max norm optimization for robust matrix recovery this paper studies the matrix completion problem under arbitrary sampling schemes we propose a new estimator incorporating both max norm and nuclear norm regularization based on which we can conduct efficient low rank matrix recovery using a random subset of entries observed with additive noise under general non uniform and unknown sampling distributions this method significantly relaxes the uniform sampling assumption imposed for the widely used nuclear norm penalized approach and makes low rank matrix recovery feasible in more practical settings theoretically we prove that the proposed estimator achieves fast rates of convergence under different settings computationally we propose an alternating direction method of multipliers algorithm to efficiently compute the estimator which bridges a gap between theory and practice of machine learning methods with max norm regularization further we provide thorough numerical studies to evaluate the proposed method using both simulated and real datasets

from one cell to the whole froth a dynamical map we investigate two and three dimensional shell structured inflatable froths which can be constructed by a recursion procedure adding successive layers of cells around a germ cell we prove that any froth can be reduced into a system of concentric shells there is only a restricted set of local configurations for which the recursive inflation transformation is not applicable these configurations are inclusions between successive layers and can be treated as vertices and edges decorations of a shell structured inflatable skeleton the recursion procedure is described by a logistic map which provides a natural classification into euclidean hyperbolic and elliptic froths froths tiling manifolds with different curvatures can be classified simply by distinguishing between those with a bounded or unbounded number of elements per shell without any a priori knowledge on their curvature a result associated with maximal orientational entropy is obtained on topological properties of natural cellular systems the topological characteristics of all experimentally known tetrahedrally close packed structures are retrieved

jamming at zero temperature and zero applied stress the epitome of disorder we have studied how two and three dimensional systems made up of particles interacting with finite range repulsive potentials jam i e develop a yield stress in a disordered state at zero temperature and zero applied stress at low packing fractions phi the system is not jammed and each particle can move without impediment from its neighbors for each configuration there is a unique jamming threshold phi c at which particles can no longer avoid each other and the bulk and shear moduli simultaneously become nonzero the distribution of phi c values becomes narrower as the system size increases so that essentially all configurations jam at the same packing fraction in the thermodynamic limit this packing fraction corresponds to the previously measured value for random close packing in fact our results provide a well defined meaning for random close packing in terms of the fraction of all phase space with inherent structures that jam the jamming threshold point j occurring at zero temperature and applied stress and at the random close packing density has properties reminiscent of an ordinary critical point as point j is approached from higher packing fractions power law scaling is found for the divergence of the first peak in the pair correlation function and in the vanishing of the pressure shear modulus and excess number of overlapping neighbors moreover near point j certain quantities no longer self average suggesting the existence of a length scale that diverges at j however point j also differs from an ordinary critical point the scaling exponents do not depend on dimension but do depend on the interparticle potential finally as point j is approached from high packing fractions the density of vibrational states develops a large excess of low frequency modes indeed at point j the density of states is a constant all the way down to zero frequency all of these results suggest that point j is a point of maximal disorder and may control behavior in its vicinity perhaps even at the glass transition

diversity of order and densities in jammed hard particle packings recently the conventional notion of random close packing has been supplanted by the more appropriate concept of the maximally random jammed mrj state this inevitably leads to the necessity of distinguishing the mrj state among the entire collection of jammed packings while the ideal method of addressing this question would be to enumerate and classify all possible jammed hard sphere configurations practical limitations prevent such a method from being employed instead we generate numerically a large number of representative jammed hard sphere configurations primarily relying on a slight modification of the lubachevsky stillinger algorithm to do so and evaluate several commonly employed order metrics for each of these packings our investigation shows that even in the large system limit jammed systems of hard spheres can be generated with a wide range of packing fractions from phi approximately 0 52 to the fcc limit phi approximately 0 74 moreover at a fixed packing fraction the variation in the order can be substantial indicating that the density alone does not uniquely characterize a packing interestingly each order metric evaluated yielded a relatively consistent estimate for the packing fraction of the maximally random jammed state phi mrj approximately 0 63 this estimate however is compromised by the weaknesses in the order metrics available and we propose several guiding principles for future efforts to define more broadly applicable metrics

low rank covariance function estimation for multidimensional functional data multidimensional function data arise from many fields nowadays the covariance function plays an important role in the analysis of such increasingly common data in this article we propose a novel

counterintuitive ground states in soft core models it is well known that statistical mechanics systems exhibit subtle behavior in high dimensions in this paper we show that certain natural soft core models such as the gaussian core model have unexpectedly complex ground states even in relatively low dimensions specifically we disprove a conjecture of torquato and stillinger who predicted that dilute ground states of the gaussian core model in dimensions 2 through 8 would be bravais lattices we show that in dimensions 5 and 7 there are in fact lower energy non bravais lattices the nearest three dimensional analog is the hexagonal close packing but it has higher energy than the face centered cubic lattice we believe these phenomena are in fact quite widespread and we relate them to decorrelation in high dimensions

comment on packing hyperspheres in high dimensional euclidean spaces it is shown that the numerical data in cond mat 0608362 are in very good agreement with the predictions of cond mat 0601573

pairwise support vector machines and their application to large scale problems pairwise classification is the task to predict whether the examples a b of a pair a b belong to the same class or to different classes in particular interclass generalization problems can be treated in this way in pairwise classification the order of the two input examples should not affect the classification result to achieve this particular kernels as well as the use of symmetric training sets in the framework of support vector machines were suggested the paper discusses both approaches in a general way and establishes a strong connection between them in addition an efficient implementation is discussed which allows the training of several millions of pairs the value of these contributions is confirmed by excellent results on the labeled faces in the wild benchmark

neighbor list collision driven molecular dynamics simulation for nonspherical hard particles we apply the algorithm presented in the first part of this series of papers to systems of hard ellipses and ellipsoids the theoretical machinery needed to treat such particles including the overlap potentials is developed in full detail we describe an algorithm for predicting the time of collision for two moving ellipses or ellipsoids we present performance results for our implementation of the algorithm demonstrating that for dense systems of very aspherical ellipsoids the novel techniques of using neighbor lists and bounding sphere complexes offer as much as two orders of magnitude improvement in efficiency over direct adaptations of traditional event driven molecular dynamics algorithms the practical utility of the algorithm is demonstrated by presenting several interesting physical applications including the generation of jammed packings inside spherical containers the study of contact force chains in jammed packings and melting the densest known equilibrium crystals of prolate spheroids

moduli space of cubic newton maps abstract in this article we study the topology and bifurcations of the moduli space m 3 of cubic newton maps it s a subspace of the moduli space of cubic rational maps carrying the riemann orbifold structure c 2 3 we prove two results the boundary of the unique unbounded hyperbolic component is a jordan arc and the boundaries of all other hyperbolic components are jordan curves the head s angle map is surjective and monotone the fibers of this map are characterized completely the first result is a moduli space analogue of the first author s dynamical regularity theorem 37 the second result confirms a conjecture of tan lei

kernelization of matrix updates when and how abstract we define what it means for a learning algorithm to be kernelizable in the case when the instances are vectors asymmetric matrices and symmetric matrices respectively we can characterize kernelizability in terms of an invariance of the algorithm to certain orthogonal transformations if we assume that the algorithm s action relies on a linear prediction then we can show that in each case the linear parameter vector must be a certain linear combination of the instances we give a number of examples of how to apply our methods in particular we show how to kernelize multiplicative updates for symmetric instance matrices

the kissing number in four dimensions the kissing number problem asks for the maximal number k n of equal size nonoverlapping spheres in n dimensional space that can touch another sphere of the same size this problem in dimension three was the subject of a famous discussion between isaac newton and david gregory in 1694 in three dimensions the problem was finally solved only in 1953 by schutte and van der waerden in this paper we present a solution of a long standing problem about the kissing number in four dimensions namely the equality fc 4 24 is proved the proof is based on a modification of delsarte s method

geometrical frustration a study of four dimensional hard spheres the smallest maximum kissing number voronoi polyhedron of three dimensional 3d euclidean spheres is the icosahedron and the tetrahedron is the smallest volume that can show up in delaunay tessellation no periodic lattice is consistent with either and hence these dense packings are geometrically frustrated because icosahedra can be assembled from almost perfect tetrahedra the terms icosahedral and polytetrahedral packing are often used interchangeably which leaves the true origin of geometric frustration unclear here we report a computational study of freezing of 4d euclidean hard spheres where the densest voronoi cluster is compatible with the symmetry of the densest crystal while polytetrahedral order is not we observe that under otherwise comparable conditions crystal nucleation in four dimensions is less facile than in three dimensions which is consistent with earlier observations m skoge et al phys rev e 74 041127 2006 we conclude that it is the geometrical frustration of polytetrahedral structures that inhibits crystallization

exactly solvable disordered sphere packing model in arbitrary dimensional euclidean spaces we introduce a generalization of the well known random sequential addition rsa process for hard spheres in d dimensional euclidean space mathbb r d we show that all of the n particle correlation functions of this nonequilibrium model in a certain limit called the ghost rsa packing can be obtained analytically for all allowable densities and in any dimension this represents the first exactly solvable disordered sphere packing model in an arbitrary dimension the fact that the maximal density ensuremath phi ensuremath infty 1 2 d of the ghost rsa packing implies that there may be disordered sphere packings in sufficiently high d whose density exceeds minkowski s lower bound for bravais lattices the dominant asymptotic term of which is 1 2 d indeed we report on a conjectural lower bound on the density whose asymptotic behavior is controlled by 2 ensuremath 0 778 65 dots d thus providing the putative exponential improvement on minkowski s 100 year old bound our results suggest that the densest packings in sufficiently high dimensions may be disordered rather than periodic implying the existence of disordered classical ground states for some continuous potentials

freezing transition of hard hyperspheres we investigate the system of d dimensional hard spheres in d dimensional space where d 3 for the fluid phase of these hyperspheres we generalize scaled particle theory to arbitrary d and furthermore use the virial expansion and the percus yevick integral equation for the crystalline phase we adopt cell theory based on elementary geometrical assumptions about close packed lattices regardless of the approximation applied and for dimensions as high as d550 we find a first order freezing transition which preempts the kirkwood second order instability of the fluid the relative density jump increases with d and a generalized lindemann rule of melting holds we have also used ideas from fundamental measure theory to obtain a free energy density functional for hard hyperspheres finally we have calculated the surface tension of a hypersphere fluid near a hard smooth hyper wall within scaled particle theory

correspondence counter example to clockwise nature of nyquist locus of stable transfer functionsrsquo horowitz and ben adam 1989 noted that the nyquist loci of stable proper real rational functions tend to be clockwise they conjectured that these loci are always clockwise it will be shown by counter example that this conjecture is false the conjecture had been proved using theorem 1 of their paper the same counterexample can be used to show that theorem 1 is incorrect

an unfitted finite element method using discontinuous galerkin in this paper we present a new approach to simulations on complex shaped domains the method is based on a discontinuous galerkin dg method using trial and test functions defined on a structured grid essential boundary conditions are imposed weakly via the dg formulation this method offers a discretization where the number of unknowns is independent of the complexity of the domain we will show numerical computations for an elliptic scalar model problem in 2 and 3 convergence rates for different polynomial degrees are studied

techniques for image enhancement and segmentation of tomographic images of porous materials abstract this article presents a three stage approach combining novel and traditional algorithms for the segmentation of images of porous and composite materials obtained from x ray tomography the first stage is an anisotropic diffusion filter which removes noise while preserving significant features the second stage applies the unsharp mask sharpening filter which enhances edges and partially reverses the smoothing that is often a consequence of tomographic reconstruction the final stage uses a combination of watershed and active contour methods for segmentation of the grey scale data for the data sets we have analysed this approach gives the highest quality results in addition it has been implemented on cluster type parallel computers and applied to cubic images comprising up to 20003 voxels

ground states and formal duality relations in the gaussian core model we study dimensional trends in ground states for soft matter systems specifically using a high dimensional version of parrinello rahman dynamics we investigate the behavior of the gaussian core model in up to eight dimensions the results include unexpected geometric structures with surprising anisotropy as well as formal duality relations these duality relations suggest that the gaussian core model possesses unexplored symmetries and they have implications for a broad range of soft core potentials

packing hyperspheres in high dimensional euclidean spaces we present a study of disordered jammed hard sphere packings in four five and six dimensional euclidean spaces using a collision driven packing generation algorithm we obtain the first estimates for the packing fractions of the maximally random jammed mrj states for space dimensions d 4 5 and 6t o be mrj0 46 0 31 and 0 20 respectively to a good approximation the mrj density obeys the scaling form mrj c1 2 d c2d 2 d where c1 2 72 and c2 2 56 which appears to be consistent with the highdimensional asymptotic limit albeit with different coefficients calculations of the pair correlation function g2r and structure factor sk for these states show that short range ordering appreciably decreases with increasing dimension consistent with a recently proposed decorrelation principle which among other things states that unconstrained correlations diminish as the dimension increases and vanish entirely in the limit d as in three dimensions where mrj0 64 the packings show no signs of crystallization are isostatic and have a power law divergence in g2r at contact with power law exponent 0 4 across dimensions the cumulative number of neighbors equals the kissing number of the conjectured densest packing close to where g2r has its first minimum additionally we obtain estimates for the freezing and melting packing fractions for the equilibrium hard sphere fluid solid transition f0 32 and m0 39 respectively for d 4 and f0 20 and m0 25 respectively for d 5 although our results indicate the stable phase at high density is a crystalline solid nucleation appears to be strongly suppressed with increasing dimension

revisiting stein s paradox multi task averaging we present a multi task learning approach to jointly estimate the means of multiple independent distributions from samples the proposed multi task averaging mta algorithm results in a convex combination of the individual task s sample averages we derive the optimal amount of regularization for the two task case for the minimum risk estimator and a minimax estimator and show that the optimal amount of regularization can be practically estimated without cross validation we extend the practical estimators to an arbitrary number of tasks simulations and real data experiments demonstrate the advantage of the proposed mta estimators over standard averaging and james stein estimation

spherical codes maximal local packing density and the golden ratio the densest local packing dlp problem in d dimensional euclidean space rd involves the placement of n nonoverlapping spheres of unit diameter near an additional fixed unit diameter sphere such that the greatest distance from the center of the fixed sphere to the centers of any of the n surrounding spheres is minimized solutions to the dlp problem are relevant to the realizability of pair correlation functions for packings of nonoverlapping spheres and might prove useful in improving upon the best known upper bounds on the maximum packing fraction of sphere packings in dimensions greater than 3 the optimal spherical code problem in rd involves the placement of the centers of n nonoverlapping spheres of unit diameter onto the surface of a sphere of radius r such that r is minimized it is proved that in any dimension all solutions between unity and the golden ratio to the optimal spherical code problem for n spheres are also solutions to the corresponding dlp problem it follows that for any packing

clockwise property of the nyquist plot with implications for absolute stability abstract in this paper the clockwise property of the nyquist plot of stable transfer functions is addressed the relationships between clockwise property and modulus and or phase monotonicity are analyzed some necessary and sufficient conditions for a class of stable transfer functions to enjoy the clockwise property are obtained these results are used to enlarge the class of systems for which the kalman conjecture on absolute stability of nonlinear systems holds

geometrical structure of disordered sphere packings the three dimensional structure of large packings of monosized spheres with volume fractions ranging between 0 58 and 0 64 has been studied with x ray computed tomography we search for signatures of organization classifying local arrangements and exploring the effects of local geometrical constrains on the global packing this study is the largest and the most accurate empirical analysis of disordered packings at the grain scale to date mapping over 380 000 sphere coordinates with precision within 0 1 of the sphere diameters we discuss topological and geometrical methods to characterize and classify these systems emphasizing the implications that local geometry can have on the mechanisms of formation of these amorphous structures some of the main results are 1 the observation that the average number of contacts increases with the volume fraction 2 the discovery that these systems have a very compact contact network 3 the finding that disordered packing can be locally more efficient than crystalline packings 4 the observation that the peaks of the radial distribution function follow power law divergences 5 the discovery that geometrical frustration plays no role in the formation of such amorphous packings

fractal heterogeneous media a method is presented for generating compact fractal disordered media by generalizing the random midpoint displacement algorithm the obtained structures are invasive stochastic fractals with the hurst exponent varying as a continuous parameter as opposed to lacunar deterministic fractals such as the menger sponge by employing the detrending moving average algorithm a carbone phys rev e 76 056703 2007 the hurst exponent of the generated structure can be subsequently checked the fractality of such a structure is referred to a property defined over a three dimensional topology rather than to the topology itself consequently in this framework the hurst exponent should be intended as an estimator of compactness rather than of roughness applications can be envisaged for simulating and quantifying complex systems characterized by self similar heterogeneity across space for example exploitation areas range from the design and control of multifunctional self assembled artificial nanostructures and microstructures to the analysis and modeling of complex pattern formation in biology environmental sciences geomorphological sciences etc

geometrical ambiguity of pair statistics point configurations point configurations have been widely used as model systems in condensed matter physics materials science and biology statistical descriptors such as the n body distribution function g n are usually employed to characterize point configurations among which the most extensively used is the pair distribution function g 2 an intriguing inverse problem of practical importance that has been receiving considerable attention is the degree to which a point configuration can be reconstructed from the pair distribution function of a target configuration although it is known that the pair distance information contained in g 2 is in general insufficient to uniquely determine a point configuration this concept does not seem to be widely appreciated and general claims of uniqueness of the reconstructions using pair information have been made based on numerical studies in this paper we present the idea of the distance space called the mathbb d space the pair distances of a specific point configuration are then represented by a single point in the mathbb d space we derive the conditions on the pair distances that can be associated with a point configuration which are equivalent to the realizability conditions of the pair distribution function g 2 moreover we derive the conditions on the pair distances that can be assembled into distinct configurations i e with structural degeneracy these conditions define a bounded region in the mathbb d space by explicitly constructing a variety of degenerate point configurations using the mathbb d space we show that pair information is indeed insufficient to uniquely determine the configuration in general we also discuss several important problems in statistical physics based on the mathbb d space including the reconstruction of atomic structures from experimentally obtained g 2 and a recently proposed decorrelation principle the degenerate configurations have relevance to open questions involving the famous traveling salesman problem

estimates of the optimal density of sphere packings in high dimensions the problem of finding the asymptotic behavior of the maximal density max of sphere packings in high euclidean dimensions is one of the most fascinating and challenging problems in discrete geometry one century ago minkowski obtained a rigorous lower bound on max that is controlled asymptotically by 1 2d where d is the euclidean space dimension an indication of the difficulty of the problem can be garnered from the fact that exponential improvement of minkowski s bound has proved to be elusive even though existing upper bounds suggest that such improvement should be possible using a statistical mechanical procedure to optimize the density associated with a test pair correlation function and a conjecture concerning the existence of disordered sphere packings s torquato and f h stillinger exp math 15 307 2006 the putative exponential improvement on max was found with an asymptotic behavior controlled by 1 2 0 77865 d using the same methods we investigate whether this exponential im

phase behavior of colloidal superballs shape interpolation from spheres to cubes the phase behavior of hard superballs is examined using molecular dynamics within a deformable periodic simulation box a superball s interior is defined by the inequality x 2q y 2q z 2q ensuremath le 1 which provides a versatile family of convex particles q ensuremath ge 0 5 with cubelike and octahedronlike shapes as well as concave particles ql0 5 with octahedronlike shapes here we consider the convex case with a deformation parameter q between the sphere point q 1 and the cube q ensuremath infty we find that the asphericity plays a significant role in the extent of cubatic ordering of both the liquid and crystal phases calculation of the first few virial coefficients shows that superballs that are visually similar to cubes can have low density equations of state closer to spheres than to cubes dense liquids of superballs display cubatic orientational order that extends over several particle lengths only for large q along the ordered high density equation of state superballs with 1lql3 exhibit clear evidence of a phase transition from a crystal state to a state with reduced long ranged orientational order upon the reduction of density for q ensuremath ge 3 long ranged orientational order persists until the melting transition the width of the apparent coexistence region between the liquid and ordered high density phase decreases with q up to q 4 0 the structures of the high density phases are examined using certain order parameters distribution functions and orientational correlation functions we also find that a fixed simulation cell induces artificial phase transitions that are out of equilibrium current fabrication techniques allow for the synthesis of colloidal superballs and thus the phase behavior of such systems can be investigated experimentally

consistency of trace norm minimization regularization by the sum of singular values also referred to as the trace norm is a popular technique for estimating low rank rectangular matrices in this paper we extend some of the consistency results of the lasso to provide necessary and sufficient conditions for rank consistency of trace norm minimization with the square loss we also provide an adaptive version that is rank consistent even when the necessary condition for the non adaptive version is not fulfilled

neighbor list collision driven molecular dynamics simulation for nonspherical hard particles i algorithmic details in this first part of a series of two papers we present in considerable detail a collision driven molecular dynamics algorithm for a system of non spherical particles within a parallelepiped simulation domain under both periodic or hard wall boundary conditions the algorithm extends previous event driven molecular dynamics algorithms for spheres and is most efficient when applied to systems of particles with relatively small aspect ratios and with small variations in size we present a novel partial update near neighbor list nnl algorithm that is superior to previous algorithms at high densities without compromising the correctness of the algorithm this efficiency of the algorithm is further increased for systems of very aspherical particles by using bounding sphere complexes bsc these techniques will be useful in any particle based simulation including monte carlo and time driven molecular dynamics additionally we allow for a non vanishing rate of deformation of the boundary which can be used to model macroscopic strain and also alleviate boundary effects for small systems in the second part of this series of papers we specialize the algorithm to systems of ellipses and ellipsoids and present performance results for our implementation demonstrating the practical utility of the algorithm

counter example to a recent result on the stability of interval matrices by s bialas abstract in bialas 1983 a condition is given which is supposedly necessary and sufficient for the stability of interval matrices we demonstrate via a counter example that the condition given is in fact not sufficient

on spectral learning in this paper we study the problem of learning a matrix w from a set of linear measurements our formulation consists in solving an optimization problem which involves regularization with a spectral penalty term that is the penalty term is a function of the spectrum of the covariance of w instances of this problem in machine learning include multi task learning collaborative filtering and multi view learning among others our goal is to elucidate the form of the optimal solution of spectral learning the theory of spectral learning relies on the von neumann characterization of orthogonally invariant norms and their association with symmetric gauge functions using this tool we formulate a representer theorem for spectral regularization and specify it to several useful example such as schatten p norms trace norm and spectral norm which should proved useful in applications

densest local sphere packing diversity general concepts and application to two dimensions two dimensional circular disk problem we find and present the putative densest packings and corresponding rminn for selected values of n up to n 348 and use this knowledge to construct such a realizability condition and an upper bound we additionally analyze the properties and characteristics of the maximally dense packings finding significant variability in their symmetries and contact networks and that the vast majority differ substantially from the triangular lattice even for large n our work has implications for packaging problems nucleation theory and surface physics

spectral ewald acceleration of stokesian dynamics for polydisperse suspensions in this work we develop the spectral ewald accelerated stokesian dynamics seasd a novel computational method for dynamic simulations of polydisperse colloidal suspensions with full hydrodynamic interactions seasd is based on the framework of stokesian dynamics sd with extension to compressible solvents and uses the spectral ewald se method lindbo and tornberg 2010 29 for the wave space mobility computation to meet the performance requirement of dynamic simulations we use graphic processing units gpu to evaluate the suspension mobility and achieve an order of magnitude speedup compared to a cpu implementation for further speedup we develop a novel far field block diagonal preconditioner to reduce the far field evaluations in the iterative solver and seasd nf a polydisperse extension of the mean field brownian approximation of banchio and brady 2003 39 we extensively discuss implementation and parameter selection strategies in seasd and demonstrate the spectral accuracy in the mobility evaluation and the overall o n log n computation scaling we present three computational examples to further validate seasd and seasd nf in monodisperse and bidisperse suspensions the short time transport properties the equilibrium osmotic pressure and viscoelastic moduli and the steady shear brownian rheology our validation results show that the agreement between seasd and seasd nf is satisfactory over a wide range of parameters and also provide significant insight into the dynamics of polydisperse colloidal suspensions

dimensional study of the caging order parameter at the glass transition the glass problem is notoriously hard and controversial even at the mean field level little is agreed upon regarding why a fluid becomes sluggish while exhibiting but unremarkable structural changes it is clear however that the process involves self caging which provides an order parameter for the transition it is also broadly assumed that this cage should have a gaussian shape in the mean field limit here we show that this ansatz does not hold by performing simulations as a function of spatial dimension d we find the cage to keep a nontrivial form quantitative mean field descriptions of the glass transition such as mode coupling theory density functional theory and replica theory all miss this crucial element although the mean field random first order transition scenario of the glass transition is qualitatively supported here and non mean field corrections are found to remain small on decreasing d reconsideration of its implementation is needed for it to result in a coherent description of experimental observations

algorithm for random close packing of spheres with periodic boundary conditions the isotropic algorithm is constructed for random close packing of equisized spheres with triply periodic boundary conditions all previously published packing methods with periodic boundaries were kinetics determined i e they contained a densification rate as an arbitrary parameter in contrast the present algorithm is kinetic independent and demonstrates an unambiguous convergence to the experimental results to suppress crystallization the main principles of our algorithm are 1 to form a contact network at an early stage and 2 retain contacts throughout the densification as far as possible the particles are allowed to swell by the numerical solution of the differential equations of densification the rhs of these equations is calculated efficiently from a linear system by a combination of conjugate gradient iterations and exact sparse matrix technology when an excessive contact occurs and one of the existing bonds should be broken to continue the densification an efficient criterion based on multidimensional simplex geometry is used for searching the separating bond the algorithm has a well defined termination point resulting in a perfect contact network with the average coordination number six for particle number n 1 and a system of normal reactions between the spheres maintaining the structure these forces are the counterpart of the algorithm and can be used to calculate small elastic particle deformations in a granular medium extensive calculations are presented for 50 n 5 400 and demonstrate very good agreement with the experimental packing density about 0 637 and structure

efficient svm training using low rank kernel representations svm training is a convex optimization problem which scales with the training set size rather than the feature space dimension while this is usually considered to be a desired quality in large scale problems it may cause training to be impractical the common techniques to handle this difficulty basically build a solution by solving a sequence of small scale subproblems our current effort is concentrated on the rank of the kernel matrix as a source for further enhancement of the training procedure we first show that for a low rank kernel matrix it is possible to design a better interior point method ipm in terms of storage requirements as well as computational complexity we then suggest an efficient use of a known factorization technique to approximate a given kernel matrix by a low rank matrix which in turn will be used to feed the optimizer finally we derive an upper bound on the change in the objective function value based on the approximation error and the number of active constraints support vectors this bound is general in the sense that it holds regardless of the approximation method

breakdown of elasticity theory for jammed hard particle packings conical nonlinear constitutive theory abstract hard particle packings have provided a rich source of outstanding theoretical problems and served as useful starting points to model the structure of granular media liquids living cells glasses and random media the nature of jammed hard particle packings is a current subject of keen interest we demonstrate that the response of jammed hard particle packings to global deformations cannot be described by linear elasticity even for small particle displacements but involves a conical nonlinear constitutive theory it is the singular nature of the hard particle potential that leads to the breakdown of linear elasticity interestingly a nonlinear theory arises because the feasible particle displacements leading to unjamming depend critically on the local spatial arrangement of the particles implying a directionality in the feasible strains that is absent in particle systems with soft potentials mathematically the set of feasible strains has a conical structure i e components of the imposed strain tensor generally obey linear inequalities the nature of the nonlinear behavior is illustrated by analyzing several specific packings finally we examine the conditions under which a packing can be considered to incompressible in the traditional sense

principal component analysis of persistent homology rank functions with case studies of spatial point patterns sphere packing and colloids abstract persistent homology while ostensibly measuring changes in topology captures multiscale geometrical information it is a natural tool for the analysis of point patterns in this paper we explore the statistical power of the persistent homology rank functions for a point pattern x we construct a filtration of spaces by taking the union of balls of radius a centred on points in x x a x x b x a the rank function k x a b r 2 a b r is then defined by k x a b rank h k x a h k x b where is the induced map on homology from the inclusion map on spaces we consider the rank functions as lying in a hilbert space and show that under reasonable conditions the rank functions from multiple simulations or experiments will lie in an affine subspace this enables us to perform functional principal component analysis which we apply to experimental data from colloids at different effective temperatures and to sphere packings with different volume fractions we also investigate the potential of rank functions in providing a test of complete spatial randomness of 2d point patterns using the distances to an empirically computed mean rank function of binomial point patterns in the unit square

contact network in nearly jammed disordered packings of hard sphere chains we present salient results of the analysis of the geometrical structure of a large fully equilibrated ensemble of nearly jammed packings of linear freely jointed chains of tangent hard spheres generated via extensive monte carlo simulations in spite the expected differences due to chain connectivity both the pair correlation function and the contact network for chain packings are found to strongly resemble those in packings of monomeric hard spheres at the maximally random jammed mrj state a remarkable finding of the present work is the tendency of chains to form closed loops at the mrj state as a consequence of chain collapse our simulations on disordered nearly jammed chain packings yield an average coordination number of 6 which fulfills the isostaticity condition and is in excellent agreement with the corresponding simulation a donev s torquato and f h stillinger phys rev e 71 011105 2005 and experimental t aste m saadatfar and t j senden phys rev e 71 061302 2005 findings for jammed packings of monatomic hard spheres an exact correspondence between the statistical mechanical ensembles of monomeric spheres and of hard sphere chains offers insights regarding the structure and topology of the contact network of hard sphere systems at the mrj state

learning multiple tasks with kernel methods we study the problem of learning many related tasks simultaneously using kernel methods and regularization the standard single task kernel methods such as support vector machines and regularization networks are extended to the case of multi task learning our analysis shows that the problem of estimating many task functions with regularization can be cast as a single task learning problem if a family of multi task kernel functions we define is used these kernels model relations among the tasks and are derived from a novel form of regularizers specific kernels that can be used for multi task learning are provided and experimentally tested on two real data sets in agreement with past empirical work on multi task learning the experiments show that learning multiple related tasks simultaneously using the proposed approach can significantly outperform standard single task learning particularly when there are many related tasks but few data per task

an efficient algorithm for testing the generic rigidity of graphs in the plane given a structure made up of n sites connected by b bars the problem of recognizing which subsets of sites form rigid units is not a trivial one because of the non local character of rigidity in central force systems even though this is a very old problem of statics no simple algorithms are available for it so the most usual approach has been to solve the elastic equations which is very time consuming for large systems recently an integer algorithm was proposed for this problem in two dimensions using matching methods from graph theory and laman s theorem for two dimensional graphs the method is relatively simple but its time complexity grows as in the worst case and almost as fast on practical cases so that an improvement is highly desirable in this paper we describe a further elaboration of that procedure which relies upon the description of the system as a collection of rigid bodies connected by bars instead of sites connected by bars sets of rigidly connected objects are replaced by a unique body and this is done recursively as more rigid connections between bodies are discovered at larger scales as a consequence of this rescaling transformation our algorithm has much improved average behaviour even when its worst case complexity remains the time complexity of the body bar algorithm is found to scale as for the randomly diluted triangular lattice while the original site bar version scales as for the same problem

convexity of a frequency response are associated with a stable entire function in this paper the authors extend the convexity results for hurwitz polynomials reported in hamann and barmish 1993 to a large class of entire functions this class includes all the hurwitz quasi polynomials the authors show that the origin may fail to be in the interior of the inner frequency response set of a hurwitz quasi polynomial and state a sufficient condition which prevents this from happening

new perspectives on k support and cluster norms we study a regularizer which is defined as a parameterized in mum of quadratics and which we call the box norm we show that the k support norm a regularizer proposed by argyriou et al 2012 for sparse vector prediction problems belongs to this family and the box norm can be generated as a perturbation of the former we derive an improved algorithm to compute the proximity operator of the squared box norm and we provide a method to compute the norm we extend the norms to matrices introducing the spectral k support norm and spectral box norm we note that the spectral box norm is essentially equivalent to the cluster norm a multitask learning regularizer introduced by jacob et al 2009a and which in turn can be interpreted as a perturbation of the spectral k support norm centering the norm is important for multitask learning and we also provide a method to use centered versions of the norms as regularizers numerical experiments indicate that the spectral k support and box norms and their centered variants provide state of the art performance in matrix completion and multitask learning problems respectively

statistical properties and shell analysis in random cellular structures we investigate the statistical properties of two dimensional random cellular systems froths in terms of their shell structure the froth is analyzed as a system of concentric layers of cells around a given central cell we derive exact analytical relations for the topological properties of the sets of cells belonging to these layers experimental observations of the shell structure of two dimensional soap froth are made and compared with the results on two kinds of voronoi constructions it is found that there are specific differences between soap froths and purely geometrical constructions in particular these systems differ in the topological charge of clusters as a function of shell number in the asymptotic values of defect concentrations and in the number of cells in a given layer we derive approximate expressions with no free parameters which correctly explain these different behaviors s1063 651x 96 13111 1

reproducing morphologies of disorderly self assembling planar molecules with static and dynamic simulation methods by matching density abstract monte carlo and molecular dynamics simulations are the two main numerical approaches to modeling molecular self assembly and ordering conceptually however each method explores different paths through the thermodynamic landscape molecular dynamics depends on the position and momentum terms monte carlo is a static set and thus the momentum term is replaced with an energy term that is dependent on the volume and entropy until now it was unclear if a stochastic process of densifying particles would have the same internal structure as morphologies produced from classical mechanics this paper provides a systematic i e statistical analysis of the outcomes of 4032 simulations for hard core circular objects as a function of the number of molecules and the boundary conditions structural classification of the resultant ensembles averaged pair correlation function bond order parameter translational order parameter and voronoi diagrams shows that stochastic and dynamic approaches do not alter the morphology of the steric molecules we conclude that when the probability density of covering area fractions are matched the ensembles produced from the two methods will show the same level of structural disorder and positional patterns the resultant morphology from both models therefore is not a product of dynamic unrest but that of the relaxation of entropic frustration from macromolecular crowding although statistically the two methods produce similar configurations nuances arise from the static and dynamic nature of modeling as a result monte carlo is slightly better suited to modeling systems when the desired morphology is represented by a metastable state molecular dynamics on the other hand is more suited to finding defects that can arise in morphologies regardless a fixed density will result in similar morphologies from both techniques driven by similar configurational entropy

local density fluctuations hyperuniformity and order metrics questions concerning the properties and quantification of density fluctuations in point patterns continue to provide many theoretical challenges the purpose of this paper is to characterize certain fundamental aspects of local density fluctuations associated with general point patterns in any space dimension d our specific objectives are to study the variance in the number of points contained within a regularly shaped window ensuremath omega of arbitrary size and to further illuminate our understanding of hyperuniform systems i e point patterns that do not possess infinite wavelength fluctuations for large windows hyperuniform systems are characterized by a local variance that grows only as the surface area rather than the volume of the window we derive two formulations for the number variance i an ensemble average formulation which is valid for statistically homogeneous systems and ii a volume average formulation applicable to a single realization of a general point pattern in the large system limit the ensemble average formulation which includes both real space and fourier representations enables us to show that a homogeneous point pattern in a hyperuniform state is at a critical point of a type with appropriate scaling laws and critical exponents but one in which the direct correlation function rather than the pair correlation function is long ranged we also prove that the non negativity of the local number variance does not add a new realizability condition on the pair correlation the volume average formulation is superior for certain computational purposes including optimization studies in which it is desired to find the particular point pattern with an extremal or targeted value of the variance we prove that the simple periodic linear array yields the global minimum value of the average variance among all infinite one dimensional hyperuniform patterns we also evaluate the variance for common infinite periodic lattices as well as certain nonperiodic point patterns in one two and three dimensions for spherical windows enabling us to rank order the spatial patterns our results suggest that the local variance may serve as a useful order metric for general point patterns contrary to the conjecture that the lattices associated with the densest packing of congruent spheres have the smallest variance regardless of the space dimension we show that for d 3 the body centered cubic lattice has a smaller variance than the face centered cubic lattice finally for certain hyperuniform disordered point patterns we evaluate the direct correlation function structure factor and associated critical exponents exactly

functional approximation and projection of stored energy functions in computational homogenization of hyperelastic materials a probabilistic perspective abstract this work is concerned with the construction of a surrogate model for the homogenized stored energy functions defining the effective behavior of nonlinear elastic microstructures here a probabilistic standpoint is adopted and allows for the definition of a nonlinear mapping between the macroscopic deformations and the homogenized potential this functional approximation is specifically obtained by means of a polynomial chaos expansion the coefficients of which are computed through a gauss legendre quadrature rule by invoking well known results related to projections in hilbert spaces closest approximations of arbitrary potentials to standard e g ogden type models are subsequently defined and characterized by appropriate residuals numerical illustrations on various microstructures are finally provided in order to discuss the relevance of the proposed framework in particular it is shown that the surrogate model compares very well with reference results at both the macroscale and the structural scale even for moderate order expansions and that the aforementioned closest approximations constitute very accurate approximation provided that the subset onto which the homogenized response is projected is properly chosen this result readily allows for nonconcurrent coupling using standard constitutive models available in commercial codes and therefore makes the approach very attractive for engineering applications

learning a robust relevance model for search using kernel methods this paper points out that many search relevance models in information retrieval such as the vector space model bm25 and language models for information retrieval can be viewed as a similarity function between pairs of objects of different types referred to as an s function an s function is specifically defined as the dot product between the images of two objects in a hilbert space mapped from two different input spaces one advantage of taking this view is that one can take a unified and principled approach to address the issues with regard to search relevance the paper then proposes employing a kernel method to learn a robust relevance model as an s function which can effectively deal with the term mismatch problem one of the biggest challenges in search the kernel method exploits a positive semi definite kernel referred to as an s kernel the paper shows that when using an s kernel the model learned by the kernel method is guaranteed to be an s function the paper then gives more general principles for constructing s kernels a specific implementation of the kernel method is proposed using the ranking svm techniques and click through data the proposed approach is employed to learn a relevance model as an extension of bm25 referred to as robust bm25 experimental results on web search and enterprise search data show that robust bm25 significantly outperforms baseline methods and can successfully tackle the term mismatch problem

sharp oracle inequalities for high dimensional matrix prediction we observe xi yi i 1n where the yi s are real valued outputs and the xi s are m t matrices we observe a new entry x and we want to predict the output y associated with it we focus on the high dimensional setting where mt n this includes the matrix completion problem with noise as well as other problems we consider linear prediction procedures based on different penalizations involving a mixture of several norms the nuclear norm the frobenius norm and the l1 norm for these procedures we prove sharp oracle inequalities using a statistical learning theory point of view a surprising fact in our results is that the rates of convergence do not depend on m and t directly the analysis is conducted without the usually considered incoherency condition on the unknown matrix or restricted isometry condition on the sampling operator moreover our results are the first to give for this problem an analysis of penalization such as nuclear norm penalization as a regularization algorithm our oracle inequalities prove that these procedures have a prediction accuracy close to the deterministic oracle one given that the reguralization parameters are well chosen

clockwise nature of nyquist locus of stable transfer functions abstract f s is a finite proper rational function with real coefficients and all its poles in the interior of the left half plane the nyquist locus of f s is the mapping f s s j with in 0 it is shown that the nyquist locus of f s is clockwise at all in 0

structure of hard hypersphere fluids in odd dimensions the structural properties of single component fluids of hard hyperspheres in odd space dimensionalities d are studied with an analytical approximation method that generalizes the rational function approximation earlier introduced in the study of hard sphere fluids s b yuste and a santos phys rev a 43 5418 1991 the theory makes use of the exact form of the radial distribution function to first order in density and extends it to finite density by assuming a rational form for a function defined in laplace space the coefficients being determined by simple physical requirements fourier transform in terms of reverse bessel polynomials constitute the mathematical framework of this approximation from which an analytical expression for the static structure factor is obtained in its most elementary form the method recovers the solution of the percus yevick closure to the ornstein zernike equation for hyperspheres at odd dimensions the present formalism allows one to go beyond by yielding solutions with thermodynamic consistency between the virial and compressibility routes to any desired equation of state excellent agreement with available computer simulation data at d 5 and d 7 is obtained

upper bounds on the minimum distance of spherical codes we use linear programming techniques to obtain new upper bounds on the maximal squared minimum distance of spherical codes with fixed cardinality functions q sub j n s are introduced with the property that q sub j n s m if and only if the levenshtein bound l sub m n s on a n s max w w is an n w s code can be improved by a polynomial of degree at least m 1 general conditions on the existence of new bounds are presented we prove that for fixed dimension n spl ges 5 there exists a constant k k n such that all levenshtein bounds l sub m n s for m spl ges 2k 1 can be improved an algorithm for obtaining new bounds is proposed and discussed

optimal asymptotic bounds for spherical designs in this paper we prove the conjecture of korevaar and meyers for each n cdt d there exists a spherical t design in the sphere s d consisting of n points where cd is a constant depending only on d

matrix completion with covariate information this article investigates the problem of matrix completion from the corrupted data when the additional covariates are available despite being seldomly considered in the matrix completion literatu

when is there a representer theorem vector versus matrix regularizers we consider a general class of regularization methods which learn a vector of parameters on the basis of linear measurements it is well known that if the regularizer is a nondecreasing function of the l2 norm then the learned vector is a linear combination of the input data this result known as the representer theorem lies at the basis of kernel based methods in machine learning in this paper we prove the necessity of the above condition in the case of differentiable regularizers we further extend our analysis to regularization methods which learn a matrix a problem which is motivated by the application to multi task learning in this context we study a more general representer theorem which holds for a larger class of regularizers we provide a necessary and sufficient condition characterizing this class of matrix regularizers and we highlight some concrete examples of practical importance our analysis uses basic principles from matrix theory especially the useful notion of matrix nondecreasing functions

learning the kernel matrix with semidefinite programming kernel based learning algorithms work by embedding the data into a euclidean space and then searching for linear relations among the embedded data points the embedding is performed implicitly by specifying the inner products between each pair of points in the embedding space this information is contained in the so called kernel matrix a symmetric and positive semidefinite matrix that encodes the relative positions of all points specifying this matrix amounts to specifying the geometry of the embedding space and inducing a notion of similarity in the input space classical model selection problems in machine learning in this paper we show how the kernel matrix can be learned from data via semidefinite programming sdp techniques when applied to a kernel matrix associated with both training and test data this gives a powerful transductive algorithm using the labeled part of the data one can learn an embedding also for the unlabeled part the similarity between test points is inferred from training points and their labels importantly these learning problems are convex so we obtain a method for learning both the model class and the function without local minima furthermore this approach leads directly to a convex method for learning the 2 norm soft margin parameter in support vector machines solving an important open problem

toy model for the mean field theory of hard sphere liquids we investigate a toy model of liquid based on simplified hypernetted chain equations in very large spatial dimension d the model does not exhibit a phase transition but several regimes of behavior when d infinity can be observed in different intervals of the density

on kissing numbers and spherical codes in high dimensions abstract we prove a lower bound of d 3 2 2 3 d on the kissing number in dimension d this improves the classical lower bound of chabauty shannon and wyner by a linear factor in the dimension we obtain a similar linear factor improvement to the best known lower bound on the maximal size of a spherical code of acute angle in high dimensions

robust algorithm to generate a diverse class of dense disordered and ordered sphere packings via linear programming we have formulated the problem of generating dense packings of nonoverlapping nontiling nonspherical particles within an adaptive fundamental cell subject to periodic boundary conditions as an optimization problem called the adaptive shrinking cell asc formulation s torquato and y jiao phys rev e 80 041104 2009 because the objective function and impenetrability constraints can be exactly linearized for sphere packings with a size distribution in d dimensional euclidean space r d it is most suitable and natural to solve the corresponding asc optimization problem using sequential linear programming slp techniques we implement an slp solution to produce robustly a wide spectrum of jammed sphere packings in r d for d 2 3 4 5 and 6 with a diversity of disorder and densities up to the respective maximal densities a novel feature of this deterministic algorithm is that it can produce a broad range of inherent structures locally maximally dense and mechanically stable packings besides the usual disordered ones such as the maximally random jammed state with very small computational cost compared to that of the best known packing algorithms by tuning the radius of the influence sphere for example in three dimensions we show that it can produce with high probability a variety of strictly jammed packings with a packing density anywhere in the wide range 0 6 0 7408 where 18 0 7408 corresponds to the density of the densest packing we also apply the algorithm to generate various disordered packings as well as the maximally dense packings for d 2 4 5 and 6 our jammed sphere packings are characterized and compared to the corresponding packings generated by the well known lubachevsky stillinger ls molecular dynamics packing algorithm compared to the ls procedure our slp protocol is able to ensure that the final packings are truly jammed produces disordered jammed packings with anomalously low densities and is appreciably more robust and computationally faster at generating maximally dense packings especially as the space dimension increases

a new approach to collaborative filtering operator estimation with spectral regularization we present a general approach for collaborative filtering cf using spectral regularization to learn linear operators mapping a set of users to a set of possibly desired objects in particular several recent low rank type matrix completion methods for cf are shown to be special cases of our proposed framework unlike existing regularization based cf our approach can be used to incorporate additional information such as attributes of the users objects a feature currently lacking in existing regularization based cf approaches using popular and well known kernel methods we provide novel representer theorems that we use to develop new estimation methods we then provide learning algorithms based on low rank decompositions and test them on a standard cf data set the experiments indicate the advantages of generalizing the existing regularization based cf methods to incorporate related information about users and objects finally we show that certain multi task learning methods can be also seen as special cases of our proposed approach

rank regularization and bayesian inference for tensor completion and extrapolation a novel regularizer of the parafac decomposition factors capturing the tensor s rank is proposed in this paper as the key enabler for completion of three way data arrays with missing entries set in a bayesian framework the tensor completion method incorporates prior information to enhance its smoothing and prediction capabilities this probabilistic approach can naturally accommodate general models for the data distribution lending itself to various fitting criteria that yield optimum estimates in the maximum a posteriori sense in particular two algorithms are devised for gaussian and poisson distributed data that minimize the rank regularized least squares error and kullback leibler divergence respectively the proposed technique is able to recover the ground truth tensor rank when tested on synthetic data and to complete brain imaging and yeast gene expression datasets with 50 and 15 of missing entries respectively resulting in recovery errors at 11 db and 15 db

restricted p isometry properties of nonconvex matrix recovery recently a nonconvex relaxation of low rank matrix recovery lmr called the schatten p quasi norm minimization 0 p 1 was introduced instead of the previous nuclear norm minimization in order to approximate the problem of lmr closer in this paper we introduce a notion of the restricted p isometry constants 0 p 1 and derive a p rip condition for exact reconstruction of lmr via schatten p quasi norm minimization in particular we determine how many random gaussian measurements are needed for the p rip condition to hold with high probability which gives a theoretical result that it needs fewer measurements with small p for exact recovery via schatten p quasi norm minimization than when p 1

spectral regularization algorithms for learning large incomplete matrices we use convex relaxation techniques to provide a sequence of regularized low rank solutions for large scale matrix completion problems using the nuclear norm as a regularizer we provide a simple and very efficient convex algorithm for minimizing the reconstruction error subject to a bound on the nuclear norm our algorithm soft impute iteratively replaces the missing elements with those obtained from a soft thresholded svd with warm starts this allows us to efficiently compute an entire regularization path of solutions on a grid of values of the regularization parameter the computationally intensive part of our algorithm is in computing a low rank svd of a dense matrix exploiting the problem structure we show that the task can be performed with a complexity of order linear in the matrix dimensions our semidefinite programming algorithm is readily scalable to large matrices for example soft impute takes a few hours to compute low rank approximations of a 106 x 106 incomplete matrix with 107 observed entries and fits a rank 95 approximation to the full netflix training set in 3 3 hours our methods achieve good training and test errors and exhibit superior timings when compared to other competitive state of the art techniques

new upper bounds for kissing numbers from semidefinite programming recently a schrijver derived new upper bounds for binary codes using semidefinite programming in this paper we adapt this approach to codes on the unit sphere and we compute new upper bounds for the kissing number in several dimensions in particular our computations give the known values for the cases

a linear programming algorithm to test for jamming in hard sphere packings jamming in hard particle packings has been the subject of considerable interest in recent years in a paper by torquato and stillinger j phys chem b 105 2001 a classification scheme of jammed packings into hierarchical categories of locally collectively and strictly jammed configurations has been proposed they suggest that these jamming categories can be tested using numerical algorithms that analyze an equivalent contact network of the packing under applied displacements but leave the design of such algorithms as a future task in this work we present a rigorous and practical algorithm to assess whether an ideal hard sphere packing in two or three dimensions is jammed according to the aforementioned categories the algorithm is based on linear programming and is applicable to regular as well as random packings of finite size with hard wall and periodic boundary conditions if the packing is not jammed the algorithm yields representative multi particle unjamming motions furthermore we extend the jamming categories and the testing algorithm to packings with significant interparticle gaps we describe in detail two variants of the proposed randomized linear programming approach to test for jamming in hard sphere packings the first algorithm treats ideal packings in which particles form perfect contacts another algorithm treats the case of jamming in packings with significant interparticle gaps this extended algorithm allows one to explore more fully the nature of the feasible particle displacements we have implemented the algorithms and applied them to ordered as well as random packings of circular disks and spheres with periodic boundary conditions some representative results for large disordered disk and sphere packings are given but more robust and efficient implementations as well as further applications e g non spherical particles are anticipated for the future

trace norm regularization reformulations algorithms and multi task learning we consider a recently proposed optimization formulation of multi task learning based on trace norm regularized least squares while this problem may be formulated as a semidefinite program sdp its size is beyond general sdp solvers previous solution approaches apply proximal gradient methods to solve the primal problem we derive new primal and dual reformulations of this problem including a reduced dual formulation that involves minimizing a convex quadratic function over an operator norm ball in matrix space this reduced dual problem may be solved by gradient projection methods with each projection involving a singular value decomposition the dual approach is compared with existing approaches and its practical effectiveness is illustrated on simulations and an application to gene expression pattern analysis

monte carlo simulation of hard hyperspheres in six seven and eight dimensions for low to moderate densities the pair correlation function of hard hyperspheres in six seven and eight dimensions is obtained from monte carlo simulations the value of the pair correlation function at contact is compared with the results from molecular dynamics calculations and a variety of theoretical approaches remarkably good agreement is found with the simple closed form equations of y song e a mason and r m stratt j phys chem 93 6916 6919 1989 the monte carlo results for the equation of state are compared with the theoretical expressions of m baus and j l colot phys rev a 36 3912 1987 m luban and j p j michels phys rev a 41 6796 1990 and high order virial expansions in addition in seven dimensions comparisons are made with the exact py solution provided by m robles m l de haro and a santos j chem phys 120 9113 2004 very good agreement was observed between theory and computer simulation in all dimensions

solving inverse problems with piecewise linear estimators from gaussian mixture models to structured sparsity a general framework for solving image inverse problems with piecewise linear estimations is introduced in this paper the approach is based on gaussian mixture models which are estimated via a maximum a posteriori expectation maximization algorithm a dual mathematical interpretation of the proposed framework with a structured sparse estimation is described which shows that the resulting piecewise linear estimate stabilizes the estimation when compared with traditional sparse inverse problem techniques we demonstrate that in a number of image inverse problems including interpolation zooming and deblurring of narrow kernels the same simple and computationally efficient algorithm yields results in the same ballpark as that of the state of the art

local minima and convergence in low rank semidefinite programming the low rank semidefinite programming problem lrsdpr is a restriction of the semidefinite programming problem sdp in which a bound r is imposed on the rank of x and it is well known that lrsdpr is equivalent to sdp if r is not too small in this paper we classify the local minima of lrsdpr and prove the optimal convergence of a slight variant of the successful yet experimental algorithm of burer and monteiro 5 which handles lrsdpr via the nonconvex change of variables x rrt in addition for particular problem classes we describe a practical technique for obtaining lower bounds on the optimal solution value during the execution of the algorithm computational results are presented on a set of combinatorial optimization relaxations including some of the largest quadratic assignment sdps solved to date

complex networks on hyperbolic surfaces abstract we explore a novel method to generate and characterize complex networks by means of their embedding on hyperbolic surfaces evolution through local elementary moves allows the exploration of the ensemble of networks which share common embeddings and consequently share similar hierarchical properties this method provides a new perspective to classify network complexity both on local and global scale we demonstrate by means of several examples that there is a strong relation between the network properties and the embedding surface

low dimensional lattices vii coordination sequences the coordination sequence s n of a lattice or net gives the number of nodes that are n bonds away from a given node s 1 is the familiar coordination number extending the work of o keeffe and ot

convexity of frequency response arcs associated with a stable polynomial associated with a polynomial p s and an interval omega contained in r is a frequency response arc this arc is obtained by sweeping the frequency omega over omega and plotting p j omega in the complex plane it is said that an arc is proper if it does not pass through the origin and if the net phase change of p j omega as omega increases over omega is no more than 180 degrees the convexity of all proper frequency response arcs associated with a hurwitz polynomial is established the ramifications and extensions of arc convexity are discussed of particular interest is the fact that the so called inner frequency response set is convex this set consists of all points which can be connected to the origin via a continuous path which does not intersect the plot of p j omega for omega in r convexity of the inner frequency response set is shown to lead to an extreme point result for robust stability of a class of feedback systems having a structured unmodeled dynamic in the feedback path an extension of the arc convexity result for an arbitrary convex root location region d is included

universality of cluster dynamics we have studied the kinetics of cluster formation for dynamical systems of dimensions up to n 8 interacting through elastic collisions or coalescence these systems could serve as possible models for gas kinetics polymerization and self assembly in the case of elastic collisions we found that the cluster size probability distribution undergoes a phase transition at a critical time which can be predicted from the average time between collisions this enables forecasting of rare events based on limited statistical sampling of the collision dynamics over short time windows the analysis was extended to l p normed spaces p 1 to allow for some amount of interpenetration or volume exclusion the results for the elastic collisions are consistent with previously published low dimensional results in that a power law is observed for the empirical cluster size distribution at the critical time we found that the same power law also exists for all dimensions n 2 8 two dimensional l p norms and even for coalescing collisions in two dimensions this broad universality in behavior may be indicative of a more fundamental process governing the growth of clusters

geometric properties of random disk packings random packings ofn 2000 rigid disks in the plane subject to periodic boundary conditions on a square primitive cell have been generated by a concurrent construction which treats all disks on an equal footing as opposed to previously investigated sequential constructions the particles start with random positions and velocities and as they move about they grow uniformly in size from points to jammed disks the collection of packings displays several striking geometric features these include for largen typically polycrystalline textures with irregular grain boundaries and linear shear fractures the packings occasionally contain monovacancies and trapped but unjammed rattler disks the latter appear to be confined to the grain boundaries the linear shear fractures preserve bond orientational order but disrupt translational order within the crystalline grains a new efficient event driven simulation algorithm is employed to generate the histories of colliding and jamming disks on a computer which can process one million floating point instructions per second the algorithm processes more than one million pairwise collisions per hour

operator valued kernels for learning from functional response data in this paper we consider the problems of supervised classification and regression in the case where attributes and labels are functions a data is represented by a set of functions and the label is also a function we focus on the use of reproducing kernel hilbert space theory to learn from such functional data basic concepts and properties of kernel based learning are extended to include the estimation of function valued functions in this setting the representer theorem is restated a set of rigorously defined infinite dimensional operator valued kernels that can be valuably applied when the data are functions is described and a learning algorithm for nonlinear functional data analysis is introduced the methodology is illustrated through speech and audio signal processing experiments

pair correlation function characteristics of nearly jammed disordered and ordered hard sphere packings we study the approach to jamming in hard sphere packings and in particular the pair correlation function g2srd around contact both theoretically and computationally our computational data unambiguously separate the narrowing d function contribution to g2 due to emerging interparticle contacts from the background contribution due to near contacts the data also show with unprecedented accuracy that disordered hard sphere packings are strictly isostatic i e the number of exact contacts in the jamming limit is exactly equal to the number of degrees of freedom once rattlers are removed for such isostatic packings we derive a theoretical connection between the probability distribution of interparticle forces pfsfd which we measure computationally and the contact contribution to g2 we verify this relation for computationally generated isostatic packings that are representative of the maximally random jammed state we clearly observe a maximum in pf and a nonzero probability of zero force shedding light on long standing questions in the granular media literature we computationally observe an unusual power law divergence in the near contact contribution to g2 persistent even in the jamming limit with exponent 0 4 clearly distinguishable from previously proposed inversesquare root divergence additionally we present high quality numerical data on the two discontinuities in the split second peak of g2 and use a shared neighbor analysis of the graph representing the contact network to study the local particle clusters responsible for the peculiar features finally we present the computational data on the contact contribution to g2 for vacancy diluted fcc crystal packings and also investigate partially crystallized packings along the transition from maximally disordered to fully ordered packings we find that the contact network remains isostatic even when ordering is present unlike previous studies we find that ordering has a significant impact on the shape of pf for small forces

random sequential addition of hard spheres in high euclidean dimensions sphere packings in high dimensions have been the subject of recent theoretical interest employing numerical and theoretical methods we investigate the structural characteristics of random sequential addition rsa of congruent spheres in d dimensional euclidean space mathbb r d in the infinite time or saturation limit for the first six space dimensions 1 ensuremath le d ensuremath le 6 specifically we determine the saturation density pair correlation function cumulative coordination number and the structure factor in each of these dimensions we find that for 2 ensuremath le d ensuremath le 6 the saturation density ensuremath phi s scales with dimension as ensuremath phi s c 1 2 d c 2 d 2 d where c 1 0 202 phantom rule 0 2em 0ex 048 and c 2 0 973 phantom rule 0 2em 0ex 872 we also show analytically that the same density scaling is expected to persist in the high dimensional limit albeit with different coefficients a byproduct of this high dimensional analysis is a relatively sharp lower bound on the saturation density for any d given by ensuremath phi s ensuremath ge d 2 1 ensuremath s 0 2 d 1 where s 0 0 1 is the structure factor at k 0 i e infinite wavelength number variance in the high dimensional limit we demonstrate that a pal asti type conjecture the saturation density in mathbb r d is equal to that of the one dimensional problem raised to the d th power cannot be true for rsa hyperspheres we show that the structure factor s k must be analytic at k 0 and that rsa packings for 1 ensuremath le d ensuremath le 6 are nearly hyperuniform consistent with the recent decorrelation principle we find that pair correlations markedly diminish as the space dimension increases up to six we also obtain kissing contact number statistics for saturated rsa configurations on the surface of a d dimensional sphere for dimensions 2 ensuremath le d ensuremath le 5 and compare to the maximal kissing numbers in these dimensions we determine the structure factor exactly for the related ghost rsa packing in mathbb r d and demonstrate that its distance from hyperuniformity increases as the space dimension increases approaching a constant asymptotic value of 1 2 our work has implications for the possible existence of disordered classical ground states for some continuous potentials in sufficiently high dimensions