A new and elementary proof is given of the recent result of Cuccagna, Pelinovsky, and Vougalter based on the variational principle for the quadratic form of a self-adjoint operator. It is the negative index theorem for a linearized $\text{NLS}$ operator in three dimensions.

We prove that, in ordered plane geometries endowed with a very weak notion of orthogonality, one can always triangulate any triangle into seven acute triangles, and, in case the given triangle is not acute, into no fewer than seven.

We establish sufficient conditions on the weight functions $u$ and $v$ for the validity of the multidimensional weighted inequality

$${{\left( \int_{E}{\Phi {{\left( {{T}_{k}}f\left( x \right) \right)}^{q}}u\left( x \right)dx} \right)}^{1/q}}\,\le \,C{{\left( \int_{E}{\Phi {{\left( f\left( x \right) \right)}^{p}}v\left( x \right)dx} \right)}^{1/p}},$$

where $0\,<\,p,\,q\,<\,\infty $, $\Phi $ is a logarithmically convex function, and ${{T}_{k}}$ is an integral operator over star-shaped regions. The condition is also necessary for the exponential integral inequality. Moreover, the estimation of $C$ is given and we apply the obtained results to generalize some multidimensional Levin–Cochran-Lee type inequalities.

In this paper, we establish several weighted ${{L}^{p}}\left( 1\,<\,p\,\infty \right)$ Hardy type inequalities related to the generalized Greiner operator by improving the method of Kombe. Then the best constants in inequalities are discussed by introducing new polar coordinates.

We establish variants of stability estimates in norms somewhat stronger than the ${{H}^{1}}$-norm under Arnold's stability hypotheses on steady solutions to the Euler equations for fluid flow on planar domains.

We discuss properties of the regular part ${{S}_{\text{reg}}}$ of a subcartesian space $S$. We show that ${{S}_{\text{reg}}}$ is open and dense in $S$ and the restriction to ${{S}_{\text{reg}}}$ of the tangent bundle space of $S$ is locally trivial.

In this paper we provide lower bounds for the dimension of various critical sets, and we point out some differential maps with high dimensional critical sets.

We construct new examples of surfaces of general type with ${{P}_{g}}\,=\,0$ and ${{K}^{2}}\,=\,5$ as ${{\mathbb{Z}}_{2}}\,\times \,{{\mathbb{Z}}_{2}}$-covers and show that they are genus three hyperelliptic fibrations with bicanonical map of degree two.

We consider semilinear periodic problems with the right-hand side nonlinearity satisfying a double resonance condition between two successive eigenvalues. Using a combination of variational and degree theoretic methods, we prove the existence of at least two nontrivial solutions.

We extend the idea of interval pattern avoidance defined by Yong and the author for ${{S}_{n}}$ to arbitrary Weyl groups using the definition of pattern avoidance due to Billey and Braden, and Billey and Postnikov. We show that, as previously shown by Yong and the author for $\text{G}{{\text{L}}_{n}}$, interval pattern avoidance is a universal tool for characterizing which Schubert varieties have certain local properties, and where these local properties hold.

Multiplicity-free algebraic geometry is the study of subvarieties $Y\,\subseteq \,X$ with the “smallest invariants” as witnessed by a multiplicity-free Chow ring decomposition of $\left[ Y \right]\,\in \,{{A}^{*}}\left( X \right)$ into a predetermined linear basis.

This paper concerns the case of Richardson subvarieties of the Grassmannian in terms of the Schubert basis. We give a nonrecursive combinatorial classification of multiplicity-free Richardson varieties, i.e., we classify multiplicity-free products of Schubert classes. This answers a question of W. Fulton.

In this paper we propose a new technical tool for analyzing representations of Hilbert ${{C}^{*}}$- product systems. Using this tool, we give a new proof that every doubly commuting representation over ${{\mathbb{N}}^{k}}$ has a regular isometric dilation, and we also prove sufficient conditions for the existence of a regular isometric dilation of representations over more general subsemigroups of $\mathbb{R}_{+}^{k}$.

In this paper we give a sufficient condition for a complete, simply connected, and strict nearly Kähler manifold of dimension 6 to be a homogeneous nearly Kähler manifold. This result was announced in a previous paper by the first author.

We prove a local, unipotent, analog of Kedlaya's theorem for the pro-$p$ part of the fundamental group of integral affine schemes in characteristic $p$.

In an $\text{H}$-closed, Urysohn space, disjoint $\text{H}$-sets can be separated by disjoint open sets. This is not true for an arbitrary H-closed space even if one of the $\text{H}$-sets is a point. In this paper, we provide a systematic study of those spaces in which disjoint $\text{H}$-sets can be separated by disjoint open sets.

This paper studies the existence and uniqueness of almost periodic solutions of nonlinear impulsive functional differential equations with infinite delay. The results obtained are based on the Lyapunov–Razumikhin method and on differential inequalities for piecewise continuous functions.

Let $f$ be a classical newform of weight 2 on the upper half-plane ${{H}^{(2)}}$, $E$ the corresponding strong Weil curve, $K$ a class number one imaginary quadratic field, and $F$ the base change of $f$ to $K$. Under a mild hypothesis on the pair $(\,f\,,\,K)$, we prove that the period ratio ${{\Omega }_{E}}/(\sqrt{\left| D \right|}{{\Omega }_{F}})$ is in $\mathbb{Q}$. Here ${{\Omega }_{F}}$ is the unique minimal positive period of $F$, and ${{\Omega }_{E}}$ the area of $E(\mathbb{C})$. The claim is a specialization to base change forms of a conjecture proposed and numerically verified by Cremona and Whitley.

Let $G$ be a graph of order $p$, let $a$, $b$, and $n$ be nonnegative integers with $1\,\le \,a\,<\,b$, and let $g$ and $f$ be two integer-valued functions defined on $V\left( G \right)$ such that $a\,\le \,g\left( x \right)\,<\,f\left( x \right)\,\le \,b$ for all $x\,\in \,V\left( G \right)$. A $\left( g,\,f \right)$-factor of graph $G$ is a spanning subgraph $F$ of $G$ such that $g\left( x \right)\,\le \,{{d}_{F}}\left( x \right)\,\le \,f\left( x \right)$ for each $x\,\in \,V\left( F \right)$. Then a graph $G$ is called $\left( g,\,f,\,n \right)$-critical if after deleting any $n$ vertices of $G$ the remaining graph of $G$ has a $\left( g,\,f \right)$-factor. The binding number $\text{bind}\left( G \right)$ of $G$ is the minimum value of $\left| {{N}_{G}}\left( X \right) \right|/\left| X \right|$ taken over all non-empty subsets $X$ of $V\left( G \right)$ such that ${{N}_{G}}\left( X \right)\,\ne \,V\left( G \right)$. In this paper, it is proved that $G$ is a $\left( g,\,f,\,n \right)$-critical graph if

$$\text{bind}\left( G \right)\,>\,\frac{\left( a\,+\,b\,-\,1 \right)\left( p\,-\,1 \right)}{\left( a\,+\,1 \right)p\,-\,\left( a\,+b \right)\,-\,bn\,+\,2}\,\text{and}\,\text{p}\ge \,\frac{\left( a\,+\,b\,-\,1 \right)\left( a\,+\,b\,-2 \right)}{a\,+\,1}\,+\,\frac{bn}{a}.$$

Furthermore, it is shown that this result is best possible in some sense.