optimality of the triangular lattice for a particle system with wasserstein interaction we prove strong crystallization results in two dimensions for an energy that arises in the theory of block copolymers the energy is defined on sets of points and their weights or equivalently on the set of atomic measures it consists of two terms the first term is the sum of the square root of the weights and the second is the quadratic optimal transport cost between the atomic measure and the lebesgue measure

the γ limit of the two dimensional ohta kawasaki energy droplet arrangement via the renormalized energy this is the second in a series of papers in which we derive a expansion for the two dimensional non local ginzburg landau energy with coulomb repulsion known as the ohta kawasaki model in connection with diblock copolymer systems in this model two phases appear which interact via a nonlocal coulomb type energy here we focus on the sharp interface version of this energy in the regime where one of the phases has very small volume fraction thus creating small droplets of the minority phase in a sea of the majority phase in our previous paper we computed the limit of the leading order energy which yields the averaged behavior for almost minimizers namely that the density of droplets should be uniform here we go to the next order and derive a next order limit energy which is exactly the coulombian renormalized energy obtained by sandier and serfaty as a limiting interaction energy for vortices in the magnetic ginzburg landau model the derivation is based on the abstract scheme of sandier serfaty that serves to obtain lower bounds for 2 scale energies and express them through some probabilities on patterns via the multiparameter ergodic theorem thus without appealing to the euler lagrange equation we establish for all configurations which have almost minimal energy the asymptotic roundness and radius of the droplets and the fact that they asymptotically shrink to points whose arrangement minimizes the renormalized energy in some averaged sense via a kind of equivalence the obtained results also yield an expansion of the minimal energy and a characterization of the zero super level sets of the minimizers for the original ohta kawasaki energy this leads to the expectation of seeing triangular lattices of droplets as energy minimizers

complexity of bezout s theorem v polynomial time we show that there are algorithms which find an approximate zero of a system of polynomial equations and which function in polynomial time on the average the number of arithmetic operations is cn 4s where n is the input size and c a universal constant

higher dimensional coulomb gases and renormalized energy functionals we consider a classical system of n charged particles in an external confining potential in any dimension d larger than 2 the particles interact via pairwise repulsive coulomb forces and the coupling parameter scales like the inverse of n mean field scaling by a suitable splitting of the hamiltonian we extract the next to leading order term in the ground state energy beyond the mean field limit we show that this next order term which characterizes the fluctuations of the system is governed by a new renormalized energy functional providing a way to compute the total coulomb energy of a jellium i e an infinite set of point charges screened by a uniform neutralizing background in any dimension the renormalization that cuts out the infinite part of the energy is achieved by smearing out the point charges at a small scale as in onsager s lemma we obtain consequences for the statistical mechanics of the coulomb gas next to leading order asymptotic expansion of the free energy or partition function characterizations of the gibbs measures estimates on the local charge fluctuations and factorization estimates for reduced densities this extends results of sandier and serfaty to dimension higher than two by an alternative approach

entry times distribution for dynamical balls on metric spaces we show that the entry and return times for dynamical balls bowen balls is exponential for systems that have an alpha mixing invariant measure with certain regularities we also show that systems modeled by young s tower has exponential entry time distribution for dynamical balls we also apply the results to conformal maps and expanding maps on the interval

local microscopic behavior for 2d coulomb gases the study of two dimensional coulomb gases lies at the interface of statistical physics and non hermitian random matrix theory in this paper we give a large deviation principle ldp for the empirical fields obtained under the canonical gibbs measure by zooming around a point in the bulk of the equilibrium measure up to the finest averaging scale n 1 2 varepsilon the rate function is given by the sum of the renormalized energy of serfaty et al weighted by the inverse temperature and of the specific relative entropy we deduce a local law which quantifies the convergence of the empirical measures of the particles to the equilibrium measure up to the finest scale

entry times distribution for mixing systems we consider the return times dynamics to bowen balls for continuous maps on metric spaces which have invariant probability measures with certain mixing properties these mixing properties are satisfied for instance by systems that allow young tower constructions we show that the higher order return times to bowen balls are in the limit poisson distributed we also provide a general result for the asymptotic behavior of the recurrence time for bowen balls for ergodic systems and those with specification

the complexity of partial derivatives abstract let l denote the nonscalar complexity in k x 1 x n we prove l x 1 x n 3l using this we determine the complexity of single power sums single elementary symmetric functions the resultant and the discriminant as root functions up to order of magnitude also we linearly reduce matrix inversion to computing the determinant

spectrum and trace invariance criterion and its statistical applications abstract necessary and sufficient conditions are established for the set of nonzero eigenvalues of the product b a to be invariant with respect to the choice of a generalized inverse b it turns out that the same conditions constitute a criterion for the invariance of trace b a the algebraic result derived is then applied to investigate some properties of canonical correlations in the context of the general gauss markov model m y x 2 v in particular the results of watson 1967 and puntanen 1987 are generalized and a new characterization of the best linear unbiased estimator of x under the model m is established

uniform energy distribution for an isoperimetric problem with long range interactions we study minimizers of a nonlocal variational problem the problem is a mathematical paradigm for the ubiquitous phenomenon of energy driven pattern formation induced by competing short and long range interactions the short range interaction is attractive and comes from an interfacial energy and the long range interaction is repulsive and comes from a nonlocal energy contribution in particular the problem is the sharp interface version of a problem used to model microphase separation of diblock copolymers a natural conjecture is that in all space dimensions minimizers are essentially periodic on an intrinsic scale however proving any periodicity result turns out to be a formidable task in dimensions larger than one in this paper we address a weaker statement concerning the distribution of energy for minimizers we prove in any space dimension that each component of the energy interfacial and nonlocal of any minimizer is uniformly distributed on cubes which are suciently large with respect to the intrinsic length scale moreover we also prove an l 1 bound on the optimal potential associated with the long range interactions this bound allows for an interesting interpretation note that the average volume fraction of the optimal pattern in a subsystem of size r fluctuates around the system average m the bound on the potential yields a rate of decay of these fluctuations as r tends to 1 this rate of decay is stronger than the one for a random checkerboard pattern in this sense the optimal pattern has less large scale variations of the average volume fraction than a pattern with a finite correlation

asymptotics of greedy energy points for a symmetric kernel k x x x u on a locally compact metric space x we investigate the asymptotic behavior of greedy k energy points a i 1 for a compact subset a c x that are defined inductively by selecting a 1 a arbitrarily and a n 1 so that n i 1 k a n l1 a i inf x a n i 1 k x a i we give sufficient conditions under which these points also known as leja points are asymptotically energy minimizing i e have energy n i j k a i aj as n that is asymptotically the same as e a n min i j k x i x j x 1 x n a and have asymptotic distribution equal to the equilibrium measure for a for the case of riesz kernels k s x y x y s s 0 we show that if a is a rectifiable jordan arc or closed curve in rp and s 1 then greedy k s energy points are not asymptotically energy minimizing in contrast to the case s 1 additional results are obtained for greedy k s energy points on a sphere for greedy best packing points the case s and for weighted riesz kernels for greedy best packing points we provide a simple counterexample to a conjecture attributed to l bos

hitting times and periodicity in random dynamics we prove quenched laws of hitting time statistics for random subshifts of finite type in particular we prove a dichotomy between the law for periodic and for non periodic points we show that this applies to random gibbs measures

asymptotics for minimal discrete energy on the sphere we investigate the energy of arrangements of n points on the surface of the unit sphere sd in rd 1 that interact through a power law potential v 1 rs where s 0 and r is euclidean distance with ed s n denoting the minimal energy for such n point arrangements we obtain bounds valid for all n for ed s n in the cases when 0 s d and 2 d s for s d we determine the precise asymptotic behavior of ed d n as n as a corollary lower bounds are given for the separation of any pair of points in an n point minimal energy configuration when s d 2 for the unit sphere in r3 d 2 we present two conjectures concerning the asymptotic expansion of e2 s n that relate to the zeta function l s for a hexagonal lattice in the plane we prove an asymptotic upper bound that supports the first of these conjectures of related interest we derive an asymptotic formula for the partial sums of l s when 0 s 2 the divergent case

fluctuation bounds for entropy production estimators in gibbs measures we give bounds for the fluctuations of estimators of the mean entropy production in gibbsian sources these bounds are valid for every n where n denotes the size length of the sample we consider two estimators which are based on waiting and hitting times

complexity of bezout s theorem iv probability of success extensions we estimate the probability that a given number of projective newton steps applied to a linear homotopy of a system of n homogeneous polynomial equations in n 1 complex variables of fixed degrees will find all the roots of the system we also extend the framework of our analysis to cover the classical implicit function theorem and revisit the condition number in this context further complexity theory is developed

the positive semidefiniteness of partitioned matrices abstract the positive semidefiniteness of a partitioned matrix is characterized in terms of its submatrices the result is applied to a variety of problems concerning lowner ordered matrices which need not be partitioned themselves

on the third critical field in ginzburg landau theory using recent results by the authors on the spectral asymptotics of the neumann laplacian with magnetic field we give precise estimates on the critical field h c 3 describing the appearance of superconductivity in superconductors of type ii furthermore we prove that the local and global definitions of this field coincide near h c 3 only a small part near the boundary points where the curvature is maximal of the sample carries superconductivity we give precise estimates on the size of this zone and decay estimates in both the normal to the boundary and parallel variables

a proof of crystallization in two dimensions many materials have a crystalline phase at low temperatures the simplest example where this fundamental phenomenon can be studied are pair interaction energies of the type where y x 2 is the position of particle x and v r is the pair interaction energy of two particles which are placed at distance r due to the mermin wagner theorem it can t be expected that at finite temperature this system exhibits long range ordering we focus on the zero temperature case and show rigorously that under suitable assumptions on the potential v which are compatible with the growth behavior of the lennard jones potential the ground state energy per particle converges to an explicit constant e where e is the minimum of a simple function on 0 furthermore if suitable dirichlet or periodic boundary conditions are used then the minimizers form a triangular lattice to the best knowledge of the author this is the first result in the literature where periodicity of ground states is established for a physically relevant model which is invariant under the euclidean symmetry group consisting of rotations and translations

the probability that a slightly perturbed numerical analysis problem is difficult we prove a general theorem providing smoothed analysis estimates for conic condition numbers of problems of numerical analysis our probability estimates depend only on geometric invariants of the corresponding sets of ill posed inputs several applications to linear and polynomial equation solving show that the estimates obtained in this way are easy to derive and quite accurate the main theorem is based on a volume estimate of varepsilon tubular neighborhoods around a real algebraic subvariety of a sphere intersected with a spherical disk of radius sigma besides varepsilon and sigma this bound depends only on the dimension of the sphere and on the degree of the defining equations

on the worst case arithmetic complexity of approximating zeros of systems of polynomials let d 1 cdots d n be positive integers let mathcal p denote the set of systems of polynomials f mathbb c n to mathbb c n that have only finitely many zeros including those at infinity and that satisfy degree f i d i for all i let 0 varepsilon leqq r it is shown for fixed d 1 cdots d n that with respect to a certain model of computation the worst case computational complexity of obtaining varepsilon approximations to at least those zeros xi satisfying xi leqq r for arbitrary f in mathcal p is theta log log r varepsilon that is to say both upper and lower bounds are proved an algorithm for proving the upper bound is introduced the number of operations required by this algorithm is o left n mathcal d 4 log mathcal d log log r varepsilon n 2 mathcal d 4 left begin array 20 c 1 sigma d i n end array right 4 right quad text where mathcal d prod limits i 1

on the lowner minus and star partial orderings of nonnegative definite matrices and their squares abstract we investigate how the ordering of two hermitian nonnegative definite matrices a and b relates to the ordering of their squares a 2 and b 2 in the sense of the lowner partial ordering the minus partial ordering and the star partial ordering the condition that a and b commute appears essential in these investigations we also give some comments on possible extensions of our results by replacing the squares a 2 and b 2 with the k th powers a k and b k

on a problem posed by steve smale the 17th of the problems proposed by steve smale for the 21st century asks for the existence of a deterministic algorithm computing an approximate solution of a system of n complex polynomials in n unknowns in time polynomial on the average in the size n of the input system a partial solution to this problem was given by carlos beltr an and luis miguel pardo who exhibited a randomized algorithm doing so in this paper we further extend this result in several directions firstly we exhibit a linear homotopy algorithm that eciently implements a nonconstructive idea of

optimal logarithmic energy points on the unit sphere we study minimum energy point charges on the unit sphere s d in r d 1 d 3 that interact according to the logarithmic potential log 1 r where r is the euclidean distance between points such optimal n point configurations are uniformly distributed as n we quantify this result by estimating the spherical cap discrepancy of optimal energy configurations the estimate is of order o n 1 d 2 essential is an improvement of the lower bound of the optimal logarithmic energy which yields the second term 1 d log n n in the asymptotical expansion of the optimal energy previously this was known for the unit sphere in r 3 only furthermore we present an upper bound for the error of integration for an equally weighted numerical integration rule q n with the n nodes forming an optimal logarithmic energy configuration for polynomials p of degree at most n this bound is cd n 1 d n d 2 p as n n 1 d 0 for continuous functions f of s d satisfying a lipschitz condition with constant c f the bound is 12dc f c d f o n 1 d 2 as n

smale s 17th problem average polynomial time to compute affine and projective solutions smale s 17th problem asks can a zero of n complex polynomial equations in n unknowns be found approximately on the average in polynomial time with a uniform algorithm we give a positive answer to this question namely we describe a uniform probabilistic algorithm that computes an approximate zero of systems of polynomial equations f cn cn performing a number of arithmetic operations which is polynomial in the size of the input on the average

some further results on hermitian matrix inequalities abstract some new results on the lowner partial ordering between certain sums products and direct products of matrices are derived proofs of these results are based on a necessary and sufficient condition for the nonnegative definiteness of the difference of two hermitian matrices which follows from the criterion for the nonnegative definiteness of a partitioned hermitian matrix due to albert 1969

on match lengths zero entropy and large deviations with application to sliding window lempel ziv algorithm the sliding window lempel ziv swlz algorithm that makes use of recurrence times and match lengths has been studied from various perspectives in information theory literature in this paper we undertake a finer study of these quantities under two different scenarios 1 zero entropy sources that are characterized by strong long term memory and 2 the processes with weak memory as described through various mixing conditions for zero entropy sources a general statement on match length is obtained it is used in the proof of almost sure optimality of fixed shift variant of lempel ziv fslz and swlz algorithms given in literature through an example of stationary and ergodic processes generated by an irrational rotation we establish that for a window of size n w a compression ratio given by o log n w n w a where a depends on n w and approaches 1 as n w is obtained under the application of fslz and swlz algorithms in addition we give a general expression for the compression ratio for a class of stationary and ergodic processes with zero entropy next we extend the study of ornstein and weiss on the asymptotic behavior of the normalized version of recurrence times and establish the large deviation property for a class of mixing processes in addition an estimator of entropy based on recurrence times is proposed for which large deviation principle is proved for sources satisfying similar mixing conditions

rigid continuation paths i quasilinear average complexity for solving polynomial systems how many operations do we need on the average to compute an approximate root of a random gaussian polynomial system beyond smale s 17th problem that asked whether a polynomial bound is possible we prove a quasi optimal bound text input size 1 o 1 this improves upon the previously known text input size frac32 o 1 bound the new algorithm relies on numerical continuation along rigid continuation paths the central idea is to consider rigid motions of the equations rather than line segments in the linear space of all polynomial systems this leads to a better average condition number and allows for bigger steps we show that on the average we can compute one approximate root of a random gaussian polynomial system of n equations of degree at most d in n 1 homogeneous variables with o n 5 d 2 continuation steps this is a decisive improvement over previous bounds that prove no better than sqrt 2 min n d continuation steps on the average

two local operators and the blue abstract for a given matrix a a matrix p such that pa a is said to be a local identity and such that p2a pa is said to be a local idempotent in the paper some simple properties of such operators are presented their relation to the best linear unbiased estimation in the general gauss markov model is also demonstrated

lowner partial ordering and space preordering of hermitian non negative definite matrices abstract if a matrix a is below a matrix b with respect to the lowner partial ordering within the set of hermitian non negative definite matrices then the columns of a are linear combinations of the columns of b in this note the class of all such combinations providing a complete characterization of the lowner partial ordering is presented in addition new results on partial ordering of squares of hermitian non negative definite matrices are given

sharp error terms and neccessary conditions for exponential hitting times in mixing processes we prove an upper bound for the error in the exponential approximation of the hitting time law of a rare event in mixing processes with exponential decay mixing processes with a summable function and for general mixing processes with a finite alphabet in the first case the bound is uniform as a function of the measure of the event in the last two cases the bound depends also on the time scale t this allows us to get further statistical properties as the ratio convergence of the expected hitting time and the expected return time a uniform bound is a consequence we present an example that shows that this bound is sharp we also prove that second moments are not necessary for having the exponential law moreover we prove a necessary condition for having the exponential limit law

multiparameter groups of measure preserving transformations a simple proof of wiener s ergodic theorem a simple proof of wiener s multiparameter ergodic theorem is given in this paper at the same time it is shown that two of the hypotheses of an existing proof can be dispensed with

1d log gases and the renormalized energy crystallization at vanishing temperature we study the statistical mechanics of a one dimensional log gas or beta ensemble with general potential and arbitrary beta the inverse of temperature according to the method we introduced for two dimensional coulomb gases in sandier and serfaty ann probab 2014 such ensembles correspond to random matrix models in some particular cases the formal limit beta infty corresponds to weighted fekete sets and is also treated we introduce a one dimensional version of the renormalized energy of sandier and serfaty commun math phys 313 3 635 743 2012 measuring the total logarithmic interaction of an infinite set of points on the real line in a uniform neutralizing background we show that this energy is minimized when the points are on a lattice by a suitable splitting of the hamiltonian we connect the full statistical mechanics problem to this renormalized energy w and this allows us to obtain new results on the distribution of the points at the microscopic scale in particular we show that configurations whose w is above a certain threshold which tends to min w as beta rightarrow infty have exponentially small probability this shows that the configurations have increasing order and crystallize as the temperature goes to zero

an application of the modular function in nonlocal variational problems using the modular function and its invariance under the action of a modular group and an heuristic reduction of a mathematical model we present a mathematical account of a hexagonal pattern selection observed in di block copolymer melts

renormalized energy concentration in random matrices we define a renormalized energy as an explicit functional on arbitrary point configurations of constant average density in the plane and on the real line the definition is inspired by ideas of sandier and serfaty from the ginzburg landau model to vortex lattice problems 2012 1d log gases and the renormalized energy 2013 roughly speaking it is obtained by subtracting two leading terms from the coulomb potential on a growing number of charges the functional is expected to be a good measure of disorder of a configuration of points we give certain formulas for its expectation for general stationary random point processes for the random matrix sine processes on the real line 1 2 4 and ginibre point process and zeros of gaussian analytic functions process in the plane we compute the expectation explicitly moreover we prove that for these processes the variance of the renormalized energy vanishes which shows concentration near the expected value we also prove that the 2 sine process minimizes the renormalized energy in the class of determinantal point processes with translation invariant correlation kernels

symmetrizers of matrices abstract a symmetrizer of a given pair of matrices a and b is defined as a matrix x for which the product axb is symmetric right and left symmetrizers of a given matrix a are defined accordingly the main results of the paper are general representations of all three types of symmetrizers the problem considered arose in connection with certain questions pertaining to admissible linear estimation in a gauss markoff model

a uniqueness result for minimizers of the 1d log gas renormalized energy abstract in 21 sandier and serfaty studied the one dimensional log gas model in particular they gave a crystallization result by showing that the one dimensional lattice z is a minimizer for the so called renormalized energy which they obtained as a limit of the n particle log gas hamiltonian for n however this minimizer is not unique among infinite point configurations for example local perturbations of z leave the renormalized energy unchanged in this paper we establish that uniqueness holds at the level of stationary point processes the only minimizer being given by averaging z over a choice of the origin in 0 1 this is proved by showing a quantitative estimate on the two point correlation function of a process in terms of its renormalized energy

on some characterizations of the star partial ordering for matrices and rank subtractivity abstract the main result is that drazin s star partial ordering a b holds if and only if a b and b a b a where a b is defined by a a a b and aa ba and where a b denotes rank subtractivity here a and b are m n complex matrices and the superscript denotes the moore penrose inverse several other characterizations of a b are given with particular emphasis on what extra condition must be added in order that rank subtractivity becomes the stronger star order a key tool is a new canonical form for rank subtractivity connections with simultaneous singular value decompositions schur complements and idempotent matrices are also mentioned

finding elliptic fekete points sets two numerical solution approaches abstract the objective of this work is to provide a methodology for approximating globally optimal fekete point configurations this problem is of interest in numerical mathematics and scientific modeling following a brief discussion of the analytical background lipschitz global optimization lgo is applied to determine i e to numerically approximate fekete point configurations next to this optimization approach an alternative strategy by formulating a set of differential algebraic equations daes of index 2 will be considered the steady states of the daes coincide with the optima of the function to be minimized illustrative numerical results with configurations of up to 150 fekete points are presented to show the viability of both approaches

fast computation of zeros of polynomial systems with bounded degree under finite precision a solution for smale s 17th problem for the case of systems with bounded degree was recently given this solution an algorithm computing approximate zeros of complex polynomial systems in average polynomial time assumed infinite precision in this paper we describe a finite precision version of this algorithm our main result shows that this version works within the same time bounds and requires a precision which on the average amounts to a polynomial amount of bits in the mantissa of the intervening floating point numbers

convexity of solutions of semilinear elliptic equations on considere le probleme elliptique suivant u f u dans u m sur ou est un domaine convexe de r 2 et m est soit une constante reelle soit on etablit que pour une bonne fonction monotone g t pout toute solution u on a g u est strictement convexe dans

a new approach to variational problems with multiple scales we introduce a new concept the young measure on micropatterns to study singularly perturbed variational problems that lead to multiple small scales depending on a small parameter e this allows one to extract in the limit e 0 the relevant information at the macroscopic scale as well as the coarsest microscopic scale say e and to eliminate all finer scales to achieve this we consider rescaled functions rx t x s e t viewed as maps of the macroscopic variable s with values in a suitable function space the limiting problem can then be formulated as a variational problem on the young measures generated by rex as an illustration we study a one dimensional model that describes the competition between formation of microstructure and highest gradient regularization we show that the unique minimizer of the limit problem is a young measure supported on sawtooth functions with a given period 2001 john wiley sons inc

characterizations of the best linear unbiased estimator in the general gauss markov model with the use of matrix partial orderings abstract under the general gauss markov model y x 2 v two new characterizations of blue x are derived involving the lowner and rank subtractivity partial orderings between the dispersion matrix of blue x and the dispersion matrix of y as particular cases of these characterizations three new criteria for the equality between olse x and blue x are given

nonexistence of a minimizer for thomas fermi dirac von weizsacker model 1 x y dx dy a nonexistence result similar to theorem 1 also holds theorem 2 there exists constant v0 0 such that the variational problem 4 inf v e does not have a minimizer if v v0 if the coulomb term is not present in 1 and 3 a standard modica mortela type result establishes a link between the two functional through gamma convergence when the volume constraint m is large this is no longer valid for the energy functionals 1 and 3 due to the coulomb term the connection between the two functionals is just on the formal level

asymptotics for hitting times essentially decreasing sequences of balls in ametric space x and identi ed to be the distribution function of the posi tive exponential law with parameter 1 in many classes of mixing systems in abadi and galves 2001 collet and galves 1993 hirata saussol andvaienti 1999 saussol 1998 and young 1999 nonexponential asymp totics have been obtained in coelho and de faria 1990 for instance theliterature on the subject is rather important and our list is incomplete for further information we refer the reader to the surveys abadi 2004 orcoelho 2000 there exists a connection between return time asymptotics and hittingtime asymptotics indeed as shown in hirata saussol and vaienti 1999 when the asymptotics for return times is exponential with parameter 1 thenso is the one for hitting times possible asymptotics for return times that is weak limits for f

large deviation principle for empirical fields of log and riesz gases we study a system of n particles with logarithmic coulomb or riesz pairwise interactions confined by an external potential we examine a microscopic quantity the tagged empirical field for which we prove a large deviation principle at speed n the rate function is the sum of an entropy term the specific relative entropy and an energy term the renormalized energy introduced in previous works coupled by the temperature we deduce a variational property of the sine beta processes which arise in random matrix theory we also give a next to leading order expansion of the free energy of the system proving the existence of the thermodynamic limit

characterizations and dispersion matrix robustness of efficiently estimable parametric functionals in linear models with nuisance parameters abstract algebraic representations dimensional expressions and characterizations are given for the subspace of functionals of the main parameters which can be estimated with full efficiency under a linear model y w z 2 v containing nuisance parameters subspaces of functionals of for which the ordinary least squares estimator is robust against an alternative dispersion matrix v are obtained and a particular subspace of such functionals is found wherein the ordinary least squares estimator is both dispersion matrix robust and robust against the presence of nuisance parameters

asymptotic behaviour of a pile up of infinite walls of edge dislocations we consider a system of parallel straight edge dislocations and we analyse its asymptotic behaviour in the limit of many dislocations the dislocations are represented by points in a plane and they are arranged in vertical walls each wall is free to move in the horizontal direction the system is described by a discrete energy depending on the one dimensional horizontal positions x i 0 of the n walls the energy contains contributions from repulsive pairwise interactions between all walls a global shear stress forcing the walls to the left and a pinned wall at x 0 that prevents the walls from leaving through the left boundary we study the behaviour of the energy as the number of walls n tends to infinity and characterise this behaviour in terms of convergence there are five different cases depending on the asymptotic behaviour of the single dimensionless parameter n corresponding to beta n ll 1 n 1 n ll beta n ll 1 and beta n gg 1 and the two critical regimes n 1 n and n 1 as a consequence we obtain characterisations of the limiting behaviour of stationary states in each of these five regimes the results shed new light on the open problem of upscaling large numbers of dislocations we show how various existing upscaled models arise as special cases of the theorems of this paper the wide variety of behaviour suggests that upscaled models should incorporate more information than just dislocation densities this additional information is encoded in the limit of the dimensionless parameter n

beurling landau densities of weighted fekete sets and correlation kernel estimates let q be a suitable real function on c an n fekete set corresponding to q is a subset z n vertical bar z nn of c which maximizes the expression pi n i infinity in this note we prove that fekete sets are in a sense maximally spread out with respect to the equilibrium measure in general our results apply only to a part of the fekete set which is at a certain distance away from the boundary of the droplet however for the potential q vertical bar z vertical bar 2 we obtain results which hold globally and we conjecture that such global results are true for a wide range of potentials c 2012 elsevier inc all rights reserved less

the optimal sink and the best source in a markov chain it is well known that the distributions of hitting times in markov chains are quite irregular unless the limit as time tends to infinity is considered we show that nevertheless for a typical finite irreducible markov chain and for nondegenerate initial distributions the tails of the distributions of the hitting times for the states of a markov chain can be ordered i e they do not overlap after a certain finite moment of time if one considers instead each state of a markov chain as a source rather than a sink then again the states can generically be ordered according to their efficiency the mechanisms underlying these two orderings are essentially different though our results can be used e g for a choice of the initial distribution in numerical experiments with the fastest convergence to equilibrium stationary distribution for characterization of the elements of a dynamical network according to their ability to absorb and transmit the substance information that is circulated over the network for determining optimal stopping moments stopping signals words when dealing with sequences of symbols etc

random monomial ideals abstract inspired by the study of random graphs and simplicial complexes and motivated by the need to understand average behavior of ideals we propose and study probabilistic models of random monomial ideals we prove theorems about the probability distributions expectations and thresholds for events involving monomial ideals with given hilbert function krull dimension and first graded betti numbers and show that our models generalize several well known random simplicial complex models finally we present several experimentally backed conjectures about regularity projective dimension strong genericity and cohen macaulayness of random monomial ideals

lower bounds for the energy of unit vector fields and applications we prove lower bounds for the dirichlet energy of a unit vector field defined in a perforated domain of r2with nonzero degree on the outer boundary in terms of the total diameter of the holes we use this to derive lower bounds and then compactness results for sequences ue of minimizers or almost minimizers of ginzburg landau functionals with coupling constant 1 e2tending to

2d coulomb gases and the renormalized energy we study the statistical mechanics of classical two dimensional coulomb gases with general potential and arbitrary the inverse of the temperature such ensembles also correspond to random matrix models in some particular cases the formal limit case corresponds to weighted fekete sets and also falls within our analysis it is known that in such a system points should be asymptotically distributed according to a macroscopic equilibrium measure and that a large deviations principle holds for this as proven by petz and hiai in advances in differential equations and mathematical physics atlanta ga 1997 1998 amer math soc and ben arous and zeitouni esaim probab statist 2 1998 123 134 by a suitable splitting of the hamiltonian we connect the problem to the renormalized energy w a coulombian interaction for points in the plane introduced in comm math phys 313 2012 635 743 which is expected to be a good way of measuring the disorder of an infinite configuration of points in the plane by so doing we are able to examine the situation at the microscopic scale and obtain several new results a next order asymptotic expansion of the partition function estimates on the probability of fluctuation from the equilibrium measure at microscale and a large deviations type result which states that configurations above a certain threshhold of w have exponentially small probability when the estimate becomes sharp showing that the system has to crystallize to a minimizer of w in the case of weighted fekete sets this corresponds to saying that these sets should microscopically look almost everywhere like minimizers of w which are conjectured to be abrikosov triangular lattices

the compound poisson distribution and return times in dynamical systems previously it has been shown that some classes of mixing dynamical systems have limiting return times distributions that are almost everywhere poissonian here we study the behaviour of return times at periodic points and show that the limiting distribution is a compound poissonian distribution we also derive error terms for the convergence to the limiting distribution we also prove a very general theorem that can be used to establish compound poisson distributions in many other settings

equalities and inequalities for inertias of hermitian matrices with applications abstract the inertia of a hermitian matrix is defined to be a triplet composed of the numbers of the positive negative and zero eigenvalues of the matrix counted with multiplicities respectively in this paper we show some basic formulas for inertias of 2 2 block hermitian matrices from these formulas we derive various equalities and inequalities for inertias of sums parallel sums products of hermitian matrices submatrices in block hermitian matrices differences of outer inverses of hermitian matrices as applications we derive the extremal inertias of the linear matrix expression a bxb with respect to a variable hermitian matrix x in addition we give some results on the extremal inertias of hermitian solutions to the matrix equation ax b as well as the extremal inertias of a partial block hermitian matrix

a lower bound for the worst case cubature error on spheres of arbitrary dimension this paper is concerned with numerical integration on the unit sphere sr of dimension r 2 in the euclidean space r 1 we consider the worst case cubature error denoted by e qm hs sr of an arbitrary m point cubature rule qm for functions in the unit ball of the sobolev space hs sr where s and show that the positive constant cs r in the estimate depends only on the sphere dimension r 2 and the index s of the sobolev space hs sr this result was previously only known for r 2 in which case the estimate is order optimal the method of proof is constructive we construct for each qm a bad function fm that is a function which vanishes in all nodes of the cubature rule and for which our proof uses a packing of the sphere sr with spherical caps as well as an interpolation result between sobolev spaces of different indices

some properties of matrix partial orderings abstract the matrix partial orderings considered are 1 the star ordering and 2 the minus ordering or rank subtractivity both in the set of m n complex matrices and 3 the lowner ordering in the set of m m matrices the problems discussed are 1 inheriting certain properties under a given ordering 2 preserving an ordering under some matrix multiplications 3 relationships between an ordering among direct or kronecker and hadamard products and the corresponding orderings between the factors involved 4 orderings between generalized inverses of a given matrix and 5 preserving or reversing a given ordering under generalized inversions several generalizations of results known in the literature and a number of new results are derived

from the ginzburg landau model to vortex lattice problems we introduce a coulombian renormalized energy w which is a logarithmic type of interaction between points in the plane computed by a renormalization we prove various of its properties such as the existence of minimizers and show in particular using results from number theory that among lattice configurations the triangular lattice is the unique minimizer its minimization in general remains open

statistics of return times a general framework and new applications in this paper we provide general estimates for the errors between the distribution of the first and more generally the k th return time suitably rescaled and the poisson law for measurable dynamical systems in the case that the system exhibits strong mixing properties these bounds are explicitly expressed in terms of the speed of mixing using these approximations the poisson law is finally proved to hold for a large class of non hyperbolic systems on the interval

hitting and return times in ergodic dynamical systems given an ergodic dynamical system x t and u c x measurable with u 0 let u u x denote the normalized hitting time of x x to u we prove that given a sequence u n with u n 0 the distribution function of the normalized hitting times to u n converges weakly to some subprobability distribution f if and only if the distribution function of the normalized return time converges weakly to some distribution function f and that in the converging case f t 0 t 1 f s ds t 0 this in particular characterizes asymptotics for hitting times and shows that the asymptotics for return times is exponential if and only if the one for hitting times is also

covering of spheres by spherical caps and worst case error for equal weight cubature in sobolev spaces we prove that the covering radius of an n point subset xnxn of the unit sphere sd rd 1sd rd 1 is bounded above by a power of the worst case error for equal weight cubature 1n x xnf x sdfd d for functions in the sobolev space wps sd where d d denotes normalized area measure on sdsd these bounds are close to optimal when s is close to d pd p our study of the worst case error along with results of brandolini et al motivate the definition of quasi monte carlo qmc design sequences for wps sd which have previously been introduced only in the hilbert space setting p 2p 2 we say that a sequence xn xn of n point configurations is a qmc design sequence for wps sd with s d ps d p provided the worst case equal weight cubature error for xnxn has order n s dn s d as n n a property that holds in particular for a sequence of spherical t designs in which each design has order tdtd points for the case p 1p 1 we deduce that any qmc design sequence xn xn for w1s sd with s ds d has the optimal covering property i e the covering radius of xnxn has order n 1 dn 1 d as n n a significant portion of our effort is devoted to the formulation of the worst case error in terms of a bessel kernel and showing that this kernel satisfies a bernstein type inequality involving the mesh ratio of xnxn as a consequence we prove that any qmc design sequence for wps sd is also a qmc design sequence for wp s sd for all 1 p s d ps s d p

dimension reduction techniques for the minimization of theta functions on lattices we consider the minimization of theta functions theta lambda alpha sum p in lambda e pi alpha p 2 amongst lattices lambda subset mathbb r d by reducing the dimension of the problem following as a motivation the case d 3 where minimizers are supposed to be either the bcc or the fcc lattices a first way to reduce dimension is by considering layered lattices and minimize either among competitors presenting different sequences of repetitions of the layers or among competitors presenting different shifts of the layers with respect to each other the second case presents the problem of minimizing theta functions also on translated lattices namely minimizing l u mapsto theta l u alpha another way to reduce dimension is by considering lattices with a product structure or by successively minimizing over concentric layers the first direction leads to the question of minimization amongst orthorhombic lattices whereas the second is relevant for asymptotics questions which we study in detail in two dimensions

the matrix inequality m b mb abstract the loewner partial ordering between nonnegative definite matrices m and b mb is considered the main result obtained generalizes a number of results known in the literature the rank subtractivity partial ordering and the drazin partial ordering between m and b mb are also discussed

quasi monte carlo rules for numerical integration over the unit sphere mathbb s 2 we study numerical integration on the unit sphere mathbb s 2 subseteq mathbb r 3 using equal weight quadrature rules where the weights are such that constant functions are integrated exactly the quadrature points are constructed by lifting a 0 m 2 net given in the unit square 0 1 2 to the sphere mathbb s 2 by means of an area preserving map a similar approach has previously been suggested by cui and freeden siam j sci comput 18 2 595 609 1997 we prove three results the first one is that the construction is almost optimal with respect to discrepancies based on spherical rectangles further we prove that the point set is asymptotically uniformly distributed on mathbb s 2 and finally we prove an upper bound on the spherical cap l 2 discrepancy of order n 1 2 log n 1 2 where n denotes the number of points this improves upon the bound on the spherical cap l 2 discrepancy of the construction by lubotzky phillips and sarnak commun pure appl math 39 s suppl s149 s186 1986 by a factor of sqrt log n numerical results suggest that the 0 m 2 nets lifted to the sphere mathbb s 2 have spherical cap l 2 discrepancy converging with the optimal order of n 3 4