nonlinear scattering theory at low energy abstract we study the scattering theory of a conservative nonlinear one parameter group of operators on a hilbert space x relative to a group of linear unitary operators under certain hypotheses the scattering operator carries a neighborhood of 0 in x into x the theory is designed to apply to the semilinear schrodinger and klein gordon equations

imposing dirichlet boundary conditions in the extended finite element method this paper is devoted to the imposition of dirichlet type conditions within the extended finite element method x fem this method allows one to easily model surfaces of discontinuity or domain boundaries on a mesh not necessarily conforming to these surfaces imposing neumann boundary conditions on boundaries running through the elements is straightforward and does preserve the optimal rate of convergence of the background mesh observed numerically in earlier papers on the contrary much less work has been devoted to dirichlet boundary conditions for the x fem or the limiting case of stiff boundary conditions in this paper we introduce a strategy to impose dirichlet boundary conditions while preserving the optimal rate of convergence the key aspect is the construction of the correct lagrange multiplier space on the boundary as an application we suggest to use this new approach to impose precisely zero pressure on the moving resin front in resin transfer moulding rtm process while avoiding remeshing the case of inner conditions is also discussed as well as two important practical cases material interfaces and phase transformation front capturing copyright 2006 john wiley sons ltd

x fem an efficient algorithm for simulation of phase transition during prostate cryosurgery abstract cryosurgery is a surgical technique to ablate abnormal tumor tissues with the help of extreme cold using cryogens such as liquid nitrogen solid carbon dioxide argon etc this therapy may be used as a primary treatment for early prostate cancers effectively during the process of freezing in cryosurgery a phase transition occurs between the frozen tumor tissues and unfrozen surrounding tissues the present study aims to showcase the efficiency of fixed grid x fem to numerically simulate the evolving phase transition front with time which further illustrates location dimensions of ice ball generated after freezing it has been observed that x fem with a fixed mesh is capable to capture phase transitions front and temperature fields accurately while traditional form of fem requires adaptive re meshing at each time step from physician point of view obtained temperature fields are really important to decide the extent of freezing location of different cryoprobes and numbers of cryoprobes to be used for a successful treatment

on the exponent of a primitive digraph abstract we characterize the equality case of the upper bound d n s n 2 for the exponent of a primitive digraph in the case s 2 where n is the number of the vertices of the digraph d and s is the length of the shortest elementary circuit of d we also answer a question about the equality case when s 1 the exponent of an n n primitive nonnegative matrix a is the same as the exponent of the associated digraph d a of a so every theorem in this paper can be translated into a theorem about nonnegative matrices

the discontinuity enriched finite element method we introduce a new methodology for modeling problems with both weak and strong discontinuities independently of the finite element discretization at variance with the extended generalized finite element method x gfem the new method named the discontinuity enriched finite element method de fem adds enriched degrees of freedom only to nodes created at the intersection between a discontinuity and edges of elements in the mesh although general the method is demonstrated in the context of fracture mechanics and its versatility is illustrated with a set of traction free and cohesive crack examples we show that de fem recovers the same rate of convergence as the standard fem with matching meshes and we also compare the new approach to x gfem

nonlinear scattering theory at low energy sequel abstract we extend the nonlinear scattering theory and apply it to schrodinger and klein gordon equations with various kinds of nonlinear terms including some nonlocal ones

a nonlinear schrodinger equation with two symmetric point interactions in one dimension we consider a time dependent one dimensional nonlinear schrodinger equation with a symmetric double well potential represented by two dirac s among our results we give an explicit formula for the integral kernel of the unitary semigroup associated with the linear part of the hamiltonian then we establish the corresponding strichartz type estimate and we prove local existence and uniqueness of the solution to the original nonlinear problem

conditional stability theorem for the one dimensional klein gordon equation the paper addresses the conditional non linear stability of the steady state solutions of the one dimensional klein gordon equation for large time we explicitly construct the center stable manifold for the steady state solutions using the modulation method of soffer and weinstein and strichartz type estimates the main difficulty in the one dimensional case is that the required decay of the klein gordon semigroup does not follow from strichartz estimates alone we resolve this issue by proving an additional weighted decay estimate and further refinement of the function spaces which allows us to close the argument in spaces with very little time decay

decay for the wave and schrodinger evolutions on manifolds with conical ends part ii let c n be a compact imbedded riemannian manifold of dimension d 1 and define the d 1 dimensional riemannian manifold m x r x w x with r 0 and smooth and the natural metric ds 2 1 r x 2 dx 2 r 2 x ds 2 we require that m has conical ends r x x o x 1 as x the hamiltonian flow on such manifolds always exhibits trapping dispersive estimates for the schrodinger evolution e it m and the wave evolution e it m are obtained for data of the form f x y n u x where y n are eigenfunctions of with eigenvalues u 2 n in this paper we discuss all cases d n 1 if n 0 there is the following accelerated local decay estimate with 0 1 e it m y n f l m c n m t d 1 2 w 1 f l 1 m where w x x and similarly for the wave evolution our method combines two main ingredients a a detailed scattering analysis of schrodinger operators of the form 2 v 2 1 4 2 u on the line where u is real valued and smooth with u l 0 3 l for all l 0 as and v 0 in particular we introduce the notion of a zero energy resonance for this class and derive an asymptotic expansion of the wronskian between the outgoing jost solutions as the energy tends to zero in particular the division into part i and part ii can be explained by the former being resonant at zero energy where the present paper deals with the nonresonant case b estimation of oscillatory integrals by non stationary phase

lp lp estimates for the schrodinger equation on the line and inverse scattering for the nonlinear schrodinger equation with a potential abstract in this paper i prove a l p l p estimate for the solutions to the one dimensional schrodinger equation with a potential in l 1 where in the generic case 3 2 and in the exceptional case i e when there is a half bound state of zero energy 5 2 i use this estimate to construct the scattering operator for the nonlinear schrodinger equation with a potential i prove moreover that the low energy limit of the scattering operator uniquely determines the potential and the coupling constant of the nonlinearity using a method that allows as well for the reconstruction of the potential and of the nonlinearity

lp continuity of wave operators in z abstract we recover for discrete schrodinger operators on the lattice z stronger analogues of the results by weder r weder the w k p continuity of the schrodinger wave operators on the line comm math phys 208 1999 507 520 and by d ancona and fanelli p d ancona l fanelli l p boundedness of the wave operator for the one dimensional schrodinger operator comm math phys 268 2006 415 438 on r

asymptotic stability of small gap solitons in nonlinear dirac equations we prove dispersive decay estimates for the one dimensional dirac operator and use them to prove asymptotic stability of small gap solitons in the nonlinear dirac equations with quintic and higher order nonlinear terms

a stable and optimally convergent generalized fem sgfem for linear elastic fracture mechanics abstract in this paper we investigate the accuracy and conditioning of the stable generalized fem sgfem and compare it with standard generalized fem gfem for a 2 d fracture mechanics problem the sgfem involves localized modifications of enrichments used in the gfem and the conditioning of the stiffness matrix in this method is of the same order as in the fem numerical experiments show that using the sgfem with only the modified heaviside functions which are used as enrichments in the gfem to approximate the solution of fracture problems in 2 d gives inaccurate results however the sgfem using an additional set of enrichment function yields accurate results while not deteriorating the conditioning of the stiffness matrix rules for the selection of the optimal set of enrichment nodes based on the definition of enrichment functions used in the sgfem are also presented this set leads to optimal convergence rates while keeping the number of degrees of freedom equal to or close to the gfem we show that it is necessary to enrich additional nodes when the crack line is located along element edges in 2 d the selection of these nodes depends on the definition of the enrichment functions at the crack discontinuity a simple and yet generic implementation strategy for the sgfem in an existing gfem xfem software is described the implementation can be used with 2 d and 3 d elements it leads to an efficient evaluation of sgfem enrichment functions

wielandt s proof of the exponent inequality for primitive nonnegative matrices the proof of the exponent inequality found in wielandt s unpublished diaries of a result announced without proof in his well known paper on nonnegative irreducible matrices a facsimile a transcription a translation and a commentary are presented

dispersion for the schrodinger equation on networks in this paper we consider the schrodinger equation on a network formed by a tree with the last generation of edges formed by infinite strips we give an explicit description of the solution of the linear schrodinger equation with constant coefficients this allows us to prove dispersive estimates which in turn are useful for solving the nonlinear schrodinger equation the proof extends also to the laminar case of positive step function coefficients having a finite number of discontinuities

a corrected xfem approximation without problems in blending elements the extended finite element method xfem enables local enrichments of approximation spaces standard finite elements are used in the major part of the domain and enriched elements are employed where special solution properties such as discontinuities and singularities shall be captured in elements that blend the enriched areas with the rest of the domain problems arise in general these blending elements often require a special treatment in order to avoid a decrease in the overall convergence rate a modification of the xfem approximation is proposed in this work the enrichment functions are modified such that they are zero in the standard elements unchanged in the elements with all their nodes being enriched and varying continuously in the blending elements all nodes in the blending elements are enriched the modified enrichment function can be reproduced exactly everywhere in the domain and no problems arise in the blending elements the corrected xfem is applied to problems in linear elasticity and optimal convergence rates are achieved copyright 2007 john wiley sons ltd

a bound on the exponent of primitivity in terms of diameter abstract an n n nonnegative matrix a is called primitive if for some positive integer k every entry in ak is positive ak 0 the exponent of primitivity of a is defined to be a min k z ak 0 where z denotes the set of positive integers two conjectures due to hartwig neumann and lin in 1984 or so are that a m 1 2 1 and a d2 1 where m is the degree of the minimal polynomial of a and d is the diameter of the directed graph of a it is well known that the latter is stronger than the former because d m 1 in a recent paper we have proved a m 1 2 1 in this paper we prove the conjecture a d2 1 on that basis

the lp lp estimate for the schrodinger equation on the half line abstract in this paper we prove the lp lp estimate for the schrodinger equation on the half line and with homogeneous dirichlet boundary condition at the origin

dispersive effects for the schrodinger equation on the tadpole graph abstract we consider the free schrodinger group e i t d 2 d x 2 on the tadpole graph r we first show that the time decay estimates l 1 r l r is in t 1 2 with a constant independent of the circumference of the circle our proof is based on an appropriate decomposition of the kernel of the resolvent further we derive a dispersive perturbation estimate which proves that the solution on the queue of the tadpole converges uniformly after compensation of the underlying time decay to the solution of the neumann half line problem as the circle shrinks to a point to obtain this result we suppose that the initial condition fulfills a high frequency cutoff

reconstruction of a potential on the line that is a priori known on the half line in this paper the authors study the inverse schrodinger scattering on the real line a method is given that allows unique reconstruction of a potential that is a priori known on the half line from the knowledge of the reflection coefficient and the bound state energies in particular no information on the forming constants is required the method is based on an appropriate trace formula and on the solution of the nonlinear ordinary differential equation that is obtained when the potential is replaced by its trace formula in the schrodinger equation the deift trubowitz approach to inverse scattering is followed the main new point is the way in which bound states are treated in addition to its mathematical interest the case when the potential is a priori known on the half line is particularly interesting in many applications one can consider for example a potential that has compact support or that it is zero on a half line

stability of propagation features under time asymptotic approximations for a class of dispersive equations abstract we consider solutions of dispersive equations on the line defined by fourier multipliers with initial data having compactly supported fourier transforms in this paper a refinement of an existing method permitting to expand time asymptotically the solution formulas is proposed here the first term of the expansion is supported in a space time cone whose origin depends explicitly on the initial datum as an important consequence of our refined method the first term inherits the mean position of the solution together with a constant variance error and a shifted time decay rate is obtained hence this refinement which takes into account both spatial and frequency information of the initial datum makes stable some propagation features under time asymptotic approximations and permits a better description of the time asymptotic behaviour of the solutions the results are achieved firstly by making apparent the cone origin in the solution formula secondly by applying precisely an adapted version of the stationary phase method with a new error bound and finally by minimising the error bound with respect to the cone origin

enrichment of enhanced assumed strain approximations for representing strong discontinuities addressing volumetric incompressibility and the discontinuous patch test summary we present a geometrically nonlinear assumed strain method that allows for the presence of arbitrary intra finite element discontinuities in the deformation map special attention is placed on the coarsemesh accuracy of these methods and their ability to avoid mesh locking in the incompressible limit given an underlying mesh and an arbitrary failure surface we first construct an enriched approximation for the deformation map with the nonlinear analogue of the extended finite element method x fem with regard to the richer space of functions spanned by the gradient of the enriched approximation we then adopt a broader interpretation of variational consistency for the construction of the enhanced strain in particular in those elements intersected by the failure surface we construct enhanced strain approximations which are orthogonal to piecewise constant stress fields contrast is drawn with existing strong discontinuity approaches where the enhanced strain variations in localized elements were constructed to be orthogonal to constant nominal stress fields importantly the present formulation gives rise to a symmetric tangent stiffness matrix even in localized elements the present modification also allows for the satisfaction of a discontinuous patch test wherein two different constant stress fields on each side of the failure surface lie in the solution space we demonstrate how the proposed modifications eliminate spurious stress oscillations along the failure surface particularly for nearly incompressible material response additional numerical examples are provided to illustrate the efficacy of the modified method for problems in hyperelastic fracture mechanics copyright c 2000 john wiley sons ltd

a hierarchical detection framework for computational contact mechanics a novel methodology consisting of three hierarchical levels is proposed for the detection phase of contact mechanics simulations the top level of the hierarchy uses kinematic information from the objects involved in the simulation to determine approximate collision times these instants then determine when the engine resumes operation for further detection by using bounding volume hierarchies the second level of detection precludes contact by computing simple exclusion tests on bounding volumes of increasing tightness when contact cannot be ruled out by using simple tests the final level of detection comes into effect by using thorough checks on finite element primitives to that purpose a robust optimization based formulation that does not rely on orthogonal projections is outlined the detection framework can be used to predict the exact collision time among finite element discretizations the performance of the proposed methodology is investigated with a set of examples

a diameter bound on the exponent of a primitive directed graph abstract a directed graph g is primitive if there exists a positive integer k such that for every pair of vertices u v g there is a walk from u to v of length k the least such k is called the exponent of g we define g k to be the directed graph having the same vertex set as g and arcs u v if and only if there is a walk in g of length k from vertex u to vertex v a well known upper bound for the exponent of a primitive directed graph g of order n is n 1 2 1 due to h wielandt in 1950 our main result is the following refinement of the wielandt bound if g is a primitive directed graph with diameter d then the exponent of g is at most d 2 1 we construct the primitive graphs for which equality is attained and we generalize the bound to the class of irreducible matrices in the course of proving the main result we find the following which is interesting in its own right if g is a primitive directed graph with diameter d then the diameter of g k is at most d for all positive integers k

modeling quasi static crack growth with the extended finite element method part i computer implementation abstract the extended finite element method x fem is a numerical method for modeling strong displacement as well as weak strain discontinuities within a standard finite element framework in the x fem special functions are added to the finite element approximation using the framework of partition of unity for crack modeling in isotropic linear elasticity a discontinuous function and the two dimensional asymptotic crack tip displacement fields are used to account for the crack this enables the domain to be modeled by finite elements without explicitly meshing the crack surfaces and hence quasi static crack propagation simulations can be carried out without remeshing in this paper we discuss some of the key issues in the x fem and describe its implementation within a general purpose finite element code the finite element program dynaflow is considered in this study and the implementation for modeling 2 d cracks in isotropic and bimaterial media is described in particular the array allocation for enriched degrees of freedom use of geometric based queries for carrying out nodal enrichment and mesh partitioning and the assembly procedure for the discrete equations are presented we place particular emphasis on the design of a computer code to enable the modeling of discontinuous phenomena within a finite element framework

dispersive estimates for higher dimensional schrodinger operators with threshold eigenvalues i the odd dimensional case we investigate l1 rn l rn dispersive estimates for the schrodinger operator h v when there is an eigenvalue at zero energy and n 5 is odd in particular we show that if there is an eigenvalue at zero energy then there is a time dependent rank one operator ft satisfying ft l1 l t 2 n2 for t 1 such that eithpac ft l1 l t 1 n2 for t 1 with stronger decay conditions on the potential it is possible to generate an operator valued expansion for the evolution taking the form eithpac h t 2 n2a 2 t 1 n2a 1 t n2a0 with a 2 and a 1 finite rank operators mapping l1 rn to l rn while a0 maps weighted l1 spaces to weighted l spaces the leading order terms a 2 and a 1 vanish when certain orthogonality conditions between the potential v and the zero energy eigenfunctions are satisfied we show that under the same orthogonality conditions the remaining t n2a0 term also exists as a map from l1 rn to l rn hence eithpac h satisfies the same dispersive bounds as the free evolution despite the eigenvalue at zero

a generalized finite element method for the simulation of three dimensional dynamic crack propagation this paper is aimed at presenting a partition of unity method for the simulation of three dimensional dynamic crack propagation the method is a variation of the partition of unity finite element method and hp cloud method in the context of crack simulation this method allows for modeling of arbitrary dynamic crack propagation without any remeshing of the domain in the proposed method the approximation spaces are constructed using a partition of unity pu and local enrichment functions the pu is provided by a combination of shepard and finite element partitions of unity this combination of pus allows the inclusion of arbitrary crack geometry in a model without any modification of the initial discretization it also avoids the problems associated with the integration of moving least squares or conventional shepard partitions of unity used in several meshless methods the local enrichment functions can be polynomials or customized functions these functions can efficiently approximate the singular fields around crack fronts the crack propagation is modeled by modifying the partition of unity along the crack surface and does not require continuous remeshings or mappings of solutions between consecutive meshes as the crack propagates in contrast with the boundary element method the proposed method can be applied to any class of problems solvable by the classical finite element method in addition the proposed method can be implemented into most finite element data bases several numerical examples demonstrating the main features and computational efficiency of the proposed method for dynamic crack propagation are presented

a 3d interface enriched generalized finite element method for weakly discontinuous problems with complex internal geometries an interface enriched generalized finite element method gfem is introduced for 3d problems with discontinuous gradient fields the proposed method differs from conventional gfem by assigning the generalized degrees of freedom to the interface nodes i e nodes generated along the interface when creating integration subdomains instead of the nodes of the original mesh a linear combination of the lagrangian shape functions in these integration subelements are then used as the enrichment functions to capture the discontinuity in the gradient field this approach provides a great flexibility for evaluating the enrichment functions including for cases where elements are intersected with multiple interfaces we show that the method achieves optimal rate of convergence with meshes which do not conform to the phase interfaces at a computational cost similar to or lower than that of conventional gfem the potential of the method is demonstrated by solving several heat transfer problems with discontinuous gradient field encountered in particulate and fiber reinforced composites and in actively cooled microvascular materials

dispersive estimate for the 1d schrodinger equation with a steplike potential abstract we prove a sharp dispersive estimate p a c u t x c t 1 2 u 0 l 1 r for the one dimensional schrodinger equation i u t u x x v x u v 0 x u 0 where 1 x 2 v l 1 r and v 0 is a step function real valued and constant on the positive and negative real axes

a new formulation for imposing dirichlet boundary conditions on non matching meshes generating matching meshes for problems with complex boundaries is often an intricate process and the use of non matching meshes appears as an appealing solution yet enforcing boundary conditions on non matching meshes is not a straightforward process especially when prescribing those of dirichlet type by combining a type of generalized finite element method gfem with the lagrange multiplier method a new technique for the treatment of essential boundary conditions on non matching meshes is introduced in this manuscript the new formulation yields a symmetric stiffness matrix and is straightforward to implement as a result the methodology makes possible the analysis of problems with the use of simple structured meshes irrespective of the problem domain boundary through the solution of linear elastic problems we show that the optimal rate of convergence is preserved for piecewise linear finite elements yet the formulation is general and thus it can be extended to other elliptic boundary value problems

kirchhoff s rule for quantum wires we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the hamiltonian to be minus the laplace operator with general boundary conditions at the vertices this results in a scattering theory with n channels the corresponding on shell s matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy e 0 is given explicitly in terms of the boundary conditions and the lengths of the internal lines it is shown to be unitary which may be viewed as the quantum version of kirchhoff s law we exhibit covariance and symmetry properties it is symmetric if the boundary conditions are real also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low energy behaviour of one theory gives the high energy behaviour of the transformed theory finally we provide a composition rule by which the on shell s matrix of a graph is factorizable in terms of the s matrices of its subgraphs all proofs use only known facts from the theory of self adjoint extensions standard linear algebra complex function theory and elementary arguments from the theory of hermitian symplectic forms

the exponent and circumdiameter of primitive digraphs abstract the exponent d of a primitive digraph d is the smallest m such that for each ordered pair of not necessarily distinct vertices u v there exists a u v walk of length m if d is the set of all cycle lengths then the circumdiameter of d denoted dc d is the maximum over all ordered pairs of not necessarily distinct vertices u v of the length of a shortest u v walk that intersects cycles of all lengths it is well known that d d dc d in which d is the frobenius schur index several new sufficient conditions and families of digraphs for which equality holds in the above upper bound are given and some families of digraphs d for which d d dc d 1 are defined additional sufficient conditions for equality in the above upper bound for d and a new upper bound for dc d are given for digraphs with large exponent that is digraphs on n vertices having d n 1 2 1 2 2 the circumdiameter and bounds on the exponent for the digraph of a leslie matrix are found

a note on the eigenvalues of a primitive matrix with large exponent abstract let a be a primitive stochastic matrix of order n 7 and exponent at least n 1 2 1 2 2 we describe the general form of the characteristic polynomial of a and prove that a must have at least 2 n 4 4 complex eigenvalues of modulus at greater than 1 2 sin n 1 2 n 1 observe that this last quantity tends to 1 as n both combinatorial and algebraic arguments are used to establish the result

an interface enriched generalized fem for problems with discontinuous gradient fields summary a new generalized fem is introduced for solving problems with discontinuous gradient fields the method relies on enrichment functions associated with generalized degrees of freedom at the nodes generated from the intersection of the phase interface with element edges the proposed approach has several advantages over conventional generalized fem formulations such as a lower computational cost easier implementation and straightforward handling of dirichlet boundary conditions a detailed convergence study of the proposed method and a comparison with the standard fem are presented for heat transfer problems the method achieves the optimal rate of convergence using meshes that do not conform to the interfaces present in the domain while achieving a level of accuracy comparable to that of the standard fem with conforming meshes various application problems are presented including the conjugate heat transfer problem encountered in microvascular materials copyright 2011 john wiley sons ltd

dispersive estimates for schrodinger operators in dimensions one and three we consider l 1 l estimates for the time evolution of hamiltonians h v in dimensions d 1 and d 3 with bound t 2 we require decay of the potentials but no regularity in d 1 the decay assumption is 1 x v x dx whereas in d 3 it is v x c 1 x 3

dispersive analysis of charge transfer models we prove dispersive estimates for the time dependent schrodinger equation with a charge transfer hamiltonian as a by product we also obtain another proof of asymptotic completeness of the wave operators for a charge transfer model established earlier by k yajima and j m graf we also consider a more general matrix non self adjoint charge transfer problem this model appears naturally in the study of nonlinear multisoliton systems and is specifically motivated by the problem of asymptotic stability of multisoliton states of a nonlinear schrodinger equation 2004 wiley periodicals inc

the extended finite element method xfem for solidification problems an enriched finite element method for the multi dimensional stefan problems is presented in this method the standard finite element basis is enriched with a discontinuity in the derivative of the temperature normal to the interface the approximation can then represent the phase interface and the associated discontinuity in the temperature gradient within an element the phase interface can be evolved without re meshing or the use of artificial heat capacity techniques the interface is described by a level set function that is updated by a stabilized finite element scheme several examples are solved by the proposed method to demonstrate the accuracy and robustness of the method copyright 2001 john wiley sons ltd

stable generalized finite element method sgfem abstract the generalized finite element method gfem is a partition of unity method pum where the trial space of standard finite element method fem is augmented with non polynomial shape functions with compact support these shape functions which are also known as the enrichments mimic the local behavior of the unknown solution of the underlying variational problem gfem has been successfully used to solve a variety of problems with complicated features and microstructure however the stiffness matrix of gfem is badly conditioned much worse compared to the standard fem and there could be a severe loss of accuracy in the computed solution of the associated linear system in this paper we address this issue and propose a modification of the gfem referred to as the stable gfem sgfem we show that sgfem retains the excellent convergence properties of gfem does not require a ramp function in the presence of blending elements and the conditioning of the associated stiffness matrix is not worse than that of the standard fem moreover sgfem is very robust with respect to the parameters of the enrichments we show these features of sgfem on several examples

from continuous to discontinuous failure in a gradient enhanced continuum damage model abstract a computational framework for the description of the combined continuous discontinuous failure in a regularised strain softening continuum is proposed the continuum is regularised through the introduction of gradient terms into the constitutive equations at the transition to discrete failure the problem fields are enhanced through a discontinuous interpolation based on the partition of unity paradigm of finite element shape functions the inclusion of internal discontinuity surfaces where boundary conditions are applied without modifications of the original finite element mesh avoids the unrealistic damage growth typical of this class of regularised continuum models combined models allow for the analysis of the entire failure process from diffuse microcracking to localised macrocracks the discretisation procedure is described in detail and numerical examples illustrate the performance of the combined continuous discontinuous approach

primitive digraphs with smallest large exponent abstract a primitive digraph d on n vertices has large exponent if its exponent d satisfies n d w n where n w n 2 2 and w n n 1 2 1 it is shown that the minimum number of arcs in a primitive digraph d on n 5 vertices with exponent equal to n is either n 1 or n 2 explicit constructions are given for fixed n even and odd for a primitive digraph on n vertices with exponent n and n 2 arcs these constructions extend to digraphs with some exponents between n and w n a necessary and sufficient condition is presented for the existence of a primitive digraph on n vertices with exponent n and n 1 arcs together with some number theoretic results this gives an algorithm that determines for fixed n whether the minimum number of arcs is n 1 or n 2

nonlinear schrodinger equations with exceptional potentials abstract we consider the cubic nonlinear schrodinger equation with an exceptional potential we obtain a sharp time decay for the global in time solution and we get the large time asymptotic profile of small solutions we prove the existence of modified scattering for this model that is linear scattering modulated by a phase our approach is based on the spectral theorem for the perturbed linear schrodinger operator and a factorization technique that allows us to control the resonant nonlinear term we make some parity assumptions in order to control the small energy behavior of the scattering coefficients and of the wave functions

on the spectral theory and dispersive estimates for a discrete schrodinger equation in one dimension based on the recent work komech et al dispersive estimates for 1d discrete schrodinger and klein gordon equations appl anal 85 1487 2006 for compact potentials we develop the spectral theory for the one dimensional discrete schrodinger operator h v n 1 n 1 2 n vn n we show that under appropriate decay conditions on the general potential and a nonresonance condition at the spectral edges the spectrum of h consists of finitely many eigenvalues of finite multiplicities and the essential absolutely continuous spectrum while the resolvent satisfies the limiting absorption principle and the puiseux expansions near the edges these properties imply the dispersive estimates eithpa c h l 2 l 2 t 3 2 for any fixed 52 and any t 0 where pa c h denotes the spectral projection to the absolutely continuous spectrum of h in addition based on the scattering theory for the discrete jost solutions and the previous results by stefanov and kevrekidis asymptotic behaviour of small