Let $J$ be an abelian variety and $A$ be an abelian subvariety of $J$ , both defined over $Q$ . Let $x$ be an element of ${{H}^{1}}\left( Q,\,A \right)$ . Then there are at least two definitions of $x$ being visible in $J$ : one asks that the torsor corresponding to $x$ be isomorphic over $Q$ to a subvariety of $J$ , and the other asks that $x$ be in the kernel of the natural map ${{H}^{1}}\left( Q,\,A \right)\,\to \,{{H}^{1}}\left( \text{Q},\,J \right)$ . In this article, we clarify the relation between the two definitions.

In this article we study the Cesàro operator

$$C\left( f \right)\left( Z \right)=\frac{1}{Z}\int_{0}^{Z}{f\left( \zeta\right)d}\zeta,$$

and its companion operator $\mathcal{T}$ on Hardy spaces of the upper half plane. We identify $\mathcal{C}$ and $\mathcal{T}$ as resolvents for appropriate semigroups of composition operators and we find the norm and the spectrum in each case. The relation of $\mathcal{C}$ and $\mathcal{T}$ with the corresponding Cesàro operators on Lebesgue spaces ${{L}^{p}}\left( \mathbb{R} \right)$ of the boundary line is also discussed.

We prove variants of Schwarz's lemma involving monotonicity properties of condenser capacity and inner radius. Also, we examine when a similar monotonicity property holds for the hyperbolic metric.

Let $\text{ }\!\!\lambda\!\!\text{ }\left( n \right)$ denote the Liouville function. Complementary to the prime number theorem, Chowlaconjectured that

*

$$\sum\limits_{n\le x}{\lambda \,\left( f\left( n \right) \right)}=o\left( x \right)$$

for any polynomial $f\left( x \right)$ with integer coefficients which is not of form $bg{{\left( x \right)}^{2}}$ .

Dufresnoy and Pisot characterized the smallest Pisot number of degree $n\,\ge \,3$ by giving explicitly its minimal polynomial. In this paper, we translate Dufresnoy and Pisot’s result to the Laurent series case. The aim of this paper is to prove that the minimal polynomial of the smallest Pisot element $\left( \text{SPE} \right)$ of degree $n$ in the field of formal power series over a finite field is given by $P\left( Y \right)\,=\,{{Y}^{n}}\,-\,\alpha X{{Y}^{n-1}}\,-{{\alpha }^{n}}$ where $\alpha $ is the least element of the finite field ${{\mathbb{F}}_{q}}\backslash \left\{ 0 \right\}$ (as a finite total ordered set). We prove that the sequence of SPEs of degree $n$ is decreasing and converges to $\alpha X$ . Finally, we show how to obtain explicit continued fraction expansion of the smallest Pisot element over a finite field.

Total embedding distributions have been known for a few classes of graphs. Chen, Gross, and Rieper computed it for necklaces, close-end ladders and cobblestone paths. Kwak and Shim computed it for bouquets of circles and dipoles. In this paper, a splitting theorem is generalized and the embedding distributions of generalized fan graphs are obtained

In this note, we first give a characterization of super weakly compact convex sets of a Banach space $X$ : a closed bounded convex set $K\,\subset \,X$ is super weakly compact if and only if there exists a ${{w}^{*}}$ lower semicontinuous seminorm $P$ with $P\,\ge \,{{\sigma }_{K}}\,\equiv \,{{\sup }_{x\in K}}\left\langle \,\cdot \,,\,x \right\rangle $ such that ${{P}^{2}}$ is uniformly Fréchet differentiable on each bounded set of ${{X}^{*}}$ . Then we present a representation theoremfor the dual of the semigroup swcc $\left( X \right)$ consisting of all the nonempty super weakly compact convex sets of the space $X$ .

We prove a result concerning power series $f\left( Z \right)\,\in \,\mathbb{C}\left[\!\left[ Z \right]\!\right]$ satisfying a functional equation ofthe form?

1

$$f\left( {{Z}^{d}} \right)\,=\,\sum\limits_{k=1}^{n}{\frac{{{A}_{k}}\left( Z \right)}{{{B}_{k}}\left( Z \right)}\,f{{\left( Z \right)}^{k}}},$$

where ${{A}_{k}}\left( Z \right),\,{{B}_{k}}\left( Z \right)\,\in \,\mathbb{C}\left[ Z \right]$ . In particular, we show that if $f\left( Z \right)$ satisfies a minimal functional equation of the above form with $n\,\ge \,2$ , then $f\left( Z \right)$ is necessarily transcendental. Towards a more complete classification, the case $n=\,1$ is also considered.

M. Hu and S. Wen considered quasisymmetrically minimal uniform Cantor sets of Hausdorff dimension 1, where at the $K$ -th set one removes from each interval $I$ a certain number ${{n}_{k}}$ of open subintervals of length ${{c}_{k}}\left| I \right|$ , leaving $\left( {{n}_{k}}\,+\,1 \right)$ closed subintervals of equal length. Quasisymmetrically Moran sets of Hausdorff dimension 1 considered in the paper are more general than uniform Cantor sets in that neither the open subintervals nor the closed subintervals are required to be of equal length.

We prove the non-existence of real hypersurfaces in complex projective space whose structure Jacobi operator is Lie $\mathbb{D}$ -parallel and satisfies a further condition.

In 1968, Galvin conjectured that an uncountable poset $P$ is the union of countably many chains if and only if this is true for every subposet $Q\,\subseteq \,P$ with size ${{\aleph }_{1}}$ . In 1981, Rado formulated a similar conjecture that an uncountable interval graph $G$ is countably chromatic if and only if this is true for every induced subgraph $H\,\subseteq \,G$ with size ${{\aleph }_{1}}$ . Todorčević has shown that Rado's conjecture is consistent relative to the existence of a supercompact cardinal, while the consistency of Galvin's conjecture remains open. In this paper, we survey and collect a variety of results related to these two conjectures. We also show that the extension of Rado's conjecture to the class of all chordal graphs is relatively consistent with the existence of a supercompact cardinal.

We establish estimates for restrictions to certain curves in ${{\mathbb{R}}^{2}}$ of the Fourier transforms of some fractal measures.

We show that the following ${{K}_{0}}$ -monoid properties of ${{C}^{*}}$ -algebras in the class $\Omega $ are inherited by simple unital ${{C}^{*}}$ -algebras in the class $TA\Omega :$ (1) weak comparability, (2) strictly unperforated, (3) strictly cancellative.

Let $g\,\mapsto \,{{g}^{*}}$ denote an involution on a group $G$ . For any (commutative, associative) ring $R$ (with 1), $*$ extends linearly to an involution of the group ring $RG$ . An element $\alpha \,\in \,RG$ is symmetric if ${{\alpha }^{*}}\,=\,\alpha $ and skew-symmetric if ${{\alpha }^{*}}\,=\,-\alpha $ . The skew-symmetric elements are closed under the Lie bracket, $[\alpha ,\,\beta ]\,=\,\alpha \beta \,-\,\beta \alpha $ . In this paper, we investigate when this set is also closed under the ring product in $RG$ . The symmetric elements are closed under the Jordan product, $\alpha \,o\,\beta \,=\,\alpha \beta \,+\beta \alpha $ . Here, we determine when this product is trivial. These two problems are analogues of problems about the skew-symmetric and symmetric elements in group rings that have received a lot of attention.

A linear Cantor set $C$ with zero Lebesgue measure is associated with the countable collection of the bounded complementary open intervals. A rearrangment of $C$ has the same lengths of its complementary intervals, but with different locations. We study the Hausdorff and packing $h$ -measures and dimensional properties of the set of all rearrangments of some given $C$ for general dimension functions $h$ . For each set of complementary lengths, we construct a Cantor set rearrangement which has the maximal Hausdorff and the minimal packing $h$ -premeasure, up to a constant. We also show that if the packing measure of this Cantor set is positive, then there is a rearrangement which has infinite packing measure.

We consider a nonlinear periodic problem driven by a nonlinear nonhomogeneous differential operator and a Carathéodory reaction term $f\left( t,\,x \right)$ that exhibits a $\left( p\,-\,1 \right)$ -superlinear growth in $x\,\in \,\mathbb{R}$ near $\pm \infty $ and near zero. A special case of the differential operator is the scalar $p$ -Laplacian. Using a combination of variational methods based on the critical point theory with Morse theory (critical groups), we show that the problem has three nontrivial solutions, two of which have constant sign (one positive, the other negative).

In this paper, we consider the Gross-Pitaevskii equation for the trapped dipolar quantum gases. We obtain the sharp criterion for the global existence and finite time blow-up in the unstable regime by constructing a variational problem and the so-called invariant manifold of the evolution flow.

In this paper we determine theHausdorff measure of noncompactness on the sequence space $n\left( \phi\right)$ of $\text{W}\text{.}\,\text{L}\text{.}\,\text{C}.$ Sargent. Further we apply the technique of measures of noncompactness to the theory of infinite systems of differential equations in the Banach sequence spaces $n\left( \phi\right)$ and $m\left( \phi\right)$ . Our aim is to present some existence results for infinite systems of differential equations formulated with the help of measures of noncompactness.

An epimorphism $\phi :\,G\,\to \,H$ of groups, where $G$ has rank $n$ , is called coessential if every (ordered) generating $n$ -tuple of $H$ can be lifted along $\phi $ to a generating $n$ -tuple for $G$ . We discuss this property in the context of the category of groups, and establish a criterion for such a group $G$ to have the property that its abelianization epimorphism $G\,\to \,{G}/{[G,G]}\;$ , where $[G,\,G]$ is the commutator subgroup, is coessential. We give an example of a family of 2-generator groups whose abelianization epimorphism is not coessential. This family also provides counterexamples to the generalized Andrews–Curtis conjecture.

Let $\left( X,\,B,\,\mu\right)$ be a $\sigma $ -finite measure space and let $H\,\subset \,{{L}^{2}}\left( X,\,\mu\right)$ be a separable reproducing kernel Hilbert space on $X$ . We show that the multiplier algebra of $H$ has property $\left( {{A}_{1}}\left( 1 \right) \right)$ .