Without the assumptions of normality and solidness, we investigate some properties of cones with some semiinterior points in normed algebras, introduce two novel notions of $S$-set and $S$-number associated with every semiinterior point in the underlying cone, give a constructive example with calculations of these new quantities, study some topological characterization of the topology induced by $\varpi$-cone metric of $\varpi$-cone metric space over Banach algebra with the help of semiinterior points instead of interior points, and then generalize some fixed point theorems of contraction type mapping defined on $\varpi$-cone metric space over Banach algebra with nonnormal and nonsolid cone.

We specify the fine spectrum of the quadruple band matrix operator $Q(r,s,t,u)$ over the sequence spaces ${\ell }_p$ and $b{v}_p$, where $1<p<\infty$. The quadruple band matrix $Q(r,s,t,u)$ is the general state of ${\mathrm{\Delta }}^3$, $D(r,0,0,s)$, $B(r,s,t)$, ${\mathrm{\Delta }}^2$, $B(r,s)$, $\mathrm{\Delta }$, right shift and Zweier matrices, where ${\mathrm{\Delta }}^3$, $B(r,s,t)$, ${\mathrm{\Delta }}^2$, $B(r,s)$ and $\mathrm{\Delta }$ are called third-order difference, triple band, second-order difference, double band (generalized difference) and difference matrix, respectively.

The aim of this paper is to deal with a fourth-order three-point boundary value problem with an iterative source term. By using a technique based on Green’s function method, the contraction mapping principle and Schauder’s fixed point theorem, we establish some sufficient conditions for proving the existence, uniqueness and continuous dependence of a bounded solution on parameters. Furthermore, two examples are provided to illustrate the main findings, which are completely new, and extend some relevant works to some degree.

We define and study a Weyl functional calculus for an $n$-tuple of commuting Paley–Wiener of $r$-exponential type elements of a Michael algebra. We show that this calculus preserves the same properties as in the Banach case. In particular, it is well defined for polynomials and gives results consistent with the natural algebraic definition. As applications, we obtain for bounded real numerical range series, two generalizations of P. Lévy theorems: the first one for Fourier series and the second for power series.

In recent years, a large class of nuclear $C^{\ast }$-algebras have been classified, modulo an assumption on the universal coefficient theorem (UCT). We think this assumption is redundant and propose a strategy for proving it. Indeed, following the original proof of the classification theorem, we propose bridging the gap between reduction theorems and examples. While many such bridges are possible, various approximate ideal structures appear quite promising.

Let $N(a,b,c,d;n)$ be the number of representations of $n$ as $a{x}^2+b{y}^2+c{z}^2+d{w}^2$ and $T(a,b,c,d;n)$ be the number of representations of $n$ as

$$a\frac{X(X+1)}{2}+b\frac{Y(Y+1)}{2}+c\frac{Z(Z+1)}{2}+d\frac{W(W+1)}{2},$$

where $a,b,c,d$ are positive integers, $x,y,z,w$ are integers, and $n,X,Y,Z,W$ are nonnegative integers. In this paper, by using theta function identities, we confirm Sun’s conjectures that $T(a,b,c,d;n)$ is a linear combination of $N(a,b,c,d;m)$ and $N(a,b,c,d;4m)$, with $m=8n+a+b+c+d$, for several values of $a,b,c,d$.

We examine various properties of positive integers selected according to the Riemann zeta distribution. That is, if $\zeta (s)={\sum }_{n\ge 1}1∕n^s$, $s>1$, then we consider the random variable $X_s$ with $P(X_s=n)=1∕(\zeta (s)n^s)$, $n\ge 1$. We derive various results such as the analog of the Erdös–Kac central limit theorem (CLT) for the number of distinct prime factors, $\omega (X_s)$, of $X_s$, as $s\searrow 1$, large and moderate deviations for $\omega (X_s)$, and a Berry–Esseen result. In addition, we prove analogs of Erdös–Delange type formulas for expectations of additive and multiplicative functions evaluated at $X_s$. We also examine various applications using Dirichlet series. Finally, we show how to derive asymptotic distributional results for an integer selected uniformly at random from $[N]=\{1,2,\dots ,N\}$ or according to the harmonic distribution from their analogous asymptotic results for $X_s$ as $s\searrow 1$.

We establish the generating function for $\overline{c{\varphi }_6}(n)$, the number of generalized Frobenius partitions of $n$ with $6$ colors whose order is $6$ under cyclic permutation of the $6$ colors. Furthermore, we find some congruences for $\overline{c{\varphi }_6}(n)$ modulo $24$.

The principal aim of this paper is to establish the uniqueness and Ulam–Hyers–Mittag–Leffler (UHML) stability of solutions for a new class of multiterm fractional time-delay differential equations in the context of the $\mathrm{\Phi }$-Caputo fractional derivative. To achieve this purpose, the generalized Laplace transform method alongside facet with properties of the Mittag–Leffler functions (M–LFs) are utilized to give a new representation formula of the solutions for the aforementioned problem. Besides that, the uniqueness of the solutions of the considered problem is also proved by applying the well-known Banach contraction principle coupled with the $\mathrm{\Phi }$-fractional Bielecki-type norm, while the $\mathrm{\Phi }$-fractional Gronwall type inequality and the Picard operator (PO) technique, combined with the abstract Gronwall lemma, are used to prove the UHML stability results for the proposed problem. Lastly, an example is offered to assure the validity of the obtained theoretical results.

An identity of Szego, and a volume calculation, heuristically suggest a simple expression for the distribution of Verblunsky coefficients with respect to the (normalized) exponential of the $S^1$ trace of the Gaussian free field. This heuristic expression is not quite correct. A proof of the correct formula has been found by Chhaibi and Najnudel (2019). Their proof uses random matrix theory and overcomes many difficult technical issues. In addition to presenting the Szego perspective, we show that the Chhaibi and Najnudel theorem implies a family of combinatorial identities (for moments of measures) which are of intrinsic interest.

The theory of integration for functions with values in a topological vector space is a bit of a messy subject. There are many different choices for the integral, most distinct from one another in general, and with no compelling reason to always adopt one definition or another as the “right” one. No attempt is made here to resolve this question of which integral is the “right” integral, but we overview a few of the common notions and prove a couple of useful properties for the notion of integrability by seminorm. For this notion of integrability, the completeness of the space of integrable functions with values in a complete locally convex space is established. This space is then easily seen to be isomorphic to the completion of the projective tensor product of the usual ${\text{L}}^1$ space of scalar functions with the vector space. Absolute continuity for this notion of integrability is also considered. Here, it is shown that the familiar properties of differentiation for absolutely continuous scalar functions holds in the vector-valued case.

Given a graph $G$, the adjacency matrix and degree diagonal matrix of $G$ are denoted by $A(G)$ and $D(G)$, respectively. In 2017, Nikiforov proposed the $A_{\sigma }$-matrix: $A_{\sigma }(G)=\sigma D(G)+(1-\sigma )A(G)$, where $\sigma \in [0,1]$. The largest eigenvalue of this novel matrix is called the $A_{\sigma }$-index of $G$. Let ${\mathcal{ℬ}}_n^{\alpha }$ be the class of $n$-vertex block graphs with independence number $\alpha$ and let $\mathcal{𝒢}(n,k)$ be another class of $n$-vertex graphs with $k$ cut edges. We show that the maximum $A_{\sigma }$-index, among all graphs $G\in {\mathcal{ℬ}}_n^{\alpha }$ (resp. $G\in \mathcal{𝒢}(n,k)$), is attained at a unique graph. It is surprising to see that in both cases, the extremal graphs are usually pineapple graphs. We use two methods to establish upper bounds on the $A_{\sigma }$-index of the corresponding extremal graphs. As a byproduct we obtain an upper bound for signless Laplacian spectral radius $q_1(G)$, when $G\in {\mathcal{ℬ}}_n^{\alpha }$.

Let $\eta$ be a nonzero complex number. Let $\mathcal{𝒜}$ and $\mathcal{ℬ}$ be two von Neumann algebras, one of which has no central abelian projections, and let $\mathrm{\Phi }:\mathcal{𝒜}\to \mathcal{ℬ}$ be not necessarily a linear bijection. For arbitrary elements $A,B\in \mathcal{𝒜}$, one can define their Jordan $\eta$-$\ast$-product in the sense of $A{◇}_{\eta }B=AB+\eta B{A}^{\ast }$. Let $p_n(X_1,X_2,\dots ,X_n)$ be the polynomial defined by $n$ indeterminates $X_1,\dots ,X_n$ and their Jordan multiple $\eta$-$\ast$-product. In this article, it is shown that $\mathrm{\Phi }$ satisfies the condition

$$\mathrm{\Phi }(p_n(A_1,A_2,\dots ,A_n))=p_n(\mathrm{\Phi }(A_1),\mathrm{\Phi }(A_2),\dots ,\mathrm{\Phi }(A_n))$$

for all $A_1,A_2,\dots ,A_n\in \mathcal{𝒜}$ if and only if one of the following statements holds true:

It is proved that every pair of sufficiently large even integers can be represented in the form of a pair of equations of eight prime cubes and 609 powers of 2, and each sufficiently large even integer is the sum of eight cubes of primes and 157 powers of 2. These results constitute refinements upon those of Z. X. Liu (2013) and of X. D. Zhao and W. X. Ge (2020).

We first define a $h$-$\text{log}$-convex function and so, we investigate some integral inequalities for $h$-$\text{log}$-convex functions that involve the classical Hermite–Hadamard inequality. In the end, by using notations of Mond–Pečarić method in operator inequalities, we extend some inequalities for operator $h$-$\text{log}$-convex functions.

Let ${\mathrm{\Sigma }}_g$ be a closed, connected, and oriented surface of genus $g\ge 24$, and let $\mathrm{\Gamma }$ be a finite index subgroup of the mapping class group $\mathrm{Mod}({\mathrm{\Sigma }}_g)$ that contains the Torelli group $\mathcal{ℐ}({\mathrm{\Sigma }}_g)$. Then any orientation-preserving $C^1$ action of $\mathrm{\Gamma }$ on the circle cannot be faithful.

We also show that if $\mathrm{\Gamma }$ is a finite index subgroup of $\mathrm{Aut}(F_n)$, when $n\ge 8$, that contains the subgroup of IA-automorphisms, then any orientation-preserving $C^1$ action of $\mathrm{\Gamma }$ on the circle cannot be faithful.

Similarly, if $\mathrm{\Gamma }$ is a finite index subgroup of $\mathrm{Out}(F_n)$, when $n\ge 8$, that contains the Torelli group ${\mathcal{𝒯}}_n$, then any orientation preserving $C^1$ action of $\mathrm{\Gamma }$ on the circle cannot be faithful.

In fact, when $n\ge 10$, any orientation-preserving $C^1$ action of a finite index subgroup of $\mathrm{Aut}(F_n)$ or $\mathrm{Out}(F_n)$ on the circle cannot be faithful.

The ring of integer-valued polynomials over a given subset $S$ of $\mathbb{Z}$ (or $\mathrm{Int}(S,\mathbb{Z}))$ is defined as the set of polynomials in $\mathbb{Q}[x]$ which maps $S$ to $\mathbb{Z}$. In factorization theory, it is crucial to check the irreducibility of a polynomial. In this article, we make Bhargava factorials our main tool to check the irreducibility of a given polynomial $f\in \mathrm{Int}(S,\mathbb{Z}))$. We also generalize our results to arbitrary subsets of a Dedekind domain.

Several new upper bounds for the Berezin number of bounded linear operators defined on reproducing kernel Hilbert spaces are given. The bounds obtained here improve on the earlier ones.

We study the pointwise boundary behavior of the $\infty$-Laplacian problem

$${\mathrm{\Delta }}_{\infty }u=b(x)f(u),u\ge 0,x\in \mathrm{\Omega },u{|}_{\partial \mathrm{\Omega }}=\infty ,$$

where $b\in C(\overline{\mathrm{\Omega }})$ is positive in $\mathrm{\Omega }$, $f\in C[0,\infty )\cap C^1(0,\infty )$ is positive and nondecreasing on $(0,\infty )$ and satisfies an Keller–Osserman type condition.

In this paper, using Nevanlinna theory we discuss the relations between the Borel directions and the uniqueness of algebroid functions. We get several uniqueness theorems of algebroid functions in angular domain which contains the Borel directions. These results extend those for meromorphic functions obtained by some scholars.

We initiate the study of relative operator entropies and Tsallis relative operator entropies in the setting of JB-algebras. We establish their basic properties and extend the operator inequalities on relative operator entropies and Tsallis relative operator entropies to this setting. In addition, we improve the lower and upper bounds of the relative operator $(\alpha ,\beta )$-entropy in the setting of JB-algebras that were established in Hilbert space operators setting by Nikoufar (2014; 2020). Though we employ the same notation as in the classical setting of Hilbert space operators, the inequalities in the setting of JB-algebras have different connotations and their proofs require techniques in JB-algebras.

We introduce Lambda–Pascal sequence spaces ${\ell }_q(G)$, $c_0(G)$, $c(G)$ and ${\ell }_{\infty }(G)$ generated by the matrix $G$ which is obtained by the product of Pascal matrix and $\mathrm{\Lambda }$-matrix. It is proved that the Lambda–Pascal sequence spaces ${\ell }_q(G)$, $c_0(G)$, $c(G)$ and ${\ell }_{\infty }(G)$ are $\mathrm{BK}$-spaces and linearly isomorphic to ${\ell }_q$, $c_0$, $c$ and ${\ell }_{\infty }$, respectively. We construct Schauder bases and obtain $\alpha$-, $\beta$- and $\gamma$-duals of the new spaces. We state and prove characterization theorems related to matrix transformation from the space ${\ell }_q(G)$ to the spaces ${\ell }_{\infty }$, $c$ and $c_0$. Finally, we determine necessary and sufficient conditions for a matrix operator to be compact from the space $c_0(G)$ to any one of the spaces ${\ell }_{\infty }$, $c$, $c_0$ or ${\ell }_1$.

We prove some supercongruences involving the Apéry polynomials

$$A_n(x)=\sum _{k=0}^n{\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)}^2{\left(\genfrac{}{}{0.0pt}{}{n+k}{k}\right)}^2{x}^k(n\in \mathbb{N}=\{0,1,\dots ,\}),$$

the generalized Domb numbers

$$D_n(A,B,C)=\sum _{k=0}^n{\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)}^A{\left(\genfrac{}{}{0.0pt}{}{2k}{k}\right)}^B{\left(\genfrac{}{}{0.0pt}{}{2n-2k}{n-k}\right)}^C(n\in \mathbb{N})\text{ and}{Q}_n=\sum _{k=0}^n\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)\left(\genfrac{}{}{0.0pt}{}{n-k}{k}\right)\left(\genfrac{}{}{0.0pt}{}{n+k}{k}\right)(n\in \mathbb{N}),$$

which were conjectured by Z.-W. Sun. For example, we show that for any prime $p>3$ and positive integer $r$ we have

$$\frac{A_{p^r}(-1)-A_{p^{r-1}}(-1)}{p^{3r}}\equiv \frac{29}{6}{B}_{p-3}(\mathrm{mod}p)\text{ and}\frac{Q_{p^r}-Q_{p^{r-1}}}{p^{3r}}\equiv -\frac{1}{9}{B}_{p-3}(\mathrm{mod}p),$$

where $B_0,B_1,B_2,\dots$ are the Bernoulli numbers. The following supercongruences hold modulo $p$:

where $\left(\frac{\cdot }{p}\right)$ denotes the Legendre symbol and $E_0,E_1,E_2,\dots$ are the Euler numbers.

We prove that, for $k=30$, every sufficiently large even integer can be represented as a sum of eight cubes of primes and $k$ powers of 2. This is an improvement of Zhao’s result for $k=169$.