gradient systems in view of information geometry abstract dualistic properties of a gradient flow on a manifold m associated with a dualistic structure g is studied from an information geometrical viewpoint some useful applications are also investigated

existence of unbounded positive solutions of quasilinear elliptic equations in two dimensional exterior domains in this paper by using the fixed point theory we obtain a new existence result for bounded positive solutions of the quasilinear elliptic equations in two dimensional exterior domains

contact geometric descriptions of vector fields on dually flat spaces and their applications in electric circuit models and nonequilibrium statistical mechanics contact geometry has been applied to various mathematical sciences and it has been proposed that a contact manifold and a strictly convex function induce a dually flat space that is used in information geometry here such a dually flat space is related to a legendre submanifold in a contact manifold in this paper contact geometric descriptions of vector fields on dually flat spaces are proposed on the basis of the theory of contact hamiltonian vector fields based on these descriptions two ways of lifting vector fields on legendre submanifolds to contact manifolds are given for some classes of these lifted vector fields invariant measures in contact manifolds and stability analysis around legendre submanifolds are explicitly given throughout this paper legendre duality is explicitly stated in addition to show how to apply these general methodologies to applied mathematical disciplines electric circuit models and some examples taken from nonequilibrium statistical mechanics are analyzed

border collision period doubling scenario using a one dimensional dynamical system representing a poincare return map for dynamical systems ofthe lorenz type we investigate the border collision period doubling bifurcation scenario in contrast to theclassical period doubling scenario this scenario is formed by a sequence of pairs of bifurcations whereby eachpair consists of a border collision bifurcation and a pitchfork bifurcation the characteristic properties of thisscenario like symmetry breaking and symmetry recovering as well as emergence of coexisting attractors andnoninvariant attractive sets are investigated doi 10 1103 physreve 70 026222 pacs number s 05 45 ac 82 40 bj 95 10 fhi introduction

stochastic dynamics of invasion and fixation we study evolutionary game dynamics in finite populations we analyze an evolutionary process which we call pairwise comparison for which we adopt the ubiquitous fermi distribution function from statistical mechanics the inverse temperature in this process controls the intensity of selection leading to a unified framework for evolutionary dynamics at all intensities of selection from random drift to imitation dynamics we derive a simple closed formula that determines the feasibility of cooperation in finite populations whenever cooperation is modeled in terms of any symmetric two person game in contrast with previous results the present formula is valid at all intensities of selection and for any initial condition we investigate the evolutionary dynamics of cooperators in finite populations and study the interplay between intensity of selection and the remnants of interior fixed points in infinite populations as a function of a given initial number of cooperators showing how this interplay strongly affects the approach to fixation of a given trait in finite populations leading to counterintuitive results at different intensities of selection

contact geometry of the pontryagin maximum principle this paper gives a brief contact geometric account of the pontryagin maximum principle we show that key notions in the pontryagin maximum principle such as the separating hyperplanes costate necessary condition and normal abnormal minimizers have natural contact geometric interpretations we then exploit the contact geometric formulation to give a simple derivation of the transversality condition for optimal control with terminal cost

information geometry of q gaussian densities and behaviors of solutions to related diffusion equations this paper presents new geometric aspects of the behaviors of solutions to the porous medium equation pme and its associated equation first we discuss thermostatistical structure with information geometry on a manifold of generalized exponential densities a dualistic relation between the two existing formalisms is elucidated next by equipping the manifold of q gaussian densities with such a structure we derive several physically and geometrically interesting properties of the solutions the manifold is proved invariant and attracting for the evolving solutions which play crucial roles in our analysis we demonstrate that the moment conserving projection of a solution coincides with a geodesic curve on the manifold further the evolutional velocities of the second moments and the convergence rate to the manifold are evaluated in terms of the bregman divergence finally we show that the self similar solution is geometrically special in the sense that it is simultaneously geodesic with respect to the mutually dual two affine connections

scaling behavior in a proportional voting process we perform a statistical analysis on the proportional elections held in brazil in october 1998 we show that the distribution of votes among candidates for the whole country follows a power law n v ensuremath propto v ensuremath ensuremath alpha with ensuremath alpha 1 00 ifmmode pm else textpm fi 0 02 extending over two orders of magnitude the voting distributions for several states of the federation also display scale invariant behavior with ensuremath alpha ensuremath approx 1 we argue that this particular voting system can be modeled as a typical multiplicative process in which the choice of the candidate is governed by a product of probabilities

border collision bifurcations an explanation for observed bifurcation phenomena recently physical and computer experiments involving systems describable by continuous maps that are nondifferentiable on some surface in phase space have revealed novel bifurcation phenomena these phenomena are part of a rich new class of bifurcations which we call border collision bifurcations a general criterion for the occurrence of border collision bifurcations is given illustrative numerical results including transitions to chaotic attractors are presented these border collision bifurcations are found in a variety of physical experiments

liouville s theorem and the canonical measure for nonconservative systems from contact geometry standard statistical mechanics of conservative systems relies on the symplectic geometry of the phase space this is exploited to derive hamilton s equations liouville s theorem and to find the canonical invariant measure in this work we analyze the statistical mechanics of a class of nonconservative systems stemming from contact geometry in particular we find out the generalized hamilton s equations liouville s theorem and the microcanonical and canonical measures invariant under the contact flow remarkably the latter measure has a power law density distribution with respect to the standard contact volume form finally we argue on the several possible applications of our results

stochastic bifurcations in the nonlinear parallel ising model we investigate the phase transitions of a nonlinear parallel version of the ising model characterized by an antiferromagnetic linear coupling and ferromagnetic nonlinear one this model arises in problems of opinion formation the mean field approximation shows chaotic oscillations by changing the couplings or the connectivity the spatial model shows bifurcations in the average magnetization similar to that seen in the mean field approximation induced by the change of the topology after rewiring short range to long range connection as predicted by the small world effect these coherent periodic and chaotic oscillations of the magnetization reflect a certain degree of synchronization of the spins induced by long range couplings similar bifurcations may be induced in the randomly connected model by changing the couplings or the connectivity and also the dilution degree of asynchronism of the updating we also examined the effects of inhomogeneity mixing ferromagnetic and antiferromagnetic coupling which induces an unexpected bifurcation diagram with a bubbling behavior as also happens for dilution

border collision bifurcations including period two to period three for piecewise smooth systems abstract we examine bifurcation phenomena for maps that are piecewise smooth and depend continuously on a parameter in the simplest case there is a surface in phase space along which the map has no derivative or has two one sided derivatives is the border of two regions in which the map is smooth as the parameter is varied a fixed point e may collide with the border and we may assume that this collision occurs at 0 a variety of bifurcations occur frequently in such situations but never or almost never occur in smooth systems in particular e may cross the border and so will exist for 0 but it may be a saddle in one case say 0 for 0 there may be a stable period 3 orbit which similarly shrinks to e0 as 0 hence one observes the following stable periodic orbits a stable period 2 orbit collapses to a point and is reborn as a stable period 3 orbit we also see analogously stable period 2 to stable period p orbit bifurcations with p 5 11 52 or period 2 to quasi periodic or even to a chaotic attractor we believe this phenomenon will be seen in many applications

the noisy voter model under the influence of contrarians abstract the influence of contrarians on the noisy voter model is studied at the mean field level the noisy voter model is a variant of the voter model where agents can adopt two opinions optimistic or pessimistic and can change them by means of an imitation herding and an intrinsic noise mechanisms an ensemble of noisy voters undergoes a finite size phase transition upon increasing the relative importance of the noise to the herding form a bimodal phase where most of the agents share the same opinion to a unimodal phase where almost the same fraction of agent are in opposite states by the inclusion of contrarians we allow for some voters to adopt the opposite opinion of other agents anti herding we first consider the case of only contrarians and show that the only possible steady state is the unimodal one more generally when voters and contrarians are present we show that the bimodal unimodal transition of the noisy voter model prevails only if the number of contrarians in the system is smaller than four and their characteristic rates are small enough for the number of contrarians bigger or equal to four the voters and the contrarians can be seen only in the unimodal phase moreover if the number of voters and contrarians as well as the noise and herding rates are of the same order then the probability functions of the steady state are very well approximated by the gaussian distribution

disorder induced phase transition in kinetic models of opinion dynamics abstract we propose a model of continuous opinion dynamics where mutual interactions can be both positive and negative different types of distributions for the interactions all characterized by a single parameter p denoting the fraction of negative interactions are considered results from exact calculation of a discrete version and numerical simulations of the continuous version of the model indicate the existence of a universal continuous phase transition at p p c below which a consensus is reached although the order disorder transition is analogous to a ferromagnetic paramagnetic phase transition with comparable critical exponents the model is characterized by some distinctive features relevant to a social system

dualistic differential geometry of positive definite matrices and its applications to related problems abstract analysis on the set of positive definite matrices is involved in many engineering problems we develop a differential geometric theory of the set of positive definite matrices by means of a specific class of connections introduced into it consequently various dualistic aspects of the set are elucidated next using the theory we derive new results and interesting geometrical interpretations for matrix approximation positive definite matrix completion and linear matrix inequality problems

technical communique a simple proof of the pontryagin maximum principle on manifolds applying the tubular neighborhood theorem we give a simple proof of the pontryagin maximum principle on a smooth manifold the idea is as follows given a control system on a manifold m we embed it into some r n and extend the control system to r n then we apply the pontryagin maximum principle on r n to the extended system and project the consequence to m

nonequilibrium transitions in complex networks a model of social interaction we analyze the nonequilibrium order disorder transition of axelrod s model of social interaction in several complex networks in a small world network we find a transition between an ordered homogeneous state and a disordered state the transition point is shifted by the degree of spatial disorder of the underlying network the network disorder favoring ordered configurations in random scale free networks the transition is only observed for finite size systems showing system size scaling while in the thermodynamic limit only ordered configurations are always obtained thus in the thermodynamic limit the transition disappears however in structured scale free networks the phase transition between an ordered and a disordered phase is restored

universality of weak selection weak selection which means a phenotype is slightly advantageous over another is an important limiting case in evolutionary biology recently it has been introduced into evolutionary game theory in evolutionary game dynamics the probability to be imitated or to reproduce depends on the performance in a game the influence of the game on the stochastic dynamics in finite populations is governed by the intensity of selection in many models of both unstructured and structured populations a key assumption allowing analytical calculations is weak selection which means that all individuals perform approximately equally well in the weak selection limit many different microscopic evolutionary models have the same or similar properties how universal is weak selection for those microscopic evolutionary processes we answer this question by investigating the fixation probability and the average fixation time not only up to linear but also up to higher orders in selection intensity we find universal higher order expansions which allow a rescaling of the selection intensity with this we can identify specific models which violate linear weak selection results such as the one third rule of coordination games in finite but large populations

a simple proof of the discrete time geometric pontryagin maximum principle on smooth manifolds we establish a geometric pontryagin maximum principle for discrete time optimal control problems on finite dimensional smooth manifolds under the following three types of constraints a constraints on the states pointwise in time b constraints on the control actions pointwise in time c constraints on the frequency spectrum of the optimal control trajectories our proof follows in spirit the path to establish geometric versions of the pontryagin maximum principle on smooth manifolds indicated in chang 2011 in the context of continuous time optimal control

contact variational integrators we present geometric numerical integrators for contact flows that stem from a discretization of herglotz variational principle first we show that the resulting discrete map is a contact transformation and that any contact map can be derived from a variational principle then we discuss the backward error analysis of our variational integrators including the construction of a modified lagrangian surprisingly this construction presents some interesting simplifications compared to the corresponding construction for symplectic systems throughout the paper we use the damped harmonic oscillator as a benchmark example to compare our integrators to their symplectic analogues

hessian information geometric formulation of hamiltonian systems and generalized toda s dual transform in this paper a class of classical hamiltonian systems is geometrically formulated this class is such that a hamiltonian can be written as the sum of a kinetic energy function and a potential energy function in addition these energy functions are assumed strictly convex for this class of hamiltonian systems hessian and information geometric formulation is given with this formulation a generalized toda s dual transform is proposed where his original transform was used in deriving his integrable lattice system then a relation between the generalized toda s dual transform and the legendre transform of a class of potential energy functions is shown as an extension of this formulation dissipation less electric circuit models are also discussed in the geometric viewpoint above

exact baker campbell hausdorff formula for the contact heisenberg algebra in this work we introduce the contact heisenberg algebra which is the restriction of the jacobi algebra on contact manifolds to the linear and constant functions we give the exact expression of its corresponding baker campbell hausdorff formula we argue that this result is relevant to the quantization of contact systems

clusters and the entropy in opinion dynamics on complex networks abstract in this work we investigate a heterogeneous population in the modified hegselmann krause opinion model on complex networks we introduce the shannon information entropy about all relative opinion clusters to characterize the cluster profile in the final configuration independent of network structures there exists the optimal stubbornness of one subpopulation for the largest number of clusters and the highest entropy besides there is the optimal bounded confidence or subpopulation ratio of one subpopulation for the smallest number of clusters and the lowest entropy however network structures affect cluster profiles indeed a large average degree favors consensus for making different networks more similar with complete graphs the network size has limited impact on cluster profiles of heterogeneous populations on scale free networks but has significant effects upon those on small world networks

some properties of conservative port contact systems the dynamics of open irreversible thermodynamic systems that is systems including both the balance equation of the energy and the entropy has been formulated as contact vector fields with generating functions depending on some external control variable and called conservative port contact systems in this paper we relate the dynamical properties of these systems equilibrium points asymptotic stability to properties of the generating functions the contact hamiltonian functions we show that the equilibrium points of the system satisfy certain conditions involving the contact hamiltonian function we also consider lyapunov s first theorem to emphasize a stability criterion for the equilibrium points in terms of this contact hamiltonian function and relate it to some thermodynamical properties these results are then related to the physical phenomena that are taking place in the system

nature of phase transitions in a probabilistic cellular automaton with two absorbing states we present a probabilistic cellular automaton with two absorbing states which can be considered a natural extension of the domany kinzel model despite its simplicity it shows a very rich phase diagram with two second order and one first order transition lines that meet at a bicritical point we study the phase transitions and the critical behavior of the model using mean field approximations direct numerical simulations and field theory the second order critical curves and the kink critical dynamics are found to be in the directed percolation and parity conservation universality classes respectively the first char21 order phase transition is put in evidence by examining the hysteresis cycle we also study the chaotic phase in which two replicas evolving with the same noise diverge using mean field and numerical techniques finally we show how the shape of the potential of the field theoretic formulation of the problem can be obtained by direct numerical simulations

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

phase transitions of extended range probabilistic cellular automata with two absorbing states we study phase transitions in a long range one dimensional cellular automaton with two symmetric absorbing states it includes and extends several other models like the ising and domany kinzel ones it is characterized by competing ferromagnetic linear and antiferromagnetic nonlinear couplings despite its simplicity this model exhibits an extremely rich phase diagram we present numerical results and mean field approximations

global existence of solutions with prescribed asymptotic behavior for second order nonlinear differential equations where the nonlinearity f satis9es certain monotonicity conditions speci9ed in the sequel we always assume without mentioning that the function f t u v is continuous in the domain d t u v t t0 u v r where t0 1 it is supposed also that all equations and inequalities involving t hold for t t0 unless otherwise speci9ed the symbols o and o have their standard meanings as t and we use the notation r def 0 eq 1 is often used for mathematical modelling of various physical chemical and biological systems and attracts constant interest of researchers a great deal of papers published during the last three decades are concerned with local and global existence of solutions of eq 1 and its numerous particular cases uniqueness of solutions

topological bifurcations in a model society of reasonable contrarians people are often divided into conformists and contrarians the former tending to align to the majority opinion in their neighborhood and the latter tending to disagree with that majority in practice however the contrarian tendency is rarely followed when there is an overwhelming majority with a given opinion which denotes a social norm such reasonable contrarian behavior is often considered a mark of independent thought and can be a useful strategy in financial markets we present the opinion dynamics of a society of reasonable contrarian agents the model is a cellular automaton of ising type with antiferromagnetic pair interactions modeling contrarianism and plaquette terms modeling social norms we introduce the entropy of the collective variable as a way of comparing deterministic mean field and probabilistic simulations bifurcation diagrams in the mean field approximation the model exhibits bifurcations and a chaotic phase interpreted as coherent oscillations of the whole society however in a one dimensional spatial arrangement one observes incoherent oscillations and a constant average in simulations on watts strogatz networks with a small world effect the mean field behavior is recovered with a bifurcation diagram that resembles the mean field one but where the rewiring probability is used as the control parameter similar bifurcation diagrams are found for scale free networks and we are able to compute an effective connectivity for such networks

existence for unbounded positive solutions of schrodinger equations in two dimensional exterior domains in this paper by using the fixed point theory under quite general conditions on the nonlinear term we obtain a existence result of unbounded positive solutions of schrodinger equations in two dimensional exterior domains

diffusion equations from master equations a discrete geometric approach in this paper continuous time master equations with finite states employed in nonequilibrium statistical mechanics are formulated in the language of discrete geometry in this formulation chains in algebraic topology are used and master equations are described on graphs that consist of vertices representing states and of directed edges representing transition matrices it is then shown that master equations under the detailed balance conditions are equivalent to discrete diffusion equations where the laplacians are defined as self adjoint operators with respect to introduced inner products an isospectral property of these laplacians is shown for non zero eigenvalues and its applications are given the convergence to the equilibrium state is shown by analyzing this class of diffusion equations in addition a systematic way to derive closed dynamical systems for expectation values is given for the case that the detailed balance conditions are not imposed master equations are expressed as a form of a continuity equation