deterministic walks in random environments deterministic walks in random environments dwre occupy an intermediate position between purely random generated by random trials and purely deterministic generated by deterministic dynamical systems e g by maps models of diffusion these models combine deterministic and probabilistic features we review general properties of dwre and demonstrate that to a large extent their dynamics and their statistics can be analyzed consecutively and separately we also show that orbits of one dimensional walks in rigid environments with non constant rigidity almost surely visit each site infinitely many times

influence of memory in deterministic walks in random media analytical calculation within a mean field approximation consider a random medium consisting of n points randomly distributed so that there is no correlation among the distances separating them this is the random link model which is the high dimensionality limit mean field approximation for the euclidean random point structure in the random link model at discrete time steps a walker moves to the nearest point which has not been visited in the last ensuremath mu steps memory producing a deterministic partially self avoiding walk the tourist walk we have analytically obtained the distribution of the number n of points explored by the walker with memory ensuremath mu 2 as well as the transient and period joint distribution this result enables us to explain the abrupt change in the exploratory behavior between the cases ensuremath mu 1 memoryless walker driven by extreme value statistics and ensuremath mu 2 walker with memory driven by combinatorial statistics in the ensuremath mu 1 case the mean newly visited points in the thermodynamic limit n 1 is just n e 2 72 dots while in the ensuremath mu 2 case the mean number n of visited points grows proportionally to n 1 2 also this result allows us to establish an equivalence between the random link model with ensuremath mu 2 and random map uncorrelated back and forth distances with ensuremath mu 0 and the abrupt change between the probabilities for null transient time and subsequent ones

classical motion in two dimensional crystals the classical motion of an electron of high enough energy in a two dimensional crystal is diffusive for many potentials with coulomb singularities a simple model of the dynamics is developed which predicts the dependence of the diffusion constantd on the particle energye in the high energy limit d e const e3 2 this diffusion law is checked for a concrete crystal by numerically integrating the hamilton equations for an ensemble of initial conditions finally this method is compared with other models of the classical dynamics in a crystal especially the sinai billiard

implied constraints and lp duals of general nonlinear programming problems a very surprising result is derived in this paper that there exists a family of lp duals for general nlp problems a general dual problem is first derived from implied constraints via a simple bounding technique it is shown that the lagrangian dual is a special case of this general dual and that other special cases turn out to be lp problems the lp duals provide a very powerful computational device but are derived using fairly strict conditions hence they can often be infeasible even if the primal nlp problem is feasible and bounded many directions for relaxing these conditions are outlined for future research a concept of local duality is also introduced for the first time akin to the concept of local optimality

equivalence of deterministic walks on regular lattices on the plane abstract we consider deterministic walks on square triangular and hexagonal two dimensional lattices in each case there is a scatterer at every lattice site that can be in one of two states that forces the walker to turn either to his her immediate right or left after the walker is scattered the scatterer changes state a lattice with an arrangement of scatterers is an environment we show that there are only two environments for which the scattering rules are injective mirrors or rotators on the three lattices on hexagonal lattices webb and cohen 2014 proved that if a walker with a given initial position and velocity moves through an environment of mirrors rotators then there is an environment of rotators mirrors through which the walker would move with the same trajectory we refer to these trajectories on mirror and rotator environments as equivalent walks we prove the equivalence of walks on square and triangular lattices and include a proof of the equivalence of walks on hexagonal lattices the proofs are based both on the geometry of the lattice and the structure of the scattering rule

derivation of duality in mathematical programming and optimization theory abstract consider the primal problem to minimize a function over a set of feasible solutions in euclidean space this paper derives a general dual problem directly from this primal problem it shows that the lagrangian conjugate and johri s formulations of duality in nonlinear programming the linear programming duality and the minimum norm duality principle in optimization theory are all special cases of this general duality formulation this explains the duality why two seemingly different problems provide the same answer by showing that there really is no duality but that the dual problems are just different forms of the primal problem all these results are derived in a simple intuitive unified manner

deterministic and random partially self avoiding walks in random media abstract consider a set of n cities randomly distributed in the bulk of a hypercube with d dimensions a walker with memory begins his route from a given city of this map and moves at each discrete time step to the nearest point which has not been visited in the preceding steps after reviewing the more interesting general results we consider one dimensional disordered media and show that the walker needs not to have full memory of its trajectory to explore the whole system it suffices to have memory of order ln n ln 2

statistical properties of dynamical systems with singularities we prove statistical properties of two dimensional hyperbolic dynamical systems with singularities bunimovich sinai and chernov proved a theorem on the subexponential decay of correlations and a central limit theorem for billiard systems in this paper we use their techniques to prove the same results for abstract systems

on the boltzmann equation for the lorentz gas we consider the boltzmann grad limit for the lorentz or wind tree model we prove that if is a fixed configuration of scatterer centers belonging to a set of full measure with respect to the poisson distribution with parameter 0 then the evolution of an initial a c particle density tends in the boltzmann grad limit to the solution of the boltzmann equation for the model as an intermediate step we prove that the process of the free path lengths and impact parameters induced by the lebesgue measure on a small region tends to a limiting independent process

grids and the arithmetics of jordan pairs abstract we prove that every 2 finitedimensional covering standard division grid of a jordan pair v over a henselian field canonically determines a norm on v this is used to classify maximal orders which contain a covering division grid of v we show that weakly separable orders always satisfy this condition and link their existence to ramification properties of v

ergodic and statistical properties of piecewise linear hyperbolic automorphisms of the 2 torus we study generic piecewise linear hyperbolic automorphisms of the 2 torus we explain why the resulting dynamical system is ergodic and mixing and prove the exponential decay of correlations

proof of the boltzmann sinai ergodic hypothesis for typical hard disk systems we consider the system of n 2 hard disks of masses m 1 m n and radius r in the flat unit torus 2 we prove the ergodicity actually the b mixing property of such systems for almost every selection m 1 m n r of the outer geometric parameters

semiprime prime and simple jordan superpairs covered by grids in this paper we continue the investigation of jordan superpairs covered by a grid which have recently been described by the second author e neher j algebra 269 2003 28 73 for such a jordan superpair we characterize semiprimeness primeness and simplicity in terms of its associated supercoordinate system

the boltzmann grad limit of the periodic lorentz gas we study the dynamics of a point particle in a periodic array of spherical scatterers and construct a stochastic process that governs the time evolution for random initial data in the limit of low scatterer density boltzmanngrad limit a generic path of the limiting process is a piecewise linear curve whose consecutive segments are generated by a markov process with memory two

statistical properties of lorentz gas with periodic configuration of scatterers in our previous paper markov partitions for some classes of dispersed billiards were constructed using these partitions we estimate the decay of velocity auto correlation function and prove the central limit theorem of probability theory and donsker s invariance principle for lorentz gas with periodic configuration of scatterers

lagrangian heuristics for the two echelon single source capacitated facility location problem facility location problems form an important class of integer programming problems with application in the distribution and transportation industries in this paper we are concerned with a particular type of facility location problem in which there exist two echelons of facilities each facility in the second echelon has limited capacity and can be supplied by only one facility or depot in the first echelon each customer is serviced by only one facility in the second echelon the number and location of facilities in both echelons together with the allocation of customers to the second echelon facilities are to be determined simultaneously we propose a mathematical model for this problem and consider six heuristics based on lagrangian relaxation for its solution to solve the dual problem we make use of a subgradient optimization procedure we present numerical results for a large suite of test problems these indicate that the lower bounds obtained from some relaxations have a duality gap which frequently is one third of the one obtained from traditional linear programming relaxation furthermore the overall solution time for the heuristics are less than the time to solve the lp relaxation

a transversal fundamental theorem for semi dispersing billiards for billiards with a hyperbolic behavior fundamental theorems ensure an abundance of geometrically nicely situated and sufficiently large stable and unstable invariant manifolds a transversal fundamental theorem has recently been suggested by the present authors to proveglobal ergodicity and then as an easy consequence the k property of semidispersing billiards in particular the global ergodicity of systems ofn 3 elastic hard balls conjectured by the celebratedboltzmann sinai ergodic hypothesis in fact the suggested transversal fundamental theorem has been successfully applied by the authors in the casesn 3 and 4 the theorem generalizes the fundamental theorem of chernov and sinai that was really the fundamental tool to obtainlocal ergodicity of semi dispersing billiards our theorem however is stronger even in their case too since its conditions are simpler and weaker moreover a complete set of conditions is formulated under which the fundamental theorem and its consequences like the zig zag theorem are valid for general semi dispersing billiards beyond the utmost interesting case of systems of elastic hard balls as an application we also give conditions for the ergodicity and consequently the k property of dispersing billiards transversality means the following instead of the stable and unstable foliations occurring in the chernov sinai formulation of the stable version of the fundamental theorem we use the stable foliation and an arbitrary nice one transversal to the stable one

conditional proof of the boltzmann sinai ergodic hypothesis we consider the system of n 2 elastically colliding hard balls of masses m1 mn and radius r on the flat unit torus mathbb t nu 2 we prove the so called boltzmann sinai ergodic hypothesis i e the full hyperbolicity and ergodicity of such systems for every selection m1 mn r of the external parameters provided that almost every singular orbit is geometrically hyperbolic sufficient i e the so called chernov sinai ansatz is true the present proof does not use the formerly developed rather involved algebraic techniques instead it employs exclusively dynamical methods and tools from geometric analysis

walks in rigid environments continuous limits we derive the continuous limits of kinetic equations for spatially discrete systems generated by the motion of a particle in a random array of scatterers the type of scatterer at a vertex changes after the r th visit of the particle to this vertex where 1 r such deterministic cellular automata belong to the class of walks in rigid environments it has been recently shown that they form the simplest dynamical models with sub diffusive diffusive and super diffusive behaviour due to the deterministic character of the dynamics the continuous limit equations obtained for these models are of the euler type rather than the diffusive type the reason for that is that the fluctuations in these models are relatively small and there is no scaling of probabilities similar for example to those in the case of biased random walk that can account for them

quadratic jordan superpairs covered by grids abstract in this paper we are proposing a theory of jordan superpairs defined over super commutative superrings our framework has two novelties we allow scalars of even and odd parity and we do not assume that 1 2 lies in our base superring to demonstrate that it is possible to work in this generality we classify jordan superpairs covered by a grid

percolation and motion in a simple random environment the authors present a particularly simple model of deterministic classical motion in a two dimensional random environment as the parameters of the model are varied a transition occurs from all trajectories being localised to some being extended they construct a mean field theory for this transition and relate the model exactly to percolation models in particular parameter ranges they point out that it is a member of a new class of site percolation analogues of direct bond percolation

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

physical insight into superdiffusive dynamics of sinai billiard through collision statistics abstract we report on distinct steady motion dynamic regimes in chaotic sinai billiard sb a numerical study on elastic reflections from the sb boundary square wall of length l and circle obstacle of radius r is carried out for different r l the research is based on the exploration of the generalized diffusion equation and on the analysis of wall collision and the circle collision distributions observed at late times the asymptotes for the diffusion coefficient d r and the corresponding diffusion exponent z r are established for all geometries the universal r independent diffusion with d 1 t 1 3 and z 1 1 5 replaces the ballistic motion regime z 0 1 attributed to square billiard r 0 geometrically this superdiffusive regime is bounded by small radii 0 r r 1 r 1 l 2 4 when both diagonal and non diagonal bleher s corridors are open in the correspondent square lattice of lorentz gas lg model the relaxation dynamics observed is ensured by the universal diffusive propagation of the regular type and the bouncing ball type orbits within the random walk scheme this superdiffusive regime is due to the levy flights between the long distant scatterers with the increase in circle radius r 1 r r 2 r 2 l 2 the diagonal corridor closes but the arc touching effects continue to bring the long living bouncing ball orbits back to the non diagonal infinite corridors this transient non universal dynamics with 1 5 z r 2 also associated with the trapping of regular orbits seems to be characteristic of non fully hyperbolic billiard systems in sb with finite horizon r r 2 all the principal corridors are closed and the interplay between square and circle boundaries generates the known chaotic dynamics attributed to the fully hyperbolic systems this is also observed through the normal brownian diffusion z 2 2 and the gaussian statistics proved for both kinds of collisions

implied constraints and a unified theory of duality in linear and nonlinear programming abstract a unified theory of duality in linear and nonlinear programming can be easily derived from the perspective of implied constraints that is constraints implied by the given constraints of the problem the key idea is to construct special types of implied constraints which will bound the objective function this simple idea motivates a complete concise and clear development of a unified theory of duality which seeks the tightest bound on the primal objective function the derivation is intuitive and geometric and leads to a more general formulation of duality in nonlinear programming than the formulations known so far duality theorems are shown to hold for this formulation in general to solve the dual the set of implied constraints on the primal objective function needs to be generated a duality gap exists if this set is only partially generated in a dual formulation this is implicity the case in formulations such as the lagrangian and surrogate duality in linear programming this set can be generated linearly resulting in the constraints of the dual problem this formulation is also equivalent to a special conjugate function the conjugate function approach to duality is formulated more for the perturbations in the parameters of the problem and has not developed this fact

recurrence properties of lorentz lattice gas cellular automata recurrence properties of a point particle moving on a regular lattice randomly occupied with scatterers are studied for strictly deterministic nondeterministic and purely random scattering rules

long time tails of the velocity autocorrelation functions for the triangular periodic lorentz gas we present numrical results on the velocity autocorrelation function vacf c t for the periodic lorentz gas on a two dimensional triangular lattice as a function of the radiusr of the hard disk scatterers on the lattice our results for the unbounded horizon case 0 r sqrt 3 4 confirm 1 t decay of the vacf for long times out to 100 times the mean free time between collisions and provide strong support for the conjecture by friedman and martin that the 1 t decay is due to long free paths along which a moving particle does not scatter up to timet even after new sets of long free paths become available forr 1 4 we continue to find good agreement between numerical results and an analytically estimated 1 t decay for the bounded horizon case sqrt 3 4 leqslant r leqslant 0 5 our numerical vacfs decay exponentially although it is difficult to discriminate among pure exponential decay exponential decay with prefactor and stretched exponential decay

a non interactive elicitation method for non linear multiattribute utility functions theory and application to agricultural economics practical implementation of multiattribute utility theory is limited partly for the lack of operative methods to elicit the parameters of the multiattribute utility function particularly when this function is not linear as a consequence most studies are restricted to linear specifications which are easier to estimate and to interpret we propose an indirect method to elicit the parameters of a non linear utility function to be compatible with the observed behaviour of decision makers rather than with their answers to direct surveys the idea rests on approaching the parameter estimation problem as a dual of the decision problem by making the observed decisions to be compatible with a rational decision making process

propagation dispersion equation a propagation dispersion equation is derived for the first passage distribution function of a particle moving on a substrate with time delays the equation is obtained as the hydrodynamic limit of the first visit equation an exact microscopic finite difference equation describing the motion of a particle on a lattice whose sites operate as time delayers the propagation dispersion equation should be contrasted with the advection diffusion equation or the classical fokker planck equation as it describes a dispersion process in time instead of diffusion in space with a drift expressed by a propagation speed with non zero bounded values the temporal dispersion coefficient is shown to exhibit a form analogous to taylor s dispersivity physical systems where the propagation dispersion equation applies are discussed

limit theorems and markov approximations for chaotic dynamical systems we prove the central limit theorem and weak invariance principle for abstract dynamical systems based on bounds on their mixing coefficients we also develop techniques of markov approximations for dynamical systems we apply our results to expanding interval maps axiom a diffeomorphisms chaotic billiards and hyperbolic attractors

polynomial identities and nonidentities of split jordan pairs abstract we show that split jordan pairs over rings without 2 torsion can be distinguished by polynomial identities with integer coefficients in particular this holds for simple finite dimensional jordan pairs over algebraically closed fields of characteristic not 2 we also generalize results of drensky and racine and of rached and racine on polynomial identities of respectively jordan algebras and jordan triple systems

johri s general dual the lagrangian dual and the surrogate dual duality formulations can be derived from a nonlinear primal optimization problem in several ways one abstract theoretical concept presented by johri is the framework of general dual problems they provide the tightest of specific bounds on the primal optimum generated by dual subproblems which relax the primal problem with respect to the objective function or to the feasible set or even to both the well known lagrangian dual and surrogate dual are shown to be special cases dominating functions and including sets which are the two relaxation devices of johri s general dual turn out to be the most general formulations of augmented lagrangian functions and augmented surrogate regions

microscopic chaos and reaction diffusion processes in the periodic lorentz gas we apply the hypothesis of microscopic chaos to diffusion controlled reaction which we study in a reactive periodic lorentz gas the relaxation rate of the reactive eigenmodes is obtained as eigenvalue of the frobenius perron operator which determines the reaction rate the cumulative functions of the eigenstates of the frobenius perron operator are shown to be generalizations of lebesgue s singular continuous functions for small enough densities of catalysts the reaction is controlled by the diffusion a random walk model of this diffusion controlled reaction process is presented which is used to study the dependence of the reaction rate on the density of catalysts

levy flights and random searches in this work we discuss some recent contributions to the random search problem our analysis includes superdiffusive levy processes and correlated random walks in several regimes of target site density mobility and revisitability we present results in the context of mean field like and closed form average calculations as well as numerical simulations we then consider random searches performed in regular lattices and lattices with defects and we discuss a necessary criterion for distinguishing true superdiffusion from correlated random walk processes we invoke energy considerations in relation to critical survival states on the edge of extinction and we analyze the emergence of levy behavior in deterministic search walks finally we comment on the random search problem in the context of biological foraging

how fast does langton s ant move the automaton known as langton s ant exhibits a propagation phase where the particle dynamics the ant produces a regular periodic pattern called highway despite the simplicity of its basic algorithm langton s ant has remained a puzzle in terms of analytical description here i show that propagation dynamics obeys a general difference equation for a class of automata which includes 1 d 2 d triangular and square lattice models in the case of langton s ant the speed of the ant in the highway c 2 52 follows exactly from the equation

origin of power law distributions in deterministic walks the influence of landscape geometry we investigate the properties of a deterministic walk whose locomotion rule is always to travel to the nearest site initially the sites are randomly distributed in a closed rectangular a l ifmmode times else texttimes fi l landscape and once reached they become unavailable for future visits as expected the walker step lengths present characteristic scales in one l ensuremath rightarrow 0 and two a l ensuremath sim l dimensions however we find scale invariance for an intermediate geometry when the landscape is a thin striplike region this result is induced geometrically by a dynamical trapping mechanism leading to a power law distribution for the step lengths the relevance of our findings in broader contexts of both deterministic and random walks is also briefly discussed

statistical properties of the periodic lorentz gas multidimensional case in 1981 bunimovich and sinai established the statistical properties of the planar periodic lorentz gas with finite horizon our aim is to extend their theory to the multidimensional lorentz gas in that case the markov partitions of the bunimovich sinai type the main tool of their theory are not available we use a crude approximation to such partitions which we call markov sieves their construction in many dimensions is essentially different from that in two dimensions it requires more routine calculations and intricate arguments we try to avoid technical details and outline the construction of the markov sieves in mostly qualitative heuristic terms hoping to carry out our plan in full detail elsewhere modulo that construction our proofs are conclusive in the end we obtain a stretched exponential bound for the decay of correlations the central limit theorem and donsker s invariance principle for multidimensional periodic lorentz gases with finite horizon

implied constraints and an alternate unified development of nonlinear programming theory abstract this paper offers an alternate unified view of nonlinear programming theory from the perspective of implied constraints optimality is identically characterized for both constrained and unconstrained problems in terms of implied constraints it is shown that there is a weaker condition than the guignard constraint qualification for the existence of finite multipliers in the karush kuhn tucker conditions surprisingly this condition does not directly qualify the constraints but instead qualifies the objective in terms of implied constraints more surprisingly the existence of the finite multipliers follows directly from this objective qualification it is not necessary for the point to be a local optimum methods for generating implied constraints are used to obtain a more general sufficient condition for local and global optimality a single unified formulation of duality shows that duality is nothing more than an effort to generate the tightest implied constraint duality theorems hold in general for this formulation convexity is not required and the existence of the duality gap in prior formulations is easily explained the algorithmic potential of this approach is highlighted by showing that the simplex method systematically tries to imply the objective from the constraints of the problem

topological dynamics of flipping lorentz lattice gas models we study the topological dynamics of the flipping mirror model of ruijgrok and cohen with one or an infinite number of particles in particular we prove the topological transitivity and topological mixing up to a natural first integral for the one particle model

lie algebras graded by 3 graded root systems and jordan pairs covered by grids let r be an irreducible root system a lie algebra l is called graded by r if l is graded with grading group the root lattice of r such that the only nonzero homogeneous subspaces of l have degree 0 or a root in r the grading is induced by the adjoint action of a split cartan subalgebra of a finite dimensional simple lie subalgebra of l with root system r and l is generated by the homogeneous subspaces of nonzero degree this class of lie algebras was introduced and studied by s berman and r moody in invent math 108 1992 where in particular a classification up to central equivalence is given in the simply laced case the doubly laced cases have recently been classified by g benkart and e zelmanov let r be a 3 graded root system i e r is not of type e 8 f 4 or g 2 in this paper lie algebras graded by r are described in a unified way without case by case considerations namely it is shown that a lie algebra l is 3 graded if and only if l is a central extension of the tits kantor koecher algebra of a jordan pair covered by a grid whose associated 3 graded root system is isomorphic to r this result is then used to classify lie algebras graded by r we give the classification of jordan pairs covered by a grid and describe their tits kantor koecher algebras one of the advantages of this approach is that it works over rings containing 1 2 and 1 3 and also for infinite root systems another application is the description of slodowy s intersection matrix algebras arising from multiply affinized cartan matrices