a locally divergence free interior penalty method for two dimensional curl curl problems an interior penalty method for certain two dimensional curl curl problems is investigated in this paper this method computes the divergence free part of the solution using locally divergence free discontinuous p 1 vector fields on graded meshes it has optimal order convergence up to an arbitrarily small epsilon for the source problem and the eigenproblem results of numerical experiments that corroborate the theoretical results are also presented

analysis of multilevel methods for eddy current problems in papers by arnold falk and winther and by hiptmair novel multigrid methods for discrete h curl elliptic boundary value problems have been proposed such problems frequently occur in computational electromagnetism particularly in the context of eddy current simulation this paper focuses on the analysis of those nodal multilevel decompositions of the spaces of edge finite elements that form the foundation of the multigrid methods it provides a significant extension of the existing theory to the case of locally vanishing coefficients and nonconvex domains in particular asymptotically uniform convergence of the multigrid method with respect to the number of refinement levels can be established under assumptions that are satisfied in realistic settings for eddy current problems the principal idea is to use approximate helmholtz decompositions of the function space h curl into an h1 regular subspace and gradients the main results of standard multilevel theory for h1 elliptic problems can then be applied to both subspaces this yields preliminary decompositions still outside the edge element spaces judicious alterations can cure this

edge elements on anisotropic meshes and approximation of the maxwell equations this paper deals with some edge element methods on a class of anisotropic meshes based on tetrahedra and prismatic pentahedral elements anisotropic local interpolation error estimates are derived for all these types of elements and for functions from classical and weighted sobolev spaces as a particular application the numerical approximation of the maxwell equations in domains with edges is investigated using nedelec s edge elements where anisotropic finite element meshes are appropriate some anisotropic regularity results of the solutions of maxwell equations on such domains are proved some numerical tests are described and inverse estimates are established both showing that our theoretical orders of convergence are optimal

impossibility of local state transformation via hypercontractivity local state transformation is the problem of transforming an arbitrary number of copies of a bipartite resource state to a bipartite target state under local operations that is given two bipartite states is it possible to transform an arbitrary number of copies of one of them into one copy of the other state under local operations only this problem is a hard one in general since we assume that the number of copies of the resource state is arbitrarily large in this paper we prove some bounds on this problem using the hypercontractivity properties of some super operators corresponding to bipartite states we measure hypercontractivity in terms of both the usual super operator norms as well as completely bounded norms

consistent relations in the method of reducibility in the cluster variation method a redundancy is noted in the set of lagrange multipliers effective fields and effective interactions used in the expressions of the reduced density matrices given by morita 4 in some problems with less symmetry 6 the expressions of the reduced density matrices without this redundancy are given an argument showing that a certain expression of the free energy in terms of the lagrange multipliers is stationary with respect to the variations of these is presented

c 0 elements for generalized indefinite maxwell equations in this paper we develop the c 0 finite element method for a generalized curlcurl grad div indefinite maxwell problem in a lipschitz domain such as nonconvex polygon for which the solution of the problem may be nonsmooth and only have the h r regularity for some r 1 the ingredients of our method are that two mass lumping l 2 projectors are applied to curl and div operators in the problem and that c 0 linear element or isoparametric bilinear element enriched with one element bubble on every triangle element or with two element bubbles on every quadrilateral element respectively is employed for each component of the nonsmooth solution due to the fact that the element bubbles can be statically eliminated at element levels our method is essentially three nodes or four nodes c 0 lagrange element method with two fortin type interpolations established a very technical duality argument is elaborated to estimate the error for the indefinite problem for the nonsmooth solution having the h r regularity where r may vary in the interval 0 1 we obtain the error bound mathcal o h r in an energy norm some numerical experiments are performed to confirm the theoretical error bounds

quantum reverse hypercontractivity its tensorization and application to strong converses in this paper we develop the theory of quantum reverse hypercontractivity inequalities and show how they can be derived from log sobolev inequalities next we prove a generalization of the stroock varopoulos inequality in the non commutative setting which allows us to derive quantum hypercontractivity and reverse hypercontractivity inequalities solely from 2 log sobolev and 1 log sobolev inequalities respectively we then prove some tensorization type results providing us with tools to prove hypercontractivity and reverse hypercontractivity not only for certain quantum superoperators but also for their tensor powers finally as an application of these results we generalize a recent technique for proving strong converse bounds in information theory via reverse hypercontractivity inequalities to the quantum setting we prove strong converse bounds for the problems of quantum hypothesis testing and classical quantum channel coding based on the quantum reverse hypercontractivity inequalities that we derive

on the use of perfectly matched layers at corners for scattering problems with sign changing coefficients we investigate in a 2d setting the scattering of time harmonic electromagnetic waves by a plasmonic device represented as a non dissipative bounded and penetrable obstacle with a negative permittivity using the t coercivity approach we first prove that the problem is well posed in the classical framework h loc 1 if the negative permittivity does not lie in some critical interval whose definition depends on the shape of the device when the latter has corners for values inside the critical interval unusual strong singularities for the electromagnetic field can appear in that case well posedness is obtained by imposing a radiation condition at the corners to select the outgoing black hole plasmonic wave that is the one which carries energy towards the corners a simple and systematic criterion is given to define what is the outgoing solution finally we propose an original numerical method based on the use of perfectly matched layers at the corners we emphasize that it is necessary to design an ad hoc technique because the field is too singular to be captured with standard finite element methods

matrix poincare φ sobolev inequalities and quantum ensembles sobolev type inequalities have been extensively studied in the frameworks of real valued functions and non commutative lp spaces and have proven useful in bounding the time evolution of classical quantum markov processes among many other applications in this paper we consider yet another fundamental setting matrix valued functions and prove new sobolev type inequalities for them our technical contributions are two fold i we establish a series of matrix poincare inequalities for separably convex functions and general functions with gaussian unitary ensembles inputs and ii we derive sobolev inequalities for matrix valued functions defined on boolean hypercubes and for those with gaussian distributions our results recover the corresponding classical inequalities i e real valued functions when the matrix has one dimension finally as an application of our technical outcomes we derive the upper bounds for a fundamental entropic quantity the holevo quantity in quantum information science since classical quantum channels are a special instance of matrix valued functions this is obtained through the equivalence between the constants in the strong data processing inequality and the sobolev inequality

solving electromagnetic eigenvalue problems in polyhedral domains with nodal finite elements a few years ago costabel and dauge proposed a variational setting which allows one to solve numerically the time harmonic maxwell equations in 3d polyhedral geometries with the help of a continuous approximation of the electromagnetic field in order to remove spurious eigenmodes their method required a parameterization of the variational formulation in order to avoid this difficulty we use a mixed variational setting instead of the parameterization which allows us to handle the divergence free constraint on the field in a straightforward manner the numerical analysis of the method is carried out and numerical examples are provided to show the efficiency of our approach

concentration of quantum states from quantum functional and transportation cost inequalities quantum functional inequalities e g the logarithmic sobolev and poincare inequalities have found widespread application in the study of the behavior of primitive quantum markov semigroups the classical counterparts of these inequalities are related to each other via a so called transportation cost inequality of order 2 tc2 the latter inequality relies on the notion of a metric on the set of probability distributions called the wasserstein distance of order 2 tc2 in turn implies a transportation cost inequality of order 1 tc1 in this paper we introduce quantum generalizations of the inequalities tc1 and tc2 making use of appropriate quantum versions of the wasserstein distances one recently defined by carlen and maas and the other defined by us we establish that these inequalities are related to each other and to the quantum modified logarithmic sobolev and poincare inequalities as in the classical case we also show that these inequalities imply certain concentration type results

hybrid electric magnetic methods in eddy current problems we discuss a method of approximation of eddy current equations in bounded space that relies on the use of the electric field as primary variable in the conductive parts and of the magnetic field in the air theoretical issues about this hybrid approach are addressed is the problem well posed and one shows that edge elements offer a straightforward way to discretize such formulations the now popular a v method is then shown to come as a variant of this formulation in such a way that it allows the use of standard nodal elements however attention is called on the fact that the numerical convergence of the latter method may be in question

the hp local discontinuous galerkin method for low frequency time harmonic maxwell equations the local discontinuous galerkin method for the numerical approximation of the time harmonic maxwell equations in a low frequency regime is introduced and analyzed topologically nontrivial domains and heterogeneous media are considered containing both conducting and insulating materials the presented method involves discontinuous galerkin discretizations of the curl curl and grad div operators derived by introducing suitable auxiliary variables and so called numerical fluxes an hp analysis is carried out and error estimates that are optimal in the meshsize h and slightly suboptimal in the approximation degree p are obtained

a least squares finite element method for the magnetostatic problem in a multiply connected lipschitz domain a new least squares finite element method is developed for the curl div magnetostatic problem in lipschitz and multiply connected domains filled with anisotropic nonhomogeneous materials in order to deal with possibly low regularity of the solution local l 2 projectors are introduced to standard least squares formulation and applied to both curl and div operators coercivity is established by adding suitable mesh dependent bilinear terms as a result any continuous finite elements lower order elements are enriched with suitable bubbles can be employed a desirable error bound is obtained bf u bf u h 0 le c bf u tilde bf u 0 where bf u h and tilde bf u are the finite element approximation and the finite element interpolant of the exact solution bf u respectively numerical tests confirm the theoretical results

the χ2 divergence and mixing times of quantum markov processes we introduce quantum versions of the 2 divergence provide a detailed analysis of their properties and apply them in the investigation of mixing times of quantum markov processes an approach similar to the one presented in the literature for classical markov chains is taken to bound the trace distance from the steady state of a quantum processes a strict spectral bound to the convergence rate can be given for time discrete as well as for time continuous quantum markov processes furthermore the contractive behavior of the 2 divergence under the action of a completely positive map is investigated and contrasted to the contraction of the trace norm in this context we analyze different versions of quantum detailed balance and finally give a geometric conductance bound to the convergence rate for unital quantum markov processes

a high order accurate parallel solver for maxwell s equations on overlapping grids a scheme for the solution of the time dependent maxwell s equations on composite overlapping grids is described the method uses high order accurate approximations in space and time for maxwell s equations written as a second order vector wave equation high order accurate symmetric difference approximations to the generalized laplace operator are constructed for curvilinear component grids the modified equation approach is used to develop high order accurate approximations that use only three time levels and have the same time stepping restriction as the second order scheme discrete boundary conditions for perfect electrical conductors and for material interfaces are developed and analyzed the implementation is optimized for component grids that are cartesian resulting in a fast and efficient method the solver runs on parallel machines with each component grid distributed across one or more processors numerical results in two and three dimensions are presented for the fourth order accurate version of the method these results demonstrate the accuracy and efficiency of the approach

an adaptive finite element method for the h ψ formulation of time dependent eddy current problems in this paper we develop an adaptive finite element method based on reliable and efficient a posteriori error estimates for the h formulation of eddy current problems with multiply connected conductors multiply connected domains are considered by making cuts the competitive performance of the method is demonstrated by an engineering benchmark problem team workshop problem 7 and a singular problem with analytic solution

hardy littlewood theory for semigroups soit t t t 0 un semi groupe symetrique sousmarkovien et soit n 2 alors on a les proprietes equivalentes a dim t t n b il existe c 2 0 s t t t f c 2 t n 2 f 1 f l 1 x t 0 en fait a b meme dans le cas n 2

relating different quantum generalizations of the conditional renyi entropy recently a new quantum generalization of the renyi divergence and the corresponding conditional renyi entropies was proposed here we report on a surprising relation between conditional renyi entropies based on this new generalization and conditional renyi entropies based on the quantum relative renyi entropy that was used in previous literature our result generalizes the well known duality relation h a b h a c 0 of the conditional von neumann entropy for tripartite pure states to renyi entropies of two different kinds as a direct application we prove a collection of inequalities that relate different conditional renyi entropies and derive a new entropic uncertainty relation

a finite element method for a curlcurl graddiv eigenvalue interface problem in this paper we propose and study a finite element method for a curlcurl graddiv eigenvalue interface problem its solution may be of piecewise non h 1 we would like to approximate such a solution in an h 1 conforming finite element space with the discretizations of both curl and div operators of the underlying eigenvalue problem in two finite element spaces the proposed method is essentially a standard h 1 conforming element method up to element bubbles which can be statically eliminated at element levels we first analyze the proposed method for the related source interface problem by establishing the stability and the error bounds we then analyze the underlying eigenvalue interface problem and we obtain the error bounds mathcal o h 2r 0 for eigenvalues which correspond to eigenfunctions in prod j 1 j h r omega j 3 hookrightarrow h r 0 omega 3 space where the piecewise regularity r and the global regularity r 0 may belong to the most interesting interval 0 1

numerical solution to the time dependent maxwell equations in axisymmetric singular domains the singular complement method in this paper we present a method to solve numerically the axisymmetric time dependent maxwell equations in a singular domain in math methods appl sci 25 2002 49 math methods appl sci 26 2003 861 the mathematical tools and an in depth study of the problems posed in the meridian half plane were exposed the numerical method and experiments based on this theory are now described here it is also the generalization to axisymmetric problems of the singular complement method that we developed to solve maxwell equations in 2d singular domains see c r acad sci paris t 330 2000 391 it is based on a splitting of the space of solutions in a regular subspace and a singular one derived from the singular solutions of the laplace problem numerical examples are finally given to illustrate our purpose in particular they show how the singular complement method captures the singular part of the solution

a discontinuous galerkin method on refined meshes for the two dimensional time harmonic maxwell equations in composite materials in this paper a discontinuous galerkin method for the two dimensional time harmonic maxwell equations in composite materials is presented the divergence constraint is taken into account by a regularized variational formulation and the tangential and normal jumps of the discrete solution at the element interfaces are penalized due to an appropriate mesh refinement near exterior and interior corners the singular behaviour of the electromagnetic field is taken into account optimal error estimates in a discrete energy norm and in the l 2 norm are proved in the case where the exact solution is singular

singularities of electromagnetic fields in polyhedral domains in this paper we investigate the singular solutions of time harmonic maxwell equations in a domain which has edges and polyhedral corners it is now well known that in the presence of non convex edges the solution fields have no square integrable gradients in general and that the main singularities are the gradients of singular functions of the laplace operator 4 5 we show how this type of result can be derived from the classical mellin analysis and how this analysis leads to sharper results concerning the singular parts which belong to h 1 for the singular functions we exhibit simple and explicit formulas based on generalized dirichlet and neumann singularities for the laplace operator these formulas are more explicit than the results announced in our note 10

quantum logarithmic sobolev inequalities and rapid mixing a family of logarithmic sobolev inequalities on finite dimensional quantum state spaces is introduced the framework of non commutative lp spaces is reviewed and the relationship between quantum logarithmic sobolev inequalities and the hypercontractivity of quantum semigroups is discussed this relationship is central for the derivation of lower bounds for the logarithmic sobolev ls constants essential results for the family of inequalities are proved and we show an upper bound to the generalized ls constant in terms of the spectral gap of the generator of the semigroup these inequalities provide a framework for the derivation of improved bounds on the convergence time of quantum dynamical semigroups when the ls constant and the spectral gap are of the same order convergence bounds on finite dimensional state spaces are particularly relevant for the field of quantum information theory we provide a number of examples where improved bounds on the mixing time of several semigroups are obtained incl

on a finite element method for solving the three dimensional maxwell equations abstract the aim of this paper is to present a method for solving the time domain three dimensional maxwell equations which can be coupled with a particle solver for this purpose maxwell s equations are reformulated as a constrained wave equation system with lagrange multipliers associated to the conditions 0 and e e 0 we approximate both the fields and the lagrange multipliers with a finite element method using a taylor hood element

continuous galerkin methods for solving the time dependent maxwell equations in 3d geometries a few years ago costabel and dauge proposed a variational setting which allows one to solve numerically the time harmonic maxwell equations in 3d geometries with the help of a continuous approximation of the electromagnetic field in this paper we investigate how their framework can be adapted to compute the solution to the time dependent maxwell equations in addition we propose some extensions such as the introduction of a mixed variational setting and its discretization to handle the constraint on the divergence of the field 2007 elsevier inc all rights reserved

on different finite element methods for approximating the gradient of the solution to the helmholtz equation abstract we consider the numerical solution of the helmholtz equation by different finite element methods in particular we are interested in finding an efficient method for approximating the gradient of the solution we first approximate the gradient by the standard ritz galerkin method as a second method a two stage method due to aziz and werschulz 1980 is presented it is shown that this method gives the same accuracy in the computed gradient and in the computed solution also in the nonconforming case finally a direct method with asymptotic error estimates is given it turns out that the presented direct method is of lowest computational complexity test examples are presented to illustrate the accuracy of the methods

discontinuous least squares finite element method for the div curl problem in this paper we consider the div curl problem posed on nonconvex polyhedral domains we propose a least squares method based on discontinuous elements with normal and tangential continuity across interior faces as well as boundary conditions weakly enforced through a properly designed least squares functional discontinuous elements make it possible to take advantage of regularity of given data divergence and curl of the solution and obtain convergence also on nonconvex domains in general this is not possible in the least squares method with standard continuous elements we show that our method is stable derive a priori error estimates and present numerical examples illustrating the method

hypercontractivity for semigroups of unital qubit channels hypercontractivity is proved for products of qubit channels that belong to self adjoint semigroups the hypercontractive bound gives necessary and sufficient conditions for a product of the form e t 1 h 1 otimes cdots otimes e t n h n to be a contraction from l p to l q where l p is the algebra of 2 n dimensional matrices equipped with the normalized schatten norm and each generator h j is a self adjoint positive semidefinite operator on the algebra of 2 dimensional matrices as a particular case the result establishes the hypercontractive bound for a product of qubit depolarizing channels

interface solutions of partial differential equations with point singularity abstract partial differential equations pdes such as those that describe linear elastic fracture mechanics admit analytic solutions with low regularity at crack tips existing numerical methods for such pdes suffer from the constraint of this low regularity and fail to deliver an optimal high order rate of convergence we approach this problem by i choosing an artificial interface to enclose the center of low regularity and ii representing the solution in the interior of artificial interface as unknown linear combination of known low regularity modes of the solution this results in an interface formulation of the original pde and the linear combination is represented in the interface conditions by enforcing the smooth component of the numerical solution in the interior domain to be approximately zero a least squares problem is obtained for the unknown coefficients the solution of this least squares problem will provide the approximate interface conditions for the numerical solution of the pde in the exterior domain the potential of our interface formulation is favorably demonstrated by numerical experiments on one and two dimensional poisson equations with low regularity solutions high order numerical solutions of unknown coefficients and pdes are obtained this proves the potential of the proposed interface formulation as the theoretical basis for solving linear elastic fracture mechanics problems we address the relations between our interface formulation and domain decomposition methods as well as a regularization strategy for the poisson boltzmann equation with singular charge density

augmented formulations for solving maxwell equations we consider augmented variational formulations for solving the static or time harmonic maxwell equations for that a term is added to the usual h curl conforming formulations it consists of a weighted l2 scalar product between the divergence of the em and the divergence of test fields in this respect the methods we present are h curl div conforming we also build mixed augmented variational formulations with either one or two lagrange multipliers to dualize the equation on the divergence and when applicable the relation on the tangential or normal trace of the field it is proven that one can derive formulations which are equivalent to the original static or time harmonic maxwell equations in the latter case spurious modes are automatically excluded numerical analysis and experiments will be presented in the forthcoming paper augmented formulations for solving maxwell equations numerical analysis and experiments in preparation

mixed discontinuous galerkin approximation of the maxwell operator we introduce and analyze a discontinuous galerkin discretization of the maxwell operator in mixed form here all the unknowns of the underlying system of partial differential equations are approximated by discontinuous finite element spaces of the same order for piecewise constant coefficients the method is shown to be stable and optimally convergent with respect to the mesh size numerical experiments highlighting the performance of the proposed method for problems with both smooth and singular analytical solutions are presented

stabilized interior penalty methods for the time harmonic maxwell equations we propose stabilized interior penalty discontinuous galerkin methods for the indefinite time harmonic maxwell system the methods are based on a mixed formulation of the boundary value problem chosen to provide control on the divergence of the electric field we prove optimal error estimates for the methods in the special case of smooth coefficients and perfectly conducting boundary using a duality approach

hypercontractivity and the logarithmic sobolev inequality for the completely bounded norm we develop the notions of hypercontractivity hc and the log sobolev ls inequality for completely bounded norms of one parameter semigroups of super operators acting on matrix algebras we prove the equivalence of the completely bounded versions of hc and ls under suitable hypotheses we also prove a version of the gross lemma which allows ls at general q to be deduced from ls at q 2

analysis of a continuous finite element method for h mathrm curl mathrm div elliptic interface problem in this paper we develop a continuous finite element method for the curlcurl grad div vector second order elliptic problem in a three dimensional polyhedral domain occupied by discontinuous nonhomogeneous anisotropic materials in spite of the fact that the curlcurl grad div interface problem is closely related to the elliptic interface problem of vector laplace operator type the continuous finite element discretization of the standard variational problem of the former generally fails to give a correct solution even in the case of homogeneous media whenever the physical domain has reentrant corners and edges to discretize the curlcurl grad div interface problem by the continuous finite element method we apply an element local l 2 projector to the curl operator and a pseudo local l 2 projector to the div operator where the continuous lagrange linear element enriched by suitable element and face bubbles may be employed it is shown that the finite element problem retains the same coercivity property as the continuous problem an error estimate mathcal o h r in an energy norm is obtained between the analytical solution and the continuous finite element solution where the analytical solution is in prod l 1 l h r omega l 3 for some r in 1 2 1 due to the domain boundary reentrant corners and edges e g nonconvex polyhedron and due to the interfaces between the different material domains in omega bigcup l 1 l omega l

hypercontractivity in noncommutative lpspaces we introduce and study a setup for hypercontractivity in the interpolating family of noncommutative lpspaces associated to a locally normal state on an inductive limitc algebra

approximations for a frustrated ising model abstract the cluster effective field approximation was proposed as a useful approximation for the study of frustrated ising models in order to examine its usefulness the ising model with frustration on the centred square lattice is studied in the square effective field approximation and the results are compared with those in the pair and in the square approximation and with the exact results the study is made in the framework of the cluster variation method

mathematical analysis and finite element simulation of a magnetized ferrite model microwave scattering by obstacles made of ferrite materials has been studied for a long time because of its important applications in radar satellite optical imaging system signal processing atmospheric science and even cloaking device design in this paper we present for the first time the rigorous mathematical analysis of a magnetized ferrite model widely used by engineers the well posedness of the model is established first then we develop a finite element time domain method for solving this model numerical stability similar to the continuous stability is proved for our scheme and the optimal error estimate in the l 2 norm is obtained numerical results are consistent with our theoretical analysis and show how wave propagates in the ferrite material

a singular field method for maxwell s equations numerical aspects for 2d magnetostatics the present paper deals with the solution of maxwell type problems by means of nodal h1 conforming finite elements in a nonconvex piecewise regular domain surrounded by a perfect conductor such a discretization cannot in general approximate the singular behavior of the electromagnetic field near reentrant corners or edges the singular field method consists of adding to the finite element discretization space some particular fields which take into account the singular behavior the latter are deduced from the singular functions associated with the scalar laplace operator the theoretical justification of this approach as well as the analysis of the convergence of the approximation are presented for a very simple model problem arising from magnetostatics in a translation invariant setting but the study can be easily extended to numerous maxwell type problems the numerical implementation of both variants is studied for a domain containing a single reentrant corner

an adaptive perfectly matched layer technique for 3 d time harmonic electromagnetic scattering problems an adaptive perfectly matched layer pml technique for solving the time harmonic electromagnetic scattering problems is developed the pml parameters such as the thickness of the layer and the fictitious medium property are determined through sharp a posteriori error estimates combined with the adaptive finite element method the adaptive pml technique provides a complete numerical strategy to solve the scattering problem in the framework of fem which produces automatically a coarse mesh size away from the fixed domain and thus makes the total computational costs insensitive to the thickness of the pml absorbing layer numerical experiments are included to illustrate the competitive behavior of the proposed adaptive method

regular decomposition and a framework of order reduced methods for fourth order problems this paper is devoted to the construction of order reduced method of fourth order problems a constructive framework is presented such that a problem on the high regularity space can be deduced to an equivalent system on three low regularity spaces which are connected by a regular decomposition corresponding to a decomposition of the regularity of the high order space the generated numerical schemes based on the deduced problems can be of lower complicacy and the framework is fit for various fourth order problems three fourth order problems are then discussed under the framework including one in two dimension and two in three dimension they are each corresponding to a regular decomposition and thus are discretised based on the discretised analogues of the regular decomposition optimal error estimates are given

quantum reverse hypercontractivity we develop reverse versions of hypercontractive inequalities for quantum channels by generalizing classical techniques we prove a reverse hypercontractive inequality for tensor products of qubit depolarizing channels we apply this to obtain a rapid mixing result for depolarizing noise applied to large subspaces and to prove bounds on a quantum generalization of non interactive correlation distillation

robust a posteriori error estimation for the maxwell equations we consider a robust a posteriori error estimator of residual type for the maxwell equations in a bounded two and three dimensional domain the continuous problem is approximated using conforming approximated spaces the main goal is to express the lower and upper bounds with respect to appropriate norms for that purpose a new interpolant of clement nedelec type is introduced and some interpolation error estimates are proved numerical tests are presented which confirm our theoretical results

computation of maxwell singular solution by nodal continuous elements abstract in this paper we propose and analyze a nodal continuous and h 1 conforming finite element method for the numerical computation of maxwell s equations with singular solution in a fractional order sobolev space h r where r may take any value in the most interesting interval 0 1 the key feature of the method is that mass lumping linear finite element l 2 projections act on the curl and divergence partial differential operators so that the singular solution can be sought in a setting of l 2 space we shall use the nodal continuous linear finite elements enriched with one element bubble in each element to approximate the singular and non h 1 solution discontinuous and nonhomogeneous media are allowed in the method some error estimates are given and a number of numerical experiments for source problems as well as eigenvalue problems are presented to illustrate the superior performance of the proposed method

time dependent maxwell s equations with charges in singular geometries this paper is devoted to the solution of the instationary maxwell equations with charges the geometry of the domain can be singular in the sense that its boundary can include reentrant corners or edges the difficulties arise from the fact that those geometrical singularities generate in their neighborhood strong electromagnetic fields the time dependency of the divergence of the electric field is addressed to tackle this problem some new theoretical and practical results are presented on curl free singular fields and on singular fields with l2 non vanishing divergence the method which allows to compute the instationary electromagnetic field is based on a splitting of the spaces of solutions into a two term direct sum first the subspace of regular fields it coincides with the whole space of solutions provided that the domain is either convex or with a smooth boundary second a singular subspace defined and characterized via the singularities of the laplace operator several numerical examples are presented to illustrate the mathematical framework this paper is the generalization of the singular complement method

exact boundary controllability of the second order maxwell system theory and numerical simulation the exact controllability of the second order time dependent maxwell equations for the electric field is addressed through the hilbert uniqueness method a two grid preconditioned conjugate gradient algorithm is employed to inverse the h u m operator and to construct the numerical control the underlying initial value problems are discretized by lagrange finite elements and an implicit newmark scheme two dimensional numerical experiments illustrate the performance of the method

a singular field method for the solution of maxwell s equations in polyhedral domains it is well known that in the case of a regular domain the solution of the time harmonic maxwell s equations allows a discretization by means of nodal finite elements this is achieved by solving a regularized problem similar to the vector helmholtz equation the present paper deals with the same problem in the case of a nonconvex polyhedron it is shown that a nodal finite element method does not approximate in general the solution to maxwell s equations but actually the solution to a neighboring variational problem involving a different function space indeed the solution to maxwell s equations presents singularities near the edges and corners of the domain that cannot be approximated by lagrange finite elements a new method is proposed involving the decomposition of the solution field into a regular part that can be treated numerically by nodal finite elements and a singular part that has to be taken into account explicitly this singular field method is presented in various situations such as electri

a simplified cluster variation method abstract a variational calculation is given of an approximation scheme which is far easier to apply than the cluster variation method in the higher approximations the curie temperature of the ising model in the square approximation in the present scheme is obtained as 2 6253 and 4 7611 respectively for the system on the square and the sc lattice these values are nearly equal to the mean values of those in the pair bethe and the square kikuchi approximation the phase diagram in the square approximation of the present scheme is compared with other approximations and the exact result for a frustrated ising model on the centred square lattice

some applications of hypercontractive inequalities in quantum information theory hypercontractive inequalities have become important tools in theoretical computer science and have recently found applications in quantum computation in this note we discuss how hypercontractive inequalities in various settings can be used to obtain fairly concise proofs of several results in quantum information theory a recent lower bound of lancien and winter on the bias achievable by local measurements which are 4 designs spectral concentration bounds for k local hamiltonians and a recent result of pellegrino and seoane sepulveda giving general lower bounds on the classical bias obtainable in multiplayer xor games

a delta regularization finite element method for a double curl problem with divergence free constraint to deal with the divergence free constraint in a double curl problem rm curl mu 1 rm curl u f and rm div varepsilon u 0 in omega where mu and varepsilon represent the physical properties of the materials occupying omega we develop a delta regularization method rm curl mu 1 rm curl u delta delta varepsilon u delta f to completely ignore the divergence free constraint rm div varepsilon u 0 we show that u delta converges to u in h rm curl omega norm as delta rightarrow 0 the edge finite element method is then analyzed for solving u delta with the finite element solution u delta h a quasioptimal error bound in the h rm curl omega norm is obtained between u and u delta h including a uniform with respect to delta stability of u delta h in the h rm curl omega norm all the theoretical analysis is done in a general setting where mu and varepsilon may be discontinuous an

multigrid in h div and h curl summary we consider the solution of systems of linear algebraic equations which arise from the finite element discretization of variational problems posed in the hilbert spaces bf h div and bf h curl in three dimensions we show that if appropriate finite element spaces and appropriate additive or multiplicative schwarz smoothers are used then the multigrid v cycle is an efficient solver and preconditioner for the discrete operator all results are uniform with respect to the mesh size the number of mesh levels and weights on the two terms in the inner products

positivity improving operators and hypercontractivity here 1 1 is the famous nelson hypercontractivity inequality 14 which has proved to be extremely useful in euclidean quantum field theory see e g 16 like 1 1 the estimate 1 2 follows very easily from the it6 calculus of brownian motion 3 here in order to make the paper self contained and to introduce the reader to some general principles of interest we submit a new proof of 1 2 by mimicing a standard proof of 1 1 we should like to add that 1 2 has at least a formal resemblance to the familiar correlation inequalities from statistical mechanics see e g 11 12 16 using 1 1 and 1 2 it is simple to obtain similar estimates for the biquantization of a linear contraction on hilbert space sect 4 now suppose a is a self adjoint operator on l 2 of a probability space and set a a maxspec a sup spec a maxspec a where spec a denotes the spectrum of a below we will state conditions that guarantee that the gap a a is proper in fact a simplified version of our main result theorem 5 3 gives a positive numerical lower bound of a a if i a is positivity preserving ii sup ilzf i1 i f 12 1 k 2 and iii inf izfllp v i l f l l 1 f 0 k 0 some 0 p 1 here i iii imply that a has a unique ground state it shall be empha

a quantitative arrow theorem arrow s impossibility theorem states that any constitution which satisfies independence of irrelevant alternatives iia and unanimity and is not a dictator has to be non transitive in this paper we study quantitative versions of arrow theorem consider n voters who vote independently at random each following the uniform distribution over the six rankings of three alternatives arrow s theorem implies that any constitution which satisfies iia and unanimity and is not a dictator has a probability of at least 6 n for a non transitive outcome when n is large 6 n is a very small probability and the question arises if for large number of voters it is possible to avoid paradoxes with probability close to 1 here we give a negative answer to this question by proving that for every epsilon 0 there exists a delta delta epsilon 0 which depends on epsilon only such that for all n and all constitutions on three alternatives if the constitution satisfies the iia condition for every pair of alternatives a b the probability that the constitution ranks a above b is at least epsilon for every voter i the probability that the social choice function agrees with a dictatorship on i at most 1 epsilon then the probability of a non transitive outcome is at least our results generalize to any number k 3 of alternatives and to other distributions over the alternatives we further derive a quantitative characterization of all social choice functions satisfying the iia condition whose outcome is transitive with probability at least 1 our results provide a quantitative statement of arrow theorem and its generalizations and strengthen results of kalai and keller who proved quantitative arrow theorems for k 3 and for balanced constitutions only i e for constitutions which satisfy for every pair of alternatives a b that the probability that the constitution ranks a above b is exactly 1 2 the main novel technical ingredient of our proof is the use of inverse hypercontractivity to show that if the outcome is transitive with high probability then there are no two different voters who are pivotal with for two different pairwise preferences with non negligible probability another important ingredient of the proof is the application of non linear invariance to lower bound the probability of a paradox for constitutions where all voters have small probability for being pivotal

a nonconforming finite element method for a two dimensional curl curl and grad div problem a numerical method for a two dimensional curl curl and grad div problem is studied in this paper it is based on a discretization using weakly continuous p 1 vector fields and includes two consistency terms involving the jumps of the vector fields across element boundaries optimal convergence rates up to an arbitrary positive epsilon in both the energy norm and the l 2 norm are established on graded meshes the theoretical results are confirmed by numerical experiments

mathematical analysis of conducting and superconducting transmission lines this paper is concerned with a mathematical study of guided propagation in the microstrip transmission lines used in microelectronics in the first part the case of a zero thickness perfectly conducting strip is considered using a regularized formulation of maxwell s equations it is shown that finding guided modes amounts to the spectral analysis of a noncompact family of self adjoint operators the existence of guided modes is then proved thanks to the min max principle in the second part we deal with the case of a zero thickness superconducting strip an asymptotic model derived from london s equation is studied and the existence of guided modes is deduced from the case of the perfectly conducting strip copyright 2000 society for industrial and applied mathematics

a hinged plate equation and iterated dirichlet laplace operator on domains with concave corners abstract fourth order hinged plate type problems are usually solved via a system of two second order equations for smooth domains such an approach can be justified however when the domain has a concave corner the bi laplace problem with navier boundary conditions may have two different types of solutions namely u 1 with u 1 u 1 h 1 and u 2 h 2 h 1 we will compare these two solutions a striking difference is that in general only the first solution obtained by decoupling into a system preserves positivity that is a positive source implies that the solution is positive the other type of solution is more relevant in the context of the hinged plate we will also address the higher dimensional case our main analytical tools will be the weighted sobolev spaces that originate from kondratiev in two dimensions we will show an alternative that uses conformal transformation next to rigorous proofs the results are illustrated by some numerical experiments for planar domains

a new and simple implementation of the element local l2 projected continuous finite element method in this paper we present a new and simple method modified from the original element local l 2 projected continuous finite element method to resolve some static electromagnetic problems in two dimension this numerical method is established on the least square process to minimize the total error s energy where the residual due to an additional element local projection is considered many numerical experiments are presented for the electrostatic problem the maxwell s equations and the maxwell s eigenvalue problem and the numerical performances are excellent with the optimal convergence orders even for the non h 1 solution

optimal hypercontractivity for fermi fields and related non commutative integration inequalities optimal hypercontractivity bounds for the fermion oscillator semigroup are obtained these are the fermion analogs of the optimal hypercontractivity bounds for the boson oscillator semigroup obtained by nelson in the process several results of independent interest in the theory of non commutative integration are established

iterative solutions for electromagnetic fields at perfectly reflective and transmissive interfaces using clifford algebra and the multidimensional cauchy integral a technique is presented for calculating the solution of maxwell s equations using a cfie based on the cauchy integral and formulated in the guise of clifford algebra this formulation has a geometric interpretation leading to an iterative method of solution which is easily proven as convergent and correct for both perfectly reflective and perfectly transmissive interfaces simple test cases involving a cubic boundary and plane wave and dipole sources are used to investigate the numerical performance of the method

a matrix reverse holder inequality abstract a matrix reverse holder inequality is given this result is a counterpart to the concavity property of matrix weighted geometric means it extends a scalar inequality due to gheorghiu and contains several kantorovich type inequalities

an adaptive multilevel method for time harmonic maxwell equations with singularities we develop an adaptive edge finite element method based on reliable and efficient residual based a posteriori error estimates for low frequency time harmonic maxwell equations with singularities the resulting discrete problem is solved by the multigrid preconditioned minres minimum residual iteration algorithm we demonstrate the efficiency and robustness of the proposed method by extensive numerical experiments for cavity problems with singular solutions which include in particular scattering over screens

numerical solution to the time dependent maxwell equations in two dimensional singular domains in this paper we present a method to solve numerically the time dependent maxwell equations in nonsmooth and nonconvex domains indeed the solution is not of regularity h1 in space in general moreover the space of h1 regular fields is not dense in the space of solutions thus an h1 conforming finite element method can fail even with mesh refinement the situation is different than in the case of the laplace problem or of the lame system for which mesh refinement or the addition of conforming singular functions work to cope with this difficulty the singular complement method is introduced this method consists of adding some well chosen test functions these functions are derived from the singular solutions of the laplace problem also the scm preserves the interesting features of the original method easiness of implementation low memory requirements small cost in terms of the cpu time to ascertain its validity some concrete problems are solved numerically