stochastic resonance in an ensemble of bistable systems under stable distribution noises and nonhomogeneous coupling in this paper stochastic resonance of an ensemble of coupled bistable systems driven by noise having an stable distribution and nonhomogeneous coupling is investigated the stable distribution considered here is characterized by four intrinsic parameters 0 2 is called the stability parameter for describing the asymptotic behavior of stable densities 1 1 is a skewness parameter for measuring asymmetry 0 is a scale parameter for measuring the width of the distribution and is a location parameter for representing the mean value it is demonstrated that the resonant behavior is optimized by an intermediate value of the diversity in coupling strengths we show that the stability parameter and the scale parameter can be well selected to generate resonant effects in response to external signals in addition the interplay between the skewness parameter and the location parameter on the resonance effects is also studied we further show that the asymmetry of a l evy stable distribution resulting from the skewness parameter and the location parameter can enhance the resonance effects both theoretical analysis and simulation are presented to verify the results of this paper

yang s system of particles and hecke algebras the graded hecke algebra has a simple realization as a certain algebra of operators acting on a space of smooth functions this operator algebra arises from the study of the root system analogue of yang s system of n particles on the real line with delta function potential it turns out that the spectral problem for this generalization of yang s system is related to the problem of finding the spherical tempered representations of the graded hecke algebra this observation turns out to be very useful for both these problems application of our technique to affine hecke algebras yields a simple formula for the formal degree of the generic iwahori spherical discrete series representations

exact partition functions for potts antiferromagnets on cyclic lattice strips abstract we present exact calculations of the zero temperature partition function of the q state potts antiferromagnet on arbitrarily long strips of the square triangular and kagome lattices with width l y 2 or 3 vertices and with periodic longitudinal boundary conditions from these in the limit of infinite length we obtain the exact ground state entropy s 0 k b ln w these results are of interest since this model exhibits nonzero ground state entropy s 0 0 for sufficiently large q and hence is an exception to the third law of thermodynamics we also include results for homeomorphic expansions of the square lattice strip the analytic properties of w q are determined and related to zeros of the chromatic polynomial for long finite strips

synchronization in weighted scale free networks with degree degree correlation abstract we study synchronization phenomena in scale free networks of asymmetrically coupled dynamical systems featuring degree degree correlation in the topology of the connection wiring we show that when the interaction is dominant from the high degree to the low degree or from the low degree to the high degree nodes synchronization is enhanced for assortative or disassortative degree degree correlations

pinning impulsive synchronization of reaction diffusion neural networks with time varying delays this paper investigates the exponential synchronization of reaction diffusion neural networks with time varying delays subject to dirichlet boundary conditions a novel type of pinning impulsive controllers is proposed to synchronize the reaction diffusion neural networks with time varying delays by applying the lyapunov functional method sufficient verifiable conditions are constructed for the exponential synchronization of delayed reaction diffusion neural networks with large and small delay sizes it is shown that synchronization can be realized by pinning impulsive control of a small portion of neurons of the network the technique used in this paper is also applicable to reaction diffusion networks with neumann boundary conditions numerical examples are presented to demonstrate the effectiveness of the theoretical results

chromatic polynomials for j h i strip graphs and their asymptotic limits abstract we calculate the chromatic polynomials p for n vertex strip graphs of the form j l 1 m h i where j and i are various subgraphs on the left and right ends of the strip whose bulk is comprised of m fold repetitions of a subgraph h the strips have free boundary conditions in the longitudinal direction and free or periodic boundary conditions in the transverse direction this extends our earlier calculations for strip graphs of the form l 1 m h i we use a generating function method from these results we compute the asymptotic limiting function w lim n p 1 n for q z this has physical significance as the ground state degeneracy per site exponent of the ground state entropy of the q state potts antiferromagnet on the given strip in the complex q plane w is an analytic function except on a certain continuous locus b in contrast to the l 1 m h i strip graphs where b i is independent of i and ii consists of arcs and possible line segments that do not enclose any regions in the q plane we find that for some j l 1 m h i strip graphs b i does depend on i and j and ii can enclose regions in the q plane our study elucidates the effects of different end subgraphs i and j and of boundary conditions on the infinite length limit of the strip graphs

intertwining operators associated to the group for any finite reflection group g on an euclidean space there is a parametrized commutative algebra of differential difference operators with as many parameters as there are conjugacy classes of reflections in g there exists a linear isomorphism on polynomials which intertwines this algebra with the algebra of partial differential operators with constant coefficients for all but a singular set of parameter values containing only certain negative rational numbers this paper constructs an integral transform implementing the intertwining operator for the group s3 the symmetric group on three objects for parameter value the transform is realized as an absolutely continuous measure on a compact subset of m2 r which contains the group as a subset of its boundary the construction of the integral formula involves integration over the unitary group u 3 associated to any finite reflection group g on an euclidean space there is a parametrized commutative algebra of differential difference operators with as many parameters as there are conjugacy classes of reflections in g it has been shown that there exists a linear isomorphism on polynomials which intertwines this algebra with the algebra of partial differential operators with constant coefficients for all but a singular set of parameter values this singular set contains only negative values and is closely linked to the zero set of the poincare series of g this paper constructs an integral transform implementing the intertwining map for the group s3 the symmetric group on three objects for positive parameter values previously this had been done only for the group z2 acting by sign change on r where the transform is a classical fractional integral the transform in this paper has its origin in the adjoint action of the unitary group u 3 on the linear space of real diagonal 3 x 3 matrices the complexification of the maximal torus this will lead to a transform realized as an absolutely continuous measure on a certain compact subset of m2 r here is a concise statement of the main result rephrased from formulas 5 1 5 6 the operator v intertwines the differential difference operators received by the editors august 22 1994 1991 mathematics subject classification primary 22e30 33c80 secondary 33c50 20b30

complex temperature phase diagrams for the q state potts model on self dual families of graphs and the nature of the q limit exact calculations of the potts model partition function z g q v have been presented for arbitrary q and temperature like variable v on self dual strip graphs g of the square lattice with fixed width l y and arbitrarily great length lx with two types of boundary conditions letting lx infinity the resultant free energy and complex temperature phase diagram have been computed including the locus b where the free energy is nonanalytic results are analyzed for widths l y 1 2 3 these results have been used to study the approach to the large q limit of b

asymptotic limits and zeros of chromatic polynomials and ground state entropy of potts antiferromagnets we study the asymptotic limiting function w g q lim n to infty p g q 1 n where p g q is the chromatic polynomial for a graph g with n vertices we first discuss a subtlety in the definition of w g q resulting from the fact that at certain special points q s the following limits do not commute lim n to infty lim q to q s p g q 1 n ne lim q to q s lim n to infty p g q 1 n we then present exact calculations of w g q and determine the corresponding analytic structure in the complex q plane for a number of families of graphs g including circuits wheels biwheels bipyramids and cyclic and twisted ladders we study the zeros of the corresponding chromatic polynomials and prove a theorem that for certain families of graphs all but a finite number of the zeros lie exactly on a unit circle whose position depends on the family using the connection of p g q with the zero temperature potts antiferromagnet we derive a theorem concerning the maximal finite real point of non analyticity in w g q denoted q c and apply this theorem to deduce that q c sq 3 and q c hc 3 sqrt 5 2 for the square and honeycomb lattices finally numerical calculations of w hc q and w sq q are presented and compared with series expansions and bounds

periodic integrable systems with delta potentials in this paper we study root system generalizations of the quantum bose gas on the circle with pair wise delta function interactions the underlying symmetry structures are shown to be governed by the associated graded algebra of cherednik s suitably filtered degenerate double affine hecke algebra acting by dunkl type differential reflection operators we use gutkin s generalization of the equivalence between the impenetrable bose gas and the free fermi gas to derive the bethe ansatz equations and the bethe ansatz eigenfunctions

desynchronization waves in small world networks a regular array of oscillators with random coupling exhibits a transition from synchronized motion to desynchronized but ordered waves as a global coupling parameter is increased due to the spread of localized instability of eigenvectors of the laplacian matrix we find that shortcuts which make a regular network small world can destroy ordered desynchronization wave patterns wave patterns in a small world network are usually destroyed gradually as the degree of regularity in the network deteriorates no ordered wave patterns are observed in scale free and random networks the formation of ordered wave patterns in a coupled oscillator network can be explained by considering the time evolution of phase in each oscillator we derive a general type of the kardar parisi zhang equation for phase evolution in a prototype oscillator network the equation demonstrates well the ordered desynchronized wave patterns found in the network with and without shortcuts our results provide a qualitative justification for the requirement of certain degree of regularity in the network for ordered wave patterns to arise

synchronization transitions on scale free neuronal networks due to finite information transmission delays we investigate front propagation and synchronization transitions in dependence on the information transmission delay and coupling strength over scale free neuronal networks with different average degrees and scaling exponents as the underlying model of neuronal dynamics we use the efficient rulkov map with additive noise we show that increasing the coupling strength enhances synchronization monotonously whereas delay plays a more subtle role in particular we found that depending on the inherent oscillation frequency of individual neurons regions of irregular and regular propagating excitatory fronts appear intermittently as the delay increases these delay induced synchronization transitions manifest as well expressed minima in the measure for spatial synchrony appearing at every multiple of the oscillation frequency larger coupling strengths or average degrees can broaden the region of regular propagating fronts by a given information transmission delay and further improve synchronization these results are robust against variations in system size intensity of additive noise and the scaling exponent of the underlying scale free topology we argue that fine tuned information transmission delays are vital for assuring optimally synchronized excitatory fronts on complex neuronal networks and indeed they should be seen as important as the coupling strength or the overall density of interneuronal connections we finally discuss some biological implications of the presented results

connection between yangian symmetry and the quantum inverse scattering method the quantum nonlinear schrodinger model with two component fermions exhibits a yangian symmetry when considered on an infinite interval we construct the generators of the yangian using dunkl operators we show an equivalence between the monodromy matrix constructed from the dunkl operators and that of the quantum inverse scattering method under the yangian algebra the space of states with a fixed particle number forms a tensor product representation of fundamental representations

families of graphs with w r g q functions that are nonanalytic at 1 q 0 denoting p g q as the chromatic polynomial for coloring an n vertex graph g with q colors and considering the limiting function w g q lim n to infty p g q 1 n a fundamental question in graph theory is the following is w r g q q 1 w g q analytic or not at the origin of the 1 q plane where the complex generalization of q is assumed this question is also relevant in statistical mechanics because w g q exp s 0 k b where s 0 is the ground state entropy of the q state potts antiferromagnet on the lattice graph g and the analyticity of w r g q at 1 q 0 is necessary for the large q series expansions of w r g q although w r is analytic at 1 q 0 for many g there are some g for which it is not for these w r has no large q series expansion it is important to understand the reason for this nonanalyticity here we give a general condition that determines whether or not a particular w r g q is analytic at 1 q 0 and explains the nonanalyticity where it occurs we also construct infinite families of graphs with w r functions that are non analytic at 1 q 0 and investigate the properties of these functions our results are consistent with the conjecture that a sufficient condition for w r g q to be analytic at 1 q 0 is that g is a regular lattice graph lambda this is known not to be a necessary condition

thin fisher zeros various authors have suggested that the loci of partition function zeros can profitably be regarded as phase boundaries in the complex temperature or field planes we obtain the fisher zeros for ising and potts models on non planar thin regular random graphs using this approach and note that the locus of fisher zeros on a bethe lattice is identical to the corresponding random graph since the number of states q appears as a parameter in the potts solution the limiting locus of chromatic zeros is also accessible

harmonic oscillator under levy noise unexpected properties in the phase space a harmonic oscillator under the influence of noise is a basic model of various physical phenomena under gaussian white noise the position and velocity of the oscillator are independent random variables which are distributed according to the bivariate gaussian distribution with elliptic level lines the distribution of phase is homogeneous none of these properties hold in the general levy case thus the level lines of the joint probability density are not elliptic the coordinate and the velocity of the oscillator are strongly dependent and this dependence is quantified by introducing the corresponding parameter width deficit the distribution of the phase is inhomogeneous and highly nontrivial

the mechanism of stochastic resonance it is shown that a dynamical system subject to both periodic forcing and random perturbation may show a resonance peak in the power spectrum which is absent when either the forcing or the perturbation is absent

improved lower bounds on the ground state entropy of the antiferromagnetic potts model we present generalized methods for calculating lower bounds on the ground state entropy per site s 0 or equivalently the ground state degeneracy per site w e s 0 k b of the antiferromagnetic potts model we use these methods to derive improved lower bounds on w for several lattices

chromatic polynomials for families of strip graphs and their asymptotic limits abstract we calculate the chromatic polynomials p gs m q and from these the asymptotic limiting functions w g s q lim n p g s q 1 n for families of n vertex graphs gs m comprised of m repeated subgraphs h adjoined to an initial graph i these calculations of w gs q for infinitely long strips of varying widths yield important insights into properties of w q for two dimensional lattices in turn these results connect with statistical mechanics since w q is the ground state degeneracy of the q state potts model on the lattice for our calculations we develop and use a generating function method which enables us to determine both the chromatic polynomials of finite strip graphs and the resultant w gs q function in the limit n from this we obtain the exact continuous locus of points b where w gs q is nonanalytic in the complex q plane this locus is shown to consist of arcs which do not separate the q plane into disconnected regions zeros of chromatic polynomials are computed for finite strips and compared with the exact locus of singularities b we find that as the width of the infinitely long strips is increased the arcs comprising b elongate and move toward each other which enables one to understand the origin of closed regions that result for the infinite 2d lattice

flow polynomials and their asymptotic limits for lattice strip graphs we present exact calculations of flow polynomials f g q for lattice strips of various fixed widths l y 4 and arbitrarily great lengths l x with several different boundary conditions square honeycomb and triangular lattice strips are considered we introduce the notion of flows per face fl in the infinite length limit we study the zeros of f g q in the complex q plane and determine exactly the asymptotic accumulation sets of these zeros in the infinite length limit for the various families of strips the function fl is nonanalytic on this locus the loci are found to be noncompact for many strip graphs with periodic or twisted periodic longitudinal boundary conditions and compact for strips with free longitudinal boundary conditions we also find the interesting feature that aside from the trivial case l y 1 the maximal point q cf where crosses the real axis is universal on cyclic and mobius strips of the square lattice for all widths for which we have calculated it and is equal to the asymptotic value q cf 3 for the infinite square lattice

influence of time delayed feedback in the firing pattern of thermally sensitive neurons we explore the dynamics of a hodgkin huxley type model for thermally sensitive neurons that exhibit intrinsic oscillatory activity the model is modified to include a feedback loop that is represented by two parameters the synaptic strength and the transmission delay time we analyze the dynamics of the neuron depending on the temperature the synaptic strength and the delay time we find parameter regions where the effect of the recurrent connexion is excitatory inducing spikes or trains of spikes and regions where it is inhibitory reducing or eliminating completely the spiking behavior we characterize the complex interplay of the intrinsic dynamics of the neuron with the recurrent feedback input and a noisy input

scale free brain functional networks functional magnetic resonance imaging is used to extract functional networks connecting correlated human brain sites analysis of the resulting networks in different tasks shows that a the distribution of functional connections and the probability of finding a link versus distance are both scale free b the characteristic path length is small and comparable with those of equivalent random networks and c the clustering coefficient is orders of magnitude larger than those of equivalent random networks all these properties typical of scale free small world networks reflect important functional information about brain states

existence and stability of almost periodic solutions for cellular neural networks with time varying delays in leakage terms on time scales abstract this paper is concerned with cellular neural networks with time varying delays in leakage terms on time scales some sufficient conditions on the existence uniqueness and global exponential stability of almost periodic solutions are established an example is presented to illustrate the feasibility and effectiveness of the obtained results

ground state entropy of potts antiferromagnets on homeomorphic families of strip graphs abstract we present exact calculations of the zero temperature partition function and ground state degeneracy per site w for the q state potts antiferromagnet on a variety of homeomorphic families of planar strip graphs g ch k1 k2 k m where k1 k2 and k describe the homeomorphic structure and m denotes the length of the strip several different ways of taking the total number of vertices to infinity by sending i m with k1 k2 and k fixed ii k1 and or k2 with m and k fixed and iii k with m and p k1 k2 fixed are studied and the respective loci of points b where w is nonanalytic in the complex q plane are determined the b s for limit i are comprised of arcs which do not enclose regions in the q plane and for many values of p and k include support for re q b for limits ii and iii is the unit circle q 1 1

integrable three body problems with two and three body interactions spin generalizations of both the elliptic calogero marchioro wolfes model and the nonlinear schrodinger model are studied these models are three body problems with two and three body potentials and mathematically related with the exceptional root system of type we construct the integrable differential difference operator the so called dunkl operator based on the infinite dimensional representation for solutions of the variant of the classical yang baxter equation by use of these operators we investigate the integrability and the scattering matrices

ground state entropy of the potts antiferromagnet on cyclic strip graphs we present exact calculations of the zero temperature partition function chromatic polynomial and the exponent of the ground state entropy s0 for the q state potts antiferromagnet on families of cyclic and twisted cyclic mobius strip graphs composed of p sided polygons our results suggest a general rule concerning the maximal region in the complex q plane to which one can analytically continue from the physical interval where s0 0 the chromatic zeros and their accumulation set exhibit the rather unusual property of including support for re q 0 and provide further evidence for a relevant conjecture

stochastic resonance on weakly paced scale free networks we study the impact of additive gaussian noise and weak periodic forcing on the dynamics of a scale free network of bistable overdamped oscillators the periodic forcing is introduced to a single oscillator and therefore acts as a pacemaker trying to impose its rhythm on the whole ensemble we show that an intermediate intensity of temporally and spatially uncorrelated noise is able to optimally assist the pacemaker in achieving this goal thus providing evidence for stochastic resonance on weakly paced scale free networks because of the inherent degree inhomogeneity of individual oscillators forming the scale free network the placement of the pacemaker within the network is thereby crucial as two extremes we consider separately the introduction of the pacemaker to the oscillator with the highest degree and to one of the oscillators having the lowest degree in both cases the coupling strength plays a crucial role since it determines to what extent the whole network will follow the pacemaker on the expense of a weaker correlation between the pacemaker and the units that are directly linked with the paced oscillator higher coupling strengths facilitate the global outreach of the pacemaker but require higher noise intensities for the optimal response in contrast lower coupling strengths and comparatively low noise intensities localize the optimal response to immediate neighbors of the paced oscillator if the pacemaker is introduced to the main hub the transition between the locally and globally optimal responses is characterized by a double resonance that postulates the existence of an optimal coupling strength for the transmission of weak rhythmic activity across scale free networks we corroborate the importance of the inhomogeneous structure of scale free networks by additionally considering regular networks of oscillators with different degrees of coupling

t 0 partition functions for potts antiferromagnets on square lattice strips with twisted periodic boundary conditions we present exact calculations of the zero temperature partition function for the q state potts antiferromagnet or equivalently the chromatic polynomial for two families of arbitrarily long strip graphs of the square lattice with periodic boundary conditions in the transverse direction and i periodic and ii twisted periodic boundary conditions in the longitudinal direction so that the strip graphs are embedded on i a torus and ii a klein bottle in the limit of infinite length we calculate the exponent of the entropy w q show it to be the same for i and ii and determine its analytic structure

structure of the partition function and transfer matrices for the potts model in a magnetic field on lattice strips we determine the general structure of the partition function of the q state potts model in an external magnetic field z g q v w for arbitrary q temperature variable v and magnetic field variable w on cyclic mobius and free strip graphs g of the square sq triangular tri and honeycomb hc lattices with width ly and arbitrarily great length lx for the cyclic case we prove that the partition function has the form z lambda l y times l x q v w sum d 0 l y tilde c d mathrm tr t z lambda l y d m where denotes the lattice type tilde c d are specified polynomials of degree d in q t z lambda l y d is the corresponding transfer matrix and m lx lx 2 for sq tri hc respectively an analogous formula is given for mobius strips while only t z lambda l y d 0 appears for free strips we exhibit a method for calculating t z lambda l y d for arbitrary ly and give illustrative examples explicit results for arbitrary ly are presented for t z lambda l y d with d ly and d ly 1 we find very simple formulas for the determinant mathrm det t z lambda l y d we also give results for self dual cyclic strips of the square lattice

ground state entropy of potts antiferromagnets on cyclic polygon chain graphs we present exact calculations of chromatic polynomials for families of cyclic graphs consisting of linked polygons where the polygons may be adjacent or separated by a given number of bonds from these we calculate the exponential of the ground state entropy w for the q state potts model on these graphs in the limit of infinitely many vertices a number of properties are proved concerning the continuous locus of nonanalyticities in w our results provide further evidence for a general rule concerning the maximal region in the complex q plane to which one can analytically continue from the physical interval where s0 0

ground state entropy in potts antiferromagnets abstract the q state potts antiferromagnet exhibits nonzero ground state entropy s0 g q 0 for sufficiently large q on a given n vertex lattice or graph g and its n limit g we present exact calculations of the zero temperature partition function z g q t 0 and w g q where s 0 k b ln w for this model on a number of families g these calculations have interesting connections with graph theory since z g q t 0 p g q where the chromatic polynomial p g q gives the number of ways of coloring the vertices of the graph g such that no adjacent vertices have the same color generalizing q from z to c we determine the accumulation set b of the zeros of p g q which constitute the continuous loci of points on which w is nonanalytic the potts antiferromagnet has a zero temperature critical point at the maximal value qc where b crosses the real q axis in particular exact solutions for w and b are given for infinitely long finite width strips of various lattices in addition to their intrinsic interest these yield insight into the approach to the 2d thermodynamic limit some corresponding results are presented for the exact finite temperature potts free energy on families of graphs finally we present rigorous upper and lower bounds on w for 2d lattices

stability and bifurcation analysis of delay induced tumor immune interaction model a modified mathematical model of kuznetsov et al 1994 representing tumor immune interaction with discrete time delay is proposed in this paper by choosing the discrete time delay as the bifurcation parameter we establish the sufficient condition for local stability of interior steady state hopf bifurcation occurs when the time delay passes a critical value the estimation of the length of delay to preserve stability has been derived moreover based on the normal form theory and center manifold theorem we establish explicit expressions to determine the direction of hopf bifurcation and the stability of bifurcating periodic solutions numerical simulations are obtained to validate our analytical findings by varying the system parameter a two parameters bifurcation diagrams are also shown by varying the system parameters which bring out the rich biological dynamics of the system

hopf bifurcation for neutral type neural network model with two delays in this paper the dynamics of a neutral neural network model with two delays is investigated the condition to ensure the stability of the zero solution of the system is decided by choosing 1 and 2 as parameters respectively then the hopf bifurcation is discussed by using the center manifold theory and normal form method introduced by hassard and kazarinoff global existence of periodic solution is studied by using the global hopf bifurcation theory finally some numerical simulations are carried out to illustrate the analytical results

pinning synchronization of directed networks with switching topologies a multiple lyapunov functions approach this paper studies the global pinning synchronization problem for a class of complex networks with switching directed topologies the common assumption in the existing related literature that each possible network topology contains a directed spanning tree is removed in this paper using tools from m matrix theory and stability analysis of the switched nonlinear systems a new kind of network topology dependent multiple lyapunov functions is proposed for analyzing the synchronization behavior of the whole network it is theoretically shown that the global pinning synchronization in switched complex networks can be ensured if some nodes are appropriately pinned and the coupling is carefully selected interesting issues of how many and which nodes should be pinned for possibly realizing global synchronization are further addressed finally some numerical simulations on coupled neural networks are provided to verify the theoretical results

ground state degeneracy of potts antiferromagnets homeomorphic classes with noncompact w boundaries abstract we present exact calculations of the zero temperature partition function z g q t 0 and ground state degeneracy w g q for the q state potts antiferromagnet on a number of families of graphs g for which generalizing q from z to c the boundary b of regions of analyticity of w in the complex q plane is noncompact passing through z 1 q 0 for these types of graphs since the reduced function w red q 1 w is nonanalytic at z 0 there is no large q taylor series expansion of w red the study of these graphs thus gives insight into the conditions for the validity of the large q expansions it is shown how such families of graphs can be generated from known families by homeomorphic expansion

notes on the structure of the function interacting gas intertwining operator in the degenerate affine hecke algebra the wavefunction of the function interacting bose gas on the infinite interval is studied by introducing the intertwining operator of the degenerate affine hecke algebra the non symmetric eigenfunction of the dunkl operator is constructed

lower bounds and series for the ground state entropy of the potts antiferromagnet on archimedean lattices and their duals we prove a general rigorous lower bound for w lambda q exp s 0 lambda q k b the exponent of the ground state entropy of the q state potts antiferromagnet on an arbitrary archimedean lattice lambda we calculate large q series expansions for the exact w r lambda q q 1 w lambda q and compare these with our lower bounds on this function on the various archimedean lattices it is shown that the lower bounds coincide with a number of terms in the large q expansions and hence serve not just as bounds but also as very good approximations to the respective exact functions w r lambda q for large q on the various lattices lambda plots of w r lambda q are given and the general dependence on lattice coordination number is noted lower bounds and series are also presented for the duals of archimedean lattices as part of the study the chromatic number is determined for all archimedean lattices and their duals finally we report calculations of chromatic zeros for several lattices these provide further support for our earlier conjecture that a sufficient condition for w r lambda q to be analytic at 1 q 0 is that lambda is a regular lattice

chromatic polynomials potts models and all that the q state potts model can be defined on an arbitrary finite graph and its partition function encodes much important information about that graph including its chromatic polynomial flow polynomial and reliability polynomial the complex zeros of the potts partition function are of interest both to statistical mechanicians and to combinatorists i give a pedagogical introduction to all these problems and then sketch two recent results a construction of a countable family of planar graphs whose chromatic zeros are dense in the whole complex q plane except possibly for the disc q 1 1 b proof of a universal upper bound on the q plane zeros of the chromatic polynomial or antiferromagnetic potts model partition function in terms of the graph s maximum degree

a free energy bound for the hopfield model the author gives a simple upper and lower bound on the free energy density of the hopfield model of size n with p stored patterns in the limit where n and p tend to infinity with p n to alpha 1 the two bounds coincide for alpha 0

higgs bundles gauge theories and quantum groups the appearance of the bethe ansatz equation for the nonlinear schrodinger equation in the equivariant integration over the moduli space of higgs bundles is revisited we argue that the wave functions of the corresponding two dimensional topological u n gauge theory reproduce quantum wave functions of the nonlinear schrodinger equation in the n particle sector this implies the full equivalence between the above gauge theory and the n particle sub sector of the quantum theory of the nonlinear schrodinger equation this also implies the explicit correspondence between the gauge theory and the representation theory of the degenerate double affine hecke algebra we propose a similar construction based on the g g gauged wzw model leading to the representation theory of the double affine hecke algebra

global asymptotic stability of stochastic bam neural networks with distributed delays and reaction diffusion terms this paper is concerned with global asymptotic stability of a class of reaction diffusion stochastic bi directional associative memory bam neural networks with discrete and distributed delays based on suitable assumptions we apply the linear matrix inequality lmi method to propose some new sufficient stability conditions for reaction diffusion stochastic bam neural networks with discrete and distributed delays the obtained results are easy to check and improve upon the existing stability results an example is also given to demonstrate the effectiveness of the obtained results

stability of synchronous oscillations in a system of hodgkin huxley neurons with delayed diffusive and pulsed coupling we study the synchronization dynamics for a system of two hodgkin huxley hh neurons coupled diffusively or through pulselike interactions by calculating the maximum transverse lyapunov exponent we found that with diffusive coupling there are three regions in the parameter space corresponding to qualitatively distinct behaviors of the coupled dynamics in particular the two neurons can synchronize in two regions and desynchronize in the third when excitatory and inhibitory pulse coupling is considered we found that synchronized dynamics becomes more difficult to achieve in the sense that the parameter regions where the synchronous state is stable are smaller numerical simulations of the coupled system are presented to validate these results the stability of a network of coupled hh neurons is then analyzed and the stability regions in the parameter space are exactly obtained

complex q zeros of the partition function of the potts model with long range interactions abstract the zeros of the partition function of the ferromagnetic q state potts model with long range interactions in the complex q plane are studied in the mean field case while preliminary numerical results are reported for the finite 1 d chains with power law decaying interactions in both cases at any fixed temperature the zeros lie on the arc shaped contours which cross the positive real axis at the value for which the given temperature is transition temperature for finite number of spins the positive real axis is free of zeros which approach to it in the thermodynamic limit the convergence exponent of the zero closest to the positive real q axis is found to have the same value as the temperature critical exponent 1

extreme fluctuations in stochastic network coordination with time delays we study the effects of uniform time delays on the extreme fluctuations in stochastic synchronization and coordination problems with linear couplings in complex networks we obtain the average size of the fluctuations at the nodes from the behavior of the underlying modes of the network we then obtain the scaling behavior of the extreme fluctuations with system size as well as the distribution of the extremes on complex networks and compare them to those on regular one dimensional lattices for large complex networks when the delay is not too close to the critical one fluctuations at the nodes effectively decouple and the limit distributions converge to the fisher tippett gumbel density in contrast fluctuations in low dimensional spatial graphs are strongly correlated and the limit distribution of the extremes is the airy density finally we also explore the effects of nonlinear couplings on the stability and on the extremes of the synchronization landscapes

exact potts model partition function on strips of the triangular lattice abstract in this paper we present exact calculations of the partition function z of the q state potts model and its generalization to real q for arbitrary temperature on n vertex strip graphs of width ly 2 and arbitrary length of the triangular lattice with free cyclic and mobius longitudinal boundary conditions these partition functions are equivalent to tutte whitney polynomials for these graphs the free energy is calculated exactly for the infinite length limit of the graphs and the thermodynamics is discussed considering the full generalization to arbitrary complex q and temperature we determine the singular locus b in the corresponding c 2 space arising as the accumulation set of partition function zeros as n in particular we study the connection with the t 0 limit of the potts antiferromagnet where b reduces to the accumulation set of chromatic zeros comparisons are made with our previous exact calculation of potts model partition functions for the corresponding strips of the square lattice our present calculations yield as special cases several quantities of graph theoretic interest

yang baxter equation in long range interacting systems we consider the su p spin chains with long range interactions and the spin generalization of the calogero sutherland models we show that their properties derive from a transfer matrix obeying the yang baxter equation we obtain the expression of the conserved quantities of the dynamical models and we diagonalise them in the spin chain case we establish the connection between the degeneracies of the spectrum and the representation theory of the yangians we use a correspondence with the dynamical models to diagonalise the hamiltonian finally we extend the previous results to the case of a trigonometric r matrix

differential difference operators associated to reflection groups there is a theory of spherical harmonics for measures invariant under a finite reflection group the measures are products of powers of linear functions whose zero sets are the mirrors of the reflections in the group times the rotation invariant measure on the unit sphere in rn a commutative set of differential difference operators each homogeneous of degree 1 is the analogue of the set of first order partial derivatives in the ordinary theory of spherical harmonics in the case of r2 and dihedral groups there are analogues of the cauchy riemann equations which apply to gegenbauer and jacobi polynomial expansions the analysis of orthogonality structures for polynomials in several variables is a problem of vast dimensions this paper is part of an ongoing program to establish a workable theory for one particular class the underlying structure is based on finite coxeter groups these are finite groups acting on euclidean space generated by reflections in the zero sets of a collection of linear functions the roots the weight functions for the orthogonality are products of powers of these linear functions restricted to the surface of the unit sphere in addition the weight function is required to be invariant under the action of the group the resulting theory has strong similarities to the theory of spherical harmonics this was established in previous papers of the author 3 4 5 most notably a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by a certain second order differential difference operator ordinary partial differentiation acts as an endomorphism on ordinary harmonic functions the use of such operators and their adjoints leads to recurrence formulas and orthogonal decompositions for harmonic polynomials in this paper we construct a commutative set of first order differentialdifference operators associated to the second order operator previously mentioned received by the editors june 1 1987 1980 mathematics subject classification 1985 revision primary 33a45 33a65 20h15 secondary 20c30 42c10 51f15

effects of levy noise in a neuronal competition model abstract the effects of non gaussian levy noise in a competitive neural model of binocular rivalry are analyzed numerically we first investigate how levy noise added to the input strength changes the period of oscillations and leads to the disappearance of the winner take all behavior noise induced alternations are then studied in terms of their statistical measures we are interested in the overlapping of the regions where the model s behavior satisfies the experimental constraints on the mean dominance duration and the coefficient of variation of the dominance durations the results indicate that changes in each of the parameters of levy noise produce changes in the overlap regions the differences with gaussian noise are emphasized finally we plot the distributions of the perceptual dominance durations for different parameters of levy noise and compare these distributions with gamma log normal and weibull distributions using the kolmogorov smirnov goodness of fit test

q plane zeros of the potts partition function on diamond hierarchical graphs we report exact results concerning the zeros of the partition function of the potts model in the complex q plane as a function of a temperature like boltzmann variable v for the m th iterate graphs dm of the diamond hierarchical lattice including the limit m in this limit we denote the continuous accumulation locus of zeros in the q planes at fixed v v0 as bq v0 we apply theorems from complex dynamics to establish the properties of bq v0 for v 1 the zero temperature potts antiferromagnet or equivalently chromatic polynomial we prove that bq 1 crosses the real q axis at i a minimal point q 0 ii a maximal point q 3 iii q 32 27 iv a cubic root that we give with the value q q1 1 638 896 9 and v an infinite number of points smaller than q1 converging to 32 27 from above similar results hold for bq v0 for any 1 0 potts ferromagnet we also provide the computer generated plots of bq v0 at various values of v0 in both the antiferromagnetic and ferromagnetic regimes and compare them to the numerically computed zeros of z d4 q v0

synchronization transition in scale free networks clusters of synchrony we study the synchronization transition in scale free networks that display power law asymptotic behaviors in their degree distributions the critical coupling strength and the order parameter critical exponent derived by the mean field approach depend on the degree exponent ensuremath lambda which implies a close connection between structural organization and the emergence of dynamical order in complex systems we also derive the finite size scaling behavior of the order parameter finding that the giant cluster of synchronized nodes is formed in different ways between scale free networks with 2l ensuremath lambda l3 and those with ensuremath lambda g3

chromatic polynomials of large triangular lattices evaluating the q colourings of a lattice is equivalent to solving the q state zero temperature antiferromagnetic potts model this has recently been done exactly for an infinite triangular lattice with q real here the results are extended to the full complex q plane giving the limiting distribution of the zeros of the chromatic polynomial the results are compared with finite lattice calculations and the occurrence of isolated real zeros converging on the beraha numbers is noted

exact chromatic polynomials for toroidal chains of complete graphs abstract we present exact calculations of the partition function of the zero temperature potts antiferromagnet equivalently the chromatic polynomial for graphs of arbitrarily great length composed of repeated complete subgraphs kb with b 5 6 which have periodic or twisted periodic boundary condition in the longitudinal direction in the lx limit the continuous accumulation set of the chromatic zeros b is determined we give some results for arbitrary b including the extrema of the eigenvalues with coefficients of degree b 1 and the explicit forms of some classes of eigenvalues we prove that the maximal point where b crosses the real axis qc satisfies the inequality qc b for 2 b the minimum value of q at which b crosses the real q axis is q 0 and we make a conjecture concerning the structure of the chromatic polynomial for klein bottle strips

scale free networks from varying vertex intrinsic fitness a new mechanism leading to scale free networks is proposed in this letter it is shown that in many cases of interest the connectivity power law behavior is neither related to dynamical properties nor to preferential attachment assigning a quenched fitness value x i to every vertex and drawing links among vertices with a probability depending on the fitnesses of the two involved sites gives rise to what we call a good get richer mechanism in which sites with larger fitness are more likely to become hubs i e to be highly connected

switching from exact scheme to nonstandard finite difference scheme for linear delay differential equation one dimensional models are important for developing demonstrating and testing new methods and approaches which can be extended to more complex systems we design for a linear delay differential equation a reliable numerical method which consists of two time splits as follows a it is an exact scheme at the early time evolution t where is the discrete value of the delay b thereafter it is a nonstandard finite difference nsfd scheme obtained by suitable discretizations at the backtrack points it is shown theoretically and computationally that the nsfd scheme is dynamically consistent with respect to the asymptotic stability of the trivial equilibrium solution of the continuous model extension of the nsfd to nonlinear epidemiological models and its good performance are tested on a numerical example

the structure of chromatic polynomials of planar triangulations and implications for chromatic zeros and asymptotic limiting quantities we present an analysis of the structure and properties of chromatic polynomials of one parameter and multi parameter families of planar triangulation graphs where is a vector of integer parameters we use these to study the ratio of to the tutte upper bound 1 n 5 where and n is the number of vertices in in particular we calculate limiting values of this ratio as n for various families of planar triangulations we also use our calculations to analyze zeros of these chromatic polynomials we study a large class of families with p 1 and p 2 and show that these have a structure of the form for p 1 where 1 q 2 2 q 3 and 3 1 and for p 2 we derive properties of the coefficients and show that has a real chromatic zero that approaches as one or more of the mi the generalization to p 3 is given further we present a one parameter family of planar triangulations with real zeros that approach 3 from below as m implications for the ground state entropy of the potts antiferromagnet are discussed

paths to globally generalized synchronization in scale free networks we apply the auxiliary system approach to study paths to globally generalized synchronization in scale free networks of identical chaotic oscillators including h enon maps logistic maps and lorentz oscillators as the coupling strength ensuremath epsilon between nodes of the network is increased transitions from partially to globally generalized synchronization and intermittent behaviors near the synchronization thresholds are found the generalized synchronization starts from the hubs of the network and then spreads throughout the whole network with the increase of ensuremath epsilon our result is useful for understanding the synchronization process in complex networks

ground state degeneracy of potts antiferromagnets cases with noncompact w boundaries having multiple points at we present exact calculations of the zero temperature partition function and ground state degeneracy per site for the q state potts antiferromagnet on a number of families of graphs for which the boundary of regions of analyticity of w in the complex q plane is noncompact and has the properties that i in the plane the point z 0 is a multiple point on and ii includes support for these families are generated by the method of homeomorphic expansion our results give further insight into the conditions for the validity of large q series expansions for the reduced function

ground state entropy of potts antiferromagnets and the approach to the two dimensional thermodynamic limit we study the ground state degeneracy per site exponent of the ground state entropy w lambda l x infty times l y q for the q state potts antiferromagnet on infinitely long strips with width l y of 2d lattices lambda with free and periodic boundary conditions in the y direction denoted fbc y and pbc y we show that the approach of w to its 2d thermodynamic limit as l y increases is quite rapid for moderate values of q and l y simeq 4 w lambda l x infty times l y q is within about 5 and cal o 10 3 of the 2d value w lambda l x infty times l y infty q for fbc y and pbc y respectively the approach of w to the 2d thermodynamic limit is proved to be monotonic non monotonic for fbc y pbc y it is noted that ground state entropy determinations on infinite strips can be used to obtain the central charge for cases with critical ground states

fixed time synchronization of markovian jump fuzzy cellular neural networks with stochastic disturbance and time varying delays abstract this paper mainly studies the fixed time synchronization of markovian jump fuzzy cellular neural networks with stochastic perturbations and time varying delays in the leakage term by designing delay dependent controllers with or without fuzzy terms constructing a suitable stochastic lyapunov functional and using matrix analysis techniques this paper derives some novel and useful sufficient conditions to guarantee the fixed time synchronization of the addressed drive response systems and the conditions are delay dependent which has less conservative results the finite time is also independent of the initial states finally numerical examples are given to illustrate the effectiveness of the proposed main results

boundary k matrix elliptic dunkl operator and quantum many body systems the infinite dimensional representation of the boundary k operator the solution of the reflection equation is considered the trigonometric k matrix is studied as a degenerate case of the elliptic operator a method to construct the elliptic dunkl operator is proposed and the relationship with the quantum many body problem is also discussed

synchronization processes in complex networks we present an extended analysis based on the dynamics towards synchronization of a system of coupled oscillators of the hierarchy of communities in complex networks in the synchronization process different structures corresponding to well defined communities of nodes appear in a hierarchical way the analysis also provides a useful connection between synchronization dynamics complex networks topology and spectral graph analysis

synchronization of oscillators through time shifted common inputs shared upstream dynamical processes are frequently the source of common inputs in various physical and biological systems however due to finite signal transmission speeds and differences in the distance to the source time shifts between otherwise common inputs are unavoidable since common inputs can be a source of correlation between the elements of multi unit dynamical systems regardless of whether these elements are directly connected with one another or not it is of importance to understand their impact on synchronization as a canonical model that is representative for a variety of different dynamical systems we study limit cycle oscillators that are driven by stochastic time shifted common inputs we show that if the oscillators are coupled time shifts in stochastic common inputs do not simply shift the distribution of the phase differences but rather the distribution actually changes as a result the best synchronization is therefore achieved at a precise intermediate value of the time shift which is due to a resonance like effect with the most probable phase difference that is determined by the deterministic dynamics

synchronization transition in degenerate optical parametric oscillators induced by nonlinear coupling in this paper the synchronization of certain degenerate optical parametric oscillators is investigated in detail complete and or partial synchronization can be reached when linear controller constructed by the real part or imaginary part of the subharmonic mode is imposed on the chaotic degenerate optical parametric oscillators with appropriate coupling intensity the lyapunov exponents under different coupling coefficients are calculated to find the critical condition for complete synchronization transition from complete synchronization to partial synchronization is observed when nonlinear coupling is introduced into the two chaotic oscillators it is found that synchronization of chaotic oscillators keeps robust when the intensity of the nonlinear coupling is less than the intensity of the linear coupling the complete synchronization state is destructed and transient period for partial synchronization is in certain delay when the intensity of the nonlinear coupling is beyond the intensity of the linear coupling

families of graphs with chromatic zeros lying on circles we define an infinite set of families of graphs which we call p wheels and denote wh p n that generalize the wheel p 1 and biwheel p 2 graphs the chromatic polynomial for wh p n is calculated and remarkably simple properties of the chromatic zeros are found i the real zeros occur at q 0 1 p 1 for n p even and q 0 1 p 2 for n p odd and ii the complex zeros all lie equally spaced on the unit circle q p 1 1 in the complex q plane in the n to infty limit the zeros on this circle merge to form a boundary curve separating two regions where the limiting function w wh p q is analytic viz the exterior and interior of the above circle connections with statistical mechanics are noted