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

kazhdan margulis theorem for invariant random subgroups abstract given a simple lie group g we show that the lattices in g are weakly uniformly discrete this is a strengthening of the kazhdan margulis theorem our proof however is straightforward considering general irs rather than lattices allows us to apply a compactness argument in terms of p m p actions we show that for every 0 there is an identity neighbourhood u g which intersects trivially the stabilizers of 1 of the points in every non atomic probability g space

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

comment on hamiltonian splitting for the vlasov maxwell equations the paper 1 by crouseilles einkemmer and faou used an incorrect poisson bracket for the vlasov maxwell equations if the correct poisson bracket is used the solution of one of the subsystems cannot be computed exactly in general as a result one cannot construct a symplectic scheme for the vlasov maxwell equations using the splitting hamiltonian method proposed in ref 1

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

the classical shallow water equations symplectic geometry the classical shallow water equations express the change with time of the height h and the velocity of a 1 dimensional fluid t x h x 0 h x h x 0 they possess an infinite number of integrals of motion due to benney 1973 and can be written in hamiltonian form relative to a symplectic structure introduced by manin 1978 the present paper deals with their complete integrability up to the advent of shocks this is proved in the small under an extra assumption satisfied by most height velocity pairs that hh only at isolated points

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

hamiltonian structure and darboux theorem for families of generalized lotka volterra systems this work is devoted to the establishment of a poisson structure for a format of equations known as generalized lotka volterra systems these equations which include the classical lotka volterra systems as a particular case have been deeply studied in the literature they have been shown to constitute a whole hierarchy of systems the characterization of which is made in the context of simple algebra our main result is to show that this algebraic structure is completely translatable into the poisson domain important poisson structures features such as the symplectic foliation and the darboux canonical representation rise as a result of rather simple matrix manipulations

torsion homology growth and cycle complexity of arithmetic manifolds let m be an arithmetic hyperbolic 3 manifold such as a bianchi manifold we conjecture that there is a basis for the second homology of m where each basis element is represented by a surface of low genus and give evidence for this we explain the relationship between this conjecture and the study of torsion homology growth

graphs groupoids and cuntz krieger algebras we associate to each locally finite directed graphgtwo locally compact groupoidsgandg the unit space ofgis the space of one sided infinite paths ing andg is the reduction ofgto the space of paths emanating from a distinguished vertex we show that under certain conditions theirc algebras are morita equivalent the groupoidc algebrac g is the cuntz krieger algebra of an infinite 0 1 matrix defined byg and that the algebrasc g contain thec algebras used by doplicher and roberts in their duality theory for compact groups we then analyse the ideal structure of these groupoidc algebras using the general theory of renault and calculate theirk theory

frobenius manifold for the dispersionless kadomtsev petviashvili equation we consider a frobenius structure associated with the dispersionless kadomtsev petviashvili equation this is done essentially by applying a continuous analogue of the finite dimensional theory in the space of schwartz functions on the line the potential of the frobenius manifold is found to be a logarithmic energy with quadratic external field following the construction of the principal hierarchy we construct a set of infinitely many commuting flows which extends the classical dkp hierarchy

representations of higher rank graph c algebras associated to λ semibranching function systems abstract in this paper we discuss a method of constructing separable representations of the c algebras associated to strongly connected row finite k graphs we begin by giving an alternative characterization of the semibranching function systems introduced in an earlier paper with an eye towards constructing such representations that are faithful our new characterization allows us to more easily check that examples satisfy certain necessary and sufficient conditions we present a variety of new examples relying on this characterization we then use some of these methods and a direct limit procedure to identify a faithful separable representation for any row finite source free k graph

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

rank combinatorial cost and homology torsion growth in higher rank lattices we investigate the rank gradient and growth of torsion in homology in residually finite groups as a tool we introduce a new complexity notion for generating sets using measured groupoids and combinatorial cost as an application we prove the vanishing of the above invariants for farber sequences of subgroups of right angled groups a group is right angled if it can be generated by a sequence of elements of infinite order such that any two consecutive elements commute most nonuniform lattices in higher rank simple lie groups are right angled we provide the first examples of uniform cocompact right angled arithmetic groups in sl n n 3 and so p q for some values of p q this is a class of lattices for which the congruence subgroup property is not known in general by using rigidity theory and the notion of invariant random subgroups it follows that both the rank gradient and the homology torsion growth vanish for an arbitrary sequence of subgroups in any right angled lattice in a higher rank simple lie group 2017

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

the perron frobenius operators invariant measures and representations of the cuntz krieger algebras for a transformation f on a measure space x we show that the perron frobenius operator of f can be written by a representation l2 x of the cuntz krieger algebra oa associated with f when f satisfies some assumption especially when oa is the cuntz algebra on and l2 x in the above is some irreducible representation of on then there is an f invariant measure on x which is absolutely continuous with respect to

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

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

prediction of evanescent coupling efficiency in two parallel silica nanowires abstract in this study the efficiency of evanescent wave coupling between two air clad silica nanowires in single mode operation is numerically predicted in time domain using the finite difference method to this end a three dimensional scheme developed for solving the maxwell s equations on staggered grids will be employed the electric and magnetic field solutions are sought subjected to the zero divergence condition or gauss s law in discrete level in addition we aim to conserve hamiltonians existing in ideal maxwell s equations at all times by adopting the explicit second order accurate symplectic partitioned runge kutta temporal scheme moreover all the spatial derivative terms in the faraday s and ampere s equations are approximated by the proposed scheme which can render not only fourth order spatial accuracy but also minimize the discrepancy between the exact and the derived numerical phase velocities

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

hamiltonian mechanics of complex fluids constitutive relations for complex fluids e g polymeric fluids are derived from the assumption that the equations that govern the time evolution of the complex fluids possess the generalised hamiltonian structure the constitutive relations consist of a configuration space kinetic equation an extended local equilibrium relation an expression for the extra stress tensor and an expression for the heat flux in the particularly interesting case of complex fluids with spatially inhomogeneous internal structure the extra stress tensor includes an extension of the eriksen stress and the heat flux includes non fourier type contributions

branching systems and representations of cohn leavitt path algebras of separated graphs abstract for each separated graph e c we construct a family of branching systems over a set x and show how each branching system induces a representation of the cohn leavitt path algebra associated with e c as homomorphisms over the module of functions in x we also prove that the abelianized cohn leavitt path algebra of a separated graph with no loops can be written as an amalgamated free product of abelianized cohn leavitt algebras that can be faithfully represented via branching systems

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

noncanonical poisson brackets for elastic and micromorphic solids this paper investigates the lagrangian to eulerian transformation approach to the construction of noncanonical poisson brackets for the conservative part of elastic solids and micromorphic elastic solids the dirac delta function links lagrangian canonical variables and eulerian state variables producing noncanonical poisson brackets from the corresponding canonical brackets specifying the hamiltonian functionals generates the evolution equations for these state variables from the poisson brackets different elastic strain tensors such as the green deformation tensor the cauchy deformation tensor and the higher order deformation tensor are appropriate state variables in poisson bracket formalism since they are quantities composed of the deformation gradient this paper also considers deformable directors to comprise the three elastic strain density measures for micromorphic solids furthermore the technique of variable transformation is also discussed when a state variable is not conserved along with the motion of the body

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 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

on the topological properties of symplectic maps in this paper we show that symplectic maps have surprising topological properties in particular we construct an interesting metric for the symplectic diffeomorphism groups which is related but not obviously to the topological properties of symplectic maps and phase space geometry we also prove a certain number of generalised symplectic fixed point theorems and give an application to a hamiltonian system

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

some geometric calculations on wasserstein space we compute the riemannian connection and curvature for the wasserstein space of a smooth compact riemannian manifold

multisymplectic preissman scheme for the time domain maxwell s equations from the bridges multisymplectic form of maxwell s equations we derive a multisymplectic preissman scheme which couples two time levels for 2 1 dimensional maxwell s equations the scheme is proven to preserve the discrete local energy exactly numerical results are reported to illustrate that the scheme is effective and it can get more precise numerical solutions than yee s scheme our numerical results can also indicate that the scheme keeps the discrete local energy and the global energy very well

geometric investigations of a vorticity model equation abstract this article consists of a detailed geometric study of the one dimensional vorticity model equation t u x 2 u x 0 h u x t r x s 1 which is a particular case of the generalized constantin lax majda equation wunsch showed that this equation is the euler arnold equation on diff s 1 when the latter is endowed with the right invariant homogeneous h 1 2 metric in this article we prove that the exponential map of this riemannian metric is not fredholm and that the sectional curvature is locally unbounded furthermore we prove a beale kato majda type blow up criterion which we then use to demonstrate a link to our non fredholmness result finally we extend a blow up result of castro cordoba to the periodic case and to a much wider class of initial conditions using a new generalization of an inequality for hilbert transforms due to cordoba cordoba

classification and casimir invariants of lie poisson brackets abstract we classify lie poisson brackets that are formed from lie algebra extensions the problem is relevant because many physical systems owe their hamiltonian structure to such brackets a classification involves reducing all brackets to a set of normal forms and is achieved partially through the use of lie algebra cohomology for extensions of order less than five the number of normal forms is small and they involve no free parameters we derive a general method of finding casimir invariants of lie poisson bracket extensions the casimir invariants of all low order brackets are explicitly computed we treat in detail a four field model of compressible reduced magnetohydrodynamics

perron frobenius operators and representations of the cuntz krieger algebras for infinite matrices abstract in this paper we extend the work of kawamura see k kawamura the perron frobenius operators invariant measures and representations of the cuntz krieger algebras j math phys 46 2005 for cuntz krieger algebras o a for infinite matrices a we generalize the definition of branching systems prove their existence for any given matrix a and show how they induce some very concrete representations of o a we use these representations to describe the perron frobenius operator associated to a nonsingular transformation as an infinite sum and under some hypothesis we find a matrix representation for the operator we finish the paper with a few examples

cout des relations d equivalence et des groupes we study a new dynamical invariant for dicrete groups the cost it is a real number in 1 1 n 1 bounded by the number of generators of the group and it is well behaved with respect to finite index subgroups namely the quantities 1 minus the cost are related by multiplying by the index the cost of every infinite amenable group equals 1 we compute it in some other situations including free products free products with amalgamation and hnn extensions over amenable groups and for direct product situations for instance the cost of the free group on n generators equals n we prove that each possible finite value of the cost is achieved by a finitely generated group it is dynamical because it relies on measure preserving free actions on probability borel spaces in most cases groups have fixed price which implies that two freely acting groups which define the same orbit partition must have the same cost it enables us to distinguish the orbit partitions of probability preserving free actions of free groups of different ranks at the end of the paper we give a mercuriale i e a list of costs of different groups the cost is in fact an invariant of ergodic measure preserving equivalence relations and is defined using graphings a treeing is a measurable way to provide every equivalence class orbit with the structure of a simplicial tree this an example of graphing not every relation admits a treeing we prove that every free action of a cost 1 non amenable group is not treeable but we prove that subrelations of treeable relations are treeable we give examples of relations which cannot be produced by an action of any finitely generated group the cost of a relation which can be decomposed as a direct product is shown to be 1 we define the notion for a relation to be a free product or an hnn extension and compute the cost for the resulting relation from the costs of the building blocks the cost is also an invariant of the pairs von neumann algebra cartan subalgebra

on graded irreducible representations of leavitt path algebras using the e algebraic branching systems various graded irreducible representations of a leavitt path k algebra l of a directed graph e are constructed the concept of a laurent vertex is introduced and it is shown that the minimal graded left ideals of l are generated by the laurent vertices or the line points leading to a detailed description of the graded socle of l following this a complete characterization is obtained of the leavitt path algebras over which every graded irreducible representation is finitely presented a useful result is that the irreducible representation v p induced by infinite paths tail equivalent to an infinite path p we call this a chen simple module is graded if and only if p is an irrational path we also show that every one sided ideal of l is graded if and only if the graph e contains no cycles supplementing the theorem of one of the co authors that every leavitt path algebra l is graded von neumann regular we show that l is graded self injective if and only if l is a graded semi simple algebra made up of matrix rings of arbitrary size over the field k or the graded field k xn x n where n n

development of a 3d staggered fdtd scheme for solving maxwell s equations in drude medium an explicit finite difference scheme is developed to solve the three dimensional maxwell s equations in drude medium our aim of developing this scheme in time domain is to compute solutions in staggered grids from the faraday s and ampere s equations the electric and magnetic field solutions are sought subject to the discrete zero divergence condition gauss s law the local conservation laws in ideal maxwell s equations are also numerically preserved all the time using the explicit second order accurate symplectic partitioned runge kutta temporal scheme the spatial derivative terms in the faraday s and ampere s equations are discretized to obtain fourth order accuracy using the developed scheme underlying the concept of minimizing the discrepancy between the exact and the derived numerical phase velocities dispersion and anisotropy errors have been much reduced through the procedure of minimizing phase velocity error in addition to performing the fundamental analysis on the proposed scheme the computational efficiency and long term accurate properties embedded in the proposed symplectic dispersion error reduction centered scheme are numerically demonstrated through several test problems investigated in ideal and drude media

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

metric properties of the group of area preserving diffeomorphisms area preserving cliffeoinorpliisms of the 2 disk which are identity near the boundary form a group d2 wllich can be equipped usin g tlhe l2nlorm on its lie algebra with a right invariant metric witll tllis metric the diameter of d2 is infinite in this paper we sl iow that d2 contains quasiisometric embeddings of any finitely generated free group and any finitely generated abelian free group

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

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

on the representations of leavitt path algebras abstract given a graph e we define e algebraic branching systems show their existence and how they induce representations of the associated leavitt path algebra we also give sufficient conditions to guarantee faithfulness of the representations associated to e algebraic branching systems and to guarantee equivalence of a given representation or a restriction of it to a representation arising from an e algebraic branching system

statistical mechanics with non integrable topological constraints self organization in knotted phase space the object of this study is the statistical mechanics of dynamical systems lacking a hamiltonian structure due to the presence of non integrable topological constraints that limit the accessible regions of the phase space focusing on the simplest three dimensional case we develop a procedure poissonization that assigns to any three dimensional non hamiltonian system an equivalent four dimensional hamiltonian system endowed with a proper time the statistical distribution is then constructed in the recovered four dimensional canonical phase space projecting in the original reference frame we show that the statistical distribution departs from standard maxwell boltzmann statistics the deviation is a function of the knottedness of the phase space which is measured by the helicity density of the topological constraint the theory is then generalized to a class of non hamiltonian systems in higher dimensions

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

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

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

numerical dispersion analysis of a multi symplectic scheme for the three dimensional maxwell s equations in this paper we study a multi symplectic scheme for three dimensional maxwell s equations in a simple medium this is a system of pdes with multi symplectic structures we prove that this multi symplectic scheme preserves the discrete version of local and global energy conservation law and the discrete divergence furthermore we extend the discussion to several dispersion properties of the multi symplectic scheme including the numerical dispersion relation the numerical group velocity the effect of large time steps and the cfl condition

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

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

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

recursive fermion system in cuntz algebra i embeddings of the car canonical anticommutation relations algebra of fermions into the cuntz algebra 2 or 2 d more generally are presented by using recursive constructions as a typical example an embedding of car onto the u 1 invariant subalgebra of 2 is constructed explicitly generalizing this construction to the case of 2 p an embedding of car onto the u 1 invariant subalgebra of 2 p is obtained restricting a permutation representation of the cuntz algebra we obtain the fock representation of car we apply the results to embed the algebra of parafermions of order p into 2 p according to green s ansatz

on the growth of l 2 invariants for sequences of lattices in lie groups we study the asymptotic behaviour of betti numbers twisted torsion and other spectral invariants of sequences of locally symmetric spaces our main results are uniform versions of the degeorge wallach theorem of a theorem of delorme and various other limit multiplicity theorems a basic idea is to adapt the notion of benjamini schramm convergence bs convergence originally introduced for sequences of finite graphs of bounded degree to sequences of riemannian manifolds and analyze the possible limits we show that bs convergence of locally symmetric spaces g k implies convergence in an appropriate sense of the normalized relative plancherel measures associated to l2 g this then yields convergence of normalized multiplicities of unitary representations betti numbers and other spectral invariants on the other hand when the corresponding lie group g is simple and of real rank at least two we prove that there is only one possible bs limit i e when the volume tends to infinity locally symmetric spaces always bs converge to their universal cover g k this leads to various general uniform results when restricting to arbitrary sequences of congruence covers of a fixed arithmetic manifold we prove a strong quantitative version of bs convergence which in turn implies upper estimates on the rate of convergence of normalized betti numbers in the spirit of sarnak xue an important role in our approach is played by the notion of invariant random subgroups for higher rank simple lie groups g we exploit rigidity theory and in particular the nevo stuck zimmer theorem and kazhdan s property t to obtain a complete understanding of the space of irs s of g 2017 department of mathematics princeton university

kesten s theorem for invariant random subgroups an invariant random subgroup of the countable group is a ran dom subgroup of whose distribution is invariant under conjuga tion by all elements of we prove that for a nonamenable invariant random subgroup h the spectral radius of every finitely supported random walk on i s strictly less than the spectral radius of the corresponding random walk on h this generalizes a result of kesten who proved this for normal subgroups as a byproduct we show that for a cayley graph g of a linear group with no amenable normal subgroups any sequence of finite quotients of g that spectrally approximates g converges to g in benjamini schramm convergence in particular this implies that infinite sequences of finite d regular ramanujan schreier graphs have essentially large girth

symplectic diffeomorphisms and the flux homomorphism let x be a connected smooth manifold without boundary we will write difff ox for the group of all compactly supported diffeomorphisms of x which preserve the symplectic form o2 topologized as usual as the direct limit of the subgroups diff x of all diffeomorphisms with support in the fixed compact set k let diff oox be its identity component the flux is locally a homomorphism from a neighbourhood of the identity in diffo ox to h x r see calabi 6 it gives rise to a global homomorphism s o from the universal cover of difffo0x to hi x f and hence to a homomorphism

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

fredholm properties of riemannian exponential maps on diffeomorphism groups we prove that exponential maps of right invariant sobolev h r metrics on a variety of diffeomorphism groups of compact manifolds are nonlinear fredholm maps of index zero as long as r is sufficiently large this generalizes the result of ebin et al geom funct anal 16 2006 for the l 2 metric on the group of volume preserving diffeomorphisms important in hydrodynamics in particular our results apply to many other equations of interest in mathematical physics we also prove an infinite dimensional morse index theorem settling a question raised by arnold and khesin topological methods in hydrodynamics springer new york 1998 on stable perturbations of flows in hydrodynamics finally we include some applications to the global geometry of diffeomorphism groups

graph c algebras branching systems and the perron frobenius operator in this paper we show how to produce a large number of representations of a graph c algebra in the space of the bounded linear operators in l2 x these representations are very concrete and in the case of graphs that satisfy condition l we use our techniques to realize the associated graph c algebra as a subalgebra of the bounded operators in l2 r we also show how to describe the perron frobenius operator of ergodic theory in terms of the representations we associate to a graph

realizations of af algebras as graph algebras exel laca algebras and ultragraph algebras abstract we give various necessary and sufficient conditions for an af algebra to be isomorphic to a graph c algebra an exel laca algebra and an ultragraph c algebra we also explore consequences of these results in particular we show that all stable af algebras are both graph c algebras and exel laca algebras and that all simple af algebras are either graph c algebras or exel laca algebras in addition we obtain a characterization of af algebras that are isomorphic to the c algebra of a row finite graph with no sinks

form invariant poisson brackets of hydrodynamic type with several spatial variables form invariant poisson brackets of hydrodynamic type with several spatial variables x1 xp are introduced for any commuting vector fields u1 um on a manifold mn the incompressibility equation for certain parameters of the poisson brackets is derived poisson brackets connected with an arbitrary dynamical system on mn are constructed a gauge transform for the poisson brackets with one spatial variable x is defined and its action onto the tensor invariants is studied a general identity between the coefficients bkij u and gij u is found