A new tautological relation of ${{\overline{M}}_{3,\,1}}$ in codimension 3 is derived and proved, using an invariance constraint from our previous work.

In Connections on a parabolic principal bundle over a curve, $I$ we defined connections on a parabolic principal bundle. While connections on usual principal bundles are defined as splittings of the Atiyah exact sequence, it was noted in the above article that the Atiyah exact sequence does not generalize to the parabolic principal bundles. Here we show that a twisted version of the Atiyah exact sequence generalizes to the context of parabolic principal bundles. For usual principal bundles, giving a splitting of this twisted Atiyah exact sequence is equivalent to giving a splitting of the Atiyah exact sequence. Connections on a parabolic principal bundle can be defined using the generalization of the twisted Atiyah exact sequence.

If $K$ is a number field with ${{n}_{k}}\,=\,\left[ k\,:\,\mathbb{Q} \right]$, and ${{\xi }_{k}}$ the symmetrized Dedekind zeta function of the field, the inequality

$$\Re \frac{\xi _{k}^{'}\left( \sigma \,+\,\text{i}t \right)}{{{\xi }_{k}}\left( \sigma \,+\,\text{i}t \right)}\,>\,\frac{\xi _{k}^{'}\left( \sigma \right)}{{{\xi }_{k}}\left( \sigma \right)}$$

for $t\,\ne \,0$ is shown to be true for $\sigma \,\ge \,1\,+\,8/n_{k}^{\frac{1}{3}}$ improving the result of Lagarias where the constant in the inequality was 9. In the case $k\,=\,\mathbb{Q}$ the inequality is extended to $\sigma \,\ge \,1$ for all $t$ sufficiently large or small and to the region $\sigma \,\ge \,1\,+\,1/\left( \log \,t\,-\,5 \right)$ for all $t\,\ne \,0$. This answers positively a question posed by Lagarias.

Let $\tau \left( n \right)$ be the Ramanujan $\text{ }\!\!\tau\!\!\text{ }$-function. We prove that for any integer $N$ with $\left| N \right|\,\ge \,2$ the diophantine equation

$$\underset{i=1}{\overset{148000}{\mathop{\sum }}}\,\,\tau \left( {{n}_{i}} \right)\,=\,N$$

has a solution in positive integers ${{n}_{1}},\,{{n}_{2}},\,.\,.\,.\,,\,{{n}_{148000}}$ satisfying the condition

$$\underset{1\le i\le 148000}{\mathop{\max }}\,{{n}_{i}}\ll |N{{|}^{2/11}}{{e}^{-c\log |N|/\log \log |N|}},$$

for some absolute constant $c\,>\,0$.

The (classical, small quantum, equivariant) cohomology ring of the grassmannian $G\left( k,\,n \right)$ is generated by certain derivations operating on an exterior algebra of a free module of rank $n$ (Schubert calculus on a Grassmann algebra). Our main result gives, in a unified way, a presentation of all such cohomology rings in terms of generators and relations. Using results of Laksov and Thorup, it also provides a presentation of the universal factorization algebra of a monic polynomial of degree $n$ into the product of two monic polynomials, one of degree $k$.

J. Elton used an application of Ramsey theory to show that if $X$ is an infinite dimensional Banach space, then ${{c}_{0}}$ embeds in $X$, ${{\ell }_{1}}$ embeds in $X$, or there is a subspace of $X$ that fails to have the Dunford–Pettis property. Bessaga and Pelczynski showed that if ${{c}_{0}}$ embeds in ${{X}^{*}}$ , then ${{\ell }_{\infty }}$ embeds in ${{X}^{*}}.$ Emmanuele and John showed that if ${{c}_{0}}$ embeds in $K\left( X,\,Y \right)$, then $K\left( X,\,Y \right)$ is not complemented in $L\left( X,\,Y \right)$. Classical results from Schauder basis theory are used in a study of Dunford–Pettis sets and strong Dunford–Pettis sets to extend each of the preceding theorems. The space ${{L}_{{{w}^{*}}}}\left( {{X}^{*}},\,Y \right)$ of ${{w}^{*}}\,-\,w$ continuous operators is also studied.

Let $R$ be a real closed field, let $X\,\subset \,{{R}^{n}}$ be an irreducible real algebraic set and let $Z$ be an algebraic subset of $X$ of codimension ≥ 2. Dubois and Efroymson proved the existence of an irreducible algebraic subset of $X$ of codimension 1 containing $Z$. We improve this dimension theorem as follows. Indicate by $\mu$ the minimum integer such that the ideal of polynomials in $R\left[ {{x}_{1}}\,,\,.\,.\,.\,,\,{{x}_{n}} \right]$ vanishing on $Z$ can be generated by polynomials of degree $\le \,\mu$. We prove the following two results: (1) There exists a polynomial $P\,\in \,R\left[ {{x}_{1}}\,,\,.\,.\,.\,,\,{{x}_{n}} \right]$ of degree $\le \,\mu +1$ such that $X\cap {{P}^{-1}}\left( 0 \right)$ is an irreducible algebraic subset of $X$ of codimension 1 containing $Z$. (2) Let $F$ be a polynomial in $R\left[ {{x}_{1}}\,,\,.\,.\,.\,,\,{{x}_{n}} \right]$ of degree $d$ vanishing on $Z$. Suppose there exists a nonsingular point $x$ of $X$ such that $F\left( x \right)\,=\,0$ and the differential at $x$ of the restriction of $F$ to $X$ is nonzero. Then there exists a polynomial $G\,\in \,R\left[ {{x}_{1}}\,,\,.\,.\,.\,,\,{{x}_{n}} \right]$ of degree $\le \,\max \{d,\,\mu \,+\,1\}$ such that, for each $t\,\in \,\left( -1,\,1 \right)\,\backslash \,\{0\}$, the set $\{x\in X|F\left( x \right)+tG\left( x \right)=0\}$ is an irreducible algebraic subset of $X$ of codimension 1 containing $Z$. Result (1) and a slightly different version of result (2) are valid over any algebraically closed field also.

We obtain a Bogomolov type of result for the affine space defined over the algebraic closure of a function field of transcendence degree 1 over a finite field.

Let $L$ be an $\text{RA}$ loop, that is, a loop whose loop ring over any coefficient ring $R$ is an alternative, but not associative, ring. Let $\ell \,\mapsto \,{{\ell }^{\theta }}$ denote an involution on $L$ and extend it linearly to the loop ring $RL$. An element $\alpha \,\in \,RL$ is symmetric if ${{\alpha }^{\theta }}\,=\,\alpha$ and skew-symmetric if ${{\alpha }^{\theta }}=-\alpha$ . In this paper, we show that there exists an involution making the symmetric elements of $RL$ commute if and only if the characteristic of $R$ is 2 or θ is the canonical involution on $L$, and an involution making the skew-symmetric elements of $RL$ commute if and only if the characteristic of $R$ is 2 or 4.

An irreducible graph manifold $M$ contains an essential torus if it is not a special Seifert manifold. Whether $M$ contains a closed essential surface of negative Euler characteristic or not depends on the difference of Seifert fibrations from the two sides of a torus system which splits $M$ into Seifert manifolds. However, it is not easy to characterize geometrically the class of irreducible graph manifolds which contain such surfaces. This article studies this problem in the case of graph link exteriors.

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 $n\,\ge \,2$.

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.

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 $\text{A}\left( \text{HF} \right)\,=\,\omega$.

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 $\mathfrak{p}\,<\,\mathfrak{t}$.

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.