaussian we prove an analogous result for normal domains of attraction in the sense of linnik

limit theorems for free multiplicative convolutions we determine the distributional behavior for products of free random variables in a general infinitesimal triangular array the main theorems in this paper extend a result for measures supported on the positive half line and provide a new limit theorem for measures on the unit circle with nonzero first moment

convergence rates for weighted sums in noncommutative probability space abstract we study convergence rates for weighted sums of pairwise independent random variables in a noncommutative probability space of which the weights are in a von neumann algebra as applications we first study convergence rates for weighted sums of random variables in the noncommutative lorentz space and second we study convergence rates for weighted sums of probability measures with respect to the free additive convolution

fuzzy disk for covering fuzzy points abstract in this paper we consider an important fuzzy version of the well known smallest covering circle problem which is also called the euclidean 1 center problem here we are given a set of points in the plane whose locations are crisp however the requirement for coverage of each point is fuzzy as represented by its grade of membership as such we qualify the points as fuzzy in the above framework we wish to find a fuzzy disk with minimum fuzzy area for covering the given fuzzy points after developing a general model certain special cases are investigated in detail polynomial algorithms are presented for the special cases

on the law of addition of random matrices normalized eigenvalue counting measure of the sum of two hermitian or real symmetric matrices a n and b n rotated independently with respect to each other by the random unitary or orthogonal haar distributed matrix u n i e a n u n b n u n is studied in the limit of large matrix order n convergence in probability to a limiting nonrandom measure is established a functional equation for the stieltjes transform of the limiting measure in terms of limiting eigenvalue measures of a n and b n is obtained and studied

decomposable multi parameter matroid optimization problems a framework for solving certain multi dimensional parametric matroid optimization problems in randomized linear time by prune and search is presented the common feature of these problems which include the multi parameter minimum spanning tree problem on planar and dense graphs is that their fixed parameter versions are solvable by tournament like algorithms whose structure is represented by a balanced decomposition tree

k center and k median problems in graded distances graded distances are generalizations of the euclidean distance on points in r1 they have been used in the study of special cases of np hard problems in this paper we study the k center and k median problems with graded distance matrices we first prove that the k center problem is polynomial time solvable when the distance matrix is graded up the rows or graded down the rows we then prove that the k median problem is np complete when the distance matrix is graded up the rows or graded down the rows an easy special case of the k median problem with graded distance matrices is also discussed

kostant polynomials and the cohomology ring for g b the schubert calculus for g b can be completely determined by a certain matrix related to the kostant polynomials introduced in section 5 of bernstein gelfand and gelfand bernstein i gelfand i gelfand s 1973 russ math surv 28 1 26 the polynomials are defined by vanishing properties on the orbit of a regular point under the action of the weyl group for each element w in the weyl group the polynomials also have nonzero values on the orbit points corresponding to elements which are larger in the bruhat order than w the main theorem given here is an explicit formula for these values the matrix of orbit values can be used to determine the cup product for the cohomology ring for g b using only linear algebra or as described by lascoux and schutzenberger lascoux a schutzenberger m p 1982 c r seances acad sci ser a 294 447 450 complete proofs of all the theorems will appear in a forthcoming paper

rate of convergence to the semi circular law a stochastic bound of order op n 1 2 for the kolmogorov distance between the spectral distribution function of an n n matrix from wigner ensemble and the distribution function of the semi circular law is obtained the result holds assuming that the twelfth moment of the entries of the matrix is uniformly bounded

limit theorems in bi free probability theory in this paper additive bi free convolution is defined for general borel probability measures and the limiting distributions for sums of bi free pairs of self adjoint commuting random variables in an infinitesimal triangular array are determined these distributions are characterized by their bi freely infinite divisibility and moreover a transfer principle is established for limit theorems in classical probability theory and voiculescu s bi free probability theory complete descriptions of bi free stability are given and fullness of planar probability distributions is studied all these results reveal one important feature about the theory of bi free probability that it parallels the classical theory perfectly well the emphasis in the whole work is not on the tool of bi free combinatorics but only on the analytic machinery

the analogues of entropy and of fisher s information measure in free probability theory i analogues of the entropy and fisher information measure for random variables in the context of free probability theory are introduced monotonicity properties and an analogue of the cramer rao inequality are proved

schubert polynomials for the classical groups the task of a theory of schubert polynomials is to produce explicit representatives for schubert classes in the cohomology ring of a flag variety and to do so in a manner that is as natural as possible from a combinatorial point of view to explain more fully let us review a special case the schubert calculus for grassmannians where one asks for the number of linear spaces of given dimension satisfying certain geometric conditions a typical problem is to find the number of lines meeting four given lines in general position in 3 space answer below for each of the four given lines the set of lines meeting it is a schubert variety in the grassmannian and we want the number of intersection points of these four subvarieties in the modem solution of this problem the schubert varieties induce canonical elements of the cohomology ring of the grassmannian called schubert classes the product of these schubert classes is the class of a point times the number of intersection points counted with appropriate multiplicities this reformulation of the problem though one of the great achievements of algebraic geometry is only part of a solution it remains to give a concrete model for the cohomology ring that makes explicit computation with schubert classes possible as it happens the cohomology rings of grassmannians can be identified with quotients of a polynomial ring so that schubert classes correspond to schur functions intersection numbers such as we are considering then turn out to be littlewood richardson coefficients for example the answer to our four lines 4 problem is the coefficient of the schur function s 2 2 in the product s 1 x or 2 for an extended treatment and history of the subject see 10 11 18 the identification of schur functions as schubert polynomials for grassmannians is a consequence of a more general and now highly developed theory of schubert polynomials for the flag varieties of the special linear groups sl n c the starting point for this more general theory is a construction of

a subexponential algorithm for abstract optimization problems an abstract optimization problem aop is a triple h phi where h is a finite set a total order on 2 h and phi an oracle that for given f subseteq g subseteq h either reports that f min f mid f subseteq g or returns a set f subseteq g with f e 2 sqrt n o sqrt 4 n ln n oracle calls n h in contrast any deterministic algorithm needs to make 2 n 1 oracle calls in the worst case the algorithm is applied to the problem of finding the distance between two n vertex or n facet convex polyhedra in d space and to the computation of the smallest ball containing n points in d space for both problems we give the first subexponential bounds in the arithmetic model of computation

strongly polynomial time and nc algorithms for detecting cycles in periodic graphs this paper is concerned with the problem of recognizing in a graph with rational vector weights associates with the edges the existence of a cycle whose total weight is the zero vector this problem is known to be equivalent to the problem of recognizing the existence of cycles in periodic dynamic graphs and to the validity of systems of recursive formulas it was previously conjectured that combinatorial algorithms exist for the cases of two and three dimensional vector weights it is shown that strongly polynomial algorithms exist for any fixed dimension d moreover these algorithms also establish membership in the class nc on the other hand it is shown that when the dimension of the weights is not fixed the problem is equivalent to the general linear programming problem under strongly polynomial and logspace reductions

levy processes in free probability this is the continuation of a previous article that studied the relationship between the classes of infinitely divisible probability measures in classical and free probability respectively via the bercovici pata bijection drawing on the results of the preceding article the present paper outlines recent developments in the theory of levy processes in free probability

the law of large numbers for free identically distributed random variables let x 1 x 2 be a sequence of free identically distributed random variables with common distribution it was shown by lindsay and pata in a more general context that a sufficient condition for the weak law of large numbers to hold for the sequence x 1 x2 is that formula math we show that this condition is necessary as well as sufficient even though the condition is identical with the corresponding one for commuting independent variables the proof of the result uses the analytical techniques of free convolution theory and it is quite different from the proof of the commutative theorem due to kolmogorov cf feller 1971 an introduction to probability theory and its applications wiley new york

one dimensional k center on uncertain data problems on uncertain data have attracted significant attention due to the imprecise nature of many measurement data in this paper we consider the k center problem on one dimensional uncertain data the input is a set p of weighted uncertain points on a real line and each uncertain point is specified by its probability density function pdf which is a piecewise uniform function i e a histogram the goal is to find a set q of k points on the line to minimize the maximum expected distance from the uncertain points of p to their expected closest points in q we present efficient algorithms for this uncertain k center problem and their running times almost match those for the deterministic k center problem

limit theorems in free probability theory i based on an analytical approach to the definition of multiplicative free convolution on probability measures on the nonnegative line and on the unit circle mathbb t we prove analogs of limit theorems for nonidentically distributed random variables in classical probability theory

on a multidimensional search technique and its application to the euclidean one centre problem the paper is divided into two main sections the first deals with a multidimensional search technique of megiddo j assoc comput mach 31 1984 pp 114 127 and suggests an improvement the second gives an application of the technique to the euclidean one centre problem in mathbb r d an algorithm of time complexity o 3 d 2 2 n is derived for this problem this improves the best previous bound even in the case d 2

two state free brownian motions in a two state free probability space a we define an algebraic two state free brownian motion to be a process with two state freely independent increments whose two state free cumulant generating function r z is quadratic note that a priori the distribution of the process with respect to the second state is arbitrary we show however that if a is a von neumann algebra the states are normal and is faithful then there is only a one parameter family of such processes moreover with the exception of the actual free brownian motion corresponding to these processes only exist for finite time