In this paper, we establish a number of theorems on the classic Diophantine equation of S. S. Pillai, ${{a}^{x}}-{{b}^{y}}=c$ , where $a,\,b$ and $c$ are given nonzero integers with $a,\,b\,\ge \,2$. In particular, we obtain the sharp result that there are at most two solutions in positive integers $x$ and $y$ and deduce a variety of explicit conditions under which there exists at most a single such solution. These improve or generalize prior work of Le, Leveque, Pillai, Scott and Terai. The main tools used include lower bounds for linear forms in the logarithms of (two) algebraic numbers and various elementary arguments.

Let $\mathbf{H}$ be the Hilbert function of some set of distinct points in ${{\mathbb{P}}^{n}}$ and let $\alpha \,=\,\alpha (\mathbf{H})$ be the least degree of a hypersurface of ${{\mathbb{P}}^{n}}$ containing these points. Write $\alpha ={{d}_{s}}+{{d}_{s-1}}+\cdot \cdot \cdot +{{d}_{1}}$ (where ${{d}_{i}}>0$ ). We canonically decompose $\mathbf{H}$ into $s$ other Hilbert functions $\text{H}\leftrightarrow \text{(}{{\text{H}'}_{s}}\text{,}...\text{,}{{\text{H}'}_{1}}\text{)}$ and show how to find sets of distinct points ${{\mathbb{Y}}_{s}},...,{{\mathbb{Y}}_{1}}$ , lying on reduced hypersurfaces of degrees ${{d}_{s}},...,{{d}_{1}}$ (respectively) such that the Hilbert function of ${{\mathbb{Y}}_{i}}$ is ${{\text{H'}}_{i}}$ and the Hilbert function of $\mathbb{Y}=\bigcup _{i=1}^{s}\,{{\mathbb{Y}}_{i}}$ is $\mathbf{H}$. Some extremal properties of this canonical decomposition are also explored.

Let $G$ be a solvable exponential Lie group. We characterize all the continuous topologically irreducible bounded representations $(T,\mathcal{U})$ of $G$ on a Banach space $\mathcal{U}$ by giving a $G$-orbit in ${{n}^{*}}$ ($\mathfrak{n}$ being the nilradical of $\mathfrak{g}$), a topologically irreducible representation of ${{L}^{1}}({{\mathbb{R}}^{n}},\,\,\omega )$ , for a certain weight $\omega $ and a certain $n\,\in \,\mathbb{N}$, and a topologically simple extension norm. If $G$ is not symmetric, i.e., if the weight $\omega $ is exponential, we get a new type of representations which are fundamentally different from the induced representations.

We prove a number of results about the stable and particularly the real ranks of tensor products of ${{C}^{*}}$ -algebras under the assumption that one of the factors is commutative. In particular, we prove the following:

1. (1) If $X$ is any locally compact $\sigma $-compact Hausdorff space and $A$ is any ${{C}^{*}}$-algebra, then $\text{RR(}{{C}_{0}}\text{(}X\text{)}\otimes A\text{)}\le \text{dim(}X\text{)+RR(}A\text{)}$ .

2. (2) If $X$ is any locally compact Hausdorff space and $A$ is any purely infinite simple ${{C}^{*}}$ -algebra, then $\text{RR(}{{C}_{0}}\text{(}X\text{)}\otimes A\text{)}\le 1$ .

3. (3) $\text{RR(}C([0,\,1]\,)\otimes \,A)\,\ge \,1$ for any nonzero ${{C}^{*}}$-algebra $A$, and $\text{sr(}C({{[0,\,1]}^{2}})\,\otimes \,A\text{)}\,\ge \,2$ for any unital ${{C}^{*}}$-algebra $A$.

4. (4) If $A$ is a unital ${{C}^{*}}$-algebra such that $\text{RR(}A\text{)}\,\text{=}\,\text{0,}\,\text{s}r\text{(}A\text{)}\,\text{=}\,\text{1}$, and ${{K}_{1}}(A)=0$ , then $\text{sr(}C([0,\,1])\,\otimes \,A\text{)}\,\text{=}\,1$.

5. (5) There is a simple separable unital nuclear ${{C}^{*}}$-algebra $A$ such that $\text{RR(}A\text{)}\,\text{=}\,\text{1}$ and $\text{sr(}C([0,\,1])\,\otimes \,A\text{)}\,\text{=}\,1$.

We prove that the operators $\int{_{\mathbb{R}_{+}^{2}}{{e}^{i{{x}^{a}}\cdot {{y}^{b}}}}\varphi (x,y)f(y)dy}$ map ${{L}^{p}}({{\mathbb{R}}^{2}})$ into itself for $p\,\in \,J\,=\,\,\left[ \frac{{{a}_{1}}+{{b}_{1}}}{{{a}_{1}}+(\frac{{{b}_{1}}r}{2})},\frac{{{a}_{1}}+{{b}_{1}}}{{{a}_{1}}+(1-\frac{r}{2})} \right]$ if ${{a}_{l}},{{b}_{l}}\ge 1$ and $\varphi (x,y)=|x-y{{|}^{-r}},0\le r<2$ , the result is sharp. Generalizations to dimensions $d\,>\,2$ are indicated.

We prove comparison theorems for the probability of life in a Lipschitz domain between Brownian motion and random walks.