We prove a topological restriction on the support of Sobolev functions.

It is shown that simple rings over which proper cyclic right modules are continuous coincide with simple right PCI-rings, introduced by Faith.

Let $(X,L)$ be a polarized manifold over the complex number field with dim$X=n$. In this paper, we consider a conjecture of M. C. Beltrametti and A. J. Sommese and we obtain that this conjecture is true if $n=3$ and ${{h}^{0}}\,(L)\,\ge \,2$, or $\dim\,\text{Bs}|L|\le 0$ for any $n\ge 3$. Moreover we can generalize the result of Sommese.

As a consequence of results due to Bourgain and Stegall, on a separable Banach space whose unit ball is not dentable, the set of norm attaining functionals has empty interior (in the norm topology). First we show that any Banach space can be renormed to fail this property. Then, our main positive result can be stated as follows: if a separable Banach space $X$ is very smooth or its bidual satisfies the ${{\mathcal{w}}^{*}}$-Mazur intersection property, then either $X$ is reflexive or the set of norm attaining functionals has empty interior, hence the same result holds if $X$ has the Mazur intersection property and so, if the norm of $X$ is Fréchet differentiable. However, we prove that smoothness is not a sufficient condition for the same conclusion.

We prove that every finite distributive lattice can be represented as the congruence lattice of a finite (planar) semimodular lattice.

It is known that a semigroup of quasinilpotent integral operators, with positive lower semicontinuous kernels, on ${{L}^{2}}\,(X,\,\mu )$, where $X$ is a locally compact Hausdorff-Lindelöf space and $\mu $ is a $\sigma $-finite regular Borel measure on $X$, is triangularizable. In this article we use the Banach lattice version of triangularizability to establish the ideal-triangularizability of a semigroup of positive quasinilpotent integral operators on $C(\mathbf{K})$ where $\mathbf{K}$ is a compact Hausdorff space.

In this paper we consider solutions to the free Schrödinger equation in $n+1$ dimensions. When we restrict the last variable to be a smooth function of the first $n$ variables we find that the solution, so restricted, is locally in ${{L}^{2}}$, when the initial data is in an appropriate Sobolev space.

We provide more characterizations of varieties with a weak difference term and of neutral varieties. We prove that a variety has a (weak) difference term (is neutral) with respect to the $\text{TC}$-commutator iff it has a (weak) difference term (is neutral) with respect to the linear commutator. We show that a variety $V$ is congruence meet semi-distributive iff $V$ is neutral, iff ${{M}_{3}}$ is not a sublattice of Con $\mathbf{A}$, for $\mathbf{A}\in V$, iff there is a positive integer $n$ such that $V{{\vDash }_{Con}}\alpha (\beta \,o\,\gamma )\le \alpha {{\beta }_{n}}$.

Over a decade ago, this author produced class number one criteria for real quadratic fields in terms of prime-producing quadratic polynomials. The purpose of this article is to revisit the problem from a new perspective with new criteria. We look at the more general situation involving arbitrary real quadratic orders rather than the more restrictive field case, and use the interplay between the various orders to provide not only more general results, but also simpler proofs.

The function $\Delta (x,N)$ as defined in the title is closely associated via $\Delta \,(N)\,=\,{{\sup }_{x}}\,|\,\Delta (x,N)|$ to several problems in the upper bound sieve. It is also known via a classical theorem of Franel that certain conjectured bounds involving averages of $\Delta (x,N)$ are equivalent to the Riemann Hypothesis. We improve the unconditional bounds which have been hitherto obtained for $\Delta (N)$ and show that these are close to being optimal. Several auxiliary results relating $\Delta (Np)$ to $\Delta (N)$, where $p$ is a prime with $p\nmid N$, are also obtained and two new conjectures stated.

We study Hausdorff continua in which every set of certain cardinality contains a subset which disconnects the space. We show that such continua are rim-finite. We give characterizations of this class among metric continua. As an application of our methods, we show that continua in which each countably infinite set disconnects are generalized graphs. This extends a result of Nadler for metric continua.

Let $H$ be a faithfully projective Hopf algebra over a commutative ring $k$. In [8, 9] we defined the Brauer group $\text{BQ}(k,H)$ of $H$ and an homomorphism $\pi $ from Hopf automorphism group $\text{Au}{{\text{t}}_{\text{Hopf}}}(H)$ to $\text{BQ}(k,H)$. In this paper, we show that the morphism $\pi $ can be embedded into an exact sequence.

In this article, making use of the second author’s criterion for exponentiality of a connected solvable Lie group, we give a rather simple necessary and sufficient condition for the semidirect product of a torus acting on certain connected solvable Lie groups to be exponential.

We show that normal and stable normal invariants of polarized homotopy equivalences of lens spaces $M=\,L({{2}^{m}};\,{{r}_{1}},...,\,{{r}_{n}})$ and $N\,=\,L({{2}^{m}};\,{{s}_{1}},...,{{s}_{n}})$ are determined by certain $\ell $-polynomials evaluated on the elementary symmetric functions ${{\sigma }_{i}}\,(r_{1}^{2},...,r_{n}^{2})$ and ${{\sigma }_{i}}(s_{1}^{2},...,s_{n}^{2})$. Each polynomial ${{\ell }_{k}}$ appears as the homogeneous part of degree $k$ in the Hirzebruch multiplicative $L$-sequence. When $n=8$, the elementary symmetric functions alone determine the relevant normal invariants.