In this paper we prove refined major-balance identities on the $321$-avoiding involutions of length $n$, respecting the leading element of permutations. The proof is based on sign-reversing involutions on the lattice paths within a $\lfloor n/2 \rfloor \times \lceil n/2 \rceil$ rectangle. Moreover, we prove affirmatively a question about refined major-balance identities on the $123$-avoiding involutions, respecting the number of descents.

Let $G$ be a nonabelian group and $n$ a natural number. We say that $G$ has a strict $n$-split decomposition if it can be partitioned as the disjoint union of an abelian subgroup $A$ and $n$ nonempty subsets $B_1, B_2, \ldots, B_n$, such that $|B_i| \gt 1$ for each $i$ and within each set $B_i$, no two distinct elements commute. We show that every finite nonabelian group has a strict $n$-split decomposition for some $n$. We classify all finite groups $G$, up to isomorphism, which have a strict $n$-split decomposition for $n = 1,2,3$. Finally, we show that for a nonabelian group $G$ having a strict $n$-split decomposition, the index $|G:A|$ is bounded by some function of $n$.

In this paper we construct pseudo projective modules which are not quasi projective over non-commutative perfect rings. To do it we construct finite dimensional quiver algebras over the field $\mathbb{Z}_2$. The modules which are constructed will have finite length three and only three nonzero proper submodules.

In this paper, we discuss the local existence and uniqueness for the Cauchy problem of semi heat equations with an initial data in the space $L^q$ on H type group $\mathbb{H}_{p}^{d}$, which has the dimension $p$ of the center, like the argument on the Euclidean space given by F. B. Weissler. That is, the Cauchy problem \[ \begin{cases} \left( \partial_t - \Delta_{\mathbb{H}_{p}^{d}} \right) u(g,t) = |u|^{r-1} u, \quad g \in \mathbb{H}_{p}^{d}, \; t \gt 0, \\ u(g,0) = u_0(g) \in L^q(\mathbb{H}_{p}^{d}) \end{cases} \] has a unique solution if $q \gt N(r-1)/2$ ($q = N(r-1)/2$) and $q \geq r$ ($q \gt r$), where $r \gt 1$ and $N = 2d+2p$ is the homogeneous dimension of $\mathbb{H}_{p}^{d}$.

General harmonic extension has no uniqueness and harmonic functions may have different non-tangential boundary values in different convergence sense. In this paper, we establish first $\beta$-harmonic functions in ultra-distribution frame. Further, we consider the characterization between Carleson measure space and boundary distribution space. For $\beta$-harmonic functions with boundary distributions, there exists no maximum value principle. We apply Meyer wavelets to introduce basic harmonic functions and basic observers. We apply Meyer wavelets and vaguelette knowledge to prove the uniqueness of $\beta$-harmonic extension and prove also that $\beta$-harmonic function converges to boundary distribution in the relative norm sense.

In this paper, some stronger forms of transitivity in a non-autonomous discrete dynamical system $(X,f_{1,\infty})$ generated by a sequence $(f_n)$ of continuous self maps converging uniformly to $f$, are studied. The concepts of thick sensitivity, ergodic sensitivity and multi-sensitivity for non-autonomous discrete dynamical systems, which are all stronger forms of sensitivity, are defined and studied. It is proved that under certain conditions, if the rate of convergence at which $(f_n)$ converges to $f$ is “sufficiently fast”, then various forms of sensitivity and transitivity for the non-autonomous system $(X,f_{1,\infty})$ and the autonomous system $(X,f)$ coincide. Also counter examples are given to support results.

When studying the multilinear PageRank problem, a system of polynomial equations needs to be solved. In this paper, we propose a modified Newton method and develop a monotone convergence theory for a third-order tensor when $\alpha \lt 1/2$. In this parameter regime, the sequence of vectors produced by the Newton-like method is monotonically increasing and converges to the solution. When $\alpha \gt 1/2$ we present an always-stochastic modified Newton iteration. Numerical results illustrate the effectiveness of this method.

Let $p(\cdot) \colon \mathbb{R}^n \to (0,\infty]$ be a variable exponent function satisfying the globally log-Hölder continuous condition and $A$ a general expansive matrix on $\mathbb{R}^n$. In this article, the authors first introduce the variable anisotropic Hardy space $H_A^{p(\cdot)}(\mathbb{R}^n)$ associated with $A$, via the non-tangential grand maximal function, and then establish its radial or non-tangential maximal function characterizations. Moreover, the authors also obtain various equivalent characterizations of $H_A^{p(\cdot)}(\mathbb{R}^n)$, respectively, by means of atoms, finite atoms, the Lusin area function, the Littlewood-Paley $g$-function or $g_{\lambda}^{\ast}$-function. As applications, the authors first establish a criterion on the boundedness of sublinear operators from $H^{p(\cdot)}_A(\mathbb{R}^n)$ into a quasi-Banach space. Then, applying this criterion, the authors show that the maximal operators of the Bochner-Riesz and the Weierstrass means are bounded from $H^{p(\cdot)}_A(\mathbb{R}^n)$ to $L^{p(\cdot)}(\mathbb{R}^n)$ and, as consequences, some almost everywhere and norm convergences of these Bochner-Riesz and Weierstrass means are also obtained. These results on the Bochner-Riesz and the Weierstrass means are new even in the isotropic case.

In the present paper, we introduce a class of $\alpha$-Carleson measures $\mathcal{C}_{\alpha,v}(\mathbb{R}^{n+1}_+)$, which is called by the vanishing $\alpha$-Carleson measures. We prove that $\mathcal{C}_{1/p,v}(\mathbb{R}^{n+1}_+)$ is just a predual of the tent space $\widetilde{T}_{\infty}^p$ ($0 \lt p \lt 1$). Furthermore, we construct the $\alpha$-Carleson measures and the vanishing $\alpha$-Carleson measures by the Campanato functions and its a subclass, respectively. Moreover, a characterization of the vanishing $\alpha$-Carleson measure by the compactness of Poisson integral is given in this paper. Finally, as some applications, we give the $(L^{2/\alpha},L^2)$ boundedness and compactness for some paraproduct operators.

Our goal in this paper is to translate results on function classes that are characterized by the property that all the Karush-Kuhn-Tucker points are efficient solutions, obtained in Euclidean spaces to Riemannian manifolds. We give two new characterizations, one for the scalar case and another for the vectorial case, unknown in this subject literature. We also obtain duality results and give examples to illustrate it.

Using a scheme for solving multiobjective optimization problems via a system of corresponding scalar problems, approximate optimality conditions for a nonconvex semi-infinite multiobjective optimization problem are established. As a new approach, the scheme is developed to study the approximate duality theorems of the problem via a pair of primal-dual scalar problems. Several $\epsilon$-duality theorems are given. Furthermore, the existence theorem for almost quasi weakly $\epsilon$-Pareto solutions of the primal problem, and the existence theorem for quasi weakly $\epsilon$-Pareto solutions of the dual problem are established without any constraint qualification.