boundedness of automorphic l functions in vertical strips the purpose of this paper is to prove the boundedness in vertical strips of finite width for all the l functions that appear in constant terms of eisenstein series under a certain natural assumption on local normalized intertwining operators as a corollary we prove the boundedness for a number of important l functions among them the symmetic cube and triple product l functions attached to cusp forms on gl 2 as well as several rankin selberg product l functions where the local assumption was alredy proved to be valid in each case under the same assumption the finiteness of poles on all of c for all of these l functions is also proved a necessary step in the proof of boundedness in vertical strips

control of noisy heteroclinic cycles abstract we develop a control strategy for strongly nonlinear dynamical systems possessing heteroclinic cycles connecting saddle points and subject to small random perturbations in the absence of control attracting cycles lead to solutions which exhibit intermittent behavior consisting of quiescent periods interrupted by rapid bursts we show that suitable feedback control can retain solutions near the saddle points and hence reduce the typical bursting rate the model studied and the form of control are motivated by our interest in reducing turbulence production in the boundary layer

an algorithmic introduction to numerical simulation of stochastic differential equations a practical and accessible introduction to numerical methods for stochastic differential equations is given the reader is assumed to be familiar with euler s method for deterministic differential equations and to have at least an intuitive feel for the concept of a random variable however no knowledge of advanced probability theory or stochastic processes is assumed the article is built around 10 matlab programs and the topics covered include stochastic integration the euler maruyama method milstein s method strong and weak convergence linear stability and the stochastic chain rule

cuspidality of symmetric powers with applications the purpose of this paper is to prove that the symmetric fourth power of a cusp form on rm gl 2 whose existence was proved earlier by the first author is cuspidal unless the corresponding automorphic representation is of dihedral tetrahedral or octahedral type as a consequence we prove a number of results toward the ramanujan petersson and sato tate conjectures in particular we establish the bound q sp 1 9 sb v for unramified hecke eigenvalues of cusp forms on rm gl 2 over an arbitrary number field this is the best bound available at present

stabilisation of a relative equilibrium of an underactuated auv on se 3 abstractthis paper presents stabilisation of a relative equilibrium of an underactuated autonomous underwater vehicle auv with three independent control inputs on lie group se 3 since se 3 can describe the evolution of a rigid body without singularities or ambiguities first by using the interconnection and damping assignment ida method through kinetic energy shaping a controller is proposed to stabilise the relative equilibrium in the sense of lyapunov stability in cotangent bundle t se 3 modulo an isotropy subgroup se 2 r when a marine vehicle immerses in an ideal fluid then for an auv submersed in a viscous fluid an asymptotical static control law consisting of three terms is designed to stabilise the relative equilibrium via the ida method which these three control terms are derived from kinetic energy shaping potential energy shaping and damping reassignment respectively the control term derived from kinetic energy shaping preserves stability of this motion modulo translations along

asymptotic trace formula for the hecke operators given integers m n and k we give an explicit formula with an optimal error term with square root cancelation for the petersson trace formula involving the mth and nth fourier coefficients of an orthonormal basis of s k left n right the weight k newforms with fixed square free level n provided that 4 pi sqrt mn k o left k frac 1 3 right moreover we establish an explicit formula with a power saving error term for the trace of the hecke operator mathcal t n on s k left n right averaged over k in a short interval by bounding the second moment of the trace of mathcal t n over a larger interval we show that the trace of mathcal t n is unusually large in the range 4 pi sqrt n k o left n frac 1 6 right as an application for any fixed prime p coprime to n we show that there exists a sequence k n of weights such that the error term of weyl s law for mathcal t p is unusually large and violates the prediction of arithmetic quantum chaos in particular this generalizes the result of gamburd et al j eur math soc 1 1 51 85 1999 theorem 1 4 with an improved exponent

stochastic lie group integrators we present lie group integrators for nonlinear stochastic differential equations with noncommutative vector fields whose solution evolves on a smooth finite dimensional manifold given a lie group action that generates transport along the manifold we pull back the stochastic flow on the manifold to the lie group via the action and subsequently pull back the flow to the corresponding lie algebra via the exponential map we construct an approximation to the stochastic flow in the lie algebra via closed operations and then push back to the lie group and then to the manifold thus ensuring that our approximation lies in the manifold we call such schemes stochastic munthe kaas methods after their deterministic counterparts we also present stochastic lie group integration schemes based on castell gaines methods these involve using an underlying ordinary differential integrator to approximate the flow generated by a truncated stochastic exponential lie series they become stochastic lie group integrator schemes if we use munthe kaas methods as the underlying ordinary differential integrator further we show that some castell gaines methods are uniformly more accurate than the corresponding stochastic taylor schemes lastly we demonstrate our methods by simulating the dynamics of a free rigid body such as a satellite and an autonomous underwater vehicle both perturbed by two independent multiplicative stochastic noise processes

stability analysis of numerical schemes for stochastic differential equations stochastic differential equations sdes represent physical phenomena dominated by stochastic processes as for deterministic ordinary differential equations odes various numerical schemes are proposed for sdes in this paper we study the stability of numerical schemes for scalar sdes with respect to the mean square norm which we call ms stability we will show some figures of the ms stability domain or regions for some numerical schemes and present numerical results which confirm it this notion is an extension of absolute stability in numerical methods for odes

perturbed homoclinic solutions in reversible 1 1 resonance vector fields we consider a smooth reversible vector field in r 4 such that the origin is a fixed point the differential at the origin has two double pure imaginary eigenvalues iq for the critical value 0 of the parameter we show by a normal form analysis that the vector field can be approximated by an integrable field in r 4 for which we know all solutions specially interesting ones are the homoclinics to 0 and homoclinics to periodic solutions depending on the sign of a leading nonlinear coefficient we prove in particular the persistence of these homoclinics for the full vector field

une preuve simple des conjectures de langlands pour gl n sur un corps p adique let f be a finite extension of p for each integer n 1 we construct a bijection from the set f 0 n of isomorphism classes of irreducible degree n representations of the absolute weil group of f onto the set f 0 n of isomorphism classes of smooth irreducible supercuspidal representations of gl n f those bijections preserve epsilon factors for pairs and hence we obtain a proof of the langlands conjectures for gl n over f which is more direct than harris and taylor s our approach is global and analogous to the derivation of local class field theory from global class field theory we start with a result of kottwitz and clozel on the good reduction of some shimura varieties and we use a trick of harris who constructs non galois automorphic induction in certain cases

asymptotic stability for a class of metriplectic systems using the framework of metriplectic systems on rn we will describe a constructive geometric method to add a dissipation term to a hamilton poisson system such that any solution starting in a neighborhood of a nonlinear stable equilibrium converges toward a certain invariant set the dissipation term depends only on the hamiltonian function and the casimir functions

on the ramanujan conjecture and finiteness of poles for certain l functions in this paper we prove a number of general results about automorphic l functions which appear in the constant terms of eisenstein series for an arbitrary quasi split connected reductive algebraic group over a number field these l functions were first introduced by langlands for chevalley groups 20 and through them he was led to make some of his deep conjectures 21 one significance of these l functions is that all the automorphic l functions studied so far are among them there are three general results first we obtain a uniform line of absolute convergence for all of these l functions theorem 5 1 a uniform estimate for the hecke eigenvalues of generic cusp forms on many absolutely simple quasi split connected reductive algebraic groups over number fields corollary 5 4 and an improvement on the best available estimate for the fourier coefficients of maass wave forms corollary 5 5 also follow next we establish a meromorphic continuation and functional equation for each of these l functions theorem 6 1 finally in theorem 6 2 we prove finiteness of poles on the whole complex plane for an important class of these l functions corollaries 6 6 through 6 10 more precisely let g be a quasi split connected reductive algebraic group over a number field f set g g a f where a f is the ring of adeles of f fix a borel subgroup b of g over f and let u be its unipotent radical let p be a maximal f parabolic subgroup of g with p d b in the context of the problems studied here nothing new will be obtained if one drops the maximality condition on p write p mn a levi decomposition n c u and let b u p m and n be the corresponding groups of adelic points for every place v of f let gv g fj similarly we have bv us pv me and nv let x xv be a generic character of u f u cf section 3 then each xv is generic let r ov 7tv be a cusp form on m we shall say 7t is

stability of poisson equilibria and hamiltonian relative equilibria by energy methods we develop a general stability theory for equilibrium points of poisson dynamical systems and relative equilibria of hamiltonian systems with symmetries including several generalisations of the energy casimir and energy momentum methods using a topological generalisation of lyapunov s result that an extremal critical point of a conserved quantity is stable we show that a poisson equilibrium is stable if it is an isolated point in the intersection of a level set of a conserved function with a subset of the phase space that is related to the topology of the symplectic leaf space at that point this criterion is applied to generalise the energy momentum method to hamiltonian systems which are invariant under non compact symmetry groups for which the coadjoint orbit space is not hausdorff we also show that a g stable relative equilibrium satisfies the stronger condition of being a stable where a is a specific group theoretically defined subset of g which contains the momentum isotropy subgroup of the relative equilibrium the results are illustrated by an application to the stability of a rigid body in an ideal irrotational fluid

comment on on the polynomial integrability of the kirchoff equations physica d 241 2012 1417 1420 abstract llibre and valls in physica d 241 2012 1417 1420 proved that if the kirchoff equations have a proper darboux polynomial with its cofactor satisfying some symmetry they have a polynomial first integral in this note we will improve this last result and obtain that if the kirchoff equations have a proper darboux polynomial they always have a polynomial first integral functionally independent of the three known ones our result improves that of llibre and valls in two aspects we drop the symmetric condition and prove that the obtained first integral is functionally independent of the known ones

dynamics of the kirchhoff equations i coincident centers of gravity and buoyancy abstract we study the kirchhoff equations for a rigid body immersed in an incompressible irrotational inviscid fluid in the case that the centers of buoyancy and gravity coincide the resulting dynamical equations form a non canonical hamiltonian system with a six dimensional phase space which may be reduced to a four dimensional two degree of freedom canonical system using the two casimir invariants of motion restricting ourselves to ellipsoidal bodies we identify several completely integrable subcases in the general case we analyze existence linear and nonlinear stability and bifurcations of equilibria corresponding to steady translations and rotations including mixed modes involving simultaneous motion along two body axes some of which we show can be stable by perturbing from the axisymmetric integrable case we show that slightly asymmetric ellipsoids are typically non integrable and we investigate their dynamics with a view to using motions along homo and heteroclinic orbits to execute specific maneuvers in autonomous underwater vehicles

a new form of the circle method and its application to quadratic forms if the coefficients r n satisfy suitable arithmetic conditions the behaviour of f will be determined by an appropriate rational approximation a q to with small values of q usually producing large values of f when lies in an interval a q a q with q small a major arc one hopes to estimate f asymptotically while if the corresponding q is large for the minor arcs one hopes that f will be small at least on average one can distinguish two different types of application of the circle method the first type following the above pattern uses one of a variety of mean value techniques hua s inequality for example to give an upper bound for the minor arc contribution results on waring s problem are generally proved this way for example the second type of application on which we shall be focusing our attention avoids the minor arcs completely and treats f asymptotically for all here one cannot take advantage of the mean value ideas used in the first type of problem the savings come instead from the availability of good estimates for complete exponential sums kloosterman s work 12 on quaternary quadratic forms and the papers of the author 5 and hooley 8 and 9 on cubic forms are of this latter type we should point out that such applications do not include any problems in which f is the generating function for a polynomial of degree 4 or more there are

optimal kinematic control of an autonomous underwater vehicle this note investigates the motion control of an autonomous underwater vehicle auv the auv is modeled as a nonholonomic system as any lateral motion of a conventional slender auv is quickly damped out the problem is formulated as an optimal kinematic control problem on the euclidean group of motions se 3 where the cost function to be minimized is equal to the integral of a quadratic function of the velocity components an application of the maximum principle to this optimal control problem yields the appropriate hamiltonian and the corresponding vector fields give the necessary conditions for optimality for a special case of the cost function the necessary conditions for optimality can be characterized more easily and we proceed to investigate its solutions finally it is shown that a particular set of optimal motions trace helical paths throughout this note we highlight a particular case where the quadratic cost function is weighted in such a way that it equates to the lagrangian kinetic energy of the auv for this case the regular extremal curves are constrained to equate to the auv s components of momentum and the resulting vector fields are the d alembert lagrange equations in hamiltonian form

optimal strong approximation for quadratic forms for a nondegenerate integral quadratic form f x1 xd in d 5 variables we prove an optimal strong approximation theorem let be a fixed compact subset of the affine quadric f x1 xd 1 over the real numbers take a small ball b of radius 0 0 finally assume that an integral vector 1 d mod m is given then we show that there exists an integral solution x x1 xd of f x n such that xi imodm and xn b provided that all the local conditions are satisfied we also show that 4 is the best possible exponent moreover for a nondegenerate integral quadratic form in four variables we prove the same result if n is odd and n r 1m 6 based on our numerical experiments on the diameter of lps ramanujan graphs and the expected square root cancellation in a particular sum that appears in remark 6 8 we conjecture that the theorem holds for any quadratic form in four variables with the optimal exponent 4

best possible rates of distribution of dense lattice orbits in homogeneous spaces this paper establishes upper and lower bounds on the speed of approximation in a wide range of natural diophantine approximation problems the upper and lower bounds coincide in many cases giving rise to optimal results in diophantine approximation which were inaccessible previously our approach proceeds by establishing more generally upper and lower bounds for the rate of distribution of dense orbits of a lattice subgroup in a connected lie or algebraic group g acting on suitable homogeneous spaces g h the upper bound is derived using a quantitative duality principle for homogeneous spaces reducing it to a rate of convergence in the mean ergodic theorem for a family of averaging operators supported on h and acting on g in particular the quality of the upper bound on the rate of distribution we obtain is determined explicitly by the spectrum of h in the automorphic representation on l 2 g we show that the rate is best possible when the representation in question is tempered and show that the latter condition holds in a wide range of examples

homoclinic orbits in reversible systems and their applications in mechanics fluids and optics abstract this survey article reviews the theory and application of homoclinic orbits to equilibria in reversible continuous time dynamical systems where the homoclinic orbit and the equilibrium are both reversible the focus is on even order reversible systems in four or more dimensions local theory generic argument and global existence theories are examined for each qualitatively distinct linearisation several recent results such as coalescence caused by non transversality and the reversible orbit flip bifurcation are covered a number of open problems are highlighted applications are reviewed to systems arising from a variety of disciplines with the aid of numerical methods three examples are presented in detail one of which is infinite dimensional

improved rectangular method on stochastic volterra equations an improved version of rectangular method irm is introduced in this paper to numerically solve the stochastic volterra equation sve we focus on studying the order of error between the numerical and exact solutions which is improved to o h furthermore an explicit form of the irm scheme is introduced and its convergence is established a numerical example has also been presented to show the feasibility of the methods

strong convergence of euler type methods for nonlinear stochastic differential equations traditional finite time convergence theory for numerical methods applied to stochastic differential equations sdes requires a global lipschitz assumption on the drift and diffusion coefficients in practice many important sde models satisfy only a local lipschitz property and since brownian paths can make arbitrarily large excursions the global lipschitz based theory is not directly relevant in this work we prove strong convergence results under less restrictive conditions first we give a convergence result for euler maruyama requiring only that the sde is locally lipschitz and that the pth moments of the exact and numerical solution are bounded for some p 2 as an application of this general theory we show that an implicit variant of euler maruyama converges if the diffusion coefficient is globally lipschitz but the drift coefficient satisfies only a one sided lipschitz condition this is achieved by showing that the implicit method has bounded moments and may be viewed as an euler maruyama approximation to a perturbed sde of the same form second we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial

suppression of bursting abstract we investigate the possibility of using a single small amplitude control input and feedback to stabilize equilibrium sets in a class of highly nonlinear o 2 symmetric dynamical systems possessing structurally stable heteroclinic cycles the leading order behavior near the equilibria is bilinear and homogeneous in the state variables while nonlinearities representing the symmetry breaking effect of the controller are crucial after a series of simplifying transformations we use ideas from optimal control theory to construct a stabilizing controller this study is motivated by the desire to stabilize the burst sweep cycle in low dimensional models of a turbulent boundary layer in the last two sections we apply the techniques to the 10 dimensional system of aubry et al 1988 kw bilinear systems dynamical systems heteroclinic cycles normal forms optimal control symmetry

on the polynomial integrability of kirchoff s equations abstract we prove that kirchoff s equations either are completely integrable or have at most four functionally independent polynomial first integrals

equilibria stability and hamiltonian hopf bifurcation of a gyrostat in an incompressible ideal fluid abstract for a gyrostat in a incompressible ideal fluid by writing kirchhoff s equations as a lie poisson system and using a non canonical hamiltonian formulation we provide the expressions of the equilibria when the gyrostatic momentum is constant with the form l 0 0 l and present necessary and sufficient conditions for the stability of some of them via the energy casimir method and the study of the linearized equations of the motion finally using a recent geometric method introduced by hanssmann and van der meer we give a sufficient condition for the existence of a non degenerate hamiltonian hopf bifurcation at those equilibria when the gyrostat is symmetric

a perspective on the numerical treatment of volterra equations abstract we discuss the properties and numerical treatment of various types of volterra and abel volterra integral and integro differential equations