### Jun 2021, Volume 16 Issue 3

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• RESEARCH ARTICLE
Jin CHENG, Yun GAO

We generalize the P(N)-graded Lie superalgebras of Martinez-Zelmanov. This generalization is not so restrictive but suffcient enough so that we are able to have a classification for this generalized P(N)-graded Lie superalgebras. Our result is that the generalized P(N)-graded Lie super-algebra L is centrally isogenous to a matrix Lie superalgebra coordinated by an associative superalgebra with a super-involution. Moreover, L is P(N)-graded if and only if the coordinate algebra R is commutative and the super-involution is trivial. This recovers Martinez-Zelmanov's theorem for type P(N). We also obtain a generalization of Kac's coordinatization via Tits-Kantor-Koecher construction. Actually, the motivation of this generalization comes from the Fermionic-Bosonic module construction.

• RESEARCH ARTICLE
Shanze GAO, Hui MA

We consider the closed orientable hypersurfaces in a wide class of warped product manifolds, which include space forms, deSitter-Schwarzschild and Reissner-Nordström manifolds. By using an integral formula or Brendle's Heintze-Karcher type inequality, we present some new characterizations of umbilic hypersurfaces. These results can be viewed as generalizations of the classical Jellet-Liebmann theorem and the Alexandrov theorem in Euclidean space.

• RESEARCH ARTICLE
Xiaoxia GUO, Wei SUN

We investigate periodic solutions of regime-switching jump diffusions. We first show the well-posedness of solutions to stochastic differential equations corresponding to the hybrid system. Then, we derive the strong Feller property and irreducibility of the associated time-inhomogeneous semigroups. Finally, we establish the existence and uniqueness of periodic solutions. Concrete examples are presented to illustrate the results.

• RESEARCH ARTICLE
Hongmei HU, Naihong HU, Limeng XIA

Based on the n-fold tensor product version of the generalized double-bosonization construction, we prove the Majid conjecture of the quantum Kac-Moody algebras version. Particularly, we give explicitly the double-bosonization type-crossing constructions of quantum Kac-Moody algebras for affine types $G2(1)$ , $E2(1)$,and Tp,q,r, and in this way, we can recover generators of quantum Kac-Moody algebras with braided groups defined by R-matrices in the related braided tensor category. This gives us a better understanding for the algebra structures themselves of the quantum Kac-Moody algebras as a certain extension of module-algebras/module-coalgebras with respect to the related quantum subalgebras of finite types inside.

• RESEARCH ARTICLE
Jing HUANG, Wenguang ZHAI, Deyu ZHANG

Let c>1 and $0<γ<1$: We study the solubility of the Diophantine inequality $|p1c+p2c+⋯+psc−N|<(logN)−1$ in Piatetski-Shapiro primes p1,p2, .., ps of the form $pj=[mγ]$ for some $m∈ℕ$, and improve the previous results in the cases s = 2, 3, 4.

• RESEARCH ARTICLE
Fucai LIN, Zhongbao TANG

Two non-discrete Hausdorff group topologies $τ$ and $δ$ on a group G are called transversal if the least upper bound $τ∨δ$ of $τ$ and $δ$ is the discrete topology. In this paper, we discuss the existence of transversal group topologies on locally pseudocompact, locally precompact, or locally compact groups. We prove that each locally pseudocompact, connected topological group satisfies central subgroup paradigm, which gives an affrmative answer to a problem posed by Dikranjan, Tkachenko, and Yaschenko [Topology Appl., 2006, 153:3338-3354]. For a compact normal subgroup K of a locally compact totally disconnected group G, if G admits a transversal group topology, then G/K admits a transversal group topology, which gives a partial answer again to a problem posed by Dikranjan, Tkachenko, and Yaschenko in 2006. Moreover, we characterize some classes of locally compact groups that admit transversal group topologies.

• RESEARCH ARTICLE
Yongtao LIU

By using the perpetual cutoff method, we prove two discrete versions of gradient estimates for bounded Laplacian on locally finite graphs with exception sets under the condition of $CDE′$(K,N). This generalizes a main result of F. Münch who considers the case of CD(K, $∞$) curvature. Hence, we answer a question raised by Münch. For that purpose, we characterize some basic properties of radical form of the perpetual cutoff semigroup and give a weak commutation relation between bounded Laplacian $Δ$ and perpetual cutoff semigroup $PtW$ in our setting.

• RESEARCH ARTICLE
Ziqi TAN, Xianping GUO

Proposition 5.5.6 (ii) in the book Markov Chains and Stochastic Stability (2nd ed, Cambridge Univ. Press, 2009) has been used in the proof of a theorem about ergodicity of Markov chains. Unfortunately, an example in this paper shows that this proposition is not always true. Thus, a correction of this proposition is provided.

• RESEARCH ARTICLE
Yue WANG, Jihong SHEN, Changjiang BU

A real symmetric tensor $A=(ai1⋯im)∈ℝ[mn]$ is copositive (resp., strictly copositive) if $Axm≥0$ (resp., $Axm>0$) for any nonzero nonnegative vector$x∈ℝn$: By using the associated hypergraph of $A$, we give necessary and sufficient conditions for the copositivity of $A$: For a real symmetric tensor $A$satisfying the associated negative hypergraph $H_(A)$ and associated positive hypergraph $H+(A)$ are edge disjoint subhypergraphs of a supertree or cored hypergraph, we derive criteria for the copositivity of $A$: We also use copositive tensors to study the positivity of tensor systems.

• RESEARCH ARTICLE
Xiaowei YANG, Shuang SONG, Huiming ZHANG

We study the law of the iterated logarithm (LIL) for the maximum likelihood estimation of the parameters (as a convex optimization problem) in the generalized linear models with independent or weakly dependent ($ρ$-mixing) responses under mild conditions. The LIL is useful to derive the asymptotic bounds for the discrepancy between the empirical process of the log-likelihood function and the true log-likelihood. The strong consistency of some penalized likelihood-based model selection criteria can be shown as an application of the LIL. Under some regularity conditions, the model selection criterion will be helpful to select the simplest correct model almost surely when the penalty term increases with the model dimension, and the penalty term has an order higher than O(log log n) but lower than O(n): Simulation studies are implemented to verify the selection consistency of Bayesian information criterion.

• RESEARCH ARTICLE
Lan YOU, Huixiang CHEN

We study Dorroh extensions of algebras and Dorroh extensions of coalgebras. Their structures are described. Some properties of these extensions are presented. We also introduce the finite duals of algebras and modules which are not necessarily unital. Using these finite duals, we determine the dual relations between the two kinds of extensions.

• RESEARCH ARTICLE
Baoquan YUAN, Ying ZHANG

We establish the global well-posedness of a strong solution to the 3D tropical climate model with damping. We prove that there exists the global and unique solution for α, β, γ satisfying one of the following three conditions: (1) $α,β≥4$; (2) $7/2≤α<4,β≥(5α+7)/(2α),γ≥7/(2α−5)$; (3) $3<α≤7/2,β,γ≥7/(2α−5)$.

• RESEARCH ARTICLE
Shaoteng ZHANG, Xiaoxiang JIAO

We study conformal minimal two-spheres immersed into the quaternionic projective space $ℍPn$ by using the twistor map. We present a method to construct new minimal two-spheres with constant curvature in $ℍPn$, based on the minimal property and horizontal condition of Veronese map in complex projective space. Then we construct some concrete examples of conformal minimal two-spheres in $ℍPn$ with constant curvature 2/n, n = 4, 5, 6, respectively. Finally, we prove that there exist conformal minimal two-spheres with constant curvature 2/n in $ℍPn$ (n≥7):