In this paper, we establish some strong laws of large numbers (SLLN) for non-independent variables under the framework of sublinear expectations. One of our main results is for blockwise m-dependent random variables and another is for orthogonal random variables, both of which are the generalization of SLLN for independent random variables in sublinear expectation spaces.
Loday introduced di-associative algebras and tri-associative algebras motivated by periodicity phenomena in algebraic K-theory. The purpose of this paper is to study the splittings of operations of di-associative algebras and tri-associative algebras. First, we introduce the notion of a quadri-dendriform algebra, which is a splitting of a di-associative algebra. We show that a relative averaging operator on dendriform algebras gives rise to a quadri-dendriform algebra. Conversely, a quadri-dendriform algebra gives rise to a dendriform algebra and a representation such that the quotient map is a relative averaging operator. Furthermore, any quadri-dendriform algebra can be embedded into an averaging dendriform algebra. Finally, we introduce the notion of six-dendriform algebras, which are a splitting of tri-associative algebras, and demonstrate that homomorphic relative averaging operators induce six-dendriform algebras.
A set of permutations is called sign-balanced if the set contains the same number of even permutations as odd permutations. Let $S_n(\sigma_1, \sigma_2, \ldots, \sigma_r)$ be the set of permutations in the symmetric group $S_n$ which avoids patterns $\sigma_1, \sigma_2, \ldots, \sigma_r$. The aim of this paper is to investigate when, for certain patterns $\sigma_1, \sigma_2, \ldots, \sigma_r$, $S_n(\sigma_1, \sigma_2, \ldots, \sigma_r)$ is sign-balanced for every integer $n>1$. We prove that for any $\{\sigma_1, \sigma_2, \ldots, \sigma_r\}\subseteq S_3$, if $\{\sigma_1, \sigma_2, \ldots, \sigma_r\}$ is sign-balanced except $\{132, 213, 231, 312\}$, then $S_n(\sigma_1, \sigma_2, \ldots, \sigma_r)$ is sign-balanced for every integer $n>1$. In addition, we give some results in the case of avoiding some patterns of length $4$.
The main goal of this paper is to establish the boundedness of bilinear strongly singular operator (T) and its commutator (T)b1,b2 on generalized Morrey spaces Mup(μ)over non-homogeneous metric measure spaces.Under assumption that the Lebesgue measurable functions u,u1 and u2 belong to Wτ for τ ∈(0,2),and u1u2=u.The authors prove that (T) is bounded from product spaces Mu1p1(μ×Mu2p2(μ)into spaces Mup(μ),where 1/p=1/p1+1/p2 with 1＜p1,p2＜∞;and also bounded from product spaces Mu1p1(μ)×Mu2p2(μ)into generalized weak Morrey spaces WMup(μ).Furthermore,the author also show that commutator (T)b1,b2generated by b1,b2 ∈RBMO(μ)and (T) is bounded from product spaces Mu1p1(μ)× Mu2p2(μ)into spaces Mup(μ).
In this paper, we introduce the notion of embedding tensor on 3-Hom-Lie algebras and naturally induce 3-Hom-Leibniz algebras. Moreover, the cohomology theory of embedding tensors on 3-Hom-Lie algebras is defined. As an application, we show that if two linear deformations of an embedding tensor on a 3-Hom-Lie algebra are equivalent, then their infinitesimals belong to the same cohomology class in the first cohomology group.
