existence of exotic waves for the nonlinear dispersive mkdv equation the nonlinear dispersive mkdv equation u t u 3 x u 2 x x x 0 is proved to be painleve integrable detailed classification of its travelling waves under certain parameter conditions are obtained by the improved qualitative analysis method abundant type of solutions are shown to exist including exotic traveling waves peaked waves compacted waves looped and cusped waves very particular composite waves having two singular points

on certain zeta functions attached to two hilbert modular forms ii the case of automorphic forms on a quaternion algebra as to the meaning of the symbols the reader is referred to the introduction of part i we recall here only that f is a totally real algebraic number field of degree n and w denotes the fourier coefficients of an elliptic modular form 52 z e wq a e2q iaz in part i we investigated d assuming that ex a n q s is an l function of a cm field with an algebraic valued hecke character in the present part ii we treat the case where x a can be obtained as eigenvalues of hecke operators t a on a quaternion algebra b over f thus we begin our study by recalling the theory of automorphic forms on the product o r of r copies of the upper half plane with respect to congruence subgroups of b x here r is the number of archimedean primes of f unramified in b then we introduce in section 2 the notion of arithmeticity or q rationality of such automorphic forms and show that the space of automorphic forms can be spanned by the arithmetic ones our main theorems will be stated in section 3 and proved in section 6 sections 4 and 5 are preliminaries to the proof the central result specialized to the case r 1 asserts that if g is a q rational eigenform such that gj t a x a g for all integral ideals a of f then d s0 for certain integers so are algebraic numbers times 7tdg g where d is an integer determined by the weight of g and is the petersson inner product

combined breathing kink modes in the fpu lattice abstract combined breather kink modes have been observed in several systems here we show that in the fpu lattice small amplitude breathing kink modes are three soliton solutions of an associated mkdv equation since this is integrable the modes can be constructed using the backlund transform as well as finding explicit solutions we consider the stability of the combined mode using variational techniques and illustrate stability for some parameter values and instability for others

discrete conservation laws and the convergence of long time simulations of the mkdv equation pseudospectral collocation methods and finite difference methods have been used for approximating an important family of soliton like solutions of the mkdv equation these solutions present a structural instability which make difficult to approximate their evolution in long time intervals with enough accuracy the standard numerical methods do not guarantee the convergence to the proper solution of the initial value problem and often fail by approaching solutions associated to different initial conditions in this frame the numerical schemes that preserve the discrete invariants related to some conservation laws of this equation produce better results than the methods which only take care of a high consistency order pseudospectral spatial discretization appear as the most robust of the numerical methods but finite difference schemes are useful in order to analyze the rule played by the conservation of the invariants in the convergence

markov chains for exploring posterior distributions several markov chain methods are available for sampling from a posterior distribution two important examples are the gibbs sampler and the metropolis algorithm in addition several strategies are available for constructing hybrid algorithms this paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues on the theoretical side results from the theory of general state space markov chains can be used to obtain convergence rates laws of large numbers and central limit theorems for estimates obtained from markov chain methods these theoretical results can be used to guide the construction of more efficient algorithms for the practical use of markov chain methods standard simulation methodology provides several variance reduction techniques and also give guidance on the choice of sample size and allocation

smooth manifolds for semilinear wave equations on r2 on the existence of almost periodic breathers abstract in this paper we prove several nonexistence and existence results for certain real solutions to semilinear wave equations in one space dimension these solutions represent time almost periodic free vibrations which decay toward a constant equilibrium solution as the spatial variable goes to infinity our nonexistence results are of two kinds we first prove that the requirements of almost periodicity in time and of spatial decay at infinity force such solutions to be trivial whenever their spectrum does not interact with the nonlinearity in a certain sense we also prove non existence results without requiring any spectral conditions but this is at the expense of having to impose more stringent conditions on the nonlinearities such as certain convexity properties in the vicinity of the equilibrium solution finally we prove an existence result for time almost periodic free vibrations whose profiles decay exponentially rapidly toward a constant equilibrium solution to accomplish this we convert the wave equation into a dynamical system in which the original spatial variable plays the role of time we then embed that dynamical system into an appropriate banach space of time almost periodic functions so that the one parameter family of stable and unstable manifolds which we construct carry the solutions that we seek the major difficulty to overcome in our construction is a small divisor problem it is related to the fact that the spectrum of the infinitesimal generator for the linearized flow is a pure point spectrum without gaps we discuss several examples and we also stress some important qualitative differences which distinguish the almost periodic case from the purely periodic one

wobbling kinks in ϕ 4 theory we present a uniform asymptotic expansion of the wobbling kink to any order in the amplitude of the wobbling mode the long range behavior of the radiation is described by matching the asymptotic expansions in the far field and near the core of the kink the complex amplitude of the wobbling mode is shown to obey a simple ordinary differential equation with nonlinear damping we confirm the t ensuremath 1 2 decay law for the amplitude which was previously obtained on the basis of energy considerations

kink soliton characterizing traffic congestion we study traffic congestion by analyzing a one dimensional traffic flow model developing an asymptotic method to investigate the long time behavior near a critical point we derive the modified korteweg de vries mkdv equation as the lowest order model there is an infinite number of kink solitons to the mkdv equation while it has been found by numerical simulations that the kink pattern arising in traffic congestion is uniquely determined irrespective of initial conditions in order to resolve this selection problem we consider higher order corrections to the mkdv equation and find that there is a kink soliton that can deform continuously with the perturbation represented by the addition of these corrections with numerical confirmation we show that this continuously deformable kink soliton characterizes traffic congestion we also discuss the relationship between traffic congestion and the slugging phenomenon in granular flow

on a class of nearly holomorphic automorphic forms with a b 1 e c and 0 n e z the point that attracts our attention is its nonholomorphic nature it is noteworthy that f is holomorphic if k 2 and also such nonholomorphy doesn t appear in the hilbert modular case therefore one may naturally ask whether this is an isolated oddity in our previous papers 10 and 13 we showed that it was not so in fact a similar eisenstein series on sp n q or on su n n of a particular weight can have the form

sampling based approaches to calculating marginal densities abstract stochastic substitution the gibbs sampler and the sampling importance resampling algorithm can be viewed as three alternative sampling or monte carlo based approaches to the calculation of numerical estimates of marginal probability distributions the three approaches will be reviewed compared and contrasted in relation to various joint probability structures frequently encountered in applications in particular the relevance of the approaches to calculating bayesian posterior densities for a variety of structured models will be discussed and illustrated

on some classes of mkdv periodic solutions we obtain exact periodic solutions of the positive and negative modified kortweg de vries mkdv equations we examine the dynamical stability of these solitary wave lattices through direct numerical simulations while the positive mkdv breather lattice solutions are found to be unstable the two soliton lattice solution of the same equation is found to be stable similarly a negative mkdv lattice solution is found to be stable we also touch upon the implications of these results for the kdv equation

breather lattice and its stabilization for the modified korteweg de vries equation we obtain an exact solution for the breather lattice solution of the modified korteweg de vries equation numerical simulation of the breather lattice demonstrates its instability due to the breather breather interaction however such multibreather structures can be stabilized through the concurrent application of ac driving and viscous damping terms

wobbling kinks in φ4 and sine gordon theory when the 4 model admits a kink solution it also admits a wobbling kink which satisfies the boundary conditions of a kink but possesses an internal degree of freedom in this paper we develop a formal perturbation series for the wobbling kink in 4 theory and give the first two terms in the series explicitly then we prove that the formal series actually is asymptotic for a rather long time o k ln 1 e for a certain k finally we construct an exact 3 soliton solution of the sine gordon equation that also has the properties of a wobbling kink for the sine gordon equation the wobbling kink seems to be mildly unstable

a systematical way to find breather lattice solutions to the positive mkdv equation in this paper dependent and independent variable transformations are introduced to solve the positive mkdv equation systematically by using knowledge of elliptic equation and jacobian elliptic functions it is shown that different kinds of solutions can be obtained to the positive mkdv equation including many kinds of breather lattice solutions

the modified kdv equation with variable coefficients exact uni bi variable travelling wave like solutions in this paper the modified korteweg de vries mkdv equation with variable coefficients vc mkdv equation is investigated via two kinds of approaches and symbolic computation on the one hand we firstly reduce the vc mkdv equation to a second order nonlinear nonhomogeneous ode using travelling wave like similarity transformation and then we obtain its many types of exact fractional solutions with one travelling wave like variable by applying some fractional transformations to the obtained nonlinear ode on the other hand we reduce the vc mkdv equation to two nonlinear pdes with variable coefficients using the anti tangent and anti hypertangent function transformations respectively and then we given its many types of exact solutions with two different travelling wave like variables by studying the obtained nonlinear pde with variable coefficients c 2008 elsevier inc all rights reserved

jamming transitions and the modified korteweg de vries equation in a two lane traffic flow abstract the two lattice models are presented to simulate the traffic flow on a two lane highway they are the lattice versions of the hydrodynamic model of traffic the one model a is described by the differential difference equation where time is a continuous variable and space is a discrete variable and the other model b is the difference equation in which both time and space variables are discrete the jamming transitions among the freely moving phase the coexisting phase and the uniform congested phase are studied by using the nonlinear analysis and the computer simulation the modified korteweg de vries mkdv equations are derived from the lattice models near the critical point the traffic jam is described by a kink antikink solution obtained from the mkdv equation it is found that the critical point the coexisting curve and the neutral stability line decrease with increasing the rate of lane changing also the computer simulation is performed for the model b it is shown that the coexisting curves obtained from the mkdv equation are consistent with the simulation result

monte carlo conditioning on a sufficient statistic summary in this paper we derive general formulae suitable for monte carlo computation of conditional expectations of functions of a random vector given a sufficient statistic the problem of direct sampling from the conditional distribution is considered in particular it is shown that this can be done by a simple parameter adjustment of the original statistical model provided the model has a certain pivotal structure a connection with a classical problem regarding fiducial and posterior distributions is pointed out

use of the gibbs sampler to obtain conditional tests with applications a random sample is drawn from a distribution which admits a minimal sufficient statistic for the parameters the gibbs sampler is proposed to generate samples called conditionally sufficient or co sufficient samples from the conditional distribution of the sample given its value of the sufficient statistic the procedure is illustrated for the gamma distribution co sufficient samples may be used to give exact tests of fit for the gamma distribution these are compared for size and power with approximate tests based on the parametric bootstrap

confidence intervals from monte carlo tests abstract monte carlo tests have exact level when the distribution of the test statistic is free of nuisance parameters confidence sets obtained by inverting such tests are also exact but may have a complicated structure the sets reduce to intervals if the sampling can be organized in a special way a sufficient condition is that the test statistic is monotone in the parameter of interest when all random drawings are kept fixed examples given include models from the one parameter exponential class a simple theory quantifying the impact of monte carlo uncertainty is also developed

nonlinear periodic waves solutions of the nonlinear self dual network equations the new classes of periodic solutions of nonlinear self dual network equations describing the breather and soliton lattices expressed in terms of the jacobi elliptic functions have been obtained the dependences of the frequencies on energy have been found numerical simulations of soliton lattice demonstrate their stability in the ideal lattice and the breather lattice instability in the dissipative lattice however the lifetime of such structures in the dissipative lattice can be extended through the application of ac driving terms

zeta functions and eisenstein series on metaplectic groups the main objectives of this paper are 1 to construct an euler product from a hecke eigenform of half integral weight 2 to prove its meromorphic continuation to the whole plane 3 to examine its relationship with the fourier coefficients of the form and 4 to apply these results to the investigations of eisenstein series of half integral weight the automorphic forms are holomorphic ones considered on the metaplectic cover m of g where g n sp n f with a totally real algebraic number field f the first three are generalizations of the corresponding results in the elliptic and hilbert modular cases obtained in 2 and 6 at the same time all four closely parallel similar results in the case of integral weight in 9 and s10 to describe our results let us take for simplicity f to be q in which case the weight k is half an odd positive integer we consider a holomorphic form f of weight k with respect to a principal congruence subgroup of sp n z of level n which is an eigenform in the sense that f it m 2 m f with a complex number 2 m where n is a multiple of 4 and t m is a hecke operator defined for each positive integer m in a natural way as our first main result we shall prove theorem 4 4 that

on certain zeta functions attached to two hilbert modular forms i the case of hecke characters to make our exposition smooth let us first introduce some notational conventions for an algebraic number field k of finite degree we denote by jk the set of all embeddings of k into c and by ik the free z module generated by the elements of jk we then put rik ik0 r and cik ik0 c if p eapau e ik with a e jk and po e z we put xp fl xa pfor 0 x e k this is meaningful for p e cik if xa are all real and positive throughout the paper we denote by d the complex upper half plane and by f a totally real algebraic number field of degree n now the zeta function to be studied in this paper when suitably specialized has the form

on differential operators attached to certain representations of classical groups classification et construction de sous espaces irreductibles de sr integrales contenant les applications polynomiales l operateur differentiel d operateurs differentiels de type arithmetique applications aux series de eisenstein

a counterexample to a claim about stochastic simulations engen lillegard 1997 presented a general method for doing monte carlo simulations conditioned on a sufficient statistic the basic idea was to adjust the parameter values in the corresponding unconditional simulation so that the actual value of the sufficient statistic is obtained and the claim was that if this adjustment is unique then the modified simulation is from the conditional distribution unfortunately the claim is not correct as shown by a counterexample

stochastic simulations conditioned on sufficient statistics summary a general method for doing monte carlo simulations conditioned on sufficient statistics is presented the basic idea is to adjust the parameter values in the corresponding unconditional simulation so that the actual value of the sufficient statistic is obtained in the case of multiple solutions to this problem the method has to be modified even if the corresponding value of the simulated variable is unique for each simulation the methods are illustrated for some simple models in which the conditional distributions are well known as a more complicated example an exact analysis of variance test in the gamma model is performed other examples are a modification of the kolmogorov goodness of fit test into an exact test and minimum variance unbiased estimation of cumulative distributions

solitary wave solutions and kink wave solutions for a generalized kdv mkdv equation abstract bifurcation method of dynamical systems is employed to investigate solitary wave solutions and kink wave solutions of the generalized kdv mkdv equation under some parameter conditions their explicit expressions are obtained

decay of bound states in a sine gordon equation with double well potentials we consider a spatially inhomogeneous sine gordon equation with a double well potential describing long josephson junctions with phase shifts we discuss the interactions of symmetric and antisymmetric bound states in the system using a multiple scale expansion we show that the modes decay algebraically in time due to the energy transfer from the discrete to the continuous spectrum in particular exciting the two modes at the same time yields an increased decay rate an external time periodic drive is shown to sustain symmetric state while it damps the antisymmetric one

the korteweg de vries hierarchy as dynamics of closed curves in the plane the korteweg de vries modified korteweg de vries and harry dym hierarchies of integrable systems are shown to be equivalent to a hierarchy of chiral shape dynamics of closed curves in the plane these purely local dynamics conserve an infinite number of global geometric properties of the curves such as perimeter and enclosed area several techniques used to study these integrable systems are shown to have simple differential geometric interpretations connections with incompressible inviscid fluid flow in two dimensions are suggested

integrable equations arising from motions of plane curves the motion of plane curves in klein geometry is studied it is shown that the kdv harry dym sawada kotera burgers the defocusing mkdv hierarchies the camassa holm and the kaup kupershmidt equation naturally arise from the motions of plane curves in sl 2 sim 2 sa 2 and a 2 geometries these local and nonlocal dynamics conserve global geometric quantities of curves such as perimeter and enclosed area motions of curves in euclidean special linear and similarity geometries corresponding to the traveling wave solutions of the mkdv kdv and burgers equations are discussed