spectral shift function for trapping energies in the semiclassical limit semiclassical asymptotics of the spectral shift function ssf for schrodinger operator is studied at trapping energies it is shown that the ssf converges to sum of a smooth function and a step function which is essentially the counting function of resonances in particular the weyl asymptotics is proved

fine structure of zoll spectra this paper is concerned with the wave equation on a zoll manifold i e a compact riemannian manifold m g all of whose geodesics are closed by a result of v guillemin such metrics exist in abundance ons2 for any odd functionfons2there exists a one parameter deformation of the canonical metricgothru zoll metrics gt eftwithft tf and as has been known for a long time dg cv 1 hoiv w 1 2 the periodicity of the geodesic flow leads to a near periodicity in the wave grouput eit precisely there exists a unitary equivalence r q 1where r q 1 0 where the spectrum ofrlies on an arithmetic progression and whereq 1is a pseudodifferential operator of order 1 hence the eigenvalues fall into clusterckaround spec r with widthso k 1 the main result of this paper is to determine the limit distribution of the eigenvalues in the clusters it was long ago shown by weinsten loc cit that when properly normalized the limit distribution has the form q 1d whered is the liouville measure ons m we complete the calculation by determining q 1 it was conjectured w 2 ku that up to universal constants q 1 r whereris the radon transform of m g and is the scalar curvature w 2 ku but it turns out there is a second term involving the jacobi fields in addition we show that the cluster projections define almost isometric minimal immersions of m g into spheres and that in the case of maximally degenerate zoll metrics the errors are of rapid decay

analytic hypoellipticity for sums of squares and the treves conjecture abstract we are concerned with the problem of real analytic regularity of the solutions of sums of squares with real analytic coefficients treves conjecture states that an operator of this type is analytic hypoelliptic if and only if all the strata in the poisson treves stratification are symplectic we produce a model operator p 1 having a single symplectic stratum and prove that it is gevrey s 0 hypoelliptic and not better see theorem 2 1 for a definition of s 0 we also show that this phenomenon has a microlocal character we point out explicitly that this is a counterexample to the sufficient part of treves conjecture and not to the necessary part which is still an open problem

computations in non commutative iwasawa theory we study special values of l functions of elliptic curves over mathbb q twisted by artin representations that factor through a false tate curve extension mathbb q mu p infty sqrt p infty m mathbb q in this setting we explain how to compute l functions and the corresponding iwasawa theoretic invariants of non abelian twists of elliptic curves our results provide both theoretical and computational evidence for the main conjecture of non commutative iwasawa theory

on the iwasawa invariants of elliptic curves suppose that e is an elliptic curve defined over q and that p is a prime where e has good ordinary reduction under the assumption that e is modular one can define nonnegative integers e alg e anal e and anal e the algebraic iwasawa invariants alg e and e are defined in terms of the structure of the p primary subgroup sele q p of the selmer group for e over the cyclotomic zp extension q of q the definition of the analytic invariants e and anal e is in terms of the p adic l function for e constructed by mazur and swinnerton dyer msd74 we will recall these definitions below our purpose in this article is to prove in certain cases that e anal e 0 and that alg e anal e these equalities together with a deep theorem of kato imply the main conjecture for e over q in this introduction we will discuss the nature of our results and give an outline of the proofs for the case of modular elliptic curves we want to point out however that the theorems proven in the text apply to modular forms as well as to elliptic curves with multiplicative reduction at p we will not attempt to state the most general versions here if k is any algebraic extension of q then the selmer group for e over k is a certain subgroup of h gk e q tors where gk gal q k the selmer group fits into an exact sequence 0 e k q z sele k xe k

a probabilistic approach to semi classical approximations abstract we study in this paper the semi classical expansion of the schrodinger equation using a probabilistic approach based on the wiener measure using almost analytic extensions we exhibit a probabilistic ansatz for the wave function we show that this ansatz approximates very well the wave function in the semi classical regime and gives the semi classical expansion under mild hypothesis on the potential at infinity and no analyticity conditions in this paper the study takes place before the caustics

analytic and gevrey hypoellipticity for a class of pseudodifferential operators in one variable abstract we study the hypoellipticity of pseudo differential operators in one variable when a positivity assumption is made on a suitable principal part we prove that the found gevrey regularity is optimal

distributed control of spatially invariant systems we consider distributed parameter systems where the underlying dynamics are spatially invariant and where the controls and measurements are spatially distributed these systems arise in many applications such as the control of vehicular platoons flow control microelectromechanical systems mems smart structures and systems described by partial differential equations with constant coefficients and distributed controls and measurements for fully actuated distributed control problems involving quadratic criteria such as linear quadratic regulator lqr h sub 2 and h sub spl infin optimal controllers can be obtained by solving a parameterized family of standard finite dimensional problems we show that optimal controllers have an inherent degree of decentralization and this provides a practical distributed controller architecture we also prove a general result that applies to partially distributed control and a variety of performance criteria stating that optimal controllers inherit the spatial invariance structure of the plant connections of this work to that on systems over rings and systems with dynamical symmetries are discussed

decay rates for the three dimensional linear system of thermoelasticity we consider the two and three dimensional system of linear thermoelasticity in a bounded smooth domain with dirichlet boundary conditions we analyze whether the energy of solutions decays exponentially uniformly to zero as t to infty first of all by a decoupling method we reduce the problem to an observability inequality for the lame system in linear elasticity and more precisely to whether the total energy of the solutions can be estimated in terms of the energy concentrated on its longitudinal component we show that when the domain is convex the decay rate is never uniform in fact the lack of uniform decay holds in a more general class of domains in which there exist rays of geometric optics of arbitrarily large length that are always reflected perpendicularly or almost tangentially on the boundary we also show that in three space dimensions the lack of uniform decay may also be due to a critical polarization of the energy on the transversal component of the displacement in two space dimensions we prove a sufficient and almost necessary condition for the uniform decay to hold in terms of the propagation of the transversal characteristic rays under the further assumption that the boundary of the domain does not have contacts of infinite order with its tangents we also give an example due to d hulin in which these geometric properties hold in three space dimensions we indicate without proof how a careful analysis of the polarization of singularities may lead to sharp sufficient conditions for the uniform decay to hold in two space dimensions we prove that smooth solutions decay polynomially in the energy space to a finite dimensional subspace of solutions except when the domain is a ball or an annulus finally we discuss some closely related controllability and spectral issues

the magnetic weyl calculus in the presence of a variable magnetic field the weyl pseudodifferential calculus must be modified the usual modification based on the minimal coupling principle at the level of the classical symbols does not lead to gauge invariant formulas if the magnetic field is not constant we present a gauge covariant quantization relying on the magnetic canonical commutation relations the underlying symbolic calculus is a deformation defined in terms of the magnetic flux through triangles of the classical moyal product

thermodynamically consistent phase field models of fracture variational principles and multi field fe implementations the computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies this can be overcome by a diffusive crack modeling based on the introduction of a crack phase field in this paper we outline a thermodynamically consistent framework for phase field models of crack propagation in elastic solids develop incremental variational principles and consider their numerical implementations by multi field finite element methods we start our investigation with an intuitive and descriptive derivation of a regularized crack surface functional that converges for vanishing length scale parameter to a sharp crack topology functional this functional provides the basis for the definition of suitable convex dissipation functions that govern the evolution of the crack phase field here we propose alternative rate independent and viscous over force models that ensure the local growth of the phase field next we define an energy storage function whose positive tensile part degrades with increasing phase field with these constitutive functionals at hand we derive the coupled balances of quasi static stress equilibrium and gradient type phase field evolution in the solid from the argument of virtual power here we consider a canonical two field setting for rate independent response and a time regularized three field formulation with viscous over force response it is then shown that these balances follow as the euler equations of incremental variational principles that govern the multi field problems these principles make the proposed formulation extremely compact and provide a perfect base for the finite element implementation including features such as the symmetry of the monolithic tangent matrices we demonstrate the performance of the proposed phase field formulations of fracture by means of representative numerical examples copyright 2010 john wiley sons ltd

scattering theory for the schrodinger equation in some external time dependent magnetic fields abstract we study the theory of scattering for a schrodinger equation in an external time dependent magnetic field in the coulomb gauge in space dimension 3 the magnetic vector potential is assumed to satisfy decay properties in time that are typical of solutions of the free wave equation and even in some cases to be actually a solution of that equation that problem appears as an intermediate step in the theory of scattering for the maxwell schrodinger ms system we prove in particular the existence of wave operators and their asymptotic completeness in spaces of relatively low regularity we also prove their existence or at least asymptotic results going in that direction in spaces of higher regularity the latter results are relevant for the ms system as a preliminary step we study the cauchy problem for the original equation by energy methods using as far as possible time derivatives instead of space derivatives

on the existence of euler lagrange orbits satisfying the conormal boundary conditions abstract let m g be a closed connected riemannian manifold l t m r be a tonelli lagrangian given two non empty closed submanifolds q 0 q 1 m and a real number k we study the existence of euler lagrange orbits with energy k connecting q 0 to q 1 and satisfying suitable boundary conditions known as conormal boundary conditions we introduce the mane critical value which is relevant for this problem and discuss existence results for supercritical and subcritical energies we also provide counterexamples showing that all the results are sharp

map image restoration and segmentation by constrained optimization the combinatorial optimization problem of map estimation is converted to one of constrained real optimization and then solved by using the augmented lagrange hopfield alh method proposed in 1984 the alh effectively overcomes instabilities that are inherent in the penalty method or the lagrange multiplier method in constrained optimization it produces good solutions with reasonable costs

muntz szasz theorems for nilpotent lie groups the classic muntz szasz theorem says that forf l2 0 1 and nk k 1 a strictly increasing sequence of positive integers 10xnjf x dx 0 j f 0 j 1 1nj we have generalized this theorem to compactly supported functions on rnand to an interesting class of nilpotent lie groups on rnwe rephrased the condition above on an integral against a monomial as a condition on the derivative of the fourier transformf this transform for compactly supportedf has an entire extension to complexn space and these derivatives are coefficients in a taylor series expansion off for nilpotent lie groups we have proven a muntz szasz theorem for the matrix coefficients of the operator valued fourier transform on groups that have a fixed polarizer for the representations in general position our work here is inspired by recent work on paley wiener theorems for nilpotent lie groups by moss j funct anal 114 1993 395 411 and park j funct anal 133 1995 211 300 who have proven paley wiener theorems on restricted classes of nilpotent lie groups lipsman and rosenberg trans amer math soc 348 1996 1031 1050 have extended these results for matrix coefficients to any connected simply connected nilpotent lie group as part of the proof of the muntz szasz theorem for matrix coefficients we construct a new basis in a nilpotent lie algebra which we call analmost strong malcev basis this new basis has many of the features of a strong malcev basis although it can be used to pass through subalgebras that are not ideals almost strong malcev bases are unique up to a fixed strong malcev basis we will also show that using almost strong malcev bases we can provide a partial answer to a question posed by corwin and greenleaf representations of nilpotent lie groups and their applications cambridge univ press cambridge uk 1990 on using additive coordinates for a cross section

persistence of embedded eigenvalues we consider conditions under which an embedded eigenvalue of a self adjoint operator remains embedded under small perturbations in the case of a simple eigenvalue embedded in continuous spectrum of multiplicity m infinity we show that in favorable situations the set of small perturbations of a suitable banach space which do not remove the eigenvalue form a smooth submanifold of codimension in we also have results regarding the cases when the eigenvalue is degenerate or when the multiplicity of the continuous spectrum is infinite

geometric primitive extraction using a genetic algorithm extracting geometric primitives from geometric sensor data is an important problem in model based vision a minimal subset is the smallest number of points necessary to define a unique instance of a geometric primitive a genetic algorithm based on a minimal subset representation is used to perform primitive extraction it is shown that the genetic approach is an improvement over random search and is capable of extracting more complex primitives than the hough transform

scattering theory in the energy space for a class of non linear wave equations we study the asymptotic behaviour in time of the solutions and the theory of scattering in the energy space for the non linear wave equation square varphi f varphi 0 in n n 3 we prove the existence of the wave operators asymptotic completeness for small initial data and forn 4 asymptotic completeness for arbitrarily large data the assumptions onf cover the case wheref behaves slightly better than a single powerp 1 4 n 2 both near zero and at infinity see 1 5 1 6 and 1 8

quantization of abelian varieties distributional sections and the transition from kahler to real polarizations abstract we study the dependence of geometric quantization of the standard symplectic torus on the choice of invariant polarization real and mixed polarizations are interpreted as degenerate complex structures using a weak version of the equations of covariant constancy and the weil brezin expansion to describe distributional sections we give a unified analytical description of the quantization spaces for all non negative polarizations the blattner kostant sternberg bks pairing maps between half form corrected quantization spaces for different polarizations are shown to be transitive and related to an action of sp 2 g r moreover these maps are shown to be unitary

the wave group on asymptotically hyperbolic manifolds abstract we show that the wave group on asymptotically hyperbolic manifolds belongs to an appropriate class of fourier integral operators then we use now standard techniques to analyze its regularized trace we prove that as in the case of compact manifolds without boundary the singularities of the regularized wave trace are contained in the set of periods of closed geodesics we also obtain an asymptotic expansion for the trace at zero

dynamics of polynomial mappings of c 2 we study the dynamics of polynomial self mappings f of c 2 we construct for a large class of mappings an invariant measure which is mixing and of maximal entropy h f max log d t f log 1 f where d t f is the topological degree of f and 1 f its first dynamical degree to achieve this we look at the meromorphic extensions of f to smooth minimal compactifications of c 2 when a good compactification is found we construct an f invariant green current t which contains many dynamical informations when d t f 1 f 1 the measure is obtained as p dd c vt where v is a partial green function defined on the support of t when 1 t t where t is a globally defined f invariant current

variation of iwasawa invariants in hida families let gq gl2 k be an absolutely irreducible modular galois representation over a finite field k of characteristic p assume further that is p ordinary and p distinguished in the sense that the restriction of to a decomposition group at p is reducible and non scalar the hida family h of is the set of all p ordinary p stabilized newforms f with mod p galois representation isomorphic to if is unramified at p then one must also fix an unramified line in and require that the ordinary line of f reduces to this fixed line these newforms are a dense set of points in a certain p adic analytic space of overconvergent eigenforms consisting of an intersecting system of branches i e irreducible components t a indexed by the minimal primes a of a certain hecke algebra to each modular form f h one may associate the iwasawa invariants an f an f alg f and alg f the analytic resp algebraic invariants are the number of zeroes of the p adic l function resp of the characteristic power series of the dual of the selmer group of f while the invariants are the exponents of the powers of p dividing the same objects in this paper we prove the following results on the behavior of these iwasawa invariants as f varies over h

equivariant asymptotics for bohr sommerfeld lagrangian submanifolds suppose given a complex projective manifold m with a fixed hodge form the bohr sommerfeld lagrangian submanifolds of m are the geometric counterpart to semi classical physical states and their geometric quantization has been extensively studied here we revisit this theory in the equivariant context in the presence of a compatible hamiltonian action of a connected compact lie group

a higher order phase field model for brittle fracture formulation and analysis within the isogeometric analysis framework phase field models based on the variational formulation for brittle fracture have recently been gaining popularity these models have proven capable of accurately and robustly predicting complex crack behavior in both two and three dimensions in this work we propose a fourth order model for the phase field approximation of the variational formulation for brittle fracture we derive the thermodynamically consistent governing equations for the fourth order phase field model by way of a variational principle based on energy balance assumptions the resulting model leads to higher regularity in the exact phase field solution which can be exploited by the smooth spline function spaces utilized in isogeometric analysis this increased regularity improves the convergence rate of the numerical solution and opens the door to higher order convergence rates for fracture problems we present an analysis of our proposed theory and numerical examples that support this claim we also demonstrate the robustness of the model in capturing complex three dimensional crack behavior

the index of scattering operators of dirac equations ii abstract the index formula of scattering operators of the previous paper of the author t matsui the index of scattering operators of dirac equations commun math phys 110 1987 553 571 is shown to hold for a wider class of potentials which contains both gauge potentials with compactly supported energy and instantontype potentials

the analysis of partial differential operators i hormander springer 1985 vol 1 391 pp vol 2 390 pp vol 3 525 pp vol 4 351 pp

peridynamic differential operator and its applications abstract the nonlocal peridynamic theory has been proven extremely robust for predicting damage nucleation and propagation in materials under complex loading conditions its equations of motion originally derived based on the principle of virtual work do not contain any spatial derivatives of the displacement components thus their solution does not require special treatment in the presence of geometric and material discontinuities this study presents an alternative approach to derive the peridynamic equations of motion by recasting navier s displacement equilibrium equations into their nonlocal form by introducing the peridynamic differential operator also this operator permits the nonlocal form of expressions for the determination of the stress and strain components the capability of this differential operator is demonstrated by constructing solutions to ordinary partial differential equations and derivatives of scattered data as well as image compression and recovery without employing any special filtering and regularization techniques

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

the spectral action for moyal planes extending a result of vassilevich we obtain the asymptotic expansion for the trace of a spatially regularized heat operator l f e t where is a generalized laplacian defined with moyal products and l f is moyal left multiplication the moyal planes corresponding to any skewsymmetric matrix being spectral triples the spectral action introduced in noncommutative geometry by chamseddine and connes is computed this result generalizes the connes lott action previously computed by gayral for symplectic

quantum classical correspondence using projection operators classical properties associated with a large phase space domain with a smooth boundary as compared to the planck constant are shown to be expressed quantum mechanically by a family of quantum projection operators with the help of known theorems in microlocal analysis under the conditions of egorov s theorem the conservation of this correspondence under classical quantum dynamics is asserted

bayesian linearized rock physics inversion abstractthe estimation of rock and fluid properties from seismic attributes is an inverse problem rock physics modeling provides physical relations to link elastic and petrophysical variables most of these models are nonlinear therefore the inversion generally requires complex iterative optimization algorithms to estimate the reservoir model of petrophysical properties we have developed a new approach based on the linearization of the rock physics forward model using first order taylor series approximations the mathematical method adopted for the inversion is the bayesian approach previously applied successfully to amplitude variation with offset linearized inversion we developed the analytical formulation of the linearized rock physics relations for three different models empirical granular media and inclusion models and we derived the formulation of the bayesian rock physics inversion under gaussian assumptions for the prior distribution of the model the application of the inversion to real

on a new method of proving gevrey hypoellipticity for certain sums of squares abstract we consider an operator being a sum of squares of vector fields it has the form p r n p x d x d y d t d x 2 x 2 p 1 d y x r d t 2 this type of operator is c hypoelliptic by hormander s theorem 18 its analytic or gevrey hypoellipticity has then been studied by a number of authors and is relevant in relation to the treves conjecture the poisson treves stratification of p includes both symplectic and non symplectic strata in this paper we show that p is gevrey p r p hypoelliptic by constructing a parametrix whose symbol belongs to some exotic classes one can also show that this number is optimal

reproducing kernel hierarchical partition of unity part ii applications in this part of the work the meshless hierarchical partition of unity proposed in 1 referred here as part i is used as a multiple scale basis in numerical computations to solve practical problems the applications discussed in the present work fall into two categories 1 a wavelet adaptivity refinement procedure and 2 a so called wavelet petrov galerkin procedure in the applications of wavelet adaptivity the hierarchical reproducing kernels are used as a multiple scale basis to compute the numerical solutions of the helmholtz equation a model equation of wave propagation problems and to simulate shear band formation in an elasto viscoplastic material a problem dictated by the presence of the high gradient deformation in both numerical experiments numerical solutions with high resolution are obtained by inserting the wavelet like basis into the primary interpolation function basis a process that may be viewed as a spectral p type refinement by using the interpolant that has synchronized convergence property as a weighting function a wavelet petrov galerkin procedure is proposed to stabilize computations of some pathological problems in numerical computations such as advection diffusion problems and stokes flow problem it offers an alternative procedure in stablized methods and also provides some insight or new interpretation of the method detailed analysis has been carried out on the stability and convergence of the wavelet petrov galerkin method copyright 1999 john wiley sons ltd

a nonlocal operator method for solving partial differential equations abstract a nonlocal operator method is proposed which is generally applicable for solving partial differential equations pdes of mechanical problems the nonlocal operator can be regarded as the integral form equivalent to the differential form in the sense of a nonlocal interaction model for solving the unknown field the variation of a nonlocal operator plays an equivalent role as the derivatives of the shape functions in the meshless methods or those of the finite element method thus it circumvents many difficulties in the calculation of shape functions and their derivatives the nonlocal operator method can consistently applied with common procedure leading to the weak forms i e the variational principle and the weighted residual method based on these the residual and the tangent stiffness matrix can be obtained with ease the nonlocal operator method is enhanced here also with an operator energy functional to satisfy the linear consistency of the field higher order nonlocal operators and higher order operator energy functional are hereby generalized a highlighted of the present method is the functional derived based on the nonlocal operator can convert the construction of residual and stiffness matrix into a series of matrix multiplications using the predefined nonlocal operators the nonlocal strong forms of different functionals can be obtained easily via the concept of support and dual support the two basic elements introduced in the paper several numerical examples of different types of pdes are presented in the end to show the effectiveness of the present method and also serve for validation

picard lefschetz theory and characters of a semisimple lie group the paper applies picard lefschetz theory to the distribution characters of infinite dimensional representations of semisimple lie groups and analyzes their asymptotic behaviour at the identity

solvability for a class of doubly characteristic differential operators on 2 step nilpotent groups since h lewy s le discovery of a smooth linear partial differential operator without solution the problem of solvability of linear differential operators has been investigated intensively by various authors this has meanwhile led most notably through the work of hdrmander maslov egorov nirenberg erves and beals fefferman but with many important contributions also by others to a very satisfactory theory in the case of principal type operators see h6 much less is known about operators whose principal symbol vanishes of second order on part of its characteristic variety i e those which admit double characteristics even in the case of real principal symbols the main objective of this article is to present a model class of such operators for which we are able to discuss solvability in a complete way our class being composed of rather special operators will certainly not capture all phenomena which might arise in the study of general doubly characteristic operators with real symbol as the results in st already indicate but it is still rich enough to exhibit several new features if one compares with operators of principal type the operators that we shall consider will be left invariant differential operators on connected simply connected 2 step nilpotent lie groups g of the form m 0 1 le ajkvjvk u j k l

dixmier traces on noncompact isospectral deformations abstract we extend the isospectral deformations of connes landi and dubois violette to the case of riemannian spin manifolds carrying a proper action of the noncompact abelian group r l under deformation by a torus action a standard formula relates dixmier traces of measurable operators to integrals of functions on the manifold we show that this relation persists for actions of r l under mild restrictions on the geometry of the manifold which guarantee the dixmier traceability of those operators

on the fokker planck boltzmann equation we consider the boltzmann equation perturbed by fokker planck type operator to overcome the lack of strong a priori estimates and to define a meaningful collision operator we introduce a notion of renormalized solution which enables us to establish stability results for sequences of solutions and global existence for the cauchy problem with large data the proof of stability and existence combines renormalization with an analysis of a defect measure

a higher order nonlocal operator method for solving partial differential equations abstract a higher order nonlocal operator method for the solution of boundary value problems is developed the proposed higher order nonlocal operator brings several advantages as compared to the original nonlocal operator method ren et al 2020 which only ensures first order convergence furthermore it can be applied to directly and efficiently obtain all partial derivatives of higher orders simultaneously without the need of using shape functions only the functionals based on the nonlocal operators termed as operator functional are needed to obtain the final discrete system of equations which significantly facilitates the implementation several numerical examples are presented to show the effectiveness and accuracy of the proposed higher order nonlocal operator method including the solution of the poisson equation in 2 5 dimensional space kirchhoff and von karman plate problems incompressible elastic materials as well as phase field modeling of fracture

toward global solution to map image restoration and segmentation using common structure of local minima abstract in this paper an iterative optimization algorithm called the comb algorithm is presented for approximating the global solution to map image restoration and segmentation the comb derives new initial configurations based on the best local minimum found so far and leads a local search towards the global minimum experimental comparisons show that the comb produces solutions of quality comparable to simulated annealing

a phase field description of dynamic brittle fracture abstract in contrast to discrete descriptions of fracture phase field descriptions do not require numerical tracking of discontinuities in the displacement field this greatly reduces implementation complexity in this work we extend a phase field model for quasi static brittle fracture to the dynamic case we introduce a phase field approximation to the lagrangian for discrete fracture problems and derive the coupled system of equations that govern the motion of the body and evolution of the phase field we study the behavior of the model in one dimension and show how it influences material properties for the temporal discretization of the equations of motion we present both a monolithic and staggered time integration scheme we study the behavior of the dynamic model by performing a number of two and three dimensional numerical experiments we also introduce a local adaptive refinement strategy and study its performance in the context of locally refined t splines we show that the combination of the phase field model and local adaptive refinement provides an effective method for simulating fracture in three dimensions

a phase field model for rate independent crack propagation robust algorithmic implementation based on operator splits the computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies this can be overcome by a diffusive crack modeling based on the introduction of a crack phase field following our recent work c miehe f welschinger m hofacker thermodynamically consistent phase field models of fracture variational principles and multi field fe implementations international journal for numerical methods in engineering doi 10 1002 nme 2861 on phase field type fracture we propose in this paper a new variational framework for rate independent diffusive fracture that bases on the introduction of a local history field it contains a maximum reference energy obtained in the deformation history which may be considered as a measure for the maximum tensile strain obtained in history it is shown that this local variable drives the evolution of the crack phase field the introduction of the history field provides a very transparent representation of the balance equation that governs the diffusive crack topology in particular it allows for the construction of a new algorithmic treatment of diffusive fracture here we propose an extremely robust operator split scheme that successively updates in a typical time step the history field the crack phase field and finally the displacement field a regularization based on a viscous crack resistance that even enhances the robustness of the algorithm may easily be added the proposed algorithm is considered to be the canonically simple scheme for the treatment of diffusive fracture in elastic solids we demonstrate the performance of the phase field formulation of fracture by means of representative numerical examples

foundational aspects of singular integrals abstract we investigate integration of classes of real valued continuous functions on 0 1 of course difficulties arise if there is a non l 1 element in the class and the hadamard finite part integral p f does not apply such singular integrals arise naturally in many contexts including pdes and singular odes the lebesgue integral as well as p f starting at zero obey two fundamental conditions i they act as antiderivatives and ii if f g on 0 a then their integrals from 0 to x coincide for any x 0 a we find that integrals from zero with the essential properties of p f plus positivity exist by virtue of the axiom of choice ac on all functions on 0 1 which are l 1 e 1 for all e 0 however this existence proof does not provide a satisfactory construction without some regularity at 0 the existence of general antiderivatives which satisfy only i and ii above on classes with a non l 1 element is independent of zf the usual zfc axioms for mathematics without ac and even of zfdc zf with the axiom of dependent choice moreover we show that there is no mathematical description that can be proved within zfc or even extensions of zfc with large cardinal hypotheses to uniquely define such an antiderivative operator such results are precisely formulated for a variety of sets of functions and proved using methods from mathematical logic descriptive set theory and analysis we also analyze p f on analytic functions in the punctured unit disk and make the connection to singular initial value problems

generalized heegner cycles and p adic rankin l series this article studies a distinguished collection of so called generalized heegner cycles in the product of a kuga sato variety with a power of a cm elliptic curve its main result is a p adic analogue of the gross zagier formula which relates the images of generalized heegner cycles under the p adic abel jacobi map to the special values of certain p adic rankin l series at critical points that lie outside their range of classical interpolation

asymptotic expansion of the trace of the heat kernel associated to the dirichlet to neumann operator abstract for a given bounded domain with smooth boundary in a smooth riemannian manifold m g by decomposing the dirichlet to neumann operator into a sum of the square root of the laplacian and a pseudodifferential operator and by applying grubb s method of symbolic calculus for the corresponding pseudodifferential heat kernel operators we establish a procedure to calculate all the coefficients of the asymptotic expansion of the trace of the heat kernel associated to dirichlet to neumann operator as t 0 in particular we explicitly give the first four coefficients of this asymptotic expansion these coefficients provide precise information regarding the area and curvatures of the boundary of the domain in terms of the spectrum of the steklov problem

a new discretization for mth laplace equations with arbitrary polynomial degrees this paper introduces new mixed formulations and discretizations for m th laplace equations of the form 1 m delta m u f for arbitrary m 1 2 3 dots based on novel helmholtz type decompositions for tensor valued functions the new discretizations allow for ansatz spaces of arbitrary polynomial degree and the lowest order choice coincides with the nonconforming fems of crouzeix and raviart for m 1 and of morley for m 2 since the derivatives are directly approximated the lowest order discretizations consist of piecewise affine and piecewise constant functions for any m 1 2 dots moreover a uniform implementation for arbitrary m is possible besides the a priori and a posteriori analysis this paper proves optimal convergence rates for adaptive algorithms for the new discretizations

canonical periods and congruence formulae the purpose of this article is to show how congruences between the fourier coefficients of hecke eigenforms give rise to corresponding congruences between the algebraic parts of the critical values of the associated l functions this study was initiated by b mazur in his fundamental work on the eisenstein ideal see maz77 and maz79 where it was made clear that congruences for analytic l values were closely related to the integral structure of certain hecke rings and cohomology groups the results of maz79 also showed that congruences were useful in the study of nonvanishing of l functions this idea was then further developed by stevens ste82 and rubin wiles rw82 the work of rubin and wiles in particular used congruences to study the behavior of elliptic curves in towers of cyclotomic fields a key ingredient here was a theorem of washington which states roughly that almost l values in certain families are nonzero modulo p this theme has recently been taken up again in the work of ono skinner osa osb james jam and kohnen koh97 while the earlier history was primarily concerned with cyclotomic twists the current emphasis is on families of twists by quadratic characters here one wants quantitative estimates for the number of quadratic twists of a given modular form which have nonvanishing l function at s 1 we continue this trend in the present work by using our general results to obtain a strong nonvanishing theorem for the quadratic twists of modular elliptic curves with rational points of order three this generalizes a beautiful example due to kevin james and provides new evidence for a conjecture of goldfeld gol79 it should however be pointed out that even the study of quadratic twists may be traced back to mazur the reader is urged to look at pages 212 213 of maz79 and especially at the footnote at the bottom of page 213 the theorems of davenport heilbronn dh71 and washington was78 which are crucial in this paper are both mentioned in mazur s article we want to begin by discussing the congruences that lie at the heart of this article thus let f anq n be an elliptic modular cuspform of level m and weight k 2 assume that f is a simultaneous eigenform for all the hecke operators and that a1 f 1 the l function associated to f is defined by the dirichlet series l s f ann s which converges for the real part of s sufficiently large and has analytic continuation to s c a fundamental theorem of shimura shi76 states that l s f enjoys the following algebraicity property

gevrey regularity for a class of sums of squares of monomial vector fields abstract analytic or gevrey hypoellipticity is proved for a class of sums of squares of vector fields having a symplectic characteristic manifold of dimension 2 and arbitrary even codimension we note that this class contains examples for which the treves stratification seems to work as well as examples for which the treves stratification does not identify properly the non symplectic stratum

the weyl type asymptotic formula for biharmonic steklov eigenvalues on riemannian manifolds abstract let be a bounded domain with c 2 smooth boundary in an n dimensional oriented riemannian manifold it is well known that for the biharmonic equation 2 u 0 in with the condition u 0 on there exists an infinite set u k of biharmonic functions in with positive eigenvalues k satisfying u k k u k 0 on in this paper by a new method we establish the weyl type asymptotic formula for the counting function of the biharmonic steklov eigenvalues k

complete determination of the singularity structure of zeta functions series of extended epstein type provide examples of non trivial zeta functions with important physical applications the regular part of their analytic continuation is seen to be a convergent or an asymptotic series their singularity structure is completely determined in terms of the wodzicki residue in its generalized form which is proven to yield the residua of all the poles of the zeta function and not just that of the rightmost pole obtainable from the dixmier trace the calculation is a very down to earth application of these powerful functional analytical methods in physics

on asymptotic properties of biharmonic steklov eigenvalues abstract in this paper by explicitly calculating the principal symbols of pseudodifferential operators we establish two weyl type asymptotic formulas with sharp remainder estimates for the counting functions of the two classes of biharmonic steklov eigenvalue problems of smooth bounded domains in a riemannian manifold

on optimal exponential decay properties of solutions to the korteweg de vries equation abstract we consider the cauchy problem associated to the korteweg de vries equation kdv and study the preservation of exponential decay of order 3 2 on the right of the x axis as time evolves more precisely for a solution of the equation which decays at t 0 as e a 0 x 3 2 for x 0 we find an optimal function a t with a 0 a 0 such that the solution decays as e a t x 3 2 for x t 0

congruence formula for certain dihedral twists in this article we prove a congruence formula for the special values of certain dihedral twists of two primitive modular forms of weight two with isomorphic residual galois representation at a prime p

dual horizon peridynamics a stable solution to varying horizons abstract in this paper we present a dual horizon peridynamics formulation which allows for simulations with dual horizon with minimal spurious wave reflection we prove the general dual property for dual horizon peridynamics based on which the balance of momentum and angular momentum in pd are naturally satisfied we also analyze the crack pattern of random point distribution and the multiple materials issue in peridynamics for selected benchmark problems we show that dh pd is less sensitive to the spatial than the original pd formulation

multidimensional extension of the generalized chowla selberg formula after recalling the precise existence conditions of the zeta function of a pseudodifferential operator and the concept of reflection formula an exponentially convergent expression for the analytic continuation of a multidimensional inhomogeneous epstein type zeta function of the general form with a the p p matrix of a quadratic form a p vector and q a constant is obtained it is valid on the whole complex s plane is exponentially convergent and provides the residua at the poles explicitly it reduces to the famous formula of chowla and selberg in the particular case p 2 q 0 some variations of the formula and physical applications are considered

moyal planes are spectral triples axioms for nonunital spectral triples extending those introduced in the unital case by connes are proposed as a guide and for the sake of their importance in noncommutative quantum field theory the spaces r2n endowed with moyal products are intensively investigated some physical applications such as the construction of noncommutative wick monomials and the computation of the connes lott functional action are given for these noncommutative hyperplanes

schwinger terms and cohomology of pseudodifferential operators we study the cohomology of the schwinger term arising in second quantization of the class of observables belonging to the restricted general linear algebra we prove that for all pseudodifferential operators in 3 1 dimensions of this type the schwinger term is equivalent to the twisted radul cocycle a modified version of the radul cocycle arising in non commutative differential geometry in the process we also show how the ordinary radul cocycle for any pair of pseudodifferential operators in any dimension can be written as the phase space integral of the star commutator of their symbols projected to the appropriate asymptotic component

reproducing kernel hierarchical partition of unity part i formulation and theory this work is concerned with developing the hierarchical basis for meshless methods a reproducing kernel hierarchical partition of unity is proposed in the framework of continuous representation as well as its discretized counterpart to form such hierarchical partition a class of basic wavelet functions are introduced based upon the built in consistency conditions the differential consistency conditions for the hierarchical kernel functions are derived it serves as an indispensable instrument in establishing the interpolation error estimate which is theoretically proven and numerically validated for a special interpolant with different combinations of the hierarchical kernels a synchronized convergence effect may be observed being different from the conventional legendre function based p type hierarchical basis the new hierarchical basis is an intrinsic pseudo spectral basis which can remain as a partition of unity in a local region because the discrete wavelet kernels form a partition of nullity these newly developed kernels can be used as the multi scale basis to solve partial differential equations in numerical computation as a p type refinement copyright 1999 john wiley sons ltd

on asymptotic expansions of twisted products the series development of the quantum mechanical twisted product is studied the series is shown to make sense as a moment asymptotic expansion of the integral formula for the twisted product either pointwise or in the distributional sense depending on the nature of the factors a condition is given that ensures convergence and is stronger than previously known results possible applications are examined