maximal automorphism groups of symmetric riemann surfaces with small genus the genus of a finite group g is the smallest genus of its cayley graphs if g has genus g 1 then by theorems of tom tucker g 168 g 1 and this inequality is strict unless g can be generated by elements a b c satisfying a2 b2 c2 ab 2 bc 3 ac 7 1 with ab and bc generating a proper subgroup of g conversely any group g of the latter sort has genus g g 168 1 and moreover is faithfully representable as a group of homeomorphisms of a compact riemann surface s of the same genus g with half the elements of g reversing the surface s orientation this paper describes all such groups g which have order less than 2 million and gives for each corresponding value of g in the range 1 g 11 905 the number of distinct possibilities for the riemann surface s the classification follows that of all hurwitz groups of order less than 1 million obtained by the same author

regular maps with almost sylow cyclic automorphism groups and classification of regular maps with euler characteristic p 2 abstract a regular map m is a cellular decomposition of a surface such that its automorphism group aut m acts transitively on the flags of m it can be shown that if a sylow subgroup p aut m has order coprime to the euler characteristic of the supporting surface then p is cyclic or dihedral this observation motivates the topic of the current paper where we study regular maps whose automorphism groups have the property that all their sylow subgroups contain a cyclic subgroup of index at most 2 the main result of the paper is a complete classification of such maps as an application we show that no regular maps of euler characteristic p 2 exist for p a prime greater than 7

pseudo real riemann surfaces and chiral regular maps a riemann surface is called pseudo real if it admits anticonformal automorphisms but no anticonformal involution in this paper we study general properties of the automorphism groups of such surfaces and their uniformizing nec groups in particular we prove that there exist pseudo real riemann surfaces of every possible genus g 2 we also study pseudo real surfaces of genus 2 and 3 further we establish a connection between pseudo real surfaces with maximal automorphism group and chiral 3 valent regular maps and use this to show there exist such surfaces for infinitely many genera by exhibiting infinite families of chiral regular maps of type 3 k for all k 7

regular tessellations of surfaces and p q 2 triangle groups the purpose of this paper is a study of the existence and classification of certain regular tessellations of surfaces and their relationship with the schwarz triangle groups of type p q 2 these tessellations are the natural generalizations to arbitrary surfaces of the spherical tessellations corresponding to the five platonic solids a regular tessellation of type p q a p q pattern for short on a surface m is a tessellation of m by p sided faces such that the valence of each vertex is q a more precise definition appears in 1 no global symmetry is assumed a p q pattern on m is said to be geometric if m admits a riemannian metric of constant curvature with respect to which the edges of the pattern are geodesic arcs of equal length and the interior angles at all vertices in all faces are 2 7 q if the curvature happens to he zero then we further insist that the area of each face be 1 it is natural to call two p q patterns on m

braces radical rings and the quantum yang baxter equation abstract non degenerate cycle sets are equivalent to non degenerate unitary set theoretical solutions of the quantum yang baxter equation we embed such cycle sets into generalized radical rings braces and study their interaction in this context we establish a galois theory between ideals of braces and quotient cycle sets our main result determines the relationship between two square free cycle sets operating transitively on each other

classification of regular maps of negative prime euler characteristic we give a classification of all regular maps on nonorientable surfaces with a negative odd prime euler characteristic equivalently on nonorientable surfaces of genus where is an odd prime a consequence of our classification is that there are no regular maps on nonorientable surfaces of genus where is a prime such that mod and

stability of invariant circles in a class of dissipative maps on considere le systeme perturbe x x by c f e x y mod 1 y by c g e x y b 1 ou f e et g e sont periodiques par rapport a x on etudie des bornes explicites reliant le parametre de dissipation b et la nature des fonctions perturbantes a la stabilite du cercle invariant

fuchsian groups and algebraic number fields given the signature of a finitely generated fuchsian group we find the minimal extension of the rationals for which there is a fuchsian group having the required signature whose matrix entries lie in this field

pseudodifferential operators on manifolds with foliated boundaries abstract let x be a smooth compact manifold with boundary for smooth foliations on the boundary of x admitting a resolution in terms of a fibration we construct a pseudodifferential calculus generalizing the fibred cusp calculus of mazzeo and melrose in particular we introduce certain symbols leading to a simple description of the fredholm operators inside the calculus when the leaves of the fibration resolving the foliation are compact we also obtain an index formula for fredholm perturbations of dirac type operators along the way we obtain a formula for the adiabatic limit of the eta invariant for invertible perturbations of dirac type operators a result of independent interest generalizing the well known formula of bismut and cheeger

on the set theoretical yang baxter equation we propose a general way of constructing set theoretical solutions of the yangbaxter equation we study the properties of the construction we also show that our construction includes the earlier ones given by weinstein xu and etingof schedlersoloviev

spatio temporal dimensionality in the overall complex dynamics of an experimental cable mass system an experimental model of an elastic cable mass hanging at in phase or out of phase vertically moving supports is considered system parameters are adjusted to produce two different conditions of multiple internal resonance nonregular dynamics are analyzed in various frequency ranges including meaningful external resonance conditions attention is devoted to characterization of system dimensionality in terms of both time and spatial complexity the aims of this paper are i to give a general overview of the richness and robustness of different quasiperiodic and homoclinic bifurcation scenarios to chaos in various regions of the control parameter space ii to characterize steady nonregular response through delay embedding technique for attractor reconstruction and proper orthogonal decomposition of spatio temporal flow and iii to identify spatial configuration variables experimental eigenfunctions contributing mostly to nonregular dynamics thus obtaining hints about possible reduced models for reproducing complex regimes system dimensionality will be evaluated both by relating the dimension of attractors to the dimension of the linear phase space and from the dominating proper orthogonal modes

affine structures of decomposable solvable groups abstract bi crossed products of groups are closely related to braces a ring like structure connected with solutions to the yang baxter equation the asymmetric product of braces which has become useful for the construction of regular affine groups and simple braces as well as several other product constructions are unified extended and simplified all bi crossed products of bicyclic braces are classified

classification of regular maps of prime characteristic revisited avoiding the gorenstein walter theorem abstract breda nedela and sira 2005 classified the regular maps on surfaces of euler characteristic p for every prime p this classification relies on three key theorems each proved using the highly non trivial characterisation of finite groups with dihedral sylow 2 subgroups due to d gorenstein and j h walter 1965 here we give new proofs of those three facts and hence the entire classification using somewhat more elementary group theory using without referring to the gorenstein walter theorem

on the numerical approximation of the rotation number abstract the starting point of this paper is a polygonal approximation of an invariant curve of a map using this polygonal approximation an approximation for the circle map the restriction of the map to the invariant curve is obtained the rotation number of the circle map is then approximated by the rotation number of the approximated circle map the error in the obtained approximate rotation number is discussed and related to the error in the polygonal approximation of the invariant curve simple algorithms for the approximation of the rotation number are described a numerical example illustrates the theory

behavior of a three torus in truncated navier stokes equations abstract the presence of a three torus in a seven mode truncation of the three dimensional navier stokes equations is investigated numerically by means of cross section and power spectra furthermore by taking advantage of particular features of the model rotation vectors circle maps and torus maps can be computed with high accuracy and used to study the dynamics in particular some interesting phenomena of partial phase locking are described in deep detail the three torus which arises via a hopf bifurcation and persists in a wide parameter range is found to break and originate a strange attractor the onset of chaos and the associated bifurcation point can be defined quite precisely

global bifurcations in rayleigh be nard convection experiments empirical maps and numerical bifurcation analysis abstract we use nonlinear signal processing techniques based on artificial neural networks to construct an empirical mapping from experimental rayleigh benard convection data in the quasiperiodic regime the data in the form of a one parameter sequence of poincare sections in the interior of a mode locked region resonance horn are indicative of a complicated interplay of local and global bifurcations with respect to the experimentally varied rayleigh number the dynamic phenomena apparent in the data include period doublings complex intermittent behavior secondary hopf bifurcations and chaotic dynamics we use the fitted map to reconstruct the experimental dynamics and to explore the associated local and global bifurcation structures in phase space using numerical bifurcation techniques we locate the stable and unstable periodic solutions calculate eigenvalues approximate invariant manifolds of saddle type solutions and identify bifurcation points this approach constitutes a promising data post processing procedure for investigating phase space and parameter space of real experimental systems it allows us to infer phase space structures which the experiments can only probe with limited measurement precision and only at a discrete number of operating parameter settings

finite groups with dihedral sylow 2 subgroups it is mainly based on characterizations of p y q and a by brauer et al 4 and by suzuki 141 on the feit thompson odd order theorem 5 used throughout this paper without reference and on glauberman s zjtheorem 6 not available when gorenstein and walter proved their fundamental classification theorem our maximal subgroup approach is adopted from 11 it was glauberman s work 71 privately communicated which revived the author s interest in the subject via 3 it led to extensive simplifications of his original proof in the remainder of this section we list some elementary general facts in addition to standard notation we write f x for 0 y and 1 s for xex xs x j a group e is called quasisimple if 1 e e and e z e is simple

arc transitive abelian regular covers of cubic graphs abstract quite a lot of attention has been paid recently to the construction of edge or arc transitive covers of symmetric graphs in most cases the approach has involved voltage graph techniques which are excellent for finding regular covers in which the group of covering transformations is either cyclic or elementary abelian or more generally homocyclic but are not so easy to use when the covering group has other forms even when it is abelian but not homocyclic in this paper a different approach is introduced that can be used more widely this new approach takes a universal group for the action of the automorphism group of the base graph and uses reidemeister schreier theory to obtain a presentation for a universal covering group and some representation theory and other methods for determining suitable quotients this approach is then used to find all arc transitive abelian regular covers of k 4 k 3 3 the cube q 3 and the petersen graph a sequel will do the same for the heawood graph

universality and self similarity in the bifurcations of circle maps abstract the bifurcation structure in a two parameter family of circle maps is considered these maps have a topological degree that may be different from one a generalization of the rotation number is given and symmetries of the bifurcations in parameter space are described continuity arguments are used to establish the existence of periodic orbits by plotting the locus of parameter values associated with superstable cycles self similar bifurcations are found these bifurcations are a generalization of the familiar period doubling cascade in maps with one extrema to two parameter maps with two extrema finally a scheme for the global organization of bifurcation in these maps is proposed

a decomposition theorem for square free unitary solutions of the quantum yang baxter equation abstract it is known that every skew polynomial ring with generating set x and binomial relations in the sense of gateva ivanova trans amer math soc 343 1994 203 is an artin schelter regular domain of global dimension x moreover every such ring gives rise to a non degenerate unitary set theoretical solution r x 2 x 2 of the quantum yang baxter equation which fixes the diagonal of x 2 gateva ivanova s conjecture talk at the international algebra conference miskolc hungary 1996 states that conversely every such solution r comes from a skew polynomial ring with binomial relations an equivalent conjecture duke math j 100 1999 169 says that the underlying set x is r decomposable we prove these conjectures and construct an indecomposable solution r with x which shows that an extension to infinite x is false

observations of order and chaos in nonlinear systems experiments on nonlinear electrical oscillators the belousov zhabotinskii reaction rayleigh benard convection and couette taylor flow have revealed several common routes to chaos that have also been found in numerical studies of models with a few degrees of freedom experimental results are presented illustrating the following transition sequences period doubling and the u sequence intermittency the periodic quasiperiodic chaotic sequence frequency locking and an alternating periodic chaotic sequence

a parallel algorithm for the computation of invariant tori in large scale dissipative systems a parallelizable algorithm to compute invariant tori of high dimensional dissipative systems obtained upon discretization of pdes is presented the size of the set of equations to be solved is only a small multiple of the dimension of the original system the sequential and parallel implementations are compared with a previous method sanchez et al 2010 11 showing that important savings in wall clock time can be achieved in order to test it a thermal convection problem of a binary mixture of fluids has been used the new method can also be applied to problems with very low rotation numbers for which the previous is not suitable this is tested in two examples of two dimensional maps

realizability of branched coverings of surfaces absrract a branched covering m n of degree d between closed surfaces determines a collection 6d of partitions of d its branch data corresponding to the set of branch points the collection of partitions must satisfy certain obvious conditions implied by the riemann hurwitz formula this paper investigates the extent to which any such finite collection 6d of partitions of d can be realized as the branch data of a suitable branched covering if n is not the 2 sphere such data can always be realized if 6d contains sufficiently many elements compared to d then it can be realized and whenever d is nonprime examples are constructed of nonrealizable data

nonresonant hopf hopf bifurcation and a chaotic attractor in neutral functional differential equations abstract an algorithm for calculating the third order normal form of a nonresonant hopf hopf singularity in a neutral functional differential equation nfde is established the van der pol equation with extended delay feedback is investigated as an nfde of second order the existence of hopf hopf bifurcation is studied and the unfolding near these critical points is given by applying this algorithm periodic solutions and quasi periodic solutions are found with the aid of the bifurcation diagram and corresponding numerical illustrations are presented with the breaking down of the 3 torus a chaotic attractor appears in this nfde of second order following the ruelle takens newhouse scenario which usually arises for an ordinary differential equation of order at least 4 this transition is shown via both theoretical and numerical approaches

the numerical approximation of the rotation number of planar maps abstract given a convex jordan curve c we present a method which computes the numerical approximation of the rotation number of a planar map whose invariant curve is c it does not require any numerical approximation of the invariant curve but just exploits the order by which the iterates of the map are generated numerical examples are reported and discussed

universal properties of the transition from quasi periodicity to chaos in dissipative systems abstract an exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi periodic regime is held at a fixed irrational value our approach is a natural extension of k a m theory to strong coupling most of our analysis is for analytic circle maps we have found a strong coupling fixed point where invertibility is lost which describes the universal features of the transition to chaos we find numerically that any two such critical maps with the same winding number are c 1 conjugate it follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency when the winding number has a periodic continued fraction our renormalization transform has a fixed point and spectra are self similar in addition for a set of non periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub space of all critical maps physically one finds singular and universal spectra that do not scale

renormalization in a circle map with two inflection points abstract an extended version of the sine circle map becomes non invertible by developing simultaneously two critical points in part of parameter space universality in the corresponding transition from quasiperiodicity to chaos manifests itself in the organisation of doubly superstable orbits four operators are needed to generate all such orbits the existence of non trivial equivalence relations for operation sequences makes the present theory different from other multi operator renormalisation approaches possible implications of the theory in higher dimensional systems are discussed

quasiperiodicity in dissipative systems a renormalization group analysis abstract dynamical systems with quasiperiodic behavior i e two incommensurate frequencies may be studied via discrete maps which show smooth continuous invariant curves with irrational winding number in this paper these curves are followed using renormalization group techniques which are applied to a one dimensional system circle and also to an area contracting map of an annulus two fixed points are found representing different types of universal behavior a trivial fixed point for smooth motion and a nontrivial fixed point the latter representsthe incipient breakup of a quasiperiodic motion with frequency ratio the golden mean into a more chaotic flow fixed point functions are determined numerically and via an e expansion and eigenvalues are calculated

on the appearance of multiple attractors in discrete food chains abstract klebanoff and hastings a klebanoff a hastings chaos in three species food chains journal of mathematical biology 32 1994 427 451 detected cases of multiple attractors in continuous food chains in this paper we discuss similar phenomena in the discrete food chains introduced by lindstrom t lindstrom on the dynamics of discrete food chains low and high frequency behavior and chaos journal of mathematical biology 45 2002 396 418 the results imply that the dynamical properties including species persistence may change due to disturbances that do not involve changes in the environmental parameters thus there are possibilities that species may be eradicated or start to oscillate at different frequencies without any changes in the environment since this is now shown to hold in both seasonal and non seasonal environments we expect that this is a rather general property of ecosystems

the index of the spinc dirac operator on the weighted projective space and the reciprocity law of the fourier dedekind sum abstract the main purpose of this paper is to give a geometric interpretation of the reciprocity law of the fourier dedekind sum given by m beck and s robins in fact the v index of the spin c dirac operator on the weighted projective space is equal to the dimension of the space of all weighted homogeneous polynomials of given degree and this equality gives precisely the beck robins reciprocity law in this equality the fourier dedekind sums appear as the localization terms of the v index of the spin c dirac operators and have a relationship to the eta invariants of lens spaces

an unstable mode selection rule frustrated optical instability due to competing boundary conditions abstract self oscillation phenomenon which occurs in a system with more than one spatial boundary condition may fall into frustration from competition between resonance conditions due to the multiple boundaries in this paper we study a nonlinear optical instability in a compound resonator which is a typical example of this phenomenon in this system for a fixed control parameter the asymptotic behaviour is just oscillation at a single frequency however as the rule which determines the oscillation frequency contains chaos the final oscillation frequency shows a very sensitive dependence on the ratio of lengths of the resonators in the compound cavity elementary number theoretic method is successfully applied to the analysis of this phenomenon and an almost complete selection rule for the oscillating mode is established the selection rule explains very well the recent experiments this can be regarded as a typical example of a system in which the final pattern itself is dynamically stable but the choice of the pattern is unstable with respect to noise in the environment

merging of resonance tongues abstract this paper considers the resonance behavior found near a saddle node hopf bifurcation in a flow the global structure of resonance tongues is examined and it is shown that in the absence of certain symmetries different resonance tongues can merge with one another the position and shape of the resonance tongues are seen to be related to the presence of homoclinic and heteroclinic bifurcations of two saddle foci the implications of these results for the normal form analysis of the saddle node hopf bifurcation are discussed

characterizing bifurcations and classes of motion in the transition to chaos through 3d tori of a continuous experimental system in solid mechanics abstract the dynamics of a continuous experimental system exhibiting quasiperiodic transition to chaos has been investigated an elastic cable mass system hanging at in phase vertically moving supports is considered with vibrational parameters in the neighbourhood of external and internal resonance conditions adjusted to be able to introduce three torus dynamics measurements of the nonregular response are made in frequency ranges including primary resonance of the first symmetric in plane mode of the cable quantitative characterisation of regular and nonregular attractors and of the configuration variables involved in the motion is made by means of delay embedding technique and proper orthogonal decomposition of spatio temporal flow attention is devoted to analyse in depth the scenario of transition to chaos exhibited by the model steady quasiperiodic motions on three tori partial and full phase locking of the flow and chaotic attractors are observed in spite of the dynamics variedness the systematic characterisation of the response allows us to set the bifurcation behaviour of the experimental system in the framework of theoretic numeric transition to chaos through three dimensional tori

hopf saddle node bifurcation for fixed points of 3d diffeomorphisms analysis of a resonance bubble the dynamics near a hopf saddle node bifurcation of fixed points of diffeomorphisms is analysed by means of a case study a two parameter model map g is constructed such that at the central bifurcation the derivative has two complex conjugate eigenvalues of modulus one and one real eigenvalue equal to 1 to investigate the effect of resonances the complex eigenvalues are selected to have a 1 5 resonance it is shown that near the origin of the parameter space the family g has two secondary hopf saddle node bifurcations of period five points a cone like structure exists in the neighbourhood formed by two surfaces of saddle node and a surface of hopf bifurcations quasi periodic bifurcations of an invariant circle forming a frayed boundary are numerically shown to occur in model g along such cantor like boundary an intricate bifurcation structure is detected near a 1 5 resonance gap subordinate quasi periodic bifurcations are found nearby suggesting the occurrence of a cascade of quasi periodic bifurcations c 2008 elsevier b v all rights reserved

chaotic motion on hamiltonian tori kam tori are well known to be that finite part of phase space in which the motion of a weakly perturbed classical hamiltonian system remains integrable moreover they are barriers in the phase space of systems with two degrees of freedom the authors show that certain hamiltonian systems contain invariant non regular tori with compressible flow using as an example an electron moving in electromagnetic fields which are periodic in space they demonstrate i that strange attractors and repellers of well known autonomous or quasi periodically time driven systems may occur on such strange tori ii that one may find barriers consisting of non regular hamiltonian tori in systems with any number of degrees of freedom the flow on such barriers is non chaotic though and iii that certain kam tori transform into non regular tori rather than breaking up when the perturbation becomes strong

the dynamics of phase locking and points of resonance in a forced magnetic oscillator abstract we report data on an experimental system a forced symmetric oscillator containing a saturable inductor with magnetic hysteresis it displays a hopf bifurcation to quasiperiodicity entrainment horns and chaos we study in detail the bifurcations and hysteresis occurring near points of resonance particularly strong resonance and show how the observed behavior can be understood using arnold s theory much of the behavior relating to the entrainment horns is explored period doubling and symmetry breaking bifurcations homoclinic bifurcations and crises and other bifurcations taking place at the horn boundaries important features of the behavior related to symmetry properties of the oscillator are studied and explained through the concept of a half cycle map the system is shown to exhibit a hopf bifurcation from a phase locked state to periodic islands similar to those found in hamiltonian systems an initialization technique is used to observe the manifolds of saddle orbits and other hidden structure an unusual differential equation model is developed which is irreversible and generates a noninvertible poincare map of the plane noninvertibility of this planar map has important effects on the behavior observed the poincare map may also be approximated through experimental measurements resulting in a planar map with parameter dependence this model gives good correspondence with the system in a region of the parameter space

the genus of compact riemann surfaces with maximal automorphism group any compact riemann surface with genus g 1 has at most 84 g 1 conformal automorphisms in this paper it is shown that there are just 32 integers g in the range 1 g 11905 for which there exist compact riemann surfaces of genus g with this maximum possible number of automorphisms the automorphism group of each such surface is isomorphic to a quotient n of the 2 3 7 triangle group x y x2 y3 xy 7 1 where n is one of 92 proper normal subgroups of with index less than 106 these normal subgroups are found using elementary group theoretic techniques

trinity symmetry and kaleidoscopic regular maps a cellular embedding of a connected graph also known as a map on an orientable surface has trinity symmetry if it is isomorphic to both its dual and its petrie dual a map is regular if for any two incident vertex edge pairs there is an automorphism of the map sending the first pair onto the second given a map m with all vertices of the same degree d for any e relatively prime to d the power map m e is formed from m by replacing the cyclic rotation of edges at each vertex on the surface with the e th power of the rotation a map is kaleidoscopic if all of its power maps are pairwise isomorphic in this paper we present a covering construction that gives infinite families of kaleidoscopic regular maps with trinity symmetry

minimal genus of abelian actions on klein surfaces with boundary for a finite group g the hurwitz problem is to determine the genera of the surfaces which admit an effective g action for a closed surface genus means the number of handles in the orientable case or the number of crosscaps in the nonorientable case for a punctured or bounded surface it means the algebraic genus which is the rank of the fundamental group as a free group or equivalently the first betti number of the surface when the surface has negative euler characteristic various uniformization theorems allow one to assume that the action is conformal with respect to some riemannian structure dianalytic with respect to some dianalytic structure i e kleinian structure see a g or isometric with respect to some complete hyperbolic structure with geodesic boundary see chap 13 of t1 when the surfaces are riemann surfaces the problem is often considered for conformal hence orientation preserving actions but in the present work we make no restriction on orientability of the surface or the action although there are some results which concern the s table hurwitz problem the determination up to finitely many cases of the genera admitting g actions k m m most attention has been focused on the determination of the minimal genus g 2 admitting a g action hurwitz h2 found the general

numerical approximation of rough invariant curves of planar maps in this paper we describe a simple algorithm for the approximation of an invariant curve gamma of a planar map f we are particularly interested in the case where gamma is attracting but not smooth results for the delayed logistic map illustrate the performance of the algorithm

solutions of the yang baxter equation associated with a left brace abstract given a left brace b a method is given to construct explicitly all the non degenerate involutive set theoretic solutions x r of the yang baxter equation such that the associated permutation group g x r is isomorphic as a left brace to b this method depends entirely on the brace structure of b

period doubling mode interactions with circular symmetry abstract a two parameter analysis of the interaction between two period doubling modes with azimuthal wavenumbers k and l l k 1 is carried out for systems with circular symmetry the problem is formulated in terms of an o 2 equivariant map on c 2 in the generic case all primary secondary and tertiary solution branches and their stability properties are classified the results depend on whether k l is odd or even when one mode bifurcates subcritically and the other supercritically the pure mode branches lose stability to a branch of reflection symmetric mixed modes which in turn can undergo a tertiary hopf bifurcation to a quasiperiodic reflection symmetric pattern we conjecture that this invariant circle can break down with increasing amplitude to produce reflection symmetric chaos additional phase instabilities may occur in which the reflection symmetry is broken and the resulting pattern drifts either clockwise or counterclockwise the results explain a number of observations by ciliberto and gollub on parametrically excited surface waves in circular geometry and imply several new predictions for such experiments in an appendix the theory is compared with previous attempts to model the experiments

link indices of hurwitz maps abstract let m be a regular map of type m n with automorphism group a smooth quotient of the ordinary 2 m n triangle group then by recent work of melekoglu and singerman we can associate with m some positive integers called the link indices of m the number of link indices is always one two or three depending on the parity of m and n in the case where m and n are both odd which happens when m is a hurwitz map of type 3 7 there is a unique link index in this paper we prove that every positive integer divisible by 2 or 3 is the link index of some hurwitz map we also find the link indices of all hurwitz maps of genus up to 10000 and of all the hurwitz maps with automorphism group psl 2 q for some q 700 and we show that every integer between 2 and 360 is the link index of some hurwitz map

topological quantum codes on compact surfaces with genus g 2 in this paper we propose a construction procedure of a class of topological quantum error correcting codes on surfaces with genus g 2 this generalizes the toric codes construction we also tabulate all possible surface codes with genus 2 5 in particular this construction reproduces the class of codes obtained when considering the embedding of complete graphs ks for s 1 mod 4 on surfaces with appropriate genus we also show a table comparing the rate of different codes when fixing the distance to 3 5

bifurcation and chaos in a periodically stimulated cardiac oscillator abstract periodic stimulation of an aggregate of spontaneously beating cultured cardiac cells displays phase locking period doubling bifurcations and aperiodic chaotic dynamics at different values of the stimulation parameters this behavior is analyzed by considering an experimentally determined one dimensional poincare or first return map a simplified version of the experimentally determined poincare map is proposed and several features of the bifurcations of this map are described

chaotic dynamics of a bouncing ball abstract a detailed study of a mapping on a two dimensional manifold is made the mapping describes a system subject to periodic forcing in particular an imperfectly elastic ball bouncing on a vibrating platform quasiperiodic motion on a one dimensional manifold is proven and observed numerically at low forcing while at higher forcing smale horseshoes are present we examine the evolution of the attracting set with changing parameter spatial structure is oganised by fixed points of the mapping and sudden changes occur by crises a new type of chaos in which a trajectory alternates between two distinct chaotic regions is described and explained in terms of manifold collisions throughout we are concerned to examine the behaviour of lyapunov exponents typical behaviour of lyapunov exponents in the quasiperiodic regime under the influence of external noise is discussed at higher forcing a certain symmetry of the attractor allows an analytic expression for the exponents to be given

chaotic attractors on a 3 torus and torus break up abstract two coupled driven van der pol oscillators can have three frequency quasiperiodic attractors which lie on a 3 torus the evidence presented in this paper indicates that the torus is destroyed when the stable and unstable manifolds of an unstable orbit become tangent furthermore no chaotic orbits lying on a torus were observed suggesting that in most cases at least in the case of this system orbits do not become chaotic before their tori are destroyed to expedite the calculations a method was developed which can be used to determine if an orbit is on a torus without actually displaying that orbit the method also described in this paper was designed specifically for our system the basic idea however could be used for studying attractors of other systems very few modifications of the method if any would be necessary when studying systems with the number of degrees of freedom equal to that of our van der pol system

computing arnol d tongue scenarios a famous phenomenon in circle maps and synchronisation problems leads to a two parameter bifurcation diagram commonly referred to as the arnol d tongue scenario one considers a perturbation of a rigid rotation of a circle or a system of coupled oscillators in both cases we have two natural parameters the coupling strength and a detuning parameter that controls the rotation number frequency ratio the typical parameter plane of such systems has arnol d tongues with their tips on the decoupling line opening up into the region where coupling is enabled and in between these arnol d tongues quasi periodic arcs in this paper we present unified algorithms for computing both arnol d tongues and quasi periodic arcs for both maps and odes the algorithms generalise and improve on the standard methods for computing these objects we illustrate our methods by numerically investigating the arnol d tongue scenario for representative examples including the well known arnol d circle map family a periodically forced oscillator caricature and a system of coupled van der pol oscillators

three coupled oscillators mode locking global bifurcations and toroidal chaos abstract many systems exhibit behaviour which can be described by three coupled oscillators provided the coupling is not too strong such systems can be modelled by maps of the two torus to itself in this paper we describe and explain those aspects of the bifurcation diagram for two parameter families of torus maps that involve change of mode locking type we introduce the concepts of partial and full mode locking for the coupled oscillators these notions correspond to the presence of one or two rational relations between the frequencies respectively numerical investigation of a particular family of torus maps reveals an intricate web of global bifurcations in order to explain the results we first show that we can approximate maps in the neighbourhood of a rational translation to arbitrary order by the time 1 map of a flow on the torus then we analyse those condimension 1 and 2 bifurcations for flows on the torus which change the set of frequency ratios we find a large variety of bifurcation diagrams in particular many involving homotopically non trivial saddle connections next we ask how the picture changes when the time 1 maps of a family of flows are perturbed to a general family of diffeomorphisms we find toroidal chaos in the neighbourhood of all these bifurcations meaning that there exist orbits which perform a pseudo random sequence of rotations in different directions around the torus the corresponding behaviour for the coupled oscillators is that the frequency ratios perform a random walk finally we show how all these ingredients can be put together to give global scenarios for bifurcation for families of torus maps which we believe to be of general applicability to physical systems with three weakly coupled modes of oscillation

the eta function of the localised dirac operator for the universal cover of sl2 r abstract recently brodzki niblo plymen and wright determined a closed explicit description of the spectrum of the dirac operator d for the universal cover of sl 2 r localised at a representations in the principal series and in the discrete series 4 using this description we introduce an eta function loc s d for the localised dirac operator and we study its analytic properties we show that the localised eta function has an analytic extension to the complex plane with simple poles at negative integers and at s 1 and it is regular at s 0 and we give a formula for loc 0 d we discuss a possible eta invariant for the dirac operator

bifurcations from an invariant circle for two parameter families of maps of the plane a computer assisted study we consider a two parameter family of maps of the plane to itself each map has a fixed point in the first quadrant and is a diffeomorphism in a neighborhood of this point for certain parameter values there is a hopf bifurcation to an invariant circle which is smooth for parameter values in a neighborhood of the bifurcation point however computer simulations show that the corresponding invariant set fails to be even topologically a circle for parameter values far from the bifurcation point this paper is an attempt to elucidate some of the mechanisms involved in this loss of smoothness and alteration of topological type

numerical computation of invariant circles of maps abstract a method for the numerical computation of invariant circles of maps is presented along with appropriate techniques for its implementation the method involves solution of a functional equation by discretization and newton iteration the resulting algorithm is applied to a map of the cylinder and some examples of bifurcations of invariant circles are illustrated generalization of the method to the computation of invariant circles in more than two dimensions as well as sections of invariant tori is discussed

attractors on an n torus quasiperiodicity versus chaos the occurrence of quasiperiodic motions in nonconservative dynamical systems is of great fundamental importance however current understanding concerning the question of how prevalent such motions should be is incomplete with this in mind the types of attractors which can exist for flows on an n torus are studied numerically for n 3 and 4 specifically nonlinear perturbations are applied to maps representing n frequency quasiperiodic attractors these perturbations can cause the original n frequency quasiperiodic attractors to bifurcate to other types of attractors our results show that for small and moderate nonlinearity the frequency of occurrence of quasiperiodic motions is as follows n frequency quasiperiodic attractors are the most common followed by n 1 frequency quasiperiodic attractors followed by periodic attractors however as the nonlinearity is further increased n frequency quasiperiodicity becomes less common ceasing to occur when the map becomes noninvertible chaotic attractors are very rare for n 3 for small to moderate nonlinearity but are somewhat more common for n 4 examination of the types of chaotic attractors that occur for n 3 reveals a rich variety of structure and dynamics in particular we see that there are chaotic attractors which apparently fill the entire n torus i e limit sets of orbits on these attractors are the entire torus furthermore these are the most common types of chaotic attractors at moderate nonlinearities

the residue of the global η function at the origin where i dvol x l denotes the riemannian measure on m using the calculus of pseudo differential operators depending upon a complex parameter which was developed by seeley 9 one can show that both q s x p and r s p can be meromorphically extended to the complex plane with simple poles at s m n u for n 0 the residues of l s x p at the poles for n 0 m are given by local formulas in the jets of the total symbol of p we define

involutive yang baxter groups in 1992 drinfeld posed the question of finding the set theoretic solutions of the yang baxter equation recently gateva ivanova and van den bergh and etingof schedler and soloviev have shown a group theoretical interpretation of involutive non degenerate solutions namely there is a one to one correspondence between involutive non degenerate solutions on finite sets and groups of i type a group g of i type is a group isomorphic to a subgroup of fa n sym n so that the projection onto the first component is a bijective map where fa n is the free abelian group of rank n and sym n is the symmetric group of degree n the projection of g onto the second component sym n we call an involutive yang baxter group iyb group this suggests the following strategy to attack drinfeld s problem for involutive non degenerate set theoretic solutions first classify the iyb groups and second for a given iyb group g classify the groups of i type with g as associated iyb group it is known that every iyb group is solvable in this paper some results supporting the converse of this property are obtained more precisely we show that some classes of groups are iyb groups we also give a non obvious method to construct infinitely many groups of i type and hence infinitely many involutive non degenerate set theoretic solutions of the yang baxter equation with a prescribed associated iyb group

rotation sets and phase locking in an electronic three oscillator system abstract the parameter space of an electronic three oscillator system is investigated and various codimension one and two bifurcations predicted by baesens guckenheimer kim and mackay are identified sampled time series from the experimental systems are recorded and analysed for partial mode locking or resonance one or two independent rational relations between the average rates of change of the angles describing the system using knowledge of where the invariant torus lies and the torus unfolding scheme of ashwin and swift examples of toroidal and annular chaos are investigated by finding bounds on the size and shape of the rotation set

counterexample to a conjecture about braces abstract we find an example of a finite solvable group in fact a finite p group in which is not possible to define a left brace structure or equivalently which is not an iyb group this answers a question posed by cedo jespers and del rio related to the yang baxter equation our argument is an improvement of an argument of rump using results about hopf galois extensions and lsa structures we explain explicitly the relation between these two areas of mathematics and brace theory hoping that it will be useful in the future

reconstructing bifurcation diagrams only from time waveforms abstract a simple algorithm is proposed for reconstructing bifurcation diagrams only from time waveforms of a dynamical system without knowing an explicit form of the dynamics and its dependence on the parameters reconstruction means that the obtained bifurcation diagram is qualitatively similar to that of the original system the algorithm consists of two steps first a nonlinear predictor is sought using a particular class of functions with many parameters second the karhunen loeve transform is used to find only significant parameters contributing the bifurcations the algorithm is tested against two families of dynamical system the henon family and the coupled logistic delayed logistic family

characters of finite groups with dihedral sylow 2 subgroups in this article we give alternate proofs of some intermediate results in the gorenstein walter classification 6 of groups with dihedral sylow 2subgroups these intermediate results will be applied in an alternate proof 1 of the entire classification by the first author our proofs rely only on elementary character theory and do not use block theory but they are motivated by the related work of brauer 2 and of gorenstein and walter 6 propositions l 41 which was obtained by block theory now we describe the situation which will be assumed in the following sections and in the three theorems stated below for elementary properties of groups with dihedral sylow 2 subgroups the reader is referred to s section 7 71

fourier methods for quasi periodic oscillations quasi periodic oscillations and invariant tori play an important role in the study of forced or coupled oscillators this paper presents two new numerical methods for the investigation of quasi periodic oscillations both algorithms can be regarded as generalizations of the averaging and the harmonic spectral balance methods the algorithms are easy to implement and require only minimal a priori knowledge of the system most importantly the methods do not depend on an a priori co ordinate transformation the methods are applied to a number of illustrative examples from non linear electrical engineering and the results show that the methods are efficient and reliable in addition these examples show that the presented algorithms can also continue through regions of sub harmonic phase locked resonance even though they are designed only for the quasi periodic case copyright 2006 john wiley sons ltd

dependencies between competition and predation and their consequences for initial value sensitivity we commence with a discussion of dependencies between various processes in discrete consumer resource models two basic models are formulated these models are closely related and differ only in terms of which dependencies are assumed the models can be regarded as hybrids of the lotka volterra model the beverton holt model and the nicholson bailey model a complete stability analysis of the equilibria is given with the conclusion that the stability properties of these models are closely related to gause type predator prey systems the numerical part of our analysis brings out remarkable qualitative differences concerning the initial value sensitivity of the oscillations of the two basic models initial value sensitivity was measured by lyapunov exponents and the observed differences cannot be brought to light by a study of limiting cases or by equilibrium analysis the actual reasons behind these results remain a delicate mathematical question properties associated directly with the time series like

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

bifurcation of periodic points of some diffeomorphisms on r3 on etudie l application f definie par f x y z ax b y z bx a y z cx dx n ez ou a b c d et e sont des nombres reels d 0 et n 1 un entier

bifurcation to infinitely many sinks this paper considers one parameter families of diffeomorphisms ft in two dimensions which have a curve of dissipative saddle periodic pointspt i e ftn pt pt and detdftn pt 1 the family is also assumed to create new homoclinic intersections of the stable and unstable manifolds ofpt as the parameter varies throught0 gavirlov and silnikov proved that if the new homoclinic intersections are created nondegenerately att0 then there is an infinite cascade of periodic sinks i e there are parameter valuestn accumulating att0 for which there is a sink of periodn gs2 sect 4 we show that this result is true for real analytic diffeomorphisms even if the homoclinic intersection is created degenerately we give computer evidence to show that this latter result is probably applicable to the henon map fora near 1 392 andb equal 0 3