a5 invariants the cohomology of l3 4 and related extensions five of the sporadic simple groups have maximal subgroups involving m 21 l 3 4 they are m 22 m 23 m c l hs and o n in this paper we study the cohomology rings of these maximal subgroups and several further groups closely associated with l 3 4

some projective modules determined by sporadic geometries abstract we determine which known sporadic geometries have projective lefschetz modules an elementary lemma on local collapsibility greatly simplifies the task of verifying projectivity the modules are analyzed when possible in terms of projective covers of individual irreducibles

vanishing of categorical obstructions for permutation orbifolds the orbifold construction a mapsto a g for a finite group g is fundamental in rational conformal field theory the construction of rep ag from rep a on the categorical level often called gauging is also prominent in the study of topological phases of matter given a non degenerate braided fusion category mathcal c with a g action the key step in this construction is to find a braided g crossed extension compatible with the action the extension theory of etingof nikshych ostrik gives two obstructions for this problem o 3 in h 3 g and o 4 in h 4 g for certain coefficients the latter of which depends on a categorical lifting of the action and is notoriously difficult to compute we show that in the case where g le s n acts by permutations on mathcal c boxtimes n both of these obstructions vanish this verifies a conjecture of muger and constitutes a nontrivial test of the conjecture that all modular tensor categories come from vertex operator algebras or conformal nets

the computation of the cohomology rings of all groups of order 128 abstract we describe the computation of the mod 2 cohomology rings of all 2328 groups of order 128 one consequence is that all groups of order less than 256 satisfy the strong form of benson s regularity conjecture

the exotic conformal galilei algebra and nonlinear partial differential equations abstract the conformal galilei algebra cga and the exotic conformal galilei algebra ecga are applied to construct partial differential equations pdes and systems of pdes which admit these algebras we show that there are no single second order pdes invariant under the cga but systems of pdes can admit this algebra moreover a wide class of nonlinear pdes exists which are conditionally invariant under cga it is further shown that there are systems of nonlinear pdes admitting ecga with the realisation obtained very recently in d martelli y tachikawa comments on galilei conformal field theories and their geometric realisation preprint arxiv 0903 5184v2 hep th 2009 moreover wide classes of nonlinear systems invariant under two different 10 dimensional subalgebras of ecga are explicitly constructed and an example with possible physical interpretation is presented

continuous subgroups of the generalized schrodinger groups some general results on the subalgebras of the lie algebra asch n of the generalized schrodinger group sch n and on the subalgebras of the lie algebra asch n of the generalized extended schrodinger group sch n have been obtained the subalgebra structure of the algebras asch n and asch n are studied with respect to inner automorphisms of the groups sch n and sch n respectively the maximal abelian subalgebras and the one dimensional subalgebras of the algebras asch n and asch n have been explicitly found the full classification of the subalgebras of the algebras asch 3 asch 3 which are nonconjugate to the subalgebras of asch 2 asch 2 respectively has been carried out

painleve analysis of the non linear schrodinger family of equations the authors apply the painleve test to the generalised derivative non linear schrodinger equation iut uxx iauu ux ibu2ux cu3u 2 where u denotes the complex conjugate of u and a b and c are real constants to determine under what conditions the equation might be completely integrable it is shown that apart from a trivial multiplicative factor this equation possesses the painleve property for partial differential equations as formulated by weiss tabor and carnevale 1983 only if c 1 4b 2b a when then this relation holds this is equivalent under a gauge transformation to the derivative non linear schrodinger equation dnls of kaup and newell which is known to be completely integrable or else to a linear equation

essential cohomology and extraspecial groups let p be an odd prime number and let g be an extraspecial pgroup the purpose of the paper is to show that g has no non zero essential mod p cohomology and in fact that h g ifp is cohen macaulay if and only if igi 27 and exp g 3

identification of an algebraic domain in two dimensions from a finite number of its generalized polarization tensors this paper aims at studying how finitely many generalized polarization tensors of an algebraic domain can be used to determine its shape precisely given a planar set with real algebraic boundary it is shown that the minimal polynomial with real coefficients vanishing on the boundary can be identified as the generator of a one dimensional kernel of a matrix whose entries are obtained from a finite number of generalized polarization tensors the size of the matrix depends polynomially on the degree of the boundary of the algebraic domain the density with respect to hausdorff distance of algebraic domains among all bounded domains invites to extend via approximation our reconstruction procedure beyond its natural context based on this a new algorithm for shape recognition classification is proposed with some strong hints about its efficiency

irreducible specht modules are signed young modules recently donkin defined signed young modules as a simultaneous generalization of young and twisted young modules for the symmetric group we show that in odd characteristic if a specht module s is irreducible then s is a signed young module thus the set of irreducible specht modules coincides with the set of irreducible signed young modules this provides evidence for our conjecture that the signed young modules are precisely the class of indecomposable self dual modules with specht filtrations the theorem is false in characteristic two

on the cohomology of galois groups determined by witt rings let f denote a field of characteristic different from two in this paper we describe the mod 2 cohomology of a galois group gf called the w group of f which is known to essentially characterize the witt ring wf of anisotropic quadratic modules over f we show that h gf f2 contains the mod 2 galois cohomology of f and that its structure will reflect important properties of the field we construct a space xf endowed with an action of an elementary abelian group e such that the computation of the cohomology of gf reduces to calculating the equivariant cohomology h e xf f2 for the case of a field which is not formally real this amounts to computing the cohomology of an explicit euclidean space form an object which is interesting in its own right we provide a number of examples and a substantial combinatorial computation for the cohomology of the universal w groups

on the spectrum of the poincare variational problem for two close to touching inclusions in 2d we study the spectrum of the poincare variational problem for two close to touching inclusions in r2 we derive the asymptotics of its eigenvalues as the distance between the inclusions tends to zero

the galilean relativistic principle and nonlinear partial differential equations the second order partial differential equations invariant under transformations of galilei rotation scale and projection are described

new non lie ansatze and exact solutions of nonlinear reaction diffusion convection equations a constructive method for obtaining new exact solutions of nonlinear evolution equations is further developed the method is based on the consideration of a fixed nonlinear partial differential equation together with an additional generating condition in the form of a linear high order ordinary differential equation using this method new non lie ansatze and exact solutions are obtained for two classes of diffusion equations with power and exponential nonlinearities which describe real processes in physics chemistry and biology the analysis of the found solutions and the relation of the proposed method to some approaches which have been suggested in several recently published papers are presented

generalized polarization tensors for shape description with each domain an infinite number of tensors called the generalized polarization tensors gpts is associated the gpts contain significant information on the shape of the domain in the recent paper ammari et al in math comput 81 367 386 2012 a recursive optimal control scheme to recover fine shape details of a given domain using gpts is proposed in this paper we show that the gpts can be used for shape description we also show that high frequency oscillations of the boundary of a domain are only contained in its high order gpts indeed we provide an original stability and resolution analysis for the reconstruction of small shape changes from the gpts by developing a level set version of the recursive optimization scheme we make the change of topology possible and show that the gpts can capture the topology of the domain we also propose an indicator of topology which could be used in some particular cases to test whether we have the correct number of connected components in the reconstructed image we provide analytical and numerical evidence that gpts can capture topology and high frequency shape oscillations the results of this paper clearly show that the concept of gpts is a very promising new tool for shape description

self similar solutions for a coupled system of nonlinear schrodinger equations this work is devoted to the study of self similar solutions of the 2 1 dimensional coupled nonlinear schrodinger equations which occur in nonlinear optics 1 psi 1 z psi 1 xx psi 1 yy eta mod psi 1 to 2 1 h mod psi 2 mod 2 psi 1 0 and square root 1 psi 2 psi 2 xx psi 2 yy eta mod psi 2 mod 2 1 h mod psi 1 psi 2 0 where psi 1 and psi 2 are complex functions h is a non vanishing real parameter eta or 1 and in or 1 the authors give the point symmetry properties of the model and calculate generic 2 1 dimensional symmetry reductions some exact and approximate solutions are obtained in particular they use a variational approach to determine and classify a set of physically relevant localized nonsingular self similar solutions

generic module theory abstract the process of restricting modules to cyclic shifted subgroups is a fundamental technique in the modular representation theory of elementary abelianp groups ifeis elementary abelian ofp rankrandkis an algebraically closed field of characteristicp then each point inkr 0 determines a cyclic shifted subgroup because the restriction of ake module to this shifted subgroup depends only upon the corresponding point in projective space it is often convenient to work with2kr 1instead ofkr 0 roughly speaking this paper shows that ifvis an irreducible subvariety of3kr 1andmis ake module then for almost all points invthe direct sum decomposition ofmis the same upon restriction moreover this decomposition is completely determined by the behavior ofmupon restriction to the cyclic shifted subgroup corresponding to the generic point ofv a similar idea provides a stratification of the rank variety ofminto a disjoint union of locally closed subspaces the closures of these subspaces are then described in terms of deformations of modules over a group of orderp

signed young modules and simple specht modules abstract by a result of hemmer every simple specht module of a finite symmetric group over a field of odd characteristic is a signed young module while specht modules are parametrized by partitions indecomposable signed young modules are parametrized by certain pairs of partitions the main result of this article establishes the signed young module labels of simple specht modules along the way we prove a number of results concerning indecomposable signed young modules that are of independent interest in particular we determine the label of the indecomposable signed young module obtained by tensoring a given indecomposable signed young module with the sign representation as consequences we obtain the green vertices green correspondents cohomological varieties and complexities of all simple specht modules and a class of simple modules of symmetric groups and extend the results of gill on periodic young modules to periodic indecomposable signed young modules

the cohomology of the lyons group and double covers of alternating groups from the classification theorem for finite simple groups we know that there are precisely eight sporadic simple groups which have z 2 as their largest elementary abelian 2 subgroup i e they have rank equal to four at the prime p 2 the two mathieu groups m22 m23 the two janko groups j2 j3 the mclaughlin group mcl the lyons group ly and two groups with much larger 2 sylow subgroups hs higman sims and co3 one of conway s groups in previous work the mod 2 cohomology rings of m22 am2 m23 m j2 and j3 cmm and mcl am3 were computed one of the main objectives of this paper is to determine the cohomology ring h ly f2 even though ly is a large and complicated group it is the largest of those in the list of above having order equal to 51 765 179 004 10 the answer turns out to be surprisingly simple we show that ly has two conjugacy classes of maximal 2 elementary subgroups one 2 with weyl group a7 and one 2 3 with weyl group l3 2 and we prove that these subgroups detect h ly in am4 we determined the ring of invariants h 24 a7 f2 d8 d12 d14 d15 1 a18 a20 a21 a24 a25 a27 a45 while it is well known that the invariants h 23 l3 2 are f2 d4 d6 d7 the dickson algebra from this we obtain

on a class of homogeneous nonlinear schrodinger equations a class of homogeneous norm conserving nonlinear wave equations of the schrodinger type is studied it is shown that those equations which derive from a lagrangian can be linearized but have no regular confined solutions whereas the equations which cannot be obtained from a local lagrangian do admit such confined solutions the latter however are unstable against small perturbations of the initial data

modulational instabilities and soliton solutions of a generalized nonlinear schrodinger equation previous forms of the generalized nonlinear schrodinger equation are ambiguously related to the corresponding nonlinear dispersion relation the authors therefore propose a hamiltonian equation which is determined in a unique way soliton solutions of the nonlinear schrodinger equations are found and modulational instabilities are investigated

analysis of plasmon resonance on smooth domains using spectral properties of the neumann poincare operator abstract we investigate in a quantitative way the plasmon resonance at eigenvalues and the essential spectrum the accumulation point of eigenvalues of the neumann poincare operator on smooth domains we first extend the symmetrization principle so that the single layer potential becomes a unitary operator from h 1 2 onto h 1 2 we then show that the resonance at the essential spectrum is weaker than that at eigenvalues it is shown that anomalous localized resonance occurs at the essential spectrum on ellipses and cloaking due to anomalous localized resonance does occur on ellipses like on the core shell structure considered in 19 it is shown that cloaking due to anomalous localized resonance does not occur at the essential spectrum on three dimensional balls

on carlson s depth conjecture in group cohomology we establish a weak form of carlson s conjecture on the depth of the mod p cohomology ring of a p group in particular duflot s lower bound for the depth is tight if and only if the cohomology ring is not detected on a certain family of subgroups the proofs use the structure of the cohomology ring as a comodule over the cohomology of the centre via the multiplication map we demonstrate the existence of systems of parameters so called polarised systems which are particularly well adapted to this comodule structure

vertices of specht modules and blocks of the symmetric group abstract this paper studies the vertices in the sense defined by j a green of specht modules for symmetric groups the main theorem gives for each indecomposable non projective specht module a large subgroup contained in one of its vertices a corollary of this theorem is a new way to determine the defect groups of symmetric groups the main theorem is also used to find the green correspondents of a particular family of simple specht modules as a corollary this gives a new proof of the brauer correspondence for blocks of the symmetric group the proof of the main theorem uses the brauer homomorphism on modules as developed by m broue together with combinatorial arguments using young tableaux

semicharacteristics bordism and free group actions in this paper we give characteristic class formulae for all semicharac teristic classes of all compact closed manifolds with finite fundamental groups these invariants are identified with elements in certain odd l groups and ex actly which elements occur is specified an appendix calculates the cohomology of the model groups needed a second appendix determines the structure of the l groups needed the euler characteristic of an odd dimensional manifold is zero a natural substitute is a semicharacteristic an alternating sum of the homology up to the middle dimension study of semicharacteristics was initiated by kervaire k who examined their role in differential topology and geometry the first semicharacteristic bordism invariant was introduced by derham it turns out k xx 2 m2k x k l diffl ff f a j i i

nonlinear galilei invariant pdes with infinite dimensional lie symmetry a class of nonlinear galilei invariant generalizations of the heat equation admitting infinite dimensional lie symmetry is presented lie symmetries and examples of exact solutions are found for some nonlinear equations generalizing the heat equation full sets of the absolute differential invariants for two non equivalent representations of the galilei algebra are constructed

some homotopy equivalences for sporadic geometries a previous work rsy90 established the projectivity of the reduced lefschetz modules of certain sporadic group geometries and the present paper continues that work in a wider context recent developments in sporadic group cohomology include some applications of rsy90 which in turn suggested treatment of a broader class of geometries recurring similarities in the proofs also led to a more unified treatment establishing the stronger result of homotopy equivalence of thep local geometry with the usual elementary poset ap g one equivalence method proceeds by means of a new closed set in a standard technique of quillen it was further observed that the larger list of simple groups now treated essentially coincides with those of characteristicp type suggesting another equivalence method via the poset bp g of radical or stubborn p subgroups in particular one finds that these sporadic groups satisfy an analogue of the borel tits theorem that normalizers ofp groups lie in simplex stabilizers still further intriguing coincidences remain to be explained

generic and maximal jordan types for a finite group scheme g over a field k of characteristic p 0 we associate new invariants to a finite dimensional kg module m namely for each generic point of the projectivized cohomological variety text proj mathop h bullet g k we exhibit a generic jordan type of m in the very special case in which g e is an elementary abelian p group our construction specializes to the non trivial observation that the jordan type obtained by restricting m via a generic cyclic shifted subgroup does not depend upon a choice of generators for e furthermore we construct the non maximal support variety g m a closed subset of text proj mathop h bullet g k which is proper even when the dimension of m is not divisible by p

the geometry and cohomology of the mathieu group m12 it is usually very complex to calculate the cohomology of a finite group g to date the most significant results have been on the symmetric and alternating groups n m m ma mu a m m the general linear groups over a finite field q1 and the extra special p groups q2 in the list above only the alternating groups and the gln f2 are simple indeed among the simple groups only a very few aside from those above have been understood it is probable that away from the characteristic p h psln fpn z q is available but even this is not obvious especially for q 2 and certainly we currently have no information on most of the families of groups of lie type for example g2 fq e6 fq e7 fq e8 fq or any of the sporadic groups but m11 and j1 about 15 years ago quillen introduced very powerful techniques that allowed for a much deeper understanding of group cohomology however it is also apparent that successful calculations over z p have only occurred when the group has a well behaved lattice of p elementary abelian subgroups formidable combinatorial problems arise in the general situation as well as necessarily complicated multiplicative relations for example the general calculation of h gln fp z p remains quite inaccessible and there has been only marginal progress since quillen s landmark results

symmetric invariants and cohomology of groups some time ago nakaoka in determined the additive structure of h z f2 around the same time the ring structure of h 274 f2 was calculated and it was assumed that the same type of simple multiplicative relations would appear for n 4 since then most work has concentrated on the embedding 2 o using the polynomial ring structure of h xo f2 in particular most of the recent work in hi was explicitly or implicitly described in m m more than a decade ago and provides no new information on the multiplicative structure of the mod 2 cohomology of finite symmetric groups in this paper we use invariant theory to detect unexpected multiplicative relations in h z f2 the complicated nature of the numerical evidence which we present explains the absence of complete calculations for n 16 the rings of invariants which we study are at the core of any computation of h 2 f2 they build up sucessively yielding relations rich in symmetry but of a highly convoluted type the following problem lies at the heart of all this let p be a polynomial ring k over f2 on n variables and let p k p then ek acts by permuting blocks of generators 1

on integrable doebner goldin equations we integrate sub families of a family of nonlinear schrodinger equations proposed by doebner and goldin in the 1 1 dimensional case which have exceptional lie symmetries since the method of integration involves non local transformations of dependent and independent variables general solutions obtained include implicitly determined functions by properly specifying one of the arbitrary functions contained in these solutions we obtain broad classes of explicit square integrable solutions the physical significance and some analytical properties of the solutions obtained are briefly discussed

sampling and reconstruction of shapes with algebraic boundaries we present a sampling theory for a class of binary images with finite rate of innovation fri every image in our model is the restriction of mathbb 1 p leq 0 to the image plane where mathbb 1 denotes the indicator function and p is some real bivariate polynomial this particularly means that the boundaries in the image form a subset of an algebraic curve with the implicit polynomial p we show that the image parameters i e the polynomial coefficients satisfy a set of linear annihilation equations with the coefficients being the image moments the inherent sensitivity of the moments to noise makes the reconstruction process numerically unstable and narrows the choice of the sampling kernels to polynomial reproducing kernels as a remedy to these problems we replace conventional moments with more stable generalized moments that are adjusted to the given sampling kernel the benefits are threefold 1 it relaxes the requirements on the sampling kernels 2 produces annihilation equations that are robust at numerical precision and 3 extends the results to images with unbounded boundaries we further reduce the sensitivity of the reconstruction process to noise by taking into account the sign of the polynomial at certain points and sequentially enforcing measurement consistency we consider various numerical experiments to demonstrate the performance of our algorithm in reconstructing binary images including low to moderate noise levels and a range of realistic sampling kernels

parameterized families of polynomials for bounded algebraic curve and surface fitting interest in algebraic curves and surfaces of high degree as geometric models or shape descriptors for different model based computer vision tasks has increased in recent years and although their properties make them a natural choice for object recognition and positioning applications algebraic curve and surface fitting algorithms often suffer from instability problems one of the main reasons for these problems is that while the data sets are always bounded the resulting algebraic curves or surfaces are in most cases unbounded in this paper the authors propose to constrain the polynomials to a family with bounded zero sets and use only members of this family in the fitting process for every even number d the authors introduce a new parameterized family of polynomials of degree d whose level sets are always bounded in particular its zero sets this family has the same number of degrees of freedom as a general polynomial of the same degree three methods for fitting members of this polynomial family to measured data points are introduced experimental results of fitting curves to sets of points in r sup 2 and surfaces to sets of points in r sup 3 are presented

sylow subgroups of symmetric and alternating groups and the vertex of s kp p 1p in characteristic p abstract we show that the sylow p subgroups of a symmetric group respectively an alternating group are characterized as the p subgroups containing all elementary abelian p subgroups up to conjugacy of the symmetric group respectively the alternating group we apply the characterization result for symmetric groups to compute the vertices of the hook specht modules associated to the partition k p p 1 p under the assumption that k 1 mod p and k 1 mod p 2

modular characteristic classes for representations over finite fields abstract the cohomology of the degree n general linear group over a finite field of characteristic p with coefficients also in characteristic p remains poorly understood for example the lowest degree previously known to contain nontrivial elements is exponential in n in this paper we introduce a new system of characteristic classes for representations over finite fields and use it to construct a wealth of explicit nontrivial elements in these cohomology groups in particular we obtain nontrivial elements in degrees linear in n we also construct nontrivial elements in the mod p homology and cohomology of the automorphism groups of free groups and the general linear groups over the integers these elements reside in the unstable range where the homology and cohomology remain mysterious

properties of nonlinear schrodinger equations associated with diffeomorphism group representations the authors recently derived a family of nonlinear schrodinger equations on r3 from fundamental considerations of generalized symmetry ih cross delta t psi h cross 2 2m del 2 psi f psi psi psi ih cross d del 2 psi mod del psi mod 2 mod psi mod 2 psi where f is an arbitrary real functional and d a real continuous quantum number these equations descriptive of a quantum mechanical current that includes a diffusive term correspond to unitary representations of the group diff m parametrized by d where m r3 is the physical space in the present paper we explore the most natural ansatz for f which is labelled by five real coefficients we discuss the invariance properties describe the stationary states and some non stationary solutions and determine the extra dissipative terms that occur in the ehrenfest theorem we identify an interesting galilean invariant subfamily whose properties we investigate including the case where the dissipative terms vanish

field theory and the cohomology of some galois groups abstract we prove that two arithmetically significant extensions of a field f coincide if and only if the witt ring wf is a group ring z n g furthermore working modulo squares with galois groups which are 2 groups we establish a theorem analogous to hilbert s theorem 90 and show that an identity linking the cohomological dimension of the galois group of the quadratic closure of f the length of a filtration on a certain module over a galois group and the dimension over f 2 of the square class group of the field holds for a number of interesting families of fields finally we discuss the cohomology of a particular galois group in a topological context

poincare s variational problem in potential theory one of the earliest attempts to rigorously prove the solvability of dirichlet s boundary value problem was based on seeking the solution in the form of a potential of double layer and this leads to an integral equation whose kernel is in general both singular and non symmetric c neumann succeeded with this approach for smoothly bounded convex domains and h poincare by a tremendous tour de force showed how to push through the analysis for domains with sufficiently smooth boundaries but no hypothesis of convexity in this work he was according to his own account guided by consideration of a variational problem involving the partition of energy of an electrostatic field induced by charges placed on the boundary of a domain more precisely the charge distributions which render stationary the energy of the field inside the domain divided by the energy of the field outside the domain unfortunately a rigorous treatment of this problem was not possible with the tools available at that time as poincare was well aware so far as we know the only one to propose a rigorous treatment of poincare s problem was t carleman in the two dimensional case in his doctoral dissertation thanks to later developments especially concerning sobolev spaces and spectral theory of operators on hilbert space we can now give a complete general and rigorous account of poincare s variational problem and that is the main object of the present paper as a by product we refine some technical aspects in the theory of symmetrizable operators and prove in any number of dimensions the basic properties of the analogue of the planar bergman schiffer singular integral equation we interpret poincare s variational principle as a non selfadjoint eigenvalue problem for the angle operator between two distinct pairs of subspaces of potentials we also prove a series of novel spectral analysis facts some of them conjectured by poincare related to the poincare neumann integral operator

on non linear partial differential equations with an infinite dimensional conditional symmetry abstract the invariance of non linear partial differential equations under a certain infinite dimensional lie algebra a n z in n spatial dimensions is studied the special case a 1 2 was introduced in j stat phys 75 1994 1023 and contains the schrodinger lie algebra sch 1 as a lie subalgebra it is shown that there is no second order equation which is invariant under the massless realizations of a n z however a large class of strongly non linear partial differential equations is found which are conditionally invariant with respect to the massless realization of a n z such that the well known monge ampere equation is the required additional condition new exact solutions are found for some representatives of this class

galilei invariant nonlinear systems of evolution equations all systems of n 1 dimensional quasilinear second order evolution equations invariant under chain of algebras ag 1 n contained in implied by ag1 1 n contained in implied by ag2 1 n are described the results obtained are illustrated by the examples of the nonlinear schrodinger equations hamilton jacobi type systems and reaction diffusion equations

symmetric and exterior powers linear source modules and representations of schur superalgebras we study modules for the general linear group over an infinite field of arbitrary characteristic which are direct summands of tensor products of exterior powers and symmetric powers of the natural module these modules which we call listing modules include the tilting modules and the injective modules for schur algebras the modules are studied via their relationship to linear source modules for symmetric groups on the one hand and simple modules for schur superalgebras on the other listing modules are parametrized by certain pairs of partitions they are used to describe by generators and relations the grothendieck ring of polynomial functors generated by the symmetric and exterior powers we also continuing work of j grabmeier describe the vertices and sources of linear source modules for symmetric groups 2000 mathematical subject classification 20g05 20c30

intertwining operators for l conformal galilei algebras and hierarchy of invariant equations the l conformal galilei algebra denoted by is a non semisimple lie algebra specified by a pair of parameters d l the algebra is regarded as a nonrelativistic analogue of the conformal algebra we derive hierarchies of partial differential equations which have invariance of the group generated by with a central extension as kinematical symmetry this is done by developing a representation theory such as verma modules singular vectors of and vector field representations for d 1 2

evolution equations invariant under two dimensional space time schrodinger group the most general second order evolution equation t f x t x x xx xx 0 invariant under the galilei galilei similitude and schrodinger groups in two dimensions is constructed a preliminary step is a classification of all possible realizations of the corresponding lie algebras of vector fields in r2 c parametrized by x t and applications of this study include the investigation of nonlinear alternatives to quantum mechanics and nonrelativistic classical field theories among the schrodinger invariant equations in particular are found integrable equations linearizable by contact transformations

the cohomology of the mclaughlin group and some associated groups the group of order 27 syl2 m22 h seems quite remarkable being the sylow 2 subgroup of three of the sporadics m22 m23 mcl as well as the simple group of lie type psu 4 3 also it is a particularly nice index two subgroup of syl2 ly we are interested in determining the mod 2 cohomology of these groups and throughout this note all cohomology groups will be taken with f2 coe cients so the coe cients will generally be suppressed we use the standard atlas see co notation we have already determined the rings h m22 h h am2 and h m23 h h m in this article we determine the rings h psu4 3 h h and h mcl h h

analytical shape recovery of a conductivity inclusion based on faber polynomials a conductivity inclusion inserted in a homogeneous background induces a perturbation in the background potential this perturbation admits a multipole expansion whose coefficients are the so called generalized polarization tensors gpts gpts can be obtained from multistatic measurements as a modification of gpts the faber polynomial polarization tensors fpts were recently introduced in two dimensions in this study we design two novel analytical non iterative methods for recovering the shape of a simply connected inclusion from gpts by employing the concept of fpts first we derive an explicit expression for the coefficients of the exterior conformal mapping associated with an inclusion in a simple form in terms of gpts which allows us to accurately reconstruct the shape of an inclusion with extreme or near extreme conductivity secondly we provide an explicit asymptotic formula in terms of gpts for the shape of an inclusion with arbitrary conductivity by considering the inclusion as a perturbation of its equivalent ellipse with this formula one can non iteratively approximate an inclusion of general shape with arbitrary conductivity including a straight or asymmetric shape numerical experiments demonstrate the validity of the proposed analytical approaches

the generalized polarization tensors for resolved imaging part i shape reconstruction of a conductivity inclusion with each lipschitz domain and material parameter an infinite number of tensors called the generalized polarization tensors gpts is associated the gpts contain significant information on the shape of the domain and its material parameter they generalize the concept of polarization tensor pt which can be seen as the first order gpt it is known that given an arbitrary shape one can find an equivalent ellipse or ellipsoid with the same pt in this paper we consider the problem of recovering finer details of the shape of a given domain using higher order polarization tensors we design an optimization approach which solves the problem by minimizing a weighted discrepancy functional in order to compute the shape derivative of this functional we rigorously derive an asymptotic expansion of the perturbations of the gpts that are due to a small deformation of the boundary of the domain our derivations are based on the theory of layer potentials we perform some numerical experiments to demonstrate the validity and the limitations of the proposed method the results clearly show that our approach is very promising in recovering fine shape details mathematics subject classification msc2000 35r30 35b30

describing complicated objects by implicit polynomials this paper introduces and focuses on two problems first is the representation power of closed implicit polynomials of modest degree for curves in 2 d images and surfaces in 3 d range data super quadrics are a small subset of object boundaries that are well fitted by these polynomials the second problem is the stable computationally efficient fitting of noisy data by closed implicit polynomial curves and surfaces the attractive features of these polynomials for vision is discussed

primitives and central detection numbers in group cohomology abstract fix a prime p given a finite group g let h g denote its mod p cohomology in the early 1990s henn lannes and schwartz introduced two invariants d 0 g and d 1 g of h g viewed as a module over the mod p steenrod algebra they showed that in a precise sense h g is respectively detected and determined by h d c g v for d d 0 g and d d 1 g with v running through the elementary abelian p subgroups of g the main goal of this paper is to study how to calculate these invariants we find that a critical role is played by the image of the restriction of h g to h c where c is the maximal central elementary abelian p subgroup of g a measure of this is the top degree e g of the finite dimensional hopf algebra h c h g f p a number that tends to be quite easy to calculate our results are complete when g has a p sylow subgroup p in which every element of order p is central using the benson carlson duality we show that in this case d 0 g d 0 p e p and a similar exact formula holds for d 1 as a bonus we learn that h e g p contains nontrivial essential cohomology reproving and sharpening a theorem of adem and karagueuzian in general we are able to show that d 0 g max e c g v v g if certain cases of benson s regularity conjecture hold in particular this inequality holds for all groups such that the difference between the p rank of g and the depth of h g is at most 2 when we look at examples with p 2 we learn that d 0 g 14 for all groups with 2 sylow subgroup of order up to 64 with equality realized when g su 3 4 en route we study two objects of independent interest if c is any central elementary abelian p subgroup of g then h g is an h c comodule and we prove that the subalgebra of h c primitives is always noetherian of krull dimension equal to the p rank of g minus the p rank of c if the depth of h g equals the rank of z g we show that the depth essential cohomology of g is nonzero reproving and extending a theorem of green and cohen macauley in a certain sense and prove related structural results