Let $\alpha$ be an irrational number and $\varphi$: $\mathbb{N} \to \mathbb{R^+}$ be a decreasing sequence tending to zero. Consider the set \[E_{\varphi}(\alpha)=\{\beta \in \mathbb{R}: \ \|n \alpha- \beta\|<\varphi(n)\ {\rm {holds\ for\ infinitely\ many}} \ n \in \mathbb{N}\}\], where $\|{\cdot}\|$ denotes the distance to the nearest integer. We show that for general error function $\varphi$, the Hausdorff dimension of $E_{\varphi}(\alpha)$ depends not only on $\varphi$, but also heavily on $\alpha$. However, recall that the Hausdorff dimension of $E_{\varphi}(\alpha)$ is independent of $\alpha$ when $\varphi(n) = n^{-\gamma}$ with $\gamma >1$.

We construct a compactly generated, totally disconnected, locally compact group whose Hecke algebra with respect to any compact open subgroup does not have a $C^*$-enveloping algebra.

We consider expansions with respect to the multi-dimensional Hermite functions and to the multi-dimensional Hermite polynomials. They are respectively eigenfunctions of the Harmonic oscillator $\cal{L}= -\Delta +|x|^2$ and of the Ornstein-Uhlenbeck operator ${\bf L} = -\Delta +2x \cdot \nabla.$ The corresponding heat semigroups and Riesz transforms are considered and results on both aspects (polynomials and functions) are obtained.

We introduce the concept of left APP-rings which is a generalization of left p.q.-Baer rings and right PP-rings, and investigate its properties. It is shown that the APP property is inherited by polynomial extensions and is a Morita invariant property.

The $\ell^{1}$-convolution algebra of a semilattice is known to have trivial cohomology in degrees 1, 2 and 3 whenever the coefficient bimodule is symmetric. We extend this result to all cohomology groups of degree $\geq 1$ with symmetric coefficients. Our techniques prove a stronger splitting result, namely that the splitting can be made natural with respect to the underlying semilattice.

We prove some theorems on locally graded groups and nilpotent-by-Chernikov groups.

Let $G$ be a compact $p$-adic analytic group and let $\Lambda_G$ be its completed group algebra with coefficient ring the $p$-adic integers $\mathbb{Z}_p$. We show that the augmentation ideal in $\Lambda_G$ of a closed normal subgroup $H$ of $G$ is localisable if and only if $H$ is finite-by-nilpotent, answering a question of Sujatha. The localisations are shown to be Auslander-regular rings with Krull and global dimensions equal to dim $H$. It is also shown that the minimal prime ideals and the prime radical of the $\mathbb{F}_p$-version $\Omega_G$ of $\Lambda_G$ are controlled by $\Omega_{\Delta^+}$, where $\Delta^+$ is the largest finite normal subgroup of $G$. Finally, we prove a conjecture of Ardakov and Brown [1].

Let $K$ be a field of characteristic zero and $A_{2}:=A_{2}(K)$ the 2$nd$–Weyl algebra over $K$. We establish a close connection between the maximal left ideals of $A_{2}$ and the simple derivations of $K[X_{1},X_{2}]$.

MAIN THEOREM. Let$d = \partial_1 + \beta\partial_2$be a simple derivation of$K[X_1, X_2]$with$\beta\in K[X_1, X_2]$. Then, there exists$\gamma\in K[X_1, X_2]$such that$d + \gamma$generates a maximal left ideal of$A_2$. More precisely, the following is true:

1. $\deg_{X_2} (\beta) \geq 2$ or $\deg_{X_2} (\beta) = 1$ and $\deg_{X_1} (\partial_2(\beta)) \geq 1$;

2. $d + gX_2$generates a maximal left ideal of$A_2$if$g \in K[X_1]\\{0\}$is such that

1. (a) $g \in -(\partial^2_2(\beta)/2)\mathbb{N}$when$\deg_{X_2} (\beta) \geq 2$,

2. (b) $\deg_{X_1} (g) < \deg_{X_1} (\partial_2(\beta))$when$\deg_{X_2} (\beta) = 1$.

As applications, we obtain large families of concrete examples of cyclic maximal left ideals of $A_2$; such examples have been rather scarce so far.

We establish some upper and lower bounds for the number of rational points of Prym varieties defined over finite fields. They are better than the usual Weil bounds valid for any abelian varieties defined over such fields.

The Hansen-Mullen Primitivity Conjecture (HMPC) (1992) asserts that, with some (mostly obvious) exceptions, there exists a primitive polynomial of degree $n$ over any finite field with any coefficient arbitrarily prescribed. This has recently been proved whenever $n \geq 9$. It is also known to be true when $n \leq 3$. We show that there exists a primitive polynomial of any degree $n\geq 4$ over any finite field with its second coefficient (i.e., that of $x^{n-2}$) arbitrarily prescribed. In particular, this establishes the HMPC when $n=4$. The lone exception is the absence of a primitive polynomial of the form $x^4+a_1x^3 +x^2+a_3x+1$ over the binary field. For $n \geq 6$ we prove a stronger result, namely that the primitive polynomial may also have its constant term prescribed. This implies further cases of the HMPC. When the field has even cardinality 2-adic analysis is required for the proofs.

The existence of multiple positive solutions is presented for the singular Dirichlet boundary value problems \[\left\{\begin{array}{@{}ll} x^{\prime\prime}+\Phi(t)\,f(t,x(t),|x'(t)|)=0,\\[3pt] x(0)=0,\ \ x(1)=0, \end{array}\right.\] using the fixed point index; here $f$ may be singular at $x=0$ and $x'=0$.

We define a subset $\mathcal Z$ of $(1,+\infty)$ with the property that for each $\alpha \in {\mathcal Z}$ there is a nonzero real number $\xi = \xi(\alpha)$ such that the integral parts $[\xi \alpha^n]$ are even for all $n \in \mathbb{N}$. A result of Tijdeman implies that each number greater than or equal to 3 belongs to $\mathcal{Z}$. However, Mahler's question on whether the number 3/2 belongs to $\mathcal{Z}$ or not remains open. We prove that the set ${\mathcal S}:=(1,+\infty) \textbackslash {\mathcal Z}$ is nonempty and find explicitly some numbers in ${\mathcal Z} \cap$ (5/4,3) and in ${\mathcal S} \cap (1,2)$.

Answering a question of Hoffmann and of Kambites, an example is exhibited of a finitely generated semigroup $S$ such that $S$ embeds in a group and $S$ is not automatic, but the universal group of $S$ is automatic.

A proper submodule $N$ of an $R$-module $M$ is called a weakly prime submodule, if for each submodule $K$ of $M$ and elements $a, b$ of $R$, $abK \subseteq N$, implies that $aK\subseteq N$ or $bK\subseteq N$. In this paper we will study weakly prime submodules and we shall compare weakly prime submodules with prime submodules.

Let $\chi(n)$ be a quadratic character modulo a prime $p$. For a fixed integer $s\ne 0$, we estimate certain exponential sums with truncated $L$-functions \[L_{s,p}(n) = \sum_{j=1}^n \frac{\chi(\,j)}{j^s}\qquad (n =1, 2, \ldots)\]. Our estimate implies certain uniformly of distribution properties of reductions of $L_{s,p}(n)$ in the residue classes modulo $p$.