inclusion pairs satisfying eshelby s uniformity property eshelby conjectured that if for a given uniform loading the field inside an inclusion is uniform then the inclusion must be an ellipse or an ellipsoid this conjecture has been proved to be true in two and three dimensions provided that the inclusion is simply connected in this paper we provide an alternative proof of cherepanov s result that an inclusion with two components can be constructed inside which the field is uniform for any given uniform loading for two dimensional conductivity or for antiplane elasticity for planar elasticity we show that the field inside the inclusion pair is uniform for certain loadings and not for others we also show that the polarization tensor associated with the inclusion pair lies on the lower hashin shtrikman bound and hence the conjecture of polya and szego is not true among nonsimply connected inclusions as a consequence we construct a simply connected inclusion which is nothing close to an ellipse but in which the field is almost uniform

reconstruction of closely spaced small inclusions in this paper we establish an explicit asymptotic formula for the steady state voltage perturbations caused by closely spaced small conductivity inhomogeneities based on this new formula we design a very effective numerical method to identify the location and some geometric features of these inhomogeneities from a finite number of boundary measurements the viability of our approach is documented by numerical examples

the cavity of the optimal shape under the shear stresses the problem of optimal shape of a single cavity in an infinite 2 d elastic domain is analyzed an elastic plane is subjected to a uniform load at infinity the cavity of the fixed area is said to be optimal if it provides the minimal energy change between the homogeneous plane and the plane with the cavity we show that for the case of shear loading the contour of the optimal cavity is not smooth but is shaped as a curved quadrilateral the shape is specified in terms of conformal mapping coefficients and explicit analytical representations for components of the dipole tensor associated with the cavity are employed we also find the exact values of angles at the corners of the optimal contour the applications include the problems of optimal design for dilute composites

independent domination in finitely defined classes of graphs we study the independent dominating set problem restricted to graph classes defined by finitely many forbidden induced subgraphs the main result is two sufficient conditions for the problem to be np hard in a finitely defined class of graphs we conjecture that those conditions are also necessary and describe several classes of graphs verifying the conjecture

determination of the insulated inclusion in conductivity problem and related eshelby conjecture abstract in the study of composites it is important to determine the shape of inclusions there is an interesting case in conductivity problem that the inclusion is insulated in present paper we first obtain the representation formula of the solution to an exterior problem and then prove that for any uniform loading such solution can be extended into the inclusion as an affine function if and only if the insulated inclusion is an ellipse or an ellipsoid we also show that an analogous result holds for the elasticity problem which is related to eshelby conjecture the main results in this paper are motivated by ammari kang kim and lee 2013 ammari kang and lim 2005 kang and milton 2008 and liu 2008

on normal subgroups which are direct products now that the classification of finite simple groups is complete it is logical to look at the extension problem an important special case to consider is when m is a minimal normal subgroup of g and both g m and m are known groups if m is abelian various techniques have been used to derive information about g indeed almost the entire theory of finite solvable groups can be said to rest upon these techniques the motivation behind the present paper was to develop techniques for dealing with the situation when m is not abelian specifically we consider the following problems 1 determine the structure of g from the structure of g m and some subgroup or subgroups of g 2 find subgroups h in g such that g hm and in particular determine whether m has a complement in g 3 determine when two subgroups h and h found in 2 are conjugate in g if m is a non abelian minimal normal subgroup of a finite group g then

the eshelby theorem and its variants for piezoelectric media we prove the eshelby theorem for an ellipsoidal piezoelectric inclusion in an infinite piezoelectric material explicit formulas for the link and polarization matrices are derived passing to the limits with respect to parameters in the corresponding equations the result is extended to cases when either the inclusion or the surrounding material is purely elastic

on representing finite lattices as intervals in subgroup lattices of finite groups let mnbe the lattice of length 2 withn 1 atoms it is an open problem to decide whether or not every such lattice or indeed whether or not every finite lattice can be represented as an interval in the subgroup lattice of some finite group we complete the work of the second author lucchini by reducing this problem to a series of questions concerning the finite non abelian simple groups

on the absence of eshelby property for non ellipsoidal inclusions it is shown that the eshelby property does not hold for any inclusion bounded by a polynomial surface of higher than the second degree or any inclusion bounded by a non convex surface inclusions bounded by segments of two or more different surfaces are also precluded the absence of the eshelby property for non ellipsoidal inclusions is then discussed

variational formulation of the equivalent eigenstrain method with an application to a problem with radial eigenstrains abstract in this paper a variational formulation of the equivalent eigenstrain method is established a functional of the hashin shtrikman type is proposed such that the solution of the equivalent eigenstrain equation is a unique minimizer of the functional moreover it is also shown that the equivalent eigenstrain equation is the euler lagrange equation of the potential energy of the inclusions an approximate solution of the equivalent eigenstrain equation is then found as a minimizer of the functional on a finite dimensional span of basic eigenstrains special attention is paid to possible symmetries of the problem the variational formulation is illustrated by determination of effective linear elastic properties in particular material with a simple cubic microstructure is considered in detail a solution for the polynomial radial basic eigenstrains approximation is found in particular for the homogeneous eigenstrain approximation the effective moduli are derived in an exact closed form

bounds on herglotz functions and fundamental limits of broadband passive quasistatic cloaking using a sum rule we derive new bounds on herglotz functions that generalize those given in bernland et al j phys a math theor 44 14 145205 2011 and gustafsson and sjoberg new j phys 12 4 043046 2010 these bounds apply to a wide class of linear passive systems such as electromagnetic passive materials among these bounds we describe the optimal ones and also discuss their meaning in various physical situations like in the case of a transparency window where we exhibit sharp bounds then we apply these bounds in the context of broadband passive cloaking in the quasistatic regime to refute the following challenging question is it possible to construct a passive cloaking device that cloaks an object over a whole frequency band our rigorous approach although limited to quasistatics gives quantitative limitations on the cloaking effect over a finite frequency range by providing inequalities on the polarizability tensor associated with the cloaking device we emphasize that our resul

monodromy groups of hurwitz type problems abstract we solve the hurwitz monodromy problem for degree 4 covers that is the hurwitz space h 4 g of all simply branched covers of p 1 of degree 4 and genus g is an unramified cover of the space p 2 g 6 of 2 g 6 tuples of distinct points in p 1 we determine the monodromy of 1 p 2 g 6 on the points of the fiber this turns out to be the same problem as the action of 1 p 2 g 6 on a certain local system of z 2 vector spaces we generalize our result by treating the analogous local system with z n coefficients 3 n in place of z 2 this in turn allows us to answer a question of ellenberg concerning families of galois covers of p 1 with deck group z n 2 s 3

regularity properties of solutions to transmission problems we show that the gradients of solutions to certain elliptic and parabolic transmission problems with internal lipschitz boundary and constant co efficients at each side of the internal boundary are square integrable along the internal boundary

satgraphs and independent domination part 1 a graph g is called a satgraph if there exists a partition a b v g such that a induces a clique possibly a 0 b induces a matching i e g b is a 1 regular subgraph possibly b 0 and there are no triangles a b b where a a and b b b we also introduce the hereditary closure of iaj denoted by hiaj hereditary satgraphs the class hiaj contains split graphs in turn hiaj is contained in the class of all 1 2 split graphs a gyarfas generalized split graphs and ramsey numbers j combin theory ser a 81 2 1998 255 261 the latter being still not characterized we characterize satgraphs in terms of forbidden induced subgraphs there exist close connections between satgraphs and the satisfiability problem sat in fact sat is linear time equivalent to finding the independent domination number in the corresponding satgraph it follows that the independent domination problem is np complete for the hereditary satgraphs in particular it is np complete for perfect graphs

groups satisfying a strong complement property abstract let g n h be a finite group where n is normal in g and h is a complement of n in g for a given generating sequence h 1 h d of h we say that n h 1 h d satisfies the strong complement property if h 1 x 1 h d x d is a complement of n in g for all x 1 x d n when d is the minimal number of elements needed to generate h and n h 1 h d satisfies the strong complement property for every generating sequence h 1 h d with length d then we say that n h satisfies the strong complement property in the case when n and h are coprime we show that n h can only satisfy the strong complement property if h is cyclic or if h acts trivially on n we give on the other hand a number of examples that show this does not need to be the case when considering the strong complement property of n h 1 h d for a given fixed generating sequence in the case when n and h are not of coprime order we give examples where n h satisfies the strong complement property and where h is not cyclic and does not act trivially on n

automorphisms of induced extensions suppose m is a normal subgroup of the group g and m is a direct product of subgroups li 1 1 the objective of the present paper is to show that the problem of finding the automorphisms of g may be reduced to that of finding certain isomorphisms of the groups g m and n k if n 1 then n k g and our theorems are trivialities thus our results are really concerned with the case when n 1 suppose si is an automorphism of g if m n and k are the images under a of m n and k respectively then a induces an isomorphism of g m onto g m and an isomorphism y of nfk onto n ik further b maps the subgroup n m onto n m 7 maps m k onto m fk and and 7 induce the same mapping of n m onto n m here we identify n m and n m with n k f mik and n k m k respectively thus the previous statement simply means that if y kn k n for ne n and n en then mn m n the thrust of the present paper is to reverse the above namely suppose fl is an isomorphism of g m onto g m for some normal subgroup m and suppose y is an isomorphism of n k onto n jk for some subgroups n and k with k a normal subgroup of n assume that k m n fl maps kim onto n m y maps m k onto m k and i and 7 induce the same mapping of n m onto n m if certain other hypotheses are satisfied the other assumptions basically stating that m n and k are like m n and k we show that there is an automorphism u of g such that z induces both i and 7

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

equations in groups that are virtually direct products abstract in this note we show that the satisfiability of equations and inequations with recognisable constraints is decidable in groups that are virtually direct products of finitely many hyperbolic groups

on characterizing the set of possible effective tensors of composites the variational method and the translation method a general algebraic framework is developed for characterizing the set of possible effective tensors of composites a transformation to the polarization problem simplifies the derivation of the hashin shtrikman variational principles and simplifies the calculation of the effective tensors of laminate materials a general connection is established between two methods for bounding effective tensors of composites the first method is based on the variational principles of hashin and shtrikman the second method due to tartar murat lurie and cherkaev uses translation operators or equivalently quadratic quasiconvex functions a correspondence is established between these translation operators and bounding operators on the relevant non local projection operator t1 an important class of bounds namely trace bounds on the effective tensors of two component media are given a geometrical interpretation after a suitable fractional linear transformation of the tensor space each bound corresponds to a tangent plane to the set of possible tensors a wide class of translation operators that generate these bounds is found the extremal translation operators in this class incorporate projections onto spaces of antisymmetric tensors these extremal translations generate attainable trace bounds even when the tensors of the two components are not well ordered in particular they generate the bounds of walpole on the effective bulk modulus the variational principles of gibiansky and cherkaev for bounding complex effective tensors are reviewed and used to derive some rigorous bounds that generalize the bounds conjectured by golden and papanicolaou an isomorphism is shown to underlie their variational principles this isomorphism is used to obtain dirichlet type variational principles and bounds for the effective tensors of general non selfadjoint problems it is anticipated that these variational principles which stem from the work of gibiansky and cherkaev will have applications in many fields of science

solutions to the polya szego conjecture and the weak eshelby conjecture eshelby showed that if an inclusion is of elliptic or ellipsoidal shape then for any uniform elastic loading the field inside the inclusion is uniform he then conjectured that the converse is true that is that if the field inside an inclusion is uniform for all uniform loadings then the inclusion is of elliptic or ellipsoidal shape we call this the weak eshelby conjecture in this paper we prove this conjecture in three dimensions in two dimensions a stronger conjecture which we call the strong eshelby conjecture has been proved if the field inside an inclusion is uniform for a single uniform loading then the inclusion is of elliptic shape we give an alternative proof of eshelby s conjecture in two dimensions using a hodographic transformation as a consequence of the weak eshelby s conjecture we prove in two and three dimensions a conjecture of polya and szego on the isoperimetric inequalities for the polarization tensors pts the polya szego conjecture asserts that the inclusion whose electrical pt has the minimal trace takes the shape of a disk or a ball

on the inapproximability of independent domination in 2p3 free perfect graphs we consider the complexity of approximation for the independent dominating set problem in 2p3 free graphs i e graphs that do not contain two disjoint copies of the chordless path on three vertices as an induced subgraph we show that if p np the problem cannot be approximated for 2p3 free graphs in polynomial time within a factor of n1 e for any constant e 0 where n is the number of vertices in the graph moreover we show that the result holds even if the 2p3 free graph is restricted to being weakly chordal and thereby perfect

on the modified green operator integral for polygonal polyhedral and other non ellipsoidal inclusions abstract a strange particularity of polyhedral inclusions and of fibres of regular polygonal cross sections has been recently stressed in the literature for respectively fully or transversally isotropic elasticity of the embedding material they have a mean and a central green operator integral both equal to the uniform one of respectively the sphere or the cylindrical fibre in using the radon transform rt method this particularity is here shown to be shared by much larger shape types in the same limits of material elasticity symmetry as a subcase even more shape types fulfill the similar particularity for material linear properties of second rank characteristic tensor such as thermal conductivity magnetic or dielectric properties when calculated using the rt method the modified green operator integral at any interior point of a bounded domain inclusion takes the form of a weighted average over an angular distribution of a single elementary operator the weight function is geometrically defined from the characteristic function of the domain and the four rank or second rank elementary operator depends on the material linear property of concern the rt method simply shows that the noticed particularity is due to matching symmetry between the inclusion shape through the weight function and the material property through the elementary operator the general geometrical characteristics of the inclusion shapes belonging to these sphere class and cylindrical fibre class are specified and some remarkable shapes of these classes are commented

solutions to the eshelby conjectures we present solutions to the eshelby conjectures based on a variational inequality we first discuss the meanings of the original eshelby s statement by fourier analysis we establish the connection between the homogeneous eshelby inclusion problem and the classic newtonian potential problem we then proceed to the solutions of the eshelby conjectures under some hypothesis on the material properties and restricted to connected inclusions with lipschitz boundary we show that one version of the eshelby conjectures is valid in all dimensions and the other version is valid in two dimensions we also show the existence of multiply connected inclusions in all dimensions and the existence of non ellipsoidal connected inclusions in three and higher dimensions such that in physical terms and in the context of elasticity some uniform eigenstress of the inclusion induces uninform strain on the inclusion we numerically calculate these special inclusions based on the finite element method

univalent σ harmonic mappings we study mappings from 2 into 2 whose components are weak solutions to the elliptic equation in divergence form div u 0 which we call harmonic mappings we prove sufficient conditions for the univalence i e injectivity of such mappings moreover we prove local bounds in bmo on the logarithm of the jacobian determinant of such univalent mappings thus obtaining the a e nonvanishing of their jacobian in particular our results apply to harmonic mapping associated with any periodic structure and therefore they play an important role in homogenization

uniform stress fields inside multiple inclusions in an elastic infinite plane under plane deformation multiple elastic inclusions with uniform internal stress fields in an infinite elastic matrix are constructed under given uniform remote in plane loadings the method is based on the sufficient and necessary condition imposed on the boundary value of a holomorphic function that guarantees the existence of the holomorphic function in a multiply connected region the unknown shape of each of the multiple inclusions is characterized by a conformal mapping this work focuses on a major large class of multiple inclusions characterized by a simple condition that covers and is much beyond the known related results reported in previous works extensive examples of multiple inclusions with or without geometrical symmetry are shown our results showed that the inclusion shapes obtained for the uniformity of internal stress fields are independent of the remote loading only when all of the multiple inclusions have the same shear modulus as that of the matrix moreover specific conditions are derived on remote loading elastic constants of the inclusions and uniform internal stress fields which guarantee the existence of multiple symmetric inclusions or multiple rotationally symmetrical inclusions with uniform internal stress fields

maximal subgroups in composite finite groups the purpose of this paper is to present a method for translating the problem of finding all maximal subgroups of finite groups into questions concerning groups that are nearly simple a finite group is called nearly simple if it has only one minimal normal subgroup and that it nonabelian and simple in view of the recently announced classification of all finite simple groups this seems to be a useful reduction though it must be acknowledged see e g scott 6 that there are still enormous obstacles on the way to understanding even just the maximal subgroups of the simple groups let g be a finite group and m a minimal normal subgroup of g the maximal subgroups of g containing m are of course in bijective correspondence with the maximal subgroups of the smaller group g m and so we need not concern ourselves with those if m is abelian the maximal subgroups of g not containing m are precisely the complements of m in g the number of conjugacy classes of these is 0 or the order of the first cohomology group g m the case of a4 nona lian and nonsimple is the principal part of this paper if neither reduction is applicable then all minimal normal subgroups of g are nonabelian simple groups this is dealt with in the entirely straightforward penultimate section of the paper these reductions are all canonical or natural in a sense which could perhaps be expressed in the language of categories and functors but here we prefer to stay with older conventions in particular we usually do not distinguish between a homomorphism and that obtained from it by restricting the codomain nevertheless the interested reader will observe that much of the strength of the results lies precisely in their canonical nature implicit as it may remain in this exposition while i do not know of any explicit statement of this reduction 114 002 l 8693 86 ie3 00

solutions to eshelby s problems of non elliptical thermal inclusions and cylindrical elastic inclusions of non elliptical cross section eshelby s inclusion problem is solved for non elliptical inclusions in the context of twodimensional thermal conduction and for cylindrical inclusions of non elliptical cross section within the framework of generalized plane elasticity first we consider a twodimensional infinite isotropic or anisotropic homogeneous medium with a non elliptical inclusion subjected to a prescribed uniform heat flux free temperature gradient eshelby s conduction tensor field and its area average are first expressed compactly in terms of two boundary integrals avoiding the usual singularity and then specified analytically for arbitrary polygonal inclusions and for inclusions characterized by the finite laurent series next we are interested in a three dimensional infinite isotropic or transversely isotropic homogeneous medium with a cylindrical inclusion of a non elliptical cross section that undergoes uniform generalized plane eigenstrains the solution to this problem is obtained by decomposing a generalized plane eigenstrain tensor into a plane strain part and an anti plane strain part exploiting the mathematical similarity between two dimensional thermal conduction and anti plane elasticity and combining the relevant results of zou et al zou et al 2010 j mech phys solids 58 346 372 doi 10 1016 j jmps 2009 11 008 with those derived in the present work for eshelby s conduction tensor field and its area average

independent domination in hereditary classes we investigate independent domination problem within hereditary classes of graphs boliac and lozin independent domination in finitely defined classes of graphs theoret comput sci 301 1 3 2003 271 284 proved some sufficient conditions for independent domination problem to be np complete within finitely defined hereditary classes of graphs they posed a question whether the conditions are also necessary we show that the conditions are not necessary since independent domination problem is np hard within 2p3 free graphs moreover we show that the problem remains np hard for a new hereditary class of graphs called hereditary 3 satgraphs we characterize hereditary 3 satgraphs in terms of forbidden induced subgraph as corollaries we prove that independent domination problem is np hard within the class of all 2p3 free perfect graphs and for k1 5 free weakly chordal graphs finally we compare complexity of independent domination problem with that of independent set problem for a hierarchy of hereditary classes recently proposed by hammer and zverovich construction of maximal stable sets with k extensions combin probab comput 13 2004 1 8 for each class in the hierarchy a maximum independent set can be found in polynomial time and the hierarchy covers all graphs however our characterization of hereditary 3 satgraphs implies that independent domination problem is np hard for almost all classes in the hierarchy this fact supports a conjecture that independent domination is harder than independent set problem within hereditary classes