In a recent paper, Miller derived a Kummer-type transformation for the generalised hypergeometric function $_{p}{{F}_{p}}(x)$ when pairs of parameters differ by unity, by means of a reduction formula for a certain Kampé de Fériet function. An alternative and simpler derivation of this transformation is obtained here by application of the well-known Kummer transformation for the confluent hypergeometric function corresponding to $p\,=\,1$

This note gives a simple cocycle-theoretic proof of the Verdier hypercovering theorem. This theorem approximates morphisms $[X,\,Y]$ in the homotopy category of simplicial sheaves or presheaves by simplicial homotopy classes of maps, in the case where $Y$ is locally fibrant. The statement proved in this paper is a generalization of the standard Verdier hypercovering result in that it is pointed (in a very broad sense) and there is no requirement for the source object $X$ to be locally fibrant.

In this paper we define lower, upper, and symmetric completeness and discuss closure of the sets in products and direct sums. In particular, we introduce suitable bases for these topologies, which leads us to investigate completeness of the direct sum and its components. Some results obtained about $X$-topologies and polars of the neighborhoods.

The groups of (linear) similarity and coincidence isometries of certain modules $\Gamma $ in $d$-dimensional Euclidean space, which naturally occur in quasicrystallography, are considered. It is shown that the structure of the factor group of similarity modulo coincidence isometries is the direct sum of cyclic groups of prime power orders that divide $d$. In particular, if the dimension $d$ is a prime number $p$, the factor group is an elementary abelian $p$-group. This generalizes previous results obtained for lattices to situations relevant in quasicrystallography.

A complex hyperbolic triangle group is a group generated by three involutions fixing complex lines in complex hyperbolic space. Our purpose in this paper is to improve a previous result and to discuss discreteness of complex hyperbolic triangle groups of type $(n,\,n,\,\infty ;\,k)$.

The goal of this note is to prove that the signed Segre forms of Griffiths’ positive vector bundles are positive.

It is shown that a Lie algebra having a nilpotent radical has a fundamental set of invariants consisting of Casimir operators. A different proof is given in the well known special case of an abelian radical. A result relating the number of invariants to the dimension of the Cartan subalgebra is also established.

Let $X$ be a smooth complex projective variety, and let $H\,\in \,\text{Pic}\left( X \right)$ be an ample line bundle. Assume that $X$ is covered by rational curves with degree one with respect to $H$ and with anticanonical degree greater than or equal to $\left( \dim\,X\,-\,1 \right)/2$. We prove that there is a covering family of such curves whose numerical class spans an extremal ray in the cone of curves $\text{NE}\left( X \right)$.

We generalize Löwner's method for proving that matrix monotone functions are operator monotone. The relation $x\,\le \,y$ on bounded operators is our model for a definition of ${{C}^{*}}$-relations being residually finite dimensional.

Our main result is a meta-theorem about theorems involving relations on bounded operators. If we can show there are residually finite dimensional relations involved and verify a technical condition, then such a theorem will follow from its restriction to matrices.

Applications are shown regarding norms of exponentials, the norms of commutators, and “positive” noncommutative $*$-polynomials.

For a manifold with boundary, the restriction of Chern's transgression form of the Euler curvature form over the boundary is closed. Its cohomology class is called the secondary Chern– Euler class and was used by Sha to formulate a relative Poincaré–Hopf theorem under the condition that the metric on the manifold is locally product near the boundary. We show that the secondary Chern–Euler form is exact away from the outward and inward unit normal vectors of the boundary by explicitly constructing a transgression form. Using Stokes’ theorem, this evaluates the boundary term in Sha's relative Poincaré–Hopf theorem in terms of more classical indices of the tangential projection of a vector field. This evaluation in particular shows that Sha's relative Poincaré–Hopf theorem is equivalent to the more classical law of vector fields.

We establish a mixed norm estimate for the Radon transform in ${{\mathbb{R}}^{2}}$ when the set of directions has fractional dimension. This estimate is used to prove a result about an exceptional set of directions connected with projections of planar sets. That leads to a conjecture analogous to a well-known conjecture of Furstenberg.

In this paper we study real hypersurfaces in a non-flat complex space form with $\eta $-parallel shape operator. Several partial characterizations of these real hypersurfaces are obtained.

The study carried out in this paper about some new examples of Banach spaces, consisting of certain valued fields extensions, is a typical non-archimedean feature. We determine whether these extensions are of countable type, have $t$-orthogonal bases, or are reflexive. As an application we construct, for a class of base fields, a norm $\left\| \,.\, \right\|$ on ${{c}_{0}}$, equivalent to the canonical supremum norm, without non-zero vectors that are $\left\| \,.\, \right\|$-orthogonal and such that there is a multiplication on ${{c}_{0}}$ making $\left( {{c}_{0}},\,\left\| \,.\, \right\| \right)$ into a valued field.

An algebra $A$ is homogeneous if the automorphism group of $A$ acts transitively on the one-dimensional subspaces of $A$. The existence of homogeneous algebras depends critically on the choice of the scalar field. We examine the case where the scalar field is the rationals. We prove that if $A$ is a rational homogeneous algebra with $\dim\,A\,>\,1$, then ${{A}^{2}}\,=\,0$.

We determine the best constants ${{C}_{p,\infty }}$ and ${{C}_{1,p}},\,1\,<\,p\,<\,\infty $, for which the following holds. If $u,v$ are orthogonal harmonic functions on a Euclidean domain such that $v$ is differentially subordinate to $u$, then

$${{\left\| v \right\|}_{p}}\le {{C}_{p}}{{,}_{\infty }}{{\left\| u \right\|}_{\infty }},\,\,\,\,\,\,\,\,\,\,\,{{\left\| v \right\|}_{1}}\le {{C}_{1,p}}{{\left\| u \right\|}_{p}}.$$

In particular, the inequalities are still sharp for the conjugate harmonic functions on the unit disc of ${{\mathbb{R}}^{2}}$. Sharp probabilistic versions of these estimates are also studied. As an application, we establish a sharp version of the classical logarithmic inequality of Zygmund.

The zero-divisor graph $\Gamma (R)$ of a commutative ring $R$ is the graph whose vertices consist of the nonzero zero-divisors of $R$ such that distinct vertices $x$ and $y$ are adjacent if and only if $xy\,=\,0$. In this paper, a characterization is provided for zero-divisor graphs of Boolean rings. Also, commutative rings $R$ such that $\Gamma (R)$ is isomorphic to the zero-divisor graph of a direct product of integral domains are classified, as well as those whose zero-divisor graphs are central vertex complete.

Let $\left( X,\,\mathcal{B},\,m,\,\tau \right)$ be a dynamical system with $\left( X,\mathcal{B},m \right)$ a probability space and $\tau $ an invertible, measure preserving transformation. This paper deals with the almost everywhere convergence in ${{\text{L}}^{1}}\left( X \right)$ of a sequence of operators of weighted averages. Almost everywhere convergence follows once we obtain an appropriate maximal estimate and once we provide a dense class where convergence holds almost everywhere. The weights are given by convolution products of members of a sequence of probability measures $\left\{ {{v}_{i}} \right\}$ defined on $\mathbb{Z}$. We then exhibit cases of such averages where convergence fails.

Various weighted ${{l}_{p}}$-norm inequalities in convolutions are derived by a simple and general principle whose ${{l}_{2}}$ version was obtained by using the theory of reproducing kernels. Applications to the Riemann zeta function and a difference equation are also considered.

In this paper we prove Chen inequalities for submanifolds of real space forms endowed with a semi-symmetric non-metric connection, i.e., relations between the mean curvature associated with a semi-symmetric non-metric connection, scalar and sectional curvatures, Ricci curvatures and the sectional curvature of the ambient space. The equality cases are considered.

In this paper, we study locally projectively flat fourth root Finsler metrics and their generalized metrics. We prove that if they are irreducible, then they must be locally Minkowskian.