Let $v$ be a henselian valuation of any rank of a field $K$ and let $\bar{v}$ be the unique extension of $v$ to a fixed algebraic closure $\overline{K}$ of $K$. In 2005, we studied properties of those pairs $\left( \theta ,\,\alpha \right)$ of elements of $\overline{K}$ with $\left[ K\left( \theta \right):K \right]\,>\,\left[ K\left( \alpha \right):K \right]$ where $\alpha $ is an element of smallest degree over $K$ such that

$$\bar{v}\left( \theta \,-\,\alpha \right)\,=\,\sup \left\{ \bar{v}\left( \theta \,-\,\beta \right)\,|\,\beta \,\in \,\bar{K},\,\left[ K\left( \beta \right):K \right]\,<\,\left[ K\left( \theta \right):K \right] \right\}\,.$$

Such pairs are referred to as distinguished pairs. We use the concept of liftings of irreducible polynomials to give a different characterization of distinguished pairs.

It is well known that the factorization properties of a domain are reflected in the structure of its group of divisibility. The main theme of this paper is to introduce a topological/graph-theoretic point of view to the current understanding of factorization in integral domains. We also show that connectedness properties in the graph and topological space give rise to a generalization of atomicity.

We prove that no proper Segal algebra of a SIN group is approximately amenable.

Consider a quartic $q$-Weil polynomial $f$. Motivated by equidistribution considerations, we define, for each prime $\ell$, a local factor that measures the relative frequency with which $f$$ \bmod \,\ell $ occurs as the characteristic polynomial of a symplectic similitude over ${{\mathbb{F}}_{\ell }}$. For a certain class of polynomials, we show that the resulting infinite product calculates the number of principally polarized abelian surfaces over ${{\mathbb{F}}_{q}}$ with Weil polynomial $f$.

The main aim of this paper is to investigate various structural properties of hyperplanes of $c$, the Banach space of the convergent sequences. In particular, we give an explicit formula for the projection constants and prove that an hyperplane of $c$ is isometric to the whole space if and only if it is 1-complemented. Moreover, we obtain the classification of those hyperplanes for which their duals are isometric to ${{\ell }_{1}}$ and give a complete description of the preduals of ${{\ell }_{1}}$ under the assumption that the standard basis of ${{\ell }_{1}}$ is weak$^{*}$-convergent.

We classify the affine actions of ${{U}_{q}}\left( sl\left( 2 \right) \right)$ on commutative polynomial rings in $m\,\ge \,1$ variables. We show that, up to scalar multiplication, there are two possible actions. In addition, for each action, the subring of invariants is a polynomial ring in either $m$ or $m\,-\,1$ variables, depending upon whether $q$ is or is not a root of 1.

The precise condition on a completely regular space $X$ for every character on $C\left( X \right)$ to be an evaluation at some point in $X$ is that $X$ be realcompact. Usually, this classical result is obtained by relying heavily on involved (and even nonconstructive) extension arguments. This note provides a direct proof that is accessible to a large audience.

Let $\Gamma$ be a connection on a smooth manifold $M$. In this paper we give some properties of $\Gamma$ by studying the corresponding Lie algebras. In particular, we compute the first Chevalley–Eilenberg cohomology space of the horizontal vector fields Lie algebra on the tangent bundle of $M$, whose the corresponding Lie derivative of $\Gamma$ is null, and of the horizontal nullity curvature space.

We examine spectral behavior of irreducible tuples that do not admit the boundary property. In particular, we prove under some mild assumption that the spectral radius of such an $m$-tuple $\left( {{T}_{1}},\,.\,.\,.\,,\,{{T}_{m}} \right)$must be the operator norm of $T_{1}^{*}\,{{T}_{1}}\,+\,.\,.\,.\,+\,T_{m}^{*}{{T}_{m}}$. We use this simple observation to ensure the boundary property for an irreducible, essentially normal, joint q-isometry, provided it is not a joint isometry. We further exhibit a family of reproducing Hilbert $\mathbb{C}\left[ {{z}_{1}},\,.\,.\,.\,,{{z}_{m}} \right]$-modules (of which the Drury–Arveson Hilbert module is a prototype) with the property that any two nested unitarily equivalent submodules are indeed equal.

In a previous paper, we proved that the 1-d periodic fractional Schrödinger equation with cubic nonlinearity is locally well-posed in ${{H}^{s}}$ for $s>1-\alpha /2$ and globally well-posed for $s>10\alpha -1/12$. In this paper we define an invariant probability measure $\mu$ on ${{H}^{s}}$ for $s<\alpha -1/2$, so that for any $\text{ }\!\!\varepsilon\!\!\text{ }>0$ there is a set $\Omega \subset {{H}^{s}}$ such that $\mu \left( {{\Omega }^{c}} \right)<\text{ }\!\!\varepsilon\!\!\text{ }$ and the equation is globally well-posed for initial data in $\Omega$. We see that this fills the gap between the local well-posedness and the global well-posedness range in an almost sure sense for $\frac{1-\alpha }{2}<\alpha -\frac{1}{2},i.e.,\alpha >\frac{2}{3}$ in an almost sure sense.

In this paper we characterize the isometries of subspaces of the little Zygmund space. We show that the isometries of these spaces are surjective and represented as integral operators. We also show that all hermitian operators on these settings are bounded

The Chowla conjecture states that if $t$ is any given positive integer, there are infinitely many prime positive integers $N$ such that $\text{Per}\left( \sqrt{N} \right)\,=\,t$, where $\text{Per}\left( \sqrt{N} \right)$ is the period length of the continued fraction expansion for $\sqrt{N}$. C. Friesen proved that, for any $k\,\in \,\mathbb{N}$, there are infinitely many square-free integers $N$, where the continued fraction expansion of $\sqrt{N}$ has a fixed period. In this paper, we describe all polynomials $Q\,\in \,{{\mathbb{F}}_{q}}\left[ X \right]$ for which the continued fraction expansion of $\sqrt{Q}$ has a fixed period. We also give a lower bound of the number of monic, non-squares polynomials $Q$ such that $\deg \,Q=\,2d$ and $Per\sqrt{Q}\,=\,t$.

We present various weighted integral inequalities for partial derivatives acting on products and compositions of functions that are applied in order to establish some new Opial-type inequalities involving functions of several independent variables. We also demonstrate the usefulness of our results in the field of partial differential equations.

Motivated by Almgren’s work on the isoperimetric inequality, we prove a sharp inequality relating the length and maximum curvature of a closed curve in a complete, simply connected manifold of sectional curvature at most −1. Moreover, if equality holds, then the norm of the geodesic curvature is constant and the torsion vanishes. The proof involves an application of the maximum principle to a function defined on pairs of points.

Let $b\,\in \,\text{BMO}\left( {{\mathbb{R}}^{n}} \right)$ and ${{T}_{\Omega }}$ be the singular integral operator with kernel $\Omega \left( x \right)/{{\left| x \right|}^{n}}$, where $\Omega$ is homogeneous of degree zero, integrable, and has mean value zero on the unit sphere ${{S}^{n-1}}$. In this paper, using Fourier transform estimates and approximation to the operator ${{T}_{\Omega }}$ by integral operators with smooth kernels, it is proved that if $b\,\in \,\text{CMO}\left( {{\mathbb{R}}^{n}} \right)$ and $\Omega$ satisfies certain minimal size condition, then the commutator generated by $b$ and ${{T}_{\Omega }}$ is a compact operator on ${{L}^{p}}\left( {{\mathbb{R}}^{n}} \right)$ for appropriate index $p$. The associated maximal operator is also considered.

Let $C$ be a curve over a complete valued field having an infinite residue field and whose skeleton is a chain of loops with generic edge lengths. We prove that any divisor on the chain of loops that is rational over the value group lifts to a divisor of the same rank on $C$, confirming a conjecture of Cools, Draisma, Robeva, and the third author.

We show that a class of semilinear Laplace–Beltrami equations on the unit sphere in ${{\mathbb{R}}^{n}}$ has infinitely many rotationally symmetric solutions. The solutions to these equations are the solutions to a two point boundary value problem for a singular ordinary differential equation. We prove the existence of such solutions using energy and phase plane analysis. We derive a Pohozaev-type identity in order to prove that the energy to an associated initial value problem tends to infinity as the energy at the singularity tends to infinity. The nonlinearity is allowed to grow as fast as ${{\left| s \right|}^{p-1}}s$ for $\left| s \right|$ large with $1\,<\,p\,<\,\left( n+5 \right)/\left( n-3 \right)$.

A magma$\left( M,\star \right)$ is a nonempty set with a binary operation. A double magma$\left( M,\star ,\bullet \right)$ is a nonempty set with two binary operations satisfying the interchange law$\left( w\star x \right)\bullet \left( y\star z \right)=\left( w\bullet y \right)\star \left( x\bullet z \right)$. We call a double magma proper if the two operations are distinct, and commutative if the operations are commutative. A double semigroup, first introduced by Kock, is a double magma for which both operations are associative. Given a non-trivial group $G$ we define a system of two magma $\left( G,\star ,\bullet \right)$ using the commutator operations $x\star y=\left[ x,y \right]\left( ={{x}^{-1}}{{y}^{-1}}xy \right)$ and $x\bullet y=\left[ y,x \right]$. We show that $\left( G,\star ,\bullet \right)$ is a double magma if and only if $G$ satisfies the commutator laws $\left[ x,y;x,z \right]=1$ and ${{\left[ w,x;y,z \right]}^{2}}=1$. We note that the first law defines the class of 3-metabelian groups. If both these laws hold in $G$, the double magma is proper if and only if there exist ${{x}_{0}},{{y}_{0}}\in G$ for which ${{\left[ {{x}_{0}},{{y}_{0}} \right]}^{2}}\ne 1$. This double magma is a double semigroup if and only if $G$ is nilpotent of class two. We construct a specific example of a proper double semigroup based on the dihedral group of order 16. In addition, we comment on a similar construction for rings using Lie commutators.

Let $G$ be a group and $\mathbb{K}\,=\,\mathbb{C}\,\text{or}\,\mathbb{R}$. In this article, as a generalization of the result of Albert and Baker, we investigate the behavior of bounded and unbounded functions $f\,:\,G\,\to \,\mathbb{K}$ satisfying the inequality

$$\left| f\left( \sum\limits_{k=1}^{n}{{{x}_{k}}} \right)\,-\,\underset{k=1}{\overset{n}{\mathop{\Pi }}}\,f\left( {{x}_{k}} \right) \right|\,\,\le \phi \left( {{x}_{2}},\,.\,.\,.\,,{{x}_{n}} \right),\,\,\,\,\,\,\forall {{x}_{1}},\,.\,.\,.\,,{{x}_{n}}\,\in \,G,$$

Where $\phi :\,{{G}^{n-1}}\,\to \,[0,\,\infty )$. Also as a a distributional version of the above inequality we consider the stability of the functional equation

$$u\,\circ \,S\,-\,\overbrace{u\,\otimes \,.\,.\,.\,\otimes \,u}^{n-\text{times}}\,=\,0,$$

where $u$ is a Schwartz distribution or Gelfand hyperfunction, $\circ$ and $\otimes$ are the pullback and tensor product of distributions, respectively, and $S\left( {{x}_{1}},\,.\,.\,.\,,{{x}_{n}} \right)\,=\,{{x}_{1}}\,+\,.\,.\,.\,+\,{{x}_{n}}$.

Let $R$ be a ring and let $g$ be an endomorphism of $R$. The additive mapping $d:\,R\,\to \,R$ is called a Jordan semiderivation of $R$, associated with $g$, if

$$d\left( {{x}^{2}} \right)=d\left( x \right)x+g\left( x \right)d\left( x \right)=d\left( x \right)g\left( x \right)+xd\left( x \right)\,\text{and}\,d\left( g\left( x \right) \right)=g\left( d\left( x \right) \right)$$

for all $x\,\in \,R$. The additive mapping $F:\,R\,\to \,R$ is called a generalized Jordan semiderivation of $R$, related to the Jordan semiderivation $d$ and endomorphism $g$, if

$$F\left( {{x}^{2}} \right)=F\left( x \right)x+g\left( x \right)d\left( x \right)=F\left( x \right)g\left( x \right)+xd\left( x \right)\,\,and\,F\left( g\left( x \right) \right)=g\left( F\left( x \right) \right)$$

for all $x\,\in \,R$. In this paper we prove that if $R$ is a prime ring of characteristic different from 2, $g$ an endomorphism of $R,\,d$ a Jordan semiderivation associated with $g,\,F$ a generalized Jordan semiderivation associated with $d$ and $g$, then $F$ is a generalized semiderivation of $R$ and $d$ is a semiderivation of $R$. Moreover, if $R$ is commutative, then $F\,=\,d$.