It is shown that the homogeneous ergodic bilinear averages with Möbius or Liouville weight converge almost surely to zero; that is, if $T$ is a map acting on a probability space $(X,\mathcal{A},\nu )$, and $a,b\in \mathbb{Z}$, then for any $f,g\in L^2\left(X\right)$, for almost all $x\in X$,

$$\frac{1}{N}{\sum }_{n=1}^N\mathit{\nu }\left(n\right)f\left(T^{an}x\right)g\left(T^{bn}x\right)\underset{N\to +\infty }{\to }0,$$ where $\mathit{\nu }$ is the Liouville function or the Möbius function. We further obtain that the convergence almost everywhere holds for the short interval with the help of Zhan’s estimation. Moreover, we establish that if $T$ is weakly mixing and its restriction to its Pinsker algebra has singular spectrum, then for any integer $k\ge 1$, for any $f_j\in L^{\infty }\left(X\right)$, $j=1,\dots ,k$, for almost all $x\in X$, we have

$$\frac{1}{N}{\sum }_{n=1}^N\mathit{\nu }\left(n\right){\prod }_{j=1}^k{f}_j\left(T^{nj}x\right)\underset{N\to +\infty }{\to }0.$$

We consider the Cauchy problem for the nonlinear Schrödinger (NLS) equation with double nonlinearities with opposite sign, with one term mass-critical and the other term mass-supercritical and energy-subcritical, which includes the well-known two-dimensional cubic-quintic NLS equation arising in the study of the boson gas with 2- and 3-body interactions. We prove global well-posedness and scattering in $H^1\left({\mathbb{R}}^d\right)$ below the threshold for nonradial data when $1\le d\le 4$.

Let $G$ be a finite group and $\mathrm{cd}\left(G\right)$ be the set of all irreducible complex character degrees of $G$ without multiplicities. The aim of this paper is to propose an extension of Huppert’s conjecture from non-Abelian simple groups to almost simple groups of Lie type. Indeed, we conjecture that if $H$ is an almost simple group of Lie type with $\mathrm{cd}\left(G\right)=\mathrm{cd}\left(H\right)$, then there exists an Abelian normal subgroup $A$ of $G$ such that $G/A\cong H$. It is furthermore shown that $G$ is not necessarily the direct product of $H$ and $A$. In view of Huppert’s conjecture, we also show that the converse implication does not necessarily hold for almost simple groups. Finally, in support of this conjecture, we will confirm it for projective general linear and unitary groups of dimension 3.

Darboux transformations are nongroup-type symmetries of linear differential operators. One can define Darboux transformations algebraically by the intertwining relation $ML=L_{1}M$ or the intertwining relation $ML=L_{1}N$ in the cases when the former is too restrictive.

Here we show that Darboux transformations for operators of the form $L={\partial }_x{\partial }_y+a{\partial }_x+b{\partial }_y+c$ (sometimes referred to as 2D Schrödinger operators or Laplace operators) are always compositions of atomic Darboux transformations of two different well-studied types of Darboux transformations, provided that the chain of Laplace transformations for the original operator is long enough.

We prove a nonexistence theorem for product-type manifolds. In particular, we show that the 4-manifold ${\Sigma }_g\times {\Sigma }_h$ obtained from the product of closed surfaces does not admit any locally conformally flat metric arising from discrete and faithful representations for genus $g\ge 2$ and $h\ge 1$.

Given a discrete subgroup $\Gamma$ of finite co-volume of $\mathbf{PGL}(2,\mathbb{R})$, we define and study parabolic vector bundles on the quotient $\Sigma$ of the (extended) hyperbolic plane by $\Gamma$. If $\Gamma$ contains an orientation-reversing isometry, then the above is equivalent to studying real and quaternionic parabolic vector bundles on the orientation cover of degree two of $\Sigma$. We then prove that isomorphism classes of polystable real and quaternionic parabolic vector bundles are in a natural bijective correspondence with the equivalence classes of real and quaternionic unitary representations of $\Gamma$. Similar results are obtained for compact-type real parabolic vector bundles over Klein surfaces.

We define three function spaces related to a Schrödinger form and its semigroup: two are spaces of excessive functions defined through the Schrödinger semigroup, and one is the space of weak subsolutions defined through the Schrödinger form. We define the maximum principle for each space and prove the equivalence of three maximum principles. Moreover, we give a necessary and sufficient condition for each maximum principle in terms of the principal eigenvalue of time-changed processes.