In this paper, by investigating the monotonicity of a function composed of $% \left( \sinh x\right) /x$ and $\cosh x$ with two parameters in $x$ on $% \left( 0,\infty \right) $, we prove serval theorems related to inequalities for hyperbolic functions, which generalize known results and establish some new and sharp inequalities. As applications, some new and sharp inequalities for bivariate means are presented.

Zhen-Hang Yang

We define a knot/link invariant using set theoretical solutions $(X,\sigma)$ of the Yang-Baxter equation and non commutative 2-cocycles. We also define, for a given $(X,\sigma)$, a universal group Unc(X) governing all 2-cocycles in $X$, and we exhibit examples of computations.

Marco A. Farinati Juliana García Galofre

Suppose that $X$ is a projective manifold whose tangent bundle $T_X$ contains a locally free strictly nef subsheaf. We prove that $X$ is isomorphic to a projective bundle over a hyperbolic manifold. Moreover, if the fundamental group $\pi_1(X)$ is virtually abelian, then $X$ is isomorphic to a projective space.

Jie Liu Wenhao Ou Xiaokui Yang

Given a homogeneous space $X = G/\Gamma$ with $G$ containing the group $H = (\mathrm{SO}(n,1))^k$. Let $x\in X$ such that $Hx$ is dense in $X$. Given an analytic curve $\phi: I=[a,b] \rightarrow H$, we will show that if $\phi$ satisfies certain geometric condition, then for a typical diagonal subgroup $A =\{a(t): t \in \mathbb{R}\} \subset H$ the translates $\{a(t)\phi(I)x: t >0\}$ of the curve $\phi(I)x$ will tend to be equidistributed in $X$ as $t \rightarrow +\infty$. The proof is based on the study of linear representations of $\mathrm{SO}(n,1)$ and $H$.

Lei Yang

Consider the Fano manifold $X$ formed by blowing up $\mathbb{P}^n$ along its linear subspace $\mathbb{P}^r$, we check the conifold conditions [3, 1] for its mirror Laurent polynomial $f$, which can imply that $X$ satisfies both Conjecture $\mathcal{O}$ and Gamma conjecture I by Galkin-Golyshev-Iritani [2].

Zongrui Yang

In this paper, we construct large subalgebras of crossed product C*-algebras of noncommutative C*-dynamics from ideals. We apply our results to study locally trivial unital $C(X)$-algebras such as mapping tori.

Xiaochun Fang N. C. Phllips Junqi Yang

We study stable rationality of conic bundles $X$ over $\mathbb{P}^1$ defined over non-closed field $k$ via the cohomology of the Galois group of finite field extension $k'/k$ with action on the geometric Picard lattice of $X$.

Kaiqi Yang

In this article, we construct complete Calabi-Yau metrics on abelian fibrations $X$ over $\mathbb{C}$. We also provide compactification for $X$ so that the compactified variety has negative canonical bundle.

Ruiming Liang Yang Zhang

A bijective map $r: X^2 \longrightarrow X^2$, where $X = \{x_1, ..., x_n \}$ is a finite set, is called a \emph{set-theoretic solution of the Yang-Baxter equation} (YBE) if the braid relation $r_{12}r_{23}r_{12} = r_{23}r_{12}r_{23}$ holds in $X^3.$ A non-degenerate involutive solution $(X,r)$ satisfying $r(xx)=xx$, for all $x \in X$, is called \emph{square-free solution}. There exist close relations between the square-free set-theoretic solutions of YBE, the semigroups of I-type, the semigroups of skew polynomial type, and the Bieberbach groups, as it was first shown in a joint paper with Michel Van den Bergh. In this paper we continue the study of square-free solutions $(X,r)$ and the associated Yang-Baxter algebraic structures -- the semigroup $S(X,r)$, the group $G(X,r)$ and the $k$- algebra $A(k, X,r)$ over a field $k$, generated by $X$ and with quadratic defining relations naturally arising and uniquely determined by $r$. We study the properties of the associated Yang-Baxter structures and prove a conjecture of the present author that the three notions: a square-free solution of (set-theoretic) YBE, a semigroup of I type, and a semigroup of skew-polynomial type, are equivalent. This implies that the Yang-Baxter algebra $A(k, X,r)$ is Poincar\'{e}-Birkhoff-Witt type algebra, with respect to some appropriate ordering of $X$. We conjecture that every square-free solution of YBE is retractable, in the sense of Etingof-Schedler.

Tatiana Gateva-Ivanova

In this paper, we study a coupled Hartree-type system given by \[ \left\{ \begin{array}{ll} -\Delta u = K_{1}(x)(|x|^{-(N-\alpha)} * K_{1}(x)v^{2^{*}_{\alpha}})v^{2^{*}_{\alpha}-1} & \text{in } \mathbb{R}^N, \\[1mm] -\Delta v = K_{2}(x)(|x|^{-(N-\alpha)} * K_{2}(x)u^{2^{*}_{\alpha}})u^{2^{*}_{\alpha}-1} & \text{in } \mathbb{R}^N, \end{array} \right. \] which is critical with respect to the Hardy-Littlewood-Sobolev inequality. Here, $N \geq 5$, $\alpha < N - 5 + \frac{6}{N-2}$, $2^{*}_{\alpha} = \frac{N + \alpha}{N - 2}$, and $(x', x'') \in \mathbb{R}^2 \times \mathbb{R}^{N-2}$. The functions $K_{1}(|x'|, x'')$ and $K_{2}(|x'|, x'')$ are bounded, nonnegative functions on $\mathbb{R}^{+} \times \mathbb{R}^{N-2}$, sharing a common, topologically nontrivial critical point. We address the challenge of establishing the nondegeneracy of positive solutions to the limiting system. By employing a finite-dimensional reduction technique and developing new local Poho\v{z}aev identities, we construct infinitely many synchronized-type solutions, with energies that can be made arbitrarily large.

Weiwei Ye Qing Guo Minbo Yang Xinyun Zhang