the hamiltonian structure for dynamic free boundary problems hamiltonian structures for 2 or 3 dimensional incompressible flows with a free boundary are determined which generalize a previous structure of zakharov for irrotational flow our poisson bracket is determined using the method of arnold namely reduction from canonical variables in the lagrangian material description using this bracket the hamiltonian form for the equations of a liquid drop with a free boundary having surface tension is demonstrated the structure of the bracket in terms of a reduced cotangent bundle of a principal bundle is explained in the case of two dimensional flows the vorticity bracket is determined and the generalized enstrophy is shown to be a casimir function this investigation also clears up some confusion in the literature concerning the vorticity bracket even for fixed boundary flows

the hamiltonian structure of general relativistic perfect fluids we show that the evolution equations for a perfect fluid coupled to general relativity in a general lapse and shift are hamiltonian relative to a certain poisson structure for the fluid variables a lie poisson structure associated to the dual of a semi direct product lie algebra is used while the bracket for the gravitational variables has the usual canonical symplectic structure the evolution is governed by a hamiltonian which is equivalent to that obtained from a canonical analysis the relationship of our hamiltonian structure with other approaches in the literature such as clebsch potentials lagrangian to eulerian transformations and its use in clarifying linearization stability are discussed

semidirect products and reduction in mechanics this paper shows how to reduce a hamiltonian system on the cotangent bundle of a lie group to a hamiltonian system in the dual of the lie algebra of a semidirect product the procedure simplifies unifies and extends work of greene guillemin holm holmes kupershmidt marsden morrison ratiu sternberg and others the heavy top compressible fluids magnetohydrodynamics elasticity the maxwell vlasov equations and multifluid plasmas are presented as examples starting with lagrangian variables our method explains in a direct way why semidirect products occur so frequently in examples it also provides a framework for the systematic introduction of clebsch or canonical variables

a note on colored homfly polynomials for hyperbolic knots from wzw models using the correspondence between chern simons theories and wess zumino witten models we present the necessary tools to calculate colored homfly polynomials for hyperbolic knots for two bridge hyperbolic knots we derive the colored homfly invariants in terms of crossing matrices of the underlying wess zumino witten model our analysis extends previous works by incorporating non trivial multiplicities for the primaries appearing in the crossing matrices so as to describe colorings of homfly invariants beyond the totally symmetric or anti symmetric representations of su n the crossing matrices directly relate to 6j symbols of the quantum group mathcal u q su n we present powerful methods to calculate such quantum 6j symbols for general n this allows us to determine previously unknown colored homfly polynomials for two bridge hyperbolic knots we give explicitly the homfly polynomials colored by the representation 2 1 for two bridge hyperbolic knots with up to eight crossings yet the scope of application of our techniques goes beyond knot theory e g our findings can be used to study correlators in wess zumino witten conformal field theories or in the limit to classical groups to determine color factors for yang mills amplitudes

optimal error estimate for energy preserving splitting schemes for maxwell s equations abstract two efficient splitting schemes are developed for 3d maxwell s equations the schemes are energy preserving and unconditionally stable while being implemented explicitly rigorous optimal error estimates are established for the proposed schemes and especially the constant in the error estimates is only o t numerical results confirm the theoretical analysis and numerical comparison with some existing methods shows the good performance of the present schemes

large n duality lagrangian cycles and algebraic knots we consider knot invariants in the context of large n transitions of topological strings in particular we consider aspects of lagrangian cycles associated to knots in the conifold geometry we show how these can be explicitly constructed in the case of algebraic knots we use this explicit construction to explain a recent conjecture relating study of stable pairs on algebraic curves with homfly polynomials furthermore for torus knots using the explicit construction of the lagrangian cycle we also give a direct a model computation and recover the homfly polynomial for this case

symplectic and multisymplectic numerical methods for maxwell s equations in this paper we compare the behaviour of one symplectic and three multisymplectic methods for maxwell s equations in a simple medium this is a system of pdes with symplectic and multisymplectic structures we give a theoretical discussion of how some numerical methods preserve the discrete versions of the local and global conservation laws and verify this behaviour in numerical experiments we also show that these numerical methods preserve the divergence furthermore we extend the discussion on dispersion for multi symplectic methods applied to pdes with one spatial dimension to include anisotropy when applying multi symplectic methods to maxwell s equations in two spatial dimensions lastly we demonstrate how varying the courant friedrichs lewy cfl number can cause the multi symplectic methods in our comparison to behave differently which can be explained by the study of backward error analysis for pdes

the splitting finite difference time domain methods for maxwell s equations in two dimensions in this paper we consider splitting methods for maxwell s equations in two dimensions a new kind of splitting finite difference time domain methods on a staggered grid is developed the corresponding schemes consist of only two stages for each time step which are very simple in computation the rigorous analysis of the schemes is given by the energy method it is proved that the scheme is unconditionally stable and convergent for the problems with perfectly conducting boundary conditions numerical dispersion analysis and numerical experiments are presented to show the efficient performance of the proposed methods furthermore the methods are also applied to solve a scattering problem successfully

energy conserved splitting finite difference time domain methods for maxwell s equations in three dimensions in this paper we develop and analyze efficient energy conserved splitting finite difference time domain fdtd schemes for solving three dimensional maxwell s equations in electromagnetic computations all proposed energy conserved splitting finite difference time domain ec s fdtd algorithms are strictly proved to be energy conserved and unconditionally stable and they can be computed efficiently rigorous convergence results are obtained for the schemes the ec s fdtdii schemes are proved to have second order in both time step and spatial steps while the ec s fdtdi schemes have second order in spatial steps and first order in time step the error estimates are optimal and especially the constant in the error estimates is proved to be only o t numerical experiments confirm the theoretical analysis results

coadjoint orbits vortices and clebsch variables for incompressible fluids this paper is a study of incompressible fluids especially their clebsch variables and vortices using symplectic geometry and the lie poisson structure on the dual of a lie algebra following ideas of arnold and others it is shown that euler s equations are lie poisson equations associated to the group of volume preserving diffeomorphisms the dual of the lie algebra is seen to be the space of vortices and kelvin s circulation theorem is interpreted as preservation of coadjoint orbits in this context clebsch variables can be understood as momentum maps the motion of n point vortices is shown to be identifiable with the dynamics on a special coadjoint orbit and the standard canonical variables for them are a special kind of clebsch variables point vortices with cores vortex patches and vortex filaments can be understood in a similar way this leads to an explanation of the geometry behind the hald beale majda convergence theorems for vorticity algorithms symplectic structures on the coadjoint orbits of a vortex patch and filament are computed and shown to be closely related to those commonly used for the kdv and the schrodinger equations respectively

splitting k symplectic methods for non canonical separable hamiltonian problems non canonical hamiltonian systems have k symplectic structures which are preserved by k symplectic numerical integrators there is no universal method to construct k symplectic integrators for arbitrary non canonical hamiltonian systems however in many cases of interest by using splitting we can construct explicit k symplectic methods for separable non canonical systems in this paper we identify situations where splitting k symplectic methods can be constructed comparative numerical experiments in three non canonical hamiltonian problems show that symmetric non symmetric splitting k symplectic methods applied to the non canonical systems are more efficient than the same order gauss methods non symmetric symplectic methods applied to the corresponding canonicalized systems for the non canonical lotka volterra model the splitting algorithms behave better in efficiency and energy conservation than the k symplectic method we construct via generating function technique in our numerical experiments the favorable energy conservation property of the splitting k symplectic methods is apparent

a rigorous derivation of the hamiltonian structure for the nonlinear schrodinger equation abstract we consider the cubic nonlinear schrodinger equation nls in any spatial dimension which is a well known example of an infinite dimensional hamiltonian system inspired by the knowledge that the nls is an effective equation for a system of interacting bosons as the particle number tends to infinity we provide a derivation of the hamiltonian structure which is comprised of both a hamiltonian functional and a weak symplectic structure for the nonlinear schrodinger equation from quantum many body systems our geometric constructions are based on a quantized version of the poisson structure introduced by marsden morrison and weinstein 24 for a system describing the evolution of finitely many indistinguishable classical particles

curve counting via stable pairs in the derived category for a nonsingular projective 3 fold x we define integer invariants virtually enumerating pairs c d where c x is an embedded curve and d c is a divisor a virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of x the resulting invariants are conjecturally equivalent after universal transformations to both the gromov witten and dt theories of x for calabi yau 3 folds the latter equivalence should be viewed as a wall crossing formula in the derived category

all loop topological string amplitudes from chern simons theory we demonstrate the equivalence of all loop closed topological string amplitudes on toric local calabi yau threefolds with computations of certain knot invariants for chern simons theory we use this equivalence to compute the topological string amplitudes in certain cases to very high degree and to all genera in particular we explicitly compute the topological string amplitudes for open image in new window 2 up to degree 12 and open image in new window 1 open image in new window 1 up to total degree 10 to all genera this also leads to certain novel large n dualities in the context of ordinary superstrings involving duals of type ii superstrings on local calabi yau three folds without any fluxes

new multisymplectic self adjoint scheme and its composition scheme for the time domain maxwell s equations in this paper we investigate euler box scheme for bridges multisymplectic form of maxwell s equations a new multisymplectic scheme is derived for maxwell s equations we prove that it is also a self adjoint scheme in time direction the multisymplecticity of composition schemes based on the new scheme is also discussed two numerical examples are proposed to indicate that the derived multisymplectic schemes are effective when used to integrate the 2 1 dimensional maxwell s equations

torus knots and the rational daha author s gorsky e oblomkov a rasmussen j shende v abstract 2014 we conjecturally extract the triply graded khovanov rozansky homology of the m n torus knot from the unique finite dimensional simple representation of the rational daha of type a rank n 1 and central character m n the conjectural differentials of gukov dunfield and the third author receive an explicit algebraic expression in this picture yielding a prescription for the doubly graded khovanov rozansky homologies we match our conjecture to previous conjectures of the first author relating knot homology to q t catalan numbers and to previous conjectures of the last three authors relating knot homology to hilbert schemes on singular curves

casimir invariants and the jacobi identity in dirac s theory of constrained hamiltonian systems we consider constrained hamiltonian systems in the framework of dirac s theory we show that the jacobi identity results from imposing that the constraints are casimir invariants regardless of whether the matrix of poisson brackets between constraints is invertible or not we point out that the proof we provide ensures the validity of the jacobi identity everywhere in the phase space and not just on the surface defined by the constraints two examples are considered a finite dimensional system with an odd number of constraints and the vlasov poisson reduction from vlasov maxwell equations

the spatial fourth order energy conserved s fdtd scheme for maxwell s equations abstract in this paper we develop a new spatial fourth order energy conserved splitting finite difference time domain method for maxwell s equations based on the staggered grids the splitting technique is applied to lead to a three stage energy conserved splitting scheme at each stage using the spatial fourth order difference operators on the strict interior nodes by a linear combination of two central differences one with a spatial step and the other with three spatial steps we first propose the spatial high order near boundary differences on the near boundary nodes which ensure the scheme to preserve energy conservations and to have fourth order accuracy in space step the proposed scheme has the important properties energy conserved unconditionally stable non dissipative high order accurate and computationally efficient we first prove that the scheme satisfies energy conversations and is in unconditional stability we then prove the optimal error estimates of fourth order in spatial step and second order in time step for the electric and magnetic fields and obtain the convergence and error estimate of divergence free as well numerical dispersion analysis and numerical experiments are presented to confirm our theoretical results

symplectic conifold transitions we introduce a symplectic surgery in six dimensions which collapses lagrangian three spheres and replaces them by symplectic two spheres under mirror symmetry it corresponds to an operation on complex 3 folds studied by clemens friedman and tian we describe several examples which show that there are either many more calabi yau manifolds e g rigid ones than previously thought or there exist symplectic calabi yaus non kahler symplectic 6 folds with c1 0 the analogous surgery in four dimensions with a generalisation to ade trees of lagrangians implies that the canonical class of a minimal complex surface contains symplectic forms if and only if it has positive square

communication complexity method for measuring nondeterminism in finite automata while deterministic finite automata seem to be well understood surprisingly many important problems concerning nondeterministic finite automata nfa s remain open one such problem area is the study of different measures of nondeterminism in finite automata and the estimation of the sizes of minimal nondeterministic finite automata in this paper the concept of communication complexity is applied in order to achieve progress in this problem area the main results are as follows 1 deterministic communication complexity provides lower bounds on the size of nfa s with bounded unambiguity applying this fact the proofs of several results about nfa s with limited ambiguity can be simplified and presented in a uniform way 2 there is a family of languages konk2 with an exponential size gap between nfa s with polynomial leaf number ambiguity and nfa s with ambiguity k this partially provides an answer to the open problem posed by b ravikumar and o ibarra 1989 siam j comput 18 1263 1282 and h leung 1998 siam j comput 27 1073 1082

the hamiltonian structure of the maxwell vlasov equations morrison 25 has observed that the maxwell vlasov and poisson vlasov equations for a collisionless plasma can be written in hamiltonian form relative to a certain poisson bracket we derive another poisson structure for these equations by using general methods of symplectic geometry the main ingredients in our construction are the symplectic structure on the co adjoint orbits for the group of canonical transformations and the symplectic structure for the phase space of the electromagnetic field regarded as a gauge theory our poisson bracket satisfies the jacobi identity whereas morrison s does not 37 our construction also shows where canonical variables can be found and can be applied to the yang mills vlasov equations and to electromagnetic fluid dynamics

the hamiltonian structure of classical chromohydrodynamics noncanonical hamiltonian structures are presented both for yang mills vlasov plasmas and for ideal fluids interacting with yang mills fields the hamiltonian structure for the yang mills vlasov system passes over to that for the yang mills fluid in the cold plasma limit the resulting hamiltonian structure is shown to correspond to a lie algebra

the differentiable structure of three remarkable diffeomorphism groups the goal of this paper is to find the lie groups for three well known lie algebras the globally hamiltonian vector fields the infinitesimal quantomorphisms and the homogeneous real valued functions of degree one on the cotangent bundle minus the zero section in all our considerations the underlying manifold is assumed to be compact without boundary before explaining and motivating our results a very brief review of the lie group structure of diffeomorphism groups is in order 9 10 18 if m n are compact boundaryless finite dimensional manifolds rn tn n the tangent bundle of n an hs map s dimm 2 from m to n has by definition all derivatives of order dim m 2 then the space ip m n of all such maps is a hilbert manifold whose tangent space at every point is the hilbert space t i h m n gelid m tn i n g f if e m is a vector bundle the same construction works for all hs sections hs e of e s dim m 2 if n ir hs m n will be denoted by c m ir let i m denote the diffeomorphisms of m of sobolev class h s l i e r t l m if and only if t is bijective and t t l m m are of class h 1 s l m is a topological group and since it is open in hs i m m it is also a hilbert manifold right multiplication r m i m r o t is c for each t y m and if t e s k l m left multiplication l t m i m l t o is c k the inversion map t t 1 in t m is only continuous the tangent maps of r and l at e the identity of a m are given by tr tel x t ox where x yys i m hs i tm the set of all h l vector fields on m the tangent space at e tr coincides with y m which is a sobolev space the bracket x y of x y f s i m is however only of class h one derivative is lost the usual bracket of vector fields is the lie algebra bracket of 5f x m i e if j

numerical analysis of a multi symplectic scheme for the time domain maxwell s equations in this paper we analyze qualitative properties of the first multi symplectic scheme for two dimensional maxwell s equations we prove that the scheme is unconditionally stable and convergent non dissipative and divergence free the numerical dispersion relation of the scheme is shown to converge to the exact dispersion relation of the maxwell equations we also present some numerical results to confirm our theoretical results

the topological vertex we construct a cubic field theory which provides all genus amplitudes of the topological a model for all non compact toric calabi yau threefolds the topology of a given feynman diagram encodes the topology of a fixed calabi yau with schwinger parameters playing the role of kahler classes of the threefold we interpret this result as an operatorial computation of the amplitudes in the b model mirror which is the quantum kodaira spencer theory the only degree of freedom of this theory is an unconventional chiral scalar on a riemann surface in this setup we identify the b branes on the mirror riemann surface as fermions related to the chiral boson by bosonization

the maxwell vlasov equations in euler poincare form low s well known action principle for the maxwell vlasov equations of ideal plasma dynamics was originally expressed in terms of a mixture of eulerian and lagrangian variables by imposing suitable constraints on the variations and analyzing invariance properties of the lagrangian as one does for the euler equations for the rigid body and ideal fluids we first transform this action principle into purely eulerian variables hamilton s principle for the eulerian description of low s action principle then casts the maxwell vlasov equations into euler poincare form for right invariant motion on the diffeomorphism group of position velocity phase space openface r 6 legendre transforming the eulerian form of low s action principle produces the hamiltonian formulation of these equations in the eulerian description since it arises from euler poincare equations this hamiltonian formulation can be written in terms of a poisson structure that contains the lie poisson bracket on the dual of a semidirect product lie algebra because of degeneracies in the lagrangian the legendre transform is dealt with using the dirac theory of constraints another maxwell vlasov poisson structure is known whose ingredients are the lie poisson bracket on the dual of the lie algebra of symplectomorphisms of phase space and the born infeld brackets for the maxwell field we discuss the relationship between these two hamiltonian formulations we also discuss the general kelvin noether theorem for euler poincare equations and its meaning in the plasma context

the diameter of the symplectomorphism group is infinite shnirelman has proved that the group of volume preversing diffeomorphisms of the cube in r 3 has finite diameter and has announced the result that this is false for the square despite the fact that shnirelman formulated his theorem only for the 3 dimensional cube his proof can be modified for the case of the group of volume preserving diffeomorphisms of any compact simply connected riemannian manifold of dimension 2 it turns out that the situation with the group of symplectic diffeomorphisms is completely different we prove that the diameter of the symplectomorphism group of any compact exact symplectic manifold necessarily with boundary is infinite

localization and gluing of topological amplitudes we develop a gluing algorithm for gromov witten invariants of toric calabi yau threefolds based on localization and gluing graphs the main building block of this algorithm is a generating function of cubic hodge integrals of special form we conjecture a precise relation between this generating function and the topological vertex at fractional framing

the hamiltonian structure of nonlinear elasticity the material and convective representations of solids rods and plates it is our belief that a thorough understanding of the mathematical underpinnings of elasticity is crucial to its analytical and numerical implementation for example in the analysis of rotating structures the coupling of the equations for geometrically inexact models obtained by linearization or other approximations with those for rotating rigid bodies can easily lead to misleading artificial softening effects that can significantly alter numerical results see simo and vuquoc 1986c especially equations 3 and 5 in this paper we consider fully nonlinear geometrically exact models for rods plates and shells which take into account shear and torsion as well as the usual bending effects in traditional rod and plate models these models can be obtained either from the three dimensional theory by a systematic use of projection methods see e g antman 1972 and naghdi 1972 or by a direct approach within the context of cosserat continuum remarkably the two approaches lead to essentially the same form of the governing field equations in the present context we have chosen as a model problem a particular rod model which may be regarded as an extension of the classical kirchhoff love model see love 1944 to include shear deformations as in

hamiltonian splitting for the vlasov maxwell equations a new splitting is proposed for solving the vlasov maxwell system this splitting is based on a decomposition of the hamiltonian of the vlasov maxwell system and allows for the construction of arbitrary high order methods by composition independent of the specific deterministic method used for the discretization of the phase space moreover we show that for a spectral method in space this scheme satisfies poisson s equation without explicitly solving it finally we present some examples in the context of the time evolution of an electromagnetic plasma instability which emphasizes the excellent behavior of the new splitting compared to methods from the literature

splitting multisymplectic integrators for maxwell s equations in the paper we describe a novel kind of multisymplectic method for three dimensional 3 d maxwell s equations splitting the 3 d maxwell s equations into three local one dimensional lod equations then applying a pair of symplectic runge kutta methods to discretize each resulting lod equation it leads to splitting multisymplectic integrators we say this kind of schemes to be lod multisymplectic scheme lod ms the discrete conservation laws convergence dispersive relation dissipation and stability are investigated for the schemes theoretical analysis shows that the schemes are unconditionally stable non dissipative and of first order accuracy in time and second order accuracy in space as a reduction we also consider the application of lod ms to 2 d maxwell s equations numerical experiments match the theoretical results well they illustrate that lod ms is not only efficient and simple in coding but also has almost all the nature of multisymplectic integrators

a proof of a conjecture of marino vafa on hodge integrals where i c1 li is the first chern class of li and j cj e is the j th chern class of the hodge bundle hodge integrals arise in the calculations of gromov witten invariants by localization techniques 14 7 the explicit evaluation of hodge integrals is a difficult problem the hodge integrals involving only classes can be computed recursively by witten s conjecture 26 proven by kontsevich 13 algorithms of computing hodge integrals are described in 2 in 23 m marino and c vafa obtained a closed formula for a generating function of certain open gromov witten invariants some of which has been reduced to hodge integrals by localization techniques which are not fully clarified mathematically this leads to a conjectural formula of hodge integrals to state this formula we introduce some notation following 28 let

cylindrically invariant flows in relativistic fluid dynamics abstract we show that partially cylindrically symmetric solutions of the motion equations of special relativistic adiabatic ideal fluid dynamics have the same noncanonical hamiltonian structure as the original motion equations

hyperkahler structures and infinite dimensional grassmannians abstract in this paper we describe an example of a hyperkahler quotient of a banach manifold by a banach lie group although the initial manifold is not diffeomorphic to a hilbert manifold not even to a manifold modelled on a reflexive banach space the quotient space obtained is a hilbert manifold which can be furthermore identified either with the cotangent space of a connected component gr res j j z of the restricted grassmannian or with a natural complexification of this connected component thus proving that these two manifolds are isomorphic hyperkahler manifolds moreover kahler potentials associated with the natural complex structure of the cotangent space of gr res j and with the natural complex structure of the complexification of gr res j are computed using kostant souriau s theory of prequantization

energy dissipation splitting finite difference time domain method for maxwell equations with perfectly matched layers abstract in this paper we develop a novel kind of energy dissipation splitting finite difference time domain scheme for solving two dimensional maxwell equations with perfectly matched layers the discrete energy dissipation law convergence dispersion relation and stability are investigated for the scheme theoretical analysis shows that the scheme is unconditionally stable and of second order accuracy both in time and space numerical experiments confirm the theoretical results

reduction theory and the lagrange routh equations reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of euler jacobi lagrange hamilton routh poincare and others the modern vision of mechanics includes besides the traditional mechanics of particles and rigid bodies field theories such as electromagnetism fluid mechanics plasma physics solid mechanics as well as quantum mechanics and relativistic theories including gravity symmetries in these theories vary from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas to subtle symmetries underlying integrable systems reduction theory concerns the removal of symmetries and their associated conservation laws variational principles along with symplectic and poisson geometry provide fundamental tools for this endeavor reduction theory has been extremely useful in a wide variety of areas from a deeper understanding of many physical theories including new variational and poisson structures to stability theory integrable systems as well as geometric phases this paper surveys progress in selected topics in reduction theory especially those of the last few decades as well as presenting new results on non abelian routh reduction we develop the geometry of the associated lagrange routh equations in some detail the paper puts the new results in the general context of reduction theory and discusses some future directions

stable pairs and the homfly polynomial given a planar curve singularity we prove a conjecture of oblomkov shende relating the geometry of its hilbert scheme of points to the homfly polynomial of the associated algebraic link more generally we prove an extension of this conjecture due to diaconescu hua soibelman relating stable pair invariants on the conifold to the colored homfly polynomial of the algebraic link our proof uses wall crossing techniques to prove a blowup identity on the algebro geometric side we prove a matching identity for the colored homfly polynomials of a link using skein theoretic techniques

the alexander polynomial of a plane curve singularity and the ring of functions on it we prove two formulae that express the alexander polynomial delta sp c of several variables of a plane curve singularity c in terms of the ring mathscr o sb c of germs of analytic functions on the curve one of them expresses delta sp c in terms of dimensions of some factors corresponding to a multi indexed filtration on the ring mathscr o sb c the other one gives the coefficients of the alexander polynomial delta sp c as euler characteristics of some explicitly described spaces complements to arrangements of projective hyperplanes

a partial differential equation with infinitely many periodic orbits chaotic oscillations of a forced beam this paper delineates a class of time periodically perturved evolution equations in a banach space whose associated poincar e map contains a smale horseshoe this implies that such systems possess periodic orbits with arbitrarily high period the method uses techniques originally due to melnikov and applies to systems of the form x f0 x f1 x t where x f0 x is hamiltonian and has a homoclinic orbit we give an example from structural mechanics sinusoidally forced vibrations of a buckled beam

non canonical poisson bracket for nonlinear elasticity with extensions to viscoelasticity the authors introduce a generalized bracket which is capable of generating the dynamical equations governing the flow of both elastic and viscoelastic media this generalized bracket is divided into two parts a noncanonical poisson bracket and a new dissipation bracket the non canonical poisson bracket is the eulerian equivalent of the canonical lagrangian poisson bracket corresponding to an ideal non dissipative continuum it is derived for a nonlinear elastic medium and then it is used to obtain the eulerian equations of motion in nonlinear elasticity valid for large deformations it is shown that the proposed non canonical bracket naturally leads to a materially objective relation involving the upper convected time derivative of a strain tensor as suggested by oldroyd 40 years ago the dissipation bracket for linear irreversible thermodynamics is next proposed in a general phenomenological circumstance so that the dissipation processes occurring in real systems viscous and relaxation phenomena can be incorporated into the hamiltonian formalism this bracket diverges from previously proposed dissipation brackets in that it uses the same generating functional i e the hamiltonian as the poisson bracket rather than an entropy functional or a dissipative potential it is shown that in combination with the choice of an appropriate hamiltonian functional the generalized bracket proposed here can generate the governing equations for many viscoelastic media including the voigt solid and the maxwell viscoelastic fluid

the hamiltonian structure of the modified and fifth order korteweg de vries equation dirac s theory of constraints is used to derive the hamiltonian structure of the modified and fifth order korteweg de vries equations a transformation to physical variables is performed on the canonical structure and is shown to be equivalent to the dirac bracket

energy conserved splitting spectral methods for two dimensional maxwell s equations in this paper the energy conserved splitting legendre galerkin method and energy conserved splitting legendre collocation method for maxwell s equations in two dimensions are proposed the schemes are energy conserved unconditionally stable and can be implemented efficiently the both methods are of second order convergence in time the high order error estimates in space for the two methods are also obtained for the ec s lc method an identity of the discrete divergence of electric fields is also obtained numerical examples are provided to confirm the theoretical analysis

new solutions of the jacobi equations for three dimensional poisson structures a systematic investigation of the skew symmetric solutions of the three dimensional jacobi equations is presented as a result three disjoint and complementary new families of solutions are characterized such families are very general thus unifying many different and well known poisson structures seemingly unrelated which now appear embraced as particular cases of a more general solution this unification is not only conceptual but allows the development of algorithms for the explicit determination of important properties such as the symplectic structure the casimir invariants and the darboux canonical form which are known only for a limited sample of poisson structures these common procedures are thus simultaneously valid for all the particular cases which can now be analyzed in a unified and more economic framework instead of using a case by case approach in addition the methods developed are valid globally in phase space thus ameliorating the usual scope of darboux reduction which is only of

knots bps states and algebraic curves we analyze relations between bps degeneracies related to labastida marino ooguri vafa lmov invariants and algebraic curves associated to knots we introduce a new class of such curves which we call extremal a polynomials discuss their special properties and determine exact and asymptotic formulas for the corresponding extremal bps degeneracies these formulas lead to nontrivial integrality statements in number theory as well as to an improved integrality conjecture which is stronger than the known m theory integrality predictions furthermore we determine the bps degeneracies encoded in augmentation polynomials and show their consistency with known colored homfly polynomials finally we consider refined bps degeneracies for knots determine them from the knowledge of super a polynomials and verify their integrality we illustrate our results with twist knots torus knots and various other knots with up to 10 crossings

on the power of randomized multicounter machines one way two counter machines represent a universal model of computation here we consider the polynomial time classes of multicounter machines with a constant number of reversals and separate the computational power of nondeterminism randomization and determinism for instance we show that polynomial time one way multicounter machines with error probability tending to zero with growing input length can recognize languages that cannot be accepted by polynomial time nondeterministic two way multicounter machines with a bounded number of reversals a similar result holds for the comparison of determinism and one sided error randomization and of determinism and las vegas randomization

higher order symmetries of the compressible one dimensional isentropic fluid equations the equations of compressible one dimensional isentropic flow are shown to have infinitely many higher order symmetries by means of recursion operators furthermore a new first order conserved functional is found to be associated with a third order symmetry via noether s theorem

exponential integrators for stochastic maxwell s equations driven by ito noise abstract this article presents explicit exponential integrators for stochastic maxwell s equations driven by both multiplicative and additive noises by utilizing the regularity estimate of the mild solution we first prove that the strong order of the numerical approximation is 1 2 for general multiplicative noise combining a proper decomposition with the stochastic fubini s theorem the strong order of the proposed scheme is shown to be 1 for additive noise moreover for linear stochastic maxwell s equation with additive noise the proposed time integrator is shown to preserve exactly the symplectic structure the evolution of the energy as well as the evolution of the divergence in the sense of expectation several numerical experiments are presented in order to verify our theoretical findings

lin constraints clebsch potentials and variational principles the poisson bracket formulation of fluid plasma and rigid body type systems has undergone considerable recent development using techniques of symmetry group reduction the relationship between this approach and that using lin constraints and clebsch potentials is established the connection is made in the setting of abstract clebsch variables as well as that of variational principles on reduced spaces variational principles for both the clebsch and reduced form such as fluids in spatial representation are derived from the standard variational principle of hamilton in material lagrangian representation using reduction theory

a formula of two partition hodge integrals jmg n where tpi ci l is the first chern class of l and xj cj e is the j th chern class of the hodge bundle the study of hodge integrals is an important part of intersection theory on mg n hodge integrals also naturally arise when one computes gromov witten invariants by localization techniques for example the following generating series of hodge integrals arises when one computes local invariants of a toric fano surface in a calabi yau 3 fold by virtual localization 33

extended kinetic theory the domain of applicability of one point kinetic theory is enlarged by admitting two point distribution functions or some of their moments as extra state variables kinetic equations governing the time evolution in the extended state space are introduced as particular realizations of the nonlinear onsager casimir equation this abstract equation collects the common structure of equations arising in the analysis of compatibility of two levels of description e g kinetic theory and equilibrium thermodynamics or kinetic theory and hydrodynamics the structure consists of a poisson bracket and two potentials the poisson bracket expresses kinematics the first potential called the thermodynamic potential generates the time evolution and links the theory with thermodynamics the second potential called the dissipative potential introduces dissipation into the time evolution that is compatible with the kinematics and thermodynamics to find a particular realization of the nonlinear onsager casimir equati

knot invariants from topological recursion on augmentation varieties using the duality between wilson loop expectation values of su n chern simons theory on s 3 and topological open string amplitudes on the local mirror of the resolved conifold we study knots on s 3 and their invariants encoded in colored homfly polynomials by means of topological recursion in the context of the local mirror calabi yau threefold of the resolved conifold we generalize the topological recursion of the remodelled b model in order to study branes beyond the class of toric harvey lawson special lagrangians as required for analyzing non trivial knots on s 3 the basic ingredients for the proposed recursion are the spectral curve given by the augmentation variety of the knot and the calibrated annulus kernel encoding the topological annulus amplitudes associated to the knot we present an explicit construction of the calibrated annulus kernel for torus knots and demonstrate the validity of the topological recursion we further argue that if an explicit form of the calibrated annulus kernel is provided for any other knot the proposed topological recursion should still be applicable we study the implications of our proposal for knot theory which exhibit interesting consequences for colored homfly polynomials of mutant knots

legendrian knots and constructible sheaves we study the unwrapped fukaya category of lagrangian branes ending on a legendrian knot our knots live at contact infinity in the cotangent bundle of a surface the fukaya category of which is equivalent to the category of constructible sheaves on the surface itself consequently our category can be described as constructible sheaves with singular support controlled by the front projection of the knot we use a theorem of guillermou kashiwara schapira to show that the resulting category is invariant under legendrian isotopies a subsequent article establishes its equivalence to a category of representations of the chekanov eliashberg differential graded algebra we also find two connections to topological knot theory first drawing a positive braid closure on the annulus the moduli space of rank n objects maps to the space of local systems on a circle the second page of the spectral sequence associated to the weight filtration on the pushforward of the constant sheaf is the colored by n triply graded khovanov rozansky homology second drawing a positive braid closure in the plane the number of points of our moduli spaces over a finite field with q elements recovers the lowest coefficient in a of the homfly polynomial of the braid closure

bps states torus links and wild character varieties a string theoretic framework is constructed relating the cohomology of wild character varieties to refined stable pair theory and torus link invariants explicit conjectural formulas are derived for wild character varieties with a unique irregular point on the projective line for this case this leads to a conjectural colored generalization of existing results of hausel mereb and wong as well as shende treumann and zaslow

on the power of las vegas ii two way finite automata abstract the investigation of the computational power of randomized computations is one of the central tasks of complexity and algorithm theory while for one way finite automata the power of different computational modes was successfully determined one does not have any nontrivial result relating the power of determinism las vegas and nondeterminism for two way finite automata the main results of this paper are as follows i if for a regular language l there exist small two way nondeterministic finite automata for both l and l then there exists a small two way las vegas finite automaton for l ii there is a quadratic gap between nondeterminism and las vegas for two way finite automata iii for every k n there is a regular language s k such that s k can be accepted by a two way las vegas finite automaton with o k states but every two way deterministic finite automaton recognizing s k has at least k 2 log 2 k states