dimer statistics of honeycomb lattices on klein bottle mobius strip and cylinder abstract dimer statistics is a central problem in statistical physics in this paper the enumerations of close packed dimers of honeycomb lattices on klein bottle mobius strip and cylinder are considered by establishing a pfaffian orientation or a crossing orientation and then computing the determinants of the skew symmetric matrices of the resulting orientation graphs we obtain explicit expressions of the number of close packed dimers of the klein bottle polyhex the mobius polyhex and the cylindrical polyhex

hamiltonian mappings and circle packing phase spaces we introduce three area preserving maps with phase space structures which resemble circle packings each mapping is derived from a kicked hamiltonian system with one of the three different phase space geometries planar hyperbolic or spherical and exhibits an infinite number of coexisting stable periodic orbits which appear to pack the phase space with circular resonances

nonlocal ginzburg landau equation for cortical pattern formation we show how a nonlocal version of the real ginzburg landau gl equation arises in a large scale recurrent network model of primary visual cortex we treat cortex as a continuous two dimensional sheet of cells that signal both the position and orientation of a local visual stimulus the recurrent circuitry is decomposed into a local part which contributes primarily to the orientation tuning properties of the cells and a long range part that introduces spatial correlations we assume that a the local network exists in a balanced state such that it operates close to a point of instability and b the long range connections are weak and scale with the bifurcation parameter of the dynamical instability generated by the local circuitry carrying out a perturbation expansion with respect to the long range coupling strength then generates a nonlocal coupling term in the gl amplitude equation we use the nonlocal gl equation to analyze how axonal propagation delays arising from the slow conduction velocities of the long range connections affect spontaneous pattern formation

extensive chaos in the nikolaevskii model we carry out a systematic study of a different type of chaos at onset soft mode turbulence based on numerical integration of the simplest one dimensional model the chaos is characterized by a smooth interplay of different spatial scales with defect generation being unimportant the lyapunov exponents are calculated for several system sizes for fixed values of the control parameter epsilon the lyapunov dimension and the kolmogorov sinai entropy are calculated and both shown to exhibit extensive and microextensive scaling the distribution functional is shown to satisfy gaussian statistics at small wave numbers and small frequency

an integrable deformation of an ellipse of small eccentricity is an ellipse the classical birkho conjecture claims that the boundary of a strictly convex integrable billiard table is necessarily an ellipse or a circle as a special case in this article we show that a version of this conjecture is true for tables bounded by small perturbations of ellipses of small eccentricity

break up of resonant invariant curves in billiards and dual billiards associated to perturbed circular tables two area preserving twist maps are associated to a smooth closed convex table the classical billiard map and the dual billiard map when the table is circular these maps are integrable and their phase spaces are foliated by invariant curves the invariant curves with rational rotation numbers are resonant and do not persist under generic perturbations of the circle we present a sufficient condition for the break up of these curves this condition is expressed directly in terms of the fourier coefficients of the perturbation it follows from a standard melnikov argument

asymptotics in a family of linked strip maps abstract we apply round off to planar rotations obtaining a one parameter family of invertible maps of a two dimensional lattice as the angle of rotation approaches 2 the fourth iterate of the map produces piecewise rectilinear motion which develops along the sides of convex polygons we characterise the dynamics which resembles outer billiards of polygons as the concatenation of so called strip maps each providing an elementary perturbation of an underlying integrable system significantly there are orbits which are subject to an arbitrarily large number of these perturbations during a single revolution resulting in the appearance of a novel discrete space version of near integrable hamiltonian dynamics we study the asymptotic regime of the limiting integrable system analytically and numerically some features of its very rich near integrable dynamics we unveil a dichotomy there is one regime in which the nonlinearity tends to zero and a second where it does not in the latter case numerical experiments suggest that the distribution of the periods of orbits is consistent with that of random dynamics in the former case the fluctuations result in an intricate structure of resonances

asymptotic dynamics of the dual billiard transformation given a strictly convex plane curve the dual billiard transformation is the transformation of its exterior defined as follows given a point x outside the curve draw a support line to it from the point and reflect x at the support point we show that the dual billiard transformation far from the curve is well approximated by the time 1 transformation of a hamiltonian flow associated with the curve

wavelet description of the nikolaevskii model we present the results of a wavelet based approach to the study of the chaotic dynamics of a one dimensional model that shows a direct transition to spatiotemporal chaos we find that the dynamics of this model in the spatiotemporally chaotic regime may be understood in terms of localized dynamics in both space and scale wave number a projection onto a daubechies basis yields a good separation of scales as shown by an examination of the contribution of different wavelet levels to the power spectrum at most scales including the most energetic ones we find essentially gaussian dynamics we also show that removal of certain wavelet modes can be carried out without altering the dynamics of the system as described by the lyapunov spectrum

the nonlinear heat equation on w random graphs for systems of coupled differential equations on a sequence of w random graphs we derive the continuum limit in the form of an evolution integral equation we prove that solutions of the initial value problems ivps for the discrete model converge to the solution of the ivp for its continuum limit these results combined with the analysis of nonlocally coupled deterministic networks in medvedev the nonlinear heat equation on dense graphs and graph limits arxiv e prints 2013 justify the continuum thermodynamic limit for a large class of coupled dynamical systems on convergent families of graphs

renormalization of polygon exchange maps arising from corner percolation we describe a family of polygon exchange transformations parameterized by points in the square 0 frac 1 2 times 0 frac 1 2 whenever and are irrational has periodic orbits of arbitrarily large period we show that for almost all parameters the polygon exchange map has the property that almost every point is periodic however there is a dense set of irrational parameters for which this fails by choosing parameters carefully the measure of non periodic points can be made arbitrarily close to full measure these results are powered by a notion of renormalization which holds in a more general setting namely we consider a renormalization of tilings arising from the corner percolation model

the boundary of a square tiling of a graph coincides with the poisson boundary answering a question of benjamini and schramm ann probab 24 3 1219 1238 1996 we show that the poisson boundary of any planar uniquely absorbing e g one ended and transient graph with bounded degrees can be realised geometrically as a circle which arises from a discrete version of riemann s mapping theorem this implies a conjecture of northshield potential anal 2 4 299 314 1993 some of our technique apply to the non planar case and might have further applications when the graph is also hyperbolic then under mild conditions we prove the equivalence of several boundary constructions

angular billiard and algebraic birkhoff conjecture abstract in this paper we introduce a new dynamical system which we call angular billiard it acts on the exterior points of a convex curve in euclidean plane in a neighborhood of the boundary curve this system turns out to be dual to the birkhoff billiard using this system we get new results on algebraic birkhoff conjecture on integrable billiards

complex ginzburg landau equation with nonlocal coupling a ginzburg landau type equation with nonlocal coupling is derived systematically as a reduced form of a universal class of reaction diffusion systems near the hopf bifurcation point and in the presence of another small parameter the reaction diffusion systems to be reduced are such that the chemical components constituting local oscillators are nondiffusive or hardly diffusive so that the oscillators are almost uncoupled while there is an extra diffusive component which introduces effective nonlocal coupling over the oscillators linear stability analysis of the reduced equation about the uniform oscillation is also carried out this revealed that new types of instability which can never arise in the ordinary complex ginzburg landau equation are possible and their physical implication is briefly discussed

3 connected planar spaces uniquely embed in the sphere we characterize those locally connected subsets of the sphere that have a unique embedding in the sphere i e those for which every homeomorphism of the subset to itself extends to a homeomorphism of the sphere this implies that if g is the closure of an embedding of a 3 connected graph in the sphere such that every 1 way infinite path in g has a unique accumulation point in g then g has a unique embedding in the sphere in particular the standard or freudenthal compactification of a 3 connected planar graph embeds uniquely in the sphere

finite time blow up and global existence for the nonlocal complex ginzburg landau equation abstract in this paper we consider the cauchy problem for a family of nonlocal complex ginzburg landau equation e i u t u 1 x n u p u p 2 u u on r n where 2 2 and r first the local well posedness in lebesgue spaces is established by a fixed point argument then we set up the h 2 regularity and derive the energy identities by constructing some invariant sets we further find some sufficient conditions on finite time blow up and global existence for solutions which in particular determines a sharp threshold of initial date when 0 in addition we estimate the lifespan of solutions as a function of and obtain the lower bound of the maximal existence time for r

phase equation with nonlinear excitation for nonlocally coupled oscillators abstract some reaction diffusion systems feature nonlocal interaction and near the point of hopf bifurcation can be represented as a system of nonlocally coupled oscillators phase of oscillations satisfies an evolution pde which takes different forms depending on the values of parameters in the simplest case the equation is effectively a diffusion equation which is excitation free however more complex forms are possible such as the nikolaevskii equation and the kuramoto sivashinsky equation incorporating linear excitation we analyse a situation when the phase equation is based on nonlinear excitation we derive conditions on the values of the parameters leading to the situation and show that the values satisfying the conditions exist

on polynomially integrable planar outer billiards and curves with symmetry property we show that every polynomially integrable planar outer convex billiard is elliptic we also prove an extension of this statement to non convex billiards

application of the kudryashov method for finding exact solutions of the high order nonlinear evolution equations abstract the application of the kudryashov method for finding exact solutions of the high order nonlinear evolution equations is considered some classes of solitary wave solutions for the families of nonlinear evolution equations of fifth sixth and seventh order are obtained the efficiency of the kudryashov method for finding exact solutions of the high order nonlinear evolution equations is demonstrated

dimer statistics on the klein bottle abstract the problem of enumerating close packed dimers or perfect matchings on a quadratic lattice embedded on the klein bottle is considered thomassen c thomassen tilings of the torus and the klein bottle and vertex transitive graphs on a fixed surface trans amer math soc 323 2 1991 605 635 characterized that there are six quadrilateral lattices embedded on the klein bottle lu and wu w t lu and f y wu close packed dimers on nonorientable surfaces phys lett a 293 2002 235 246 had obtained a expression for the number of close packed dimers on one of them in this paper we investigate four other embeddings and obtain explicit expressions of the numbers of close packed dimers and free energy per dimer by enumerating pfaffians

anomalous scaling on a spatiotemporally chaotic attractor the nikolaevskiy model for pattern formation with continuous symmetry exhibits spatiotemporal chaos with strong scale separation extensive numerical investigations of the chaotic attractor reveal unexpected scaling behavior of the long wave modes surprisingly the computed amplitude and correlation time scalings are found to differ from the values obtained by asymptotically consistent multiple scale analysis however when higher order corrections are added to the leading order theory of matthews and cox the anomalous scaling is recovered

detecting free splittings in relatively hyperbolic groups we describe an algorithm which determines whether or not a group which is hyperbolic relative to abelian groups admits a nontrivial splitting over a finite group

polygonal invariant curves for a planar piecewise isometry we investigate a remarkable new planar piecewise isometry whose generating map is a permutation of four cones for this system we prove the coexistence of an infinite number of periodic components and an uncountable number of transitive components the union of all periodic components is an invariant pentagon with unequal sides transitive components are invariant curves on which the dynamics are conjugate to a transitive interval exchange the restriction of the map to the invariant pentagonal region is the first known piecewise isometric system for which there exist an infinite number of periodic components but the only aperiodic points are on the boundary of the region the proofs are based on exact calculations in a rational cyclotomic field we use the system to shed some light on a conjecture that pwis can possess transitive invariant curves that are not smooth

a 4 dimensional kleinian group we give an example of a 4 dimensional kleinian group which is finitely generated but not finitely presented and is a subgroup of a cocompact kleinian group

dynamics of reaction diffusion patterns controlled by asymmetric nonlocal coupling as a limiting case of differential advection a one component bistable reaction diffusion system with asymmetric nonlocal coupling is derived as a limiting case of a two component activator inhibitor reaction diffusion model with differential advection the effects of asymmetric nonlocal couplings in such a bistable reaction diffusion system are then compared to the previously studied case of a system with symmetric nonlocal coupling we carry out a linear stability analysis of the spatially homogeneous steady states of the model and numerical simulations of the model to show how the asymmetric nonlocal coupling controls and alters the steady states and the front dynamics in the system in a second step a third fast reaction diffusion equation is included which induces the formation of more complex patterns a linear stability analysis predicts traveling waves for asymmetric nonlocal coupling in contrast to a stationary turing patterns for a system with symmetric nonlocal coupling these findings are verified by direct numerical integration of the full equations with nonlocal coupling

dynamics of kaleidoscopic maps abstract first we introduce a certain class of piecewise affine elliptic rotation maps on t 2 called the kaleidoscopic maps and describe its importance and then we concentrate our efforts on a special case when the rotation angle of a kaleidoscopic map is n n z for the special case we answer the conjectures regarding the periodicity and the singularity structure of such kaleidoscopic dynamics in the process we prove the partial riddling of the regular orbits that gives rise to the classification of periodic sets and estimate the hausdorff dimension of the singular set finally we study the dynamics of such kaleidoscopic maps restricted within the singular set and answer conjectures concerning the chaos the local chaos and the ergodicity with respect to the normalized hausdorff measure of the singular set

caustics and evolutes for convex planar domains les caustiques pour l application de la boule de billard sur une ellipse satisfont une certaine propriete d evolution on definit un operateur reliant la courbure d une caustique a la courbure de la frontiere pour cette application dans des domaines planaires convexes lisses et on l utilise pour deduire une equation qui caracterise ces courbes satisfaisant la propriete d evolution comme ellipses

billiard map and rigid rotation abstract can a billiard map be locally conjugated to a rigid rotation we prove that the answer to this question is positive in the category of formal series we also present numerical evidence that for good rotation angles the answer is also positive in an analytic category

noise induced turbulence in nonlocally coupled oscillators we demonstrate that nonlocally coupled limit cycle oscillators subject to spatiotemporally white gaussian noise can exhibit a noise induced transition to turbulent states after illustrating noise induced turbulent states with numerical simulations using two representative models of limit cycle oscillators we develop a theory that clarifies the effective dynamical instabilities leading to the turbulent behavior using a hierarchy of dynamical reduction methods we determine the parameter region where the system can exhibit noise induced turbulent states which is successfully confirmed by extensive numerical simulations at each level of the reduction

groups quasi isometric to right angled artin groups we characterize groups quasi isometric to a right angled artin group raag g with finite outer automorphism group in particular all such groups admit a geometric action on a cat 0 cube complex that has an equivariant fibering over the davis building of g this characterization will be used in forthcoming work of the first author to give a commensurability classification of the groups quasi isometric to certain raags

tilings of the torus and the klein bottle and vertex transitive graphs on a fixed surface we describe all regular tiings of the torus and the klein bottle we apply this to describe for each orientable respectively nonorientable surface s all but finitely many vertex transitive graphs which can be drawn on s but not on any surface of smaller genus respectively crosscap number in particular we prove the conjecture of babai that for each g 3 there are only finitely many vertex transitive graphs of genus g in fact they all have order 2 there are only finitely many groups that act on the surface of genus g we also derive a nonorientable version of hurwitz theorem

chemical turbulence equivalent to nikolavskii turbulence we find evidence that a certain class of reaction diffusion rd systems can exhibit chemical turbulence equivalent to nikolaevskii turbulence we study an extended complex ginzburg landau cgl equation derived from this class of rd systems first we show numerically that the power spectrum of this cgl equation in the neighborhood of a codimension two turing char21 benjamin feir point is qualitatively quite similar to that of the nikolaevskii equation then we demonstrate that the nikolaevskii equation can in fact be obtained from this cgl equation through a phase reduction procedure

three dimensional manifolds kleinian groups and hyperbolic geometry 1 a conjectural picture of 3 manifolds a major thrust of mathematics in the late 19th century in which poincare had a large role was the uniformization theory for riemann surfaces that every conformai structure on a closed oriented surface is represented by a riemannian metric of constant curvature for the typical case of negative euler characteristic genus greater than 1 such a metric gives a hyperbolic structure any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions the exceptional cases the sphere and the torus have spherical and euclidean structures three manifolds are greatly more complicated than surfaces and i think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 or more except perhaps for the fact that so many 3 manifolds are beautiful the situation has changed so that i feel fairly confident in proposing the

critical exponents of nikolaevskii turbulence we study the spatial power spectra of nikolaevskii turbulence in one dimensional space first we show that the energy distribution in wave number space is extensive in nature then we demonstrate that when varying a particular parameter the spectrum becomes qualitatively indistinguishable from that of kuramoto sivashinsky turbulence next we derive the critical exponents of turbulent fluctuations finally we argue that in some previous studies parameter values for which this type of turbulence does not appear were mistakenly considered and we resolve inconsistencies obtained in previous studies

diffusion speed in piecewise smooth billiards a classical consequence of kam theory is that if is a smooth strictly convex curve then neither inner nor outer billiard orbits can accumulate on on the other hand several authors observed that this result is false in general if is not smooth or if it is not strictly convex in this paper we consider piecewise smooth strictly convex curves with curvature jump discontinuities and estimate the diffusion speed for those curves namely we show that both inner and outer billiard maps for have orbits which approach with speed 1 m 2

analogues of cayley graphs for topological groups we define for a compactly generated totally disconnected locally compact group a graph called a rough cayley graph that is a quasi isometry invariant of the group this graph carries information about the group structure in an analogous way to the ordinary cayley graph for a finitely generated group with this construction the machinery of geometric group theory can be applied to topological groups this is illustrated by a study of groups where the rough cayley graph has more than one end and a study of groups where the rough cayley graph has polynomial growth

on the definition of word hyperbolic groups formal languages based on multiplication tables of finitely generated groups are investigated and used to give a linguistic characterization of word hyperbolic groups

global synchronization in two dimensional lattices of discrete belousov zhabotinsky oscillators abstract macroscopic synchronization was investigated experimentally in the belousov zhabotinsky reaction with a two dimensional lattice arrangement of coupled chemical oscillators the natural frequency of each oscillator was distributed randomly and the spacing distance d between beads was varied where the coupling strength between two oscillators was controlled by changing d at small d beyond a certain critical value the oscillators synchronized spontaneously with each other global synchronization in the lattice was described as a function of d the critical coupling strength for global synchronization depends on the total number of oscillators when noisy light was applied uniformly to whole oscillators the order of synchronization was maximized at optimal noise intensity noise synchronization was observed

the complexity of grigorchuk groups with application to cryptography abstract the turing complexity of the word problems of a class of groups introduced by grigorchuk 1985 is examined in particular it is shown that such problems of permutation groups of the infinite complete binary tree yield natural complete sets that separate time and space complexity classes if they are distinct a refinement of savitch s translation theorem as well as a similar result restricted for time complexity follow new families of nonfinitely presented groups are shown to have word problems uniformly solvable in simultaneous logspace and quadratic time a new family of public key cryptosystems based on these word problems is constructed

introducing symplectic billiards abstract in this article we introduce a simple dynamical system called symplectic billiards as opposed to usual birkhoff billiards where length is the generating function for symplectic billiards symplectic area is the generating function we explore basic properties and exhibit several similarities but also differences of symplectic billiards to birkhoff billiards