In this paper, we study rational approximations for certain algebraic power series over a finite field.
We obtain results for irrational elements of strictly positive degree
satisfying an equation of the type
\begin{equation}
\alpha=\displaystyle\frac{A\alpha^{q}+B}{C\alpha^{q}}
\end{equation}
where $(A, B, C)\in
(\mathbb{F}_{q}[X])^{2}\times\mathbb{F}_{q}^{\star}[X]$.
In particular,
we will give, under some conditions on the polynomials $A$, $B$
and $C$, well approximated elements satisfying this equation.

We introduce and investigate the notion of envelope dimension of
commutative rings and modules over them. In particular, we show that
the envelope dimension of a ring, $R$, is equal to that of the
$R$-module $R^{(\mathbb{N})}$. Also we prove that the Krull dimension of a
ring is no more than its envelope dimension and characterize
Noetherian rings for which these two dimensions are equal. Moreover we
generalize and study the concept of simplified radical formula for
modules, which
we defined in an earlier paper.

For any finite Galois extension $K$ of $\mathbb Q$
and any conjugacy class $C$ in $\operatorname {Gal}(K/\mathbb Q)$,
we show that there exist infinitely many Carmichael numbers
composed solely of primes for which the associated class of Frobenius
automorphisms is $C$. This result implies that for every natural
number $n$ there are infinitely many Carmichael numbers of the form
$a^2+nb^2$ with $a,b\in\mathbb Z $.

It is a well-known fact, that the greatest ambit for
a topological group $G$ is the Samuel compactification of $G$ with
respect to the right uniformity on $G.$ We apply the original
description by Samuel from 1948 to give a simple computation of the
universal minimal flow for groups of automorphisms of uncountable
structures using Fraïssé theory and Ramsey theory. This work
generalizes some of the known results about countable structures.

Let $b\gt 1$ be an integer. We prove that for almost all $n$, the sum of the
digits in base $b$ of the numerator of the Bernoulli number $B_{2n}$
exceeds $c\log n$, where $c:=c(b)\gt 0$ is some constant depending on
$b$.

In this work we introduce a class of discrete groups containing
subgroups of abstract translations and dilations, respectively. A
variety of wavelet systems can appear as $\pi(\Gamma)\psi$, where $\pi$ is
a unitary representation of a wavelet group and $\Gamma$ is the abstract
pseudo-lattice $\Gamma$. We prove a condition in order that a Parseval
frame $\pi(\Gamma)\psi$ can be dilated to an orthonormal basis of the
form $\tau(\Gamma)\Psi$ where $\tau$ is a super-representation of
$\pi$. For a subclass of groups that includes the case where the
translation subgroup is Heisenberg, we show that this condition
always holds, and we cite familiar examples as applications.

Let $X$ be a compact metric space. A lower bound for the radius of
comparison of the C*-algebra $\operatorname{C}(X)$ is given in terms of
$\operatorname{dim}_{\mathbb{Q}} X$, where $\operatorname{dim}_{\mathbb{Q}} X $ is
the cohomological dimension with rational coefficients. If
$\operatorname{dim}_{\mathbb{Q}} X =\operatorname{dim} X=d$, then the
radius of comparison of the C*-algebra $\operatorname{C}(X)$ is $\max\{0, (d-1)/2-1\}$ if $d$ is odd, and must be either $d/2-1$ or $d/2-2$ if $d$ is even (the possibility of $d/2-1$ does occur, but we do not know if the possibility of $d/2-2$ also can occur).

In this paper, the dimension function of a self-affine generalized scaling set associated with an $n\times n$ integral expansive dilation $A$ is studied. More specifically, we consider the dimension function of an $A$-dilation generalized scaling set $K$ assuming that $K$ is a self-affine tile satisfying $BK = (K+d_1) \cup (K+d_2)$, where $B=A^t$, $A$ is an $n\times n$ integral expansive matrix with $\lvert \det A\rvert=2$, and $d_1,d_2\in\mathbb{R}^n$. We show that the dimension function of $K$ must be constant if either $n=1$ or $2$ or one of the digits is $0$, and that it is bounded by $2\lvert K\rvert$ for any $n$.

The Mahler measure of a nonzero $n$-variable polynomial $P$ is the integral of
$\log|P|$ on the unit $n$-torus. A result of Boyd and Lawton says that
the Mahler measure of a multivariate polynomial is the limit of Mahler
measures of univariate polynomials. We prove the analogous
result for different extensions of Mahler measure such as generalized
Mahler measure (integrating the maximum of $\log|P|$ for possibly
different $P$'s),
multiple Mahler measure (involving products of $\log|P|$ for possibly
different $P$'s), and higher Mahler measure (involving $\log^k|P|$).

In this paper we study uniqueness of entire functions
sharing a non-zero finite value with linear differential polynomials
and address a result of W. Wang and P. Li.

Let $a_1, \cdots, a_9$ be non-zero integers and $n$ any integer. Suppose
that $a_1+\cdots+a_9 \equiv n( \textrm{mod}\,2)$ and $(a_i, a_j)=1$ for $1 \leq i \lt j \leq 9$.
In this paper we prove that (i) if $a_j$ are not all of the same sign, then the above cubic
equation has prime solutions satisfying
$p_j \ll |n|^{1/3}+\textrm{max}\{|a_j|\}^{14+\varepsilon};$
and (ii) if all $a_j$ are positive and $n \gg \textrm{max}\{|a_j|\}^{43+\varepsilon}$, then the cubic
equation $a_1p_1^3+\cdots +a_9p_9^3=n$ is soluble in primes $p_j$.
This result is the extension of the linear and quadratic relative problems.

This paper gives a new upper bound for the essential dimension and the
essential 2-dimension of the split simply connected group of type
$E_7$ over a field of characteristic not 2 or 3. In particular,
$\operatorname{ed}(E_7) \leq 29$, and $\operatorname{ed}(E_7;2) \leq 27$.

We prove weak-type $(1,1)$ estimates for compositions of maximal
operators with singular integrals. Our main object of interest is the
operator $\Delta^*\Psi$ where $\Delta^*$ is Bourgain's maximal
multiplier operator and $\Psi$ is the sum of several modulated
singular integrals; here our method yields a significantly improved
bound for the $L^q$ operator norm when $1 \lt q \lt 2.$ We also consider
associated variation-norm estimates.

We identify the quantum limits of scattering states
for the modular surface. This is obtained through the study of quantum
measures of non-holomorphic Eisenstein series away from the critical
line. We provide a range of stability for the quantum unique
ergodicity theorem of Luo and Sarnak.

This paper provides an erratum to Y. N. Petridis,
N. Raulf, and M. S. Risager, ``Quantum Limits
of Eisenstein Series and Scattering States.'' Canad. Math. Bull., published
online 2012-02-03, http://dx.doi.org/10.4153/CMB-2011-200-2.

Let $1 \lt p_1 \leq p_2 \leq p_3 \leq \dots$ be an infinite sequence
$\mathcal{P}$ of real numbers for which $p_i \to \infty$, and associate to
this sequence the Beurling zeta function $\zeta_{\mathcal{P}}(s):=
\prod_{i=1}^{\infty}(1-p_i^{-s})^{-1}$. Suppose that for some constant
$A\gt 0$, we have
$\zeta_{\mathcal{P}}(s) \sim A/(s-1)$, as $s\downarrow 1$. We prove that
$\mathcal{P}$ satisfies an analogue of a classical theorem of Mertens:
$\prod_{p_i \leq x}(1-1/p_i)^{-1} \sim A \e^{\gamma} \log{x}$, as
$x\to\infty$.
Here $\e = 2.71828\ldots$ is the base of the natural logarithm and
$\gamma = 0.57721\ldots$ is the usual Euler--Mascheroni constant. This
strengthens a recent theorem of Olofsson.

We obtain an asymptotic formula for the number
of square-free integers in $N$ consecutive values
of polynomials on average over integral
polynomials of degree at most $k$ and of
height at most $H$, where $H \ge N^{k-1+\varepsilon}$
for some fixed $\varepsilon\gt 0$.
Individual results of this kind for polynomials of degree $k \gt 3$,
due to A. Granville (1998),
are only known under the $ABC$-conjecture.

We show that any exceptional non-trivial Dehn surgery on a twist knot, except the trefoil,
yields a $3$-manifold whose fundamental group is left-orderable.
This is a generalization of a result of Clay, Lidman and Watson, and
also gives a new supporting evidence for a conjecture of Boyer, Gordon and Watson.

We prove that a connected, countable dense homogeneous space is
$n$-homogeneous for every $n$, and strongly 2-homogeneous provided it
is locally connected. We also present an example of a connected and
countable dense homogeneous space which is not strongly
2-homogeneous. This answers Problem 136 of Watson in the Open Problems
in Topology Book in the negative.

Using the Dadarlat isomorphism, we give a characterization for the
Kasparov product of $C^*$-algebra extensions. A certain relation
between $KK(A, \mathcal q(B))$ and $KK(A, \mathcal q(\mathcal k B))$ is also considered when
$B$ is not stable and it is proved that $KK(A, \mathcal q(B))$ and
$KK(A, \mathcal q(\mathcal k B))$ are not isomorphic in general.