the singular set of solutions to non differentiable elliptic systems we estimate the hausdorff dimension of the singular set of solutions to elliptic systems of the type if the vector fields a and b are holder continuous with respect to the variable x with exponent then the hausdorff dimension of the singular set of any weak solution is at most n 2

a discrete geometric approach to solving time independent schrodinger equation the time independent schrodinger equation stems from quantum theory axioms as a partial differential equation this work aims at providing a novel discrete geometric formulation of this equation in terms of integral variables associated with precise geometric elements of a pair of three dimensional interlocked grids one of them based on tetrahedra we will deduce in a purely geometric way a computationally efficient discrete counterpart of the time independent schrodinger equation in terms of a standard symmetric eigenvalue problem moreover boundary and interface conditions together with non homogeneity and anisotropy of the media involved are accounted for in a straightforward manner this approach yields to a sensible computational advantage with respect to the finite element method where a generalized eigenvalue problem has to be solved instead such a modeling tool can be used for analyzing a number of quantum phenomena in modern nano structured devices where the accounting of the real 3d geometry is a crucial issue

singularities of energy minimizing maps from the ball to the sphere we study maps from the unit ball b in r3 to the unit sphere s2 in r3 which minimize dirichlet s energy integral varepsilon varphi int b nabla varphi 2 dv

baker campbell hausdorff relations for the orthosymplectic group osp 1 2 baker campbell hausdorff relations are presented for the connected supergroup associated with the orthosymplectic algebra osp 1 2

new coherent states of the lie superalgebra osp 1 2 r a new kind of coherent states css for lie superalgebra osp l 2 r are introduced some properties of these states are discussed it is shown that they can encompass the glauber css su 1 1 css and squeezed states within a common formalism the d algebra differential operator realization of the osp 1 2 r generators on the css and their diagonal projectors are constructed

optimal partial regularity for sub elliptic systems with dini continuous coefficients under the superquadratic natural growth the paper is concerned with nonlinear sub elliptic systems with dini continuous coefficients under super quadratic m 2 natural growth conditions in the heisenberg group a general criterion for weak solutions to be regular in the neighborhood of a given point is established by a generalization of method of a harmonic approximation in particular our result is optimal in the sense that in the case of holder continuous coefficients we obtain directly the optimal holder exponent for horizontal gradient of weak solutions on its regular set

representations of osp 2 1 and the metaplectic representation shift operator techniques are used to treat the irreducible representations of the superalgebra osp 2 1 apart from obtaining the well known gradestar dispin representations which arise when the even part is the compact su 2 algebra the case when the star conditions on the even part are those satisfied by the noncompact su 1 1 algebra is also treated in this case no gradestar representations arise and the star representations are found to consist of the direct sum of two discrete series representations of su 1 1 one of these representations can be realized in terms of functions of a single complex variable and turns out to be a simple example of a metaplectic representation

homotopy classes in sobolev spaces and the existence of energy minimizing maps consider the problem of finding maps f m n that are stationary for an energy functional such as fm idfl p where m and n are compact riemannian manifolds and p l such maps may be found by minimizing the functional but if we minimize among all maps from m to n then the minimum is 0 and is attained only by constant maps thus in order to find nontrivial stationary maps we would like to use the topology of m and n to define classes of maps from m to n in which we can minimize the functional for instance one could try to minimize among maps in a given homotopy class but this is not possible in general unless p dimm since a minimizing sequence of mappings in one homotopy class can converge in the appropriate weak topology to a map in another homotopy class however in this paper we show that it is possible to minimize among maps fwhose restrictions to a lower dimensional skeleton of a triangulation of m belong to a given homotopy class to state the results precisely we need to refer to certain sobolev spaces and norms we will assume without loss of generality that m and n are submanifolds of euclidean spaces r m and r respectively and we let lip m n denote the space of lipschitz maps from m to n we define li p m r to be the space of all functions fe lp m r n such that there exist functions

dirichlet and neumann boundary value problems for yang mills connections the yang mills equations have been extensively studied on compact riemannian manifolds without boundary a substantial amount of work has also been done on non compact examples with a boundary component at infinity see 5 10 11 14 15 here we define and describe a solution of dirichlet and neumann boundary value problems for yang mills connections on a compact four dimensional manifold with smooth boundary the equations in the interior are the full yang mills equations we do not restrict ourselves to self dual or anti self dual solutions in the case where the structure group in the gauge theory is abelian our results are contained in the hodge de rham theory for manifolds with boundary see 6 9 4 3 in the general case our results describe a non linear version of this theory the base manifold m of the theory is a compact orientable four dimensional manifold with smooth boundary m we assume that m is endowed with a riemannian metric we consider a principal fiber bundle p over m with structure group g a compact lie group which will have a metric invariant under the adjoint action in the usual example this comes from trace the unknown in the problem is the connection a i e a differential form in p ad g where g is the lie algebra of g there are two natural boundary value problems one of them requires fixing the tangential part of the connection a on the bundle restricted to m if m is locally described as x 0 this amounts to prescribing a1 x dx a2 x dx 2 a3 x dx 3 on m the connection is fixed in the sense that a particular choice is fixed in the gauge equivalence class of a on the boundary the yang mills equations are satisfied in the interior this corresponds to the dirichlet problem in the other boundary value problem we consider a is free at the boundary but the normal part of the curvature is set equal to zero we call this the neumann problem for the dirichlet problem we consider a sequence of l1 connections that minimizes the yang mills functional with prescribed admissible tangential boundary data we show that there exists a weak limiting connection a by finding local good gauges the limiting connection is only locally defined we show that a solves the yang mills equations and satisfies the boundary conditions up to

smoothness of harmonic maps for hypoelliptic diffusions harmonic maps are viewed as maps sending a fixed diffusion to manifold valued martingales under a convexity condition we prove that the continuity of real valued harmonic functions implies the continuity of harmonic maps then we prove with a probabilistic method that continuous harmonic maps are smooth under hormander s condition the proof relies on the study of martingales with values in the tangent bundle

liquid crystals and the monotonicity formula abstract we give numerical evidence that the monotonicity formula holds for the liquid crystal functional and study part of the associated bifurcation diagram

state of the art eigensolvers for electronic structure calculations of large scale nano systems the band edge states determine optical and electronic properties of semiconductor nano structures which can be computed from an interior eigenproblem we study the reliability and performance of state of the art iterative eigensolvers on large quantum dots and wires focusing on variants of preconditioned cg lanczos and davidson methods one davidson variant the gd k olsen method is identified to be as reliable as the commonly used preconditioned cg while consistently being between two and three times faster

a fourier wachspress method for solving helmholtz s equation in three dimensional layered domains in this paper we present a fast direct method for solving poisson s or helmholtz s equation in three dimensional layered domains the method combines a fourier method for two dimensions and a variant of wachspress s method in the third dimension the resulting scheme is capable of efficiently creating solutions that are highly accurate even when the coefficients defining the layered structure are discontinuous or extreme mesh refinement is used

the group with grassmann structure uosp 1 2 the finite dimensional representations of the lie superalgebraosp 1 2 and the group with grassmann structureosp 1 2 have been studied the explicit expression of the projection operator of the superalgebraosp 1 2 has been found the operator permits an arbitrary finite dimensional representation to be expanded in the components multiple to the irreducible ones the clebsch gordan coefficients for the tensor product of two arbitrary irreducible representations have been obtained the matrix elements of the irreducible representations of the groupuosp 1 2 the analoque of the compact form of the grouposp 1 2 are studied the explicit form of these matrix elements the differential equations satisfied by them and the integral of their product have been found

path integral for new coherent states of lie superalgebra osp 1 2 r a path integral formulation in the representation of new coherent states for the lie superalgebra osp l 2 r is introduced by the use of the completeness relation of the new coherent states a path integral expression for the transition amplitude between two osp l 2 r new coherent states for a hamiltonian which is linear in the generators of the superalgebra are obtained in the classical limit the equations of motion for the system are derived

a rayleigh chebyshev procedure for finding the smallest eigenvalues and associated eigenvectors of large sparse hermitian matrices a procedure is presented for finding a number of the smallest eigenvalues and their associated eigenvectors of large sparse hermitian matrices the procedure a modification of an inverse subspace iteration procedure uses adaptively determined chebyshev polynomials to approximate the required application of the inverse operator on the subspace the method is robust converges with acceptable rapidity and can easily handle operators with eigenvalues of multiplicity greater than one numerical results are shown that demonstrate the utility of the procedure

partial regularity for discontinuous sub elliptic systems with subquadratic growth in the heisenberg group abstract we consider nonlinear sub elliptic systems with vmo coefficients in the heisenberg group and prove partial holder continuity for discontinuous sub elliptic problems with subquadratic growth by a generalization of the technique of a harmonic approximation the model case is the sub elliptic p laplace system with vmo coefficients in the form i 1 2 n x i a i 1 x u 2 p 2 2 x i u 0 where a i v m o and 1 p 2

path integrals and supercoherent states for the real supergroup osp 1 2 r with both its compact and noncompact versions supercoherent states are introduced with a method close to the one by perelomov for the even subgroups su 2 or su 1 1 these states labeled by a complex c number and grassmann variable minimize the uncertainty of the quadratic casimir operator of the group a path integral formalism is developed for the transition amplitude between supercoherent states for a hamiltonian linear in the generators of the superalgebra which leads to a super riccati equation finally in the classical limit the canonical equations of motion are derived which involve a generalized super poisson bracket

unitary representations of non compact supergroups we give a general theory for the construction of oscillator like unitary irreducible representations uirs of non compact supergroups in a super fock space this construction applies to all non compact supergroupsg whose coset spaceg k with respect to their maximal compact subsupergroupk is hermitean supersymmetric we illustrate our method with the example of su m p n q by giving its oscillator like uirs in a particle state basis as well as supercoherent state basis the same class of uirs can also be realized over the super hilbert spaces of holomorphic functions of az variable labelling the coherent states

on the regularity of harmonic functions and spherical harmonic maps defined on lattices abstract a growth lemma for certain discrete symmetric laplacians defined on a lattice z d zd rd with spacing is proved the lemma implies a de giorgi theorem that the harmonic functions for these laplacians are equi holder continuous 0 these results are then applied to establish regularity properties for the harmonic maps defined on zd and taking values in an n dimensional sphere sn uniform in questions of the convergence 0 and the dirichlet problem for these discrete harmonic maps are also addressed

path integrals for quantum algebras and the classical limit coherent states path integral formalism for the simplest quantum algebras q oscillator suq 2 and suq 1 1 is introduced in the classical limit canonical structure is derived with a modified symplectic and riemannian metric non constant deformation induced curvature for the phase spaces is obtained

complementary geometric formulations for electrostatics summary the simultaneous use of a pair of complementary discrete formulations for electrostatic boundary value problems bvps allows to accurately compute electromagnetic quantities such as capacitance or electrostatic force with a minimum computational effort in fact the two formulations provide the upper and lower bounds for these quantities and their averages result quite close to the exact solution even for extremely coarse meshes despite the potential benefit to the many three dimensional large scale applications taking advantage of this feature is not exploited in practice due to theoretical difficulties in the potential design the aim of this paper is to fill this gap by rigorously introducing a pair of three dimensional complementary geometric formulations to solve electrostatic bvps on domains covered by conformal polyhedral meshes in particular an original formulation based on a vector potential is introduced by using cohomology theory with integer coefficients it is shown how the so called thick links are needed which are representatives of the second cohomology group generators of the dielectric region two easy to implement graph theoretic algorithms to automatically find such a basis with optimal computational complexity are described some benchmark problems are presented to show how the simultaneous use of both formulations yields to a sensible computational advantage therefore solvers based on complementary formulations should be embedded in the next generation of electromagnetic computer aided engineering cae softwares copyright 2010 john wiley sons ltd

coherent states for the noncompact supergroups osp 2 2n r an analysis of the coherent states for the noncompact supergroup osp 2 2n r is presented in contrast to osp 1 2n r both typical and atypical representations have to be considered the measure of integration in general for osp 2 2n r coherent states is calculated it is then used to construct the decomposition of unity for the special case of osp 2 2 r it is found however that the typical and atypical representations of osp 2 2 r have to be treated separately it is verified that the coherent states for osp 2 2 r are closest to classical in the sense of perelomov

c1 partial regularity for sub elliptic systems with dini continuous coefficients in carnot groups abstract in this paper we consider nonlinear sub elliptic systems with dini continuous coefficients for the case 1 m 2 in carnot groups based on a generalization of the technique of a harmonic approximation introduced by duzaar grotowski kronz and a sobolev poincare type inequality established in carnot groups a c 1 partial regularity result for weak solutions of sub elliptic systems with natural growth terms is established in particular our result is optimal in the sense that in the case of holder continuous coefficients it yields directly the optimal holder exponent for horizontal gradients of weak solutions on its regular set

partial regularity for sub elliptic systems with dini continuous coefficients involving natural growth terms in the heisenberg group abstract we consider nonlinear sub elliptic systems with dini continuous coefficients in divergence form in the heisenberg group based on a generalization of the method of a harmonic approximation introduced by duzaar and steffen partial regularity of weak solutions for sub elliptic systems under natural growth conditions is established in particular our result is optimal in the sense that in the case of holder continuous coefficients we obtain directly the optimal holder exponent for the horizontal gradients of weak solutions on the regular set

path integral formalism for suq 2 coherent states a path integral formulation in the representation of coherent states for the quantum group suq 2 is introduced an expression for the transition amplitude connecting two suq 2 coherent states is constructed and the corresponding canonical equations of motion are also derived

supergroups factorizations through matrix realization using a matrix realization generic elements of the supergroups u m n and osp m 2n are obtained through the exponentiation of the corresponding super lie algebra elements the emphasis is put on the contribution of the odd part the application to the factorization problem is given and the relationship with nonlinear differential superequations is clarified

harmonic maps from noncompact riemannian manifolds with non negative ricci curvature outside a compact set in this paper we prove the uniqueness and existence of harmonic maps of finite energy from a complete noncompact riemannian manifold m g with sobolev constant s2 m 0 and ricci curvature ric m 0 outside some compact subset into a complete manifold of nonpositive curvature or a regular ball in particular we prove the uniqueness and existence of bounded harmonic functions on m g

partial regularity for discontinuous sub elliptic systems with vmo coefficients involving controllable growth terms in heisenberg groups abstract in this paper we consider discontinuous sub elliptic systems with vmo coefficients for the controllable growth case of p 2 in the heisenberg group based on a generalization of the technique of a harmonic approximation with the superquadratic growth case a partial holder continuity result for weak solutions of discontinuous sub elliptic systems with vmo coefficients is established in particular the primary model covered by our analysis is the non degenerate p sub laplace system involving super quadratic controllable growth terms

interior partial regularity for nonlinear elliptic systems with dini continuous coefficients for the case 1 m 2 in this article we consider the interior regularity for weak solutions of nonlinear elliptic systems with divergence type and dini continuous coefficients under subquadratic growth conditions we adopt another a harmonic approximation technique and prove the interior partial regularity of weak solutions the proof yields directly the optimal holder exponent for the derivative of the weak solutions on the regular set

self consistent field calculations using chebyshev filtered subspace iteration the power of density functional theory is often limited by the high computational demand in solving an eigenvalue problem at each self consistent field scf iteration the method presented in this paper replaces the explicit eigenvalue calculations by an approximation of the wanted invariant subspace obtained with the help of well selected chebyshev polynomial filters in this approach only the initial scf iteration requires solving an eigenvalue problem in order to provide a good initial subspace in the remaining scf iterations no iterative eigensolvers are involved instead chebyshev polynomials are used to refine the subspace the subspace iteration at each step is easily five to ten times faster than solving a corresponding eigenproblem by the most efficient eigen algorithms moreover the subspace iteration reaches self consistency within roughly the same number of steps as an eigensolver based approach this results in a significantly faster scf iteration

regularity of harmonic maps with prescribed singularities in this paper we studied the regularity problem for harmonic maps into hyperbolic spaces with prescribed singularities along codimension two submanifolds this is motivated from one of hawking s conjectures on the uniqueness of kerr solutions among all axially symmetric asymptotically flat stationary solutions to the vacuum einstein equation in general relativity

partially constrained boundary conditions with energy minimizing mappings on considere la regularite a la frontiere pour une application minimisant l energie u m n pour deux types de conditions aux limites frontiere libre et condition limite partiellement contrainte

biaxial escape in nematics at low temperature abstract in the present work we study minimizers of the landau de gennes free energy in a bounded domain r 3 we prove that at low temperature minimizers do not vanish even for topologically non trivial boundary conditions this is in contrast with a simplified ginzburg landau model for superconductivity studied by bethuel brezis and helein merging this with an observation of canevari we obtain as a corollary the occurrence of biaxial escape the tensorial order parameter must become strongly biaxial at some point in in particular while it is known that minimizers cannot be purely uniaxial we prove the much stronger and physically relevant fact that they lie in a different homotopy class

compactness theorems for coupled yang mills fields in this paper we consider coupled yang mills fields on vector bundle e over compact riemannian manifold m under appropriate conditions on the curvature and the higgs field two compactness theorems are proved

efficient solution of the schroedinger poisson equations in layered semiconductor devices in this paper we present several mathematical models that can be used to create approximate solutions of the three dimensional schroedinger poisson equation in layered semiconductor devices a general algorithmic strategy that can be used to create efficient solution procedures for each of these models is described computational results demonstrating the accuracy and efficiency that can be obtained with the use of these models is presented

an l p regularity theory for harmonic maps motivated by the harmonic map heat flow we consider maps between riemannian manifolds such that the tension field belongs to an lp space under an appropriate smallness condition a certain degree of regularity follows for suitable solutions of the harmonic map heat flow we have a partial regularity result as a consequence

parabolic theory as a high dimensional limit of elliptic theory the aim of this article is to show how certain parabolic theorems follow from their elliptic counterparts this technique is demonstrated through new proofs of five important theorems in parabolic unique continuation and the regularity theory of parabolic equations and geometric flows specifically we give new proofs of an l 2 carleman estimate for the heat operator and the monotonicity formulas for the frequency function associated to the heat operator the two phase free boundary problem the flow of harmonic maps and the mean curvature flow the proofs rely only on the underlying elliptic theorems and limiting procedures belonging essentially to probability theory in particular each parabolic theorem is proved by taking a high dimensional limit of the related elliptic result

time ordering techniques and solution of differential difference equation appearing in quantum optics time ordering techniques based on the magnus expansion and the wei norman algebraic procedure are discussed and their relevance and usefulness to quantum optics are stressed

optimal partial regularity of second order nonlinear elliptic systems with dini continuous coefficients for the superquadratic case abstract we consider the regularity for weak solutions of second order nonlinear elliptic systems with dini continuous coefficients for the superquadratic case under natural growth condition and obtain a general criterion for a weak solution to be regular in the neighborhood of a given point in particular the proof yields directly the optimal holder exponent for the derivative of the weak solutions on the regular set