invariants of 3 manifolds via link polynomials and quantum groups the aim of this paper is to construct new topological invariants of compact oriented 3 manifolds and of framed links in such manifolds our invariant of a link in a closed oriented 3 manifold is a sequence of complex numbers parametrized by complex roots of 1 for a framed link in s 3 the terms of the sequence are equale to the values of the suitably parametrized jones polynomial of the link in the corresponding roots of 1 in the case of manifolds with boundary our invariant is a sequence of finite dimensional complex linear operators this produces from each root of unity q a 3 dimensional topological quantum field theory

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

direct error in constitutive equation formulation for plane stress inverse elasticity problem we present a new computational formulation for inverse problems in elasticity with full field data the formulation is a variant of an error in the constitutive equation formulation but allows direct solution for the modulus field and accommodates discontinuous strain fields the development of the formulation is motivated by the relatively poor performance of current direct formulations reported so far in literature in dealing with discontinuities in the strain and material property distribution the formulation relies on minimizing the error in the constitutive equation and a momentum equation constraint numerical results on model problems show that the formulation is capable handling discontinuous and noisy strain fields and also converging with mesh refinement for continuous and discontinuous material property distributions the application to reconstruct the elastic modulus distribution in solid breast tumors is shown

combined arrival time imaging and time reversal for scatterer identification abstract the computational inverse problem of identifying a scatterer in a time dependent wave field is considered the wave speed of the background medium and the wave source are assumed to be known wave measurements possibly noisy are given at chosen discrete points in space sensor locations and time the goal is to find scatterer parameters such as location size and shape the computational solution procedure consists of two steps in the first step a standard fast arrival time imaging ati algorithm is employed this results in a rough image which provides possible regions for the location of the scatterer in the second step an optimization scheme based on time reversal tr is used to determine the location size and shape of the scatterer the preliminary ati step has the effect of reducing considerably the search space for the tr optimization also proposed here is an improved definition of the objective function used for the optimization which tends to eliminate spurious solutions numerical experiments based on a finite element discretization in space and an explicit newmark time stepping show the identification capability of the proposed scheme for a model problem involving the linear scalar wave equation in a bounded domain two types of scatterers are considered a crack with a known orientation whose location and size are sought and a rectangular scatterer whose location aspect ratio and size are sought the performance of the scheme in the presence of measurement noise is also demonstrated

coordinatization of triangulated jordan systems we obtain partial coordinatizations of any quadratic jordan triple system containing a triangle e u e i e a split jo ke ku 0 ke z hz k such that e e ez is invertible and u is faithful in the sense that l x e u 0 x 0 this is automatic if j is nondegenerate or t e k we identify a hermitian matrix subsystem jh j e du j2 e2 e h d d 7t and a clifford subsystem j k no k e j q s c we show that a simple j must be all hermitian or all clifford coincides with j or j yielding a shorter proof osborn s capacity 2 theorem for jordan algebras we also can coordinatize most unital bimodules for hermitian matrix systems the impetus for this study came from two sources in presenting a selfcontained treatment of jordan structure theory n jacobson 4 sought a shorter proof of the osborn capacity 2 theorem the proof is not short for linear jordan algebras 2 and is quite involved for quadratic jordan algebras 3 in coordinatizing jordan triple systems the second author needed a version of the capacity 2 theorem making no semisimplicity restriction on the ambient system and without assuming that the tripotents e are division tripotents lo by avoiding semisimplicity assumptions such a formulation also coordinatizes unital bimodules in the triangulated case one cannot expect as precise a coordinatization as for rank n 3 in the latter case if j is a jordan algebra then jz

nearly h1 optimal finite element methods abstract we examine the problem of finding the h 1 projection onto a finite element space of an unknown field satisfying a specified boundary value problem solving the projection problem typically requires knowing the exact solution we circumvent this issue and obtain a petrov galerkin formulation which achieves h 1 optimality requiring weighting functions to be defined locally on the element level permits only approximate h 1 optimality in multi dimensional configurations we investigate the relation between our formulation and other stabilized fem formulations we show in particular that our formulation leads to a derivation of the supg method in special cases the present formulation reduces to that of residual free bubbles finally we present guidelines for obtaining the petrov weight functions and include a numerical example for the helmholtz equation

lie algebras graded by 3 graded root systems and jordan pairs covered by grids let r be an irreducible root system a lie algebra l is called graded by r if l is graded with grading group the root lattice of r such that the only nonzero homogeneous subspaces of l have degree 0 or a root in r the grading is induced by the adjoint action of a split cartan subalgebra of a finite dimensional simple lie subalgebra of l with root system r and l is generated by the homogeneous subspaces of nonzero degree this class of lie algebras was introduced and studied by s berman and r moody in invent math 108 1992 where in particular a classification up to central equivalence is given in the simply laced case the doubly laced cases have recently been classified by g benkart and e zelmanov let r be a 3 graded root system i e r is not of type e 8 f 4 or g 2 in this paper lie algebras graded by r are described in a unified way without case by case considerations namely it is shown that a lie algebra l is 3 graded if and only if l is a central extension of the tits kantor koecher algebra of a jordan pair covered by a grid whose associated 3 graded root system is isomorphic to r this result is then used to classify lie algebras graded by r we give the classification of jordan pairs covered by a grid and describe their tits kantor koecher algebras one of the advantages of this approach is that it works over rings containing 1 2 and 1 3 and also for infinite root systems another application is the description of slodowy s intersection matrix algebras arising from multiply affinized cartan matrices

lie tori of type b2 and graded simple jordan structures covered by a triangle abstract we classify two classes of b 2 graded lie algebras which have a second compatible grading by an abelian group a graded simple torsion free and b division graded our results describe the centreless cores of a class of affine reflection lie algebras hence apply in particular to the centreless cores of extended affine lie algebras the so called lie tori for which we recover results of allison gao and faulkner our classification b extends a recent result of benkart and yoshii both classifications are consequences of a new description of jordan algebras covered by a triangle which correspond to these lie algebras via the tits kantor koecher construction the jordan algebra classifications follow from our results on graded triangulated jordan triple systems they generalize work of mccrimmon and the first author as well as the osborn mccrimmon capacity 2 theorem in the ungraded case

profinite completions of burnside type quotients of surface groups using quantum representations of mapping class groups we prove that profinite completions of burnside type surface group quotients are not virtually prosolvable in general further we construct infinitely many finite simple characteristic quotients of surface groups

the kauffman skein algebra of a surface at sqrt 1 we study the structure of the kauffman algebra of a surface with parameter equal to sqrt 1 we obtain an interpretation of this algebra as an algebra of parallel transport operators acting on sections of a line bundle over the moduli space of flat su 2 connections over the surface we analyse the asymptotics of traces of curve operators in tqft in non standard regimes where the root of unity parametrizing the tqft accumulates to a root of unity we interpret the case of sqrt 1 in terms of parallel transport operators

modified error in constitutive equations mece approach for ultrasound elastography a partial differential equation constrained optimization approach is presented for reconstructing mechanical properties e g elastic moduli the proposed method is based on the minimization of an error in constitutive equations functional augmented with a least squares data misfit term referred to as mece for modified error in constitutive equations the main theme of this paper is to demonstrate several key strengths of the proposed method on experimental data in addition some illustrative examples are provided where the proposed method is compared with a common shear wave elastography swe approach to this end both synthetic data generated with transient finite element simulations as well as ultrasonically tracked displacement data from an acoustic radiation force arf experiment are used in a standard elasticity phantom the results indicate that the mece approach can produce accurate shear modulus reconstructions with significantly less bias than swe

adjoint weighted equation for inverse problems of incompressible plane stress elasticity the adjoint weighted equation awe formulation is applied to incompressible isotropic plane stress elasticity for the solution of inverse problems for the strong form of the problem to have a solution a restrictive strain compatibility condition must hold in the awe variational formulation on the other hand a much milder sufficient condition is that uniaxial stress states be precluded making the method more robust to noisy data we prove that the formulation leads to a stable converging numerical method and show through computations that the method performs well

a fourier series based virtual fields method for the identification of 2 d stiffness distributions the virtual fields method vfm is a powerful technique for determining stiffness distributions of a sample based on measured full field strain data the advantage of the vfm over many other methods is its ability to solve inverse problems of this type without any iteration a key step in any application of the vfm is the selection of the virtual fields several techniques are based on the use of polynomials of spatial variables either on the whole domain or in a piecewise form and the material properties are considered as having single values homogeneous within the domain the first attempt to parameterise the material properties as a function of spatial variables was proposed in for reconstruction of the stiffness map of a plate with impact damage in this paper we retain the basic concepts underlying the vfm but approach the parameterisation of the material properties in the spatial frequency rather than spatial domain by performing a 3 d fourier series expansion of the stiffness distribution over the region of interest furthermore the virtual fields are not selected as polynomials of spatial variables as in the previous vfm literature but from a set of simple cosine or sine functions of different spatial frequencies the abbreviation f vfm will be used to denote the vfm in which both a fourier series is used for the material property parameterization and cosine sine functions for the virtual fields the f vfm was developed originally for 2 d geometries here it is extended to volumetric datasets resulting for example from measurements with digital volume correlation or phase contrast magnetic resonance imaging

identification of the coefficient in elliptic equations this paper seeks to identify the space distributed diffusion coefficient a in the steady state diffusion equation operatorname div a operatorname grad u f in a bounded region of the real plane u f and the flux at the boundary are observed this problem is solved by a least square method choosing for the cost function the equation error defined in a natural way by the riesz representation theorem the method is stable if a smoothing term is added to the cost function tikhonov regularisation this paper shows that selection strategies of the smoothing coefficient exist that insure convergence under the only condition that a be sufficiently smooth the case is also considered where a is piecewise smooth and it is shown how to extend the previous formulation in both cases computational results are presented

a computationally efficient approach for inverse material characterization combining gappy pod with direct inversion abstract an approach for computationally efficient inverse material characterization from partial field response measurements that combines the gappy proper orthogonal decomposition pod machine learning technique with a physics based direct inversion strategy is presented and evaluated gappy pod is used to derive a data reconstruction tool from a set of potential system response fields that are generated from available a priori information regarding the potential distribution of the unknown material properties then the gappy pod technique is applied to reconstruct the full spatial distribution of the system response from whatever portion of the response field has been measured with the chosen system testing method lastly a direct inversion strategy is presented that is derived from the equations governing the system response i e physics of the system which utilizes the full field response reconstructed by gappy pod to produce an estimate of the spatial distribution of the unknown material properties the direct inversion technique is a particularly computationally efficient inversion technique requiring a cost equivalent to a single numerical analysis therefore the majority of the computational expense of the presented approach is the one time potential response generation for the gappy pod technique which leads to an approach that is substantially computationally efficient overall two numerically simulated examples are shown in which the elastic modulus distribution was characterized based on partial field displacement response measurements both static and dynamic the inversion procedure was shown to have the capability to efficiently provide accurate estimates to material property distributions from partial field response measurements the direct inversion with gappy pod response estimation was also shown to be substantially tolerant to noise in comparison to the direct inversion given measured full field response lastly a physical example regarding elastography of an arterial construct from ultrasound imaging response measurements is shown to validate the practical applicability of the direct inversion approach with gappy pod response reconstruction

semiprime prime and simple jordan superpairs covered by grids in this paper we continue the investigation of jordan superpairs covered by a grid which have recently been described by the second author e neher j algebra 269 2003 28 73 for such a jordan superpair we characterize semiprimeness primeness and simplicity in terms of its associated supercoordinate system

residual based stabilized formulation for the solution of inverse elliptic partial differential equations abstract we consider an inverse problem where the forward problem is that of linear plane stress elasticity or equivalently that of linear heat hydraulic conduction we demonstrate that the linearized version of the saddle point problem obtained from the minimization problem inherits some stability from the forward elliptic problem in particular it is stable for the response variable and the lagrange multiplier but not for the material property field this lack of stability implies that we are unable to prove optimal convergence with mesh refinement for the overall problem we overcome this difficulty by adding to the saddle point problem a residual based term that provides sufficient stability and prove optimal convergence in an energy like norm we verify these estimates through simple numerical examples we note that while we have considered a specific model for an inverse elliptic problem in this manuscript similar ideas could be developed for a broad class of inverse elliptic problems

a 3d domain decomposition approach for the identification of spatially varying elastic material parameters summary the post treatment of 3d displacement fields for the identification of spatially varying elastic material parameters is a large inverse problem that remains out of reach for massive 3d structures we explore here the potential of the constitutive compatibility method for tackling such an inverse problem provided an appropriate domain decomposition technique is introduced in the method described here the statically admissible stress field that can be related through the known constitutive symmetry to the kinematic observations is sought through minimization of an objective function which measures the violation of constitutive compatibility after this stress reconstruction the local material parameters are identified with the given kinematic observations using the constitutive equation here we first adapt this method to solve 3d identification problems and then implement it within a domain decomposition framework which allows for reduced computational load when handling larger problems copyright 2015 john wiley sons ltd

quadratic jordan superpairs covered by grids abstract in this paper we are proposing a theory of jordan superpairs defined over super commutative superrings our framework has two novelties we allow scalars of even and odd parity and we do not assume that 1 2 lies in our base superring to demonstrate that it is possible to work in this generality we classify jordan superpairs covered by a grid

toeplitz operators in tqft via skein theory topological quantum field theory associates to a punctured surface a level r and colors c in 1 r 1 at the marked points a finite dimensional hermitian space vr c curves on act as hermitian operator t r on these spaces in the case of the punctured torus and the 4 times punctured sphere we prove that the matrix elements of t r have an asymptotic expansion in powers of 1 r and we identify the two first terms using trace functions on representation spaces of the surface in su2 we conjecture a formula for the general case then we show that the curve operators are toeplitz operators on the sphere in the sense that t r rf r r where r is the toeplitz projector and f r is an explicit function on the sphere which is smooth away from the poles using this formula we show that under some assumptions on the colors associated to the marked points the sequence t r is a toeplitz operator in the usual sense with principal symbol equal to the trace function and with subleading term explicitly computed we use this result and semi classical analysis in order to compute the asymptotics of matrix elements of the representation of the mapping class group of on vr c we recover in this way the result of tw05 on the asymptotics of the quantum 6j symbols and treat the case of the punctured s matrix we conclude with some partial results when is a genus 2 surface without marked points

the symplectic nature of fundamental groups of surfaces soit 7b g la categorie des groupes de lie g avec une forme bilineaire symetrique non singuliere b sur l algebre de lie soit le groupe fondamental d une surface orientee close de genre 1 et soit 7b s la categorie des out espaces symplectiques alors la correspondance g b hom g g b definit un foncteur 7b g 7b s ou b est la structure symplectique sur hom g g definie par b

central extensions of lie algebras graded by finite root systems lie algebras graded by finite irreducible reduced root systems have been classified up to central extensions by berman and moody benkart and zelmanov and neher in this paper we determine the central extensions of these lie algebras and hence describe them completely up to isomorphism

the structure of strongly prime quadratic jordan algebras abstract we extend the structure theory for strongly prime jordan algebras of arbitrary dimension to quadratic algebras of characteristic 2 we show that such j are either 1 hermitian forms h 0 a j h q a lying between an ample subspace of hermitian elements in a prime associative algebra a and those in its martindale ring of quotients q a 2 clifford forms with a scalar extension j j which is an algebra j j q 1 of a nondegenerate quadratic from q with basepoint 1 or 3 albert forms with a scalar extension j j which is a 27 dimensional split albert algebra j h 3 k of 3 3 hermitian matrices over an 8 dimensional split octonion algebra k as a consequence the simple jordan algebras of arbitrary dimension are either hermitian h 0 a for simple a or clifford forms in j q 1 or are albert algebras in particular our methods give an idempotent free classification of all simple finite dimensional algebras which are i special the key to this characterization is the existence of zel manov polynomials in the free special jordan algebra which are both hermitian polynomials whose values always look like hermitian elements and at the same time clifford polynomials nontrivial on 3 3 hermitian matrices hence on special algebras with 3 interconnected idempotents if a zel manov polynomial does not vanish on j then j is of hermitian type and if it does vanish on j then j is of clifford type one example of such a polynomial is q48 p16 x1 y1 z1 w1 p16 x2 y2 z2 w2 p16 x3 y3 z3 w3 for p16 d2x y z 2 dx y w dx y w dx y z x y z thus the structure of jordan algebras depends very directly on polynomial identities and commutators

toric structures on the moduli space of flat connections on a riemann surface volumes and the moment map abstract in earlier papers we constructed a hamiltonian torus action on an open dense set in the moduli space of flat su 2 connections on a compact riemann surface where the dimension of the torus is half the dimension of the moduli space this torus action shows that this set can be viewed symplectically as a noncompact toric variety the number of integral points of the moment map for the torus action turns out to be identical to the verlinde dimension d g k as an application we furnish a new proof of the relation between the large k limit of d g k and the volume of the moduli space from our point of view this relation follows from the equality between the symplectic volume of a toric variety and the euclidean volume of the image of the moment map similar considerations are shown to give rise to the volumes of moduli spaces of parabolic bundles on a riemann surface knowledge of these volumes has been shown to allow a proof of the verlinde formula for the dimension of the space of holomorphic sections of line bundles on this space

asymptotic faithfulness of the quantum su n representations of the mapping class groups we prove that the sequence of projective quantum su n representations of the mapping class group of a closed oriented surface obtained from the projective flat su n verlinde bundles over teichm uller space is asymptotically faithful that is the intersection over all levels of the kernels of these representations is trivial whenever the genus is at least 3 for the genus 2 case this intersection is exactly the order 2 subgroup generated by the hyper elliptic involution in the case of even degree and n 2 otherwise the intersection is also trivial in the genus 2 case

adjoint weighted variational formulation for the direct solution of inverse problems of general linear elasticity with full interior data we describe a novel variational formulation of inverse elasticity problems given interior data the class of problems considered is rather general and includes as special cases plane deformations compressibility and incompressiblity in isotropic materials 3d deformations and anisotropy the strong form of this problem is governed by equations of pure advective transport the variational formulation is based on a generalization of the adjoint weighted variational equation awe formulation originally developed for flow of a passive scalar we describe how to apply awe to various cases and prove several properties we prove that the galerkin discretization of the awe formulation leads to a stable convergent numerical method and prove optimal rates of convergence the numerical examples demonstrate optimal convergence of the method with mesh refinement for multiple unknown material parameters graceful performance in the presence of noise and robust behavior of the method when the target solution is c c0 or discontinuous copyright 2009 john wiley sons ltd

asymptotic properties of the quantum representations of the mapping class group we establish two results on the large level limit of projective quantum representations of surface mapping class groups obtained by quantizing moduli spaces of flat su n bundle with hitchin s connection first we prove that these projective representations lift to asymptotic representations second we show that under an infinitesimal rigidity assumption the characters of these representations have an asymptotic expansion the leading order term of this expansion agrees with the formula derived heuristically by e witten in quantum field theory and the jones polynomial

regularized virtual fields method for mechanical properties identification of composite materials abstract this paper presents a new material parameters identification method based on the virtual fields method vfm where a regularization term coming from a micromechanical model has been added in this regularized virtual fields method rvfm the system of equations associated to the vfm is solved within a constrained optimization framework the method was applied to virtual scanning electron microscope in situ tests aiming at determining the local properties of uni directional composites loaded in the transverse direction and for which the strain fields were obtained through digital image correlation dic the method was tested for composites with various volume fractions and properties contrasts and under different boundary conditions the potentials and limitations of the algorithm in the presence of noisy images were investigated and compared to that of the vfm and the other finite element based methods the corresponding results indicate that the proposed rvfm is more efficient than the other identification techniques in terms of both accuracy and computational time an optimum size of region of interest was also determined by taking into account dic magnification requirements and based on the accuracy of the identified parameters this kind of study can be very useful for determining a priori the level of resolution in the imaging system required for a favorable accuracy of the identified parameters

diagonalization in jordan pairs as every student of linear algebra knows a rectangular matrix over a division ring can be diagonalized by elementary row and column operations similarly there are normal forms for alternating and hermitian matrices which can be achieved by simultaneous row and column operations since matrices of this kind form the main examples of jordan pairs it is natural to ask whether similar results hold in the jordan setting we show that this is in fact the case study the obstruction to diagonalizability the defect and also prove that nondegenerate jordan pairs admit a rank function sharing many properties with classical matrix rank let v v v be a jordan pair and s e e a set of orthogonal idempotents the s diagonal elements are those in xi 1 v e now suppose that v is nondegenerate and satisfies dcc on principal inner ideals an element x e v a i is called diagonalizabfe if x is s diagonal for some s consisting of division idempotents to see that for matrices this is the same as the usual notion of diagonalizability suppose that v has in addition act on principal inner ideals then v contains a frame that is a finite set f of orthogonal division idempotents such that v f 0 for rectangular or hermitian matrices over division rings f can be taken to consist of the diagonal matrix units e eii whereas for alternating matrices over a field e eli1 2ie zi the jordan analogues of elementary row and column operations are the inner automorphisms b v e v e and v e v e where e e f which generate the group of f elementary automorphisms of v 6 any set of orthogonal division idempotents can be transformed up to association into f by an f elementary automorphism 6 th 21 it follows that x is diagonalizable if and only if cp x is f diagonal for some elementary automorphism cp of v this shows the equivalence with the usual definition

solution of the time harmonic viscoelastic inverse problem with interior data in two dimensions summary we consider the problem of determining the distribution of the complex valued shear modulus for an incompressible linear viscoelastic material undergoing infinitesimal time harmonic deformation given the knowledge of the displacement field in its interior in particular we focus on the two dimensional problems of anti plane shear and plane stress these problems are motivated by applications in biomechanical imaging where the material modulus distributions are used to detect and or diagnose cancerous tumors we analyze the well posedness of the strong form of these problems and conclude that for the solution to exist the measured displacement field is required to satisfy rather restrictive compatibility conditions we propose a weak or a variational formulation and prove the existence and uniqueness of solutions under milder conditions on measured data this formulation is derived by weighting the original pde for the shear modulus by the adjoint operator acting on the complex conjugate of the weighting functions for this reason we refer to it as the complex adjoint weighted equation cawe we consider a straightforward finite element discretization of these equations with total variation regularization and test its performance with synthetically generated and experimentally measured data we find that the cawe method is in general less diffusive than a corresponding least squares solution and that the total variation regularization significantly improves its performance in the presence of noise copyright 2012 john wiley sons ltd

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100