Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica Feed

On multiplicative intersections between the Pell and Jacobsthal numbers

Tue, 03 Nov 2026 00:00:00 GMT

We determine all integer triples (k, m, n) satisfying the Diophantine equations Pk = JmJn and Jk = PmPn, thereby characterizing Pell numbers that are products of two Jacobsthal numbers, and vice versa. In addition, we study the intersection of the Pell and Jacobsthal sequences and show that the only common values occur when m = 1, corresponding to the integers 1 and 5.

On a Pascal-type theorem for lines in ℙ3

Sat, 19 Sep 2026 00:00:00 GMT

We prove a three-dimensional Pascal-type theorem for configurations of lines on a smooth quadric surface in ℙ3. Starting from two triples of skew lines in the two rulings of the quadric, we show that the associated grid of nine intersection points determines natural triples of planes whose residual intersection lines are coplanar. This construction recovers the classical Pascal line after taking a plane section of the quadric. We further study the six diagonal planes associated with permutations of the grid points, showing that they split into two pencils with skew polar axes, and describe a canonical involutive mirror construction on the grid. We work over the field of complex numbers ℂ.

On an Erdős–Straus type problem

Mon, 20 Jul 2026 00:00:00 GMT

Motivated by the Erdős–Straus conjecture, we study whether the Euler totient ratio φ(n)n \frac{\varphi(n)}{n} can be expressed as a sum of three unit fractions, φ(n)n=1x+1y+1z,x,y,z∈ℕ. \frac{\varphi(n)}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}, \quad x, y, z \in \mathbb{N}. We prove that such a representation is impossible for prime powers pk with p ≥ 11, and for integers of the form pk1 qk2 with p ≥ 5 and q ≥ 15p. In contrast, for 2 ≤ p ≤ 7 the representation exists for all pk. Moreover, for every integer r ≥ 3, there are infinitely many integers n with ω(n)=r for which no such representation exists. Further, if a representation holds, we obtain the bounds 2≤x≤9+3eγlog log n,nxφ(n)x−n≤y≤2nxφ(n)x−n. 2 \le x \le 9 + 3e^{\gamma} \log \log n, \quad \frac{nx}{\varphi(n)x - n} \le y \le \frac{2nx}{\varphi(n)x - n}. In addition, if x≥3(n-1)(eγlog log n+3)n x \geq \frac{3(n-1)(e^{\gamma} \log \log n + 3)}{n} , then rad (n)≤42r-42r-13 (n) \leq \frac{4^{2^{r}} - 4^{2^{r-1}}}{3} , where γ is Euler’s constant and r = ω(n).

Report of Meeting: 21st International Conference on Functional Equations and Inequalities, Będlewo, Poland, September 14–20, 2025

Tue, 10 Mar 2026 00:00:00 GMT

Local convergence analysis of a bi-parametric iterative method in ℝ or ℂ

Tue, 10 Mar 2026 00:00:00 GMT

In this manuscript, we accelerate the local convergence of a third-order biparametric iterative method in ℝ or ℂ by assuming that the first-order Fréchet derivative satisfies the Generalized continuity condition. We extend this analysis by using the Hölder continuity condition, which allows us to solve more numerical problems. Our study also shows the sizes of the convergence balls, the smallest error bounds that can be computed, and the fact that the answer is unique. Several math tests show that this third-order method gives better results than the midpoint method established by I.K. Argyros and S. George [4]. This method solves problems that earlier studies have not been able to solve.

On Pólya-Szegö integral inequalities using k-Hilfer fractional derivative

Thu, 05 Mar 2026 00:00:00 GMT

The main aim of this paper is to use the k-Hilfer fractional derivative to derive certain Pólya-Szegö fractional integral inequalities. Further fractional integral inequalities are obtained. The results presented here extend and generalize various existing inequalities associated with the Riemann-Liouville, Caputo, Saigo, and Hilfer fractional operators. The findings contribute to the growing theory of fractional calculus and offer potential tools for the analysis of fractional differential and integral equations.

Asymptotic behavior of generalized self-similar solutions for a nonlinear hybrid problem of porous medium equations

Wed, 31 Dec 2025 00:00:00 GMT

The present paper investigates the asymptotic behavior of positive generalized self-similar solutions for a nonlinear hybrid problem involving nth-order derivative porous medium equations. We provide sufficient conditions for the existence and uniqueness of weak solutions that have compact support and dynamic characteristics. Furthermore, we establish the behavior of these solutions by examining a specific set of variables and their signs, which must meet certain conditions to determine whether the solutions exist globally or locally in time.

Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica 25 (2025)

Tue, 23 Dec 2025 00:00:00 GMT

We consider ℓp-direct sums (1 ≤ p < ∞) and c0-direct sums of countably many normed spaces and find the dual of these spaces. We characterize the support functionals of arbitrary elements in these spaces to characterize smoothness and approximate smoothness, both locally and globally. These results let us answer the Chmieliński, Khurana, and Sain question raised in [4] on the existence of a non-approximately smooth normed space whose every element is smooth. We also characterize Birkhoff-James orthogonality and its pointwise symmetry in these spaces.

Notes on Jordan type of an Artinian algebra

Fri, 28 Nov 2025 00:00:00 GMT

These notes were made for the Preparatory School on Lefschetz Properties, held from 6th to 10th May 2024, in Kraków, ahead of the conference “Lefschetz properties in algebra, geometry, topology, and combinatorics”. They are a short introduction to Jordan type of an Artinian algebra, and collect basic results known so far, with examples to illustrate them along the way.

Some application of Grunsky coefficients in the theory of univalent functions

Wed, 26 Nov 2025 00:00:00 GMT

Let function f be normalized, analytic and univalent in the unit disk 𝔻 = {z : |z| < 1} and f(z)=z+∑n=2∞anzn f\left( z \right) = z + \sum\nolimits_{n = 2}^\infty {{a_n}{z^n}} . Using a method based on Grusky coefficients we study several problems over that class of univalent functions: upper bounds of the special case of the generalized Zalcman conjecture |a2a3 − a4|, of the third logarithmic coefficient, and of the second Hankel determinant for the logarithmic coefficients.

A general Hardy-Hilbert-type integral inequality theorem

Sat, 01 Nov 2025 00:00:00 GMT

This article presents a unified framework that extends the scope of two existing theorems on multivariate Hardy-Hilbert-type integral inequalities. Key to this extension is the use of two additional adjustable parameters that increase flexibility and generality. The framework also has the originality of including the incomplete lower gamma function in the integral definitions governed by a parameter. Detailed proofs are given, mainly based on the Laplace transform, the generalized Young inequality, the generalized Hölder integral inequality and changes of variables. This article thus provides a new comprehensive foundation for future research in generalized multivariate integral inequalities.

Lefschetz Properties in Algebra, Geometry and Combinatorics: Notes for the preparatory school

Fri, 11 Jul 2025 00:00:00 GMT

These notes represent background and supplementary material for our course in the Preparatory School for the conference “Lefschetz Properties in Algebra, Geometry and Combinatorics.” The school was held in Kraków in May, 2024 and the conference took place in June, 2024.

Lefschetz properties through a topological lens

Sat, 28 Jun 2025 00:00:00 GMT

These notes were prepared for the Lefschetz Preparatory School, a graduate summer course held in Krakow, May 6–10, 2024. They present the story of the algebraic Lefschetz properties from their origin in algebraic geometry to some recent developments in commutative algebra. The common thread of the notes is a bias towards topics surrounding the algebraic Lefschetz properties that have a topological flavor. These range from the Hard Lefschetz Theorem for cohomology rings to commutative algebraic analogues of these rings, namely artinian Gorenstein rings, and topologically motivated operations among such rings.

Application of sine-Gordon model in description of Josephson junction – present state of affairs

Wed, 12 Mar 2025 00:00:00 GMT

Josephson junctions find applications for construction of various devices in measuring, transmitting, receiving, and amplifying systems as well as classical and quantum computing devices. Some of the existing and future applications are related to shape engineering, which allows for adjustment of the parameters of the junction to the needs of various applications. The article includes review of both a description of the applications and theoretical considerations devoted to the Josephson junctions and their properties.

λ-statistical convergence of double sequences of functions via difference operators

Mon, 23 Dec 2024 00:00:00 GMT

The paper aims to investigate λ-statistical convergence using modulus function and a generalized difference operator for double sequences of functions for order γ ∈ (0, 1]. Further, we prove that the statistical convergence in the newly formed sequence spaces is not well defined for γ > 1. Finally, we examine relevant inclusion relations concerning λ-statistical convergence and strongly λ-summable in the environment of the newly defined classes of double sequences of functions. Some interesting examples related to the examined results are also discussed in this paper.

Centering toric arrangements of maximal rank

Sat, 23 Nov 2024 00:00:00 GMT

The homotopy type of the complement manifold of a complexified toric arrangement has been investigated by d’Antonio and Delucchi in a paper that shows the minimality of such topological space. In this work we associate to a given toric arrangement a matrix that represents the arrangement over the integers. Then, we consider the family of toric arrangements for which this matrix has maximal rank. Our goal is to prove, by means of basic linear algebra arguments, that the complement manifold of the toric arrangements that belong to this family is diffeomorphic to that of centered toric arrangements and thus it is a minimal topological space, too.

δµ-connectedness in a µ-proximity space

Wed, 17 Jul 2024 00:00:00 GMT

In this paper we introduce the notion of δµ-connectedness on a µ-proximity space. It has been proved that δµ-connectedness can be characterized by δµ-continuous functions. We initiate the idea of δµ-chain and establish some results related to this. The concepts of δµ-component and δµ-quasi component have been introduced and their interrelation has been studied.

Rings with centrally-extended Jordan epimorphisms

Thu, 28 Mar 2024 00:00:00 GMT

The aim of this article is to introduce the concept of centrally-extended Jordan epimorphisms and proving that if R is a non-commutative prime ring (∗-ring) of characteristic not two, and G is a CE-Jordan epimorphism such that [G(x), x] ∈ Z(R) ([G(x), x∗] ∈ Z(R)) for all x ∈ R, then R is an order in a central simple algebra of dimension at most 4 over its center or there is an element λ in the extended centroid of R such that G(x) = λx (G(x) = λx∗) for all x ∈ R.

Some theoretical results on fractional-order continuous information measures

Sat, 03 Feb 2024 00:00:00 GMT

By rewriting the differential entropy in a form of a differ-integral function’s limit, and deforming the ordinary derivative to a fractional-order one, we derive in this paper a novel generalized fractional-order differential entropy along with its related information measures. When the order of fractional differentiation α → 1, the ordinary Shannon’s differential entropy is recovered, which corresponds to the results from first-order ordinary differentiation.

On maximizing line arrangements in the complex plane

Sat, 03 Feb 2024 00:00:00 GMT

In this note we provide a complete classification of weak combinatorics of the so-called maximizing line arrangements in the complex projective plane.