In the Musielak–Orlicz-type spaces ${\mathcal{𝒮}}_{\mathrm{M}}$, exact Jackson-type inequalities are obtained in terms of best approximations of functions and the averaged values of their generalized moduli of smoothness. The values of Kolmogorov, Bernstein, linear, and projective widths in ${\mathcal{𝒮}}_{\mathrm{M}}$ are found for classes of periodic functions defined by certain conditions on the averaged values of the generalized moduli of smoothness.

Let $\mathcal{\mathscr{H}}$ be a separable Hilbert space. Recently, the concept of $K$-$g$-frame was introduced as a special generalization of $g$-Bessel sequences. In this paper, we point out some gaps in the proof of some existent results concerning $K$-$g$-frame. We present examples to indicate that these results are not necessarily valid. Then we remove the gaps and provide some desired conclusions. In this respect, we deal with Schur–Horn problem, which characterizes sequences ${\left\{\parallel f_n{\parallel }^2\right\}}_{n=1}^{\infty }$, for all frames ${\left\{f_n\right\}}_{n=1}^{\infty }$ with the same frame operator. We introduce the concept of synthesis related frames. Finally, as the main result, we investigate around Schur–Horn problem, for the case where $\mathcal{\mathscr{H}}$ is finite dimensional. In fact, we prove that two frames have the same frame operator if and only if they are synthesis related.

Let ${\left\{{\phi }_i\left(z;\alpha \right)\right\}}_{i=0}^{\infty }$, corresponding to $\alpha \in \left(-1,1\right)$, be orthonormal Geronimus polynomials. We study asymptotic behavior of the expected number of real zeros, say ${\mathbb{𝔼}}_n\left(\alpha \right)$, of random polynomials

$$P_n\left(z\right):=\sum _{i=0}^n{\eta }_i{\phi }_i\left(z;\alpha \right),$$

where ${\eta }_0,\dots ,{\eta }_n$ are i.i.d. standard Gaussian random variables. When $\alpha =0$, ${\phi }_i\left(z;0\right)=z^i$ and $P_n\left(z\right)$ are called Kac polynomials. In this case it was shown by Wilkins that ${\mathbb{𝔼}}_n\left(0\right)$ admits an asymptotic expansion of the form

$${\mathbb{𝔼}}_n\left(0\right)\sim \frac{2}{\pi }log\left(n+1\right)+\sum _{p=0}^{\infty }{A}_p{\left(n+1\right)}^{-p}$$

(Kac himself obtained the leading term of this expansion). In this work we obtain a similar expansion of $\mathbb{𝔼}\left(\alpha \right)$ for $\alpha \ne 0$. As it turns out, the leading term of the asymptotics in this case is $\left(1∕\pi \right)log\left(n+1\right)$.

V. Nall proved that connectedness is preserved under inverse limits if the bounding functions are unions of functions with connected images. We show that for such functions the projections from the graph onto domain are confluent and we investigate relationships between functions satisfying this or similar conditions with confluence or openness of projections.

In this paper, we introduce the concept of $L_p$-mixed radial Blaschke–Minkowski homomorphisms for star bodies. Further, associated with dual quermassintegrals, we give an affirmative answer and two negative answers of the Busemann–Petty problem for the $L_p$-mixed radial Blaschke–Minkowski homomorphisms.

We study the $d$-dimensional ($d=2,3$) tropical climate model with only the dissipation of the first baroclinic model of the velocity $-\eta \mathrm{\Delta }v$. By choosing a class of special initial data $\left(u_0,v_0,{𝜃}_0\right)$ whose $H^s\left({ℝ}^d\right)$ norm can be arbitrarily large, we obtain the global smooth solution of $d$-dimensional tropical climate model.

Recently, various interesting binomial sums involving Catalan numbers have been derived and computed. Inspired from them, several binomial sums involving Catalan numbers are evaluated in closed forms using a hypergeometric series approach.

A pure-injective module $M$ is said to be pi-indigent if its subinjectivity domain is smallest possible, namely, consisting of exactly the absolutely pure modules. A module $M$ is called subinjective relative to a module $N$ if for every extension $K$ of $N$, every homomorphism $N\to M$ can be extended to a homomorphism $K\to M$. The subinjectivity domain of the module $M$ is defined to be the class of modules $N$ such that $M$ is $N$-subinjective. Basic properties of the subinjectivity domains of pure-injective modules and of pi-indigent modules are studied. The structure of a ring over which every simple, uniform, or indecomposable pure-injective module is injective or subinjective relative only to the smallest possible family of modules is investigated.

Let $\alpha$ be a nonzero algebraic integer of degree $d$ with conjugates ${\alpha }_1=\alpha ,\dots ,{\alpha }_d$. It is well known that $S_k\left(\alpha \right)={\sum }_{i=1}^d{\alpha }_i^k$. Here, we define $S_k^{\prime }\left(\alpha \right)={\sum }_{i=1}^d|{\alpha }_i{|}^k$ and $s_k^{\prime }\left(\alpha \right)=S_k^{\prime }\left(\alpha \right)∕d$. Then, we focus our attention on the case $k=2$ when $\alpha$ is an algebraic integer all of whose conjugates lie in a sector $|argz|\le 𝜃$, . We compute the greatest lower bound $c\left(𝜃\right)$ of ${s_2}^{\prime }\left(\alpha \right)$ for $𝜃$ belonging to eleven subintervals of $\left[0,\frac{\pi }{2}\right)$. Moreover, among these subintervals, there are twice two consecutive and complete subintervals. We use the method of explicit auxiliary functions for which the involved polynomials are found by our recursive algorithm.

In the present paper, we propose a hybrid operator of integral type and estimate some direct results. As a further modification we provide a new generalization of exponential type for which we studied the approximation order in a weighted space.

We give a sufficient condition for the $kG$-Scott module with vertex $P$ to remain indecomposable under taking the Brauer construction for any subgroup $Q$ of $P$ as $k\left[Q{C}_G\left(Q\right)\right]$-module, where $k$ is a field of characteristic $2$, and $P$ is a wreathed $2$-subgroup of a finite group $G$. This generalizes results for the cases where $P$ is abelian and some others. The motivation of this paper is that the Brauer indecomposability of a $p$-permutation bimodule ($p$ is a prime) is one of the key steps in order to obtain a splendid stable equivalence of Morita type by making use of the gluing method that then can possibly lift to a splendid derived equivalence.

We introduce the notion of quadratic Killing normal Jacobi operator (simply, Killing normal Jacobi operator) and its geometric meaning for real hypersurfaces in the complex quadric. In addition, we give a nonexistence theorem for Hopf real hypersurfaces with quadratic Killing normal Jacobi operator in the complex quadric.

We define the notion of the generalized alternating hyperharmonic numbers, and show that Euler sums of the generalized alternating hyperharmonic numbers can be expressed in terms of linear combinations of classical (alternating) Euler sums.

We are concerned with the Cauchy problem of the multidimensional inhomogeneous incompressible heat conducting Navier–Stokes flows with fractional dissipation. We show that the problem admits a unique global strong solution. Note that the initial data can be arbitrarily large and the initial density can contain vacuum states.

Motivated by the recent research of congruences and $q$-congruences, we provide two different $q$-analogues of the (G.2) supercongruence of Van Hamme through the “creative microscoping” method, which was recently devised by Guo and Zudilin. It is a remarkable fact that this is the first time giving $q$-analogues of (G.2). In addition, we propose a conjecture on a partial $q$-analogue of Swisher’s Dwork-type supercongruence (G.3).

Let $G$ be a connected graph of order $n$. The variation of the Randić index of $G$ is defined as

$$R^{\prime }\left(G\right)=\sum _{uv\in E\left(G\right)}\frac{1}{max\left\{d\left(u\right),d\left(v\right)\right\}},$$

where the summation goes over all edges $uv$ of $G$ and $d\left(u\right)$ is the degree of the vertex $u$ in $G$. In this paper, among all bicyclic graphs of order $n$, the minimum and maximum values for $R^{\prime }$ are determined, respectively.

We develop techniques for determining certain structural properties of inverse limits on compacta. In particular, we show if easily observable properties of the bonding mappings are present, then one can identify nested sequences of subcompacta of the inverse limit space whose members are copies of subcompacta of the factor spaces, and a sequence of retractions of the inverse limit space onto members of this nested sequence that converges uniformly to the identity mapping. We discuss applications to inverse limits on continua, and in the special case of a continuum $X$ that admits such a sequence of retractions onto arcs, we establish properties that describe the nature of proper subcontinua of $X$.

We look at methods for solving the Diophantine equation $a{x}^2+bxy+c{y}^2+dx+ey+f=0$ for which $\mathrm{\Delta }=b^2-4ac>0$ and $\mathrm{\Delta }$ is not a square. The methods we use transform this equation to one of the form $A{X}^2+BXY+C{Y}^2=N$. We give upper limits on the number of solutions to the latter equation that need to be reviewed to determine all solutions to the original equation. These upper limits are substantially smaller than those generally given in the literature. We also discuss ways to compactly represent all solutions to the original equation.

We discuss the existence and the uniqueness of solution of the fractional differential equation in $\left(0,1\right)$ satisfying the boundary conditions on $0$ and $1$. Using the Schauder fixed point theorem, we give the asymptotic behavior of such solution.

We establish the equivariant Cartan homotopy formula for the crossed product of $DG$-algebra obtained by a finite group action.

Using a generalization of the moment problem and the extremal properties of spectral measures corresponding to the selfadjoint extensions of a regular symmetric operator, we study point mass perturbations of spectral measures. We obtain general results for a wide class of operators and apply them to the analysis of point mass perturbations of spectral measures pertaining to Bessel and generalized Schrödinger operators.

We discuss Arens regularity of the projective tensor product of Banach algebras consisting of Schatten class operators. More precisely, exploiting Ülger’s biregularity criterion, we prove that for any Hilbert space $\mathcal{\mathscr{H}}$, the Banach algebras $S_p\left(\mathcal{\mathscr{H}}\right){\otimes }^{\gamma }{S}_q\left(\mathcal{\mathscr{H}}\right)$ and $B\left(S_2\left(\mathcal{\mathscr{H}}\right)\right){\otimes }^{\gamma }{S}_2\left(\mathcal{\mathscr{H}}\right)$ are not Arens regular for every pair $1\le p,q\le 2$; whereas, when $\mathcal{\mathscr{H}}$ is separable and $S_2\left(\mathcal{\mathscr{H}}\right)$ is equipped with the Schur product, we show that $S_2\left(\mathcal{\mathscr{H}}\right)$ is completely continuous and that $S_2\left(\mathcal{\mathscr{H}}\right){\otimes }^{\gamma }{S}_2\left(\mathcal{\mathscr{H}}\right)$ is Arens regular.

Let $K$ be a number field with ring of integers $R$ and $L=K\left(\alpha \right)$ where $\alpha$ satisfies the irreducible polynomial $P\left(X\right)=X^{3^n}+a{X}^{3^s}-b$ of $R\left[X\right]$ $\left(n,s\in \mathbb{N},n>s\right)$. We give necessary and sufficient conditions that involve only $a$, $b$, $s$, $n$ to study the monogeneity of $L$ over $K$. We also present some applications, giving integral bases of some extensions of degree $2\cdot 3^n$, as well, we give their absolute discriminant.

Graham and Pollak showed that the vertices of any connected graph $G$ can be assigned $t$-tuples with entries in $\left\{0,a,b\right\}$, called addresses, such that the distance between any two vertices can be determined from their addresses. In this paper we determine the minimum value of such $t$, called squashed-cube dimension for the total graph $T\left(\mathrm{\Gamma }\left({\mathbb{Z}}_{2^n{p}^m}\right)\right)$ where $n,m\ge 1$ and $p\ge 3$ is a prime number.

The continuous Bessel wavelet transform is extended to distributions in $H_{\mu }^{{}^{\prime }}\left(0,\infty \right)$ and obtained continuity results. Boundedness of continuous Bessel wavelet transform is investigated in a generalized Sobolev space, Besov space and Triebel–Lizorkin space.

We use the loop algebra to establish two nonisospectral problems. Two nonisospectral WKI hierarchies associated with the nonisospectral problems are derived by means of the symbolic computation by maple. It follows that the corresponding integrable Hamiltonian systems are investigated based on the Tu scheme. Additionally, we deduce the third nonisospectral WKI hierarchy based on the Lenard recursion equations. By reducing the third nonisospectral WKI hierarchy, we obtain the generalized WKI equation.

Let $V_n^{\alpha }$ be the Banach space of holomorphic functions on the open unit disk $\mathbb{𝔻}$ in the complex plane consisting of those $f$ such that and $V_{n,0}^{\alpha }$ be the closed subspace of $V_n^{\alpha }$ consisting of those $f$ for which ${lim}_{\left|z\right|\to 1}{\left(1-|z{|}^2\right)}^{\alpha }\left|f^{\left(n\right)}\left(z\right)\right|=0$, where $n$ is any nonnegative integer and $\alpha >0$. We give boundedness characterizations, norm estimates and essential norm estimates of weighted composition operators $W_{\psi ,\phi }:V_n^{\alpha }\to V_m^{\beta }$ and $W_{\psi ,\phi }:V_{n,0}^{\alpha }\to V_{m,0}^{\beta }$, respectively, where $W_{\psi ,\phi }f\left(z\right)=\psi \left(z\right)f\left(\phi \left(z\right)\right)$. As a corollary, we characterize the compactness of $W_{\psi ,\phi }$. Specifically, our characterizations involve not only the classical Julia–Carathéodory type condition, but also the powers ${\phi }^k$. In addition, our results extend several well-known results in the literature.