Statistical convergence in Lebesgue spaces is considered in this paper. A criterion for statistical convergence is given. It is shown that the known Tauberian theorems for scalar case are valid in this case, too.

We describe totally geodesic subalgebras of a metric 2-step nilpotent Lie algebra $\n$. We prove that a totally geodesic subalgebra of $\n$ is either abelian and flat or can be decomposed as a direct sum determined by the curvature transformation. In addition, we give conditions under which a totally geodesic submanifold of a simply connected 2-step nilpotent Lie group is a totally geodesic subgroup. We follow Eberlein's 1994 paper in which he imposes the condition of nonsingularity on $\n$. We remove this restriction and illustrate the distinction between the nonsingular case and the unrestricted case.

A space of pseudoquotients, $\mathcal{B}(X,S)$, is defined as equivalence classes of pairs $(x,f)$, where $x$ is an element of a non-empty set $X$, $f$ is an element of $S$, a commutative semigroup of injective maps from $X$ to $X$, and $(x,f) \sim (y,g)$ if $gx=fy$. In this note, we consider a generalization of this construction where the assumption of commutativity of $S$ is replaced by Ore type conditions. As in the commutative case, $X$ can be identified with a subset of $\mathcal{B}(X,S)$, and $S$ can be extended to a group, $G$, of bijections on $\mathcal{B}(X,S)$. We introduce a natural topology on $\mathcal{B}(X,S)$ and show that all elements of $G$ are homeomorphisms on $\mathcal{B}(X,S)$.

For a continuous surjection $T:X \to X$ on a compact metric space $X$ and a unimodular continuous weight on $X$, we consider a weighted composition operator $U_{T,w}$ on the Banach space $C(X)$ of complex-valued continuous functions on $X$ with the sup norm. The set ${\mathcal W}_{T}$ of all weights $w$, for which the operator $U_{T,w}$ has an eigenvalue with a unimodular eigenfunction, forms a topological abelian group. The group ${\mathcal W}_{T}$ admits a homomorphism $W_T$ to the first integral \v{C}ech cohomology of the space $X$. The image and the kernel of $W_T$ carry topological and ergodic aspects of the dynamics $T$. A concrete description of $\operatorname{Im}W_{T}$ and $\operatorname{Ker}W_{T}$ is given for positively expansive eventually-onto open maps (under an assumption on the induced homomorphism of the first \v{C}ech cohomology) and minimal rotations on tori.

We study the boundary behavior of Hardy space functions on a domain in $\CC^n$. We prove Fatou-type theorems using a new paradigm that avoids some of the difficult calculations in earlier work. Our results apply to a strictly broader class of domains.

We show that there are $20$ elliptic fibrations, up to isomorphism, on the modular $K3$ surface associated to the modular group $\Gamma_{1}(7)$.

In this paper, we deal with new equivalent conditions for the localized weak Radon-Nikod\'ym property in dual Banach space related to set-valued operators. First, we introduce the geometric definition of the weak Radon-Nikod\'ym property and the weakly fragmented set-valued operator. Next, using the weakly fragmented mapping, we reveal the relation between the weak Radon-Nikod\'ym property and the weakly single-valued operator. Finally, using this relation and the concept of the exposed point, the main theorem is given together with some applications.

Let $R\rightarrow S$ be a ring homomorphism. We analyze the relationship between the precovers (preenvelopes) in the category of right $R$-modules and the counterparts in the category of right $S$-modules. Some applications are also given.

We study different concepts of the convergence of sequences with multidimensional indices. Emphasis is placed on the use of Etemadi's subsequence method in the study of almost sure convergence of random fields. An improvement of the SLLN for pairwise independent random variables is also obtained.

The problem of the extendibility of Diophantine triples is closely connected with the Mordell-Weil group of the associated elliptic curve. In this paper, we examine Diophantine triples $\{k-1,k+1,c_l(k)\}$ and prove that the torsion group of the associated curves is $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ for $l=3,4$ and $l\equiv 1$ or $2 \pmod{4}$. Additionally, we prove that the rank is greater than or equal to 2 for all $l\ge2$. This represents an improvement of previous results by Dujella, Peth\H{o} and Najman, where cases $k=2$ and $l\le3$ were considered.

An elementary proof of Fell's theorem on models of homogeneous $C^*$-algebras is presented. A spectral theorem and a functional calculus for finite systems of elements which generate homogeneous $C^*$-algebras are proposed.

By using interpolation with a function parameter, we establish a moment inequality for sums of independent random variables in Lorentz spaces $\Lambda^p(\varphi)$. These estimates generalize Rosenthal inequalities in the Lorentz-Zygmund spaces $L^{p,q}(\log L)^{\gamma}$ as well as Lorentz spaces $L^{p,q}$.

We show that an inequality related to Newton's inequality provides one more relation between skewness and kurtosis. This also gives simple and alternative proofs of the bounds for skewness and kurtosis.

Let $G$ be a finite group. We denote by $\rho(G)$ the set of primes which divide some character degrees of $G$ and by $\sigma(G)$ the largest number of distinct primes which divide a single character degree of $G$. We show that $|\rho(G)|\leq 2\sigma(G)+1$ when $G$ is an almost simple group. For arbitrary finite groups $G$, we show that $|\rho(G)|\leq 2\sigma(G)+1$ provided that $\sigma(G)\leq 2$.

We provide an almost sure bound for oscillation rates of kernel density estimators for stationary processes, under the predictive dependence measures which are directly related to the data-generating mechanisms of the underlying processes. We also discuss moduli of continuity of the kernel density estimators.

A steady state Poisson-Nernst-Planck (PNP) system is studied both analytically and numerically with particular attention on I-V relations of ion channels. Assuming the dielectric constant $\varepsilon$ is small, the PNP system can be viewed as a singularly perturbed system. Due to the special structures of the zeroth order inner and outer systems, one is able to derive more explicit expressions of higher order terms in asymptotic expansions. For the case of zero permanent charge, under the assumption of electro-neutrality at both ends of the channel, our result concerning the I-V relation for two oppositely charged ion species is that the third order correction is \textit{cubic} in $V$, and, furthermore (Theorem \ref{3rd order}), up to the third order, the cubic I-V relation has \textit{three distinct real roots} (except for a very degenerate case) which corresponds to the bi-stable structure in the FitzHugh-Nagumo simplification of the Hodgkin-Huxley model. Three numerical experiments are conducted to check the cubic-like feature of the I-V curve, study the boundary value effect on the I-V relation and investigate the permanent charge effect on the I-V curve, respectively.

In Lie sphere geometry, a cycle in $\RR^n$ is either a point or an oriented sphere or plane of codimension $1$, and it is represented by a point on a projective surface $\Omega\subset \PP^{n+2}$. The Lie product, a bilinear form on the space of homogeneous coordinates $\RR^{n+3}$, provides an algebraic description of geometric properties of cycles and their mutual position in $\RR^n$. In this paper, we discuss geometric objects which correspond to the intersection of $\Omega$ with projective subspaces of $\PP^{n+2}$. Examples of such objects are spheres and planes of codimension~$2$ or more, cones and tori. The algebraic framework which Lie geometry provides gives rise to simple and efficient computation of invariants of these objects, their properties and their mutual position in $\RR^n$.