scalar curvature and hammocks scalar curvature is the simplest generalization of gaussian curvature to higher dimensions however there are many questions open with regard to its relation to other geometric quantities and topology here we will prove and illustrate some features of scalar curvature in higher dimensions related to a general hammock effect for scalar curvature namely the one sided affinity for curvature decreasing deformations the first one is concerned with some prescribed decrease of the scalar curvature scal g of some riemannian metric g on a given manifoldmn of dimension 3 we denote the e neighborhood of some set u with respect to g by ue

the quermassintegral inequalities for k convex starshaped domains abstract we give a proof of the isoperimetric inequality for quermassintegrals of non convex starshaped domains using a result of gerhardt c gerhardt flow of nonconvex hypersurfaces into spheres j differential geometry 32 1990 299 314 and urbas j urbas on the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures math z 205 1990 355 372 on an expanding geometric curvature flow

comparing k populations with linear rank statistics abstract there are many nonparametric tests available for the analysis of k independent samples for example the kruskal wallis test wilcoxon rank sum test when k 2 is widely used for detecting location differences and mood or ansari bradley or sigel tukey rank statistics are often suggested for detecting scale differences if more general alternatives are of interest then one can use omnibus tests such as the generalized kolmogorov smirnov and cramer von mises statistics see conover 1980 secs 6 3 and 6 4 or kiefer 1959 unfortunately significant results from these omnibus tests are hard to interpret instead i would like to propose some new tests based on a 4 k table of linear rank statistics that are simple to interpret because the four rows of the table are aimed at specific alternatives location scale skewness and kurtosis row summary statistics consist of weighted sums of squares of the individual entries the first two of these are the kruskal wallis statistic and mood s k sa

a new mass for asymptotically flat manifolds abstract in this paper we introduce a mass for asymptotically flat manifolds by using the gauss bonnet curvature we first prove that the mass is well defined and is a geometric invariant if the gauss bonnet curvature is integrable and the decay order satisfies n 4 3 motivated by an elegant idea of lam 35 we then show a positive mass theorem for this new mass for asymptotically flat graphs over r n moreover we obtain various penrose type inequalities in this case in these penrose type inequalities we establish relationship between the new mass and various geometric integrals

the mass of asymptotically hyperbolic manifolds motivated by certain problems in general relativity and riemannian geometry we study manifolds which are asymptotic to the hyperbolic space in a certain sense it is shown that an invariant the so called total mass can be defined unambiguously a positive mass theorem is established by using the spinor method

an elliptic analogue of the multiple gamma function a hierarchy of functions including the elliptic gamma function is introduced it can be interpreted as an elliptic analogue of the multiple gamma function and its trigonometric limit coincides with a q analogue of the multiple gamma function some properties of the functions are considered

the modular properties and the integral representations of the multiple elliptic gamma functions abstract we show the modular properties of the multiple elliptic gamma functions which are an extension of those of the theta function and the elliptic gamma function the modular property of the theta function is known as jacobi s transformation and that of the elliptic gamma function was provided by felder and varchenko in this paper we deal with the multiple sine functions since the modular properties of the multiple elliptic gamma functions result from the equivalence between two ways to represent the multiple sine functions as infinite products we also derive integral representations of the multiple sine functions and the multiple elliptic gamma functions we introduce correspondences between the multiple elliptic gamma functions and the multiple sine functions

gluing nekrasov partition functions in this paper we summarise the localisation calculation of 5d super yang mills on simply connected toric sasaki einstein se manifolds we show how various aspects of the computation including the equivariant index the asymptotic behaviour and the factorisation property are governed by the combinatorial data of the toric geometry we prove that the perturbative partition function on a simply connected se manifold corresponding to an n gon toric diagram factorises to n copies of perturbative part zero instanton sector of the nekrasov partition function this leads us to conjecture a prescription for the computation of the complete partition function by gluing n copies of the full nekrasov partition functions this work is a generalisation of some earlier computation carried out on yp q manifolds whose moment map cone has a quadrangle base and our result is valid for manifolds whose moment map cones have pentagon base hexagon base etc the algorithm we used for dealing with general cones may also be of independent interest

a definition of total energy momenta and the positive mass theorem on asymptotically hyperbolic 3 manifolds i two sets of asymptotically hyperbolic initial data are defined which correspond to the spatial infinity in asymptotically ads spacetimes and to the null infinity in asymptotically minkowski spacetimes respectively the positive mass theorem involving the total energy the total linear momentum and the total angular momentum is established for these initial data sets

an equation of monge ampere type in conformal geometry and four manifolds of positive ricci curvature we formulate natural conformally invariant conditions on a 4 manifold for the existence of a metric whose schouten tensor satisfies a quadratic inequality this inequality implies that the eigenvalues of the ricci tensor are positively pinched

the elliptic gamma function and sl 3 z z3 abstract the elliptic gamma function is a generalization of the euler gamma function and is associated to an elliptic curve its trigonometric and rational degenerations are the jackson q gamma function and the euler gamma function respectively the elliptic gamma function appears in baxter s formula for the free energy of the eight vertex model and in the hypergeometric solutions of the elliptic qkzb equations in this paper the properties of this function are studied in particular we show that elliptic gamma functions are generalizations of automorphic forms of g sl 3 z z 3 associated to a non trivial class in h3 g z

components of pearson s phi squared distance measure for the k sample problem abstract a general approach to the k sample problem for continuous random variables based on a generalization of pearson s classical 2 measure is presented tests are developed by considering an analysis of variance type of orthogonal decomposition of a distance measure for a k population goodness to fit hypothesis estimates of some of these components are shown to provide test statistics for the hypothesis that the k populations have a common distribution several classical parametric and nonparametric tests are found to be quadratic functions of these component estimates in addition a new class of test statistic is derived that possesses aspects of both parametric and nonparametric methods

multiple elliptic gamma functions associated to cones abstract we define generalizations of the multiple elliptic gamma functions and the multiple sine functions associated to good rational cones we explain how good cones are related to collections of s l r z elements and prove that the generalized multiple sine and multiple elliptic gamma functions enjoy infinite product representations and modular properties determined by the cone this generalizes the modular properties of the elliptic gamma function studied by felder and varchenko and the results about the usual multiple sine and elliptic gamma functions found by narukawa

k sample anderson darling tests abstract two k sample versions of an anderson darling rank statistic are proposed for testing the homogeneity of samples their asymptotic null distributions are derived for the continuous as well as the discrete case in the continuous case the asymptotic distributions coincide with the k 1 fold convolution of the asymptotic distribution for the anderson darling one sample statistic the quality of this large sample approximation is investigated for small samples through monte carlo simulation this is done for both versions of the statistic under various degrees of data rounding and sample size imbalances tables for carrying out these tests are provided and their usage in combining independent one or k sample anderson darling tests is pointed out the test statistics are essentially based on a doubly weighted sum of integrated squared differences between the empirical distribution functions of the individual samples and that of the pooled sample one weighting adjusts for the possibly different s

twisting with a flip the art of pestunization we construct mathcal n 2 supersymmetric yang mills theory on 4d manifolds with a killing vector field with isolated fixed points it turns out that for every fixed point one can allocate either instanton or anti instanton contributions to the partition function and that this is compatible with supersymmetry the equivariant donaldson witten theory is a special case of our construction we present a unified treatment of pestun s calculation on s 4 and equivariant donaldson witten theory by generalizing the notion of self duality on manifolds with a vector field we conjecture the full partition function for a mathcal n 2 theory on any 4d manifold with a killing vector using this new notion of self duality to localize a supersymmetric theory is what we call pestunization

test statistics derived as components of pearson s phi squared distance measure abstract a general approach to the comparison of two absolutely continuous distributions is presented based on pearson s 2 measure although no simple estimates are available for 2 it can be decomposed into components all of which are easily estimated asymptotic distribution theory is derived for the component estimates and the form of the components is considered for several standard problems it is demonstrated that the components of 2 approach provides a unified approach to the development of tests for a variety of problems and that it is useful for generating new test procedures the components of 2 approach is based on what is termed a comparison density that is given two absolutely continuous distributions f and h with densities f and h the comparison density is defined as d u f h 1 u h h 1 u where h 1 is the h quantile function since d must be uniform under the hypothesis f h an overall measure of the validity of the hypothesis is provided by 2 1 0 d u 1 2 du by u

first order analytic difference equations and integrable quantum systems we present a new solution method for a class of first order analytic difference equations the method yields explicit minimal solutions that are essentially unique special difference equations give rise to minimal solutions that may be viewed as generalized gamma functions of hyperbolic trigonometric and elliptic type euler s gamma function being of rational type we study these generalized gamma functions in considerable detail the scattering and weight functions u and w functions associated to various integrable quantum systems can be expressed in terms of our generalized gamma functions we obtain detailed information on these u and w functions exploiting the difference equations they satisfy

proof of the riemannian penrose inequality using the positive mass theorem we prove the riemannian penrose conjecture an important case of a con jecture 41 made by roger penrose in 1973 by defining a new flow of metrics this flow of metrics stays inside the class of asymptotically flat riemannian 3 manifolds with nonnegative scalar curvature which contain minimal spheres in particular if we consider a riemannian 3 manifold as a totally geodesic submanifold of a space time in the context of general relativity then outermost minimal spheres with total area a correspond to apparent horizons of black holes contributing a massa 16 scalar cur vature corresponds to local energy density at each point and the rate at which the metric becomes flat at infinity corresponds to total mass also called the adm mass the riemannian penrose conjecture then states that the total mass of an asymptotically flat 3 manifold with nonnegative scalar curvature is greater than or equal to the mass contributed by the black holes the flow of metrics we define continuously evolves the original 3 metric to a schwarzschild 3 metric which represents a spherically symmetric black hole in vacuum we define the flow such that the area of the minimal spheres which flow outward and hence the mass contributed by the black holes in each of the metrics in the flow is constant and then use the positive mass theorem to show that the total mass of the metrics is nonincreasing then since the total mass equals the mass of the black hole in a schwarzschild metric the riemannian penrose conjecture follows we also refer the reader to the beautiful work of huisken and ilmanen 30 who used inverse mean curvature flows of surfaces to prove that the total mass is at least the mass contributed by the largest black hole in sections 1 and 2 we motivate the problem discuss important quantities like total mass and horizons of black holes and state the positive mass theorem and the penrose conjecture for riemannian 3 manifolds in section 3 we give the proof of the riemannian penrose