non deterministic polynomial optimization problems and their approximations abstract a unified and general framework for the study of nondeterministic polynomial optimization problems npop is presented and some properties of npop s are investigated a characterization of npop s with regard to their approximability properties is given by proving necessary and sufficient conditions for two approximability schemes known approximability results are shown to fit within the general frame developed in the paper finally npop s are classified and studied with regard to the possibility or impossibility of reducing certain types of npop s to other types in a sense specified in the text

covariance regularization by thresholding this paper considers regularizing a covariance matrix of p variables estimated from n observations by hard thresholding we show that the thresholded estimate is consistent in the operator norm as long as the true covariance matrix is sparse in a suitable sense the variables are gaussian or sub gaussian and log p n 0 and obtain explicit rates the results are uniform over families of covariance matrices which satisfy a fairly natural notion of sparsity we discuss an intuitive resampling scheme for threshold selection and prove a general cross validation result that justifies this approach we also compare thresholding to other covariance estimators in simulations and on an example from climate data 1 introduction estimation of covariance matrices is important in a number of areas of statistical analysis including dimension reduction by principal component analysis pca classification by linear or quadratic discriminant analysis lda and qda establishing independence and conditional independence relations in the context of graphical models and setting confidence intervals on linear functions of the means of the components in recent years many application areas where these tools are used have been dealing with very high dimensional datasets and sample sizes can be very small relative to dimension examples include genetic data brain imaging spectroscopic imaging climate data and many others it is well known by now that the empirical covariance matrix for samples of size n from a p variate gaussian distribution np p is not a good estimator of the population covariance if p is large many results in random matrix theory illustrate this from the classical mary law 29 to the more recent work of johnstone and his students on the theory of the largest eigenvalues 12 23 30 and associated eigenvectors 24 however with the exception of a method for estimating the covariance spectrum 11 these probabilistic results do not offer alternatives to the sample covariance matrix alternative estimators for large covariance matrices have therefore attracted a lot of attention recently two broad classes of covariance estimators have emerged those that rely on a natural ordering among variables and assume that variables

proof of a conjecture concerning the hadamard powers of inverse m matrices abstract the main result of this paper is the following if a aij is an inverse m matrix a r a ij r denotes the rth hadamard power of a then a r is again an inverse m matrix for any real number r 1 this settles a conjecture proposed by wang et al b y wang x p zhang f z zhang on the hadamard product of inverse m matrices linear algebra appl 305 2000 23 31 affirmatively naturally it shows that the conjecture of neumann m neumann a conjecture concerning the hadamard product of inverses of m matrices linear algebra appl 285 1998 277 290 is also valid

a conjecture concerning the hadamard product of inverses of m matrices abstract we conjecture that for an n n matrix a which is an inverse of an m matrix the hadamard product a a is also an inverse of an m matrix we have checked this conjecture without failure on many many examples but here we show that for quite a few well known classes of inverses of m matrices the conjecture is true it is known that the more general conjecture that when a and b are n n inverses of m matrices then a b is also an inverse of an m matrix is false however here too we are able to display some classes of inverses of m matrices which are closed under taking hadamard products

temps de vie des solutions regulieres des equations d euler compressibles axisymetriques en dimension deux nous considerons le systeme des equations d euler isentropiques en dimension deux pour des donnees initiales invariantes par rotation et perturbations de taille e d un etat de repos on etablit un equivalent du temps de viet e de la solution classique lim 2 t 2 de plus on donne pour t leqq frac a 2 varepsilon 2 a tau une estimation de la vraie solution en calculant la taille de son ecart a une solution approchee construite dans un precedent travail

matrix positivity preservers in fixed dimension i a classical theorem of i j schoenberg characterizes functions that preserve positivity when applied entrywise to positive semidefinite matrices of arbitrary size obtaining similar characterizations in fixed dimension is intricate in this note we provide a solution to this problem in the polynomial case as consequences we derive tight linear matrix inequalities for hadamard powers of positive semidefinite matrices and a sharp asymptotic bound for the matrix cube problem involving hadamard powers

controllability gramian submatrices for a network consensus model abstract principal submatrices of the controllability gramian and their inverses are examined for a network consensus model with inputs at a subset of network nodes several properties of the gramian submatrices and their inverses including dominant eigenvalues and eigenvectors diagonal entries and sign patterns are characterized by exploiting the special doubly nonnegative structure of the matrices in addition majorizations for these properties are obtained in terms of cutsets in the network s graph based on the diffusive form of the model the asymptotic long time horizon structure of the controllability gramian is also analyzed the results on the gramian are used to study metrics for target control of the network consensus model

hub discovery in partial correlation graphs one of the most important problems in large scale inference problems is the identification of variables that are highly dependent on several other variables when dependence is measured by partial correlations these variables identify those rows of the partial correlation matrix that have several entries with large magnitudes i e hubs in the associated partial correlation graph this paper develops theory and algorithms for discovering such hubs from a few observations of these variables we introduce a hub screening framework in which the user specifies both a minimum partial correlation rho and a minimum degree delta to screen the vertices the choice of rho and delta can be guided by our mathematical expressions for the phase transition correlation threshold rho c governing the average number of discoveries they can also be guided by our asymptotic expressions for familywise discovery rates under the assumption of large number p of variables fixed number n of multivariate samples and weak dependence under the null hypothesis that the dispersion covariance matrix is sparse these limiting expressions can be used to enforce familywise error constraints and to rank the discoveries in order of increasing statistical significance for n ll p the computational complexity of the proposed partial correlation screening method is low and is therefore highly scalable thus it can be applied to significantly larger problems than previous approaches the theory is applied to discovering hubs in a high dimensional gene microarray dataset

a threshold of ln n for approximating set cover given a collection f of subsets of s 1 n setcover is the problem of selecting as few as possiblesubsets from f such that their union covers s and maxk cover is the problem of selecting k subsets from f such that their union has maximum cardinality both these problems arenp hard we prove that 1 o 1 ln n is a threshold below which setcover cannot be approximated efficiently unless np has slightlysuperpolynomial time algorithms this closes the gap up to low orderterms between the ratio of approximation achievable by the greedyalogorithm which is 1 o 1 lnn and provious results of lund and yanakakis that showed hardness ofapproximation within a ratio of log 2 n 2s0 72 ln n for max k cover we show an approximationthreshold of 1 1 e up tolow order terms under assumption that p np

vulnerability of network synchronization processes a minimum energy perspective we study the vulnerability of a linear network synchronization process to intrusion at a single network component from a graph theoretic perspective specifically we model the intruder as seeking to move the state of the synchronization process to an undesirable value or set via a local actuation we measure the network vulnerability in terms of the minimum or expected minimum actuation energy required of the adversary to achieve the goal which is related to the inverse of the controllability gramian for the process and statistics defined thereof e g its trace and determinant using explicit formulas for the gramian inverse and its associated statistics together with algebraic graph theory concepts we then develop structural and graph theoretic characterizations of the energy based vulnerability measures

the critical exponent conjecture for powers of doubly nonnegative matrices abstract doubly nonnegative matrices arise naturally in many setting including markov random fields positively banded graphical models and in the convergence analysis of markov chains in this short note we settle a recent conjecture by c r johnson et al charles r johnson brian lins olivia walch the critical exponent for continuous conventional powers of doubly nonnegative matrices linear algebra appl 435 9 2011 2175 2182 by proving that the critical exponent beyond which all continuous conventional powers of n by n doubly nonnegative matrices are doubly nonnegative is exactly n 2 we show that the conjecture follows immediately by applying a general characterization from the literature we prove a stronger form of the conjecture by classifying all powers preserving doubly nonnegative matrices and proceed to generalize the conjecture for broad classes of functions we also provide different approaches for settling the original conjecture

probabilistic checking of proofs a new characterization of np we give a new characterization of np the class np contains exactly those languages l for which membership proofs a proof that an input x is in l can be verified probabilistically in polynomial time using logarithmic number of random bits and by reading sublogarithmic number of bits from the proof we discuss implications of this characterization specifically we show that approximating clique and independent set even in a very weak sense is np hard

sharp bounds on davenport schinzel sequences of every order one of the longest standing open problems in computational geometry is bounding the complexity of the lower envelope of n univariate functions each pair of which crosses at most s times for some fixed s this problem is known to be equivalent to bounding the length of an order s davenport schinzel sequence namely a sequence over an n letter alphabet that avoids alternating subsequences of the form a b a b with length s 2 these sequences were introduced by davenport and schinzel in 1965 to model a certain problem in differential equations and have since been applied to bound the running times of geometric algorithms data structures and the combinatorial complexity of geometric arrangements let s n be the maximum length of an order s ds sequence over n letters what is s asymptotically this question has been answered satisfactorily hart and sharir 1986 agarwal et al 1989 klazar 1999 nivasch 2010 when s is even or s 3 however since the work of agarwal et al in the mid 1980s there has been a persistent gap in our understanding of the odd orders in this work we effectively close the problem by establishing sharp bounds on davenport schinzel sequences of every order s our results reveal that contrary to one s intuition s n behaves essentially like s 1 n when s is odd this refutes conjectures by alon et al 2008 and nivasch 2010

a best possible heuristic for the k center problem in this paper we present a 2 approximation algorithm for the k center problem with triangle inequality this result is best possible since for any 2 the existence of approximation algorithm would imply that p np it should be noted that no approximation algorithm for any constant has been reported to date linear programming duality theory provides interesting insight to the problem and enables us to derive in o e log e time a solution with value no more than twice the k center optimal value a by product of the analysis is an o e algorithm that identifies a dominating set in g2 the square of a graph g the size of which is no larger than the size of the minimum dominating set in the graph g the key combinatorial object used is called a strong stable set and we prove the np completeness of the corresponding decision problem

generalized strichartz inequalities for the wave equation strichartz inequalities for the wave equation u f are estimates of the solution u of the cauchy problem for that equation in the form of space time integral norms in terms of similar norms of the inhomogeneity f and of suitable norms of the initial data

on the hardness of approximating minimization problems we prove results indicating that it is hard to compute efficiently good approximate solutions to the graph coloring set covering and other related minimization problems specifically there is an e 0 such that graph coloring cannot be approximated with ratio n e unless p np set covering cannot be approximated with ratio c log n for any c 1 4 unless np is contained in dtime n poly log n similar results follow for related problems such as clique cover fractional chromatic number dominating set and others

a matrix subadditivity inequality for f a b and f a f b abstract in 1999 ando and zhan proved a subadditivity inequality for operator concave functions we extend it to all concave functions given positive semidefinite matrices a b and a non negative concave function f on 0 f a b f a f b for all symmetric norms in particular for all schatten p norms the case f t t is connected to some block matrix inequalities for instance the operator norm inequality a x x b max x b x for any partitioned hermitian matrix

improved bounds for the union of locally fat objects in the plane we show that for any gamma 0 the combinatorial complexity of the union of n locally gamma fat objects of constant complexity in the plane is frac n gamma 4 2 o log n for the special case of gamma fat triangles the bound improves to o n log n frac n gamma log 2 frac 1 gamma

inverse m matrices ii this is an update of the 1981 survey by the first author in the meantime a considerable amount has been learned about the very special structure of the important class of inverse m matrices developments since the earlier survey are emphasized but we have tried to be somewhat complete and some results have not previously been published some proofs are given where appropriate and references are given for others after some elementary preliminaries results are grouped by certain natural categories

monotonicity for entrywise functions of matrices abstract we characterize real functions f on an interval for which the entrywise matrix function a ij f a ij is positive monotone and convex respectively in the positive semidefiniteness order fractional power functions are exemplified and related weak majorizations are shown

on the 2d isentropic euler system with unbounded initial vorticity abstract this paper is devoted to the study of the low mach number limit for the 2d isentropic euler system associated to ill prepared initial data with slow blow up rate on log e 1 we prove in particular the strong convergence to the solution of the incompressible euler system when the vorticity belongs to some weighted bmo spaces allowing unbounded functions the proof is based on the extension of the result of 6 to a compressible transport model

generalizations of m matrices which may not have a nonnegative inverse abstract generalizations of m matrices are studied including the new class of gm matrices the matrices studied are of the form si b with b having the perron frobenius property but not necessarily being nonnegative results for these classes of matrices are shown which are analogous to those known for m matrices also various splittings of a gm matrix are studied along with conditions for their convergence

complete characterization of hadamard powers preserving loewner positivity monotonicity and convexity abstract entrywise powers of symmetric matrices preserving positivity monotonicity or convexity with respect to the loewner ordering arise in various applications and have received much attention recently in the literature following fitzgerald and horn 1977 8 it is well known that there exists a critical exponent beyond which all entrywise powers preserve positive definiteness similar phenomena have also recently been shown by hiai 2009 to occur for monotonicity and convexity in this paper we complete the characterization of all the entrywise powers below and above the critical exponents that are positive monotone or convex on the cone of positive semidefinite matrices we then extend the original problem by fully classifying the positive monotone or convex powers in a more general setting where additional rank constraints are imposed on the matrices we also classify the entrywise powers that are super sub additive with respect to the loewner ordering finally we extend all the previous characterizations to matrices with negative entries our analysis consequently allows us to answer a question raised by bhatia and elsner 2007 regarding the smallest dimension for which even extensions of the power functions do not preserve loewner positivity

retaining positive definiteness in thresholded matrices abstract positive definite p d matrices arise naturally in many areas within mathematics and also feature extensively in scientific applications in modern high dimensional applications a common approach to finding sparse positive definite matrices is to threshold their small off diagonal elements this thresholding sometimes referred to as hard thresholding sets small elements to zero thresholding has the attractive property that the resulting matrices are sparse and are thus easier to interpret and work with in many applications it is often required and thus implicitly assumed that thresholded matrices retain positive definiteness in this paper we formally investigate the algebraic properties of p d matrices which are thresholded we demonstrate that for positive definiteness to be preserved the pattern of elements to be set to zero has to necessarily correspond to a graph which is a union of complete components this result rigorously demonstrates that except in special cases positive definiteness can be easily lost we then proceed to demonstrate that the class of diagonally dominant matrices is not maximal in terms of retaining positive definiteness when thresholded consequently we derive characterizations of matrices which retain positive definiteness when thresholded with respect to important classes of graphs in particular we demonstrate that retaining positive definiteness upon thresholding is governed by complex algebraic conditions

a property concerning the hadamard powers of inverse m matrices abstract the main result of this paper is the following if a a ij is an inverse m matrix a k a ij k denotes the k th hadamard power of a then a k is an inverse m matrix for any positive integer k 1 in the case k 2 this settles a conjecture of neumann linear algebra appl 285 1998 278 affirmatively

a framework for exponential time hypothesis tight algorithms and lower bounds in geometric intersection graphs we give an algorithmic and lower bound framework that facilitates the construction of subexponential algorithms and matching conditional complexity bounds it can be applied to intersection graphs

norm inequalities related to operator monotone functions let a b be positive semidefinite matrices and cdot any unitarily invariant norm on the space of matrices we show f a f b geq f a b for any non negative operator monotone function f t on 0 infty and g a g b leq g a b for non negative increasing function g t on 0 infty with g 0 0 and g infty infty whose inverse function is operator monotone

regular articlegeneralized strichartz inequalities for the wave equation we make a synthetic exposition of the generalized strichartz inequalities for the wave equation obtained in 6 together with the limiting cases recently obtained in 13 with as simple proofs as possible the proofs combine stationary phase estimates dyadic decompositions the hardy littlewood inequality or the holder inequality in time and abstract duality and interpolation arguments

a distance based approach to strong target control of dynamical networks this paper deals with controllability of dynamical networks it is often unfeasible or unnecessary to fully control large scale networks which motivates the control of a prescribed subset of agents of the network this specific form of output control is known under the name target control we consider target control of a family of linear control systems associated with a network and provide both a necessary and a sufficient topological condition under which the network is strongly targeted controllable furthermore a leader selection algorithm is presented to compute leader sets achieving target control

a survey of model reduction by balanced truncation and some new results balanced truncation is one of the most common model reduction schemes in this note we present a survey of balancing related model reduction methods and their corresponding error norms and also introduce some new results five balancing methods are studied 1 lyapunov balancing 2 stochastic balancing 3 bounded real balancing 4 positive real balancing and 5 frequency weighted balancing for positive real balancing we introduce a multiplicative type error bound moreover for a certain subclass of positive real systems a modified positive real balancing scheme with an absolute error bound is proposed we also develop a new frequency weighted balanced reduction method with a simple bound on the error system based on the frequency domain representations of the system gramians two numerical examples are illustrated to verify the efficiency of the proposed methods

hadamard powers of some positive matrices abstract positivity properties of the hadamard powers of the matrix 1 x i x j for distinct positive real numbers x 1 x n and the matrix cos i j n are studied in particular it is shown that the n n matrix 1 x i x j r is positive semidefinite if and only if r is a nonnegative integer or r n 2 and for every odd integer n 3 the n n matrix cos i j n r is positive semidefinite if and only if r is a nonnegative even integer or r n 3

on the hadamard product of inverse m matrices abstract we investigate the hadamard product of inverse m matrices and present two classes of inverse m matrices that are closed under the hadamard multiplication in the end we give some inequalities on the fan product of m matrices and schur complements

formation of singularities in three dimensional compressible fluids presented are several results on the formation of singularities in solutions to the three dimensional euler equations for a polytropic ideal fluid under various assumptions on the initial data in particular it is shown that a localized fluid which is initially compressed and outgoing on average will develop singularities regardless of the size of the initial disturbance

singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids many interesting problems in classical physics involve the limiting behavior of quasilinear hyperbolic systems as certain coefficients become infinite using classical methods the authors develop a general theory of such problems this theory is broad enough to study a wide variety of interesting singular limits in compressible fluid flow and magneto fluid dynamics including new constructive local existence theorems for the time singular limit equations in particular the authors give an entirely self contained classical proof of the convergence of solutions of the compressible fluid equations to their incompressible limits as the mach number becomes small the theory depends upon a balance between certain inherently nonlinear structural conditions on the matrix coefficients of the system together with appropriate initialization procedures similar results are developed also for the compressible and incompressible navier stokes equations with periodic initial data independent of the viscosity coefficients as they tend to zero

small size eps nets for axis parallel rectangles and boxes we show the existence of varepsilon nets of size o left frac 1 varepsilon log log frac 1 varepsilon right for planar point sets and axis parallel rectangular ranges the same bound holds for points in the plane and fat triangular ranges and for point sets in boldsymbol r 3 and axis parallel boxes these are the first known nontrivial bounds for these range spaces our technique also yields improved bounds on the size of varepsilon nets in the more general context considered by clarkson and varadarajan for example we show the existence of varepsilon nets of size o left frac 1 varepsilon log log log frac 1 varepsilon right for the dual range space of fat regions and planar point sets where the regions are the ground objects and the ranges are subsets stabbed by points plugging our bounds into the technique of bronnimann and goodrich or of even rawitz and shahar we obtain improved approximation factors computable in expected polynomial time by a randomized algorithm for the hitting set or the set cover problems associated with the corresponding range spaces

geometric packing under nonuniform constraints we study the problem of discrete geometric packing here given weighted regions say in the plane and points with capacities one has to pick a maximum weight subset of the regions such that no point is covered more than its capacity we provide a general framework and an algorithm for approximating the optimal solution for packing in hypergraphs arising out of such geometric settings using this framework we get a flotilla of results on this problem and also on its dual where one wants to pick a maximum weight subset of the points when the regions have capacities for example for the case of fat triangles of similar size we show an o 1 approximation and prove that no ptas is possible

inverse m matrices abstract this is an attempt at a comprehensive expository study of those nonnegative matrices which happen to be inverses of m matrices and is aimed at an audience conversant with basic ideas of matrix theory a theme is the parallels anddifferences between the class of m matrices and the class of inverse m matrices among the primary tools used are diagonal multiplications the neumann expansion and the form of the inverse of a partitioned matrix some of the results seem not to have appeared before and several unsolved problems are mentioned

on the union of fat tetrahedra in three dimensions we show that the combinatorial complexity of the union of n fat tetrahedra in 3 space i e tetrahedra all of whose solid angles are at least some fixed constant of arbitrary sizes is o n2 e for any e 0 the bound is almost tight in the worst case thus almost settling a conjecture of pach et al 2003 our result extends in a significant way the result of pach et al 2003 for the restricted case of nearly congruent cubes the analysis uses cuttings combined with the dobkin kirkpatrick hierarchical decomposition of convex polytopes in order to partition space into subcells so that on average the overwhelming majority of the tetrahedra intersecting a subcell behave as fat dihedral wedges in as an immediate corollary we obtain that the combinatorial complexity of the union of n cubes in r3 having arbitrary side lengths is o n2 e for any e 0 again significantly extending the result of pach et al 2003 finally our analysis can easily be extended to yield a nearly quadratic bound on the complexity of the union of arbitrarily oriented fat triangular prisms whose cross sections have arbitrary sizes in r3

the critical exponent for continuous conventional powers of doubly nonnegative matrices abstract we prove that there exists an exponent beyond which all continuous conventional powers of n by n doubly nonnegative matrices are doubly nonnegative we show that this critical exponent cannot be less than n 2 and we conjecture that it is always n 2 as it is with hadamard powering we prove this conjecture when n 6 and in certain other special cases we establish a quadratic bound for the critical exponent in general

the incompressible limit of the non isentropic euler equations we study the euler equations for slightly compressible fluids that is after rescaling the limits of the euler equations of fluid dynamics as the mach number tends to zero in this paper we consider the general non isentropic equations and general data we first prove the existence of classical solutions for a time independent of the small parameter then on the whole space d we prove that the solution converges to the solution of the incompressible euler equations

almost principal minors of inverse m matrices abstract it is well known that if an inverse m matrix has a 0 entry then it must be reducible and thus have many more 0 entries this property is actually a special case of a deeper phenomenon that might be loosely described as relations among vanishing almost principal minors in an inverse m matrix this phenomenon encompasses both minors of nested dimension a certain loose monotonicity and minors of the same size in loosely related positions this phenomenon is limited to almost principal minors and where possible converses and examples are given to show the limit of the extent of this phenomenon it is also shown that if one almost principle minor is contained in another then the magnitude of the former is larger than that of the latter

tight lower bounds for the size of epsilon nets reference epfl article 190901view record in web of science record created on 2013 12 09 modified on 2017 05 12

large scale correlation screening this article addresses the problem of screening for variables with high correlations in high dimensional data in which there can be many fewer samples than variables we focus on threshold based correlation screening methods for three related applications screening for variables with large correlations within a single treatment autocorrelation screening screening for variables with large cross correlations over two treatments cross correlation screening and screening for variables that have persistently large autocorrelations over two treatments persistent correlation screening the novelty of correlation screening is that it identifies a smaller number of variables that are highly correlated with others compared with identifying a number of correlation parameters correlation screening suffers from a phase transition phenomenon as the correlation threshold decreases the number of discoveries increases abruptly we obtain asymptotic expressions for the mean number of discoveries and the phase t

the complexity of the union of alpha beta covered objects an alpha beta covered object is a simply connected planar region c with the property that for each point p in partial c there exists a triangle contained in c and having p as a vertex such that all its angles are at least alpha 0 and all its edges are at least beta cdot rm diam c long this notion extends that of fat convex objects we show that the combinatorial complexity of the union of n alpha beta covered objects of constant description complexity is o lambda s 2 n log 2n log log n where s is the maximum number of intersections between the boundaries of any pair of given objects and lambda s n denotes the maximum length of an n s davenport schinzel sequence our result extends and improves previous results concerning convex alpha fat objects

potentials of random walks on trees abstract in this article we characterize inverse m matrices and potentials whose inverses are supported on trees in the symmetric case we show they are a hadamard product of tree ultrametric matrices generalizing a result by gantmacher and krein 12 done for inverse tridiagonal matrices we also provide an algorithm that recognizes when a positive matrix w has an inverse m matrix supported on a tree this algorithm has quadratic complexity we also provide a formula to compute w 1 which can be implemented with a linear complexity finally we also study some stability properties for hadamard products and powers

semialgebraic range reporting and emptiness searching with applications in a typical range emptiness searching resp reporting problem we are given a set p of n points in mathbb r d and we wish to preprocess it into a data structure that supports efficient range emptiness resp reporting queries in which we specify a range sigma which in general is a semialgebraic set in mathbb r d of constant description complexity and we wish to determine whether p cap sigma emptyset or to report all the points in p cap sigma range emptiness searching and reporting arise in many applications and have been treated by matousek comput geom theory appl 2 1992 pp 169 186 in the special case where the ranges are half spaces bounded by hyperplanes as shown in matousek s work the two problems are closely related and they have solutions for the case of half spaces with similar performance bounds in this paper we extend the analysis to general semialgebraic ranges and show how to adapt matousek s technique without the need to linearize the ranges into a higher dimensional space this yields more efficient solutions to several useful problems and we demonstrate the new technique in four applications with the following results i an algorithm for ray shooting amid balls in mathbb r 3 which uses o n storage and o n preprocessing we use the notation o n gamma to mean an upper bound of the form c varepsilon n gamma varepsilon which holds for any varepsilon 0 where c varepsilon is a constant that depends on varepsilon and answers a query in o n 2 3 time improving the previous bound of o n 3 4 ii an algorithm that preprocesses in o n time a set p of n points in mathbb r 3 into a data structure with o n storage so that for any query line ell or for that matter any simply shaped convex set the point of p farthest from ell can be computed in o n 1 2 time this in turn yields an algorithm that computes the largest area triangle spanned by p in time o n 26 11 as well as nontrivial algorithms for computing the largest perimeter or largest height triangle spanned by p iii an algorithm that preprocesses in o n time a set p of n points in mathbb r 2 into a data structure with o n storage so that for any query alpha fat triangle delta we can determine in o 1 time whether delta cap p is empty alternatively we can report in o 1 o k time the points of delta cap p where k delta cap p iv an algorithm that preprocesses in o n time a set p of n points in mathbb r 2 into a data structure with o n storage so that given any query semidisk c or a circular cap larger than a semidisk we can determine in o 1 time whether c cap p is empty or report the k points in c cap p in o 1 o k time adapting the recent techniques of b aronov and s har peled siam j comput 38 2008 pp 899 921 b aronov s har peled and m sharir on approximate halfspace range counting and relative epsilon approximations in proceedings of the 23rd acm symposium comput geom 2007 pp 327 336 b aronov and m sharir siam j comput 39 2010 pp 2704 2725 we can turn our solutions into efficient algorithms for approximate range counting with small relative error for the cases mentioned above our technique is closely related to the notions of nearest or farthest neighbor generalized voronoi diagrams and of the union or intersection of geometric objects where sharper bounds on the combinatorial complexity of decompositions of complements of these structures yield faster range emptiness searching or reporting algorithms

computing coverage kernels under restricted settings abstract given a set b of d dimensional boxes i e axis aligned hyperrectangles a minimum coverage kernel is a subset of b of minimum size covering the same region as b computing it is np hard but as for many similar np hard problems e g box cover and orthogonal polygon covering the problem becomes solvable in polynomial time under restrictions on b we show that computing minimum coverage kernels remains np hard even when restricting the graph induced by the input to a highly constrained class of graphs alternatively we present two polynomial time approximation algorithms for this problem one deterministic with an approximation ratio within o log n and one randomized with an improved approximation ratio within o lg opt with high probability

the incompressible limit of solutions of the two dimensional compressible euler system with degenerating initial data using strichartz estimates it is possible to pass to the limit in the weakly compressible 2 d euler system when the mach number tends to zero even if the initial data are not uniformly smooth more precisely their norms in sobolev spaces embedded in c1 can be allowed to grow as small powers of 1 this leads to results of convergence to solutions of the incompressible euler system whose regularity is critical such as vortex patches or yudovich solutions 2000 wiley periodicals inc

on the power of multi prover interactive protocols abstract we look at complexity issues of interactive proof systems with multiple provers separated from each other this model developed by ben or et al 1988 allows the verifier to play the provers off each other we show this model equivalent to an alternative interactive proof system model using oracles as provers we also show that every language accepted by these models lies in nondeterministic exponential time we exhibit a relativized world where a co np language does not have multiple prover interactive proofs finally we show a simple example that one cannot parallelize multiple prover protocols as easily as the single prover model

protocols design and uncontrollable topologies construction for multi agent networks this note investigates the controllability issues of multi agent systems where each node contains generic linear dynamics first a neighbor based control protocol is proposed under which it is shown that the controllability of a multi agent system is solely decided by its communication topology structure we then further consider the effects of communication topology on the controllability from a graph theory point of view by explicitly constructing topologies the construction exhibits a partition of the designed graph with identified leaders under which the system is shown to be uncontrollable

fat triangles determine linearly many holes the authors show that for every fixed delta 0 the following holds if f is a union of n triangles all of whose angles are at least delta then the complement of f has o n connected components and the boundary of f consists of o n log log n straight segments where the constants of proportionality depend on delta this latter complexity becomes linear if all triangles are of roughly the same size or if they are all infinite wedges