We give explicit formulas for the geodesic growth series of a right-angled Coxeter group based on a link-regular graph that does not contain 4-cliques.

An abstract theory of Fourier series in locally convex topological vector spaces is developed. An analog of Fejér’s theorem is proved for these series. The theory is applied to distributional solutions of Cauchy–Riemann equations to recover basic results of complex analysis. Some classical results of function theory are also shown to be consequences of the series expansion.

Stabilization of fractional-order nonlinear systems is studied. This paper investigates the problem of global Mittag Leffler stabilization by state and output-feedback for a family of nonlinear Caputo fractional-order systems written in triangular form satisfying linear growth conditions. By using a Lyapunov function, global Mittag-Leffler stability of the closed-loop systems is achieved. Finally, simulation results dealing with a typical bioreactor example are given to illustrate that the proposed design procedures are very efficient and simple.

Finding an effective formula for describing the discriminant of a quadrinomial (a formula which can be easily computed for high values of degrees of quadrinomials) is a difficult problem. In 2018, Otake and Shaska, using advanced matrix operations, found an explicit expression for $\mathrm{\Delta }(x^n+t(x^2+ax+b))$. In this paper, we focus on deriving similar results, taking advantage of an alternative elementary approach, for quadrinomials of the form $x^n+a{x}^k+bx+c$, where $k\in \{2,3,n-1\}$. Moreover, we make some notes about $\mathrm{\Delta }(x^{2n}+a{x}^n+b{x}^l+c)$, where .

We prove that a nilpotent subgroup $H$ of a finite solvable group $G$ has order at most $|V{|}^{\beta }∕2$ if $V$ is a faithful and completely reducible $G$-module, where $\beta =\text{ln}(32)∕\text{\hspace{0.17em}ln}(9)$. We also find related bounds for nilpotent subgroups of odd order in a solvable linear group. We then further generalize these results to certain chief factors of an arbitrary linear group.

We deal with the existence of positive solutions for nonlinear fractional differential equations whose nonlinearity contains the first-order derivative in a Banach space $E$. Under the superlinear and sublinear growth of nonlinear $f$, which contains the first-order derivative, we obtain the existence of positive solutions. Our discussion is based on the fixed point index theory of condensing mapping in cones.

A fractional two-variable Hardy–Sobolev inequality and its corresponding best constant are considered in this paper. As an application, we establish existence and uniqueness results of a ground state solution to a system with fractional critical Hardy–Sobolev exponents.

This paper establishes a criterion of weighted compactness for Calderón type commutators and their oscillatory operators with polynomial phases. As applications, the compactness in weighted Lebesgue spaces for such operators with rough kernels are obtained. These results are new even in the unweighted cases.

This paper is concerned with the characterization for boundedness of the generalized Hausdorff operator on weighted spaces of integrable functions. As corollaries, the description of the generalized Hausdorff operator on $L_1(w)$ is presented for different cases of $w$. Finally, we conclude with the range of the operator.

An integral Jensen–Mercer inequality for weights satisfying conditions for the reversed Jensen–Steffensen inequality is proved here. Several integral inequalities involving more than one monotonic functions with reversed Jensen–Steffensen conditions are proved as well. Furthermore, a couple of general companion inequalities related to the integral Jensen–Mercer inequality with reversed Jensen–Steffensen conditions are presented. Applications for the generalization of weighted Ky Fan’s inequality, classical power mean and classical arithmetic, geometric and harmonic mean inequalities involving bounded variation are also given.

Let $(n,q)$ denote the greatest common divisor of positive integers $n$ and $q$, and let $f$ denote the characteristic function of squarefull numbers. Then we shall consider the asymptotic formula for sum of the modified squarefull numbers, which is ${\sum }_{n\le y}{\sum }_{q\le x}{\sum }_{d|(n,q)}df(q∕d)$ with any positive real numbers $x$ and $y$.

We consider the long-time asymptotic behavior for the Cauchy problem of the integrable coupled Hirota equations with $3\times 3$ matrix spectral problem on the line with the initial value belonging to the Schwartz space. The precise asymptotic formulas and uniform error estimates are presented based on the nonlinear steepest descent analysis of an associated matrix Riemann–Hilbert problem.

The aim of this paper is to give the matching BiHom-types of various algebraic structures, such as Rota–Baxter associative (or Lie) algebras, (tri)dendriform algebras, pre-Lie algebras, associative post-Lie algebras, compatible Lie algebras, Zinbiel algebras, compatible associative algebras, polarized associative Yang–Baxter equations, etc., and also to investigate the relations among them.

We prove new Grüss type inequalities for the Berezin symbol of some operator products. As biproducts, we find new relations between the Berezin number, the Berezin norm and some new quantities for operators acting on the reproducing kernel Hilbert space. In particular, we prove for arbitrary bounded linear operator $A$ that

$$\text{ber}(A^2)\le \frac{1}{4}\left((\text{ber}(A^4))+\parallel A^2{\parallel }_{\text{Ber}}^2\right),$$

where $\text{ber}(\cdot )$ and $\parallel \cdot {\parallel }_{\text{Ber}}$ denote, respectively, the Berezin number and the Berezin norm of operator $A$.

We consider modifications of 3D Navier–Stokes equations involving unbounded delays in $\mathrm{\Omega }\subset {ℝ}^3$. First, we show the existence and uniqueness of weak solutions by the Galerkin approximations method and energy method. Second, we prove the existence and uniqueness of stationary solutions by the Brouwer fixed point theorem. Finally, we study the stability of stationary solutions by the classical directly approach and construction of Lyapunov function. We also give a sufficient condition for the polynomial stability and polynomial decay of the stationary solutions in some special cases of unbounded variable delays.

We give a relation between the Hilbert multiplicity and the irreducible multiplicity. Here, the irreducible function is defined by the socle dimension of residue modules that define the Hilbert function. As an application, we characterize Ulrich modules in terms of the irreducible multiplicity.

Considered herein is the Cauchy problem of the two-component Camassa–Holm-type system with higher-order nonlinearity. Based on the local well-posedness results of this problem, we establish that the solution map $z_0\mapsto z(t)$ of this problem in the periodic case is not uniformly continuous in Besov spaces $B_{p,r}^s(\mathbb{𝕋})\times B_{p,r}^{s-1}(\mathbb{𝕋})$ with and $B_{2,1}^{3∕2}(\mathbb{𝕋})\times B_{2,1}^{1∕2}(\mathbb{𝕋})$ through the method of approximate solutions. This phenomenon reveals that the local well-posedness results of the solutions to this problem in the corresponding Besov spaces cannot be obtained by a sole contraction principle argument.

We study the existence results for the fractional Kirchhoff-type Schrödinger–Poisson system

where ${(-\mathrm{\Delta })}^s$ is the fractional Laplacian, $s,t\in (0,1)$, ${[u]}_s$ is the Gagliardo seminorm of $u$, $M:{ℝ}_0^{+}\to {ℝ}^{+}$ is a continuous function satisfying certain assumptions, $V:{ℝ}^3\to {ℝ}^{+}$ is continuous and $f:{ℝ}^3\times ℝ\to ℝ$ satisfies a Carathéodory condition without the Ambrosetti and Rabinowitz condition. With the help of Morse theory, we obtain the existence of nontrivial solutions for the above problem. Furthermore, by appealing to the new version of Clark’s theorem due to Liu and Wang (Ann. Inst. H. Poincaré C Anal. Non Linéaire 32:5 (2015), 1015–1037), we establish the existence of infinitely many solutions for the above system.

By reformulating the Gauss summation theorem with a free variable $x$ so that both sides of the equality can be expanded into Maclaurin series, we evaluate, in closed form, many different infinite series containing harmonic-like numbers and squared central binomial coefficients.

We construct an infinite matrix $\mathfrak{𝔖}=(s_{nk})$ defined by

for all $n,k=1,2,3,\dots$, where $S(n)$ stands for the sum of the positive divisors of $n$, and introduce sequence spaces ${\ell }_p(\mathfrak{𝔖})$, $c_0(\mathfrak{𝔖})$, $c(\mathfrak{𝔖})$ and ${\ell }_{\infty }(\mathfrak{𝔖})$ by employing the matrix $\mathfrak{𝔖}$, where . We construct Schauder bases and compute $\alpha$-, $\beta$- and $\gamma$- duals of the newly constructed spaces. We state and prove characterization theorems related to matrix transformation from the spaces ${\ell }_p(\mathfrak{𝔖})$, $c_0(\mathfrak{𝔖})$, $c(\mathfrak{𝔖})$ and ${\ell }_{\infty }(\mathfrak{𝔖})$ to the spaces ${\ell }_{\infty }$, $c$, $c_0$ and ${\ell }_1$. Finally, we determine essential conditions for compactness of a matrix operator from the sequence space $X\in \left\{{\ell }_p(\mathfrak{𝔖}),c_0(\mathfrak{𝔖}),c(\mathfrak{𝔖}),{\ell }_{\infty }(\mathfrak{𝔖})\right\}$ to any of the sequence spaces ${\ell }_{\infty }$, $c$, $c_0$ or ${\ell }_1$.

We prove that the local multilinear fractional maximal operator is bounded from $L^{p_1}(\mathrm{\Omega })\times \cdots \times L^{p_m}(\mathrm{\Omega })$ to $W^{1,q}(\mathrm{\Omega })$ when and $1∕q=1∕p_1+\cdots +1∕p_m$ with , which was previously unknown for the case . In addition, we introduce and investigate the Sobolev regularity properties of the local multilinear strong maximal operator, as well as its fractional variant. Several new pointwise estimates for the weak gradients of the above maximal functions will be established when $\overrightarrow{f}=(f_1,\dots ,f_m)$, with each $f_j\in W^{1,p_j}(\mathrm{\Omega })$ for some $p_j\in (1,\infty )$. As applications, we obtain certain boundedness for the above operators in the first-order Sobolev spaces and the Sobolev spaces with zero boundary values.