Abstract In a paper from 2021, Albrecht Böttcher, Juanita Gasca, Sergei M. Grudsky, and Anatoli V. Kozak gave a precise and complete description of all types of the limit Schmidt–Spitzer sets for tetradiagonal Toeplitz matrices. In this paper, we consider one of these possible cases, when the limit set consists of two analytic arcs that join at one point forming a cusp. For this family of Toeplitz matrices, we provide asymptotic formulas for every eigenvalue as the order of the matrix tends to infinity. Our analysis provides a theoretical understanding of the structural behavior of the eigenvalues, while the obtained formulas enable high-precision calculation of the eigenvalues.

Abstract In this article, totally positive Riordan arrays are studied. Some Riordan arrays that were already known to be totally positive but given in terms of A - and Z -sequence are presented in an explicit form. Some of them are connected to Narayana polynomials. With use of these matrices and properties of total positivity, some other totally positive Riordan arrays are found.

Abstract We work with the weighted poly-Bergman spaces defined on the unit disk, which are subspaces of the weighted $$L^2$$ L 2 space over the same domain. Particularly, we take interest in the extended Fock space formalism theory applied to this case and generalize some of the second author’s results from the standard to the weighted case. Using properties of the elements of an orthonormal basis for the weighted $$L^2$$ L 2 space, comprising orthogonal polynomials referred to as disk polynomials, we express a variety of operators, including pure isometries, in a basis-independent manner, hence completing a description to the extended Fock space corresponding to the ladder operators defined on the space.

Abstract We study the global in time existence of small solutions for subcritical fractional modified Korteweg–de Vries equation $$\begin{aligned} \left\{ \begin{array}{c} \partial _{t}u+\frac{1}{\alpha }\left| \partial _{x}\right| ^{\alpha -1}\partial _{x}u=t^{\nu }\partial _{x}\left( u^{3}\right) ,\text t>0\textbf{,}x\in \mathbb {R},\\ u\left( 0,x\right) =u_{0}\left( x\right) , x\in \mathbb {R}\textbf{,} \end{array} \right. \end{aligned}$$ ∂ t u + 1 α ∂ x α - 1 ∂ x u = t ν ∂ x u 3 , t > 0 , x ∈ R , u 0 , x = u 0 x , x ∈ R , where $$\alpha \in \left( \frac{3}{2},3\right) $$ α ∈ 3 2 , 3 and $$\nu \in \left( 0,\nu _{\alpha }\right) ,$$ ν ∈ 0 , ν α , $$\nu _{\alpha }=\frac{1}{24}$$ ν α = 1 24 for $$\frac{3}{2} <\alpha \le \frac{32}{11},$$ 3 2 < α ≤ 32 11 , $$\nu _{\alpha }=\frac{1}{3}\left( \frac{4}{\alpha }-\frac{5}{4}\right) $$ ν α = 1 3 4 α - 5 4 for $$\frac{32}{11}\le \alpha <3$$ 32 11 ≤ α < 3 , solutions u and the initial data $$u_{0}$$ u 0 are the real-valued functions. We remark that $$\nu >0$$ ν > 0 means that equation is subcritical in the sense of the large time asymptotic behavior of solutions. We assume that the initial data have an analytic extension on the sector and are small. Then we find the large time asymptotics of the solutions.