Let $⟨X,\tau ⟩$ be a metrizable topological space and let $⟨Y,\rho ⟩$ be a metric space. Let $\mathrm{\Omega }$ be a family of bounded continuous functions from $X$ to $Y$. We show that the family is Lipschitzian with respect to some compatible metric on $X$ if and only if the family can be written as a countable union of pointwise equicontinuous subfamilies. From this, we easily characterize those families of continuous functions between metrizable spaces that are Lipschitzian with respect to appropriately chosen metrics on the domain and target space.

We decompose an operator associated to a right-focal boundary value problem, whose fixed points are solutions of the boundary value problem, into multiple fixed point problems. We provide conditions for the original boundary value problem to have a solution that can be found by iteration using the decomposition.

We focus on the $({\mu }_1,{\mu }_2)$-pseudoasymptotically Bloch $\tau$-periodicity and its applications. Firstly, we define a new notion of $({\mu }_1,{\mu }_2)$-pseudoasymptotically Bloch $\tau$-periodic functions and some fundamental properties. Then, we apply the obtained results to investigate the existence and uniqueness of $({\mu }_1,{\mu }_2)$-pseudoasymptotically Bloch $\tau$-periodic mild solutions to semilinear evolution equation in Banach spaces. Finally, we gave an application that facilitates the work.

This article deals with certain star variants of the Scheepers property. We introduce and study the star-$K$-Scheepers property and corresponding game. The relationships between the game corresponding to the star-$K$-Scheepers property and other variants of the Scheepers games are investigated. Furthermore, the relation among the winning strategies of the players in such games are represented in an implication diagram. We present certain observations in terms of Alster covers and the Alexandroff duplicates. In addition, we examine a few preservation-like properties in this context. Some open problems are also posed.

We shall give a generalization of Ni and Pan’s $q$-congruence, originally conjectured by Guo, on certain sums involving central $q$-binomial coefficients.

In domain theory, powerdomains play an important role in modeling the semantics of nondeterministic functional programming languages. We extend the notion of powerdomains to the cartesian closed category of directed spaces. We define the notion of upper and lower powerspace of a directed space by the way of free algebras. We show that the upper and lower powerspaces of any directed space exist and give their concrete structures. Moreover, the upper and lower functors preserve the continuity of directed spaces. The upper and lower powerspaces of c-spaces (resp. FS-spaces) are c-spaces (resp. FS-spaces).

A partition is a cubic partition if its even parts come in two colors (blue and red). Reti defined the rank of a cubic partition as the difference between the number of even parts in blue color and the number of even parts in red color. Motivated by the works on inequalities of rank and crank for certain partitions proved by Andrews and Lewis, and Chern, Fu, Tang and Wang, we prove some inequalities for $N^{\prime }(r,m,n)$, which count the number of cubic partitions of $n$ whose rank is congruent to $r$ modulo $m$. More precisely, we establish the generating functions for $N^{\prime }(r,m,n)$ and determine the signs of the differences $N^{\prime }(r,m,n)-N^{\prime }(s,m,n)$ with $m\in \{2,3,4,6\}$ and $0\le r<s\le m-1$ by utilizing $q$-series techniques in this paper.

Let ${\{z_n\}}_{n\ge 0}$ be a log-convex sequence with $z_{n+1}-z_n>0$ for $n\ge 0$. We give several sufficient conditions for the log-balancedness of ${\{z_{n+1}-z_n\}}_{n\ge 0}$, where ${\{z_n\}}_{n\ge 0}$ satisfies a three-term (four-term) recurrence. Then, we apply these results to a series of combinatorial numbers such as Motzkin numbers, middle trinomial coefficients numbers, the Fine numbers, and so on.

The present paper deals with nonreal eigenvalues of singular indefinite Sturm–Liouville boundary value problems with limit-circle type nonoscillation endpoints. An estimate of upper bounds on nonreal eigenvalues for the singular indefinite eigenvalue problems associated to the separated self-adjoint boundary conditions with nonprincipal solutions is obtained.

We present a new proof of the well-known formula for the rank of the inclusion matrix by constructing a $k{\mathcal{𝒮}}_n$-module spanned by the columns of this matrix and calculating its dimension. This gives a representation theoretic interpretation of this formula as the sum of dimensions of Specht modules.

We study the $k(\ge 1)$-connected stable rank and the $k$-homotopy stabilization rank ( J. Funct. Anal. 255 (2008), 3303–3328) and their rational homotopy variants of AF algebras. We prove that, for each odd integer $k$, the rational $k$-connected stable rank (resp. the rational $k$-homotopy stabilization rank) of an AF algebra is equal to the $k$-connected stable rank (resp. the $k$-homotopy stabilization rank) and also characterize the condition that the (rational) $k$-connected stable rank of an AF algebra $A$ is at most $m$ in terms of the Bratteli diagram of $A$. These ranks of AF algebras for even integer $k$ are also studied. They are $k$-connected stable rank-counterparts of the (rational) $K$-stability theorem for AF algebras due to Seth and Vaidyanathan ( New York J. Math. 26 (2020), 931–949). Our proof applies their proof scheme and results.

This paper deals with the controllability for a class of nonautonomous neutral differential equations of fractional order with infinite delay in an abstract space. The semigroup theory of bounded linear operators, fractional calculus, and stochastic analysis techniques have been implemented to achieve the main result. We prove the existence of a mild solution and controllability of the system by using the theory of measure of noncompactness, fixed point theorems, and a $k$-set contractive mapping. An example is given to demonstrate the effectiveness of the abstract result.

The main goal of this article is to introduce the idea of $({\alpha }_{\perp }-F)$-convex contraction in the context of orthogonal metric spaces and to provide some novel fixed point results in said spaces. Additionally, we offer a case study to illustrate the originality of the outcomes. As an application of our key finding, we investigate the solution of a nonlinear Volterra integral equation.

Motivated essentially by the success of applications of the Mittag-Leffler function and its generalizations in science and engineering and $k$-calculus, we propose here a unification of certain $k$-generalizations of the Mittag-Leffler function including $k$-generalizations of the Saxena–Nishimoto function, the Bessel–Maitland function, the Dotsenko function, the elliptic function, etc. We obtain the order and type, asymptotic estimate, a differential equation, eigenfunction property and double series relation for the proposed unification. As a specialization, a generalized $k$-Konhauser polynomial is considered for which the series inequality relations and inverse series relations are obtained.

The Szegő limit theorem by Fedele and Gebert for matrices of the type identity minus Hankel matrix is proved for the special case $𝟙-\frac{\beta }{\pi }{H}_{N,\alpha }$ where $H_{N,\alpha }$ is the $N\times N$-Hilbert matrix, $\alpha \ge \frac{1}{2}$, and $\beta \in ℂ$. The proof uses operator theoretic tools and a reduction to the classical Kac–Akhiezer theorem for the Carleman operator. Thereby, the validity of the theorem for this special Hankel matrix can be extended from $\left|\beta \right|<1$ to $\beta \in ℂ\setminus ]1,\infty [$. The bound on the correction term is improved to $O(1)$ instead of $o(\text{ln}(N))$ for $\beta \in ℂ\setminus [1,\infty [$. The limit case $\beta =1$ is derived directly from the asymptotics for general $\beta$.

We investigate the Cauchy problem of the time-space nonlocal diffusion equation which describes tempered Lévy flight. The time derivative is defined in the Caputo sense and the spatial derivative is taken as a generalization of the fractional Laplacian. Representations and asymptotic behaviors for fundamental solutions of the nonlocal diffusion equation are considered. Then, we use the fundamental solution to obtain the representation formula of solutions of the Cauchy problem. Lastly, the quantitative decay rates for solutions are proved by employing the Fourier analysis technique.

We construct Jacobi cusp forms by computing the adjoint of a certain differential operator with respect to the Petersson scalar product in the case of Jacobi forms and Hermitian Jacobi forms. Jacobi cusp forms constructed by this method involve special values of certain Dirichlet series attached to them.

We study Brown’s definition of the probabilistic zeta function of a finite lattice as a generalization of that of a finite group. We propose a natural alternative or extension that may be better suited for nonatomistic lattices. The probabilistic zeta function admits a general Dirichlet series expression, which unlike for groups, need not be ordinary. We compute the function for several examples of finite lattices, establishing a connection with the Stirling numbers of the second kind in the case of the divisibility lattice. Furthermore, in the context of moving from groups to lattices, we are interested in lattices with probabilistic zeta function given by ordinary Dirichlet series. In this regard, we focus on partition lattices and $d$-divisible partition lattices. Using the prime number theorem, we show that the probabilistic zeta functions of the latter typically fail to be ordinary Dirichlet series.