The main purpose of this paper is to establish the elliptic gradient estimate for the heat equation on compact Riemannian manifold with control on integral Ricci curvature. We also derived the volume comparison theorem under the new integral Ricci curvature condition which extended Petersen-Wei's volume comparison theorem.
Quantum states are positive operators with unit traces on a Hilbert space. The set of all quantum states is convex. In the paper, we give a characterization of maps leave the maximal eigenvalue of convex combinations of quantum states.
In this article, the refined estimates of all homogeneous expansions for a subclass of biholomorphic spirallike mappings which have a concrete parametric representation on the unit ball in complex Banach spaces and the unit polydisk in $\mathbb{C}^n$ are mainly established. In particular, the result is sharp for the ($k+1$)-th homogeneous expansion. Meanwhile the estimates of all homogeneous expansions for a subclass of $k$-fold symmetric biholomorphic spirallike mappings which have parametric representation on the unit ball in complex Banach spaces and the unit polydisk in $\mathbb{C}^n$ are also given, and the result is sharp for the ($k+1$)-th homogeneous expansion as well. Our obtained results include many known results in some prior literatures.
We get an explicit and recursive representation for high order moments of additive functionals for discrete-time single death chains. Meanwhile, the polynomial convergence and a central limit theorem are investigated.
For $n=2$, Bian and Liu in[Algebra Colloquium, 2017, 24(2):297-308] provided a presentation for little $q$-Schur algebra in terms of generators and relations. It is more difficult for the case $n>2$. In this paper, we give a presentation for little $q$-Schur algebra $u_k(3,3)$ at odd roots.
We define the T-DMP inverse and T-CMP inverse of third-order F-square tensors by using the T-Moore-Penrose inverse, T-Drazin inverse and T-core-nilpotent decomposition theorem of tensors via the T-product. Then, we present some characterizations and properties. Finally, the Cayley-Hamilton theorem of third-order tensors is extended to T-Drazin inverses and T-DMP inverses. Examples are also given to illustrate these theorems.
A subgroup $H$ of a finite group $G$ is a BNA-subgroup of $G$ if either $H^{x}=H$ or $x\in \langle H, H^{x}\rangle $ for all $x\in G$. A finite group $G$ is called a CBNA-group if its all cyclic subgroups of order prime or $4$ are BNA-subgroups of $G$. The main aim of this paper is to investigate the structure of CBNA-groups, and the groups whose all proper subgroups are CBNA-groups are classified completely.
In general, the properties of modules over a triangular matrix ring $T=\left(\begin{smallmatrix}A & U \\0 & B\end{smallmatrix}\right)$ are studied via modules over diagonal "small rings" $A$ and $B$. However, we use model structures on the category of $T$-modules to characterize the stable categories $\underline{\mathcal{GP}}(A)$, $\underline{\mathcal{GP}}(B)$ of Gorenstein projective modules over $A$ and $B$. To this end, we introduce two subcategories of Gorenstein $T$-modules, and obtain two corresponding complete cotorsion pairs. Moreover, cotorsion pairs of modules are lifted to $T$-complexes, and the equivalences and recollements of homotopy categories of complexes are studied.
In this paper, we study $(C,\varepsilon)$-super subdifferentials of set-valued maps. First, we introduce a notion of $(C,\varepsilon)$-super efficient point of a set. Some properties and equivalent characterizations of the $(C,\varepsilon)$-super efficient points are presented. Scalarization theorems of the set-valued optimization problem are obtained in the sense of $(C,\varepsilon)$-super efficiency. Second, we define $(C,\varepsilon)$-subdifferentials of set-valued maps and research the existence conditions of $(C,\varepsilon)$-subdifferentials. Moreau-Rockafellar type theorems characterized by $(C,\varepsilon)$-subdifferentials are also established. Finally, as the applications, we establish some optimality conditions of the set-valued optimization problem involving the $(C,\varepsilon)$-super subdifferentials. The results obtained in this paper unify and generalize some results characterized by the super subdifferentials or $\varepsilon$-super subdifferentials of the set-valued maps in the literature.
Let $(Z_n)$ be a supercritical branching process with immigration in an independent and identically distributed random environment. Based on the structure of $Z_n$, using related results on random walks and technique of measure change, we establish a Cramér's large deviation expansion for $\log Z_n$.
Combing Wintner, Delange, Ushiroya and Tóth's works from 1976 to 2017, we have that the multi-variable arithmetic functions defined on integer ring can be expanded through the Ramanujan sums. This is an analogue of the Fourier expansion for periodic functions in the classical analysis. In this paper we further investigate the properties of Ramanujan sums in the polynomial ring $\mathbb{F}_{q}{[T]}$, and show that the multi-variable arithmetic functions defined on $\mathbb{F}_{q}{[T]}$ can also be expanded through the polynomial Ramanujan sums and the unitary polynomial Ramanujan sums.
We introduce a new algorithm for solving pseudomonotone variational inequality problems in Banach spaces. We prove that the sequence generated by this algorithm converges strongly an element of solutions for variational inequality problems under some suitable conditions imposed on the parameters. The results obtained in this paper extend and improve many recent ones in the literature.
In Ringel-Hall algebra of Dynkin type, the set $S$ of isomorphism classes of indecomposable modules forms a minimal Gröbner-Shirshov basis of the ideal $\hbox{Id}(S)$ generated by the set $S$, and the corresponding irreducble elements forms a PBW basis of the corresponding Ringel-Hall algbera. Our aim is to generalize this result to the derived Hall algebra of type $G_2$. First, we compute the skew commutator relations between the isomorphism classes of indecomposable objects in the bounded derived category of type $G_2$ by using the Auslander-Reiten quiver of the bounded derived category of type $G_2$. Then, we prove that the compositions between these skew commutator relations are trivial. Finally, we construct a PBW basis of the derived Hall algebra of type $G_2.$
In this paper, conformal biderivations and automorphism groups of two classes of Schrödinger-Virasoro type Lie conformal algebras ${\rm TSV}(a,b)$ and ${\rm TSV}(c)$ are completely determined, respectively. The results for the Lie conformal algebras $W(a,b)$ are also obtained as a corollary.
In this paper, a new conception called perfect permutation will be introduced. We focus on its algebraic properties and construction methods. The main result is that there exists a perfect permutation of order n when 2n + 1 is a prime. Furthermore, we use perfect permutations to construct cyclic spatially balanced Latin squares and symmetric spatially balanced Latin squares both of which are widely used in experimental designs.
In this paper, a class of infinite dimensional complete Lie algebras is obtained by extending the one dimensional homogeneous derivations of Loop algebras over simple Lie algebras. It is proved that every 2-local homogeneous derivation of this kind of infinite dimensional complete Lie algebra is a derivation.