For an analytic curve $\gamma :\,\left( a,\,b \right)\,\to \,\mathbb{C}$, the set of values approached by $\gamma \left( t \right)$, as $t\,\searrow \,\,a$ and as $t\,\nearrow \,b$ can be any two continua of $\mathbb{C}\,\cup \,\left\{ \infty \right\}$.

For smooth functions ${{a}_{1}}\,,\,{{a}_{2}}\,,\,{{a}_{3}}\,,\,{{a}_{4}}\,$ on a quaternion Heisenberg group, we characterize the existence of solutions of the partial differential operator system ${{X}_{1}}f\,=\,{{a}_{1}},\,{{X}_{2}}f=\,{{a}_{2}},\,{{X}_{3}}f\,=\,{{a}_{3}},\,\text{and}\,{{X}_{4}}f\,=\,{{a}_{4}}$. In addition, a formula for the solution function $f$ is deduced, assuming solvability of the system.

In this note, we collect various properties of Seifert homology spheres from the viewpoint of Dehn surgery along a Seifert fiber. We expect that many of these are known to various experts, but include them in one place, which we hope will be useful in the study of concordance and homology cobordism.

We make some elementary observations concerning subcritically Stein fillable contact structures on 5-manifolds. Specifically, we determine the diffeomorphism type of such contact manifolds in the case where the fundamental group is finite cyclic, and we show that on the 5-sphere, the standard contact structure is the unique subcritically fillable one. More generally, it is shown that subcritically fillable contact structures on simply connected 5-manifolds are determined by their underlying almost contact structure. Along the way, we discuss the homotopy classification of almost contact structures.

In this paper, we prove that the singular integral defined by

${{T}_{\Omega ,a}}f(x)=\text{p}\text{.}\text{v}\text{.}{{\int }_{{{\mathbb{R}}^{d}}}}\frac{\Omega (x-y)}{|x-y{{|}^{d}}}\cdot {{m}_{x,y}}a\cdot f(y)dy$is bounded on ${{L}^{p}}({{\mathbb{R}}^{d}})$ for $1\,<\,p\,<\,\infty $ and is of weak type (1,1), where $\Omega \,\in L\text{lo}{{\text{g}}^{+}}L({{S}^{d-1}})$ and ${{m}_{x,y}}a\,=:\,\int{_{0}^{1}}\,a(sx\,+\,(1\,-\,s)y)ds$, with $a\,\in \,{{L}^{\infty }}({{\mathbb{R}}^{d}})\,$ satisfying some restricted conditions.

We showbymeans of an example in ${{\mathbb{C}}^{3}}$ that Gromov’s theoremon the presence of attached holomorphic discs for compact Lagrangianmanifolds is not true in the subcritical real-analytic case, even in the absence of an obvious obstruction, i.e., polynomial convexity.

For an analytic function $f$ on the unit disk $\mathbb{D}$, we show that the ${{L}^{2}}$ integral mean of $f$ on $\text{c}\,\text{}\,\text{ }\!\!|\!\!\text{ z }\!\!|\!\!\text{ }\,\text{}\,\text{r}$ with respect to the weighted area measure ${{\left( 1\,-\,|z{{|}^{2}} \right)}^{\alpha }}dA\left( z \right)$ is a logarithmically convex function of $r$ on $\left( c,\,1 \right)$, where $-3\,\le \,\alpha \,\le \,0\,\text{and}\,\text{c}\,\in \,[\,0,\,1)$. Moreover, the range $[-3,\,0]$ for $\alpha $ is best possible. When $c\,=\,0$, our arguments here also simplify the proof for several results we obtained in earlier papers.

The study of chaotic vibration for multidimensional PDEs due to nonlinear boundary conditions is challenging. In this paper, we mainly investigate the chaotic oscillation of a two-dimensional non-strictly hyperbolic equation due to an energy-injecting boundary condition and a distributed self-regulating boundary condition. By using the method of characteristics, we give a rigorous proof of the onset of the chaotic vibration phenomenon of the zD non-strictly hyperbolic equation. We have also found a regime of the parameters when the chaotic vibration phenomenon occurs. Numerical simulations are also provided.

The classical result of Nevanlinna states that two nonconstantmeromorphic functions on the complex plane having the same images for five distinct values must be identically equal to each other. In this paper, we give a similar uniqueness theorem for the Gauss maps of complete minimal surfaces in Euclidean four-space.

We give a characterization of ${{C}^{*}}$-normed algebras, among certain involutive normed ones. This is done through the existence of enough specific positive functionals. The same question is also examined in some non normed (topological) algebras.

A self-avoiding polygon is a lattice polygon consisting of a closed self-avoiding walk on a square lattice. Surprisingly little is known rigorously about the enumeration of self-avoiding polygons, although there are numerous conjectures that are believed to be true and strongly supported by numerical simulations. As an analogous problemto this study, we considermultiple self-avoiding polygons in a confined region as a model for multiple ring polymers in physics. We find rigorous lower and upper bounds for the number ${{p}_{m\times n}}$ of distinct multiple self-avoiding polygons in the $m\,\times \,n$ rectangular grid on the square lattice. For $m\,=\,2,\,{{p}_{2\times n}}\,=\,{{2}^{n-1}}\,-1$. And for integers $m,\,n\,\ge \,3$,

$${{2}^{m+n-3}}\left( \frac{17}{10} \right){{\,}^{\left( m-2 \right)\left( n-2 \right)}}\,\le \,{{p}_{m\times n}}\,\le \,{{2}^{m+n-3}}\left( \frac{31}{16} \right){{\,}^{\left( m-2 \right)\left( n-2 \right)}}.$$

In this paper we establish the endpoint estimates and Hardy type estimates for the Riesz transform associated with the generalized Schrödinger operator.

The aim of this article is to give a proof of the Jordan Curve Theorem via complex analysis.

Given two monic polynomials $f$ and $g$ with coefficients in a number field $K$, and some $\alpha \,\in \,K$, we examine the action of the absolute Galois group $Gal\left( \bar{K}/K \right)$ on the directed graph of iterated preimages of $\alpha $ under the correspondence $g\left( y \right)\,=\,f\left( x \right)$, assuming that $\deg \left( f \right)\,>\,\deg \left( g \right)$ and that $\gcd \left( \deg \left( f \right),\deg \left( g \right) \right)\,=1$. If a prime of $K$ exists at which $f$ and $g$ have integral coefficients and at which $\alpha $ is not integral, we show that this directed graph of preimages consists of finitely many $Gal\left( \bar{K}/K \right)$-orbits. We obtain this result by establishing a $p$-adic uniformization of such correspondences, tenuously related to Böttcher’s uniformization of polynomial dynamical systems over $\mathbb{C}$, although the construction of a Böttcher coordinate for complex holomorphic correspondences remains unresolved.

In this paper, the bounded properties of oscillatory hyper-Hilbert transformalong certain plane curves $\gamma \left( t \right)$,

$${{T}_{\alpha ,\beta }}f\left( x,\,y \right)\,=\,\int_{0}^{1}{f\left( x\,-\,t,\,y\,-\,\gamma \left( t \right) \right){{e}^{i{{t}^{-\beta }}}}\frac{\text{d}t}{{{t}^{1}}+\alpha }}$$

are studied. For general curves, these operators are bounded in ${{L}^{2}}\left( {{\mathbb{R}}^{2}} \right)$ if $\beta \,\ge \,3\alpha $. Their boundedness in ${{L}^{p}}\left( {{\mathbb{R}}^{2}} \right)$ is also obtained, whenever $\beta \,\ge \,3\alpha $ and $\frac{2\beta }{2\beta -3\alpha }\,<\,p\,<\,\frac{2\beta }{3\alpha }$.

Motivated by a question of A. Skalski and P. M. Sołtan (2016) about inner faithfulness of S. Curran’s map of extending a quantum increasing sequence to a quantum permutation, we revisit the results and techniques of T. Banica and J. Bichon (2009) and study some group-theoretic properties of the quantum permutation group on points. This enables us not only to answer the aforementioned question in the positive for the case where $n\,=\,4,\,k\,=\,2$, but also to classify the automorphisms of $S_{4}^{+}$, describe all the embeddings ${{O}_{-1}}(2)\,\subset \,S_{4}^{+}$ and show that all the copies of ${{O}_{-1}}(2)$ inside $S_{4}^{+}$are conjugate. We then use these results to show that the converse to the criterion we applied to answer the aforementioned question is not valid.

On a real hypersurface $M$ in a complex two-plane Grassmannian ${{G}_{2}}\left( {{\mathbb{C}}^{m+2}} \right)$ we have the Lie derivation $\mathcal{L}$ and a differential operator of order one associated with the generalized Tanaka–Webster connection ${{\widehat{\mathcal{L}}}^{\left( k \right)}}$. We give a classification of real hypersurfaces $M$ on ${{G}_{2}}\left( {{\mathbb{C}}^{m+2}} \right)$ satisfying $\widehat{\mathcal{L}}_{\xi }^{\left( k \right)}\,S\,=\,{{\mathcal{L}}_{\xi }}S$, where $\xi$ is the Reeb vector field on $M$ and $s$ the Ricci tensor of $M$.

We give a consistent example of a zero-dimensional separable metrizable space $Z$ such that every homeomorphism of ${{Z}^{\omega }}$ acts like a permutation of the coordinates almost everywhere. Furthermore, this permutation varies continuously. This shows that a result of Dow and Pearl is sharp, and gives some insight into an open problem of Terada. Our example $Z$ is simply the set of ${{\omega }_{1}}$ Cohen reals, viewed as a subspace of ${{2}^{\omega }}$.

Let $R$ be a ring. A map $f\,:\,R\,\to \,R$ is additive if $f(a\,+\,b)\,=\,f(a)\,+\,f(b)$ for all elements $a$ and $b$ of $R$. Here, a map $f\,:\,R\,\to \,R$ is called unit-additive if $f(u\,+\,v)\,=\,f(u)\,+\,f(v)$ for all units $u$ and $v$ of $R$. Motivated by a recent result of $\text{Xu}$, $\text{Pei}$ and $\text{Yi}$ showing that, for any field $F$, every unit-additive map of ${{\mathbb{M}}_{n}}(F)$ is additive for all $n\,\ge \,2$, this paper is about the question of when every unit-additivemap of a ring is additive. It is proved that every unit-additivemap of a semilocal ring $R$ is additive if and only if either $R$ has no homomorphic image isomorphic to ${{\mathbb{Z}}_{2}}\,\text{or}\,R/J(R)\,\cong \,{{\mathbb{Z}}_{2}}\,$ with $2\,=\,0$ in $R$. Consequently, for any semilocal ring $R$, every unit-additive map of ${{\mathbb{M}}_{n}}(R)$ is additive for all $n\,\ge \,2$. These results are further extended to rings $R$ such that $R/J(R)$ is a direct product of exchange rings with primitive factors Artinian. A unit-additive map $f$ of a ring $R$ is called unithomomorphic if $f(uv)\,=\,f(u)f(v)$ for all units $u$, $v$ of $R$. As an application, the question of when every unit-homomorphic map of a ring is an endomorphism is addressed.

Let $R$ be an $n!$-torsion free semiprime ring with involution $*$ and with extended centroid $C$, where $n\,>\,1$ is a positive integer. We characterize $a\,\in \,K$, the Lie algebra of skew elements in $R$, satisfying ${{(\text{a}{{\text{d}}_{a}})}^{n}}\,=\,0$ on $K$. This generalizes both Martindale and Miers’ theorem and the theorem of Brox et al. In order to prove it we first prove that if $a,\,b\,\in \,R$ satisfy ${{(\text{a}{{\text{d}}_{a}})}^{n}}\,=\,\text{a}{{\text{d}}_{b}}$ on $R$, where either $n$ is even or $b\,=\,0$, then ${{(a\,-\,\lambda )}^{[(n+1)/2]}}\,=\,0$ for some $\lambda \,\in \,C$.