the radial volume derivative and the critical boundary displacement for cavitation we study the displacement boundary value problem of minimizing the total energy e mathbf u stored in a nonlinearly elastic body occupying a spherical domain b in its reference configuration over possibly discontinuous radial deformations mathbf u of the body subject to affine boundary data mathbf u mathbf x lambda mathbf x for mathbf x in partial b for a given value of lambda we define what we call the radial volume derivative at lambda denoted by g lambda which measures the stability or instability of the underlying homogeneous deformation mathbf u h lambda mathbf x equiv lambda mathbf x to the formation of holes we give conditions under which the critical boundary displacement lambda mathrm crit for radial cavitation is the unique solution of g lambda 0 moreover we prove that the radial volume derivative g lambda can be approximated by the corresponding volume derivative for a punctured ball b epsilon containing a pre existing cav

mappings of least dirichlet energy and their hopf differentials the paper is concerned with mappings h colon mathbb x begin array ll rm onto longrightarrow end array mathbb y between planar domains having least dirichlet energy the existence and uniqueness up to a conformal change of variables in mathbb x of the energy minimal mappings is established within the class overline fancyscript h 2 mathbb x mathbb y of strong limits of homeomorphisms in the sobolev space fancyscript w 1 2 mathbb x mathbb y a result of considerable interest in the mathematical models of nonlinear elasticity the inner variation of the independent variable in mathbb x leads to the hopf differential h z overline h bar z rm d z otimes rm d z and its trajectories for a pair of doubly connected domains in which mathbb x has finite conformal modulus we establish the following principle a mapping h in overline fancyscript h 2 mathbb x mathbb y is energy minimal if and only if its hopf differential is analytic in mathbb x and real along partial mathbb x in general the energy minimal mappings may not be injective in which case one observes the occurrence of slits in mathbb x cognate with cracks slits are triggered by points of concavity of mathbb y they originate from partial mathbb x and advance along vertical trajectories of the hopf differential toward mathbb x where they eventually terminate so no crosscuts are created

a locking free fem for cavitation computation in nearly incompressible nonlinear elasticity abstract a reduced selective integration finite element method coupled with a gradient flow method is introduced for the cavitation computation in nearly incompressible nonlinear elasticity numerical experiments show that for the iso parametric biquadratic finite element approximation of the displacement field the method is locking free and efficient for the nearly incompressible cavitation computation and has the same optimal convergence rates as in the compressible case for the elastic energy as well as the deformation and stress field if and only if the reduced integration is applied to the bulk energy term using no more than 2 gaussian quadrature nodes in each dimension

a numerical method for detecting singular minimizers of multidimensional problems in nonlinear elasticity in this paper we describe and analyse a numerical method that detects singular minimizers and avoids the lavrentiev phenomenon for three dimensional problems in nonlinear elasticity this method extends to three dimensions the corresponding one dimensional method of ball and knowles

weighted complex projective 2 designs from bases optimal state determination by orthogonal measurements we introduce the problem of constructing weighted complex projective 2 designs from the union of a family of orthonormal bases if the weight remains constant across elements of the same basis then such designs can be interpreted as generalizations of complete sets of mutually unbiased bases being equivalent whenever the design is composed of d 1 bases in dimension d we show that for the purpose of quantum state determination these designs specify an optimal collection of orthogonal measurements using highly nonlinear functions on abelian groups we construct explicit examples from d 2 orthonormal bases whenever d 1 is a prime power covering dimensions d 6 10 and 12 for example where no complete sets of mutually unbiased bases have thus far been found

symmetric informationally complete positive operator valued measures a new computer study we report on a new computer study of the existence of d2 equiangular lines in d complex dimensions such maximal complex projective codes are conjectured to exist in all finite dimensions and are the underlying mathematical objects defining symmetric informationally complete measurements in quantum theory we provide numerical solutions in all dimensions d 67 and moreover a putatively complete list of weyl heisenberg covariant solutions for d 50 a symmetry analysis of this list leads to new algebraic solutions in dimensions d 24 35 and 48 which are given together with algebraic solutions for d 4 15 and 19

on critical points of the σ2 energy over a space of measure preserving maps abstract let x r n be a bounded lipschitz domain and consider the 2 energy functional f 2 u x x u u 2 d x over the space of admissible maps a x u w 1 4 x r n u x x det u 1 for l n a e in x a good measure of how much a map u stretches areas of 2 dimensional sub manifolds of the domain x is the norm of u u 2 t x 2 t x analogously to u 2 the dirichlet energy density that is a measure of length s stretching these kinds of functionals also were arisen as a physical model describing the strong interactions of quantum fields which was introduced by t skyrme in 1961 in this paper we introduce a class of maps referred to as generalised twists and examine them in connection with the euler lagrange equations associated with f 2 x over a x in particular we present a novel characterisation of all twist solutions and this points at a surprising discrepancy between even and odd dimensions which follows very closely to the ideas that have been introduced by the second author in his recent paper shahrokhi dehkordi and taheri 2009 17 indeed we show that in even dimensions the latter system of equations admits infinitely many smooth solutions modulo isometries amongst such maps in odd dimensions this number reduces to one the result relies on a careful analysis of the full versus the restricted euler lagrange equations we investigate various qualitative properties of these solutions in view of a remarkably interesting previously unknown explicit formula

the lie algebraic significance of symmetric informationally complete measurements examples of symmetric informationally complete positive operator valued measures sic povms have been constructed in every dimension 67 however it remains an open question whether they exist in all finite dimensions a sic povm is usually thought of as a highly symmetric structure in quantum state space however its elements can equally well be regarded as a basis for the lie algebra gl d c in this paper we examine the resulting structure constants which are calculated from the traces of the triple products of the sic povm elements and which it turns out characterize the sic povm up to unitary equivalence we show that the structure constants have numerous remarkable properties in particular we show that the existence of a sic povm in dimension d is equivalent to the existence of a certain structure in the adjoint representation of gl d c we hope that transforming the problem in this way from a question about quantum state space to a question about lie algebras may help to make the existence problem tractable

maximally informative ensembles for sic povms in dimension 3 in order to find out for which initial states of the system the uncertainty of the measurement outcomes will be minimal one can look for the minimizers of the shannon entropy of the measurement in case of group covariant measurements this question becomes closely related to the problem of how informative the measurement is in the sense of its informational power namely the orbit under group action of the entropy minimizer corresponds to a maximally informative ensemble of equiprobable elements we give a characterization of such ensembles for three dimensional group covariant weyl heisenberg symmetric informationally complete positive operator valued measures sic povms in both geometric and algebraic terms it turns out that a maximally informative ensemble arises from the input state orthogonal to a subspace spanned by three linearly dependent vectors defining a sic povm geometrically or from an eigenstate of a certain weyl matrix algebraically

weak continuity and lower semicontinuity results for determinants weak continuity properties of minors and lower semicontinuity properties of functionals with polyconvex integrands are addressed in this paper in particular it is shown that if un is bounded in open image in new window and if u bv n are such that un u in l1 n and

radially symmetric motions of nonlinearly viscoelastic bodies under live loads this paper treats radially symmetric motions of nonlinearly viscoelastic circular cylindrical and spherical shells subjected to the live loads of centrifugal force and time dependent hydrostatic pressures the governing equations are exact versions of those for 3 dimensional continuum mechanics so shell does not connote an approximate via some shell theory these motions are governed by quasilinear third order parabolic hyperbolic equations having but one independent spatial variable the principal part of such a partial differential equation is determined by a general family of nonlinear constitutive equations the presence of strains in two orthogonal directions requires a careful treatment of constitutive restrictions that are physically natural and support the analysis the interaction of geometrically exact formulations the compatible use of general constitutive equations for material response and the presence of live loads show how these factors play crucial roles in the behavior of solutions in particular for different kinds of live loads there are thresholds separating materials that produce qualitatively different dynamical behavior the analysis using classical methods covers infinite time blowup for cylindrical shells subject to centrifugal forces infinite time blowup for cylindrical shells subject to steady and time dependent hydrostatic pressures finite time blowup for spherical shells subject to steady and time dependent hydrostatic pressures and the preclusion of total compression this paper concludes with a sketch using some modern methods of the existence of regular solutions until the time of blowup

regularity properties of deformations with finite energy the purpose of this paper is to point out some regularity properties of a class of functions which play an important role in nonlinear elasticity

harmonic maps with defects two problems concerning maps with point singularities from a domain c 3 tos2 are solved the first is to determine the minimum energy of when the location and topological degree of the singularities are prescribed in the second problem is the unit ball and g is given on we show that the only cases in whichg x x minimizes the energy isg const org x rx withr a rotation extensions of these problems are also solved e g points are replaced by holes 3 s2 is replaced by n sn 1 or by n pn 1 the latter being appropriate for the theory of liquid crystals

on sic povms in prime dimensions the generalized pauli group and its normalizer the clifford group have a rich mathematical structure which is relevant to the problem of constructing symmetric informationally complete povms sic povms to date almost every known sic povm fiducial vector is an eigenstate of a canonical unitary in the clifford group i show that every canonical unitary in prime dimensions p 3 lies in the same conjugacy class of the clifford group and give a class representative for all such dimensions it follows that if even one such sic povm fiducial vector is an eigenvector of such a unitary then all of them are for a given such dimension i also conjecture that in all dimensions d the number of conjugacy classes is bounded above by 3 and depends only on dmod9 and i support this claim with computer computations in all dimensions 48

spherically symmetric two phase deformations and phase transition zones this paper is concerned with characterization and stability assessment of two phase spherically symmetric deformations that can be supported by a nonlinear elastic isotropic material we study general properties of equilibrium two phase spherically symmetric deformations then we specialize to phase transformations of a solid sphere that is subjected to an all round tension pressure two material models are used to demonstrate a variety of transformation behaviours and some common features for both materials we construct phase transition zones ptzs formed in the space of principal stretches by those which can exist adjacently to an equilibrium interface then we demonstrate how the ptz can be used for the prediction of the number of two phase spherically symmetric solutions and study how the deformation field associated with each solution is related to the ptz we show that even in the simplest case of one interface the solution is not unique two equilibrium two phase solutions as well as one uniform one phase solution are found under the same boundary conditions for the three solutions we construct their load deformation diagrams and compare the associated total energies the stability of the two phase states with respect to radial and small wavelength perturbations is also examined we observe how unstable solutions are related with the ptz

mappings of finite distortion monotonicity and continuity we study mappings f f1 fn rn in the sobolev space w loc r n where is a connected open subset of rn with n 2 thus for almost every x we can speak of the linear transformation d f x rn rn called differential of f at x its norm is defined by d f x sup d f x h h sn 1 we shall often identify d f x with its matrix and denote by j x f det d f x the jacobian determinant thus using the language of differential forms we can write

distinguishability of quantum states under restricted families of measurements with an application to quantum data hiding we consider the problem of ambiguous discrimination of two quantum states when we are only allowed to perform a restricted set of measurements let the bias of a povm be defined as the total variational distance between the outcome distributions for the two states to be distinguished the performance of a set of measurements can then be defined as the ratio of the bias of this povm and the largest bias achievable by any measurements we first provide lower bounds on the performance of various povms acting on a single system such as the isotropic povm and spherical 2 and 4 designs and show how these bounds can lead to certainty relations furthermore we prove lower bounds for several interesting povms acting on multipartite systems such as the set of local povms povms which can be implemented using local operations and classical communication locc separable povms and finally povms for which every bipartition results in a measurement having positive partial transpose ppt in particular our results show that a scheme of terhal et al for hiding data against local operations and classical communication 31 has the best possible dimensional dependence

radial symmetry of p harmonic minimizers it is still not known if the radial cavitating minimizers obtained by ball philos trans r soc lond a 306 557 611 1982 and subsequently by many others are global minimizers of any physically reasonable nonlinearly elastic energy this quotation is from sivaloganathan and spector ann inst henri poincare anal non lineaire 25 1 201 213 2008 and seems to be still accurate the model case of the p harmonic energy is considered here we prove that the planar radial minimizers are indeed the global minimizers provided we prescribe the admissible deformations on the boundary in the traction free setting however even the identity map need not be a global minimizer

a numerical method for detecting singular minimizers a numerical method for computing minimizers in one dimensional problems of the calculus of variations is described such minimizers may have unbounded derivatives even when the integrand is smooth and regular in such cases because of the lavrentiev phenomenon standard finite element methods may fail to converge to a minimizer the scheme proposed is shown to converge to an absolute minimizer and is tested on an example the effect of quadrature is analyzed the implications for higher dimensional problems and in particular for fracture in nonlinear elasticity are discussed

variations on a theme of heisenberg pauli and weyl the parentage between weyl pairs the generalized pauli group and the unitary group is investigated in detail we start from an abstract definition of the heisenberg weyl group on the field and then switch to the discrete heisenberg weyl group or generalized pauli group on a finite ring the main characteristics of the latter group an abstract group of order d3 noted pd are given conjugacy classes and irreducible representation classes or equivalently lie algebra of dimension d3 associated with pd leaving the abstract sector a set of weyl pairs in dimension d is derived from a polar decomposition of su 2 closely connected to angular momentum theory then a realization of the generalized pauli group pd and the construction of generalized pauli matrices in dimension d are revisited in terms of weyl pairs finally the lie algebra of the unitary group u d is obtained as a subalgebra of the lie algebra associated with pd this leads to a development of the lie algebra of u d in a basis consisting of d2 generalized pauli matrices in the case where d is a power of a prime integer the lie algebra of su d can be decomposed into d 1 cartan subalgebras

digital image restoration using spatial interaction models this paper is concerned with developing fast nonrecursive algorithms for the minimum mean squared error restoration of degraded images the degradation is assumed to be due to a space invariant periodic nonseparable known point spread function and additive white noise our basic approach is to represent the images by a class of spatial interaction models namely the simultaneous autoregressve models and the conditional markov models defined on toroidal lattices and develop minimum mean squared error restoration algorithms using these models the restoration algorithms are optimal if the parameters characterizing the interaction models are exactly known however in practice the parameters are estimated from the images by using spatial interaction models we develop restoration algorithms that do not require the availability of the original image or its prototype the specific structure of the underlying lattice enables the implementation of the filters using fast fourier transform fft computations several restoration examples are given

a dual parametric finite element method for cavitation in nonlinear elasticity a dual parametric finite element method is introduced in this paper for the computation of singular minimizers in the 2d cavitation problem in nonlinear elasticity the method overcomes the difficulties such as the mesh entanglement and material interpenetration generally encountered in the finite element approximation of problems with extremely large expansionary deformation numerical experiments show that the method is highly efficient in the computation of cavitation problems numerical experiments are also conducted on discrete problems without the radial symmetry to show the validity of the method to more general settings and the potential of its application to the study of mechanism of cavity nucleation in nonlinear elastic materials

symmetric informationally complete quantum measurements we consider the existence in arbitrary finite dimensions d of a positive operator valued measure povm comprised of d2 rank one operators all of whose operator inner products are equal such a set is called a symmetric informationally complete povm sic povm and is equivalent to a set of d2 equiangular lines in cd sic povms are relevant for quantum state tomography quantum cryptography and foundational issues in quantum mechanics we construct sic povms in dimensions two three and four we further conjecture that a particular kind of group covariant sic povm exists in arbitrary dimensions providing numerical results up to dimension 45 to bolster this claim

unknown quantum states the quantum de finetti representation we present an elementary proof of the quantum de finetti representation theorem a quantum analog of de finetti s classical theorem on exchangeable probability assignments this contrasts with the original proof of hudson and moody z wahrschein verw geb 33 343 1976 which relies on advanced mathematics and does not share the same potential for generalization the classical de finetti theorem provides an operational definition of the concept of an unknown probability in bayesian probability theory where probabilities are taken to be degrees of belief instead of objective states of nature the quantum de finetti theorem in a closely analogous fashion deals with exchangeable density operator assignments and provides an operational definition of the concept of an unknown quantum state in quantum state tomography this result is especially important for information based interpretations of quantum mechanics where quantum states like probabilities are taken to be states of knowledge rather than

multispectral random field models for synthesis and analysis of color images multispectral extensions to the traditional gray level simultaneous autoregressive sar and markov random field mrf models are considered furthermore a new image model is proposed the pseudo markov model which retains the characteristics of the multispectral markov model yet admits to a simplified parameter estimation method these models are well suited to analysis and modeling of color images for each model considered procedures are developed for parameter estimation and image synthesis experimental results based on known image models and natural texture samples substantiate the validity of thee results

order 3 symmetry in the clifford hierarchy we investigate the action of the first three levels of the clifford hierarchy on sets of mutually unbiased bases comprising the ivanovic mutually unbiased base mub and the alltop mubs vectors in the alltop mubs exhibit additional symmetries when the dimension is a prime number equal to 1 modulo 3 and thus the set of all alltop vectors splits into three clifford orbits these vectors form configurations with so called zauner subspaces eigenspaces of order 3 elements of the clifford group highly relevant to the sic problem we identify alltop vectors as the magic states that appear in the context of fault tolerant universal quantum computing wherein the appearance of distinct clifford orbits implies a surprising inequivalence between some magic states

void nucleation and growth for a class of incompressible nonlinearly elastic materials abstract in this paper a bifurcation problem for a solid sphere subjected to uniform tensile dead loading p 0 at its boundary is examined within the framework of finite elastostatics the sphere is composed of a particular class of homogeneous isotropic incompressible nonlinearly elastic materials namely those of power law type one solution to the problem for all values of p 0 corresponds to a homogeneous state in which the sphere remains undeformed while stressed however for sufficiently large values of p 0 there is in addition a second possible configuration involving an internal traction free spherical cavity the dependence on constitutive parameters of the critical load at which bifurcation occurs is examined as well as the subsequent void growth the stress distribution after cavitation occurs is also described the results are obtained in closed analytic form

an angular momentum approach to quadratic fourier transform hadamard matrices gauss sums mutually unbiased bases the unitary group and the pauli group the construction of unitary operator bases in a finite dimensional hilbert space is reviewed through a nonstandard approach combining angular momentum theory and representation theory of su 2 a single formula for the bases is obtained from a polar decomposition of su 2 and is analyzed in terms of cyclic groups quadratic fourier transforms hadamard matrices and generalized gauss sums weyl pairs generalized pauli operators and their application to the unitary group and the pauli group naturally arise in this approach

neohookean deformations of annuli in the higher dimensional euclidean space abstract let n 2 be an integer and assume that a x r n 1 x r and a y r n 1 y r be two annuli in euclidean space r n assume that f a a resp r a a be the class of all orientation preserving resp radial homeomorphisms h a a in the sobolev space w 1 n a a which keep the boundary circles in the same order in this paper we extended the corresponding results of iwaniec and onninen which was published in math ann vol 348 2010

tomographically complete sets of orthonormal bases in finite systems quantum systems where the position and momentum are in the ring d is an odd integer are considered a tomographically complete set of bases i e a set of bases such that probabilities from tomography experiments corresponding to these bases can be used for the calculation of an arbitrary density matrix is considered such a set of bases is constructed by considering a factorization of a system with dimension d p1p2 where p1 p2 are prime numbers in terms of two subsystems with dimensions p1 and p2 appropriate combination of the mutually unbiased bases in the subsystems leads to a tomographically complete set of bases in the full system this set leads to probabilities along all the lines through the origin 0 0 in the phase space with exactly d points it is shown that the number of these lines is d the dedekind function the general theory is exemplified with examples

the convergence of regularized minimizers for cavitation problems in nonlinear elasticity consider a nonlinearly elastic body which occupies the region omega subset mathbb r m m 2 3 in its reference state and which is held in tension under prescribed boundary displacements on partial omega let bf x 0 in omega be any fixed point in the body it is known from variational arguments that for sufficiently large boundary displacements there may exist discontinuous weak solutions of the equilibrium equations corresponding to a hole forming at bf x 0 in the deformed body this is the phenomenon of cavitation for each epsilon 0 define the regularized domains omega epsilon omega backslash overline b epsilon bf x 0 which contain a preexisting hole of radius epsilon 0 centered on bf x 0 now consider the corresponding mixed displacement traction problem on omega epsilon in which the boundary partial omega is subject to the same boundary displacements and the deformed cavity surface i e the image of partial b epsilon is requi

quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity in this paper we prove the uniqueness of the solution to certain simple displacement boundary value problems in the nonlinear theory of homogeneous hyperelasticity for a body occupying a star shaped reference configuration rn whose boundary is subjected to an affine deformation i e there exists a constant n n matrix f and a constant n vector b such that x fx b for all x we consider all smooth equilibrium configurations satisfying this boundary condition clearly the homogeneous deformation x fx b for all x in bar omega is one such solution our aim is to prove under suitable hypotheses that it is the only such solution

on two dimensional markov spectral estimation we give systematic methods for two dimensional 2 d spectral estimation from raw data using gaussian markov random field mrf models the mrf models considered in this paper characterize the statistical dependence of the observation y s on its neighbors in all directions due to the modeling assumption the 2 d spectrum is an explicit function of the parameters of the mrf model thus the spectral estimation problem reduces to that of estimating the appropriate structure and the parameters of the mrf model a tractable algorithm for maximum likelihood ml estimation of parameters is obtained by using doubly periodic boundary conditions for the given data asymptotically consistent decision rules are used to choose the appropriate neighbor set of the mrf model the mrf spectral estimate obtained using the doubly periodic boundary conditions has the property that its inverse discrete fourier transforms are in perfect agreement with the periodic sample correlation values of the given observations over a symmetric window identical to the structure of the mrf model using this result and the fact that the 2 d maximum entropy spectral mes estimate has a structure similar to the mrf power spectrum we show that the mrf spectral estimate developed in this paper is a good approximation to the mes estimate this result provides a reasonable basis i e maximum likelihood on which to justify the use of the mes estimate with correlation estimates obtained from a finite piece of data samples

cartesian currents weak diffeomorphisms and existence theorems in nonlinear elasticity in this paper we introduce some new classes of functions among these a class of weak diffeomorphisms in these classes we prove by direct methods the existence of minimizers for several kinds of variational integrals in particular we prove the existence of one to one orientation preserving maps that minimize suitable energies associated with hyperelastic materials the minimizers are also proved to satisfy equilibrium equations finally radial deformations are discussed in connection with cavitation

markov random field texture models we consider a texture to be a stochastic possibly periodic two dimensional image field a texture model is a mathematical procedure capable of producing and describing a textured image we explore the use of markov random fields as texture models the binomial model where each point in the texture has a binomial distribution with parameter controlled by its neighbors and number of tries equal to the number of gray levels was taken to be the basic model for the analysis a method of generating samples from the binomial model is given followed by a theoretical and practical analysis of the method s convergence examples show how the parameters of the markov random field control the strength and direction of the clustering in the image the power of the binomial model to produce blurry sharp line like and blob like textures is demonstrated natural texture samples were digitized and their parameters were estimated under the markov random field model a hypothesis test was used for an objective assessment of goodness of fit under the markov random field model overall microtextures fit the model well the estimated parameters of the natural textures were used as input to the generation procedure the synthetic microtextures closely resembled their real counterparts while the regular and inhomogeneous textures did not

statistical mechanical approach to image processing the basic frameworks and techniques of the bayesian approach to image restoration are reviewed from the statistical mechanical point of view first a few basic notions in digital image processing are explained to convince the reader that statistical mechanics has a close formal similarity to this problem second the basic formulation of the statistical estimation from the observed degraded image by using the bayes formula is demonstrated the relationship between bayesian statistics and statistical mechanics is also listed particularly it is explained that some correlation inequalities on the nishimori line of the random spin model also play an important role in bayesian image restoration third the framework of bayesian image restoration for binary images by means of the ising model is reviewed some practical algorithms for binary image restoration are given by employing the mean field and the bethe approximations finally bayesian image restoration for a grey level image using the gaussian model is reviewed and the gaussian model is extended to a more practical probabilistic model by introducing the line state to treat the effects of edges the line state is also extended to quantized values

extremal mappings of finite distortion the theory of mappings of finite distortion has arisen out of a need to extend the ideas and applications of the classical theory of quasiconformal mappings to the degenerate elliptic setting where one finds concrete applications in non linear elasticity and the calculus of variations in this paper we initiate the study of extremal problems for mappings with finite distortion and extend the theory of extremal quasiconformal mappings by considering integral averages of the distortion function instead of its supremum norm for instance we show the following suppose that f o is a homeomorphism of the circle with f o 1 in cal w 1 2 2 then there is a unique extremal extension to the disk which is a real analytic diffeomorphism with non vanishing jacobian determinant the condition on f o is sharp classically the mapping f o is assumed to be quasisymmetric then there is an extremal quasiconformal mapping with boundary values f o but it is not always unique and it is seldom smooth indeed even when f o is quasisymmetric the cal l 1 minimiser for the distortion function will almost never be quasiconformal we further find that there are many new and unexpected phenomena concerning existence uniqueness and regularity for these extremal problems where the functionals are polyconvex but typically not convex these seem to differ markedly from phenomena observed when studying multi well type functionals

tight informationally complete quantum measurements we introduce a class of informationally complete positive operator valued measures which are in analogy with a tight frame as close as possible to orthonormal bases for the space of quantum states these measures are distinguished by an exceptionally simple state reconstruction formula which allows painless quantum state tomography complete sets of mutually unbiased bases and symmetric informationally complete positive operator valued measures are both members of this class the latter being the unique minimal rank one members recast as ensembles of pure quantum states the rank one members are in fact equivalent to weighted 2 designs in complex projective space these measures are shown to be optimal for quantum cloning and linear quantum state tomography

quantum systems with finite hilbert space galois fields in quantum mechanics a galois quantum system in which the position and momentum take values in the galois field gf pl is considered it is comprised of l component systems which are coupled in a particular way and is described by a certain class of hamiltonians displacements in the gf pl gf pl phase space and the corresponding heisenberg weyl group are studied symplectic transformations are shown to form the sp 2 gf pl group wigner and weyl functions are defined and their properties are studied frobenius symmetries which are based on frobenius automorphisms in the theory of galois fields are a unique feature of these systems for l 2 if they commute with the hamiltonian there are constants of motion which are discussed an analytic representation in the l sheeted complex plane provides an elegant formalism that embodies the properties of frobenius transformations the difference between a galois quantum system and other finite quantum systems where the position and momentum take values in the ring is discussed

hyperelastic deformations and total combined energy of mappings between annuli abstract we consider the so called combined energy of a deformation between two concentric annuli and minimize it provided that it keep order of the boundaries it is an extension of the corresponding result of euclidean energy it is interesting that the minimizers are certain radial mappings and they exist if and only if the annulus on the image domain is not too thin provided that the original annulus is fixed this in turn implies a nitsche type phenomenon next we consider the combined distortion and obtain certain related results which are dual to the results for combined energy which also involve some nitsche type phenomenon the main part of the paper is concerned with the total combined energy a certain integral operator defined as a convex linear combination of the combined energy and combined distortion of diffeomorphisms between two concentric annuli a 1 r and b 1 r first we construct radial minimizers of total combined energy then we prove that those radial minimizers are absolute minimizers on the class of all mappings between the annuli under certain constraint this extends the main result obtained by iwaniec and onninen in 9

optimizing quantum process tomography with unitary 2 designs we show that weighted unitary 2 designs define optimal measurements on the system ancilla output state for ancilla assisted process tomography of unital quantum channels examples include complete sets of mutually unbiased unitary operator bases each of these specifies a minimal series of optimal orthogonal measurements general quantum channels are also considered

sic povms and clifford groups in prime dimensions we show that in prime dimensions not equal to 3 each group covariant symmetric informationally complete positive operator valued measure sic povm is covariant with respect to a unique heisenberg weyl hw group moreover the symmetry group of the sic povm is a subgroup of the clifford group hence two sic povms covariant with respect to the hw group are unitarily or antiunitarily equivalent if and only if they are on the same orbit of the extended clifford group in dimension 3 each group covariant sic povm may be covariant with respect to three or nine hw groups and the symmetry group of the sic povm is a subgroup of at least one of the clifford groups of these hw groups respectively there may exist two or three orbits of equivalent sic povms for each group covariant sic povm depending on the order of its symmetry group we then establish a complete equivalence relation among group covariant sic povms in dimension 3 and classify inequivalent ones according to the geometric phases associated with fiducial vectors finally we uncover additional sic povms by regrouping of the fiducial vectors from different sic povms which may or may not be on the same orbit of the extended clifford group

the nitsche conjecture the nitsche conjecture is deeply rooted in the theory of doubly connected minimal surfaces however it is commonly formulated in slightly greater generality as a question of existence of a harmonic homeomorphism between circular annuli h a a r r onto a r r a in the early 1960s while attempting to describe all doubly connected minimal graphs over a given annulus a j c c nitsche observed that their conformal modulus cannot be too large then he conjectured in terms of isothermal coordinates even more a harmonic homeomorphism h a onto a exists if an only if r r 1 2 r r r r this fascinating and engaging problem remained open for almost a half of a century in the present paper we provide among further generalizations an affirmative answer to his conjecture

null lagrangians weak continuity and variational problems of arbitrary order abstract we consider the problem of minimizing integral functionals of the form i u f x k u x dx where r p u r and k u denotes the set of all partial derivatives of u with orders k the method is based on a characterization of null lagrangians l ku depending only on derivatives of order k applications to elasticity and other theories of mechanics are given

on conservation laws and necessary conditions in the calculus of variations it is well known from the work of noether that every variational symmetry of an integral functional gives rise to a corresponding conservation law in this paper we prove that each such conservation law arises directly as the euler lagrange equation for the functional on taking suitable variations around a minimizer

a general framework for frequency domain multi channel signal processing the authors provide a general framework for performing processing of stationary multichannel mc signals that is linear shift invariant within channel and shift varying across channels emphasis is given to the restoration of degraded signals it is shown that by utilizing the special structure of semiblock circulant and block diagonal matrices mc signal processing can be easily carried out in the frequency domain the generalization of many frequency domain single channel sc signal processing techniques to the mc case is presented it is shown that in mc signal processing each frequency component of a signal and system is presented respectively by a small vector and a matrix of size equal to the number of channels while in sc signal processing each frequency component in both cases is a scalar

neohookean deformations of annuli existence uniqueness and radial symmetry let mathbb x x in mathbb r 2 r x r and mathbb y y in mathbb r 2 r ast y r ast be annuli in the plane we consider the class fancyscript f mathbb x mathbb y of all orientation preserving homeomorphisms h mathbb x overset textnormal tiny onto longrightarrow mathbb y in the sobolev space fancyscript w 1 2 mathbb x mathbb y which keep the boundary circles in the same order this means that lim x searrow r h x r ast and lim x nearrow r h x r ast we study the neohookean energy integral mathcal e h int limits mathbb x dh 2 phi rm det dh quad rm for h in fancyscript f mathbb x mathbb y quad quad quad 1 where phi in fancyscript c infty 0 infty is positive and strictly convex we assume in addition that the function psi z 1 ddot phi z and its derivative extend continuously to 0 with 0 0 then we prove

on irregular weak solutions of the energy momentum equations irregular mappings that are weak solutions of the energy momentum equations are presented one example is discontinuous at a countable number of points while the other is c 1 but not c 2 these mappings are not solutions of the usual euler lagrange equations

error analysis of a dual parametric bi quadratic fem in cavitation computation in elasticity the orientation preservation conditions and approximation errors of a dual parametric bi quadratic finite element method for the computation of both radially symmetric and general nonsymmetric cavity solutions in nonlinear elasticity are analyzed the analytical results allow us to establish based on an error equidistribution principle an optimal meshing strategy for the method in cavitation computation numerical results are in good agreement with the analytical results

energy estimates and cavity interaction for a critical exponent cavitation model we consider the minimization of in a perforated domain of among maps that are incompressible det and invertible and satisfy a dirichlet boundary condition u g on if the volume enclosed by g is greater than any such deformation u is forced to map the small holes be ai onto macroscopically visible cavities which do not disappear as e 0 we restrict our attention to the critical exponent p n where the energy required for cavitation is of the order of and the model is suited therefore for an asymptotic analysis v1 vm denote the volumes of the cavities in the spirit of the analysis of vortices in ginzburg landau theory we obtain estimates for the renormalized energy showing its dependence on the size and the shape of the cavities on the initial distance between the cavitation points a1 am and on the distance from these points to the outer boundary based on those estimates we conclude for the case of two cavities that either the cavities prefer to be spherical in shape and well separated or to be very close to each other and appear as a single equivalent round cavity this is in agreement with existing numerical simulations and is reminiscent of the interaction between cavities in the mechanism of ductile fracture by void growth and coalescence 2012 wiley periodicals inc

myriad radial cavitating equilibria in nonlinear elasticity it is shown that every bounded strictly increasing smooth positive function of sufficiently slow growth is the jacobian of a radial hole creating equilibrium deformation for an appropriately constructed compressible nonlinearly elastic energy

svd based modeling for image texture classification using wavelet transformation this paper introduces a new model for image texture classification based on wavelet transformation and singular value decomposition the probability density function of the singular values of wavelet transformation coefficients of image textures is modeled as an exponential function the model parameter of the exponential function is estimated using maximum likelihood estimation technique truncation of lower singular values is employed to classify textures in the presence of noise kullback leibler distance kld between estimated model parameters of image textures is used as a similarity metric to perform the classification using minimum distance classifier the exponential function permits us to have closed form expressions for the estimate of the model parameter and computation of the kld these closed form expressions reduce the computational complexity of the proposed approach experimental results are presented to demonstrate the effectiveness of this approach on the entire 111 textures from brodatz database the experimental results demonstrate that the proposed approach improves recognition rates using a lower number of parameters on large databases the proposed approach achieves higher recognition rates compared to the traditional subband energy based approach the hybrid imm svm approach and the ggd based approach

a meshing strategy for a quadratic iso parametric fem in cavitation computation in nonlinear elasticity abstract the approximation properties of a quadratic iso parametric finite element method for a typical cavitation problem in nonlinear elasticity in two dimensions are analyzed more precisely 1 the finite element interpolation errors are established in terms of the mesh parameters 2 a mesh distribution strategy based on an error equi distribution principle is given 3 the convergence of finite element cavity solutions is proved numerical experiments show that in fact the optimal convergence rate can be achieved by the numerical cavity solutions

the road to equal norm parseval frames abstract the construction of equal norm parseval frames is fundamental for many applications of frame theory we present a construction method based on a system of ordinary differential equations which generates a flow on the set of parseval frames that converges to equal norm parseval frames we developed this method to address a question posed by vern paulsen how close is a nearly equal norm nearly parseval frame to an equal norm parseval frame the distance estimate derived here can be used to substantiate numerically found approximate constructions of equal norm parseval frames the estimate is valid for a fairly general class of frames requiring that the dimension of the hilbert space and the number of frame vectors is relatively prime in addition we re phrase our distance estimate to show that certain projection matrices which are nearly constant on the diagonal are close in hilbert schmidt norm to ones which have a constant diagonal

uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity introduction 97 constitutive assumptions 99 0 the stored energy function and weak equilibrium solutions 101 1 properties of radial equilibrium solutions 105 2 uniqueness of solutions to boundary value problems for shells 111 3 uniqueness of cavitating solutions 117 4 existence of radial equilibria 121 5 asymptotic behaviour of equilibria for shells 128 6 concluding remarks 132 7 appendix 133 references 135

regularity of the inverse of a planar sobolev homeomorphism let be a domain suppose that f w1 1loc r2 is a homeomorphism such that df x vanishes almost everywhere in the zero set of j f we show that f 1 w1 1loc f r2 and that df 1 y vanishes almost everywhere in the zero set of sharp conditions to quarantee that f 1 w1 q f r2 for some 1 q 2 are also given

the contracting curve density algorithm fitting parametric curve models to images using local self adapting separation criteria the task of fitting parametric curve models to the boundaries of perceptually meaningful image regions is a key problem in computer vision with numerous applications such as image segmentation pose estimation object tracking and 3 d reconstruction in this article we propose the contracting curve density ccd algorithm as a solution to the curve fitting problem the ccd algorithm extends the state of the art in two important ways first it applies a novel likelihood function for the assessment of a fit between the curve model and the image data this likelihood function can cope with highly inhomogeneous image regions because it is formulated in terms of local image statistics the local image statistics are learned on the fly from the vicinity of the expected curve they provide therefore locally adapted criteria for separating the adjacent image regions these local criteria replace often used predefined fixed criteria that rely on homogeneous image regions or specific edge properties the second contribution is the use of blurred curve models as efficient means for iteratively optimizing the posterior density over possible model parameters these blurred curve models enable the algorithm to trade off two conflicting objectives namely heaving a large area of convergence and achieving high accuracy we apply the ccd algorithm to several challenging image segmentation and 3 d pose estimation problems our experiments with rgb images show that the ccd algorithm achieves a high level of robustness and sub pixel accuracy even in the presence of severe texture shading clutter partial occlusion and strong changes of illumination

maximum likelihood estimation methods for multispectral random field image models considers the problem of estimating parameters of multispectral random field rf image models using maximum likelihood ml methods for images with an assumed gaussian distribution analytical results are developed for multispectral simultaneous autoregressive msar and markov random field mmrf models which lead to practical procedures for calculating ml estimates although previous work has provided least squares methods for parameter estimation the superiority of the ml method is evidenced by experimental results provided in this work the effectiveness of multispectral rf models using ml estimates in modeling color texture images is also demonstrated

estimation and choice of neighbors in spatial interaction models of images some aspects of statistical inference for a class of spatial interaction models for finite images are presented primarily the simultaneous autoregressive sar models and conditional markov cm models each of these models is characterized by a set of neighbors a set of coefficients and a noise sequence of specified characteristics we are concerned with two problems the estimation of the unknown parameters in both sar and cm models and the choice of an appropriate model from a class of such competing models assuming gaussian distributed variables we discuss maximum likelihood ml estimation methods in general the ml scheme leads to nonlinear optimization problems to avoid excessive computation an iterative scheme is given for sar models which gives approximate ml estimates in the gaussian case and reasonably good estimates in some non gaussian situations as well likewise for cm models an easily computable consistent estimate is given the asymptotic mean squared error mse of this estimate for a four neighbor cm model is shown tn be substantially less than the mse of the popular coding estimate asymptotically consistent decision rules are given for choosing an appropriate sar or cm model the usefulness of the estimation scheme and the decision rule for the choice of neighbors is illustrated by using synthetic patterns synthetic patterns obeying known sar and cm models are generated and the models corresponding to true and several competing neighbor sets are fitted the estimation scheme yields estimates close to the parameters of the true models and the decision rule for the choice of neighbors picks up the true model from the class of competing models

maximal smoothness of solutions to certain euler lagrange equations from nonlinear elasticity we investigate the maximal smoothness of stationary states for the multiple integral such variational problems are motivated by the study of nonlinear elasticity assuming certain structure conditions for and given a stationary state we derive an a priori l p estimate for for any p and where as a consequence we show that a c 1 stationary state necessarily satisfies det and is of class c 2 in nevertheless singular stationary states do exist we construct a nonsmooth c 1 solution for a particular in two dimensions such that det in and det vanishes at precisely one point in

conditional sic povms in this paper we examine a generalization of the symmetric informationally complete povms sic povms are the optimal measurements for full quantum tomography but if some parameters of the density matrix are known then the optimal sic povm should be orthogonal to a subspace this gives the concept of the conditional sic povm the existence is not known in general but we give a result in the special case when the diagonal is known of the density matrix

on the symmetry of energy minimising deformations in nonlinear elasticity ii compressible materials let a bf x in mathbb r n a 0 be the region occupied by an incompressible nonlinearly hyperelastic material in its reference configuration deformations bf u it a rightarrow mathbb r n are therefore isochoric maps and so satisfy the incompressibility constraint rm det nabla bf u 1 the displacement and the mixed displacement zero traction boundary value problems are considered here a displacement boundary condition of the form bf u bf x sigma bf x is prescribed on one of the boundary components where 0 is a given constant and the displacement on the remaining boundary component is either prescribed in the same form in the case of the pure displacement boundary value problem or left unspecified in the case of the mixed boundary value problem in this paper isoperimetric arguments are used to prove that the radially symmetric solutions to these problems are global energy minimisers in various classes of possibly non symmetric isochoric deformations of the region a

an existence theory for nonlinear elasticity that allows for cavitation in this paper the existence of minimizers in nonlinear elasticity is established under assumptions on the stored energy that permit the formation of new holes in the body such cavities have been observed in experiments on elastomers and a mathematical theory for radially symmetric cavities has been developed by ball here the full three dimensional problem is considered and an additional physically motivated energy term that is proportional to the area of the boundary of the deformed body is included the minimizers lie in a subclass of those maps in w1 p 2 p 3 that are one to one almost everywhere and preserve orientation roughly speaking this subclass consists of those maps in which cavities in one part of the body are not filled by material from other parts of the body such maps are shown to be much more regular than expected in particular some ideas of sverak are used to show that each map in this subclass has a representative which is continuous outside a set of hausdorff dimension 3 p and that this representative also satisfies lusin s condition n i e it maps lebesgue null sets onto such sets it is also shown that the distributional jacobian of such a map is a measure which is the sum of a measure that is absolutely continuous with respect to lebesgue measure and at most a countable number of dirac measures

spectra of phase point operators in odd prime dimensions and the extended clifford group we analyze the role of the extended clifford group in classifying the spectra of phase point operators within the framework laid out by gibbons et al phys rev a 70 062101 2004 for setting up wigner distributions on discrete phase spaces based on finite fields to do so we regard the set of all the discrete phase spaces as a symplectic vector space over the finite field auxiliary results include a derivation of the conjugacy classes of esl 2 fn

symmetric informationally complete positive operator valued measures and the extended clifford group we describe the structure of the extended clifford group defined to be the group consisting of all operators unitary and antiunitary which normalize the generalized pauli group or weyl heisenberg group as it is often called we also obtain a number of results concerning the structure of the clifford group proper i e the group consisting just of the unitary operators which normalize the generalized pauli group we then investigate the action of the extended clifford group operators on symmetric informationally complete positive operator valued measures or sic povms covariant relative to the action of the generalized pauli group we show that each of the fiducial vectors which has been constructed so far including all the vectors constructed numerically by renes et al is an eigenvector of one of a special class of order 3 clifford unitaries this suggests a strengthening of a conjecture of zauner s we give a complete characterization of the orbits and stability groups in dimensions 2 7 fina

surveying points in the complex projective plane fields institute ontario the perimeter insti tute ontario and tulane university new orleans s abramsky d appleby r blume kohout d brody h brown s flammia c fuchs l hardy and h zhu for stimulating discussions sms acknowledges support arising from visits to the university of nijmegen the university of sofia and the university of turin and thanks n lora lamia for helpful comments j armstrong

hyperelastic deformations of smallest total energy throughout this article mathbb x and mathbb y will be nonempty bounded domains in mathbb r n n geqq 2 the term deformation of mathbb x subset mathbb r n onto mathbb y subset mathbb r n refers to an orientation preserving homeomorphism h mathbb x overset textnormal tiny onto longrightarrow mathbb y in the sobolev class fancyscript w 1 1 mathbb x mathbb y whose inverse f mathbb y overset textnormal tiny onto longrightarrow mathbb x lies in fancyscript w 1 1 mathbb y mathbb x the general law of hyperelasticity asserts that there exists an energy integral mathcal e mathbb x h int mathbb x e x h dh rm dx such that the elastic deformations have the smallest energy we assume here that e is conformally coerced and polyconvex some additional regularity conditions are also imposed under those conditions we establish the existence and global invertibility of the minimizers the key tools in obtaining an extremal deformation h mathbb x overset textnormal tiny onto longrightarrow mathbb y regardless of its boundary values are the free lagrangians finding suitable free lagrangians and using them for a specific stored energy function e is truly a work of art we have done it here for the so called total harmonic energy and a pair of annuli in the plane in fact this challenging problem illustrates rather clearly the strength of the concept of free lagrangians