We show that every possible metric associated with critical (\gamma =2) Liouville quantum gravity (LQG) induces the same topology on the plane as the Euclidean metric. More precisely, we show that the optimal modulus of continuity of the critical LQG metric with respect to the Euclidean metric is a power of 1/\log (1/|·|). Our result applies to every possible subsequential limit of critical Liouville first passage percolation, a natural approximation scheme for the LQG metric which was recently shown to be tight.

We study the asymptotic stability of equilibrium states with positive (and variable) temperature gradient to the Boussinesq system without thermal conduction in the strip domain {ℝ}^2\times (0,1). It is shown that a unique global-in-time solution exists if the initial data is close enough to such an equilibrium state with suitable boundary conditions. Moreover, as time goes to infinity, the solution converges to the corresponding equilibrium state with explicit decay rates. Such a result reflects the well-known Rayleigh–Taylor stability phenomenon in the fluid motion.

A Lie algebra \mathfrak{g} is considered generalized reductive if it is a direct sum of a semisimple Lie algebra and a commutative radical. This paper extends the BGG category \mathcal{O} over complex semisimple Lie algebras to the category {\mathcal{O}}^{\prime } over complex generalized reductive Lie algebras. Then, we preliminarily research the highest weight modules and the projective modules in this new category {\mathcal{O}}^{\prime }, and generalize some conclusions for the classical case. Also, we investigate the associated varieties with respect to the irreducible modules in {\mathcal{O}}^{\prime } and obtain a result that extends Joseph’s result on the associated varieties for reductive Lie algebras. Finally, we study the center of the universal enveloping algebra U(\mathfrak{g}) and independently provide a new proof of a theorem by Ou–Shu–Yao for the center in the case of enhanced reductive Lie algebras.

In this paper we firstly prove that the CD_p curvature condition always satisfies for p\ge 2 on any connected locally finite graph. We show this property does not hold for . We also derive a lower bound for the first nonzero eigenvalue of the p-Laplace operator on a connected finite graph with the CD_p(m, K) condition for the case that and K > 0.

The purpose of this paper is to study the boundedness for a large class of multi-sublinear operators T_α,m generated by multilinear fractional integral operator and their commutators T_{\alpha ,m,i}^b(i=1,\dots ,m) on product generalized mixed Morrey spaces M_{{\overrightarrow{q}}_1}^{{\varphi }_1}\left({ℝ}^n\right)\times ...\times M_{q_m\to }^{{\varphi }_m}\left({ℝ}^n\right). We find sufficient conditions on (ϕ₁, ..., ϕ_m, ϕ) which ensure the boundedness of T_α,m from M_{\overrightarrow{q_1}}^{{\varphi }_1}\left({ℝ}^n\right)\times ...\times M_{q_m\to }^{{\varphi }_m}\left({ℝ}^n\right)\text{to}{M}_{\overrightarrow{p}}^{\varphi }\left({ℝ}^n\right). Moreover, we also give sufficient conditions for the boundedness of T_{\alpha ,m,i}^b from M_{\overrightarrow{q_1}}^{{\varphi }_1}\left({ℝ}^n\right)\times ...\times M_{q_m\to }^{{\varphi }_m}\left({ℝ}^n\right)\text{to}{M}_{\overrightarrow{p}}^{\varphi }\left({ℝ}^n\right). As applications, the boundedness for multi-sublinear fractional maximal operator, multilinear fractional integral operator and their commutators on product generalized mixed Morrey spaces is established.

In this article, we mainly study the critical points of solutions to the Laplace equation with Dirichlet boundary conditions in an exterior do-main in ℝ². Based on the fine analysis about the structures of connected components of the super-level sets \{x\in {ℝ}^2\backslash \mathrm{\Omega }:u(x)>t\} and sub-level sets for some t, we get the geometric distributions of interior critical point sets of solutions. Exactly, when Ω is a smooth bounded simply connected domain, {u|}_{\mathrm{\partial }\mathrm{\Omega }}=\psi (x), {lim}_{|x|\to \mathrm{\infty }}u(x)=-\mathrm{\infty } and \psi (x) has K local maximal points on ∂Ω, we deduce that {\sum }_{i=1}^l{m}_i\le K, where m₁, ..., m_l are the multiplicities of interior critical points x₁, ..., x_l of solution u respectively. In addition, when \psi (x) has only K global maximal points and K equal local minima relative to {ℝ}^2\backslash \mathrm{\Omega } on ∂Ω, we have that {\sum }_{i=1}^l{m}_i=K. Moreover, when Ω is a domain consisting of l disjoint smooth bounded simply connected domains, we deduce that {\sum }_{x_i\in \mathrm{\Omega }}{m}_i+\frac{1}{2}{\sum }_{x_j\in \mathrm{\partial }\mathrm{\Omega }}{m}_j=l-1, and the critical points are contained in the convex hull of the l simply connected domains.

In this paper, all Lie bialgebra structures on the derivation Lie algebra W over a rank d\ge 3 quantum torus associated to q are considered, where q is a d\times d matrix with all the entries being roots of unity. They are shown to be triangular coboundary. As a byproduct, it is also proved that the first cohomology group H^1(W,W \otimes W) is trivial.

In this paper, we study the finite dimensional modules over indefinite Kac–Moody Lie algebras. We prove that any Kac–Moody Lie algebra with indecomposable indefinite Cartan matrix has no non-trivial finite dimensional simple module. This result would be indispensable for researching finite dimensional modules over GIM Lie algebras.

In this paper, we introduce the Bowen polynomial entropy and study the multifractal spectrum of the local polynomial entropies for arbitrary Borel probability measures.

Both Adams spectral sequence and Adams–Novikov spectral sequence converge to the stable homotopy groups of sphere π_*(S). Suppose an element x in the E₂-term of the Adams–Novikov spectral sequence converges to a homotopy element in π_*(S). In this paper we determine that the algebraic representative \tilde{x} in the E₂-term of the Adams spectral sequence converges to the same homotopy element under the conditions related to the Novikov weight and homological dimension.