reformulation of the covering and quantizer problems as ground states of interacting particles it is known that the sphere packing problem and the number variance problem closely related to an optimization problem in number theory can be posed as energy minimizations associated with an infinite number of point particles in d dimensional euclidean space r d interacting via certain repulsive pair potentials we reformulate the covering and quantizer problems as the determination of the ground states of interacting particles in r d that generally involve single body two body three body and higher body interactions this is done by linking the covering and quantizer problems to certain optimization problems involving the void nearest neighbor functions that arise in the theory of random media and statistical mechanics these reformulations which again exemplify the deep interplay between geometry and physics allow one now to employ theoretical and numerical optimization techniques to analyze and solve these energy minimization problems the covering and quantizer problems have relevance in numerous applications including wireless communication network layouts the search of high dimensional data parameter spaces stereotactic radiation therapy data compression digital communications meshing of space for numerical analysis and coding and cryptography among other examples in the first three space dimensions the best known solutions of the sphere packing and number variance problems or their dual solutions are directly related to those of the covering and quantizer problems but such relationships may or may not exist for d 4 depending on the peculiarities of the dimensions involved our reformulation sheds light on the reasons for these similarities and differences we also show that disordered saturated sphere packings provide relatively thin economical coverings and may yield thinner coverings than the best known lattice coverings in sufficiently large dimensions in the case of the quantizer problem we derive improved upper bounds on the quantizer error using sphere packing solutions which are generally substantially sharper than an existing upper bound in low to moderately large dimensions we also demonstrate that disordered saturated sphere packings yield relatively good quantizers finally we remark on possible applications of our results for the detection of gravitational waves

new results on stability analysis for systems with discrete distributed delay the integral inequality technique is widely used to derive delay dependent conditions and various integral inequalities have been developed to reduce the conservatism of the conditions derived in this study a new integral inequality was devised that is tighter than existing ones it was used to investigate the stability of linear systems with a discrete distributed delay and a new stability condition was established the results can be applied to systems with a delay belonging to an interval which may be unstable when the delay is small or nonexistent three numerical examples demonstrate the effectiveness and the smaller conservatism of the method

model construction rule reduction and robust compensation for generalized form of takagi sugeno fuzzy systems this paper presents a systematic procedure of fuzzy control system design that consists of fuzzy model construction rule reduction and robust compensation for nonlinear systems the model construction part replaces the nonlinear dynamics of a system with a generalized form of takagi sugeno fuzzy systems which is newly developed by us the generalized form has a decomposed structure for each element of a sub i and b sub i matrices in consequent parts the key feature of this structure is that it is suitable for constructing if then rules and reducing the number of if then rules the rule reduction part provides a successive procedure to reduce the number of if then rules furthermore we convert the reduction error between reduced fuzzy models and a system to model uncertainties of reduced fuzzy models the robust compensation part achieves the decay rate controller design guaranteeing robust stability for the model uncertainties finally two examples demonstrate the utility of the systematic procedure developed

generalized caristi s fixed point theorems by bae and others abstract in this paper we extend the generalized caristi s fixed point theorems proved by bae j math anal appl 284 2003 690 697 and others

dense packings of polyhedra platonic and archimedean solids understanding the nature of dense particle packings is a subject of intense research in the physical mathematical and biological sciences the preponderance of previous work has focused on spherical particles and very little is known about dense polyhedral packings we formulate the problem of generating dense packings of nonoverlapping nontiling polyhedra within an adaptive fundamental cell subject to periodic boundary conditions as an optimization problem which we call the adaptive shrinking cell asc scheme this optimization problem is solved here using a variety of multiparticle initial configurations to find the dense packings of each of the platonic solids in three dimensional euclidean space mathbb r 3 except for the cube which is the only platonic solid that tiles space we find the densest known packings of tetrahedra icosahedra dodecahedra and octahedra with densities 0 823 dots 0 836 dots 0 904 dots and 0 947 dots respectively it is noteworthy that the densest tetrahedral packing possesses no long range order unlike the densest tetrahedral packing which must not be a bravais lattice packing the densest packings of the other nontiling platonic solids that we obtain are their previously known optimal bravais lattice packings we also derive a simple upper bound on the maximal density of packings of congruent nonspherical particles and apply it to platonic solids archimedean solids superballs and ellipsoids provided that what we term the asphericity ratio of the circumradius to inradius is sufficiently small the upper bounds are relatively tight and thus close to the corresponding densities of the optimal lattice packings of the centrally symmetric platonic and archimedean solids our simulation results rigorous upper bounds and other theoretical arguments lead us to the conjecture that the densest packings of platonic and archimedean solids with central symmetry are given by their corresponding densest lattice packings this can be regarded to be the analog of kepler s sphere conjecture for these solids the truncated tetrahedron is the only non centrally symmetric archimedean solid the densest known packing of which is a non lattice packing with density at least as high as 23 24 0 958 text 333 dots we discuss the validity of our conjecture to packings of superballs prisms and antiprisms as well as to high dimensional analogs of the platonic solids in addition we conjecture that the optimal packing of any convex congruent polyhedron without central symmetry generally is not a lattice packing finally we discuss the possible applications and generalizations of the asc scheme in predicting the crystal structures of polyhedral nanoparticles and the study of random packings of hard polyhedra

optimality and uniqueness of the leech lattice among lattices we prove that the leech lattice is the unique densest lattice in r24 the proof combines human reasoning with computer verification of the properties of certain explicit polynomials we furthermore prove that no sphere packing in r24 can exceed the leech lattice s density by a factor of more than 1 1 65 10 30 and we give a new proof that es is the unique densest lattice in r8

on delay dependent approach for robust stability and stabilization of t s fuzzy systems with constant delay and uncertainties this paper investigates robust stability analysis and stabilization of delay and uncertain systems approximated by a takagi sugeno t s fuzzy model an innovative approach is proposed to develop delay dependent stability criteria of the systems which makes use of less redundant information to construct lyapunov function employs an integral equation method to handle the cross product terms and alleviates the requirements of the bounding technique and model transformations that have been popularly adopted in many existing references this leads to significant improvement in the stability performance with far fewer unknown variables in the stability computation from the derived stability criteria a new memoryless state feedback control is further developed the controller gain and the maximum allowable delay bound of the closed loop control system can be obtained simultaneously by solving an optimization problem numerical examples are also given to demonstrate the theoretical results

an exact method for the stability analysis of time delayed linear time invariant lti systems a general class of linear time invariant systems with time delay is studied recently they attracted considerable interest in the systems and control community the complexity arises due to the exponential type transcendental terms in their characteristic equation the transcendentality brings infinitely many characteristic roots which are cumbersome to elaborate as evident from the literature a number of methodologies have been suggested with limited ability to assess the stability in the parametric domain of time delay this study offers an exact structured and robust methodology to bring a closure to the question at hand ultimately we present a unique explicit analytical expression in terms of the system parameters which not only reveals the stability regions pockets in the domain of time delay but it also declares the number of unstable characteristic roots at any given pocket the method starts with the determination of all possible purely imaginary resonant characteristic roots for any positive time delay to achieve this a simplifying substitution is used for the transcendental terms in the characteristic equation it is proven that the number of such resonant roots for a given dynamics is finite each one of these roots is created by infinitely many time delays which are periodically distributed an interesting property is also claimed next that the root crossing directions at these locations are invariant with respect to the delay and dependent only on the crossing frequency these two unique findings facilitate a simple and practical stability method which is the highlight of the work

delay induced stability of vector second order systems via simple lyapunov functionals it is well known that some important classes of systems e g inverted pendulums oscillators double integrators that cannot be stabilized by a static output feedback may be stabilized by inserting an artificial time delay in the feedback static output feedback controllers have advantages over observer based controllers in the presence of uncertainties in the system matrices and or uncertain input output delays where the observer based design becomes complicated the existing lyapunov based methods that may treat the case of stabilizing delays and that lead to stability conditions in terms of linear matrix inequalities lmis suffer from high dimensionality of the resulting lmis with a large number of decision variables in this paper we suggest simple lyapunov functionals for vector second order systems with stabilizing delays that lead to reduced order lmis with a small number of decision variables moreover differently from the existing methods we show that the presented lmis are always feasible for small enough delays

microstructural degeneracy associated with a two point correlation function and its information content a two point correlation function provides a crucial yet an incomplete characterization of a microstructure because distinctly different microstructures may have the same correlation function in an earlier letter gommes jiao and torquato phys rev lett 108 080601 2012 we addressed the microstructural degeneracy question what is the number of microstructures compatible with a specified correlation function we computed this degeneracy i e configurational entropy in the framework of reconstruction methods which enabled us to map the problem to the determination of ground state degeneracies here we provide a more comprehensive presentation of the methodology and analyses as well as additional results since the configuration space of a reconstruction problem is a hypercube on which a hamming distance is defined we can calculate analytically the energy profile of any reconstruction problem corresponding to the average energy of all microstructures at a given hamming distance from a ground state the steepness of the energy profile is a measure of the roughness of the energy landscape associated with the reconstruction problem which can be used as a proxy for the ground state degeneracy the relationship between this roughness metric and the ground state degeneracy is calibrated using a monte carlo algorithm for determining the ground state degeneracy of a variety of microstructures including realizations of hard disks and poisson point processes at various densities as well as those with known degeneracies e g single disks of various sizes and a particular crystalline microstructure we show that our results can be expressed in terms of the information content of the two point correlation functions from this perspective the a priori condition for a reconstruction to be accurate is that the information content expressed in bits should be comparable to the number of pixels in the unknown microstructure we provide a formula to calculate the information content of any two point correlation function which makes our results broadly applicable to any field in which correlation functions are employed

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

complete stability robustness of third order lti multiple time delay systems a unique procedure is presented in this paper for a complete stability robustness of the third order lti multiple time delay systems lti mtds the uniqueness of the treatment is simply due to the fact that there is no comparable methodology presently in the literature the end result of this procedure is an exhaustive and precise determination of the stable regions in the domain of time delays the backbone of the method is a novel framework called the cluster treatment of characteristic roots ctcr ctcr is constructed over two fundamental propositions the first proposition claims the existence of a bounded number of so called kernel curves where the only imaginary characteristic roots occur the second proposition is on an interesting directional invariance property of the crossing tendencies of these imaginary roots for simplicity of conveyance and without loss of generality the number of time delays is taken as two in this document the new methodology is expandable to higher order dynamics with more time delays than two as the authors intend to demonstrate in future publications publications

wirtinger based integral inequality application to time delay systems in the last decade the jensen inequality has been intensively used in the context of time delay or sampled data systems since it is an appropriate tool to derive tractable stability conditions expressed in terms of linear matrix inequalities lmis however it is also well known that this inequality introduces an undesirable conservatism in the stability conditions and looking at the literature reducing this gap is a relevant issue and always an open problem in this paper we propose an alternative inequality based on the fourier theory more precisely on the wirtinger inequalities it is shown that this resulting inequality encompasses the jensen one and also leads to tractable lmi conditions in order to illustrate the potential gain of employing this new inequality with respect to the jensen one two applications on time delay and sampled data stability analysis are provided

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

new family of tilings of three dimensional euclidean space by tetrahedra and octahedra abstract it is well known that two regular tetrahedra can be combined with a single regular octahedron to tile complete fill three dimensional euclidean space this structure was called the octet truss by buckminster fuller it was believed that such a tiling which is the delaunay tessellation of the face centered cubic fcc lattice and its closely related stacking variants are the only tessellations of that involve two different regular polyhedra here we identify and analyze a unique family comprised of a noncountably infinite number of periodic tilings of whose smallest repeat tiling unit consists of one regular octahedron and six smaller regular tetrahedra we first derive an extreme member of this unique tiling family by showing that the holes in the optimal lattice packing of octahedra obtained by minkowski over a century ago are congruent tetrahedra this tiling has 694 distinct concave i e nonconvex repeat units 24 of which possess central symmetry and hence is distinctly different and combinatorically richer than the fcc tetrahedra octahedra tiling which only has two distinct tiling units then we construct a one parameter family of octahedron packings that continuously spans from the fcc to the optimal lattice packing of octahedra we show that the holes in these packings except for the two extreme cases are tetrahedra of two sizes leading to a family of periodic tilings with units composed four small tetrahedra and two large tetrahedra that contact an octahedron these tilings generally possess 2 068 distinct concave tiling units 62 of which are centrally symmetric

on the relation between the weak palais smale condition and coercivity given by zhong abstract in this paper we discuss zhong s result that the weak palais smale condition implies coercivity under some assumption given in c k zhong a generalization of ekeland s variational principle and application to the study of the relation between the weak p s condition and coercivity nonlinear anal 29 1997 1421 1431 we also give a simple proof of zhong s result further we generalize the result of caklovic li and willem l caklovic s j li m willem a note on palais smale condition and coercivity differential integral equations 3 1990 799 800

viscosity solutions of hamilton jacobi equations with unbounded nonlinear terms abstract the aim of this paper is to introduce a new definition of viscosity solutions for hamilton jacobi equations in infinite dimensions with unbounded nonlinear terms we prove existence and uniqueness results under usual assumptions regarding the bounded part of the hamiltonian

efficient linear programming algorithm to generate the densest lattice sphere packings finding the densest sphere packing in d dimensional euclidean space r d is an outstanding fundamental problem with relevance in many fields including the ground states of molecular systems colloidal crystal structures coding theory discrete geometry number theory and biological systems numerically generating the densest sphere packings becomes very challenging in high dimensions due to an exponentially increasing number of possible sphere contacts and sphere configurations even for the restricted problem of finding the densest lattice sphere packings in this paper we apply the torquato jiao packing algorithm which is a method based on solving a sequence of linear programs to robustly reproduce the densest known lattice sphere packings for dimensions 2 through 19 we show that the tj algorithm is appreciably more efficient at solving these problems than previously published methods indeed in some dimensions the former procedure can be as much as three orders of magnitude faster at finding the optimal solutions than earlier ones we also study the suboptimal local density maxima solutions inherent structures or extreme lattices to gain insight about the nature of the topography of the density landscape

packing tiling and covering with tetrahedra it is well known that three dimensional euclidean space cannot be tiled by regular tetrahedra but how well can we do in this work we give several constructions that may answer the various senses of this question in so doing we provide some solutions to packing tiling and covering problems of tetrahedra our results suggest that the regular tetrahedron may not be able to pack as densely as the sphere which would contradict a conjecture of ulam the regular tetrahedron might even be the convex body having the smallest possible packing density

anomalous local coordination density fluctuations and void statistics in disordered hyperuniform many particle ground states we provide numerical constructions of one dimensional hyperuniform many particle distributions that exhibit unusual clustering and asymptotic local number density fluctuations growing more slowly than the volume of an observation window but faster than the surface area hyperuniformity defined by vanishing infinite wavelength local density fluctuations provides a quantitative metric of global order within a many particle configuration and signals the onset of an inverted critical point in which the direct correlation function becomes long ranged by targeting a specified form of the structure factor at small wavenumbers s k k for 0 1 using collective density variables we are able to tailor the form of asymptotic local density fluctuations while simultaneously measuring the effect of imposing weak and strong constraints on the available degrees of freedom within the system this procedure is equivalent to finding the possibly disordered classical ground state of an interacting many particle system with up to four body interactions even in one dimension the long range effective interactions induce clustering and nontrivial phase transitions in the resulting ground state configurations we provide an analytical connection between the fraction of constrained degrees of freedom within the system and the disorder order phase transition for a class of target structure factors by examining the realizability of the constrained contribution to the pair correlation function our results explicitly demonstrate that disordered hyperuniform many particle ground states and therefore also point distributions with substantial clustering can be constructed we directly relate the local coordination structure of our point patterns to the distribution of the void space external to the particles and we provide a scaling argument for the configurational entropy analogous to spin frustated system of the disordered ground states by emphasizing the intimate connection between geometrical constraints on the particle distribution and structural regularity our work has direct implications for higher dimensional systems including an understanding of the appearance of hyperuniformity and quasi long range pair correlations in maximally random strictly jammed packings of hard spheres

exact results for the jammed state of binary mixtures of superdisks on the plane abstract by analytical and numerical methods we investigate the late stage deposition of binary mixtures of oriented superdisks on a plane superdisks are objects bounded by lame curves x 2 p y 2 p 1 where deformation parameter p controls their size and shape for deposition of single type superdisks the maximum packing and jamming densities are known to be nonanalytic at p 0 5 for binary mixtures of superdisks we discover that nonanalyticities form a locus of points separating phase diagram of shape combinations into regions with different excluded area constructions an analytical expression for this phase boundary and exact constructions of the excluded areas are presented the corresponding saturation coverages are obtained by extensive numerical monte carlo simulations

note on a general palais smale condition abstract in this paper we establish a qualitative deformation theorem for a locally lipschitz function satisfying a general palais smale type condition then we show that with some assumptions this compactness condition implies the coercivity of the function

centroidal voronoi tessellations applications and algorithms a centroidal voronoi tessellation is a voronoi tessellation whose generating points are the centroids centers of mass of the corresponding voronoi regions we give some applications of such tessellations to problems in image compression quadrature finite difference methods distribution of resources cellular biology statistics and the territorial behavior of animals we discuss methods for computing these tessellations provide some analyses concerning both the tessellations and the methods for their determination and finally present the results of some numerical experiments

asymptotically necessary and sufficient conditions for stability and performance in fuzzy control applications of polya s theorem stability and performance requirements in fuzzy control of takagi sugeno systems are usually stated as fuzzy summations i e sums of terms related to lyapunov functions which are weighted by membership functions or products of them this paper presents an application to fuzzy control of polya s theorems on positive forms on the standard simplex the result provides a set of progressively less conservative sufficient conditions for proving positivity of fuzzy summations such conditions are less and less conservative as a complexity parameter n increases particular cases of such conditions are those in c h fang y s liu s w kau l hong c h lee a new lmi based approach to relaxed quadratic stabilization of t s fuzzy control systems ieee trans fuzzy systems 14 2006 286 397 x liu q zhang new approaches to h controller designs based on fuzzy observers for t s fuzzy systems via lmi automatica 39 9 2003 1571 1582 with n 2 and 3 respectively the proposed conditions are asymptotically exact i e necessary and sufficient when n tends to infinity or equivalently when a tolerance parameter tends to zero

polytopes of polynomials with zeros in a prescribed set in the publication by a c bartlett c v holot and h lin proc amer contr conf minneapolis mn 1987 a fundamental result is established on the zero locations of a family of polynomials it is shown that the zeros of a polytope p of nth order real polynomials are contained in a simply connected set d if and only if the zeros of all polynomials along the edges of p are contained in d the present authors are motivated by the fact that the requirement of simple connectedness of d may be too restrictive and applications such as dominant pole assignment and filter design where the separation of zeros is required they extend the edge criterion of bartlett et al to handle any set d whose complement d sup c has the following property every point d in d sup c lies on some continuous path which remains within d sup c and is unbounded this requirement is typically verified by inspection and allows for a large class of disconnected sets

hyperuniformity quasi long range correlations and void space constraints in maximally random jammed particle packings i polydisperse spheres hyperuniform many particle distributions possess a local number variance that grows more slowly than the volume of an observation window implying that the local density is effectively homogeneous beyond a few characteristic length scales previous work on maximally random strictly jammed sphere packings in three dimensions has shown that these systems are hyperuniform and possess unusual quasi long range pair correlations decaying as r 4 resulting in anomalous logarithmic growth in the number variance however recent work on maximally random jammed sphere packings with a size distribution has suggested that such quasi long range correlations and hyperuniformity are not universal among jammed hard particle systems in this paper we show that such systems are indeed hyperuniform with signature quasi long range correlations by characterizing the more general local volume fraction fluctuations we argue that the regularity of the void space induced by the constraints of saturation and strict jamming overcomes the local inhomogeneity of the disk centers to induce hyperuniformity in the medium with a linear small wave number nonanalytic behavior in the spectral density resulting in quasi long range spatial correlations scaling with r d 1 in d euclidean space dimensions a numerical and analytical analysis of the pore size distribution for a binary maximally random jammed system in addition to a local characterization of the n particle loops governing the void space surrounding the inclusions is presented in support of our argument this paper is the first part of a series of two papers considering the relationships among hyperuniformity jamming and regularity of the void space in hard particle packings

generalized distance and existence theorems in complete metric spaces abstract in this paper we first introduce the concept of distance on a metric space which is a generalized concept of both w distance and tataru s distance we also improve the generalizations of the banach contraction principle caristi s fixed point theorem ekeland s variational principle and the nonconvex minimization theorem according to takahashi further we discuss the relation between w distance and tataru s distance

improved reconstructions of random media using dilation and erosion processes by using the most sensitive two point correlation functions introduced to date we reconstruct the microstructures of two phase random media with heretofore unattained accuracy such media arise in a host of contexts including porous and composite media ecological structures biological media and astrophysical structures the aforementioned correlation functions are special cases of the so called canonical n point correlation function h n and generalize the ones that have been recently employed to advance our ability to reconstruct complex microstructures y jiao f h stillinger and s torquato proc natl acad sci 106 17634 2009 the use of these generalized correlation functions is tantamount to dilating or eroding a reference phase of the target medium and incorporating the additional topological information of the modified media thereby providing more accurate reconstructions of percolating filamentary and other topologically complex microstructures we apply our methods to a multiply connected donut medium and a dilute distribution of cracks a set of essentially zero measure demonstrating improved accuracy in both cases with implications for higher dimensional and biconnected two phase systems the high information content of the generalized two point correlation functions suggests that it would be profitable to explore their use to characterize the structural and physical properties of not only random media but also molecular systems including structural glasses

quasi polynomial based robust stability of time delay systems can be less conservative than lyapunov krasovskii approaches in this article stability of uncertain time delay unforced systems is analyzed through a generalized characteristic quasi polynomial which is exactly rewritten as a polytope whose interpolating functions exhibit mutual dependence it is shown that an adequate combination of kharitonov like results and polya like relaxations leads to progressively better results at the price of more demanding computational burden stability methodologies are developed and proved advantageous when compared with traditional lyapunov based approaches both for structured and unstructured uncertainty

new upper bounds on sphere packings i we continue the study of the linear programming bounds for sphere packing introduced by cohn and elkies we use theta series to give another proof of the principal theorem and present some related results and conjectures this article is in the arxiv as arxiv math mg 0110010

divide and concur a general approach to constraint satisfaction many difficult computational problems involve the simultaneous satisfaction of multiple constraints that are individually easy to satisfy these constraints might be derived from measurements as in tomography or diffractive imaging interparticle interactions as in spin glasses or a combination of sources as in protein folding we present a simple geometric framework to express and solve such problems and apply it to two benchmarks in the first application 3sat a boolean satisfaction problem the resulting method exhibits similar performance scaling as a leading context specific algorithm walksat in the second application sphere packing the method allowed us to find improved solutions to some old and well studied optimization problems based upon its simplicity and observed efficiency we argue that this framework provides a competitive alternative to stochastic methods such as simulated annealing

optimization of lattices for quantization a training algorithm for the design of lattices for vector quantization is presented the algorithm uses a steepest descent method to adjust a generator matrix in the search for a lattice whose voronoi regions have minimal normalized second moment the numerical elements of the found generator matrices are interpreted and translated into exact values experiments show that the algorithm is stable in the sense that several independent runs reach equivalent lattices the obtained lattices reach as low second moments as the best previously reported lattices or even lower specifically we report lattices in nine and ten dimensions with normalized second moments of 0 0716 and 0 0708 respectively and nonlattice tessellations in seven and nine dimensions with 0 0727 and 0 0711 which improves on previously known values the new nine and ten dimensional lattices suggest that conway and sloane s 1993 conjecture on the duality between the optimal lattices for packing and quantization might be false a discussion of the application of lattices in vector quantizer design for various sources uniform and nonuniform is included

asymptotic quantization error of continuous signals and the quantization dimension extensions of the limiting qnanfizafion error formula of bennet are proved these are of the form d s k n f n beta b where n is the number of output levels d s k n f is the s th moment of the metric distance between quantizer input and output beta b 0 k s beta is the signal space dimension and f is the signal distribution if a suitably well behaved k dimensional signal density f x exists b b s k int f rho x dx 1 rho rho k s k and b s k does not depend on f for k 1 s 2 this reduces to bennett s formula if f is the cantor distribution on 0 1 0 and this k equals the fractal dimension of the cantor set 12 13 random quantization optimal quantization in the presence of an output information constraint and quantization noise in high dimensional spaces are also investigated

subrahmanyam s fixed point theorem abstract we give a sufficient and necessary condition for the convergence of the sequence of successive approximations to a fixed point which is the conclusion of subrahmanyam s fixed point theorem

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

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

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