In this paper, firstly, we use the bosonic oscillators to construct a two-para-meter deformed Virasoro algebra, which is a non-multiplicative Hom-Lie algebra. Sec-ondly, a non-trivial Hopf structure related to the two-parameter deformed Virasoro algebra is presented, that is, we construct a new two-parameter quantum group.

In this paper, we study the group extension of a Tambara-Yamagami cate-gory which has Frobenius-Perron dimension 2pq, where p,q are prime numbers. We prove that there are two possible category types when p≠q, and five possible category types when p = q.

In this paper, we first give the concept of B-statistical uniform integrability with respect to Cesaro matrix (B-CUI), which is weaker than Cesaro uniform integra-bility. Then we establish some B-statistical convergence theorems for random sequence under the condition of B-CUI, which generalizes the outcomes of some known results. Finally, for some special kinds of triangular array or pairwise independent sequences of random variables, similar results are also derived without conditions of B-CUI.

In this paper, we develop an efficient meshless technique for solving numer-ical solutions of the three-dimensional time-fractional extended Fisher-Kolmogorov (TF-EFK) equation. Firstly, the L2-1_σ formula on a general mesh is used to discretize the Caputo fractional derivative, and then a weighted average technique at two neigh-boring time levels is adopted to implement the time discretization of the TF-EFK equa-tion. After applying this time discretization, the generalized finite difference method (GFDM) is introduced for the space discretization to solve the fourth-order nonlinear algebra system generated from the TF-EFK equation with an arbitrary domain. Nu-merical examples are investigated to validate the performance of the proposed mesh-less GFDM in solving the TF-EFK equation in high dimensions with complex domains.

In this paper, we study some basic analytic properties of a sequence of func-tions $\left\{S_{n}^{\mu, \sigma}\right\}$ that is directly derived in an adaptive algorithm originating from the clas-sical score-based secretary problem. More specifically, we show that: 1. the uniqueness of maximum points of the function sequence $\left\{S_{n}^{\mu, \sigma}\right\}$; 2. the maximum point sequence of $\left\{S_{n}^{\mu, \sigma}\right\}$ monotone increases to infinity as n tends to infinity. All of the proofs are elementary but nontrivial.

Modeling and predicting mortality rates are crucial for managing and mit- igating longevity risk in pension funds. To address the impacts of extreme mortality events in forecasting, researchers suggest directly fitting a heavy-tailed distribution to the residuals in modeling mortality indexes. Since the true mortality indexes are unobserved, this fitting relies on the estimated mortality indexes containing measure- ment errors, leading to estimation biases in standard inferences within the actuarial literature. In this paper, the empirical characteristic function (ECF) technique is em- ployed to fit heavy-tailed distributions to the time series residuals of mortality indexes and normal distributions to the measurement errors. Through a simulation study, we empirically validate the consistency of our proposed method and demonstrate the im- portance and challenges associated with making inferences in the presence of mea- surement errors. Upon analyzing publicly available mortality datasets, we observe instances where mortality indexes may follow highly heavy-tailed distributions, even exhibiting an infinite mean. This complexity adds a layer of difficulty to the statistical inference for mortality models.

Let E be a self-similar set satisfying the open set condition. Zhou and Feng posed an open problem in 2004 as follows: x∈E, under what conditions is there a set U_x containing x with $\left|U_{x}\right|>0$ such that $\bar{D}_{C}^{s}(E, x)=\frac{\mathcal{H}^{s}\left(E \cap U_{x}\right)}{\left|U_{x}\right|^{s}}$? The aim of this paper is to present a solution of this problem. Under the assumption that there exists a nonempty convex open set containing E and satisfying the requirement of the open set condition, it is proved that if x ∈ E and the upper convex density of E at x equals 1, then there exists a convex set U_x containing x with $\left|U_{x}\right|>0$ such that $\bar{D}_{C}^{s}(E, x)= \frac{\mathcal{H}^{s}\left(E \cap U_{x}\right)}{\left|U_{x}\right|^{s}}$. Finally, as an application of this result, an equivalent condition for $E_{0}=E$ is given, where $E_{0}=\left\{x \in E \mid \bar{D}_{C}^{s}(E, x)=1\right\}$.

In this paper, for given mass c > 0, we study the existence of normalized solutions to the following nonlinear Kirchhoff equation $\left\{\begin{array}{l} \left(a+b \int_{\mathbb{R}^{3}}\left[|\nabla u|^{2}+V(x) u^{2}\right] d x\right)[-\Delta u+V(x) u]=\lambda u+\mu|u|^{q-2} u+|u|^{p-2} u, \quad \text { in } \mathbb{R}^{3} \\ \int_{\mathbb{R}^{3}}|u|^{2} d x=c^{2} \end{array}\right.$ where $a>0, b>0, \lambda \in \mathbb{R}, 5<q<p<6, \mu>0$ and V is a continuous non-positive function vanishing at infinity. Under some mild assumptions on V, we prove the existence of a mountain pass normalized solution via the minimax principle.

In this paper, we establish the endpoint estimate for an oscillatory multiplier associated with wave equations on the torus, which extends the results of Fan and Sun. In addition, we obtain a more general result for sublinear operators on compact measure spaces.