zeta function determinant of the laplace operator on the d dimensional ball we present a direct approach for the calculation of functional determinants of the laplace operator on balls dirichlet and robin boundary conditions are considered using this approach formulas for any value of the dimension d of the ball can be obtained quite easily explicit results are presented here for dimensionsd 2 3 4 5 and 6

the intersection homology d module in finite characteristic for y a closed normal subvariety of codimension c of a smooth variety x brylinski and kashiwara showed that the local cohomology module c y x x contains a unique simple x submodule denoted by y x in this paper the analogous result is shown for x and y defined over a perfect field of finite characteristic moreover a local construction of y x is given relating it to the theory of tight closure from the construction one obtains a criterion for the x simplicity of c y x x

characteristic cycles of perverse sheaves and milnor fibers we show that the milnor monodromy and the milnor numbers naturally appear in the characteristic cycles of irreducible perverse sheaves having the singularities of a complex hypersurface we find also a new numerical constraint on the milnor monodromy for general hypersurface singularities

functional determinants on spheres and sectors the scalar functional determinants on sectors of the two dimensional disc and spherical cap are determined for arbitrary angles rational factors of the whole sphere and hemisphere expressions are also given in low dimensions for both the ordinary and the conformal laplacian

the general decomposition theory of su 2 gauge potential topological structure and bifurcation of su 2 chern density by means of the geometric algebra the general decomposition of su 2 gauge potential on the sphere bundle of a compact and oriented four dimensional manifold is given using this decomposition theory the su 2 chern density has been studied in detail it shows that the su 2 chern density can be expressed in terms of the function one can also find that the zero points of the vector fields are essential to the topological properties of a manifold it is shown that there exists the crucial case of branch process at the zero points based on the implicit function theorem and the taylor expansion the bifurcation of the chern density is detailed in the neighborhoods of the bifurcation points of it is pointed out that since the chern density is a topological invariant the sum topological chargers of the branches will remain constant during the bifurcation process

bifurcation in lp rn for a semilinear elliptic equation keywords lsp p bifurcation constrained minimization sobolev spaces nonlinear eigenvalue problem semilinear spectrum linearisation variational structure reference ana article 1988 001doi 10 1112 plms s3 57 3 511 record created on 2008 12 10 modified on 2016 08 08

a palais smale approach to lane emden equations we consider the unbounded domain problems u u u p 2 u in u 0i n a ndu 0o n w here is an unbounded domain in r n 2 2 and 2 for n 2 the existence of a ground state solution to the problems is greatly affected by the shape of the domain to determine the existence of the solutions in a general domain remains a challenge task for the flat interior flask domain that consists a strip and a ball attached to the bottom of the strip previous results have asserted the existence of a ground state solution when the diameter of the ball is greater than a positive constant however the existence of the solutions when the diameter of the ball equals to the width of the strip is still an important open question this article resolves the open question partially by considering a variation of the flat interior flask domain which is formed by attaching a stretched ball to the bottom of the strip

conformal transformations and the effective action in the presence of boundaries the conformal properties of the heat kernel expansion are used to determine the local form of the coefficients in a manifold with boundary the conformal transformation of the effective action is obtained a novel derivation of the boundary term in the gauss bonnet chern theorem is detailed

analytic surgery of the zeta function in this paper we study the asymptotic behavior in the sense of meromorphic functions of the zeta function of a laplace type operator on a closed manifold when the underlying manifold is stretched in the direction normal to a dividing hypersurface separating the manifold into two manifolds with infinite cylindrical ends we also study the related problem on a manifold with boundary as the manifold is stretched in the direction normal to its boundary forming a manifold with an infinite cylindrical end such singular deformations fall under the category of analytic surgery developed originally by hassell comm anal geom 6 255 289 1998 hassell et al comm anal geom 3 115 222 1995 and mazzeo and melrose geom funct anal 5 14 75 1995 in the context of eta invariants and determinants

heat kernel coefficients of the laplace operator on the d dimensional ball we present a very quick and powerful method for the calculation of heat kernel coefficients it makes use of rather common ideas as integral representations of the spectral sum mellin transforms non trivial commutation of series and integrals and skillful analytic continuation of zeta functions on the complex plane we apply our method to the case of the heat kernel expansion of the laplace operator on a d dimensional ball with either dirichlet neumann or in general robin boundary conditions the final formulas are quite simple using this case as an example we illustrate in detail our scheme which serves for the calculation of an in principle arbitrary number of heat kernel coefficients in any situation when the basis functions are known we provide a complete list of new results for the coefficients b3 b10 corresponding to the d dimensional ball with all the mentioned boundary conditions and d 3 4 5

intersection homology ii in 19 20 we introduced topological invariants ih x called intersection homology groups for the study of singular spaces x these groups depend on the choice of a perversity p a perversity is a function from 2 3 to the non negative integers such that both c and c 2 c are positive and increasing functions of c 2 1 the group ihr is defined for spaces x called pseudomanifolds a pseudomanifold of dimension n is a space that admits a stratification x x n x n z x n 3 x 1 x 0

topological structure of gauss bonnet chern density and its topological current the decomposition of spin connection on the sphere bundle of a compact and oriented even dimensional riemannian manifold is detailed it is shown that the gauss bonnet chern form can be expressed in terms of a smooth vector field and taken the form of the function the topological current of the gauss bonnet chern theorem and its topological structure are discussed

functional determinants for laplacians on annuli and elliptical regions in this article we consider the functional determinant on an annulus and elliptic regions with a variety of boundary conditions known results for the annulus are rederived and extended using contour integration techniques conformal transformations are then used to relate these results to ellipsoidal annuli providing very explicit answers similarly for the ellipse answers are related to those for a disc but explicit results can only be found for small eccentricity

witt spaces a geometric cycle theory for ko homology at odd primes introduction this paper studies a class of stratified piecewiselinear pseudomanifolds which we call witt spaces characterized by natural local intersection homology conditions 11 we compute the cobordism groups by introducing an invariant taking values in the witt group of symmetric bilinear forms over the rationals w q these pseudomanifolds solve a problem posed by d sullivan 25 to construct a class of p l cycles with signature which represent the connected version of ko homology at odd primes ko 0d z 1 2 the result was one of the first applications of the intersection homology theory of goresky and macpherson 11 specifically the rational intersection homology groups of witt spaces satisfy a poincare duality theorem to each witt space x we associate a p l invariant w x with values in w q this invariant generalizes the signature of manifolds and satisfies cobordism invariance additivity and a product formula adapting classical surgery spherical modification to this setting we prove that w x determines the cobordism class of x and obtain an explicit description of the witt cobordism groups the only nontrivial groups occur in dimensions 4k and for k 0 they are w q sullivan 25 27 has shown that a class of p l pseudomanifolds equipped with such an invariant possesses canonical orientations in ko 0 z 1 2 the orientations induce a natural transformation it of homology theories from witt space bordism to ko 0 z 1 2 the structure of w q is known 14 and we conclude that it is an equivalence at odd primes in this paper we study the cobordism theory of a class of stratified p l pseudomanifolds by means of the recent intersection homology theory of goresky and macpherson 11 goresky and macpherson were interested in finding a class of spaces with cobordism invariant signature for the purpose of extending the hirzebruch l class to the setting of manifolds with

conformal invariance and the regularised one loop effective action a unique family gn 1 of conformal invariants polynomial in the extrinsic curvature of a hypersurface embedded in an n dimensional riemannian manifold m is constructed in the quantum theory of a conformally coupled scalar field on a manifold m with boundary delta m the counterterm that regularises the one loop effective action contains members of gn 1 integrated over delta m the counterterms through n 4 are given explicitly and a derivation of the n 4 boundary contribution is given based on a flat space result

smeared heat kernel coefficients on the ball and generalized cone we consider smeared zeta functions and heat kernel coefficients on the bounded generalized cone in arbitrary dimensions the specific case of a ball is analyzed in detail and used to restrict the form of the heat kernel coefficients an on smooth manifolds with boundary supplemented by conformal transformation techniques it is used to provide an effective scheme for the calculation of the an as an application the complete a5 2 coefficient is given

whitney stratified chains and cochains this paper contains the technical constructions necessary for a geometric cycle definition of cohomology and homology in the context of whitney stratifications cup and cap products are interpreted as the transverse intersection of geometric cocycles and cycles