In this paper we study Hankel transforms and Hankel convolution operators on spaces of entire functions of finite order and their duals.

The main results deal with conditions for the validity of the weighted convolution inequality ${{\sum }_{n\in \mathbb{Z}}}|{{b}_{n}}\,{{\sum }_{k\in \mathbb{Z}}}{{a}_{n-{{k}^{x}}k}}{{|}^{p}}\,\le \,{{C}^{p\,}}\,{{\sum }_{k\in \mathbb{Z}}}\,{{\left| {{x}_{k}} \right|}^{p}}$ when $p\,\ge \,1$ .

We study Calabi–Yau manifolds constructed as double coverings of ${{\mathbb{P}}^{3}}$ branched along an octic surface. We give a list of 87 examples corresponding to arrangements of eight planes defined over $\mathbb{Q}$ . The Hodge numbers are computed for all examples. There are 10 rigid Calabi–Yau manifolds and 14 families with ${{h}^{1,2}}\,=\,1.$ The modularity conjecture is verified for all the rigid examples.

A continuumis said to be Suslinian if it does not contain uncountably many mutually exclusive nondegenerate subcontinua. We prove that Suslinian continua are perfectly normal and rim-metrizable. Locally connected Suslinian continua have weight atmost ${{\omega }_{1}}$ and under appropriate set-theoretic conditions are metrizable. Non-separable locally connected Suslinian continua are rim-finite on some open set.

We find an infinite family of projective monomial curves all of which have $h$ -vector with no negative values and are not Cohen–Macaulay.

The integers coprime to $n$ are called the totatives of $n$ . D. H. Lehmer and Paul Erdős were interested in understanding when the number of totatives between $in/k$ and $\left( i\,+1 \right)n/k$ are $1/k\text{th}$ of the total number of totatives up to $n$ . They provided criteria in various cases. Here we give an “if and only if” criterion which allows us to recover most of the previous results in this literature and to go beyond, as well to reformulate the problem in terms of combinatorial group theory. Our criterion is that the above holds if and only if for every odd character $\chi \,\left( \bmod \,\kappa\right)\,\left( \text{where}\,\kappa :=k/\gcd \left( k,\,n/{{\Pi }_{p|n}}p \right) \right)$ there exists a prime $p={{p}_{\chi }}$ dividing $n$ for which $\chi \left( p \right)=1.$

We state and prove an important special case of Suslin reciprocity that has found significant use in the study of algebraic cycles. An introductory account is provided of the regulator and norm maps on Milnor ${{K}_{2}}$ -groups (for function fields) employed in the proof.

Let $X$ be a projective smooth variety over a field $k$ . In the first part we show that an indecomposable element in $C{{H}^{2}}\left( X,\,1 \right)$ can be lifted to an indecomposable element in $C{{H}^{3}}\left( {{X}_{K}},\,2 \right)$ where $K$ is the function field of 1 variable over $k$ . We also show that if $X$ is the self-product of an elliptic curve over $\mathbb{Q}$ then the $\mathbb{Q}$ -vector space of indecomposable cycles $CH_{ind}^{3}{{\left( {{X}_{\mathbb{C}}},\,2 \right)}_{\mathbb{Q}}}$ is infinite dimensional.

In the second part we give a new definition of the group of indecomposable cycles of $C{{H}^{3}}\left( X,\,2 \right)$ and give an example of non-torsion cycle in this group.

We give a particularly elementary solution to the following well-known problem. What is the number of $k$ -subsets $X\subseteq {{I}_{n}}=\left\{ 1,2,3,\ldots ,n \right\}$ satisfying “no two elements of $X$ are adjacent in the circular display of ${{I}_{n}}$ ”? Then we investigate a new generalization (multiple cyclic choices without adjacencies) and apply it to enumerating a class of 3-line latin rectangles.

The index theory considered in this paper, a generalisation of the classical Fredholm index theory, is obtained in terms of a non-finite trace on a unital ${{C}^{*}}$ -algebra. We relate it to the index theory of M. Breuer, which is developed in a von Neumann algebra setting, by means of a representation theorem. We show how our new index theory can be used to obtain an index theorem for Toeplitz operators on the compact group $\text{U}\left( 2 \right)$ , where the classical index theory does not give any interesting result.

We establish a sharp Fourier restriction estimate for a measure on a $k$ -surface in ${{\mathbb{R}}^{n}}$ , where $n\,=\,k\left( k\,+\,3 \right)/2$

It is proved that every adjacency preserving continuous map on the vector space of real matrices of fixed size, is either a bijective affine tranformation of the form $A\,\mapsto \,PAQ\,+\,R$ , possibly followed by the transposition if the matrices are of square size, or its range is contained in a linear subspace consisting of matrices of rank at most one translated by some matrix $R$ . The result extends previously known theorems where the map was assumed to be also injective.

Let $R$ be a commutative Noetherian integral domain with field of fractions $Q$ . Generalizing a forty-year-old theorem of E. Matlis, we prove that the $R$ -module $Q/R$ (or $Q$ ) has Krull dimension if and only if $R$ is semilocal and one-dimensional. Moreover, if $X$ is an injective module over a commutative Noetherian ring such that $X$ has Krull dimension, then the Krull dimension of $X$ is at most 1.

This paper studies the integration of inclusion of subdifferentials. Under various verifiable conditions, we obtain that if two proper lower semicontinuous functions $f$ and $g$ have the subdifferential of $f$ included in the $\gamma $ -enlargement of the subdifferential of $g$ , then the difference of those functions is $\gamma $ -Lipschitz over their effective domain.

We define a uniform structure on the set of discrete sets of a locally compact topological space on which a locally compact topological group acts continuously. Then we investigate the completeness of these uniform spaces and study these spaces by means of topological dynamical systems.

It is proved here that a ring $R$ is right pseudo-Frobenius if and only if $R$ is a right Kasch ring such that the second right singular ideal is injective.