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

$$\alpha =\frac{A{{\alpha }^{q}}+B}{C{{\alpha }^{q}}},$$

where $\left( A,B,C \right)\,\in \,{{\left( {{\mathbb{F}}_{q}}\left[ X \right] \right)}^{2}}\times \mathbb{F}_{q}^{*}\left[ X \right]$. In particular, under some conditions on the polynomials $A,\,B$ and $C$, we will give well approximated elements satisfying this equation.

Let $G$ be a graph. The minimum number of colors needed to color the edges of $G$ is called the chromatic index of $G$ and is denoted by $X'\left( G \right)$. It is well known that $\Delta \left( G \right)\,\le \,\mathcal{X}'\left( G \right)\,\le \Delta \left( G \right)\,+\,1$, for any graph $G$, where $\Delta \left( G \right)$ denotes the maximum degree of $G$. A graph $G$ is said to be class 1 if ${\mathcal{X}}'\left( G \right)\,=\,\Delta \left( G \right)$ and class 2 if ${\mathcal{X}}'\left( G \right)\,=\,\Delta \left( G \right)\,+\,1$. Also, ${{G}_{\Delta }}$ is the induced subgraph on all vertices of degree $\Delta \left( G \right)$. Let $f:\,V\left( G \right)\,\to \mathbb{N}$ be a function. An $f$ -coloring of a graph $G$ is a coloring of the edges of $E\left( G \right)$ such that each color appears at each vertex $v\,\in \,V\left( G \right)$ at most $f\left( v \right)$ times. The minimum number of colors needed to $f$ -color $G$ is called the $f$ -chromatic index of $G$ and is denoted by ${{{\mathcal{X}}'}_{f}}\left( G \right)$. It was shown that for every graph $G,\,{{\Delta }_{f}}\,\left( G \right)\,\le \,{{\mathcal{X}}^{\prime }}_{f}\left( G \right)\,\le \,{{\Delta }_{f}}\,\left( G \right)\,+\,1$, where ${{\Delta }_{f}}\left( G \right)\,=\,{{\max }_{v\in \left( G \right)}}\,\left\lceil {{{d}_{G}}\left( v \right)}/{f\left( v \right)}\; \right\rceil $. A graph $G$ is said to be $f$ -class 1 if ${{\mathcal{X}}^{\prime }}_{f}\left( G \right)\,=\,{{\Delta }_{f}}\left( G \right)$, and $f$ -class 2, otherwise. Also, ${{G}_{{{\Delta }_{f}}}}$ is the induced subgraph of $G$ on $\left\{ v\,\in \,V\left( G \right)\,:\,{{{d}_{G}}\left( V \right)}/{f\left( v \right)}\;\,=\,{{\Delta }_{f}}\left( G \right) \right\}$. Hilton and Zhao showed that if ${{G}_{\Delta }}$ has maximum degree two and $G$ is class 2, then $G$ is critical, ${{G}_{\Delta }}$ is a disjoint union of cycles and $\delta \left( G \right)\,=\,\Delta \left( G \right)-1$, where $\delta \left( G \right)$ denotes the minimum degree of $G$, respectively. In this paper, we generalize this theorem to $f$ -coloring of graphs. Also, we determine the $f$ -chromatic index of a connected graph $G$ with $\left| {{G}_{{{\Delta }_{f}}}} \right|\,\le \,4$.

We provide a proof of a conjecture by Jakobson, Nadirashvili, and Toth stating that on an $n$-dimensional flat torus ${{\mathbb{T}}^{n}}$, and the Fourier transform of squares of the eigenfunctions ${{\left| \varphi \lambda \right|}^{2}}$ of the Laplacian have uniform ${{l}^{n}}$ bounds that do not depend on the eigenvalue $\lambda $. The proof is a generalization of an argument by Jakobson, et al. for the lower dimensional cases. These results imply uniform bounds for semiclassical limits on ${{\mathbb{T}}^{n+2}}$. We also prove a geometric lemma that bounds the number of codimension-one simplices satisfying a certain restriction on an $n$-dimensional sphere ${{S}^{n}}\,\left( \text{ }\!\!\lambda\!\!\text{ } \right)$ of radius $\sqrt{\lambda }$, and we use it in the proof.

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.

Inspired by Aldous' conjecture for the spectral gap of the interchange process and its recent resolution by Caputo, Liggett, and Richthammer, we define an associated order $\prec $ on the irreducible representations of ${{S}_{n}}$. Aldous' conjecture is equivalent to certain representations being comparable in this order, and hence determining the “Aldous order” completely is a generalized question. We show a few additional entries for this order.

For every subnormal $m$-variable weighted shift $S$ (with bounded positive weights), there is a corresponding positive Reinhardt measure $\mu $ supported on a compact Reinhardt subset of ${{\mathbb{C}}^{m}}$. We show that, for $m\,\ge \,2$, the dimensions of the 1-st cohomology vector spaces associated with the Koszul complexes of $S$ and its dual $\widetilde{S}$ are different if a certain radial function happens to be integrable with respect to μ (which is indeed the case with many classical examples). In particular, $S$ cannot in that case be similar to $\widetilde{S}$. We next prove that, for $m\,\ge \,2$, a Fredholm subnormal $m$-variable weighted shift $S$ cannot be similar to its dual.

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 ${{\mathbb{R}}^{\left( \mathbb{N} \right)}}$. We also 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.

A complete characterization of valuation rings closed for a holomorphic derivation is given, following an idea of Seidenberg, in dimension 2.

We introduce and study some new function spaces on Riemann surfaces. For certain parameter values these spaces coincide with the classical Dirichlet space, $\text{BMOA}$, or the recently defined ${{\text{Q}}_{p}}$ space. We establish inclusion relations that generalize earlier known inclusions between the above-mentioned spaces.

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.

For any finite Galois extension $K$ of $\mathbb{Q}$ and any conjugacy class $C$ in $\text{Gal}\left( {K}/{\mathbb{Q}}\; \right)$, 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 ${{a}^{2}}\,+\,n{{b}^{2}}$ with $a,\,b\,\in \,\mathbb{Z}$.

Let $\text{ }\!\!\lambda\!\!\text{ }\left( n \right)$ denote the Liouville function. Complementary to the prime number theorem, Chowla conjectured 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 $bg{{\left( x \right)}^{2}}$.

We give a characterization of hypercyclic abelian group $\mathcal{G}$ of affine maps on ${{\mathbb{C}}^{n}}$. If $\mathcal{G}$ is finitely generated, this characterization is explicit. We prove in particular that no abelian group generated by $n$ affine maps on ${{\mathbb{C}}^{n}}$ has a dense orbit.

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 Fraϊssé theory and Ramsey theory. This work generalizes some of the known results about countable structures

In 1961, J. Barrett showed that if the first conjugate point ${{\eta }_{1}}\left( a \right)$ exists for the differential equation ${{\left( r\left( x \right){y}'' \right)}^{\prime \prime }}=p\left( x \right)y$, where $r\left( x \right)\,>\,0$ and $p\left( x \right)\,>\,0$, then so does the first systems-conjugate point ${{\hat{\eta }}_{1}}\left( a \right)$. The aim of this note is to extend this result to the general equation with middle term ${{\left( q\left( x \right){y}' \right)}^{\prime }}$ without further restriction on $q\left( x \right)$, other than continuity.

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.

We show that there is a bijective correspondence between the polystable parabolic principal $G$-bundles and solutions of the Hermitian-Einstein equation.

Let $\left( R,\,m \right)$ be a non-zero commutative Noetherian local ring (with identity) and let $M$ be a non-zero finitely generated $R$-module. In this paper for any $\mathfrak{p}\,\in \,\text{Spec}\left( R \right)$ we show that

1

$$\text{injdi}{{\text{m}}_{{{R}_{\mathfrak{p}}}}}\,H_{\mathfrak{p}{{R}_{\mathfrak{p}}}}^{i-\dim\left( {R}/{\mathfrak{p}}\; \right)}\left( {{M}_{\mathfrak{p}}} \right)$$

1

$$\text{f}{{\text{d}}_{{{R}_{\mathfrak{p}}}}}H_{\mathfrak{p}}^{i-\dim\left( {R}/{\mathfrak{p}}\; \right)}\left( {{M}_{\mathfrak{p}}} \right)$$

are bounded from above by $\text{injdi}{{\text{m}}_{R}}\,H_{\text{m}}^{i}\left( M \right)$ and $\text{f}{{\text{d}}_{R}}\,H_{\text{m}}^{i}\left( M \right)$ respectively, for all integers $i\,\ge \,\dim\left( {R}/{\mathfrak{p}}\; \right)$.

Let $b\,>\,1$ 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 $c\,:=\,c\left( b \right)\,>\,0$ is some constant depending on $b$.