fixed points koebe uniformization and circle packings a domain in the riemann sphere hat mathbb c is called a circle domain if every connected component of its boundary is either a circle or a point in 1908 p koebe ko1 posed the following conjecture known as koebe s kreisnormierungsproblem any plane domain is conformally homeomorphic to a circle domain in hat mathbb c when the domain is simply connected this is the content of the riemann mapping theorem

coherent states associated with the wavefunctions and the spectrum of the isotonic oscillator classes of coherent states are presented by replacing the labelling parameter z of klauder perelomov type coherent states by confluent hypergeometric functions with specific parameters temporally stable coherent states for the isotonic oscillator hamiltonian are presented and these states are identified as a particular case of the so called mittag leffler coherent states

complex dynamics of duopoly game with heterogeneous players a further analysis of the output model abstract in this paper an output duopoly game with heterogeneous players is analyzed in order to study the influence of players different behavior on the dynamics of game two types of players are considered which are bounded rationality expectation and naive expectation player with naive expectation chooses an output level based on the market price of previous period while player with bounded rationality adjusts his output adaptively following a bounded rationality adjustment process based on a local estimate of the marginal profit of previous period the game model is also based on the assumption that demand and cost function are nonlinear the existence of equilibrium points and its local stability of the output game are investigated the complex dynamics bifurcations and chaos are displayed by numerical experiment numerical methods also show that the long run average profit achieved by player adopting naive expectation is higher than that achieved by player using self adaptive adjustment measure although players use similar production methods

recurrence of random walk traces we show that the edges crossed by a random walk in a network form a recurrent graph a s in fact the same is true when those edges are weighted by the number of crossings

estimates of transition densities for brownian motion on nested fractals we obtain upper and lower bounds for the transition densities of brownian motion on nested fractals compared with the estimate on the sierpinski gasket the results require the introduction of a new exponent dj related to the shortest path metric and chemical exponent on nested fractals further holder order of the resolvent densities sample paths and local times are obtained the results are obtained using the theory of multi type branching processes

green s function for a schrodinger operator and some related summation formulas summation formulas are obtained for products of associated lagurre polynomials by means of the green s function k for the hamiltonian h0 d2 dx2 x2 ax 2 a 0 k is constructed by an application of a mercer type theorem that arises in connection with integral equations the approach introduced in this paper may be useful for the construction of wider classes of generating function

analytic and geometric background of recurrence and non explosion of the brownian motion on riemannian manifolds we provide an overview of such properties of the brownian motion on complete non compact riemannian manifolds as recurrence and non explosion it is shown that both properties have various analytic characterizations in terms of the heat kernel green function liouville properties etc on the other hand we consider a number of geometric conditions such as the volume growth isoperimetric inequalities curvature bounds etc which are related to recurrence and non explosion

the dynamics of a business game a 2d piecewise smooth nonlinear map abstract in this paper we give a proper investigation of the problem of remanufacturing duopoly game 1 a smooth piecewise map is introduced to describe such game a rich analysis either analytical or numerical is presented in this paper in order to investigate the stability of the fixed points of the game s model our obtained results show that those fixed points lose its stability due to flip and neimark sacker border bifurcations furthermore we study the stability of those points when there are periodic cycles with points jump from region to other region

matrix elements for a generalized spiked harmonic oscillator closed form expressions for the singular potential integrals m x n are obtained with respect to the gol dman and krivchenkov eigenfunctions for the singular potential bx2 a x2 b 0 a 0 the formulas obtained are generalizations of those found earlier by use of the odd solutions of the schrodinger equation with the harmonic oscillator potential aguilera navarro et al j math phys 31 99 1990

uniform infinite planar triangulations the existence of the weak limit as n of the uniform measure on rooted triangulations of the sphere with n vertices is proved some properties of the limit are studied in particular the limit is a probability measure on random triangulations of the plane

variational analysis for a generalized spiked harmonic oscillator a variational analysis is presented for the generalized spiked harmonic oscillator hamiltonian operator d2 dx 2 bx 2 a x 2 x where is a real positive parameter the formalism makes use of a basis provided by exact solutions of schrodinger s equation for the gol dman and krivchenkov hamiltonian and the corresponding matrix elements that were previously found for all the discrete eigenvalues the method provides bounds which improve as the dimension d of the basis set is increased extension to the n dimensional case in arbitrary angular momentum subspaces is also presented by minimizing over the free parameter a we are able to reduce substantially the number of basis functions needed for a given accuracy

complex dynamics analysis for a duopoly stackelberg game model with bounded rationality in view of the effect of differences between plan products and actual products a duopoly stackelberg model of competition on output is formulated the firms announce plan products sequentially in planning phase and act simultaneously in production phase backward induction is used to solve subgame nash equilibrium the equilibrium outputs and equilibrium profits are affected by cost coefficients for the duopoly stackelberg model a nonlinear dynamical system which describes the time evolution with bounded rationality is analyzed the equilibria of the corresponding discrete dynamical systems are investigated the local stability analysis has been carried out the stability of nash equilibrium gives rise to complex dynamics as some parameters of the model are varied numerical simulations were used to show bifurcation diagram stability region and chaos it is also shown that the state variables feedback and parameter variation method can be used to keep the system from instability and chaos

on hyperbolic graphs induced by iterated function systems abstract for any contractive iterated function system ifs including the moran systems we show that there is a natural hyperbolic graph on the symbolic space which yields the holder equivalence of the hyperbolic boundary and the invariant set of the ifs this completes the previous studies 16 20 30 by eliminating superfluous conditions and admits more classes of sets e g the moran sets we also show that the bounded degree property of the graph can be used to characterize certain separation properties of the ifs open set condition weak separation condition the bounded degree property is particularly important when we consider random walks on such graphs this application and the other application to lipschitz equivalence of self similar sets will be discussed

separators for sphere packings and nearest neighbor graphs a collection of n balls in d dimensions forms a k ply system if no point in the space is covered by more than k balls we show that for every k ply system there is a sphere s that intersects at most o k 1 d n 1 1 d balls of and divides the remainder of into two parts those in the interior and those in the exterior of the sphere s respectively so that the larger part contains at most 1 1 d 2 n balls this bound of o k 1 d n 1 1 d is the best possible in both n and k we also present a simple randomized algorithm to find such a sphere in o n time our result implies that every k nearest neighbor graphs of n points in d dimensions has a separator of size o k 1 d n 1 1 d in conjunction with a result of koebe that every triangulated planar graph is isomorphic to the intersection graph of a disk packing our result not only gives a new geometric proof of the planar separator theorem of lipton and tarjan but also generalizes it to higher dimensions the separator algorithm can be used for point location and geometric divide and conquer in a fixed dimensional space

spiked harmonic oscillators a complete variational treatment is provided for a family of spiked harmonic oscillator hamiltonians h d2 dx2 bx2 x b 0 0 for arbitrary 0 a compact topological proof is presented that the set s n of known exact solutions for 2 constitutes an orthonormal basis of the hilbert space l2 0 closed form expressions are derived for the matrix elements of h with respect to s these analytical results and the inclusion of a further free parameter facilitate optimized variational estimation of the eigenvalues of h to high accuracy

tripoly stackelberg game model this paper is devoted to introduce and study a stackelberg game consisting of three competed firms the three firms are classified as a leader which is the first firm and the other two firms are called the followers a linear inverse demand function is used in addition a quadratic cost based on an actual and announced quantities is adopted based on bounded rationality a three dimensional discrete dynamical system is constructed for the system the backward induction is used to solve the system and to get nash equilibrium the obtained results are shown that nash equilibrium is unique and its stability is affected by the systems parameters by which the system behaves chaotically due to bifurcation and chaos appeared some numerical experiments are performed to portrays such chaotic behavior a control scheme is used to return the system back to its stability state and is supported by some simulations

negatively curved manifolds elliptic operators and the martin boundary consider a riemannian manifold m of dimension n 2 and assume that m is complete simply connected and that its sectional curvatures are bounded between two negative constants in a recent paper 7 m anderson and r schoen have proved that the martin boundary of m with respect to the laplace beltrami operator a is naturally homeomorphic to soo m the sphere at infinity of m here we shall first establish a similar result for a wide class of elliptic operators on m giving at the same time a new proof of the theorem of anderson and schoen our proof relies heavily on several concepts of potential theory but on the other hand avoids the quite difficult calculations of 7 and perhaps makes the result more understandable roughly speaking we show that if y is a second order elliptic operator defined over m and has a uniformly nice behaviour on each ball of radius 1 in m then dq m the y martin boundary of m is s m provided that the following condition holds there is a positive y ci superharmonic function on m for some positive if instead we only assume the existence of the green function for y a minimal requirement when studying the y martin boundary then d m may be very different from so m even for a rotation invariant operator y on the unit ball of rn equipped

estimates of heat kernels for non local regular dirichlet forms in this paper we present new heat kernel upper bounds for a certain class of non local regular dirichlet forms on metric measure spaces including fractal spaces we use a new purely analytic method where one of the main tools is the parabolic maximum principle we deduce off diagonal upper bound of the heat kernel from the on diagonal one under the volume regularity hypothesis restriction of the jump kernel and the survival hypothesis as an application we obtain two sided estimates of heat kernels for non local regular dirichlet forms with finite effective resistance including settings with the walk dimension greater than 2

dirichlet forms and associated heat kernels on the cantor set induced by random walks on trees abstract transient random walk on a tree induces a dirichlet form on its martin boundary which is the cantor set the procedure of the inducement is analogous to that of the douglas integral on s 1 associated with the brownian motion on the unit disk in this paper those dirichlet forms on the cantor set induced by random walks on trees are investigated explicit expressions of the hitting distribution harmonic measure and the induced dirichlet form on the cantor set are given in terms of the effective resistances an intrinsic metric on the cantor set associated with the random walk is constructed under the volume doubling property of with respect to the intrinsic metric asymptotic behaviors of the heat kernel the jump kernel and moments of displacements of the process associated with the induced dirichlet form are obtained furthermore relation to the noncommutative riemannian geometry is discussed

a new method to control chaos in an economic system in this paper the method to control chaos by using phase space compression is applied to economic systems because of economic significance of state variable in economic dynamical systems the values of state variables are positive due to capacity constraints and financial constraints we can control chaos by adding upper bound or lower bound to state variables in economic dynamical systems which is different from the chaos stabilization in engineering or physics systems the knowledge about system dynamics and the exact variety of parameters are not needed in the application of this control method so it is very convenient to apply this method two kinds of chaos in the dynamic duopoly output systems are stabilized in a neighborhood of an unstable fixed point by using the chaos controlling method the results show that performance of the system is improved by controlling chaos in practice owing to capacity constraints financial constraints and cautious responses to uncertainty in the world the firm often restrains the output advertisement expenses research cost etc to confine the range of these variables fluctuation this shows that the decision maker uses this method unconsciously in practice

on bertrand duopoly game with differentiated goods utility function of the form u q 1 q 2 q 1 q 2 is introduced new dynamical bertrand games are derived stability conditions of equilibrium points are discussed chaos and bifurcation behavior are studied the paper investigates a dynamic bertrand duopoly with differentiated goods in which boundedly rational firms apply a gradient adjustment mechanism to update their price in each period the demand functions are derived from an underlying ces utility function we investigate numerically the dynamical properties of the model we consider two specific parameterizations for the ces function and study the nash equilibrium and its local stability in the models the general finding is that the nash equilibrium becomes unstable as the speed of adjustment increases the nash equilibrium loses stability through a period doubling bifurcation and the system eventually becomes chaotic either through a series of period doubling bifurcations or after a neimark sacker bifurcation

analysis of the dynamics of cournot team game with heterogeneous players in this paper we study a dynamical system of a two team cournot game played by a team consisting of two firms with bounded rationality and a team consisting of one firm with naive expectation the equilibrium solutions and the conditions of their locally asymptotic stability are studied it is demonstrated that as some parameters in the model are varied the stability of the equilibrium will get lost and many such complex behaviors as the period bifurcation chaotic phenomenon periodic windows strange attractor and unpredictable trajectories will occur the great influence of the model parameters on the speed of convergence to the equilibrium is also shown with numerical analysis

harmonic functions on planar and almost planar graphs and manifolds via circle packings the circle packing theorem is used to show that on any bounded valence transient planar graph there exists a non constant harmonic bounded dirichlet function if p is a bounded circle packing in bbb r 2 whose contacts graph is a bounded valence triangulation of a disk then with probability 1 the simple random walk on p converges to a limit point moreover in this situation any continuous function on the limit set of p extends to a continuous harmonic function on the closure of the contacts graph of p that is this dirichlet problem is solvable we define the notions of almost planar graphs and manifolds and show that under the assumptions of transience and bounded local geometry these possess non constant harmonic bounded dirichlet functions let us stress that an almost planar graph is not necessarily roughly isometric to a planar graph

chaotic dynamics in nonlinear duopoly game with heterogeneous players in this study we investigate the dynamics of a nonlinear discrete time duopoly game where the players have heterogeneous expectations two players with different expectations are considered one is boundedly rational and the other thinks with adaptive expectations the stability conditions of the equilibria are discussed we show how the dynamics of the game depend on the model parameters we demonstrate that as some parameters of the game are varied the stability of nash equilibrium is lost through period doubling bifurcation the chaotic features are justified numerically via computing lyapunov exponents sensitive dependence on initial conditions and the fractal dimension

on fuzzifications of discrete dynamical systems let x denote a locally compact metric space and f x x be a continuous map in the 1970s zadeh presented an extension principle helping us to fuzzify the dynamical system x f i e to obtain a map f for the space of fuzzy sets on x we extend an idea mentioned in p diamond a pokrovskii chaos entropy and a generalized extension principle fuzzy sets syst 61 1994 277 283 to generalize zadeh s original extension principle in this paper we study basic properties of so called g fuzzifications such as their continuity properties we also show that for any g fuzzification i a uniformly convergent sequence of uniformly continuous maps on x induces a uniformly convergent sequence of fuzzifications on the space of fuzzy sets and ii a conjugacy resp a semi conjugacy between two discrete dynamical systems can be extended to a conjugacy resp a semi conjugacy between fuzzified dynamical systems throughout this paper we consider different topological structures in the space of fuzzy sets namely the sendograph endograph and levelwise topologies

new results on stable region of nash equilibrium of output game model we obtains that the real stable region of nash equilibrium of output game model is smaller than that in general because of the prisoners dilemma players would not select some of the values of parameters though those points of parameters orbit will tend to the nash equilibrium point that is players strategies are unstable in those values in order to discuss this problem we introduce a duopoly output dynamical model we use numerical simulation to analyze the initial output and the decisive parameters influences on the speed of the system s evolution convergence to nash equilibrium and we measure performances of the model in various period states and chaotic state by using the index of average profits it is found that the optimal strategy locates in the stable region of nash equilibrium and aggregate profits are not sensitive to initial output in certain neighborhood of nash equilibrium and the decisive parameters in their majority of stable region of nash equilibrium neither impact of the decisive parameters on the speed of the initial output convergent to nash equilibrium and the prisoners dilemma in game theory are the main reason yielding uneven parts in both sides of 3 dimensions simulation figure the projection of the even parts of the 3 dimensions simulation on the plane of parameters is the real stable region of the nash equilibrium point it is necessary to ascertain the stable region of nash equilibrium point by combining the jury condition with some performance indices in economic systems such as average profit aggregate profits aggregate sales revenues and so on we may determine the factual stable region by using the indifference character of utilities of players in the traditional stable region given by jury s condition we also obtain that players should pay attention to the performance of economic entity in various period states

stochastic perturbations for a duopoly stackelberg model abstract in this paper a duopoly stackelberg model of competition on output with stochastic perturbations is proposed sufficient conditions for the existence and the uniqueness of global positive solution of the stochastic system are derived for the stochastic stackelberg model ssm the authors obtained some sufficient conditions for the globally asymptotical stability of the nash equilibrium when ignoring the stochastic perturbations sufficient criteria for the corresponding deterministic stackelberg model dsm is also obtained in addition by applying milstein s numerical algorithm we solve the stochastic differential model then plot the time series diagram which can help us deduce the change regulation of the output of two firms in the competition game finally some discussions are given for the economic significance for the model

energy bounds for a class of singular potentials and some related series perturbation expansions up to third order for the generalized spiked harmonic oscillator hamiltonians h d2 dx2 x2 a x2 x a 0 2 1 1 2 1 4a and small values of the coupling 0 are developed upper and lower bounds for the eigenvalues are computed by the procedure of burrows et al 1987 j phys a math gen 20 889 97 for assessing the accuracy of a truncated perturbation expansion closed form sums for some related perturbation double infinite series then immediately follow as a result of this investigation

a dirichlet principle for non reversible markov chains and some recurrence theorems we extend the dirichlet principle to non reversible markov processes on countable state spaces we present two variational formulas for the solution of the poisson equation or equivalently for the capacity between two disjoint sets as an application we prove a some recurrence theorems in particular we show the recurrence of two dimensional cycle random walks under a second moment condition on the winding numbers

the components of the wired spanning forest are recurrent we show that a s all of the connected components of the wired spanning forest are recurrent proving a conjecture of benjamini lyons peres and schramm our analysis relies on a simple martingale involving the effective conductance between the endpoints of an edge in a uniform spanning tree we believe that this martingale is of independent interest and will find further applications in the study of uniform spanning trees and forests

tree indexed random walks on groups and first passage percolation suppose that i i d random variables are attached to the edges of an infinite tree when the tree is large enough the partial sumss along some of its infinite paths will exhibit behavior atypical for an ordinary random walk this principle has appeared in works on branching random walks first passage percolation and rwre on trees we establish further quantitative versions of this principle which are applicable in these settings in particular different notions of speed for such a tree indexed walk correspond to different dimension notions for trees finally if the labeling variables take values in a group then properties of the group e g polynomial growth or a nontrivial poisson boundary are reflected in the sample path behavior of the resulting tree indexed walk

boundary theory on the hata tree abstract we prove that for a certain markov chain on the symbolic space of the hata tree k the martin boundary m is homeomorphic to the trunk of the hata tree and the minimal martin boundary is the post critical set 1 2 1 2 which corresponds to the three vertices of the trunk moreover the class of p harmonic functions on m coincides with kigami s class of harmonic functions on k

on the trace of random walks on random graphs we study graph theoretic properties of the trace of a random walk on a random graph we show that for any e 0 there exists c 1 such that the trace of the simple random walk of length 1 e n lnn on the random graph g g n p for p c lnn n is with high probability hamiltonian and lnn connected in the special case p 1 i e when g kn we show a hitting time result according to which with high probability exactly one step after the last vertex has been visited the trace becomes hamiltonian and one step after the last vertex has been visited for the k th time the trace becomes 2k connected

random walks and induced dirichlet forms on self similar sets abstract let k be a self similar set satisfying the open set condition following kaimanovich s elegant idea 25 it has been proved that on the symbolic space x of k a natural augmented tree structure e exists it is hyperbolic and the hyperbolic boundary h x with the gromov metric is holder equivalent to k in this paper we consider certain reversible random walks with return ratio 0 1 on x e we show that the martin boundary m can be identified with h x and k with this setup and a device of silverstein 41 we obtain precise estimates of the martin kernel and the naim kernel in terms of the gromov product moreover the naim kernel turns out to be a jump kernel satisfying the estimate where is the hausdorff dimension of k and depends on for suitable the kernel defines a regular non local dirichlet form on k this extends the results of kigami 27 concerning random walks on certain trees with cantor type sets as boundaries see also 5

analysis of nonlinear triopoly game with heterogeneous players a nonlinear triopoly game with heterogeneous players is presented we consider three types of players boundedly rational adaptive and naive a triopoly game is modelled by a three dimensional discrete dynamical system the stability conditions of the equilibrium points are analyzed numerical simulations are used to show bifurcation diagrams phase portraits sensitive dependence on initial conditions and fractal dimension the chaotic behavior of the model has been stabilized on the nash equilibrium point by the use of the pyragas delay feedback control method

crossing estimates and convergence of dirichlet functions along random walk and diffusion paths let x n be a transient reversible markov chain and let f be a function on the state space with finite dirichlet energy we prove crossing inequalities for the process f x n n geq 1 and show that it converges almost surely and in l 2 analogous results are also established for reversible diffusions on riemannian manifolds

on a recursive construction of dirichlet form on the sierpinski gasket abstract let n denote the n th level sierpinski graph of the sierpinski gasket k we consider for any given conductance a 0 b 0 c 0 on 0 the dirichlet form e on k obtained from a recursive construction of compatible sequence of conductances a n b n c n on n n 0 we prove that there is a dichotomy situation either a 0 b 0 c 0 and e is the standard dirichlet form or a 0 b 0 c 0 or the two symmetric alternatives and e is a non self similar dirichlet form independent of a 0 b 0 the second situation has been studied in 13 9 as a one dimensional asymptotic diffusion the analytic approach here is more direct and yields sharper results in particular for the spectral property we give a precise estimate of the eigenvalue distribution of the associated laplacian which improves a similar result in 9

percolation in the hyperbolic plane the purpose of this paper is to study percolation in the hyperbolic plane and in transitive planar graphs that are quasi isometric to the hyperbolic plane

denker sato type markov chains on self similar sets in denker and sato potential anal 14 211 232 2001 publ rims 35 769 794 1999 math nachr 241 32 55 2002 studied certain markov chain on the symbolic spaces of the sierpinski gasket sg they showed that the martin boundary is homeomorphic to sg and used the potential theory on the martin boundary to induce a harmonic structure on sg in this paper we consider a more general denker sato type markov chain associated with self similar sets k with the open set condition the chain is defined on the augmented tree of the symbolic space such tree was introduced by kaimanovich it is hyperbolic in the sense of gromov kaimanovich in random walks on sierpinski graphs hyperbolicity and stochastic homogenization birhauser basel 2003 lau and wang in indiana univ math j 58 1777 1795 2009 we show that the martin boundary the hyperbolic boundary and the self similar set k are homeomorphic the hitting distribution of the chain is also obtained

markov processes and nonsymmetric dirichlet forms without regularity we study right markov processes in weak duality satisfying the sector condition of silverstein dirichlet space techniques are employed but without the customary c0 regularity hypothesis nonetheless the potential theory of such processes is quite amenable in particular it is proved that semipolar sets are quasi polar this extends results of silverstein and lejan

heat kernels on metric measure spaces and an application to semilinear elliptic equations we consider a metric measure space m d p and a heat kernel p t x v on m satisfying certain upper and lower estimates which depend on two parameters a and we show that under additional mild assumptions these parameters are determined by the intrinsic properties of the space m d namely a is the hausdorff dimension of this space whereas called the walk dimension is determined via the properties of the family of besov spaces w 2 on m moreover the parameters a and are related by the inequalities 2 a 1 we prove also the embedding theorems for the space w 2 2 and use them to obtain the existence results for weak solutions to semilinear elliptic equations on m of the form u f x u g x where is the generator of the semigroup associated with p t the framework in this paper is applicable for a large class of fractal domains including the generalized sierpinski carpet in r n

on reflected dirichlet spaces reflecting diffusion processes on smooth domains in euclidean space are well understood silverstein in 22 and 23 developed two variant procedures for constructing the reflected processes for a general class of symmetric hunt processes from a dirichlet space point of view a direct approach is given in this paper and these two variant procedures are shown to yield the same result only the techniques of martingales and ordinary markov processes are used

planar stochastic hyperbolic triangulations pursuing the approach of angel and ray ann probab 2015 we introduce and study a family of random infinite triangulations of the full plane that satisfy a natural spatial markov property these new random lattices naturally generalize angel and schramm s uniform infinite planar triangulation uipt and are hyperbolic in flavor we prove that they exhibit a sharp exponential volume growth are non liouville and that the simple random walk on them has positive speed almost surely we conjecture that these infinite triangulations are the local limits of uniform triangulations whose genus is proportional to the size an artistic representation of a random 3 connected triangulation of the plane with hyperbolic flavor

unimodular hyperbolic triangulations circle packing and random walk we show that the circle packing type of a unimodular random plane triangulation is parabolic if and only if the expected degree of the root is six if and only if the triangulation is amenable in the sense of aldous and lyons 1 as a part of this we obtain an alternative proof of the benjamini schramm recurrence theorem 19 secondly in the hyperbolic case we prove that the random walk almost surely converges to a point in the unit circle that the law of this limiting point has full support and no atoms and that the unit circle is a realisation of the poisson boundary finally we show that the simple random walk has positive speed in the hyperbolic metric

non intersection of transient branching random walks let g be a cayley graph of a nonamenable group with spectral radius rho 1 it is known that branching random walk on g with offspring distribution mu is transient i e visits the origin at most finitely often almost surely if and only if the expected number of offspring overline mu satisfies overline mu le rho 1 benjamini and muller groups geom dyn 6 231 247 2012 conjectured that throughout the transient supercritical phase 1 overline mu le rho 1 and in particular at the recurrence threshold overline mu rho 1 the trace of the branching random walk is tree like in the sense that it is infinitely ended almost surely on the event that the walk survives forever this is essentially equivalent to the assertion that two independent copies of the branching random walk intersect at most finitely often almost surely we prove this conjecture along with several other related conjectures made by the same authors a central contribution of this work is the introduction of the notion of local unimodularity which we expect to have several further applications in the future

an oligopoly model with best response and imitation rules abstract imitation based behaviors are considered in economics with significant contributions in reference to homogeneous populations where players are characterized by the same decisional processes see for example 42 48 however the presence of imitation behaviors is detected in experimental oligopolies coexisting with rational like rules this motivates us to consider an heterogeneous population where best responders and imitators coexist and compete in a deterministic oligopoly framework the model we formulate is characterized by two stationary states specifically the cournot nash equilibrium and a further production level at which best responders are inactive and imitators produce at the marked clearing price the heterogeneities among players give to the model a nonlinear structure influence the stability properties of the cournot nash equilibrium and give rise to complex dynamic scenarios we found that the imitators relative fraction have an ambiguous role in determining the stability properties of the cournot nash equilibrium and provided intermediate values of the population size its variations may cause the occurrence of both flip and neimark sacker bifurcations chaotic dynamics and coexistent attractors characterized by not connected basins may also be observed we finally note that certain dynamic regimes described by the model are provided by analogous features as those characterizing experimental outcomes and several experiments can be reproduced with different parameters sets

discrete approximation of symmetric jump processes on metric measure spaces in this paper we give general criteria on tightness and weak convergence of discrete markov chains to symmetric jump processes on metric measure spaces under mild conditions as an application we investigate discrete approximation for a large class of symmetric jump processes we also discuss some application of our results to the scaling limit of random walk in random conductance

random walks and harmonic functions on infinite planar graphs using square tilings we study a wide class of transient planar graphs through a geometric model given by a square tiling of a cylinder for many graphs the geometric boundary of the tiling is a circle and is easy to describe in general the simple random walk on the graph converges with probability 1 to a point in the geometric boundary we obtain information on the harmonic measure and estimates on the rate of convergence this allows us to extend results we previously proved for triangulations of a disk

rigidity of infinite circle packings the nerve of a packing is a graph that encodes its combinatorics the vertices of the nerve correspond to the packed sets and an edge occurs between two vertices in the nerve precisely when the corresponding sets of the packing intersect

the dynamics of a quantum bertrand duopoly with differentiated products and heterogeneous expectations abstract based on the bounded rationality and naive expectation the paper sets up a dynamic model of quantum bertrand duopoly game with differentiated products by applying quantum game theory we analyze the effect of quantum entanglement on stability of the quantum equilibrium points and complex dynamic behaviors the result shows that the quantum entanglement can enhance the stability of the system the adjustment speeds of bounded rational player can lead to chaotic characteristics of the system the quantum entanglement can control the system s chaos numerical simulations demonstrate the chaotic features via bifurcation largest lyapunov exponent strange attractors fractal dimensions and sensitivity to initial conditions

chaotic dynamics in cournot duopoly model with bounded rationality based on relative profit delegation maximization abstract in this paper we maximize the utility function under budget constraint in an oligopoly market we consider a cournot duopoly model with boundedly rational players adopting the gradient adjustment mechanism to update their productions the dynamics of cournot duopoly model with relative profit delegation maximization is analyzed we explore the local stability of nash equilibrium point and the occurrence of complexity behaviors additionally the chaotic dynamics of this game via flip bifurcations and neimark sacker bifurcations are researched numerical simulations are presented to show bifurcation diagrams sensitive dependence on initial conditions strange attractors and chaos as to the fractal structure emerged in evolution process the fractal dimension is calculated further analysis of the simulation results indicates that the chaotic behavior of the system can be stabilized on the nash equilibrium point by using state variables feedback and parameter variation method