on the contribution of degenerate periodic trajectories to the wave trace this paper is concerned with the wave trace z t of a selfadjoint elliptic pseudodifferential operator on a compact manifold it is devoted to the contribution of degenerate periodic trajectories of the associated hamiltonian flow to the singularities of z t given a periodic trajectory and a point on it we obtain a trace formula which relates microlocally the trace of the unitary group of a in a neighborhood of to an oscillatory integral the phase and the amplitude of which are written in terms of the poincare map of for suitable isolated but degenerate this enables us to obtain complete asymptotic expansions applying already known results about asymptotics of oscillatory integrals as an application we obtain lower bounds on the resonances close to the real axis for the laplace beltrami operator in r n n 2 for suitable compactly supported perturbations of the euclidean metric

local discontinuous galerkin methods for the stokes system in this paper we introduce and analyze local discontinuous galerkin methods for the stokes system for a class of shape regular meshes with hanging nodes we derive a priori estimates for the l2 norm of the errors in the velocities and the pressure we show that optimal order estimates are obtained when polynomials of degree k are used for each component of the velocity and polynomials of degree k 1 for the pressure for any k ge1 we also consider the case in which all the unknowns are approximated with polynomials of degree k and show that although the orders of convergence remain the same the method is more efficient numerical experiments verifying these facts are displayed

weak and strong solutions for the incompressible navier stokes equations with damping in this paper we show that the cauchy problem of the navier stokes equations with damping u 1u 0 has global weak solutions for any 1 global strong solution for any 7 2 and that the strong solution is unique for any 7 2 5

a semi classical trace formula at a non degenerate critical level abstract we study the semi classical trace formula at a critical energy level for a h pseudodifferential operator whose principal symbol has a unique non degenerate critical point for that energy this leads to the study of hamiltonian systems near equilibrium and near the non zero periods of the linearized flow the contributions of these periods to the trace formula are expressed in terms of degenerate oscillatory integrals the new results obtained are formulated in terms of the geometry of the energy surface and the classical dynamics on this surface

unified hybridization of discontinuous galerkin mixed and continuous galerkin methods for second order elliptic problems we introduce a unifying framework for hybridization of finite element methods for second order elliptic problems the methods fitting in the framework are a general class of mixed dual finite element methods including hybridized mixed continuous galerkin nonconforming and a new wide class of hybridizable discontinuous galerkin methods the distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements since the associated matrix is sparse symmetric and positive definite these methods can be efficiently implemented moreover the framework allows in a single implementation the use of different methods in different elements or subdomains of the computational domain which are then automatically coupled finally the framework brings about a new point of view thanks to which it is possible to see how to devise novel methods displaying very localized and simple mortaring techniques as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom

an extension of fourier analysis for the n torus in the magnetic field and its application to spectral analysis of the magnetic laplacian we solved the schrodinger equation for a particle in a uniform magnetic field in the n dimensional torus we obtained a complete set of solutions for a broad class of problems the torus tn rn is defined as a quotient of the euclidean space rn by an arbitrary n dimensional lattice the lattice is not necessary either cubic or rectangular the magnetic field is also arbitrary however we restrict ourselves within potential free problems the schrodinger operator is assumed to be the laplace operator defined with the covariant derivative we defined an algebra that characterizes the symmetry of the laplacian and named it the magnetic algebra we proved that the space of functions on which the laplacian acts is an irreducible representation space of the magnetic algebra in this sense the magnetic algebra completely characterizes the quantum mechanics in the magnetic torus we developed a new method for fourier analysis for the magnetic torus and used it to solve the eigenvalue problem of the laplacian

on the flexibility of agglomeration based physical space discontinuous galerkin discretizations in this work we show that the flexibility of the discontinuous galerkin dg discretization can be fruitfully exploited to implement numerical solution strategies based on the use of elements with very general shapes thanks to the freedom in defining the mesh topology we propose a new h adaptive technique based on agglomeration coarsening of a fine mesh the possibility to enhance the error distribution over the computational domain is investigated on a poisson problem with the goal of obtaining a mesh independent discretization the main building block of our dg method consists of defining discrete polynomial spaces directly on physical frame elements for this purpose we orthonormalize with respect to the l 2 product a set of monomials relocated in a specific element frame and we introduce an easy way to reduce the cost related to numerical integration on agglomerated meshes to complete the dg formulation for second order problems two extensions of the br2 scheme to arbitrary polyhedral grids including an estimate of the stabilization parameter ensuring the coercivity property are here proposed

existence of a solution of the wave equation with nonlinear damping and source terms abstract we study the nonlinear wave equation involving the nonlinear damping term u t u t m 1 and a source term of type u u p 1 for 1 p m we prove a global existence theorem with large initial data for 1 m p a blow up result is established for sufficiently large initial data

band asymptotics in two dimensions let x be a rank one symmetric space e g s rp cp and let a be its laplace beltrami operator it is known that the spectrum of a is discrete and consists of a sequence of eigenvalues a k k a k 0 1 2 a 1 1 each occurring with multiplicity n b n l k o k 1 2 n being the dimension of x and a and b being positive integers which depend only on x let q be a zeroth order self adjoint pseudodifferential operator if we perturb a by adding q to it the eigenvalues 1 1 break up into bands

monodromy in the quantum spherical pendulum in this article we show that monodromy in the quantum spherical pendulum can be interpreted as a maslov effect i e as multi valuedness of a certain generating function of the quantum energy levels

piecewise solenoidal vector fields and the stokes problem nonconforming finite element approximations to solutions of the stokes equations are constructed optimal rates of convergence are proved for the velocity and pressure approximations for the pressure approximation c 0 piecewise polynomial functions are used the class of vector fields used to approximate the velocity field have piecewise polynomial components discontinuous across interelement boundaries on each triangle these vector fields satisfy the incompressibility condition pointwise it is shown that these piecewise solenoidal vector fields possess optimal approximation properties to smooth solenoidal vector fields on domains with curved boundaries

inviscid limit for damped and driven incompressible navier stokes equations in mathbb r 2 we consider the zero viscosity limit of long time averages of solutions of damped and driven navier stokes equations in mathbb r 2 we prove that the rate of dissipation of enstrophy vanishes stationary statistical solutions of the damped and driven navier stokes equations converge to renormalized stationary statistical solutions of the damped and driven euler equations these solutions obey the enstrophy balance

on the uniqueness of strong solution to the incompressible navier stokes equations with damping abstract in this paper we show that the cauchy problem of the incompressible navier stokes equations with damping u 1 u 0 has global strong solution for any 3 and the strong solution is unique when 3 5 this improves earlier results

discontinuous galerkin methods for the stokes equations with nonlinear damping term on general meshes abstract this paper presents the interior penalty discontinuous galerkin ipdg methods to solve the stationary stokes equations with nonlinear damping term the ipdg discrete schemes are established on general meshes the corresponding consistency and stabilization of those schemes are proved subsequently we analyze the existence boundedness and uniqueness of the discrete solutions then the optimal error estimates are derived in the l 2 norm and h 1 like dg norm for the velocity variable and l 2 norm for the pressure variable respectively finally some numerical experiments in two dimensional are reported to demonstrate the theoretical results and show the robustness of our discrete schemes

homogeneous quantization and multiplicities of group representations let b be a compact manifold a cone over b is a principal r bundle x with base b let a x a x be the mapping associated with the action of a r on x x is called a symplectic cone if it possesses a symplectic form such that a a a compact lie group g is said to act in a homogeneous fashion on x if it acts on x in such a way that both and the principal bundle structure are preserved it is known that to such an action one can associate in a fairly canonical way a representation of g on a hilbert space h see 3 in this paper we propose a symplectic recipe for the multiplicities with which h decomposes into g irreducibles and show that this recipe is correct generically

laplace beltrami operator on a riemann surface and equidistribution of measures we consider the laplace beltrami operator on a compact riemann surface of a constant negative curvature for any eigenvalue of the laplace beltrami operator there is an associated sequence of measures on the riemann surface these measures naturally appear in quantum chaos type questions in the theory of electro magnetic flow on a riemann surface the main result of the paper is the claim that this sequence of measures has the liouville measure as the weak limit we prove a quantitative version of this equidistribution claim

an interface fitted mesh generator and virtual element methods for elliptic interface problems abstract a simple and efficient interface fitted mesh generation algorithm which can produce a semi structured interface fitted mesh in two and three dimensions quickly is developed in this paper elements in such interface fitted meshes are not restricted to simplices but can be polygons or polyhedra especially in 3d the polyhedra instead of tetrahedra can avoid slivers virtual element methods are applied to solve elliptic interface problems with solutions and flux jump conditions algebraic multigrid solvers are used to solve the resulting linear algebraic system numerical results are presented to illustrate the effectiveness of our method

semiclassical mechanics of symmetry reduction the author discusses semiclassical approximations that are adapted to given symmetry classes in quantum mechanics arbitrary abelian symmetries and also rotational symmetry are treated semiclassical approximations are derived for the projected propagator and energy dependent green s function associated with a given irreducible representation of the symmetry group from these they derive trace formulae analogous to the usual trace formula that determine the energy levels in a given symmetry class in terms of classical orbits

the remainder in weyl s law for heisenberg manifolds we prove that the error term r in weyl s law is o 5 6 for certain three dimensional heisenberg manifolds we also show that the l 2 norm of the weyl error term integrated over the moduli space of left invariant heisenberg metrics is 3 4 we conjecture that r o 3 4 i s a sharp deterministic upper bound for heisenberg three manifolds

a semi classical trace formula for schrodinger operators in the case of a critical energy level let 2 2 v x be a schrodinger operator on rn with smooth potentialv x as x the spectrum of is discrete and one can study the asymptotic of the smoothed spectral density e k ek e as 0 here ek k nis the spectrum of and c 0 r we shall investigate the case whereeis a critical value of the symbolhof and extending the work of brummelhuis paul and uribe in 3 we will prove the existence of a full asymptotic expansion for in terms of and ln and compute the leading coefficient we will consider new weyl type estimates for the counting function nec k n ek e

two level mixed finite element methods for the navier stokes equations with damping abstract in this paper the mixed finite element methods are considered for the navier stokes equations with damping optimal error estimates of the h 1 norm and l 2 norm for the velocity and the l 2 norm for the pressure are obtained moreover two kinds of two level methods based on the oseen correction and newton correction are presented finally two numerical examples are implemented to confirm the analysis

geodesic restrictions for the casimir operator abstract we consider the hyperbolic casimir operator c defined on the tangent sphere bundle sy of a compact hyperbolic riemann surface y we prove a non trivial bound on the l 2 norm of the restriction of eigenfunctions of c to certain natural hypersurfaces in sy the result that we obtain goes beyond known sharp local bounds of l hormander

the trace formula for vector bundles ii fedn x e i i j k is a tempered distributional function of t the classical trace formula says that the singular support of ii is contained in the length spectrum of x moreover under suitable hypotheses on geodesic flow the trace formula gives considerable information about the singularities in ii see dg and c there is a fairly straightforward and not terribly interesting generalization of the trace formula to vector bundles see for instance the introduction to dg we will be concerned in this article with a much more subtle generalization inspired by recent articles of hogreve potthoff and schrader hps and schrader and taylor st in communications in mathematical physics let g be a compact lie group and ir p x a principle g bundle with connection given a finite dimensional unitary representation p of g we will denote by ep the vector bundle over x associated with p and by dp the associated connection now consider a ladder pe e 1 2 of irreducible representations of g this means that the maximal weight of pe is e times the maximal weight of p for given e let afc e 1 2 o be the spectrum of the laplace operator on c epe ae d pedpe e

discontinuous galerkin methods of the non selfadjoint steklov eigenvalue problem in inverse scattering abstract in this paper we apply discontinuous galerkin methods to the non selfadjoint steklov eigenvalue problem arising in inverse scattering the variational formulation of the problem is non selfadjoint and does not satisfy h1 elliptic condition by using the spectral approximation theory of compact operators we prove the spectral approximation and optimal convergence order for the eigenvalues finally some numerical experiments are reported to show that the proposed numerical schemes are efficient for real and complex steklov eigenvalues

superclose and superconvergence of finite element discretizations for the stokes equations with damping a class of conforming mixed finite element methods fems are applied to the stokes equations with damping with the help of the prior estimates for the exact and approximated solutions and the appropriate choice of the parameters a n and r appeared in the problem the superclose and superconvergence results for the velocity in h 1 norm and the pressure in l 2 norm are derived to our best knowledge this is the first superconvergence analysis of conforming mixed fems for this kind nonlinear problem

a small scale density of states formula let m g be a c compact riemann manifold with classical hamiltonian h c t m assume that the corresponding quantization p1 op h is quantum completely integrable we establish an microlocal weyl law on short spectral intervals of size 2 e e 0 for various families of operators p1u u i containing p1 both in the mean and pointwise a e for u i the microlocalization refers to a small tubular neighbourhood of a non degenerate stable periodic bicharacteristic t m 0

toeplitz quantization of kahler manifolds ang gl n n limits for general compact kahler manifolds it is shown that both toeplitz quantization and geometric quantization lead to a well defined by operator norm estimates classical limit this generalizes earlier results of teh authors and klimek and lesniewski obtained for the tours and higher genus riemann surfaces respectively we thereby arrive at an approximation of the poisson algebra by a sequence of finitedimensional matrix algebrasgl n n

a weak galerkin finite element method for second order elliptic problems this paper introduces a finite element method by using a weakly defined gradient operator over generalized functions the use of weak gradients and their approximations results in a new concept called discrete weak gradients which is expected to play an important role in numerical methods for partial differential equations this article intends to provide a general framework for managing differential operators on generalized functions as a demonstrative example the discrete weak gradient operator is employed as a building block in the design of numerical schemes for a second order elliptic problem in which the classical gradient operator is replaced by the discrete weak gradient the resulting numerical scheme is called a weak galerkin wg finite element method it can be seen that the weak galerkin method allows the use of totally discontinuous functions in the finite element procedure for the second order elliptic problem an optimal order error estimate in both a discrete h 1 and l 2 norms are established for the corresponding weak galerkin finite element solutions a superconvergence is also observed for the weak galerkin approximation

on the small scale mass concentration of modes let be commuting jointly elliptic pseudodifferential operators on a compact manifold x of dimension n d suppose is the limit set of the bicharacteristic flow of the classical hamiltonian p 1 restricted to the variety we discuss the corresponding concentration of mass as for a subsequence of joint eigenfunctions of the p j s with eigenvalues sufficiently close to

a proof of the gutzwiller semiclassical trace formula using coherent states decomposition the gutzwiller trace formula links the eigenvalues of the schrodinger operator as planck s constant goes to zero the semiclassical regime with the closed orbits of the corresponding classical mechanical system gutzwiller gave a heuristic proof of this trace formula using the feynman integral representation for the propagator of later using the theory of fourier integral operators mathematicians gave rigorous proofs of the formula in various settings here we show how the use of coherent states allows us to give a simple and direct proof

a semi classical trace formula at a totally degenerate critical level we study the semi classical trace formula at a critical energy level for an h pseudo differential operator on open image in new window n whose principal symbol has a totally degenerate critical point for that energy this problem is studied for a large time behavior and under the hypothesis that the principal symbol of the operator has a local extremum at the critical point

stabilized low order finite elements for stokes equations with damping abstract in this paper we consider the stationary stokes equations with damping low order mixed finite element spaces are used to approximate the velocity and the pressure and a local pressure projection stabilization method is used for the pairs to overcome the lack of the inf sup condition the stability of this method is proved and the optimal order error estimates are derived by some nonlinear analysis techniques at last two numerical examples are implemented to test the stability and effectiveness of the proposed method

incompressible finite elements via hybridization part ii the stokes system in three space dimensions in this paper we introduce a new and efficient way to compute exactly divergence free velocity approximations for the stokes equations in two space dimensions we begin by considering a mixed method that provides an exactly divergence free approximation of the velocity and a continuous approximation of the vorticity we then rewrite this method solely in terms of the tangential fluid velocity and the pressure on mesh edges by means of a new hybridization technique this novel formulation bypasses the difficult task of constructing an exactly divergence free basis for velocity approximations moreover the discrete system resulting from our method has fewer degrees of freedom than the original mixed method since the pressure and the tangential velocity variables are defined just on the mesh edges once these variables are computed the velocity approximation satisfying the incompressibility condition exactly as well as the continuous numerical approximation of the vorticity can at once be obtained locally moreover a discontinuous numerical approximation of the pressure within elements can also be obtained locally we show how to compute the matrix system for our tangential velocity pressure formulation on general meshes and present in full detail such computations for the lowest order case of our method

discrete functional analysis tools for discontinuous galerkin methods with application to the incompressible navier stokes equations two discrete functional analysis tools are established for spaces of piecewise polynomial functions on general meshes i a discrete counterpart of the continuous sobolev embeddings in both hilbertian and non hilbertian settings ii a compactness result for bounded sequences in a suitable discontinuous galerkin norm together with a weak convergence property for some discrete gradients the proofs rely on techniques inspired by the finite volume literature which differ from those commonly used in finite element analysis the discrete functional analysis tools are used to prove the convergence of discontinuous galerkin approximations of the steady incompressible navier stokes equations two discrete convective trilinear forms are proposed a non conservative one relying on temam s device to control the kinetic energy balance and a conservative one based on a nonstandard modification of the pressure

on a quantum chaos theorem of r schrader and m taylor abstract we extend results of r schrader and m taylor on the semi classical asymptotics of eigenfunctions of a quantum hamiltonian for a particle in a gauge field our method is based on the guillemin sternberg uribe theory of fuzzy ladders and reductions of hamiltonian systems with symmetry

circular symmetry and the trace formula let m be a compact n dimensional manifold without boundary and let p be a positive first order elliptic self adjoint classical pseudodifferential operator on lz m the hilbert space of half densities on m the spectrum of p is discrete and accumulates only at we denote the eigenvalues with multiplicities by 2o 21 2 recall that the singularities of the inverse fourier transform of the spectral measure