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Karol Gryszka

Pedagogical University of Cracow, Institute of Mathematics, Adjunct

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University of California, Santa Barbara

Antioch University

University of British Columbia

University of Zagreb

University of British Columbia

Kansas State University

Armando Marques-Guedes

UNL - New University of Lisbon

Prof. Dr. Raffaele Pisano, MSc, HDR (Habil.)

Université des Sciences et Technologies de Lille (Lille-1)

Patrick Blessinger

St. John's University

DAMIAN K O F I MEREKU

University Of Education, Winneba

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Papers by Karol Gryszka

$Research paper thumbnail of The identity $7^{3n}+7^{3n+1}=(2\cdot 7^n)^3$ and its generalizations$

The identity $7^{3n}+7^{3n+1}=(2\cdot 7^n)^3$ and its generalizations

Annales Universitates Paedagogicae Cracoviensis. Studia ad Didacticam Mathematicae Pertinentia, Dec 30, 2023

Starting with the identity 7 3n +7 3n+1 = (2•7 n ) 3 and its sibling, we prove that for any posit... more Starting with the identity 7 3n +7 3n+1 = (2•7 n ) 3 and its sibling, we prove that for any positive integer m, the Diophantine equation x n + x n+k = z m has infinitely many solutions in nonzero integers x, z, n and k. We show that in case k > 1 the solutions come from Catalan's Conjecture. We also solve three similar Diophantine equations.

General convolution sums involving Fibonacci m-step numbers

arXiv (Cornell University), Mar 20, 2024

In this paper, using a generating function approach, we derive several new convolution sum identi... more In this paper, using a generating function approach, we derive several new convolution sum identities involving Fibonacci m-step numbers. As special instances of the results derived herein, we will get many new and known results involving Fibonacci, Tribonacci, Tetranacci and Pentanacci numbers. In addition, we establish some general results providing insights into the inner structure of such convolutions. Finally, some mixed convolutions involving Fibonacci m-step numbers and Jacobsthal and Pell numbers will be stated.

A note on generating identities for multiplicative arithmetic functions

Contributions to mathematics, Dec 31, 2023

In this article, it is shown that many of the identities involving multiplicative arithmetic func... more

Convex functions and approximate Birkhoff–James orthogonality

Aequationes Mathematicae, Oct 8, 2023

For the well developed notion of approximate Birkhoff-James orthogonality, in a real or complex n... more For the well developed notion of approximate Birkhoff-James orthogonality, in a real or complex normed linear space, we formulate a new characterization. It can be derived from other, already known, characterizations as well as obtained in a more elementary and direct way, on the basis of some simple inequalities for real convex functions.

Proximality and transitivity in relation to points that are asymptotic to themselves

Turkish Journal of Mathematics

Pre-Grüss and Grüss-Ostrowski like inequalities in Banach spaces

Mathematical Inequalities & Applications, 2022

For a given Banach space and its dual space we investigate a Chebyshev type functional. We derive... more For a given Banach space and its dual space we investigate a Chebyshev type functional. We derive a pre-Grüss inequality for the functional. We discuss various variants of assumptions leading to this inequality. To do so, we employ some superquadratic as well as convex control functions in order to weaken the classical Dragomir's condition. Next, we establish a corresponding Grüss-Ostrowski like inequalities for the space L p [a,b] .

Proof Without Words: Sum of Fibonacci Numbers and Beyond

The Mathematical Intelligencer, Jun 22, 2022

A Brief Panorama of Period-Like Motions in Dynamical Systems

Sciendo eBooks, Dec 31, 2021

In the article we draw our attention on the selection of existing notions of non-periodic motions... more In the article we draw our attention on the selection of existing notions of non-periodic motions that generalize periodic motions. These are from classic recurrence to slightly less known and recent ones such as near periodicity or G-asymptotic periodicity. We provide mutual relations between all of them and fill the existing gaps that arise in their investigation.

From biased coin to any discrete distribution

Periodica Mathematica Hungarica, Oct 29, 2020

In this note we construct an algorithm generating any discrete distribution with an arbitrary coi... more In this note we construct an algorithm generating any discrete distribution with an arbitrary coin (and, as a result, with arbitrary initial distribution). The coin need not be fair and the target distribution can be supported on a countable set. Keywords Discrete probability measure • discrete distribution • expected value • coin flip • biased coin • simulation of distribution

Identities Arising from Binomial-Like Formulas Involving Divisors of Numbers

Annales Mathematicae Silesianae, Jul 26, 2023

In this article, we derive a great number of identities involving the ω function counting distinc... more

Asymptotic period in dynamical systems in metric spaces

Colloquium Mathematicum, 2015

Semi-smooth Points in Some Classical Function Spaces

Results in Mathematics, Nov 24, 2021

The investigations of the smooth points in the spaces of continuous function were started by Bana... more The investigations of the smooth points in the spaces of continuous function were started by Banach in 1932 considering function space C(Ω). Singer and Sundaresan extended the result of Banach to the space of vector valued continuous functions C(T , E), where T is a compact metric space. The aim of this paper is to present a description of semi-smooth points in spaces of continuous functions C0(T , E) (instead of smooth points). Moreover, we also find necessary and sufficient condition for semi-smoothness in the general case.

Generalized orthogonality equations in finite-dimensional normed spaces

Annals of Functional Analysis, Feb 28, 2023

Let X , Y be real normed spaces and let ρ + , ρ − be norm derivatives. In this work, we solve a s... more Let X , Y be real normed spaces and let ρ + , ρ − be norm derivatives. In this work, we solve a system of functional equations ρ + (f (x), f (y)) = g(x)ρ + (x, y), ρ − (f (x), f (y)) = g(x)ρ − (x, y), with unknown functions f : X → Y , g : X → R. Moreover, we give partial answer to open problem posed in 2010.

Lefschetz numbers and asymptotic periods

Wydawnictwo Uniwersytetu Łódzkiego eBooks, 2022

In this note we prove several results linking Lefschetz numbers with asymptotic behaviour of the ... more

Hausdorff Dimension is onto

K o ł o M a te ma tyk ó w U n i w e r sy tetu P e d a g o g ic z ne go w K r a k o w ie

On the $$L^1$$ and pointwise divergence of continuous functions

Periodica Mathematica Hungarica, Nov 4, 2022

For a sequence of continuous functions f 1 , f 2 ,. .. : I → R (I is a fixed closed interval) wit... more For a sequence of continuous functions f 1 , f 2 ,. .. : I → R (I is a fixed closed interval) with f 1 ≤ f 2 ≤ • • • define a set I f := x ∈ I : lim n→∞ f n (x) = +∞. We study the properties of the family of all admissible I f-s and the family of all admissible I f-s under the additional assumption lim n→∞ y x f n (t) dt = +∞ for all x, y ∈ I with x < y. The origin of this problem is the limit behaviour of quasiarithmetic means.

On Weak Asymptotic Periodicity

International Journal of Bifurcation and Chaos, Feb 1, 2020

We introduce the asymptotic property associated with recurrence-like behavior of orbits in dynami... more We introduce the asymptotic property associated with recurrence-like behavior of orbits in dynamical systems in general metric spaces. We define a notion of weak asymptotic periodicity and determine its elementary properties and relations including the invariance by topological conjugacy. We use the equicontinuity and the topology of the space to describe necessary and sufficient conditions for the existence of such a behavior.

On asymptotically periodic-like motions in flows

Annales Universitatis Paedagogicae Cracoviensis, Dec 1, 2018

We study three properties associated to the recurrence of orbits in flows: asymptotic periodicity... more We study three properties associated to the recurrence of orbits in flows: asymptotic periodicity, positive asymptotic periodicity and G-asymptotic periodicity. We determine which implications between these notions hold and which do not. We also show how these notions are related to Lyapunov stability.

Identities Arising from Binomial-Like Formulas Involving Divisors of Numbers

Annales Mathematicae Silesianae

In this article, we derive a great number of identities involving the ω function counting distinc... more

Lefschetz numbers and asymptotic periods

Wydawnictwo Uniwersytetu Łódzkiego eBooks, 2022

In this note we prove several results linking Lefschetz numbers with asymptotic behaviour of the ... more

$Research paper thumbnail of The identity $7^{3n}+7^{3n+1}=(2\cdot 7^n)^3$ and its generalizations$

The identity $7^{3n}+7^{3n+1}=(2\cdot 7^n)^3$ and its generalizations

Annales Universitates Paedagogicae Cracoviensis. Studia ad Didacticam Mathematicae Pertinentia, Dec 30, 2023

Starting with the identity 7 3n +7 3n+1 = (2•7 n ) 3 and its sibling, we prove that for any posit... more Starting with the identity 7 3n +7 3n+1 = (2•7 n ) 3 and its sibling, we prove that for any positive integer m, the Diophantine equation x n + x n+k = z m has infinitely many solutions in nonzero integers x, z, n and k. We show that in case k > 1 the solutions come from Catalan's Conjecture. We also solve three similar Diophantine equations.

General convolution sums involving Fibonacci m-step numbers

arXiv (Cornell University), Mar 20, 2024

In this paper, using a generating function approach, we derive several new convolution sum identi... more In this paper, using a generating function approach, we derive several new convolution sum identities involving Fibonacci m-step numbers. As special instances of the results derived herein, we will get many new and known results involving Fibonacci, Tribonacci, Tetranacci and Pentanacci numbers. In addition, we establish some general results providing insights into the inner structure of such convolutions. Finally, some mixed convolutions involving Fibonacci m-step numbers and Jacobsthal and Pell numbers will be stated.

A note on generating identities for multiplicative arithmetic functions

Contributions to mathematics, Dec 31, 2023

In this article, it is shown that many of the identities involving multiplicative arithmetic func... more

Convex functions and approximate Birkhoff–James orthogonality

Aequationes Mathematicae, Oct 8, 2023

For the well developed notion of approximate Birkhoff-James orthogonality, in a real or complex n... more For the well developed notion of approximate Birkhoff-James orthogonality, in a real or complex normed linear space, we formulate a new characterization. It can be derived from other, already known, characterizations as well as obtained in a more elementary and direct way, on the basis of some simple inequalities for real convex functions.

Proximality and transitivity in relation to points that are asymptotic to themselves

Turkish Journal of Mathematics

Pre-Grüss and Grüss-Ostrowski like inequalities in Banach spaces

Mathematical Inequalities & Applications, 2022

For a given Banach space and its dual space we investigate a Chebyshev type functional. We derive... more For a given Banach space and its dual space we investigate a Chebyshev type functional. We derive a pre-Grüss inequality for the functional. We discuss various variants of assumptions leading to this inequality. To do so, we employ some superquadratic as well as convex control functions in order to weaken the classical Dragomir's condition. Next, we establish a corresponding Grüss-Ostrowski like inequalities for the space L p [a,b] .

Proof Without Words: Sum of Fibonacci Numbers and Beyond

The Mathematical Intelligencer, Jun 22, 2022

A Brief Panorama of Period-Like Motions in Dynamical Systems

Sciendo eBooks, Dec 31, 2021

In the article we draw our attention on the selection of existing notions of non-periodic motions... more In the article we draw our attention on the selection of existing notions of non-periodic motions that generalize periodic motions. These are from classic recurrence to slightly less known and recent ones such as near periodicity or G-asymptotic periodicity. We provide mutual relations between all of them and fill the existing gaps that arise in their investigation.

From biased coin to any discrete distribution

Periodica Mathematica Hungarica, Oct 29, 2020

In this note we construct an algorithm generating any discrete distribution with an arbitrary coi... more In this note we construct an algorithm generating any discrete distribution with an arbitrary coin (and, as a result, with arbitrary initial distribution). The coin need not be fair and the target distribution can be supported on a countable set. Keywords Discrete probability measure • discrete distribution • expected value • coin flip • biased coin • simulation of distribution

Identities Arising from Binomial-Like Formulas Involving Divisors of Numbers

Annales Mathematicae Silesianae, Jul 26, 2023

In this article, we derive a great number of identities involving the ω function counting distinc... more

Asymptotic period in dynamical systems in metric spaces

Colloquium Mathematicum, 2015

Semi-smooth Points in Some Classical Function Spaces

Results in Mathematics, Nov 24, 2021

The investigations of the smooth points in the spaces of continuous function were started by Bana... more The investigations of the smooth points in the spaces of continuous function were started by Banach in 1932 considering function space C(Ω). Singer and Sundaresan extended the result of Banach to the space of vector valued continuous functions C(T , E), where T is a compact metric space. The aim of this paper is to present a description of semi-smooth points in spaces of continuous functions C0(T , E) (instead of smooth points). Moreover, we also find necessary and sufficient condition for semi-smoothness in the general case.

Generalized orthogonality equations in finite-dimensional normed spaces

Annals of Functional Analysis, Feb 28, 2023

Let X , Y be real normed spaces and let ρ + , ρ − be norm derivatives. In this work, we solve a s... more Let X , Y be real normed spaces and let ρ + , ρ − be norm derivatives. In this work, we solve a system of functional equations ρ + (f (x), f (y)) = g(x)ρ + (x, y), ρ − (f (x), f (y)) = g(x)ρ − (x, y), with unknown functions f : X → Y , g : X → R. Moreover, we give partial answer to open problem posed in 2010.

Lefschetz numbers and asymptotic periods

Wydawnictwo Uniwersytetu Łódzkiego eBooks, 2022

In this note we prove several results linking Lefschetz numbers with asymptotic behaviour of the ... more

Hausdorff Dimension is onto

K o ł o M a te ma tyk ó w U n i w e r sy tetu P e d a g o g ic z ne go w K r a k o w ie

On the $$L^1$$ and pointwise divergence of continuous functions

Periodica Mathematica Hungarica, Nov 4, 2022

For a sequence of continuous functions f 1 , f 2 ,. .. : I → R (I is a fixed closed interval) wit... more For a sequence of continuous functions f 1 , f 2 ,. .. : I → R (I is a fixed closed interval) with f 1 ≤ f 2 ≤ • • • define a set I f := x ∈ I : lim n→∞ f n (x) = +∞. We study the properties of the family of all admissible I f-s and the family of all admissible I f-s under the additional assumption lim n→∞ y x f n (t) dt = +∞ for all x, y ∈ I with x < y. The origin of this problem is the limit behaviour of quasiarithmetic means.

On Weak Asymptotic Periodicity

International Journal of Bifurcation and Chaos, Feb 1, 2020

We introduce the asymptotic property associated with recurrence-like behavior of orbits in dynami... more We introduce the asymptotic property associated with recurrence-like behavior of orbits in dynamical systems in general metric spaces. We define a notion of weak asymptotic periodicity and determine its elementary properties and relations including the invariance by topological conjugacy. We use the equicontinuity and the topology of the space to describe necessary and sufficient conditions for the existence of such a behavior.

On asymptotically periodic-like motions in flows

Annales Universitatis Paedagogicae Cracoviensis, Dec 1, 2018

We study three properties associated to the recurrence of orbits in flows: asymptotic periodicity... more We study three properties associated to the recurrence of orbits in flows: asymptotic periodicity, positive asymptotic periodicity and G-asymptotic periodicity. We determine which implications between these notions hold and which do not. We also show how these notions are related to Lyapunov stability.

Identities Arising from Binomial-Like Formulas Involving Divisors of Numbers

Annales Mathematicae Silesianae

In this article, we derive a great number of identities involving the ω function counting distinc... more

Lefschetz numbers and asymptotic periods

Wydawnictwo Uniwersytetu Łódzkiego eBooks, 2022

In this note we prove several results linking Lefschetz numbers with asymptotic behaviour of the ... more