Limit theorems for numbers satisfying a class of triangular arrays

The paper extends the investigations of limit theorems for numbers satisfying a class of triangular arrays, defined by a bivariate linear recurrence with bivariate linear coefficients. We obtain the partial differential equation and special analytical expressions for the numbers using a semi-exponential generating function. We apply the results to prove the asymptotic normality of special classes of the numbers and specify the convergence rate to the limiting distribution. We demonstrate that the limiting distribution is not always Gaussian.

On homotopy nilpotency

We review established and recent results on the homotopy nilpotence of spaces. In particular, the homotopy nilpotency of the loop spaces \(\Omega(G/K)\) of homogenous spaces \(G/K\) for a compact Lie group \(G\) and its closed homotopy nilpotent subgroup \(K \lt G\) is discussed.

Hilbert \(C^{*}\)-modules in which all relatively strictly closed submodules are complemented

We give the characterization and description of all full Hilbert modules and associated algebras having the property that each relatively strictly closed submodule is orthogonally complemented. A strict topology is determined by an essential closed two-sided ideal in the associated algebra and a related ideal submodule. It is shown that these are some modules over hereditary algebras containing the essential ideal isomorphic to the algebra of (not necessarily all) compact operators on a Hilbert space. The characterization and description of that broader class of Hilbert modules and their associated algebras is given. As auxiliary results we give properties of strict and relatively strict submodule closures, the characterization of orthogonal closedness and orthogonal complementing property for single submodules, relation of relative strict topology and projections, properties of outer direct sums with respect to the ideals in \(\ell_\infty\) and isomorphisms of Hilbert modules, and we prove some properties of hereditary algebras and associated hereditary modules with respect to the multiplier algebras, multiplier Hilbert modules, corona algebras and corona modules.

On the Ramanujan-Nagell type Diophantine equation \(Dx^2+k^n=B\)

In this paper, we prove that the Ramanujan-Nagell type Diophantine equation \(Dx^2+k^n=B\) has at most three nonnegative integer solutions \((x, n)\) for \(k\) a prime and \(B, D\) positive integers.

Continuity of generalized Riesz potentials for double phase functionals with variable exponents

In this note, we discuss the continuity of generalized Riesz potentials \( I_{\rho}f\) of functions in Morrey spaces \(L^{\Phi,\nu(\cdot)}(G)\) of double phase functionals with variable exponents.

Determinants of some pentadiagonal matrices

In this paper we consider pentadiagonal \((n+1)\times(n+1)\) matrices with two subdiagonals and two superdiagonals at distances \(k\) and \(2k\) from the main diagonal where \(1\le k \lt 2k\le n\). We give an explicit formula for their determinants and also consider the Toeplitz and “imperfect” Toeplitz versions of such matrices. Imperfectness means that the first and last \(k\) elements of the main diagonal differ from the elements in the middle. Using the rearrangement due to Egerváry and Szász we also show how these determinants can be factorized.

Groups \(S_n \times S_m\) in construction of flag-transitive block designs

In this paper, we observe the possibility that the group \(S_{n}\times S_{m}\) acts as a flag-transitive automorphism group of a block design with point set \(\{1,\ldots ,n\}\times \{1,\ldots ,m\},4\leq n\leq m\leq 70\). We prove the equivalence of that problem to the existence of an appropriately defined smaller flag-transitive incidence structure. By developing and applying several algorithms for the construction of the latter structure, we manage to solve the existence problem for the desired designs with \(nm\) points in the given range. In the vast majority of the cases with confirmed existence, we obtain all possible structures up to isomorphism.

Approximation of nilpotent orbits for simple Lie groups

We propose a systematic and topological study of limits \(\lim_{\nu\to 0^+}G_\mathbb{R}\cdot(\nu x)\) of continuous families of adjoint orbits for a non-compact simple real Lie group \(G_\mathbb{R}\). This limit is always a finite union of nilpotent orbits. We describe explicitly these nilpotent orbits in terms of Richardson orbits in the case of hyperbolic semisimple elements. We also show that one can approximate minimal nilpotent orbits or even nilpotent orbits by elliptic semisimple orbits. The special cases of \(\mathrm{SL}_n(\mathbb{R})\) and \(\mathrm{SU}(p,q)\) are computed in detail.

Semiflows and intrinsic shape in topological spaces

In this paper we apply the intrinsic approach to shape to study attractors in topological spaces.

Regularity of a weak solution to a linear fluid-composite structure interaction problem

In this manuscript, we deal with the regularity of a weak solution to the fluid-composite structure interaction problem introduced in [12]. The problem describes a linear fluid-structure interaction between an incompressible, viscous fluid flow, and an elastic structure composed of a cylindrical shell supported by a mesh-like elastic structure. The fluid and the mesh-supported structure are coupled via the kinematic and dynamic boundary coupling conditions describing continuity of velocity and balance of contact forces at the fluid-structure interface. In [12], it is shown that there exists a weak solution to the described problem. By using the standard techniques from the analysis of partial differential equations we prove that such a weak solution possesses an additional regularity in both time and space variables for initial and boundary data satisfying the appropriate regularity and compatibility conditions imposed on the interface.