finiteness of conical algorithms with ω subdivisions in this paper the problem of finding the global optimum of a concave function over a polytope is considered a well known class of algorithms for this problem is the class of conical algorithms in particular the conical algorithm based on the so called subdivision strategy is considered it is proved that for any given accuracy e 0 this algorithm stops in a finite time by returning an e optimal solution for the problem while it is convergent for e 0

global validity of the boltzmann equation for two and three dimensional rare gas in vacuum erratum and improved result we point out an error in our earlier papers 1 and 2 and present a more direct and natural proof which although based on the same physical ideas of the previous ones saves and actually improves the validity results for the boltzmann equation given in 1 and 2

the zeros of derivatives of entire functions and the polya wiman conjecture an entire function f x which assumes only real values on the real axis is said to be a real entire function thus if f is a real entire function then its maclaurin coefficients are all real and consequently the zeros of f are symmetrically located with respect to the real axis one of the intuitive principles underlying the theory of distribution of zeros of successive derivatives of real entire functions was formulated by polya p4 in his celebrated survey article entitled on the zeros of the derivatives of a function and its analytic character in this work p4 p 181 polya states the real axis seems to exert an influence on the complex zeros of f n z it seems to attract these zeros when the order is less than 2 and it seems to repel them when the order is greater than 2 indeed the first confirmation of this principle was made by alander a2 see also p6lya p2 who in 1930 showed that the polya wiman conjecture is true if the order of f x is less than 2 3 in wil and wi2 wiman established the validity of this conjecture for functions f x of order at most 1 while in 1937 polya p3 proved it for functions of order less than 4 3 a general survey of these results and related conjectures may be found in p2 and p4 for more recent surveys of this and related areas of research see boas b2 and prather pr it seems that no progress has been made on the p6lya wiman conjecture since the publication of polya s 4 3 theorem in 1937 our proof of the conjecture is self contained and does not rely on the results cited above in section 2 we will 1 introduce some definitions and notations 2 state some of the properties of functions in the laguerre p6lya class and 3 prove the polya wiman conjecture theorem 3 in the proof of theorem 3 we

an interacting particle model with compact hierarchical structure abstract we present a model interacting particle system with a population of fixed size in which particles wander randomly in space and pairs interact at a rate determined by a reaction kernel with finite range the pairwise interaction randomly selects one of the particles the victim and instantly transfers it to the position of the other the killer thus maintaining the total number the special feature of the model is that it possesses a closed hierarchical structure in which the statistical moments of the governing master equation lead to closed equations for the reduced distribution functions the concentration pair correlation function and so on in one spatial dimension we show that persistent spatial correlations clusters arise in this model and we characterize the dynamics in terms of analytical properties of the pair correlation function as the range of the reaction kernel is increased the dynamics varies from an ensemble of largely independent random walkers at small range to tightly bound clusters with longer range reaction kernels

normal conical algorithm for concave minimization over polytopes a new conical algorithm is developed for finding the global minimum of a concave function over a polytope to ensure faster convergence and overcome some major drawbacks of previous conical algorithms a normal rather than exhaustive subdivision process is used

the derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many particle systems the chemotaxis equations are a well known system of partial differential equations describing aggregation phenomena in biology in this paper they are rigorously derived from an interacting stochastic many particle system where the interaction between the particles is rescaled in a moderate way as population size tends to infinity the novelty of this result is that in all previous applications of this kind of limiting procedure the principal part of the system is assumed to fulfill an ellipticity condition which is not satisfied in our case new techniques which deal with this difficulty are presented

kinetic hierarchy and propagation of chaos in biological swarm models we consider two models of biological swarm behavior in these models pairs of particles interact to adjust their velocities one to each other in the first process called bdg they join their average velocity up to some noise in the second process called cl one of the two particles tries to join the other one s velocity this paper establishes the master equations and bbgky hierarchies of these two processes it investigates the infinite particle limit of the hierarchies at large time scales it shows that the resulting kinetic hierarchy for the cl process does not satisfy propagation of chaos numerical simulations indicate that the bdg process has similar behavior to the cl process c 2012 elsevier b v all rights reserved

spectral gap for kac s model of boltzmann equation we consider a random walk on s n 1 the standard sphere of dimension n 1 generated by random rotations on randomly selected coordinate planes i j with 1 i j n this dynamic was used by marc kac as a model for the spatially homogeneous boltzmann equation we prove that the spectral gap on s n 1 is n 1 up to a constant independent of n

the eigenvalues of kac s master equation in this paper we derive formulae for the eigenvalues and spectral gap of the master equation for general collision kernels we prove a conjecture of mark kac s on the existence of a spectral gap independent of the number of particles we relate the eigenvalues to the nonlinear eigenvalues that occur in the exact solutions of model boltzmann equations due to m ernst

a macroscopic model for a system of swarming agents using curvature control in this paper we study the macroscopic limit of a new model of collective displacement the model called ptwa is a combination of the vicsek alignment model vicsek et al in phys rev lett 75 6 1226 1229 1995 and the persistent turning walker ptw model of motion by curvature control degond and motsch in j stat phys 131 6 989 1021 2008 gautrais et al in j math biol 58 3 429 445 2009 the ptw model was designed to fit measured trajectories of individual fish gautrais et al in j math biol 58 3 429 445 2009 the ptwa model persistent turning walker with alignment describes the displacements of agents which modify their curvature in order to align with their neighbors the derivation of its macroscopic limit uses the non classical notion of generalized collisional invariant introduced in degond and motsch in math models methods appl sci 18 1 1193 1215 2008 the macroscopic limit of the ptwa model involves two physical quantities the density and the mean velocity of individuals it is a system of hyperbolic type but is non conservative due to a geometric constraint on the velocity this system has the same form as the macroscopic limit of the vicsek model degond and motsch in math models methods appl sci 18 1 1193 1215 2008 the vicsek hydrodynamics but for the expression of the model coefficients the numerical computations show that the numerical values of the coefficients are very close the vicsek hydrodynamic model appears in this way as a more generic macroscopic model of swarming behavior as originally anticipated

the diffusion limit of transport equations derived from velocity jump processes in this paper we study the diffusion approximation to a transport equation that describes the motion of individuals whose velocity changes are governed by a poisson process we show that under an appropriate scaling of space and time the asymptotic behavior of solutions of such equations can be approximated by the solution of a diffusion equation obtained via a regular perturbation expansion in general the resulting diffusion tensor is anisotropic and we give necessary and sufficient conditions under which it is isotropic we also give a method to construct approximations of arbitrary high order for large times in a second paper part ii we use this approach to systematically derive the limiting equation under a variety of external biases imposed on the motion depending on the strength of the bias it may lead to an anisotropic diffusion equation to a drift term in the flux or to both our analysis generalizes and simplifies previous derivations that lead to the classical patlak keller segel alt

emergent behavior in flocks we provide a model for both continuous and discrete time describing the evolution of a flock our model is parameterized by a constant beta capturing the rate of decay which in our model is polynomial of the influence between birds in the flock as they separate in space our main result shows that when beta 1 2 convergence of the flock to a common velocity is guaranteed while for betages1 2 convergence is guaranteed under some condition on the initial positions and velocities of the birds only

a new approach to quantitative propagation of chaos for drift diffusion and jump processes this paper is devoted the study of the mean field limit for many particle systems undergoing jump drift or diffusion processes as well as combinations of them the main results are quantitative estimates on the decay of fluctuations around the deterministic limit and of correlations between particles as the number of particles goes to infinity to this end we introduce a general functional framework which reduces this question to the one of proving a purely functional estimate on some abstract generator operators consistency estimate together with fine stability estimates on the flow of the limiting nonlinear equation stability estimates then we apply this method to a boltzmann collision jump process for maxwell molecules to a mckean vlasov drift diffusion process and to an inelastic boltzmann collision jump process with stochastic thermal bath to our knowledge our approach yields the first such quantitative results for a combination of jump and diffusion processes

sums of products of functions of x times functions of y we study functions u of the form u x y nk 1ak x bk y mostly by means of two functional determinants wn u and dn u a sample result is that if u has the form on an open subset of a region g on which u is real analytic then u must have this form with possibly different ak and bk on all of g we also study the representability of u x y in the form on a region g in which u is either n times differentiable or is merely continuous the question whether the limit of functions of the form is again of the form is treated moreover we study those functions of the form that are solutions of some classical partial differential equations

state transitions and the continuum limit for a 2d interacting self propelled particle system we study a class of swarming problems wherein particles evolve dynamically via pairwise interaction potentials and a velocity selection mechanism we find that the swarming system undergoes various changes of state as a function of the self propulsion and interaction potential parameters in this paper we utilize a procedure which connects a class of individual based models to their continuum formulations and determine criteria for the validity of the latter h stability of the interaction potential plays a fundamental role in determining both the validity of the continuum approximation and the nature of the aggregation state transitions we perform a linear stability analysis of the continuum model and compare the results to the simulations of the individual based one c 2007 elsevier b v all rights reserved

kinetic models for topological nearest neighbor interactions we consider systems of agents interacting through topological interactions these have been shown to play an important part in animal and human behavior precisely the system consists of a finite number of particles characterized by their positions and velocities at random times a randomly chosen particle the follower adopts the velocity of its closest neighbor the leader we study the limit of a system size going to infinity and under the assumption of propagation of chaos show that the limit kinetic equation is a non standard spatial diffusion equation for the particle distribution function we also study the case wherein the particles interact with their k closest neighbors and show that the corresponding kinetic equation is the same finally we prove that these models can be seen as a singular limit of the smooth rank based model previously studied in 10 the proofs are based on a combinatorial interpretation of the rank as well as some concentration of measure arguments

exhaustive nondegenerate conical processes for concave minimization on convex polytopes an exhaustive and nondegenerate cone splitting process is defined and an algorithm stated for minimizing a quasi concave function on a bounded convex polytope described by a system of linear inequalities the algorithm crucially splits upon vertices and it is shown that this class of algorithms converges finitely to an optimal solution