On the remainder of the semialgebraic Stone–Cěch compactification of a semialgebraic set

In this work we analyze some topological properties of the remainder $?M:={\beta }_{\mathrm{s}}^{?}M?M$ of the semialgebraic Stone–Cěch compactification ${\beta }_{\mathrm{s}}^{?}M$ of a semialgebraic set $M?{\mathbb{R}}^m$ in order to ‘distinguish’ its points from those of M. To that end we prove that the set of points of ${\beta }_{\mathrm{s}}^{?}M$ that admit a metrizable neighborhood in ${\beta }_{\mathrm{s}}^{?}M$ equals $M_{\mathrm{lc}}\cup ({\mathrm{Cl}}_{{\beta }_{\mathrm{s}}^{?}M}({\stackrel{￣}{M}}_{\le 1})?{\stackrel{￣}{M}}_{\le 1})$ where $M_{\mathrm{lc}}$ is the largest locally compact dense subset of M and ${\stackrel{￣}{M}}_{\le 1}$ is the closure in M of the set of 1-dimensional points of M. In addition, we analyze the properties of the sets $\stackrel{?}{?}M$ and $\stackrel{?}{?}M$ of free maximal ideals associated with formal and semialgebraic paths. We prove that both are dense subsets of the remainder ?M and that the differences $?M?\stackrel{?}{?}M$ and $\stackrel{?}{?}M?\stackrel{?}{?}M$ are also dense subsets of ?M. It holds moreover that all the points of $\stackrel{?}{?}M$ have countable systems of neighborhoods in ${\beta }_{\mathrm{s}}^{?}M$.     

Minimal unit vector fields with respect to Riemannian natural metrics

Let $(M,g)$ be a Riemannian manifold. We denote by $\stackrel{?}{G}$ an arbitrary Riemannian g-natural metric on the unit tangent sphere bundle $T_{1}M$, such metric depends on four real parameters satisfying some inequalities. The Sasaki metric, the Cheeger–Gromoll metric and the Kaluza–Klein metrics are special Riemannian g-natural metrics. In literature, minimal unit vector fields have been already investigated, considering $T_{1}M$ equipped with the Sasaki metric ${\stackrel{?}{G}}_S$ [12]. In this paper we extend such characterization to an arbitrary Riemannian g-natural metric $\stackrel{?}{G}$. In particular, the minimality condition with respect to the Sasaki metric ${\stackrel{?}{G}}_S$ is invariant under a two-parameters deformation of the Sasaki metric. Moreover, we show that a minimal unit vector field (with respect to $\stackrel{?}{G}$) corresponds to a minimal submanifold. Then, we give examples of minimal unit vector fields (with respect to $\stackrel{?}{G}$). In particular, we get that the Hopf vector fields of the unit sphere, the Reeb vector field of a K-contact manifold, and the Hopf vector field of a quasi-umbilical hypersurface with constant principal curvatures in a Kähler manifold, are minimal unit vector fields (with respect to $\stackrel{?}{G}$).     

Global well-posedness and decay estimates of strong solutions to a two-phase model with magnetic field

In this paper, we consider the Cauchy problem for a two-phase model with magnetic field in three dimensions. The global existence and uniqueness of strong solution as well as the time decay estimates in $H^2({\mathbb{R}}^3)$ are obtained by introducing a new linearized system with respect to $(n^{\gamma }?{\stackrel{?}{n}}^{\gamma },n?\stackrel{?}{n},P?\stackrel{?}{P},u,H)$ for constants $\stackrel{?}{n}\ge 0$ and $\stackrel{?}{P}>0$, and doing some new a priori estimates in Sobolev Spaces to get the uniform upper bound of $(n?\stackrel{?}{n},n^{\gamma }?{\stackrel{?}{n}}^{\gamma })$ in $H^2({\mathbb{R}}^3)$ norm.     

On Fano and weak Fano Bott–Samelson–Demazure–Hansen varieties

Let G be a simple algebraic group over the field of complex numbers. Fix a maximal torus T and a Borel subgroup B of G containing T. Let w be an element of the Weyl group W of G, and let $Z(\stackrel{?}{w})$ be the Bott–Samelson–Demazure–Hansen (BSDH) variety corresponding to a reduced expression $\stackrel{?}{w}$ of w with respect to the data $(G,B,T)$.In this article we give complete characterization of the expressions $\stackrel{?}{w}$ such that the corresponding BSDH variety $Z(\stackrel{?}{w})$ is Fano or weak Fano. As a consequence we prove vanishing theorems of the cohomology of tangent bundle of certain BSDH varieties and hence we get some local rigidity results.     

Bipartite algebraic graphs without quadrilaterals

Let ${\mathbb{P}}^s$ be the $s$-dimensional complex projective space, and let $X,Y$ be two non-empty open subsets of ${\mathbb{P}}^s$ in the Zariski topology. A hypersurface $H$ in ${\mathbb{P}}^s\times {\mathbb{P}}^s$ induces a bipartite graph $G$ as follows: the partite sets of $G$ are $X$ and $Y$, and the edge set is defined by $\overline{u}～\overline{v}$ if and only if $(\overline{u},\overline{v})\in H$. Motivated by the Turán problem for bipartite graphs, we say that $H\cap (X\times Y)$ is $(s,t)$-grid-free provided that $G$ contains no complete bipartite subgraph that has $s$ vertices in $X$ and $t$ vertices in $Y$. We conjecture that every $(s,t)$-grid-free hypersurface is equivalent, in a suitable sense, to a hypersurface whose degree in $\overline{y}$ is bounded by a constant $d=d(s,t)$, and we discuss possible notions of the equivalence.We establish the result that if $H\cap (X\times {\mathbb{P}}^2)$ is $(2,2)$-grid-free, then there exists $F\in \mathbb{C}[\overline{x},\overline{y}]$ of degree $\le 2$ in $\overline{y}$ such that $H\cap (X\times {\mathbb{P}}^2)=\{F=0\}\cap (X\times {\mathbb{P}}^2)$. Finally, we transfer the result to algebraically closed fields of large characteristic.     

Online sum-paintability: The slow-coloring game

The slow-coloring game is played by Lister and Painter on a graph $G$. On each round, Lister marks a nonempty subset $M$ of the uncolored vertices, scoring $\left|M\right|$ points. Painter then gives a color to a subset of $M$ that is independent in $G$. The game ends when all vertices are colored. Painter and Lister want to minimize and maximize the total score, respectively. The best score that each player can guarantee is the sum-color cost of $G$, written $\stackrel{?}{s}\left(G\right)$. The game is an online variant of online sum list coloring.We prove $\frac{\left|V\left(G\right)\right|}{2\alpha \left(G\right)}+\frac{1}{2}\le \frac{\stackrel{?}{s}\left(G\right)}{\left|V\left(G\right)\right|}\le max\left\{\frac{\left|V\left(H\right)\right|}{\alpha \left(H\right)}:H?G\right\}$, where $\alpha \left(G\right)$ is the independence number, and we study when equality holds in the bounds. We compute $\stackrel{?}{s}\left(G\right)$ for graphs with $\alpha \left(G\right)=2$. Among $n$-vertex trees, we prove that $\stackrel{?}{s}$ is minimized by the star and maximized by the path. We also study $\stackrel{?}{s}\left(K_{r,s}\right)$.     

Two dimensional Mellin transform in Quantum Calculus

In this article, we introduce the two dimensional Mellin transform M_(f)(s, t),give some properties, establish the Paley-Wiener theorem and Plancherel formula, present the Hausdorff-Young inequality, and find several applications for the two dimensional Mellin transform.     

On a class of damped vibration problems with obstacles

The main purpose of this paper is to study the following damped vibration problem (1.1)$?\stackrel{?}{x}=g\left(t\right)\stackrel{?}{x}+f(t,x)$ satisfying (1.2)$x\left(0\right)?x\left(2\pi \right)=\stackrel{?}{x}\left(0\right)?\stackrel{?}{x}\left(2\pi \right)=0\text{,}x\left(t\right)\ge 0\text{,}?t\in R$(1.3)$\stackrel{?}{x}\left(t_0^{?}\right)=?\stackrel{?}{x}\left(t_0^{+}\right)\text{,}\text{if}x\left(t_0\right)=0\text{.}$ The variational principles are given and some existence and multiplicity results of nonzero periodic solutions satisfying (1.1), (1.2), (1.3) are obtained.