classical solutions of the quantum yang baxter equation the classical analogue is developed here for part of the construction in which knot and link invariants are produced from representations of quantum groups whereas previous work begins with a quantum group obtained by deforming the multiplication of functions on a poisson lie group we work directly with a poisson lie groupg and its associated symplectic groupoid the classical analog of the quantumr matrix is a lagrangian submanifold open image in new window in the cartesian square of the symplectic groupoid for any symplectic leafs ing open image in new window induces a symplectic automorphism ofs s which satisfies the set theoretic yang baxter equation when combined with the flip map exchanging components and suitably implanted in each cartesian powers n generates a symplectic action of the braid groupb n ons n application of a symplectic trace formula to the fixed point set of the action of braids should lead to link invariants but work on this last step is still in progress

a period doubling bifurcation with slow parametric variation and additive noise slow sinusoidal modulation of a control parameter can maintain a low period orbit into parameter regions where the low period orbit is locally unstable and a higherperiod orbit would normally occur whether or not a bifurcation to higher period becomes evident during the modulation depends on the competing effects of stabilization by the modulation and destabilization by inherent very low level system noise a transition often rapid from a locally unstable period 1 orbit to period 2 for example can be triggered by noise the competing effects are examined here for a period doubling bifurcation of a general unimodal map a nested set of three matched asymptotic expansions a triple deck is used to describe the combined period 1 and period 2 response the resulting solution gives estimates of whether and where an apparent period doubling bifurcation occurs typical period 1 stability boundaries are obtained that include the effect of the amplitude and frequency of the variation the noise level in the system and the allowable maximum threshold level of period 2 response

curves of genus 2 with split jacobian we say that an algebraic curve has split jacobian if its jacobian is isogenous to a product of elliptic curves if x is a curve of genus 2 and f x t e a map from x to an elliptic curve then x has split jacobian it is not true that a complement to e in the jacobian of x is uniquely determined but under certain conditions there is a canonical choice of elliptic curve e and algebraic f x e and we give an algorithm for finding that curve the construction works in any characteristic other than two applications of the algorithm are given to give explicit examples in characteristics 0 and 3 o introduction we say that a curve has split jacobian if its jacobian is isogenous to a product of elliptic curves in the later half of the nineteenth century a considerable body of work was done on the reduction of abelian integrals to elliptic krazer 1 gives a summary of the results obtained stated geometrically the results are particular families of algebraic curves of genus 2 with maps of degree 2 3 or 4 to elliptic curves such curves have split jacobian the general question of split jacobian curves and particularly those of genus 2 is of interest for several reasons split jacobian curves often have the maximal number of points over finite fields e g the examples of moret bailley 2 are one parameter families of curves of genus 2 over fields of order p2 whose jacobians are isomorphic to the square of the supersingular elliptic curare and which have maximal numbers of points over fields of order p2n n 2 split jacobian curves of genus 2 have also been used to exhibit nonisomorphic curves with the same jacobian vide 3 4 the approach in these papers is through the algebraic geometry of abelian varieties and the constructions are therefore far from explicit consider the following let x be a curve of genus 2 and f x e a map from x to an elliptic curve the jacobian of x is therefore isogenous to a product of e and another elliptic curve e problem find e e g what is its j invariant it is not clear nor even true vide t shioda 7 that e is uniquely determined however under certain conditions there is a canonical choice of complement and we give an algorithm for finding that curve our aim is to provide explicit equations for the curves and the maps between them we obtain a fairly complete combinatorial characterization of the splitting of the jacobians of curves of genus 2 the splitting is characterized by the degree of the map 5 above jacobi generalizing an example of legendre gave the complete solution for degree 2 given any involution of pl and three points not fixed by the involution the curve of genus 2 which has its 6 weierstrass points above the three points and their images under the involution maps to the two elliptic curves represented as double covers of the quotient of pl by the involution ramified at received by the editors july 20 1987 1980 mathematicx xubject laxsijication 1985 revitsion primary 14h40 llg10 ff 1988 american mathematical society 0002 9947 88 1 00 25 per page

domain formation in transitions with noise and a time dependent bifurcation parameter the characteristic size for spatial structure that emerges when the bifurcation parameter in model partial differential equations is slowly increased through its critical value depends logarithmically on the size of added noise numerics and analysis are presented for the real ginzburg landau and swift hohenberg equa tions

grassmann extensions of yang baxter maps in this paper we show that there are explicit yang baxter yb maps with darboux lax representation between grassman extensions of algebraic varieties motivated by some recent results on noncommutative extensions of darboux transformations we first derive a darboux matrix associated with the grassmann extended derivative nonlinear schrodinger dnls equation and then we deduce novel endomorphisms of grassmann varieties which possess the yb property in particular we present ten dimensional maps which can be restricted to eight dimensional yb maps on invariant leaves related to the grassmann extended nls and dnls equations we consider their vector generalisations

torsion points on curves and p adic abelian integrals theorem a let f c j be an albanese morphism defined over a number field k of a s mooth curve of genus g into its jacobian suppose j has potential complex multiplication let s denote the set of primes p of k satisfying i p does not divide 2 or 3 ii k q is unramified at p iii c has good ordinary reduction over k then the set t of torsion points of a defined over an algebraic closure of k which lie on the image of c is defined over an extension of k unramified above s moreover to pg

3d compatible ternary systems and yang baxter maps according to shibukawa ternary systems defined on quasi groups and satisfying certain conditions provide a way of constructing dynamical yang baxter yb maps after noting that these conditions can be interpreted as three dimensional compatibility of equations on quad graphs we investigate when the associated dynamical yb maps are in fact parametric yb maps in some cases these maps can be obtained as reductions of higher dimensional maps through compatible constraints conversely parametric yb maps on quasi groups with an invariance condition give rise to three dimensional compatible systems the application of this method on spaces with certain quasi group structures provides new examples of multi parametric yb maps and three dimensional compatible systems

examples of torsion points on genus two curves we describe a method that sometimes determines all the torsion points lying on a curve of genus two defined over a number field and embedded in its jacobian using a weierstrass point as base point we then apply this to the examples y2 x5 x y2 x5 5x3 x and y2 y x5

tetrahedron maps and symmetries of three dimensional integrable discrete equations a relationship between the tetrahedron equation for maps and the consistency property of integrable discrete equations on z3 is investigated our approach is a generalization of a method developed in the context of yang baxter maps based on the invariants of symmetry groups of the lattice equations the method is demonstrated by a case by case analysis of the octahedron type lattice equations classified recently leading to some new examples of tetrahedron maps and integrable coupled lattice equations a relationship between the tetrahedron equation for maps and the consistency property of integrable discrete equations on z3 is investigated our approach is a generalization of a method developed in the context of yang baxter maps based on the invariants of symmetry groups of the lattice equations the method is demonstrated by a case by case analysis of the octahedron type lattice equations classified recently leading to some new examples of tetrahedron maps and integrable coupled lattice equations

classification of integrable equations on quad graphs the consistency approach a classification of discrete integrable systems on quad graphs i e on surface cell decompositions with quadrilateral faces is given the notion of integrability laid in the basis of the classification is the three dimensional consistency this property yields among other features the existence of the discrete zero curvature representation with a spectral parameter for all integrable systems of the obtained exhaustive list the so called three leg forms are found this establishes lagrangian and symplectic structures for these systems and the connection to discrete systems of the toda type on arbitrary graphs generalizations of these ideas to the three dimensional integrable systems and to the quantum context are also discussed

explicit auto transformations of integrable chains a construction scheme for explicit auto transformations of integrable discrete differential equations chains is presented these transformations are rather convenient to obtain the exact solutions for chains as well as associated partial differential systems on the other hand they exemplify new integrable discrete mappings their group properties are also of great interest the scheme is illustrated by several examples of integrable systems which contains the nonlinear schrodinger system and the landau lifshits model

poisson yang baxter maps with binomial lax matrices a construction of multidimensional parametric yang baxter maps is presented the corresponding lax matrices are the symplectic leaves of first degree matrix polynomials equipped with the sklyanin bracket these maps are symplectic with respect to the reduced symplectic structure on these leaves and provide examples of integrable mappings an interesting family of quadrirational symplectic yb maps on c4 c4 with 3 3 lax matrices is also presented

darboux transformations finite reduction groups and related yang baxter maps in this paper we construct yang baxter yb maps using darboux matrices which are invariant under the action of finite reduction groups we present six dimensional yb maps corresponding to darboux transformations for the nonlinear schrodinger nls equation and the derivative nonlinear schrodinger dnls equation these yb maps can be restricted to four dimensional yb maps on invariant leaves the former are completely integrable and they also have applications to a recent theory of maps preserving functions with symmetries fordy a and kassotakis a 2013 j phys a math theor 46 205201 we give a six dimensional yb map corresponding to the darboux transformation for a deformation of the dnls equation we also consider vector generalizations of the yb maps corresponding to the nls and dnls equations

affine linear and d4 symmetric lattice equations symmetry analysis and reductions we consider lattice equations on which are autonomous affine linear and possess the symmetries of the square some basic properties of equations of this type are derived as well as a sufficient linearization condition and a conservation law a systematic analysis of the lie point and the generalized three and five point symmetries is presented it leads to the generic form of the symmetry generators of all the equations in this class which satisfy a certain non degeneracy condition finally symmetry reductions of certain lattice equations to discrete analogs of the painleve equations are considered

torsion points on modular curves let n 23 be a prime number in this paper we prove a conjecture of coleman kaskel and ribet about the valued points of the modular curve x 0 n which map to torsion points on j 0 n via the cuspidal embedding we give some generalizations to other modular curves and to noncuspidal embeddings of x 0 n into j 0 n

periodic points on veech surfaces and the mordell weil group over a teichmuller curve periodic points are points on veech surfaces whose orbit under the group of affine diffeomorphisms is finite we characterize those points as being torsion points if the veech surfaces is suitably mapped to its jacobian or an appropriate factor thereof for a primitive veech surface in genus two we show that the only periodic points are the weierstras points and the singularities

pseudo regular oscillations induced by external noise an examination of the effect of noise on a general system at a saddle node bifurcation has revealed that in the limit of weak noise the probability density of the time to pass through the saddle node has a universal shape the specific kinetics of the particular system serving only to set the time scale this probability density is displayed and its salient features are explicated in the case of a saddle node bifurcation leading to relaxation oscillations this analysis leads to the prediction of the existence of noise induced oscillations which appear much less random than might at first be expected the period of these oscillations has a well defined nonzero most probable value the inverse of which is a noise induced frequency this frequency can be detected as a peak in power spectra from numerical simulations of such a system this is the first case of the prediction and detection of a noise induced frequency of which the authors are aware

stochastic calculus application to dynamic bifurcations and threshold crossings for the dynamic pitchfork bifurcation in the presence of white noise the statistics of the last time at zero are calculated as a function of the noise level and the rate of change of the parameter the threshold crossing problem used for example to model the firing of a single cortical neuron is considered concentrating on quantities that may be experimentally measurable but have so far received little attention expressions for the statistics of pre threshold excursions occupation density and last crossing time of zero are compared with results from numerical generation of paths

modular equations for hyperelliptic curves we dene modular equations describing the torsion subgroups of the jacobian of a hyperelliptic curve over a nite base eld we prove factorization properties that extend the well known results used in atkin s improvement of schoof s genus 1 point counting algorithm

chaos and the effect of noise in a model of three wave mode coupling abstract the equations governing the nonlinear interaction of three resonant waves are studied in the limit that one wave is slightly unstable and the other two modes more heavily damped it is shown that the dynamics can be described in terms of a bimodal one dimensional map which allows bifurcation sequences etc to be simply constructed the effect of small additive noise can be studied in the same framework it is shown that even at very low levels the effect of noise can be extremely important in determining the period and amplitude of the oscillations

noise and slow fast dynamics in a three wave resonance problem recent research on the dynamics of certain fluid dynamical instabilities shows that when there is a slow invariant manifold subject to fast timescale instability the dynamics are extremely sensitive to noise the behaviour of such systems can be described in terms of a one dimensional map and previous work has shown how the effect of noise can be modelled by a simple adjustment to the map here we undertake an in depth investigation of a particular set of equations using the methods of stochastic integration we confirm the prediction of the earlier studies that the noise becomes important when mu log epsilon o 1 where mu is the small timescale ratio and epsilon is the noise level in addition we present detailed information about the statistics of the solution when the noise is a dominant effect the analytical results show excellent agreement with numerical simulations

yang baxter maps and symmetries of integrable equations on quad graphs a connection between the yang baxter relation for maps and the multidimensional consistency property of integrable equations on quad graphs is investigated the approach is based on the symmetry analysis of the corresponding equations it is shown that the yang baxter variables can be chosen as invariants of the multiparameter symmetry groups of the equations we use the classification results by adler bobenko and suris to demonstrate this method some new examples of yang baxter maps are derived in this way from multifield integrable equations

tetrahedron equations and the relativistic s matrix of straight strings in 2 1 dimensions the quantums matrix theory of straight strings infinite one dimensional objects like straight domain walls in 2 1 dimensions is considered thes matrix is supposed to be purely elastic and factorized the tetrahedron equations which are the factorization conditions are investigated for the special two colour model the relativistic three strings matrix which apparently satisfies this tetrahedron equation is proposed

yang baxter maps and multi field integrable lattice equations a variety of yang baxter maps are obtained from integrable multi field equations on quad graphs a systematic framework for investigating this connection relies on the symmetry groups of the equations the method is applied to lattice equations introduced by adler and yamilov which are related to the nonlinear superposition formulae for the backlund transformations of the nonlinear schrodinger system and specific ferromagnetic models

predictability of noise controlled dynamics abstract noise controlled dynamics runs counter to usual intuition the larger the noise the more regular the solutions we present numerical and analytical results for a set of three stochastic partial differential equations in one space dimension motivated by the intermittent destabilization of tall thin convection cells by horizontal shear time series are predictable in the sense that they follow limit cycles with a small variation in amplitude from cycle to cycle closer inspection reveals that the amplitude is determined by very small amounts of noise

noise induced order a new noise effect on chaos in one dimensional mappings is reported the transition from chaotic behavior to ordered behavior induced by external noise is observed in a certain class of one dimensional mappings this transition is clearly shown in terms of the lyapunov number entropy power spectrum and the nature of orbits

on discrete 2d integrable equations of higher order we study two dimensional discrete integrable equations of order 1 with respect to one independent variable and m with respect to another one a generalization of the multidimensional consistency property is proposed for this type of equations the examples are related to the backlund darboux transformations for the lattice equations of bogoyavlensky type

a noncommutative discrete potential kdv lift in this paper we construct a grassmann extension of a yang baxter map which first appeared in the work of kouloukas and papageorgiou j phys a math theor 42 404012 2009 and can be considered as a lift of the discrete potential korteweg de vries dpkdv equation this noncommutative extension satisfies the yang baxter equation and it admits a 3 3 lax matrix moreover we show that it can be squeezed down to a novel system of lattice equations which possesses a lax representation and whose bosonic limit is the dpkdv equation finally we consider commutative analogs of the constructed yang baxter map and its associated quad graph system and we discuss their integrability

on a non abelian hirota miwa equation a generalization of the hirota miwa equation to an abstract non abelian associative algebra is considered this system is integrable in the sense that it arises as compatibility condition for a linear system and has solutions constructed by means of the application of an arbitrary number of darboux transformations these solutions are in general expressed in terms of quasideterminants

integrable lattice equations with vertex and bond variables we present integrable lattice equations on a two dimensional square lattice with coupled vertex and bond variables in some of the models the vertex dynamics is independent of the evolution of the bond variables and one can write the equations as non autonomous yang baxter maps we also present a model in which the vertex and bond variables are fully coupled integrability is tested with algebraic entropy as well as multidimensional consistency