Some aspects of dimension theory for topological groups

We discuss dimension theory in the class of all topological groups. For locally compact topological groups there are many classical results in the literature. Dimension theory for non-locally compact topological groups is mysterious. It is for example unknown whether every connected (hence at least 1-dimensional) Polish group contains a homeomorphic copy of $[0,1]$. And it is unknown whether there is a homogeneous metrizable compact space the homeomorphism group of which is 2-dimensional. Other classical open problems are the following ones. Let $G$ be a topological group with a countable network. Does it follow that $\dim G=\mathrm{ind}G=\mathrm{Ind}G$? The same question if $X$ is a compact coset space. We also do not know whether the inequality $\dim (G\times H)\le \dim G+\dim H$ holds for arbitrary topological groups $G$ and $H$ which are subgroups of $\sigma$-compact topological groups. The aim of this paper is to discuss such and related problems. But we do not attempt to survey the literature.     

Improving the Clark–Suen bound on the domination number of the Cartesian product of graphs

A long-standing Vizing’s conjecture asserts that the domination number of the Cartesian product of two graphs is at least as large as the product of their domination numbers; one of the most significant results related to the conjecture is the bound of Clark and Suen, $\gamma (G\square H)\ge \gamma \left(G\right)\gamma \left(H\right)∕2$, where $\gamma$ stands for the domination number, and $G\square H$ is the Cartesian product of graphs $G$ and $H$. In this note, we improve this bound by employing the 2-packing number $\rho \left(G\right)$ of a graph $G$ into the formula, asserting that $\gamma (G\square H)\ge (2\gamma \left(G\right)?\rho \left(G\right))\gamma \left(H\right)∕3$. The resulting bound is better than that of Clark and Suen whenever $G$ is a graph with $\rho \left(G\right)<\gamma \left(G\right)∕2$, and in the case $G$ has diameter 2 reads as $\gamma (G\square H)\ge (2\gamma \left(G\right)?1)\gamma \left(H\right)∕3$.     

On two Diophantine inequalities over primes

Let $1<c<37∕18,c\ne 2$ and $N$ be a sufficiently large real number. In this paper, we prove that, for almost all $R\in (N,2N]$, the Diophantine inequality $|p_1^c+p_2^c+p_3^c?R|<{log}^{?1}N$ is solvable in primes $p_1,p_2,p_3$. Moreover, we also investigate the problem of six primes and prove that the Diophantine inequality $|p_1^c+p_2^c+p_3^c+p_4^c+p_5^c+p_6^c?N|<{log}^{?1}N$ is solvable in primes $p_1,p_2,p_3,p_4,p_5,p_6$ for sufficiently large real number $N$.     

Existence of incomplete canonical Kirkman packing designs

For $u,v$ positive integers with $u\equiv v\equiv 4\left(mod6\right)$, let ICKPD$(u,v)$ denote a canonical Kirkman packing of order $u$ missing one of order $v$. In this paper, it is shown that the necessary condition for existence of an ICKPD$(u,v)$, namely $u\ge 3v+4$, is sufficient with a definite exception $(u,v)=(16,4)$, and except possibly when $v>76$, $v\equiv 4\left(mod12\right)$ and $u\in \{3v+4,3v+10\}$.     

Some results on Vizing’s conjecture and related problems

The well-known conjecture of Vizing on the domination number of Cartesian product graphs claims that for any two graphs $G$ and $H$, $\gamma \left(G\square H\right)\ge \gamma \left(G\right)\gamma \left(H\right)$. We disprove its variations on independent domination number and Barcalkin–German number, i.e. Conjectures 9.6 and 9.2 from the recent survey Bre?ar et al. (2012) [4]. We also give some extensions of the double-projection argument of Clark and Suen (2000) [8], showing that their result can be improved in the case of bounded-degree graphs. Similarly, for rainbow domination number we show for every $k\ge 1$ that ${\gamma }_{rk}\left(G\square H\right)\ge \frac{k}{k+1}\gamma \left(G\right)\gamma \left(H\right)$, which is closely related to Question 9.9 from the same survey. We also prove that the minimum possible counterexample to Vizing’s conjecture cannot have two neighboring vertices of degree two.     

Parametric CR-umbilical locus of ellipsoids in C2

For every real numbers $a?1$, $b?1$ with $(a,b)\ne (1,1)$, the curve parametrized by $\theta \in \mathbb{R}$ valued in ${\mathbb{C}}^2?{\mathbb{R}}^4$

$\gamma :\theta ?(x(\theta )+\sqrt{?1}y(\theta ),u(\theta )+\sqrt{?1}v(\theta ))$

with components:

$\begin{array}{c}x(\theta ):=\sqrt{\frac{a?1}{a(ab?1)}}\cos ?\theta ,y(\theta ):=\sqrt{\frac{b(a?1)}{ab?1}}\sin ?\theta ,\hfill \\ u(\theta ):=\sqrt{\frac{b?1}{b(ab?1)}}\sin ?\theta ,v(\theta ):=?\sqrt{\frac{a(b?1)}{ab?1}}\cos ?\theta ,\hfill \end{array}$

has image contained in the CR-umbilical locus:

$\gamma (\mathbb{R})?\mathsf{UmbCR}\left({\mathsf{E}}_{a,b}\right)?{\mathsf{E}}_{a,b}$

of the ellipsoid ${\mathsf{E}}_{a,b}?{\mathbb{C}}^2$ of equation $a{x}^2+y^2+b{u}^2+v^2=1$, where the CR-umbilical locus of a Levi nondegenerate hypersurface $M^3?{\mathbb{C}}^2$ is the set of points at which the Cartan curvature of M vanishes.     

Fast local searches and updates in bounded universes

Given a bounded universe $\{0,1,\dots ,U?1\}$, we show how to perform predecessor searches in $O(\log \log \mathrm{\Delta })$ expected time, where Δ is the difference between the element being searched for and its predecessor in the structure, while supporting updates in $O(\log \log \mathrm{\Delta })$ expected amortized time, as well. This unifies the results of traditional bounded universe structures (which support predecessor searches in $O(\log \log U)$ time) and hashing (which supports membership queries in $O(1)$ time). We also show how these results can be applied to approximate nearest neighbour queries and range searching.     

On the Brezis–Nirenberg problem on S3, and a conjecture of Bandle–Benguria

We consider the following Brezis–Nirenberg problem on ${\mathbf{S}}^3$

$?{\mathrm{\Delta }}_{{\mathbf{S}}^3}u=\lambda u+u^5\text{in}D,u>0\text{in}D\text{and}u=0\text{on }?D,$

where D is a geodesic ball on ${\mathbf{S}}^3$ with geodesic radius ${\theta }_1$, and ${\mathrm{\Delta }}_{{\mathbf{S}}^3}$ is the Laplace–Beltrami operator on ${\mathbf{S}}^3$. We prove that for any $\lambda <?\frac{3}{4}$ and for every ${\theta }_1<\pi$ with $\pi ?{\theta }_1$ sufficiently small (depending on λ), there exists bubbling solution to the above problem. This solves a conjecture raised by Bandle and Benguria [J. Differential Equations 178 (2002) 264–279] and Brezis and Peletier [C. R. Acad. Sci. Paris, Ser. I 339 (2004) 291–394]. To cite this article: W. Chen, J. Wei, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Ternary Kloosterman sums modulo 4

Garaschuk and Lisoněk (2008) in [3] characterised ternary Kloosterman sums modulo 4, leaving the cases $K(a)\equiv 1(\mathrm{mod}4)$ and $K(a)\equiv 3(\mathrm{mod}4)$ as open problems. In this paper we complete the characterisation using well-known theorems on Gauss sums and Kloosterman sums. We also give the number of elements satisfying these congruences.