The present paper mainly gives some applications of Berezin type symbols on the Dirichlet space of unit ball. We study the solvability of some Riccati operator equations of the form XAX+ XB−CX= Drelated to harmonic Toeplitz operators on the Dirichlet space. Especially, the invariant subspaces of Toeplitz operators are also considered.

Let n≥3.The complex Lie algebra, which is attached to a unit form $q({x}_{\mathrm{1}},{x}_{\mathrm{2}},\dots ,{x}_{n})={\displaystyle {\sum}_{i=\mathrm{1}}^{n}{x}_{i}^{\mathrm{2}}}+{\displaystyle {\sum}_{\mathrm{1}\le i\le j\le n}{(-\mathrm{1})}^{j-i}}{x}_{i}{x}_{j}$ and defined by generators and generalized Serre relations, is proved to be a finite-dimensional simple Lie algebra of type ${\mathbb{A}}_{n}$,and realized by the Ringel-Hall Lie algebra of a Nakayama algebra of radical square zero. As its application of the realization, we give the roots and a Chevalley basis of the simple Lie algebra.

We study the asymptotic behavior for log-determinants of two unitary but non-Hermitian random matrices: the spherical ensembles A^{−1}B, where A and B are independent complex Ginibre ensembles and the truncation of circular unitary ensembles. The corresponding Berry-Esseen bounds and Cramér type moderate deviations are established. Our method is based on the estimates of corresponding cumulants. Numerical simulations are also presented to illustrate the theoretical results.

Operator self-similar processes, as an extension of self-similar processes, have been studied extensively. In this work, we study limit theorems for functionals of Gaussian vectors. Under some conditions, we determine that the limit of partial sums of functionals of a stationary Gaussian sequence of random vectors is an operator self-similar process.

We generalize the discrete Yamabe flow to αorder. This Yamabe flow deforms the α-order curvature to a constant. Using this new flow, we manage to find discrete α-quasi-Einstein metrics on the triangulations of ${\mathbb{S}}^{3}$.

The Brownian rough path is the canonical lifting of Brownian motion to the free nilpotent Lie group of order 2. Equivalently, it is a process taking values in the algebra of Lie polynomials of degree 2, which is described explicitly by the Brownian motion coupled with its area process. The aim of this article is to compute the finite dimensional characteristic functions of the Brownian rough path in ${?}^{d}$ and obtain an explicit formula for the case when d = 2.

We calculate the sharp bounds for some q-analysis variants of Hausdorff type inequalities of the form$$\int \begin{array}{l}+\infty \\ \mathrm{0}\end{array}}{\left({\displaystyle \int \begin{array}{l}+\infty \\ \mathrm{0}\end{array}}{\displaystyle \frac{\varphi (t)}{t}}f{\displaystyle \frac{x}{t}}{{\displaystyle d}}_{q}t\right)}^{p}{{\displaystyle d}}_{q}x\le {{\displaystyle C}}_{\varphi}{\displaystyle \int \begin{array}{l}b\\ \mathrm{0}\end{array}}{f}^{p}(t){{\displaystyle d}}_{q}t.$$As applications, we obtain several sharp q-analysis inequalities of the classical positive integral operators, including the Hardy operator and its adjoint operator, the Hilbert operator, and the Hardy-Littlewood-Pólya operator.

Let W be the Weyl group of type F_{4}: We explicitly describe a nite set of basic braid I_{*}-transformations and show that any two reduced I_{*}-expressions for a given involution in W can be transformed into each other through a series of basic braid I_{*}-transformations. Our main result extends the earlier work on the Weyl groups of classical types (i.e., A_{n}; B_{n}; and D_{n}).

We study the stationary Wigner equation on a bounded, onedimensional spatial domain with inflow boundary conditions by using the parity decomposition of L. Barletti and P. F. Zweifel [Transport Theory Statist. Phys., 2001, 30(4-6): 507–520]. The decomposition reduces the half-range, two-point boundary value problem into two decoupled initial value problems of the even part and the odd part. Without using a cutoff approximation around zero velocity, we prove that the initial value problem for the even part is well-posed. For the odd part, we prove the uniqueness of the solution in the odd L^{2}-space by analyzing the moment system. An example is provided to show that how to use the analysis to obtain the solution of the stationary Wigner equation with inflow boundary conditions.

A total-colored path is total rainbow if its edges and internal vertices have distinct colors. A total-colored graph G is total rainbow connected if any two distinct vertices are connected by some total rainbow path. The total rainbow connection number of G, denoted by trc(G), is the smallest number of colors required to color the edges and vertices of G in order to make G total rainbow connected. In this paper, we investigate graphs with small total rainbow connection number. First, for a connected graph G, we prove that $\mathrm{trc}(G)=\mathrm{3}?\mathrm{if}\left(\begin{array}{l}n-\mathrm{1}\\ \mathrm{2}\end{array}\right)+\mathrm{1}\le \left|E(G)\right|\le \left(\begin{array}{l}n\\ \mathrm{2}\end{array}\right)-\mathrm{1},$ and $\mathrm{trc}(G)=\mathrm{6}?\mathrm{if}\left(\begin{array}{l}n-\mathrm{2}\\ \mathrm{2}\end{array}\right)+\mathrm{2}\le .$ Next, we investigate the total rainbow connection numbers of graphs G with $\left|V(G)\right|=n,$ diam$(G)\ge \mathrm{2},$ and clique number $\omega (G)=n-s?\mathrm{for}?\mathrm{1}?\le s\le ?\mathrm{3}$. In this paper, we find Theorem 3 of [Discuss. Math. Graph Theory, 2011, 31(2): 313–320] is not completely correct, and we provide a complete result for this theorem.

A k-total coloring of a graph G is a mapping φ: V (G) ∪ E(G) →{1, 2, . . . , k} such that no two adjacent or incident elements in V (G) ∪ E(G) receive the same color. Let f(v) denote the sum of the color on the vertex v and the colors on all edges incident with v. We say that φ is a k-neighbor sum distinguishing total coloring of G if f(u) ≠ f(v) for each edge uv ∈ E(G). Denote ${{\displaystyle X}}_{\mathrm{\Sigma}}^{\text{'}\text{'}}(G)$ the smallest value k in such a coloring of G. Pilśniak andWoźniak conjectured that for any simple graph with maximum degree Δ(G), ${{\displaystyle X}}_{\Sigma}^{\text{'}\text{'}}(G)\le \mathrm{\Delta}(G)+\mathrm{3}$. In this paper, by using the famous Combinatorial Nullstellensatz, we prove that for K_{4}-minor free graph G with Δ(G)≥5, ${{\displaystyle X}}_{\mathrm{\Sigma}}^{\text{'}\text{'}}(G)=\mathrm{\Delta}(G)+\mathrm{1}$ if G contains no two adjacent Δ-vertices, otherwise, ${{\displaystyle X}}_{\mathrm{\Sigma}}^{\text{'}\text{'}}(G)=\mathrm{\Delta}(G)+\mathrm{2}$.

We investigate Lie bialgebra structures on the derivation Lie algebra over the quantum torus. It is proved that, for the derivation Lie algebra W over a rank 2 quantum torus, all Lie bialgebra structures on W are the coboundary triangular Lie bialgebras. As a by-product, it is also proved that the first cohomology group H^{1}(W,W ⊗W) is trivial.

We consider the oscillatory integral operator ${{\displaystyle T}}_{\alpha ,m}f(x)={\int}_{{?}^{n}}{\mathrm{e}}^{\mathrm{i}({{\displaystyle x}}_{\mathrm{1}}^{{\alpha}_{1}}{{\displaystyle y}}_{\mathrm{1}}^{m}+?+{{\displaystyle x}}_{n}^{{\alpha}_{n}}{{\displaystyle y}}_{n}^{m})}f(y)\mathrm{d}y$, where the function f is a Schwartz function. In this paper, the restriction theorem on ${S}^{n-\mathrm{1}}$ for this operator is obtained. Moreover, we obtain a necessary condition which ensures validity of the restriction theorem.

For a cubic algebraic extension K of ℚ, the behavior of the ideal counting function is considered in this paper. More precisely, let a_{K}(n) be the number of integral ideals of the field K with norm n, we prove an asymptotic formula for the sum ${\sum}_{{{\displaystyle n}}_{\mathrm{1}}^{\mathrm{2}}+{{\displaystyle n}}_{\mathrm{2}}^{\mathrm{2}}\le x}}{{\displaystyle a}}_{K}({{\displaystyle n}}_{\mathrm{1}}^{\mathrm{2}}+{{\displaystyle n}}_{\mathrm{2}}^{\mathrm{2}}).$

Anisotropy is a common attribute of the nature, which shows different characterizations in different directions of all or part of the physical or chemical properties of an object. The anisotropic property, in mathematics, can be expressed by a fairly general discrete group of dilations {A^{k} : k ∈ Z}, where A is a real n × n matrix with all its eigenvalues λ satisfy |λ|>1. The aim of this article is to study a general class of anisotropic function spaces, some properties and applications of these spaces. Let ϕ: R^{n}×[0,∞) →[0,∞) be an anisotropic p-growth function with p ∈ (0, 1]. The purpose of this article is to find an appropriate general space which includes weak Hardy space of Fefferman and Soria, weighted weak Hardy space of Quek and Yang, and anisotropic weak Hardy space of Ding and Lan. For this reason, we introduce the anisotropic weak Hardy space of Musielak-Orlicz type ${{\displaystyle H}}_{A}^{\phi ,\infty}({?}^{n})$ and obtain its atomic characterization. As applications, we further obtain an interpolation theorem adapted to ${{\displaystyle H}}_{A}^{\phi ,\infty}({?}^{n})$ and the boundedness of the anisotropic Calderón-Zygmund operator from ${{\displaystyle H}}_{A}^{\phi ,\infty}({?}^{n})$ to ${L}^{\phi ,\infty}({?}^{n})$. It is worth mentioning that the superposition principle adapted to the weak Musielak-Orlicz function space, which is an extension of a result of E. M. Stein, M. Taibleson and G. Weiss, plays an important role in the proofs of the atomic decomposition of ${{\displaystyle H}}_{A}^{\phi ,\infty}({?}^{n})$ and the interpolation theorem.