an algorithm for a constrained weber problem we present an algorithm for solving the weber problem when the set of feasible locations is the union of a finite number of convex polygons

on the multifractal nature of fully developed turbulence and chaotic systems it is generally argued that the energy dissipation of three dimensional turbulent flow is concentrated on a set with non integer hausdorff dimension recently in order to explain experimental data it has been proposed that this set does not possess a global dilatation invariance it can be considered to be a multifractal set the authors review the concept of multifractal sets in both turbulent flows and dynamical systems using a generalisation of the beta model

least squares metric unidimensional scaling of multivariate linear models the squared error loss function for the unidimensional metric scaling problem has a special geometry it is possible to efficiently find the global minimum for every coordinate conditioned on every other coordinate being held fixed this approach is generalized to the case in which the coordinates are polynomial functions of exogenous variables the algorithms shown in the paper are linear in the number of parameters they always descend and at convergence every coefficient of every polynomial is at its global minimum conditioned on every other parameter being held fixed convergence is very rapid and monte carlo tests show the basic procedure almost always converges to the overall global minimum

lp stability of solutions of boltzmann equation with external force in soft potentials in this paper we present a uniform lp stability theory for the solutions to boltzmann equation with small or large external forces in vacuum in soft potentials we assume that the initial data are sufficiently small and a fast decay in space and velocity our stability analysis is divided into two parts for small external forces we first consider the time derivative of the pth power of lp norm of the distance between two mild solutions f and g then define nonlinear functionals and apply the gronwall s inequality to obtain the lp stability for large external forces we make use of the constructive assumptions on external force minkowski s inequality and gronwall s inequality to derive the lp stability estimate

decay of the two species vlasov poisson boltzmann system specialized research fund for the doctoral program of higher education 20120121120023 natural science foundation of fujian province of china 2012j05011 national natural science foundation of china 11201389

chaos in the hamiltonian mean field model rst show that the largest lyapunov exponent remains strictly positive in the innite size limit converging to its asymptotic value with 1 lnn corrections we then elucidate the scaling laws ruling the behavior of this asymptotic value in the critical region separating the ordered clustered phase and the disordered phase present at high energy densities we also show that the full spectrum of lyapunov exponents consists of a bulk component converging to the zero value taken by a test oscillator forced by the mean eld plus subextensive bands of o lnn exponents taking nite values we nally investigate the robustness of these results by studying a 2d extension of the hmf model where each particle is endowed with 4 degrees of freedom thus allowing the emergence of chaos at the level of single particle altogether these results illustrate the subtle eects of global or long range coupling and the importance of the order in which the innite time and innite size limits are taken for an innite size hmf system represented by the vlasov equation no chaos is present while chaos exists and subsists for any nite system size

boltzmann diffusive limit beyond the navier stokes approximation given a normalized maxwellian and n 1 we establish the global in time validity of a diffusive expansion 0 1 f e t x v ef 1 t x v e 2 f 2 t x v e n f e n t x v for a solution f e to the rescaled boltzmann equation diffusive scaling 0 2 e 1 f e v x fe 1 eq f e f e inside a periodic box t 3 we assume that in the initial expansion 0 1 at t 0 the fluid parts of these f m 0 x v have arbitrary divergence free velocity fields as well as temperature fields for all 1 m n while f 1 0 x v has small amplitude in h 2 for m 2 these f m t x v are determined by a sequence of linear navier stokes fourier systems iteratively more importantly the remainder f e n t x v is proven to decay in time uniformly in e via a unified nonlinear energy method in particular our results lead to an error estimate for f 1 t x v the well known navier stokes fourier approximation and beyond the collision kernel q includes hard sphere the cutoff inverse power as well as the coulomb interactions

a crossover in the scaling law of the lyapunov exponent the authors consider products of random matrices appearing in the study of schrodinger operators or dynamical systems they show how a crossover in the scaling law of the lyapunov exponent takes place when the standard deviation of the process increases

intermittency in random optical layers at total reflection the authors show that the reflection coefficient r of a succession of n random optical layers has a well defined but sample dependent limit when n to infinity the probability distribution or r at total reflection is found to depend only on imr xi 2 3 where xi 2 is the noise variance the analytical expressions appearing in this localisation problem are found to be the same as the universal functions describing intermittency with noise

lyapunov exponents for products of rectangular real complex and quaternionic ginibre matrices we study the joint density of eigenvalues for products of independent rectangular real complex and quaternionic ginibre matrices in the limit where the number of matrices tends to infinity it is shown that the joint probability density function for the eigenvalues forms a permanental point process for all three classes the moduli of the eigenvalues become uncorrelated and log normal distributed while the distribution for the phases of the eigenvalues depends on whether real complex or quaternionic ginibre matrices are considered in the derivation for a product of real matrices we explicitly use the fact that all eigenvalues become real when the number of matrices tends to infinity finally we compare our results with known results for the lyapunov exponents as well as numerical simulations

energy method for boltzmann equation abstract a basic simple energy method for the boltzmann equation is presented here it is based on a new macro micro decomposition of the boltzmann equation as well as the h theorem this allows us to make use of the ideas from hyperbolic conservation laws and viscous conservation laws to yield the direct energy method as an illustration we apply the method for the study of the time asymptotic nonlinear stability of the global maxwellian states previous energy method starting with grad and finishing with ukai involves the spectral analysis and regularity of collision operator through sophisticated weighted norms

classical solutions to the boltzmann equation for molecules with an angular cutoff an important class of collision kernels in the boltzmann theory are governed by the inverse power law in which the intermolecular potential between two particles is an inverse power of their distance under the grad angular cutoff assumption global in time classical solutions near maxwellians are constructed in a periodic box for all soft potentials with 3 0

a conjunctive parallelogram model for pick any n data this paper proposes a multidimensional generalization of coombs 1964 parallelogram model for pick any n data which result from each of a number of subjects having selected a number of objects s he likes most from a prespecified set ofn objects in the model persons and objects are represented in a low dimensional space defined by a set of ordinal variables with a prespecified number of categories objects are represented as points and persons as intervals on each dimension a conjunctive combination rule is assumed implying that a person selects an object if and only if the object is within the subject s interval on each dimension an algorithm for fitting the model to a data set is presented and evaluated in a simulation study the model is illustrated with data on preferences regarding holiday trips

a multidimensional scaling model for the size weight illusion the kinds of individual differences in perceptions permitted by the weighted euclidean model for multidimensional scaling e g indscal are much more restricted than those allowed by tucker s three mode multidimensional scaling tmmds model or carroll s idiosyncratic scaling idioscal model although in some situations the more general models would seem desirable investigators have been reluctant to use them because they are subject to transformational indeterminacies which complicate interpretation in this article we show how these indeterminacies can be removed by constructing specific models of the phenomenon under investigation as an example of this approach a model of the size weight illusion is developed and applied to data from two experiments with highly meaningful results the same data are also analyzed using indscal of the two solutions only the one obtained by using the size weight model allows examination of individual differences in the strength of the illusion indscal can not represent such differences in this sample however individual differences in illusion strength turn out to be minor hence the indscal solution while less informative than the size weight solution is nonetheless easily interpretable

classical representation of the one dimensional anderson model a new approach is applied to the one dimensional anderson model by using a two dimensional hamiltonian map for a weak disorder this approach allows for a simple derivation of correct expressions for the localization length both at the centre and at the edge of the energy band where standard perturbation theory fails approximate analytical expressions for strong disorder are also obtained

scaling law and asymptotic distribution of lyapunov exponents in conservative dynamical systems with many degrees of freedom the authors study by numerical means the infinite product of 2n 2n conservative random matrices which mimics the chaotic behaviour of hamiltonian systems with n 1 degrees of freedom made of weakly nearest neighbour coupled oscillators the maximum lyapunov exponent lambda 1 exhibits a power law behaviour as a function of the coupling constant epsilon lambda 1 approximately epsilon beta with either beta 1 2 or beta 2 3 depending on the probability distribution of the matrix elements these power laws do not depend on n and moreover increasing n lambda 1 rapidly tends to an asymptotic value lambda 1 which only depends on epsilon and on the kind of probability distribution chosen for building up the matrices they also compute the spectrum of the lyapunov exponents and show that is has a thermodynamic limit of large n this suggests the existence of a kolmogorov entropy per degree of freedom proportional to lambda 1

the landau equation in a periodic box the landau equation which was proposed by landau in 1936 is a fundamental equation to describe collisions among charged particles interacting with their coulombic force in this article global in time classical solutions near maxwellians are constructed for the landau equation in a periodic box our result also covers a class of generalized landau equations which describes grazing collisions in a dilute gas

a mean field approximation for the maximum lyapunov exponent of bernoulli trials of matrices the author considers the infinite products of d d random matrices xi which are taken according to a bernoulli distribution i e xi a with probability 1 p and xi b with probability the maximum lyapunov exponent is estimated by selecting a suitable periodic sequence one need only then compute the eigenvalues of one particular matrix which is obtained as a finite product of matrices a and b he also discusses the cases in which the method fails because of fluctuation effects

the boltzmann equation besov spaces and optimal time decay rates in rxn abstract we prove that k th order derivatives of perturbative classical solutions to the hard and soft potential boltzmann equation without the angular cut off assumption in the whole space r x n with n 3 converge in large time to the global maxwellian with the optimal decay rate of o t 1 2 k n 2 n r in the l x r l v 2 norm for any 2 r these results hold for any 0 n 2 as long as initially f 0 b 2 l v 2 in the hard potential case we prove faster decay results in the sense that if p f 0 b 2 l v 2 and i p f 0 b 2 1 l v 2 for n 2 n 2 2 then the solution decays the global maxwellian in l v 2 l x 2 with the optimal large time decay rate of o t 1 2

exponential decay of the landau equation with potential forces abstract the cauchy problem of the landau equation with potential forces on torus is investigated the global existence of solutions with the symmetry of origin and the exponential convergence rate in time to the steady state are obtained for any initially smooth periodic origin symmetric small perturbation

optimal time decay of the boltzmann equation with frictional force in this paper we use the combination of energy method and fourier analysis to obtain the optimal time decay of the boltzmann equation with frictional force towards equilibrium precisely speaking we decompose the equation into macroscopic and microscopic partitions and perform the energy estimation then we construct a special solution operator to a linearized equation without source term and use fourier analysis to obtain the optimal decay rate to this solution operator finally combining the decay rate with the energy estimation for nonlinear terms the optimal decay rate to the boltzmann equation with frictional force is established

some properties of location problems with block and round norms the point objective problem and the weber problem are two well known formulations for locating a new facility with respect to a set of fixed facilities when locations are represented as points on a plane the point objective problem is a multiple objective formulation of minimizing the distance from a variable point to each of the fixed points similarly the weber problem is a single objective formulation of minimizing the sum of transportation costs between the variable point and the fixed points where transportation cost is a function of distance generalizing solution properties for these problems from distance measures given by the euclidean rectilinear ip and one infinity norms this paper develops solution properties under the broad classes of distance measures given by block and round norms for the point objective problem we show that i the efficient set for all round norms is the convex hull of the set of fixed points and ii the efficient set under a block norm tends to the convex hull for a sequence of block norms approaching a round norm for the weber problem we prove that i an optimal location for any block norm may be found in a finite set of intersection points belonging to the convex hull and ii this set tends to the convex hull for a sequence of block norms approaching a round norm finally we use these results to propose a synthesis of some of the main properties in continuous and network location theory

kolmogorov sinai entropy and lyapunov spectrum of a one dimensional p 4 lattice model abstract we investigate numerically the lyapunov spectrum and kolmogorov sinai entropy of chains of particles moving in double well potentials coupled by linear springs the relation to a dynamical phase transition is discussed

global well posedness for the fokker planck boltzmann equation in besov chemin lerner type spaces abstract in this paper motivated by 16 we use the littlewood paley theory to establish some estimates on the nonlinear collision term which enable us to investigate the cauchy problem of the fokker planck boltzmann equation when the initial data is a small perturbation of the maxwellian equilibrium state under the grad s angular cutoff assumption the unique global solution for the hard potential case is obtained in the besov chemin lerner type spaces c 0 l 2 b 2 r s with 1 r 2 and s 3 2 or s 3 2 and r 1 besides we also obtain the uniform stability of the dependence on the initial data

some relations between dimension and lyapounov exponents we consider differentiable maps and compact invariant sets we introduce dimensional quantities related to the ergodic invariant measures and prove some simple relations

nonlinear stability of rarefaction waves for the boltzmann equation it is well known that the boltzmann equation is related to the euler and navier stokes equations in the field of gas dynamics the relation is either for small knudsen number or for dissipative waves in the time asymptotic sense in this paper we show that rarefaction waves for the boltzmann equation are time asymptotic stable and tend to the rarefaction waves for the euler and navier stokes equations our main tool is the combination of techniques for viscous conservation laws and the energy method based on micro macro decomposition of the boltzmann equation the expansion nature of the rarefaction waves and the suitable microscopic version of the h theorem are essential elements of our analysis

on the cauchy problem for the boltzmann equation in the whole space global existence and uniform stability in lξ2 hxn abstract based on a refined energy method in this paper we prove the global existence and uniform in time stability of solutions in the space l 2 h x n to the cauchy problem for the boltzmann equation around a global maxwellian in the whole space r 3 compared with the solution space used by the spectral analysis and the classical energy method the velocity weight functions or time derivatives need not be included in the norms of l 2 h x n which is realized by introducing some temporal interactive energy functionals to estimate the macroscopic dissipation rate the key proof is carried out in terms of the macroscopic equations together with the local conservation laws it is also found that the perturbed macroscopic variables actually satisfy the linearized compressible navier stokes equations with remaining terms only related to the microscopic part

power law scaling of the top lyapunov exponent of a product of random matrices a sequence of i i d matrix valued random variables left x n right cdot x n left begin array 20 c 1 d 0 1 end array right with probabilityp and x n left begin array 20 c 1 a varepsilon b varepsilon c varepsilon 1 a varepsilon end array right with probability 1 p is considered leta e a0e o e c e c0e o e lime 0b e oa0 c0 e 0 andb e 0 for all e 0 it is shown show that the top lyapunov exponent of the matrix productxnxn 1 x1 limn 1 n n xnxn 1 x1 satisfies a power law with an exponent 1 2 that is lime 0 ln ln e 1 2

exponential decay for soft potentials near maxwellian we consider both soft potentials with angular cutoff and landau collision kernels in the boltzmann theory inside a periodic box we prove that any smooth perturbation near a given maxwellian approaches zero at the rate of e lambda t p for some 0 and 0 p 1 our method is based on an unified energy estimate with appropriate exponential velocity weight our results extend the classical result caflisch of 2 to the case of very soft potential and coulomb interactions and also improve the recent almost exponential decay results by 5 14

scaling law for the maximum lyapunov characteristic exponent of infinite product of random matrices the authors present a simple derivation for the scaling behaviour of the maximum lyapunov characteristic exponent lambda of infinite product of symplectic random matrices the considered random matrices depend on a parameter epsilon and lambda 0 for epsilon 0 they obtain lambda varies as epsilon beta with either beta 1 2 or beta 2 3 depending on the probability distribution of the matrix elements the results are in agreement with a previous numerical simulation

stability of contact discontinuity for the boltzmann equation the boltzmann equation which describes the time evolution of a large number of particles through the binary collision in statistics physics has close relation to the systems of fluid dynamics that is euler equations and navier stokes equations as for a basic wave pattern to euler equations we consider the nonlinear stability of contact discontinuities to the boltzmann equation even though the stability of the other two nonlinear waves i e shocks and rarefaction waves has been extensively studied there are few stability results on the contact discontinuity because unlike shock waves and rarefaction waves its derivative has no definite sign and decays slower than a rarefaction wave moreover it behaves like a linear wave in a nonlinear setting so that its coupling with other nonlinear waves reveals a complicated interaction mechanism based on the new definition of contact waves to the boltzmann equation corresponding to the contact discontinuities for the euler equations we succeed in obtaining the time asymptotic stability of this wave pattern with a convergence rate in our analysis an intrinsic dissipative mechanism associated with this profile is found and used for closing the energy estimates

equivalence of kinetic theory and random matrix approaches to lyapunov spectra of hard sphere systems abstract in the study of chaotic behaviour of systems of many hard spheres lyapunov exponents of small absolute values exhibit interesting characteristics leading to speculations about connections to non equilibrium statistical mechanics analytical approaches to these exponents so far can be divided into two groups macroscopically oriented approaches using kinetic theory or hydrodynamics and more microscopically oriented random matrix approaches in quasi one dimensional systems in this paper i present an approach using random matrices and weak disorder expansion in an arbitrary number of dimensions correlations between subsequent collisions of a particle are taken into account it is shown that the results are identical to those of a previous approach based on an extended enskog equation i conclude that each approach has its merits and provides different insights into the approximations made which include the stoszahlansatz the continuum limit and the long wavelength approximation the comparison also gives insight into possible connections between lyapunov exponents and fluctuations

global well posedness in spatially critical besov space for the boltzmann equation the unique global strong solution in the chemin lerner type space to the cauchy problem on the boltzmann equation for hard potentials is constructed in a perturbation framework such a solution space is of critical regularity with respect to the spatial variable and it can capture the intrinsic properties of the boltzmann equation for the proof of global well posedness we develop some new estimates on the nonlinear collision term through the littlewood paley theory

statistical mechanical formulation of lyapunov exponents we show how the lyapunov exponents of a dynamic system can in general be expressed in terms of the free energy of a non hermitian quantum many body problem this puts their study as a problem of statistical mechanics whose intuitive concepts and techniques of approximation can hence be borrowed

on the classical solutions of the boltzmann equation on etudie le probleme aux valeurs initiales pour l equation de boltzmann on montre que des solutions au sens classique existent pour tout temps si la donnee initiale est assez proche de l equilibre on se restreint a un domaine cubique et on considere seulement la condition aux limites periodique

contact discontinuity with general perturbations for gas motions the contact discontinuity is one of the basic wave patterns in gas motions the stability of contact discontinuities with general perturbations for the navier stokes equations and the boltzmann equation is a long standing open problem general perturbations of a contact discontinuity may generate diffusion waves which evolve and interact with the contact wave to cause analytic difficulties in this paper we succeed in obtaining the large time asymptotic stability of a contact wave pattern with a convergence rate for the navier stokes equations and the boltzmann equation in a uniform way one of the key observations is that even though the energy norm of the deviation of the solution from the contact wave may grow at the rate it can be compensated by the decay in the energy norm of the derivatives of the deviation which is of the order of thus this reciprocal order of decay rates for the time evolution of the perturbation is essential to close the a priori estimate containing the uniform bounds of the l8 norm on the lower order estimate and then it gives the decay of the solution to the contact wave pattern

the vlasov maxwell boltzmann system near maxwellians perhaps the most fundamental model for dynamics of dilute charged particles is described by the vlasov maxwell boltzmann system in which particles interact with themselves through collisions and with their self consistent electromagnetic field despite its importance no global in time solutions weak or strong have been constructed so far it is shown in this article that any initially smooth periodic small perturbation of a given global maxwellian which preserves the same mass total momentum and reduced total energy 22 leads to a unique global in time classical solution for such a master system the construction is based on a recent nonlinear energy method with a new a priori estimate for the dissipation the linear collision operator l not its time integration is positive definite for any solutionf t x v with small amplitude to the vlasov maxwell boltzmann system 8 and 12 as a by product such an estimate also yields an exponential decay for the simpler vlasov poisson boltzmann system 24

stability of the boltzmann equation with potential forces on torus abstract in this paper we are concerned with the stability of solutions to the cauchy problem of the boltzmann equation with potential forces on torus it is shown that the natural steady state with the symmetry of origin is asymptotically stable in the sobolev space with exponential rate in time for any initially smooth periodic origin symmetric small perturbation which preserves the same total mass momentum and mechanical energy for the non symmetric steady state it is also shown that it is stable in l 1 norm for any initial data with the finite total mass mechanical energy and entropy

finite time lyapunov exponents for products of random transformations i show how continuous products of random transformations constrained by a generic group structure can be studied by using iwasawa s decomposition into angular diagonal and shear degrees of freedom in the case of a gaussian process a set of variables adapted to the iwasawa decomposition and still having a gaussian distribution is introduced and used to compute the statistics of the finite time lyapunov spectrum of the process the variables also allow to show the exponential freezing of the shear degrees of freedom which contain information about the lyapunov eigenvectors

principles for the design of billiards with nonvanishing lyapunov exponents we introduce a large class of billiards with convex pieces of the boundary which have nonvanishing lyapunov exponents

hx vs stability of the boltzmann equation and landau equation with soft potentials based on a unified energy method the uniform in time stability of solutions in the space h x v s is constructed for the both soft potential with angular cutoff and landau collision kernels in the boltzmann theory in the whole space

exact lyapunov exponent for infinite products of random matrices despite significant work since the original paper by h furstenberg 1963 explicit formulae for lyapunov exponents of infinite products of random matrices are available only in a very few cases in this work we give a rigorous explicit formula for the lyapunov exponent for some binary infinite products of random 2 2 real matrices all these products are constructed using only two types of matrices a and b which are chosen according to a stochastic process the matrix a is singular namely its determinant is zero this formula is derived by using a particular decomposition for the matrix b which allows us to write the lyapunov exponent as a sum of convergent series the key point is the computation of all the integer powers of b which is achieved by a suitable change of frame the computation then follows by looking at each of the special types of b hyperbolic parabolic and elliptic finally we show with an example that the lyapunov exponent is a discontinuous function of the given parameter

billiards correlation functions we discuss various experiments on the time decay of velocity autocorrelation functions in billiards we perform new experiments and find results which are compatible with an exponential mixing hypothesis first put forward by friedman and martin fm they do not seem compatible with the stretched exponentials believed in spite of fm and more recently of chernov to describe the mixing the analysis leads to several byproducts we obtain information about the normal diffusive nature of the motion and we consider the probability distribution of the number of collisions in timetm astm finding a strong dependence on some geometric characteristics of the locus of the billiard obstacles

numerical study of a billiard in a gravitational field abstract billiards have always been used as models for mechanical systems in this paper we describe a very simple billiard which over a range of one continous parameter only exhibits the characteristics of hamiltonian systems having two degrees of freedom and a discontinuity the relationship between this billiard and the well known one dimensional self gravitating system with n 3 is given this billiard consists of a mass point moving in a symmetric wedge of angle 2 under the influence of a constant gravitational field for 45 complete chaos is obtained suggesting k system behavior the general properties of the mapping and some numerical results obtained are discussed of special interest are invariant curves which cross a line of discontinuity and a new universality class for lyapunov numbers

euclidean distance geometry and applications euclidean distance geometry is the study of euclidean geometry based on the concept of distance this is useful in several applications where the input data consist of an incomplete set of distances and the output is a set of points in euclidean space realizing those given distances we survey the theory of euclidean distance geometry and its most important applications with special emphasis on molecular conformation problems

optimal large time behavior of the vlasov maxwell boltzmann system in the whole space in this paper we study the large time behavior of classical solutions to the two species vlasov maxwell boltzmann system in the whole space input amssym bbb r 3 the existence of global in time nearby maxwellian solutions is known from strain in 2006 however the asymptotic behavior of these solutions has been a challenging open problem building on our previous work on time decay for the simpler vlasov poisson boltzmann system we prove that these solutions converge to the global maxwellian with the optimal decay rate of o t 3 2 3 2r in the l l norm for any 2 r if initial perturbation is smooth enough and decays in space velocity fast enough at infinity moreover some explicit rates for the electromagnetic field tending to 0 are also provided 2011 wiley periodicals inc

finding shortest paths in the plane in the presence of barriers to travel for any lp norm abstract this paper develops an efficient algorithm for the computation of the shortest paths between given sets of points origins and destinations in the plane when these paths are constrained not to cross any of a finite set of polygonal open or closed barriers it is proved that when distances are measured by an 1p norm with 1 it is also shown that optimal solutions when distances are measured according to the rectilinear or max norm i e lp norm with p 1 or p can be deduced from the results of the algorithm

viscosity for a periodic two disk fluid an existence proof we express the momentum current stress tensor for a periodic fluid with two hard disks per unit cell in terms of a single particle billiard we establish a central limit theorem for the time integrated stress tensor and thereby prove the existence of a strictly positive shear and bulk viscosity

on the cauchy problem for the nonlinear boltzmann equation global existence uniqueness and asymptotic stability the analysis of the initial value problem for the nonlinear boltzmann equation is considered in this paper a theorem defining global existence and uniqueness for initial data which decay at infinity with an inverse power law is the main result of this work and is obtained by suitable application of fixed point theorems in banach spaces the theorem also defines the asymptotic stability of the solutions

using block norms for location modeling in formulating a continuous location model with facilities represented as points in rn e g typically in the plane one must characterize the distance between two points as a function of their coordinates two criteria in selecting a distance function are 1 to obtain good approximations of actual distances and 2 to obtain a mathematical model of the location problem that is easy to solve in this paper we show how a class of norms with polygonal contours called block norms can yield attractive choices as distance functions with respect to these criteria in particular we consider the following relevant properties of block norms they generalize the concepts of rectilinear or city block travel they are dense in the set of all norms they have interesting travel interpretations in the plane they can be expressed as a sum of the absolute values of linear functions they often give better approximations to actual highway distances than the most frequently used family of norms the lp norms and finally they yield linear programming formulations of certain facility location problems i e the weber problem and the rawls problem

reconstructing distances among objects from their discriminability we describe a principled way of imposing a metric representing dissimilarities on any discrete set of stimuli symbols handwritings consumer products x ray films etc given the probabilities with which they are discriminated from each other by a perceiving system such as an organism person group of experts neuronal structure technical device or even an abstract computational algorithm in this procedure one does not have to assume that discrimination probabilities are monotonically related to distances or that the distances belong to a predefined class of metrics such as minkowski discrimination probabilities do not have to be symmetric the probability of discriminating an object from itself need not be a constant and discrimination probabilities are allowed to be 0 s and 1 s the only requirement that has to be satisfied is regular minimality a principle we consider the defining property of discrimination for ordered stimulus pairs a b b is least frequently discriminated from a if and only if a is least frequently discriminated from b regular minimality generalizes one of the weak consequences of the assumption that discrimination probabilities are monotonically related to distances the probability of discriminating a from a should be less than that of discriminating a from any other object this special form of regular minimality also underlies such traditional analyses of discrimination probabilities as multidimensional scaling and cluster analysis

an analytic estimate of the maximum lyapunov exponent in products of tridiagonal random matrices in this work we present a theoretical and numerical study of the behaviour of the maximum lyapunov exponent in products of random tridiagonal matrices in the limit of small coupling and small fluctuations such a problem is directly motivated by the investigation of coupled map lattices in a regime where the chaotic properties are quite robust and yet a complete understanding has still not been reached we derive some approximate analytic expressions by introducing a suitable continuous time formulation of the evolution equation as a first result we show that the perturbation of the lyapunov exponent due to the coupling depends only on a single scaling parameter which in the case of strictly positive multipliers is the ratio of the coupling strength with the variance of local multipliers an explicit expression for the lyapunov exponent is obtained by mapping the original problem onto a chain of nonlinear langevin equations which are eventually reduced to a single stochastic equation the probability distribution of this dynamical equation provides an excellent description for the behaviour of the lyapunov exponent finally multipliers with random signs are also considered finding that the lyapunov exponent again depends on a single scaling parameter which however has a different expression

statistical applications of linear assignment a comprehensive statistical framework is presented which encompasses a wide range of existing nonparametric methods the basic strategy referred to as linear assignment la depends on a simple index of correspondence defined between two object sets that have been matched in somea priori manner in this broad sense la can be interpreted as a general correlational technique a variety of extensions are discussed along with the attendant problems of significance testing and the construction of normalized descriptive indices

the vlasov poisson boltzmann system in the whole space the hard potential case abstract this paper is concerned with the cauchy problem on the vlasov poisson boltzmann system for hard potentials in the whole space when the initial data is a small perturbation of a global maxwellian a satisfactory global existence theory of classical solutions to this problem together with the corresponding temporal decay estimates on the global solutions is established our analysis is based on time decay properties of solutions and a new time velocity weight function which is designed to control the large velocity growth in the nonlinear term for the case of non hard sphere interactions

stochastic modeling of a billiard in a gravitational field power law behavior of lyapunov exponents we consider the motion of a point particle billiard in a uniform gravitational field constrained to move in a symmetric wedge shaped region the billiard is reflected at the wedge boundary the phase space of the system naturally divides itself into two regions in which the tangent maps are respectively parabolic and hyperbolic it is known that the system is integrable for two values of the wedge half angle 1 and 2 and chaotic for 1 2 we study the system at three levels of approximation first where the deterministic dynamics is replaced by a random evolution second where in addition the tangent map in each region is replaced by its average and third where the tangent map is replaced by a single global average we show that at all three levels the lyapunov exponent exhibits power law behavior near 1 and 2 with exponents 1 2 and 1 respectively we indicate the origin of the exponent 1 which has not been observed in unaccelerated billiards

optimal decay estimates on the linearized boltzmann equation with time dependent force and their applications although the decay in time estimates of the semi group generated by the linearized boltzmann operator without forcing have been well established there is no corresponding result for the case with general external force this paper is mainly concerned with the optimal decay estimates on the solution operator in some weighted sobolev spaces for the linearized boltzmann equation with a time dependent external force no time decay assumption is made on the force the proof is based on both the energy method through the macro micro decomposition and the lp lq estimates from the spectral analysis the decay estimates thus obtained are applied to the study on the global existence of the cauchy problem to the nonlinear boltzmann equation with time dependent external force and source precisely for space dimension n 3 the global existence and decay rates of solutions to the cauchy problem are obtained under the condition that the force and source decay in time with some rates this time decay restriction can be removed for space dimension n 5 moreover the existence and asymptotic stability of the time periodic solution are given for space dimension n 5 when the force and source are time periodic with the same period

was euclid an unnecessarily sophisticated psychologist a survey of the current state of multidimensional scaling using the city block metric is presented topics include substantive and theoretical issues recent algorithmic developments and their implications for seemingly straightforward analyses isometries with other metrics links to graph theoretic models and future prospects

the boltzmann equation with frictional force for soft potentials in the whole space abstract this paper is concerned with the cauchy problem of the boltzmann equation with frictional force for both cutoff and non cutoff soft potentials in the whole space and our main purpose is to establish its global solvability result near a given global maxwellian and to deduce the temporal convergence rates of such a global solution toward the global maxwellian when initial perturbation is sufficiently small the analysis is based on the time weighted energy method building also upon the recent studies of the cutoff vlasov poisson boltzmann system 14 15 the non cutoff vlasov poisson boltzmann system 8 and non cutoff vlasov maxwell boltzmann system 9

a kinetic flocking model with diffusion we study the stability of the equilibrium states and the rate of convergence of solutions towards them for the continuous kinetic version of the cucker smale flocking in presence of diffusion whose strength depends on the density this kinetic equation describes the collective behavior of an ensemble of organisms animals or devices which are forced to adapt their velocities according to a certain rule implying a final configuration in which the ensemble flies at the mean velocity of the initial configuration our analysis takes advantage both from the fact that the global equilibrium is a maxwellian distribution function and on the contrary to what happens in the cucker smale model ieee trans autom control 52 852 862 2007 the interaction potential is an integrable function precise conditions which guarantee polynomial rates of convergence towards the global equilibrium are found

local predictability and nonhyperbolicity through finite lyapunov exponent distributions in two degrees of freedom hamiltonian systems received 13 september 2007 revised manuscript received 5 november 2008 published 4 december 2008by using nite lyapunov exponent distributions we get insight into both the local and global properties ofa dynamical ow including its nonhyperbolic behavior several distributions of nite lyapunov exponentshave been computed in two prototypical four dimensional phase space hamiltonian systems they have beencomputed calculating the growth rates of a set of orthogonal axes arbitrarily pointed at given intervals weanalyze how such distributions serve or not for tracing the orbit nature and local ow properties such as theunstable dimension variability as the axes are allowed or not to tend to the largest stretching direction therelationship between the largest and closest to zero exponent distribution is analyzed it shows a linear depen dency at short intervals related to the number of degrees of freedom of the system finally the hyperbolicityindexes associated to the shadowing times are calculated they provide interesting information at very localscales even when there are no gaussian distributions and the values cannot be regarded as random variables doi 10 1103 physreve 78 066204 pacs number s 05 45 a

kolmogorov sinai entropy lyapunov exponents and mean free time in billiard systems we perform new experiments on the kolmogorov sinai entropy lyapunov exponents and the mean free time in billiards we study their dependence on the geometry of the scatterers made up of two interpenetrating square lattices each one with circular scatterers with different radius we find in particular that the above quantities are continuous functions of the ratio of the scatterer radius however it seems that their derivative is discontinuous around the radius ratio which separates the diffusive and nondiffusive types of geometries

the boltzmann equation global existence for a rare gas in an infinite vacuum solutions of the boltzmann equation are proved to exist globally in time under conditions that include the case of a finite volume of gas in an infinite vacuum when the mean free path of the gas is large enough it is also proved as might be expected in this case that the density of the gas at each point in space goes to zero as time goes to infinity

generalized lyapunov exponents in high dimensional chaotic dynamics and products of large random matrices we study the behavior of the generalized lyapunov exponents for chaotic symplectic dynamical systems and products of random matrices in the limit of large dimensionsd for products of random matrices without any particular structure the generalized lyapunov exponents become equal in this limit and the value of one of the generalized lyapunov exponents is obtained by simple arguments on the contrary for random symplectic matrices with peculiar structures and for chaotic symplectic maps the generalized lyapunov exponents remains different ford indicating that high dimensionality cannot always destroy intermittency

stability of the vlasov poisson boltzmann system in r3 abstract we construct global in time classical solutions to the cauchy problem for the 2 species vlasov poisson boltzmann system near maxwellian in the whole space r 3 moreover the convergence rate in time to the global maxwellian and the uniform in time stability of solutions are also obtained through energy method

a new energy method for the boltzmann equation an energy method for the boltzmann equation was proposed by liu yang and yu physica d 188 178 192 2004 based on the decomposition of the boltzmann equation and its solution around the local maxwellian the main idea is to rewrite the boltzmann equation as a fluid type dynamics system with the nonfluid component appearing in the source terms coupled with an equation for the time evolution of the nonfluid component in this paper we will elaborate this method and our main observation is that the microscopic projection of the local maxwellian with respect to a given global maxwellian is not linear but quadratic based on this and by analyzing the fluid type system using the analytic techniques for the system of conservation laws we can indeed control the conserved quantities u and 12u2 e of the boltzmann equation by the microscopic projection of the solution of the boltzmann equation with respect to the global maxwellian which is sufficient to deduce the energy estimates for the solution of

characterisation of intermittency in chaotic systems the authors discuss the characterisation of intermittency in chaotic dynamical systems by means of the time fluctuations of the response to a slight perturbation on the trajectory a set of exponents is introduced which generalise the maximum lyapunov characteristic exponent the link with the statistical mechanics formalism is emphasised and they show that the exponents are connected to a free energy formally defined for chaotic systems they perform some analytical computations in simple cases and give a few numerical examples

on the cauchy problem for boltzmann equations global existence and weak stability we study the large data cauchy problem for boltzmann equations with general collision kernels we prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in l to a solution from this stability result we deduce global existence of a solution to the cauchy problem our method relies upon recent compactness results for velocity averages a new formulation of the boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions it allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations

global existence of classical solutions to the vlasov poisson boltzmann system the time evolution of the distribution function for the charged particles in a dilute gas is governed by the vlasov poisson boltzmann system when the force is self induced and its potential function satisfies the poisson equation in this paper we give a satisfactory global existence theory of classical solutions to this system when the initial data is a small perturbation of a global maxwellian moreover the convergence rate in time to the global maxwellian is also obtained through the energy method the proof is based on the theory of compressible navier stokes equations with forcing and the decomposition of the solutions to the boltzmann equation with respect to the local maxwellian introduced in 23 and elaborated in 31

distribution of characteristic exponents in the thermodynamic limit the existence of the thermodynamic limit for the spectrum of the lyapunov characteristic exponents is numerically investigated for the fermi pasta ulam p model we show that the shape of the spectrum for energy density well above the equipartition threshold e allows the kolmogorov sinai entropy to be expressed simply in terms of the maximum exponent i the presence of a power law behaviour e is investigated the analogies with similar results obtained from the dynamics of symplectic random matrices seem to indicate the possibility of interpreting chaotic dynamics in terms of some universal properties

power law behavior of lyapunov exponents in some conservative dynamical systems abstract three billiards whose border depends on a parameter e are considered for e 0 they are integrable while for e 0 they are known to be k systems the maximal lyapunov exponent 1 is numerically computed and a power law behavior 1 e is found with 1 2 for all billiards a related abstract problem is then considered precisely the case of an infinite product of conservative 2 2 random matrices which are perturbations of commuting parabolic ones analogous computations give here two power laws with 1 2 or 2 3 depending on the probability law used to construct random matrices all these phenomena seem to have an universal character a heuristic mean field theory is also proposed based on the formal interpretation of the onset of stochasticity in our dynamical systems as a phase transition

the tunneling method for global optimization in multidimensional scaling this paper focuses on the problem of local minima of the stress function it turns out that unidimensional scaling is particularly prone to local minima whereas full dimensional scaling with euclidean distances has a local minimum that is global for intermediate dimensionality with euclidean distances it depends on the dissimilarities how severe the local minimum problem is for city block distances in any dimensionality many different local minima are found a simulation experiment is presented that indicates under what conditions local minima can be expected we introduce the tunneling method for global minimization and adjust it for multidimensional scaling with general minkowski distances the tunneling method alternates a local search step in which a local minimum is sought with a tunneling step in which a different configuration is sought with the same stress as the previous local minimum in this manner successively better local minima are obtained and experimentation so far shows that the last one is often a global minimum

boltzmann equation micro macro decompositions and positivity of shock profiles we introduce an elementary energy method for the boltzmann equation based on a decomposition of the equation into macroscopic and microscopic components the decomposition is useful for the study of time asymptotic stability of nonlinear waves the wave location is determined by the macroscopic equation the microscopic component has an equilibrating property the coupling of macroscopic and microscopic components gives rise naturally to the dissipations similar to those obtained by the chapman enskog expansion our main result is the establishment of the positivity of shock profiles for the boltzmann equation this is shown by the time asymptotic approach and the maximal principle for the collision operator

universal power law tails for singularity dominated strong fluctuations integrals of the type i t identical to integral integral dxdy delta t h x y are considered where h is a function from an ensemble labelled by parameters i can represent circulation times of fluid particles in the plane orbital periods or semiclassical densities of states for one dimensional hamiltonian systems spectral densities for two dimensional crystals or the strength of a wave pulse produced by propagation of a deformed step discontinuity as h varies over the ensemble i develops strong fluctuations associated with legendre singularities for unrestricted functions h the probability distribution p i describing the fluctuations of i decays according to a universal i 9 law obtained by a scaling argument involving arnold s classification of catastrophes if h is only quadratic in y p varies as i 10 if the integral defining i is one dimensional p varies as i 3 if it is three dimensional i decays no faster than i 48 the probability distribution of i identical to mod delta i delta t mod decays as i 2

lyapunov exponents and dimensions of chaotic neural networks dynamical properties of a network model composed of n continuous elements with randomly chosen asymmetrical couplings and reduced connectivity are studied the authors calculate numerically lyapunov exponents and dimensions for chaotic and non chaotic states of the network the dependence of the parameters can be described analytically in good agreement with the numerical data

scaling laws for all liapunov exponents models and measurements we introduce simple diamond models of random symplectic matrices in order to study the scaling laws of all liapunov exponents these universal properties appear in physical problems that are modeled by transfer matrices dynamical systems random potentials random fields etc numerical experiments for the general case are in agreement with the results derived from the models

on the effects of an uncertainty on the evolution law in dynamical systems abstract we show that a small perturbation of the evolution law of chaotic dynamical systems is practically equivalent to the finite precision on the knowledge of the initial condition in spite of the short predictability time t p on a single trajectory we found that the statistical properties are not sensitive to small changes in the evolution law or initial probability distribution this feature holds also for correlation functions at a delay larger than t p we discuss the roundoff effects on integrable systems