recursive calculation of transformation factors in terms of primitive factors the authors present an algorithm for recursively calculating coupling recoupling induction and similar coefficients in terms of the primitive coefficients the algorithm is shown to be complete for any compact group they use the 6 j symbol as an example for describing their algorithm but the algorithm applies to a wide class of group theoretic transformation factors

a note on colored homfly polynomials for hyperbolic knots from wzw models using the correspondence between chern simons theories and wess zumino witten models we present the necessary tools to calculate colored homfly polynomials for hyperbolic knots for two bridge hyperbolic knots we derive the colored homfly invariants in terms of crossing matrices of the underlying wess zumino witten model our analysis extends previous works by incorporating non trivial multiplicities for the primaries appearing in the crossing matrices so as to describe colorings of homfly invariants beyond the totally symmetric or anti symmetric representations of su n the crossing matrices directly relate to 6j symbols of the quantum group mathcal u q su n we present powerful methods to calculate such quantum 6j symbols for general n this allows us to determine previously unknown colored homfly polynomials for two bridge hyperbolic knots we give explicitly the homfly polynomials colored by the representation 2 1 for two bridge hyperbolic knots with up to eight crossings yet the scope of application of our techniques goes beyond knot theory e g our findings can be used to study correlators in wess zumino witten conformal field theories or in the limit to classical groups to determine color factors for yang mills amplitudes

detecting inclusions in electrical impedance tomography without reference measurements we develop a new variant of the factorization method that can be used to detect inclusions in electrical impedance tomography from either absolute current to voltage measurements at a single nonzero frequency or from frequency difference measurements this eliminates the need for numerically simulated reference measurements at an inclusion free body and thus greatly improves the method s robustness against forward modeling errors e g in the assumed body s shape

a regularized sampling method for solving three dimensional inverse scattering problems the inverse scattering problem under consideration is to determine the shape of an obstacle in re 3 from a knowledge of the time harmonic incident acoustic wave and the far field pattern of the scattered wave with frequency in the resonance region a method for solving this nonlinear and improperly posed problem is presented which is based on solving a linear integral equation of the first kind and avoids the use of nonlinear optimization methods numerical examples are given showing the practicality of this new approach

boundary layer techniques for solving the helmholtz equation in the presence of small inhomogeneities we consider solutions to the helmholtz equation in two and three dimensions based on layer potential techniques we provide for such solutions a rigorous systematic derivation of complete asymptotic expansions of perturbations resulting from the presence of diametrically small inhomogeneities with constitutive parameters different from those of the background medium it is expected that our results will find important applications for developing effective algorithms for reconstructing small dielectric inhomogeneities from boundary measurements

a boundary integral equation method for an inverse problem related to crack detection this paper discusses an application of a boundary integral equation method biem to an inverse problem of determining the shape and the location of cracks by boundary measurements suppose that a given body contains an interior crack the shape and the location of which are unknown on the exterior boundary of this body one carries out measurements which are interpreted mathematically as prescribing dirichlet data and measuring the corresponding neumann data or vice versa for a field governed by laplace s equation the inverse problem considered here attempts to determine the geometry of the crack from these experimental data we propose to solve this problem by minimizing the error of a certain boundary integral equation bie the process of this minimization however is shown to require solutions of certain are proposed several 2d and 3d numerical examples are given in order to test the performance of the present method

a music type algorithm for detecting internal corrosion from electrostatic boundary measurements we establish an asymptotic representation formula for the steady state current perturbations caused by internal corrosive boundary parts of small surface measure based on this formula we design a non iterative method of music multiple signal classification type for localizing the corrosive parts from voltage to current observations we perform numerical experiments to test the viability of the algorithm and the results clearly demonstrate that the algorithm works well even in the presence of relatively high noise ratios

music type electromagnetic imaging of a collection of small three dimensional inclusions in this paper we consider the localization of a collection of small three dimensional bounded homogeneous inclusions via time harmonic electromagnetic means typically using arrays of electric or magnetic dipole transmitters and receivers with given polarization s at some distance from the collection possibly also lying in the far field the inclusions somewhat apart or closely spaced are buried within a homogeneous medium and are of arbitrary contrast of permittivity conductivity and permeability vis ag vis this embedding medium the problem is formulated as an inverse scattering problem for the full maxwell equations and it involves a robust asymptotic modeling of the multistatic response matrix no specific application is studied at this stage but characterization of obstacles in subsoils nondestructive evaluation of man made structures and medical imaging are primary fields of application envisaged the proposed approach uses a music multiple signal classification type algorithm and it yields fast numbering accurate localization and estimates of the electromagnetic and geometric parameters polarization tensors of the inclusions the mathematical machinery is detailed first some specific attention being given to triaxial ellipsoidal inclusions and degenerate spherical shapes for the latter known results are retrieved then the viability of this algorithm which would be easily extended to planarly layered environments by introduction of their green s functions is documented by a variety of numerical results from synthetic noiseless and severely noisy field data

an asymptotic factorization method for inverse electromagnetic scattering in layered media we consider the inverse problem to reconstruct the number and the positions of a collection of finitely many small perfectly conducting scatterers buried within the lower halfspace of an unbounded two layered background medium from near field measurements of time harmonic electromagnetic fields for this purpose we first study the direct scattering problem and derive an asymptotic expansion of the scattered field as the size of the scatterers tends to zero integral equation methods and a factorization of the corresponding near field measurement operator are applied to prove this expansion in the second part of this work we use the asymptotic expansion to justify a noniterative reconstruction algorithm which is a combination of factorization methods and music type methods we illustrate the feasibility of this method by a numerical example

a shape topological control problem for nonlinear crack defect interaction the antiplane variational model we consider the shape topological control of a singularly perturbed variational inequality the geometry dependent state problem that we address in this paper concerns a heterogeneous medium with a micro object defect and a macro object crack modeled in two dimensions the corresponding nonlinear optimization problem subject to inequality constraints at the crack is considered within a general variational framework for the reason of asymptotic analysis singular perturbation theory is applied resulting in the topological sensitivity of an objective function representing the release rate of the strain energy in the vicinity of the nonlinear crack the antiplane strain energy release rate is expressed by means of the mode iii stress intensity factor that is examined with respect to small defects such as microcracks holes and inclusions of varying stiffness the result of shape topological control is useful either for arrests or rise of crack growth

enhanced multilevel linear sampling methods for inverse scattering problems we develop two enhanced techniques for the multilevel linear sampling method mlsm proposed in 32 for inverse scattering problems under some practical situations the mlsm suffers certain undesirable breakage cells problem we propose to avoid the curse of breakage cells by incorporating expanding and searching techniques the new techniques are shown to significantly improve the robustness of the mlsm and meanwhile they possess the same optimal computational complexity as the mlsm numerical experiments are presented to illustrate the promising features of the enhanced mlsms

reconstruction of an impedance cylinder at oblique incidence from the far field data consider scattering of obliquely incident plane electromagnetic waves by an impedance cylinder embedded in a homogeneous background medium under some assumptions the direct scattering problem is governed by a coupled system of the two dimensional helmholtz equations for x 3 components of the electric field and the magnetic field in this paper we study a corresponding inverse scattering problem which is to reconstruct an unknown scattering object from measurement data of the x 3 component of only the electric field a reconstruction scheme called the linear sampling method is established with rigorous mathematical analysis due to the coupled oblique derivative boundary condition some new characteristics appear in our analysis numerical examples are also presented to illustrate the feasibility of the reconstruction algorithm

a hybrid quantitative method for inverse scattering of multiple dielectric objects a hybrid method is proposed to locate the boundaries and determine the dielectric constants of multiple homogeneous scatterers by postprocessing the results of the linear sampling method lsm the hybrid method is composed of three steps combining the capabilities of the lsm the multiple signal classification music algorithm and the generalized multipole technique gmt initially a new measure of the noise level of the multistatic response msr matrix is proposed subsequently the lsm is applied using the tikhonov regularization then the scattered field is back propagated into the investigation domain using the gmt where in order to ensure the stability a music based approach is exploited to determine the optimum reconstructing multipole sources the case of limited scattered field data is also addressed by introducing a modification of the papoulis gerchberg algorithm pga finally the boundary and the dielectric constant of each scatterer are obtained simultaneously by solving an inverse boundary value problem through an optimization the proposed hybrid approach is capable of efficiently approximating the dielectric constant of multiple homogeneous dielectric scatterers in a parallel manner the effectiveness of the proposed method and the accuracy of the solutions are demonstrated by applying the method to both numerical and measured data

two numerical methods for recovering small inclusions from the scattering amplitude at a fixed frequency in this paper two noniterative algorithms for locating small electromagnetic inclusions from the scattering amplitude at a fixed frequency are developed in particular a variety of numerical results is presented to highlight their potential and their limitations

the imaging of small perturbations in an anisotropic media abstract in this paper we employ asymptotic analysis to determine information about small volume defects in a known anisotropic scattering medium from far field scattering data the location of the defects is reconstructed via the music algorithm from the range of the multi static response matrix derived from the asymptotic expansion of the far field pattern in the presence of small defects since the same data determines the transmission eigenvalues corresponding to the perturbed media we investigate how the presence of the defects changes the transmission eigenvalues and use this information to recover the strength of the small defects we provide convergence results on transmission eigenvalues as the size of the defects tends to zero as well as derive the first correction term in the asymptotic expansion of the simple transmission eigenvalues numerical examples are presented to show the viability of our imaging method

recent developments in inverse acoustic scattering theory we survey some of the highlights of inverse scattering theory as it has developed over the last 15 years with emphasis on uniqueness theorems and reconstruction algorithms for time harmonic acoustic waves included in our presentation are numerical experiments using real data and numerical examples of the use of inverse scattering methods to detect buried objects

optimization algorithm for reconstructing interface changes of a conductivity inclusion from modal measurements we propose an original optimization approach for reconstructing interface changes of a conductivity inclusion from measurements of eigenvalues and eigenvectors associated with the transmission problem for the laplacian let be a smooth domain and d be an inclusion contained in whose boundary is also assumed to be smooth shape deformation of d causes a perturbation of modal parameters the aim of this work is to show how this information can be used to reconstruct the unknown deformation for doing so we rigorously derive an asymptotic formula for the perturbations in the eigenvalues of the transmission problem for the laplacian that are due to small deformations of the interface of an inclusion based on this formula we design an efficient reconstruction algorithm from modal measurements our algorithm consists on minimizing a functional whose minimizer yields certain geometric properties of the unknown inclusion it naturally follows from a key identity that is in some sense dual to the asymptotic formula numerical experiments showing the viability of our algorithm are presented our results in this paper extend those established in the context of small volume inclusions as well as those for the conductivity interface problem in fact on one hand in a series of recent papers 5 3 1 2 we have derived high order asymptotic expansions of the eigenvalue perturbations due to the presence of small inclusions and used them for locating the inclusions and identifying some of their geometric features on the other hand in 4 we have derived highorder terms in the asymptotic expansions of the boundary perturbations of steady state voltage potentials resulting from small perturbations of the shape of a conductivity inclusion based on these derivations we have designed an effective algorithm to determine some geometric features of the shape perturbation of the inclusion based on boundary measurements

the topological asymptotic for pde systems the elasticity case the aim of the topological sensitivity analysis is to obtain an asymptotic expansion of a design functional with respect to the creation of a small hole in this paper such an expansion is obtained and analyzed in the context of linear elasticity for general functionals and arbitrary shaped holes by using an adaptation of the adjoint method and a domain truncation technique the method is general and can be easily adapted to other linear pdes and other types of boundary conditions

symmetric and unitary group representations i duality theory an extension of the schur weyl duality connecting the representations of the symmetric and unitary groups is given the schur weyl basis is constructed using annihilation and creation operators three factorisation lemmas are derived their importance lies in the fact that they relate the phase freedoms within the racah wigner algebra of the symmetric groups and the unitary groups extensions of the regge symmetries are also given these are expressed in five duality relations

reconstructing acoustic obstacles by planar and cylindrical waves in this paper we develop a novel method of reconstructing acoustic obstacles in r2 which follows a similar spirit of the linear sampling method originated by colton and kirsch the reconstruction scheme makes use of the near field measurements encoded into the boundary dirichlet to neumann map or the neumann to dirichlet map both the plane waves and cylindrical waves are shown to meet the reconstruction purpose rigorous mathematical justification of the reconstruction scheme is established the mapping properties of the newly introduced function operators involved in the reconstruction scheme are established these results are of significant mathematical interests for their own sake moreover due to the distinct properties of the function operators the indictor function in the proposed reconstruction scheme exhibits completely different behaviors from those having been established for the indictor function in the original linear sampling method for inverse scattering problems numerical experiments

probing for electrical inclusions with complex spherical waves let a physical body in 2 or 3 be given assume that the electric conductivity distribution inside consists of conductive inclusions in a known smooth background further assume that a subset is available for boundary measurements it is proved using hyperbolic geometry that certain information about the location of the inclusions can be exactly recovered from static electric measurements on more precisely given a ball b with center outside the convex hull of and satisfying b boundary measurements on with explicitly given dirichlet data are enough to determine whether b intersects the inclusion an approximate detection algorithm is introduced based on the theory numerical experiments in dimension two with simulated noisy data suggest that the algorithm finds the inclusion free domain near and is robust against measurement noise 2007 wiley periodicals inc

some 6 j symbols for symplectic and orthogonal groups by cerkaski s method the method devised by cerkaski 1987 has been used for the group sp 2n to find some 6 j symbols for which a single multiplicity index is required this approach possesses some advantages over that based on racah s techniques for so 3 the formulae for the 6 j symbols are comparatively simple and they automatically exhibit the symmetries of jucys 1973 elementary means are used to show that the eigenvalues of casimir s operator c2 for sp 2n are identical to those of c2 for o 2d if the authors take d n and replace the irreps of sp 2n by their transposed counterparts for o 2d where d is integral or half integral this enables the formulae for sp 2n to be used to generate expressions for some 6 j symbols for the orthogonal groups o 2d and so 2d illustrations are provided by so 2l 1 and o 4

all pairs almost shortest paths let g v e be an unweighted undirected graph on n vertices a simple argument shows that computing all distances in g with an additive one sided error of at most 1 is as hard as boolean matrix multiplication building on recent work of aingworth et al siam j comput 28 1999 pp 1167 1181 we describe an ot min n 3 2 m 1 2 n 7 3 time algorithm apasp2 for computing all distances in g with an additive one sided error of at most 2 algorithm apasp2 is simple easy to implement and faster than the fastest known matrix multiplication algorithm furthermore for every even k 2 we describe an tilde o min n 2 2 k 2 m 2 k 2 n 2 2 3k 2 time algorithm apaspk for computing all distances in g with an additive one sided error of at most k we also give an tilde o n 2 time algorithm bf apasp infty for producing stretch 3 estimated distances in an unweighted and undirected graph on n vertices no constant stretch factor was previously achieved in tilde o n 2 time we say that a weighted graph f v e k emulates an unweighted graph g v e if for every u v in v we have delta g u v le delta f u v le delta g u v k we show that every unweighted graph on n vertices has a 2 emulator with tilde o n 3 2 edges and a 4 emulator with tilde o n 4 3 edges these results are asymptotically tight finally we show that any weighted undirected graph on n vertices has a 3 spanner with tilde o n 3 2 edges and that such a 3 spanner can be built in tilde o mn 1 2 time we also describe an tilde o n m 2 3 n time algorithm for estimating all distances in a weighted undirected graph on n vertices with a stretch factor of at most 3

a direct impedance tomography algorithm for locating small inhomogeneities impedance tomography seeks to recover the electrical conductivity distribution inside a body from measurements of current flows and voltages on its surface in its most general form impedance tomography is quite ill posed but when additional a priori information is admitted the situation changes dramatically in this paper we consider the case where the goal is to find a number of small objects inhomogeneities inside an otherwise known conductor taking advantage of the smallness of the inhomogeneities we can use asymptotic analysis to design a direct i e non iterative reconstruction algorithm for the determination of their locations the viability of this direct approach is documented by numerical examples

multilevel linear sampling method for inverse scattering problems a novel multilevel algorithm is presented for implementing the widely used linear sampling method in inverse obstacle scattering problems the new method is shown to possess asymptotically optimal computational complexity for an n times n sampling mesh in mathbb r 2 or an n times n times n sampling mesh in mathbb r 3 the proposed algorithm requires one to solve only mathcal o n n 1 far field equations for a mathbb r n problem n 2 3 and this is in sharp contrast to the original version of the method which needs to solve n n far field equations numerical experiments are presented to illustrate the promising feature of the algorithm in significantly reducing the computational cost of the linear sampling method

music algorithm for location searching of dielectric anomalies from s parameters using microwave imaging abstract motivated by the biomedical engineering used in early stage breast cancer detection we investigated the use of multiple signal classification music algorithm for location searching of small anomalies using s parameters we considered the application of music to functional imaging where a small number of dipole antennas are used our approach is based on the application of born approximation or physical factorization we analyzed cases in which the anomaly is respectively small and large in relation to the wavelength and the structure of the left singular vectors is linked to the nonzero singular values of a multi static response msr matrix whose elements are the s parameters using simulations we demonstrated the strengths and weaknesses of the music algorithm in detecting both small and extended anomalies

locating several small inclusions in impedance tomography from backscatter data this paper investigates backscatter data for the inverse obstacle problem in impedance tomography when the obstacles are small it is shown that under this circumstance the backscatter data are close approximations of a rational function that has second or fourth order poles at the locations of the obstacles furthermore a numerical method is presented to locate the obstacles via the poles of certain laurent pade approximations numerical experiments explore the potential of this algorithm also for extended obstacles taking into consideration that the problem is severely ill posed

detecting interfaces in a parabolic elliptic problem from surface measurements assuming that the heat capacity of a body is negligible outside certain inclusions the heat equation degenerates to a parabolic elliptic interface problem in this work we aim to detect these interfaces from thermal measurements on the surface of the body we deduce an equivalent variational formulation for the parabolic elliptic problem and give a new proof of the unique solvability based on lions s projection lemma for the case that the heat conductivity is higher inside the inclusions we develop an adaptation of the factorization method to this time dependent problem in particular this shows that the locations of the interfaces are uniquely determined by boundary measurements the method also yields to a numerical algorithm to recover the inclusions and thus the interfaces we demonstrate how measurement data can be simulated numerically by a coupling of a finite element method with a boundary element method and finally we present some numerical results for the inverse problem

the symmetric group algebraic formulas for some sf 6j symbols and sf sf1 sf2 3jm symbols explicit rank dependent expressions have been obtained for some symmetric group sf 6j symbols and some sf sf1 sf2 3jm symbols using butler s recursion method a key point in deriving these results is the use of the reduced notation introduced by murnaghan to label irreps various symmetries of the 6j and 3jm symbols have been imposed these include the complex conjugation permutation and transpose conjugation we incorporate a new symmetry that arises from the occurrence of the two isomorphic direct product groups sf1 sf2 and sf2 sf1 as subgroups of sf in relation to the tables of 6j and 3jm symbols presented a discussion is given of the symmetric group unitary group duality

on the k shortest simple paths problem in weighted directed graphs we present the first approximation algorithm for finding the k shortest simple paths connecting a pair of vertices in a weighted directed graph that breaks the barrier of mn it is deterministic and has a running time of o k m sqrt n n 3 2 log n where m is the number of edges in the graph and n is the number of vertices let s t in v the length of the i th simple path from s to t computed by our algorithm is at most frac 3 2 times the length of the i th shortest simple path from s to t the best algorithms for computing the exact k shortest simple paths connecting a pair of vertices in a weighted directed graph are due to yen management sci 17 1970 1971 pp 712 716 and lawler management sci 18 1971 1972 pp 401 405 the running time of their algorithms using modern data structures is o k mn n 2 log n both algorithms are from the early 70s although this problem and other variants of the k shortest path problem has drawn a lot of attention during the last three and a half decades the o k mn n 2 log n bound is still unbeaten

a new optimal control approach for the reconstruction of extended inclusions the aim of this paper is to propose a new regularized optimal control formulation for recovering an extended inclusion from boundary measurements our approach provides an optimal representation of the shape of the inclusion it guarantees local lipschitz stability for the reconstruction problem some numerical experiments are performed to demonstrate the validity and the limitations of the proposed reconstruction method

identification of small inhomogeneities asymptotic factorization we consider the boundary value problem of calculating the electrostatic potential for a homogeneous conductor containing finitely many small insulating inclusions we give a new proof of the asymptotic expansion of the electrostatic potential in terms of the background potential the location of the inhomogeneities and their geometry as the size of the inhomogeneities tends to zero such asymptotic expansions have already been used to design direct i e noniterative reconstruction algorithms for the determination of the location of the small inclusions from electrostatic measurements on the boundary e g music type methods our derivation of the asymptotic formulas is based on integral equation methods it demonstrates the strong relation between factorization methods and music type methods for the solution of this inverse problem

calculation of primitive 6 j symbols the 6 j symbols of a group are independent of the subgroup chain chosen to define the basis states the authors present an improved algorithm for calculating the primitive 6 j symbols for a compact group with a faithful irrep by recursive building up using only the kronecker product rules and the general properties of 6 j previously one has sometimes needed to search for the useful equations by systematically trying all equations which involve unknown 6 j the authors show that the primitive 6 j may all be easily solved in terms of a subclass the core 6 j they discuss how the core 6 j can usually be solved proving that the method is complete for so3 the authors conjecture that the algorithm is complete for all groups

imaging of location and geometry for extended targets using the response matrix in this paper we study how to image both the location and the shape of extended targets using the response matrix obtained from inter element response of an active array of transducers in particular the time reversal technique is used for efficient initial localization of the target and the level set method is used for shape reconstruction we then show how to couple the location estimation and shape reconstruction in a complementary way to improve accuracy for range estimation resolution analysis for active arrays in remote sensing regime is also presented we illustrate with numerical experiments which show that the method is capable of imaging objects with complicated shapes and that the method is robust with respect to noisy data

symmetries of group subgroup transformation factors for coupling subduction and induction group subgroup gh transformation theory uses g transformation coefficients and gh transformation factors to describe the relationship between the various bases of representation space of any group decomposed into irreducible representation spaces of the group and its subgroups the representation spaces may be formed by the processes of coupling subduction or induction the authors further develop this gh transformation theory with the study of the various symmetries complex conjugation transposition and associativity of the gh transformation factor emphasising the origins of each symmetry and a hierarchy within these symmetries a method of calculation is given for certain transformation factors the results and methods presented here generalise those of the racah wigner coupling algebra

identification of small inhomogeneities of extreme conductivity by boundary measurements a theorem on continuous dependence we consider an electrostatic problem for a conductor consisting of finitely many small inhomogeneities of extreme conductivity embedded in a spatially varying reference medium firstly we establish an asymptotic formula for the voltage potential in terms of the reference voltage potential the location of the inhomogeneities and their geometry secondly we use this representation formula to prove a lipschitz continuous dependence estimate for the corresponding inverse problem this estimate bounds the difference in the location and in certain geometric properties of two sets of inhomogeneities by the difference in the boundary voltage potentials corresponding to a fixed current distribution

3jm and 6j tables for some bases of su6 and su3 tables of 6j symbols for su6 and su3 and tables of 3jm factors for su6 contains implies su2 su3 su3 contains implies u1 su2 and su3 contains implies so3 are presented the tables are computer produced using a program that implements the building up principle in a general form these tables are useful for calculations in high energy nuclear and solid state physics some other tabulations contain errors and none uses all the symmetries available the n independence of sun results is discussed by using the various symmetric group unitary group duality relations

identifying scattering obstacles by the construction of nonscattering waves there are many methods for identifying the shape and location of scatterers from far field data we take the view that the connections between algorithms are more illuminating than their differences particularly with regard to the linear sampling method d colton and a kirsch inverse problems 12 1996 pp 383 393 the point source method r potthast point sources and multipoles in inverse scattering theory chapman hall london uk 2001 and the music algorithm a j devaney ieee trans antennas and propagation 53 2005 pp 1600 1610 using the first two techniques we show that for a scatterer with dirichlet boundary conditions there is a nontrivial incident field that does not generate a scattered field this incident field written as an expansion of eigenfunctions of the far field operator is used in the music algorithm to image the shape and location of the obstacle as those points z where the incident field is orthogonal to the far field pattern due to a point source located at

the linear sampling method for solving the electromagnetic inverse scattering problem we consider the inverse scattering problem of determining the shape of an obstacle in mathbb r from knowledge of the time harmonic incident electromagnetic wave and the far field pattern of the scattered wave with frequency in the resonance region the approach used is the linear sampling method which does not require a priori knowledge of either the boundary condition or the connectivity of the scattering obstacle numerical examples are given for the case of both a simply connected perfect conductor and multiply connected obstacles satisfying an impedance boundary condition on the boundary of one component and satisfying a perfect conducting boundary condition on the boundary of another component

coupling subduction and induction transformations for group representations the basis of a representation space of any group may be chosen according to its decomposition into representations of the group and a subgroup chain coupling recoupling subduction and resubduction factors are each shown to arise as a result of schur s lemmas and a particular group subgroup structure when induction and mackey s subgroup theorem are considered new transformation factors occur those of induction and reinduction key properties of these factors are given together with their relationship with double coset matrix elements

asymptotic imaging of perfectly conducting cracks in this paper we consider cracks with dirichlet boundary conditions we first derive an asymptotic expansion of the boundary perturbations that are due to the presence of a small crack based on this formula we design a noniterative approach for locating a collection of small cracks in order to do so we construct a response matrix from the boundary measurements the location and the length of the crack are estimated respectively from the projection onto the noise space and the first significant singular value of the response matrix indeed the direction of the crack is estimated from the second singular vector we then consider an extended crack with dirichlet boundary conditions we rigorously derive an asymptotic expansion for the boundary perturbations that are due to a shape deformation of the crack to reconstruct an extended crack from many boundary measurements we develop two methods for obtaining a good guess several numerical experiments show how the proposed techniques for imaging small cracks as well as those for obtaining good initial guesses toward reconstructing an extended crack behave

subcubic equivalences between path matrix and triangle problems we say an algorithm on n n matrices with integer entries in l m mr or n node graphs with edge weights from l m m is truly subcubic if it runs in o n3 poly log m time for some 0 we define a notion of subcubic reducibility and show that many important problems on graphs and matrices solvable in o n3 time are equivalent under subcubic reductions namely the following weighted problems either all have truly subcubic algorithms or none of them do the all pairs shortest paths problem on weighted digraphs apsp detecting if a weighted graph has a triangle of negative total edge weight listing up to n2 99 negative triangles in an edge weighted graph finding a minimum weight cycle in a graph of non negative edge weights the replacement paths problem on weighted digraphs finding the second shortest simple path between two nodes in a weighted digraph checking whether a given matrix defines a metric verifying the correctness of a matrix product over the min semiring finding a maximum subarray in a given matrix therefore if apsp cannot be solved in n3 e time for any e 0 then many other problems also need essentially cubic time in fact we show generic equivalences between matrix products over a large class of algebraic structures used in optimization verifying a matrix product over the same structure and corresponding triangle detection problems over the structure these equivalences simplify prior work on subcubic algorithms for all pairs path problems since it now suffices to give appropriate subcubic triangle detection algorithms other consequences of our work are new combinatorial approaches to boolean matrix multiplication over the or and semiring abbreviated as bmm we show that practical advances in triangle detection would imply practical bmm algorithms among other results building on our techniques we give two improved bmm algorithms a derandomization of the combinatorial bmm algorithm of bansal and williams focs 09 and an improved quantum algorithm for bmm

on the topological derivative in shape optimization in this paper the topological derivative for an arbitrary shape functional is defined examples are provided for elliptic equations and the elasticity system in the plane the topological derivative can be used for solving shape optimization problems in structural mechanics

algebraic expressions for some so n racah coefficients in this paper the algebraic expressions for so n racah coefficients involving the coupling of three irreducible representations 1 0 over dot m 1 m 2 m 3 0 over dot are given by using the building up process their properties are discussed in detail c 1998 american institute of physics s0022 2488 98 01811 8

fast estimation of diameter and shortest paths without matrix multiplication in the recent past there has been considerable progress in devising algorithms for the all pairs shortest paths apsp problem running in time significantly smaller than the obvious time bound of o n3 unfortunately all the new algorithms are based on fast matrix multiplication algorithms that are notoriously impractical our work is motivated by the goal of devising purely combinatorial algorithms that match these improved running times our results come close to achieving this goal in that we present algorithms with a small additive error in the length of the paths obtained our algorithms are easy to implement have the desired property of being combinatorial in nature and the hidden constants in the running time bound are fairly small our main result is an algorithm which solves the apsp problem in unweighted undirected graphs with an additive error of 2 in time o n 2 5 sqrt log n this algorithm returns actual paths and not just the distances in addition we give more efficient algorithms with running time footnotesize o n 1 5 sqrt k log n n 2 log 2 n for the case where we are only required to determine shortest paths between k specified pairs of vertices rather than all pairs of vertices the starting point for all our results is an o m sqrt n log n algorithm for distinguishing between graphs of diameter 2 and 4 and this is later extended to obtaining a ratio 2 3 approximation to the diameter in time o m sqrt n log n n 2 log n unlike in the case of apsp our results for approximate diameter computation can be extended to the case of directed graphs with arbitrary positive real weights on the edges

3jm factors and basis functions for d h and c v the results of earlier work by butler and reid for the pure rotation double group dinfinity are used to obtain algebraic formulae for the 3jm factors of the branching dinfinity h contains implies cinfinity v other branchings involving these groups are trivially related to branchings involving dinfinity the transformations of the group chains o3 contains implies dinfinity h contains implies cinfinity h and o3 contains implies dinfinity h contains implies cinfinity v contains implies cinfinity to the standard angular momentum basis are also given

multistatic response matrix of a 3 d inclusion in half space and music imaging recent work on the retrieval of 3 d bounded dielectric and or magnetic inclusions in free space is extended to burial in a half space though emphasis is on the case of a single inclusion enabling the derivation of closed form mathematical results in illuminating fashion the approach extends to the case of an unknown number of well separated inclusions within an asymptotic field formulation derived from exact contrast source vector integral formulations satisfied by the time harmonic fields and using proper reciprocity relationships of the dyadic green s functions the multistatic response matrix of the inclusion is constructed from the leading order term of the fields its singular value structure is analyzed in detail for a dielectric or a magnetic contrast or both contrasts this is performed in the case of a unique electric dipole array operated in the transmit receive mode at a single frequency a multiple signal classification type algorithm follows from the decomposition yielding a cost functional the magnitude of which peaks at the inclusion center numerical results illustrate the above as a function of the geometric and electromagnetic parameters of the configuration imaging of a spherical inclusion is in particular investigated from severely noisy synthetic data as well as of two inclusions embedded within a non conductive or conductive half space

strengthened linear sampling method with a reference ball by adding a reference ball as an extra artificial obstacle component to the underlying scattering system we propose a simple but robust and effective technique to choose the crucial cut off value required in the linear sampling method lsm for inverse scattering problems the reference ball technique causes few extra computational costs to the lsm but brings in a practically important byproduct i e it eliminates the interior eigenvalue problem automatically a well known barrier when applying the lsm some mathematical justifications of the new technique are provided and numerical experiments are also presented to demonstrate its feasibility and effectiveness

high precision identification of an object optimality conditions based concept of imaging a class of inverse problems for the identification of an unknown geometric object from given measurements is considered a concept for object imaging based on optimality conditions and level sets is introduced which provides high resolution properties of the identification problem and stability to discretization and noise errors as a specific case the identification of the center of a test object of arbitrary shape and unknown boundary conditions from d boundary measurements in d spatial dimensions in the context of the helmholtz equation is described in detail for analysis and numerical realization methods from topology optimization generalized singular perturbations endowed with variational techniques and a petrov galerkin enrichment within generalized fem are used

stability and uniqueness for the crack identification problem this paper deals with the identifiability of nonsmooth defects by boundary measurements and the stability of their detection we introduce and analyze a new pointwise regularity concept at the boundary of an open set which turns out to play a crucial role in the identifiability of defects by two boundary measurements as a consequence we prove the unique identifiability for a large class of closed sets including sets with an infinite number of connected components of positive capacity and totally disconnected sets in order to rigorously justify numerical approximation results of defects by optimal design methods we prove a geometric stability result of the defect identification process without any a priori smoothness assumptions

algebraic formulas for some nontrivial un 6j symbols and umn um un 3jm symbols we give tables of algebraic formulas for some nontrivial 6j symbols and 3jm symbols of the unitary groups the tables demonstrate that the building up method can be used successfully to obtain the rank dependence of unitary group j and jm symbols to emphasize the rank dependent nature of this calculation we have employed the composite young tableaux notation or back to back notation to label the unitary group irreps in using this notation the transpose conjugate symmetry of the corresponding composite young diagram leads to a new symmetry of the unitary group 6j and 3jm symbols the transposition of the groups um and un gives rise to a further symmetry of the 3jm symbols of umn um un

finding the k shortest paths we give algorithms for finding the k shortest paths not required to be simple connecting a pair of vertices in a digraph our algorithms output an implicit representation of these paths in a digraph with n vertices and m edges in time o m n log n k we can also find the k shortest paths from a given source s to each vertex in the graph in total time o m n log n kn we describe applications to dynamic programming problems including the knapsack problem sequence alignment maximum inscribed polygons and genealogical relationship discovery

approximate distance oracles let g v e be an undirected weighted graph with vvv n and vev m let k 1 be an integer we show that g v e can be preprocessed in o kmn1 k expected time constructing a data structure of size o kn1p1 k such that any subsequent distance query can be answered approximately in o k time the approximate distance returned is of stretch at most 2k 1 that is the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k 1 a 1963 girth conjecture of erdos implies that n1p1 k space is needed in the worst case for any real stretch strictly smaller than 2kp1 the space requirement of our algorithm is therefore essentially optimal the most impressive feature of our data structure is its constant query time hence the name oracle previously data structures that used only o n1p1 k space had a query time of n1 k our algorithms are extremely simple and easy to implement efficiently they also provide faster constructions of sparse spanners of weighted graphs and improved tree covers and distance labelings of weighted or unweighted graphs

electrostatic backscattering by insulating obstacles we introduce and analyze backscattering data for a three dimensional obstacle problem in electrostatics in particular we investigate the asymptotic behavior of these data as i the measurement point goes to infinity and ii the obstacles shrink to individual points we also provide numerical simulations of these data