International Journal of Mathematics

For a measure $\mu$ on ${ℝ}^n$ with density that has a positive degree of homogeneous and a positive degree of concavity, we study the Orlicz–Minkowski problem for such general measure $\mu$ and show the existence of the even Orlicz–Minkowski problem for $\mu$. We also introduce the general Orlicz geominimal $\mu$-measures and show the existence of the Orlicz-Petty bodies for the general measure $\mu$, and hence prove the continuity of the general Orlicz geominimal $\mu$-measure.

We present new families of weighted homogeneous and Newton nondegenerate line singularities that satisfy the Zariski multiplicity conjecture.

The classification of irreducible weak modules over the Virasoro vertex operator algebra $V(c,0)$ is obtained in this paper. As one of the main results, we also classify all irreducible weak modules over the simple Virasoro vertex operator algebras $L(c^{\prime },0)$ for $c^{\prime }\ne c_{p,q}$ $(=1-\frac{6{(p-q)}^2}{pq})$, where $p,q\ge 2$ are coprime positive integers.

We show that for any two $n\times n$ matrices $X$ and $Y$ we have the inequality $s_j^2(I+XY)\le {\lambda }_j((I+X^{\ast }X)(I+Y^{\ast }Y)),$ where $s_j(T)$ and ${\lambda }_j(T)$ denote the decreasingly ordered singular values and eigenvalues of $T$. As an application, we show that for $2n\times 2n$ real positive definite matrices the symplectic eigenvalues $d_j,$ under some special conditions, satisfy the inequality $d_j(A+B)\ge d_j(A)+d_1(B)$.

A class of homomorphisms between generalized Verma modules that have an unusual degeneracy is identified. Homomorphisms in this class are called deficient homomorphisms. A family of maximally deficient homomorphisms is constructed. A necessary condition on a parabolic subalgebra is identified for the associated category of generalized Verma modules to admit deficient homomorphisms.

We determine the group structure of the homotopy set whose target is the automorphism group of the Cuntz algebra ${{𝒪}}_{n+1}$ for finite $n$ in terms of K-theory. We show that there is an example of a space for which the homotopy set is a noncommutative group, and hence, the classifying space of the automorphism group of the Cuntz algebra for finite $n$ is not an H-space. We also make an improvement of Dadarlat’s classification of continuous fields of the Cuntz algebras in terms of vector bundles.

In this paper, we study weighted $p$-harmonic forms on smooth metric measure space $(M,g,e^{-f}dv)$ with a weighted Sobolev or a weighted Poincaré inequality. When $f$ is constant, we derive a splitting theorem for Kähler manifolds with maximal bottom spectrum for the $p$-Laplacian. For general $f,$ we also obtain various splitting and vanishing theorems when the weighted curvature operator of $M$ is bounded below. As applications, we conclude Liouville property for weighted $p$-harmonic functions and $p$-harmonic maps.

Given a holomorphic conic bundle without sections, we show that the orders of finite groups acting by its fiberwise bimeromorphic transformations are bounded. This provides an analog of a similar result obtained by Bandman and Zarhin for quasi-projective conic bundles.

We show that two approaches to equivariant deformation of C${}^{\ast }$-algebras by actions of negatively curved Kählerian Lie groups, one based on oscillatory integrals and the other on quantizations maps defined by dual $2$-cocycles, are still equivalent despite the nonunitarity of our $2$-cocycles.