Let $R$ be an associative ring with unity. Then $R$ is said to be a right McCoy ring when the equation $f\left( x \right)g\left( x \right)\,=\,0$ (over $R\left[ x \right]$), where $0\,\ne \,f\left( x \right)$, $g\left( x \right)\,\in \,R\left[ x \right]$, implies that there exists a nonzero element $c\,\in \,R$ such that $f\left( x \right)c\,=\,0$. In this paper, we characterize some basic ring extensions of right McCoy rings and we prove that if $R$ is a right McCoy ring, then $R\left[ x \right]/\left( {{x}^{n}} \right)$ is a right McCoy ring for any positive integer $n\,\ge \,2$.

On a compact Kähler manifold $X$ with a holomorphic 2-form $\alpha$, there is an almost complex structure associated with α. We show how this implies vanishing theorems for the Gromov–Witten invariants of $X$. This extends the approach used by Parker and the author for Kähler surfaces to higher dimensions.

A Halin graph is a graph $H\,=\,T\,\cup \,C$, where $T$ is a tree with no vertex of degree two, and $C$ is a cycle connecting the end-vertices of $T$ in the cyclic order determined by a plane embedding of $T$. In this paper, we define classes of generalized Halin graphs, called $k$-Halin graphs, and investigate their Hamiltonian properties.

We obtain new characterizations for Bergman spaces with standard weights in terms of Lipschitz type conditions in the Euclidean, hyperbolic, and pseudo-hyperbolic metrics. As a consequence, we prove optimal embedding theorems when an analytic function on the unit disk is symmetrically lifted to the bidisk.

We consider the pushout of embedding functors in Cat, the category of small categories. We show that if the embedding functors satisfy a 3-for-2 property, then the induced functors to the pushout category are also embeddings. The result follows from the connectedness of certain associated slice categories. The condition is motivated by a similar result for maps of semigroups. We show that our theorem can be applied to groupoids and to inclusions of full subcategories. We also give an example to show that the theorem does not hold when the property only holds for one of the inclusion functors, or when it is weakened to a one-sided condition.

Let $f\in \,{{L}_{2\pi }}$ be a real-valued even function with its Fourier series $\frac{{{a}_{0}}}{2}\,+\,\sum _{n=1}^{\infty }\,{{a}_{n}}\,\cos \,nx$, and let ${{S}_{n}}\left( f,x \right)$, $n\,\,\ge \,\,1$, be the $n$-th partial sum of the Fourier series. It is well known that if the nonnegative sequence $\{{{a}_{n}}\}$ is decreasing and ${{\lim }_{n\to \infty }}\,{{a}_{n}}\,=\,0$, then

$$\underset{n\to \infty }{\mathop{\lim }}\,{{\left\| f-{{S}_{n}}\left( f \right) \right\|}_{L}}=0\text{ifanyonlyif}\underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}}\log n=0.$$

We weaken the monotone condition in this classical result to the so-called mean value bounded variation (MVBV) condition. The generalization of the above classical result in real-valued function space is presented as a special case of the main result in this paper, which gives the ${{L}^{1}}$-convergence of a function $f\in {{L}_{2\pi }}$ in complex space. We also give results on ${{L}^{1}}$-approximation of a function $f\in {{L}_{2\pi }}$ under the MVBV condition.

We investigate the following version of the circle covering problem in strictly convex (normed or) Minkowski planes: to cover a circle of largest possible diameter by $k$ unit circles. In particular, we study the cases $k\,=\,3$, $k\,=\,4$, and $k\,=\,7$. For $k\,=\,3$ and $k\,=\,4$, the diameters under consideration are described in terms of side-lengths and circumradii of certain inscribed regular triangles or quadrangles. This yields also simple explanations of geometric meanings that the corresponding homothety ratios have. It turns out that basic notions from Minkowski geometry play an essential role in our proofs, namely Minkowskian bisectors, $d$-segments, and the monotonicity lemma.

In the work presented below the classical subject of orthogonal polynomials on the unit circle is discussed in the matrix setting. An explicit matrix representation of the matrix valued orthogonal polynomials in terms of the moments of the measure is presented. Classical recurrence relations are revisited using the matrix representation of the polynomials. The matrix expressions for the kernel polynomials and the Christoffel–Darboux formulas are presented for the first time.

We prove there exist exponentially decaying generalized eigenfunctions on a blow-up of the Sierpinski gasket with boundary. These are used to show a Borel-type theorem, specifically that for a prescribed jet at the boundary point there is a smooth function having that jet.

The paper studies modular reduction techniques for abstract regular and chiral polytopes, with two purposes in mind: first, to survey the literature about modular reduction in polytopes; and second, to apply modular reduction, with moduli given by primes in $\mathbb{Z}\left[ \tau \right]$ (with $\tau$ the golden ratio), to construct new regular 4-polytopes of hyperbolic types $\{3,\,5,\,3\}$ and $\{5,\,3,\,5\}$ with automorphism groups given by finite orthogonal groups.

Falbel has shown that four pairwise distinct points on the boundary of a complex hyperbolic 2-space are completely determined, up to conjugation in $\text{PU}\left( 2,\,1 \right)$, by three complex cross-ratios satisfying two real equations. We give global geometrical coordinates on the resulting variety.

We say that a function $h\,:\,\mathbb{R}\,\to \,\mathbb{R}$ is a Hamel function $(h\,\in \,\text{HF)}$ if $h$, considered as a subset of ${{\mathbb{R}}^{2}},$ is a Hamel basis for ${{\mathbb{R}}^{2}}.$ We show that $\text{A}\left( \text{HF} \right)\,\ge \,\omega$, i.e., for every finite $F\,\subseteq \,{{\mathbb{R}}^{\mathbb{R}}}$ there exists $f\,\in \,{{\mathbb{R}}^{\mathbb{R}}}$ such that $f\,+\,F\,\subseteq \,\text{HF}$. From the previous work of the author it then follows that $\text{A}\left( \text{HF} \right)\,=\,\omega$.

We prove that for every $k\,>\,1$, there exist $k$-fold coverings of the plane (i) with strips, (ii) with axis-parallel rectangles, and (iii) with homothets of any fixed concave quadrilateral, that cannot be decomposed into two coverings. We also construct for every $k\,>\,1$ a set of points $P$ and a family of disks $D$ in the plane, each containing at least $k$ elements of $P$, such that, no matter how we color the points of $P$ with two colors, there exists a disk $D\,\in \,D$ all of whose points are of the same color.

Let $\text{q}\,\text{=}\,\text{2,}\,\text{3}$ and $f\left( X,\,Y \right)$, $g\left( X,\,Y \right)$, $h\left( X \right)$ be polynomials with integer coefficients. In this paper we deal with the curve $f{{\left( X,\,Y \right)}^{\text{q}}}\,=\,h\left( X \right)g\left( X,\,Y \right)$ , and we show that under some favourable conditions it is possible to determine all of its rational points.

Let $K\,\subset \,{{\mathbb{R}}^{n+1}}$ be a convex body of class ${{C}^{2}}$ with everywhere positive Gauss curvature. We show that there exists a positive number $\delta \left( K \right)$ such that for any $\delta \,\in \,\left( 0,\,\delta \left( K \right) \right)$ we have $\text{Vol}\left( {{K}_{\delta }} \right)\,\cdot \,\text{Vol}\left( {{\left( {{K}_{\delta }} \right)}^{*}} \right)\,\ge \,\text{Vol}\left( K \right)\,\cdot \,\text{Vol}\left( {{K}^{*}} \right)\,\ge \,\text{Vol}\left( {{K}^{\delta }} \right)\,\cdot \,\text{Vol}\left( {{\left( {{K}^{\delta }} \right)}^{*}} \right)$, where ${{K}_{\delta }}$, ${{K}^{\delta }}$ and ${{K}^{*}}$ stand for the convex floating body, the illumination body, and the polar of $K$, respectively. We derive a few consequences of these inequalities.

We prove in $\text{ZFC}$ that if $\text{cf}\left( \text{ }\lambda \text{ } \right)>{{\aleph }_{0}}$ and ${{2}^{\text{cf}\left( \text{ }\!\!\lambda\!\!\text{ } \right)}}\,<\,\text{ }\!\!\lambda\!\!\text{ }$, then $\text{ }\!\!\lambda\!\!\text{ }\,\to \,{{\left( \text{ }\!\!\lambda\!\!\text{ ,}\,\omega \,\text{+}\,\text{1} \right)}^{2}}$.

Dealing with the cardinal invariants $\mathfrak{p}$ and $\mathfrak{t}$ of the continuum, we prove that $\mathfrak{m}\,=\,\mathfrak{p}\,=\,{{\aleph }_{2}}\,\Rightarrow \,\mathfrak{t}\,=\,{{\aleph }_{2}}$ . In other words, if $\text{M}{{\text{A}}_{{{\aleph }_{1}}}}$ (or a weak version of this) holds, then (of course ${{\aleph }_{2}}\,\le \,\mathfrak{p}\,\le \,\mathfrak{t}$ and) $\mathfrak{p}\,=\,\,{{\aleph }_{2}}\,\Rightarrow \,\mathfrak{p}\,=\,\mathfrak{t}$ . The proof is based on a criterion for $\mathfrak{p}\,<\,\mathfrak{t}$.

The solutions to Hilbert's Fourth Problem in the regular case are projectively flat Finsler metrics. In this paper, we consider the so-called $\left( \alpha ,\,\beta \right)$-metrics defined by a Riemannian metric $\alpha$ and a 1-form $\beta$, and find a necessary and sufficient condition for such metrics to be projectively flat in dimension $n\,\ge \,3$.

By employing the limit set dichotomy for essentially strongly order-preserving semiflows and the assumption that limit sets have infima and suprema in the state space, we prove a generic quasi-convergence principle implying the existence of an open and dense set of stable quasi-convergent points. We also apply this generic quasi-convergence principle to a model for biochemical feedback in protein synthesis and obtain some results about the model which are of theoretical and realistic significance.

$\text{A}$ ring $R$ is said to be $n$-clean if every element can be written as a sum of an idempotent and $n$ units. The class of these rings contains clean rings and $n$-good rings in which each element is a sum of $n$ units. In this paper, we show that for any ring $R$, the endomorphism ring of a free $R$-module of rank at least 2 is 2-clean and that the ring $B\left( R \right)$ of all $\omega \,\times \,\omega$ row and column-finite matrices over any ring $R$ is 2-clean. Finally, the group ring $R{{C}_{n}}$ is considered where $R$ is a local ring.