doubly stochastic matrices with equal subpermanents abstract it has been conjectured that if a is a doubly stochastic n n matrix such that per a i j pera for all i j then either a j n then n n matrix with each entry equal to 1 n or up to permutations of rows and columns a 1 2 i n p n where p n is a full cycle permutation matrix this conjecture is proved

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

information cascade with marginal stability in a network of chaotic elements abstract a newly discovered cascade process of clusters is studied in a network of chaotic elements it is shown that the splitting of clusters and the synchronization of elements are balanced in a class of partially ordered states where marginal stability is sustained over an interval of the bifurcation parameters the partition information creation in bit space shows an avalanche process of information which leads to the anomalous behavior of power spectra roughly fitted by a power law form of the wavenumber lyapunov spectra have accumulation at null exponents analogous with those studied in fluid turbulence models

geometrical frustration in two dimensions idealizations and realizations of a hard disk fluid in negative curvature we examine a simple hard disk fluid with no long range interactions on the two dimensional space of constant negative gaussian curvature the hyperbolic plane this geometry provides a natural mechanism by which global crystalline order is frustrated allowing us to construct a tractable one parameter model of disordered monodisperse hard disks we extend free area theory and the virial expansion to this regime deriving the equation of state for the system and compare its predictions with simulations near an isostatic packing in the curved space additionally we investigate packing and dynamics on triply periodic negatively curved surfaces with an eye toward real biological and polymeric systems

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

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

new integral estimates for deformations in terms of their nonlinear strains if u is a bi lipschitzian deformation of a bounded lipschitz domain in ln n 2 we show that the lp norm p 1 p n of a certain nonlinear strain function e u associated with u dominates the distance in lq q np n p if p n from u to a suitably chosen rigid motion of n this work extends that of f john who proved corresponding estimates for p 1 under the hypothesis that u has uniformly small strain we also obtain a bound for the oscillation of du in l2 these estimates are apparently the first to apply with no a priori pointwise hypotheses upon the strain of u in 3 the integral int limits omega e u 2d 3 is dominated by typical hyperelastic energy functionals proposed in the literature for modeling the behavior of rubber thus the case n 3 p 2 gives the first general bound for the deformations of such materials in terms of the associated nonlinear elastic work

spatiotemporal chaos in a coupled map lattice with unstable couplings we study the complex spatiotemporal behaviour of a coupled map lattice with a one humped chaotic map and an unstable laplacian coupling bifurcations are numerically investigated and interpreted using low dimensional approximations corresponding to the relevant degrees of freedom of the infinite dimensional system varying the control parameter we find different phases in the chaotic domain such as localized chaos or propagating chaos spatiotemporal intermittency and transient chaos according to our results unstable coupling leads to a number of new features including the effects that i the first bifurcation becomes discontinuous and ii the chaotic regime sets in sooner than for a single map

lyapunov analysis and information flow in coupled map lattices abstract lyapunov analysis for the coupled map lattices is presented the co moving lyapunov exponent is calculated which is related with the propagation of the disturbance in space the propagation speed agrees with the zero crossing point of the co moving lyapunov exponent the co moving mutual information flow is introduced which shows the selective transmission of the information at some speed lyapunov spectra and vectors are calculated spatial structures of the vectors are investigated possible analogy with the anderson localization problem is discussed

pattern selection at high nonlinearity in a diffusively coupled logistic lattice a different pattern selection regime is shown to exist in regions of high nonlinearity in the diffusively coupled logistic lattice with nearest neighbor interaction for this regime the phase diagram of several prominent attractors is determined the results indicate that the spatiotemporal chaos in some of the regions so far believed to be areas of fully developed spatiotemporal chaos is of a transient nature

pattern dynamics in spatiotemporal chaos pattern selection diffusion of defect and pattern competition intermettency abstract a class of coupled map lattices is investigated as a model for the spatiotemporal chaos as the parameters are changed phase transitions among the patterns are observed pattern selection through the chaotic motion of domain is observed chaos is suppressed by the selection selection of the zigzag pattern is studied where localized defects which separate the two anti phase domains are found they change chaotically in time and move randomly in space the diffusion of defect is related with its chaotic motion collapse of zigzag pattern and the defect turbulence are studied in connection with the crisis in high dimensional dynamical systems pattern selection with longer wavelengths is found in the stronger coupling regime transition from pattern selection phase to fully developed turbulence occurs via the intermittency selective flicker like noise and the pareto distribution of lifetime of pattern are observed at the intermittency regime for the quantitative characterization of each phase pattern distributions pattern transition matrix static and dynamical entropies lifetime distribution of pattern spatiotemporal power spectra and lyapunov spectra are calculated phase transition from the pattern selection to fully developed turbulence is investigated with the use of these quantifiers where some order parameters and critical indeces are introduced

notes on egoritsjev s proof of the van der waerden conjecture abstract a solution to the van der waerden conjecture on the minimum of the permanent of doubly stochastic matrices has recently been found by g p ergoritsjev in russian because of the wide interest in this conjecture it seems useful to make the solution more accessible professor van lint has kindly consented to have his notes including a direct proof of an inequality available in the literature only in a more general form published in this journal ed r a b

complex spatiotemporal patterns in two lattice models with instability abstract in this paper we consider two lattice models i the discretized version of a simple nonlinear partial differential equation pde with a singular term and ii a coupled map lattice cml with unstable coupling both systems are defined by simple iteration rules involving the nearest neighbour grid points of the corresponding one dimensional lattice we show numerically that these models result in an apparently stochastic spatiotemporal behaviour and exhibit interesting regimes as the function of the parameters of the models in particular we find that by varying the relative weight b of the singular term in the pde the total width of the solution scales with b the cml leads to an intermittent behaviour between structured and turbulent regimes

deterministic and stochastic surface growth with generalized nonlinearity the scaling behavior of the interface width for the generalized kardar parisi zhang kpz equation ensuremath partial h ensuremath partial t ensuremath nu mathrm ensuremath nabla 2 h ensuremath lambda ensuremath vert ensuremath nabla h mathrm ensuremath vert mathrm ensuremath mu ensuremath eta x t is studied in two dimensions as a function of ensuremath mu with and without the additive noise term ensuremath eta in the case of additive noise the scaling of the surface width is found for all ensuremath mu to be the same as for the ordinary kpz equation ensuremath mu 2 in contrast to a previous conjecture this appears to be due to the combined action of the nonlinearity ensuremath vert ensuremath nabla h mathrm ensuremath vert mathrm ensuremath mu and the noise under renormalization which together induce a ensuremath vert ensuremath nabla h mathrm ensuremath vert 2 term for the deterministic case corresponding to the smoothing of an initially rough interface with roughness exponent ensuremath alpha good agreement with the scaling relation z min 2 ensuremath alpha 1 ensuremath mu ensuremath mu is obtained for ensuremath mu ensuremath ge 1 however for ensuremath mu 1 an instability is observed which leads to a fluctuating grooved surface for an asymptotically large system the roughness exponent is 1 and the growth exponent is approximately equal to 1 2 the evolution of two slightly different surfaces is studied in order to determine a lyapunov exponent characterizing this instability

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

a matrix of permanents abstract if a is a matrix of order n n 2 n 3 denote by the n n matrix whose i j th entry is zero if i j and if i j is the permanent of the submatrix of a obtained by deleting its ith and jth rows it is shown that if a is a nonnegative n n 2 matrix then is nonsingular if and only if a has no zero submatrix of n 1 lines this is used to give precise consequences of the occurrence of equality in alexandroff s inequality

the cauchy problem for ut δu u q with q a positive real number the nonlinear partial differential equation in the title of the paper arises in the study of the growth of surfaces in that context it is known as the generalized deterministic kpz equation the paper is concerned with the initial value problem for the equation under the assumption that the initial data function is bounded and continuous results on the existence uniqueness and regularity of solutions are obtained