rss_2.0Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica FeedSciendo RSS Feed for Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica Universitatis Paedagogicae Cracoviensis. Studia Mathematica Feed new existence results for fractional partial random nonlocal differential equations with delay<abstract> <title style='display:none'>Abstract</title> <p>The present paper deals with some existence results for the Darboux problem of partial fractional random differential equations with finite delay. The arguments are based on a random fixed point theorem with stochastic domain combined with the measure of noncompactness. An illustration is given to show the applicability of our results.</p> </abstract>ARTICLEtrue absolute summability of infinite series and Fourier series<abstract> <title style='display:none'>Abstract</title> <p>In the present paper, a theorem on <italic>θ</italic> −|<italic>T; δ</italic>|<sub>k</sub> summability method of an infinite series is proved, and also by using this method, a result on summability of a trigonometric Fourier series is obtained.</p> </abstract>ARTICLEtrueàro double sequence space derived by -analog<abstract> <title style='display:none'>Abstract</title> <p>This study includes the new Banach space <inline-formula id="j_aupcsm-2023-0009_eq_002"><alternatives><graphic xmlns:xlink="" xlink:href="graphic/j_aupcsm-2023-0009_eq_002.png"/><mml:math xmlns:mml="" display="inline" id="M2"><mml:msubsup><mml:mover accent="true"><mml:mi>ℒ</mml:mi><mml:mo>˜</mml:mo></mml:mover><mml:mi>s</mml:mi><mml:mi>q</mml:mi></mml:msubsup></mml:math><tex-math>$$\tilde {\cal L}_s^q$$</tex-math></alternatives></inline-formula> designed as the domain in 𝓛<sub><italic>s</italic></sub> space of the 4d (4-dimensional) <italic>q</italic>-Cesàro matrix obtained as the <italic>q</italic>-analog of the well-known 4d Cesàro matrix. After showing the completeness of the aforementioned space, giving some inclusion relations, determining the fundamental set of this space and calculating the duals, finally, some matrix transformations related to the new space were characterized.</p> </abstract>ARTICLEtrue concepts for groupoids<abstract> <title style='display:none'>Abstract</title> <p>In this paper, for getting more results in groupoids, we consider a set and introduce the notion of a right (left) independent subset of a groupoid, and it is studied in detail. As a corollary of these properties, the following important result is proved: for any groupoid, there is a maximal right (left) independent subset.</p> <p>Moreover, the notion of strongly right (left) independent subset is considered. It is proved that there exists a groupoid having a strongly right independent 2-set. Finally, we discuss the notion of dynamic elements with independence.</p> </abstract>ARTICLEtrue Hardy type integral inequalities with two parameters of summation<abstract> <title style='display:none'>Abstract</title> <p>In the present work, some Hardy-type integral inequalities were proved for two parameters of summation <italic>q</italic> ≤ <italic>p</italic> &lt; 0 and <italic>p</italic> &lt; 0, <italic>q</italic> &gt; 0. In addition, some two-sided estimates are obtained.</p> </abstract>ARTICLEtrue recognition in the Cantor cube<abstract> <title style='display:none'>Abstract</title> <p>Based on the notion of thin sets introduced recently by T. Banakh, Sz. Głąb, E. Jabłońska and J. Swaczyna we deliver a study of the infinite single-message transmission protocols. Such protocols are associated with a set of admissible messages (i.e. subsets of the Cantor cube <inline-formula id="j_aupcsm-2023-0006_eq_001"><alternatives><graphic xmlns:xlink="" xlink:href="graphic/j_aupcsm-2023-0006_eq_001.png"/><mml:math xmlns:mml="" display="inline" id="M1"><mml:mrow><mml:msubsup><mml:mi>ℤ</mml:mi><mml:mn>2</mml:mn><mml:mi>ω</mml:mi></mml:msubsup></mml:mrow></mml:math><tex-math><?CDATA \begin{eqnarray}{{\mathbb{Z}}}_{2}^{\omega }\end{eqnarray}?></tex-math></alternatives></inline-formula>).</p> <p>Using Banach-Mazur games we prove that all protocols detecting errors are Baire spaces and generic (in particular maximal) ones are not neither Borel nor meager.</p> <p>We also show that the Cantor cube can be decomposed to two thin sets which can be considered as the infinite counterpart of the parity bit. This result is related to so-called xor-sets defined by D. Niwiński and E. Kopczyński in 2014.</p> </abstract>ARTICLEtrue generalized Jordan derivations in rings<abstract> <title style='display:none'>Abstract</title> <p>In this article, we introduce the notion of centrally-extended generalized Jordan derivations and characterize the structure of a prime ring (resp. *-prime ring) <italic>R</italic> that admits a centrally-extended generalized Jordan derivation <italic>F</italic> satisfying [<italic>F</italic>(<italic>x</italic>), <italic>x</italic>] ∈ <italic>Z</italic>(<italic>R</italic>) (resp. [<italic>F</italic>(<italic>x</italic>), <italic>x</italic>*] ∈ <italic>Z</italic>(<italic>R</italic>)) for all <italic>x</italic> ∈ <italic>R</italic>.</p> </abstract>ARTICLEtrue results of self-similar solutions of the space-fractional diffusion equation involving the generalized Riesz-Caputo fractional derivative<abstract> <title style='display:none'>Abstract</title> <p>In this paper, we have discussed the problem of existence and uniqueness of solutions under the self-similar form to the space-fractional diffusion equation. The space-fractional derivative which will be used is the generalized Riesz-Caputo fractional derivative. Based on the similarity variable <italic>η</italic>, we have introduced the equation satisfied by the self-similar solutions for the aforementioned problem. To study the existence and uniqueness of self-similar solutions for this problem, we have applied some known fixed point theorems (i.e. Banach’s contraction principle, Schauder’s fixed point theorem and the nonlinear alternative of Leray-Schauder type).</p> </abstract>ARTICLEtrue related to the Deformed Hankel Transform<abstract> <title style='display:none'>Abstract</title> <p>The main result of the paper is the proof of the equivalence theorem for a <italic>K</italic>-functional and a modulus of smoothness for the Deformed Hankel Transform. Before that, we introduce the <italic>K</italic>-functional associated to the Deformed Hankel Transform.</p> </abstract>ARTICLEtrue and sufficient conditions for the inclusion relation between two summability methods<abstract> <title style='display:none'>Abstract</title> <p>In this paper, a general theorem gives necessary and sufficient conditions for the inclusion relation between <italic>φ</italic> – |<italic>A</italic>, <italic>β</italic> ; <italic>δ</italic>|<sub><italic>k</italic></sub> and <italic>φ</italic> – |<italic>B</italic>, <italic>β</italic> ; <italic>δ</italic>|<sub><italic>k</italic></sub> methods is proved.</p> </abstract>ARTICLEtrue points on the hyperelliptic curves = +<abstract> <title style='display:none'>Abstract</title> <p>We give an algebraic description of the set of algebraic points of degree at most <italic>d</italic> over ℚ on hyperelliptic curves <italic>y</italic><sup>2</sup> = <italic>x</italic><sup>5</sup> + <italic>n</italic><sup>2</sup>.</p> </abstract>ARTICLEtrue Traczyk’s BCK-sequences<abstract> <title style='display:none'>Abstract</title> <p>BCK-sequences and <italic>n</italic>-commutative BCK-algebras were introduced by T. Traczyk, together with two related problems. The first one, whether BCK-sequences are always prolongable. The second one, if the class of all <italic>n</italic>-commutative BCK-algebras is characterised by one identity. W. A. Dudek proved that the answer to the former question is positive in some special cases, e.g. when BCK-algebra is linearly ordered. T. Traczyk showed that the answer to the latter is a˚rmative for <italic>n</italic> = 1, 2. Nonetheless, by providing counterexamples, we proved that the answers to both those open problems are negative.</p> </abstract>ARTICLEtrue extensive note on various fractional-order type operators and some of their effects to certain holomorphic functions<abstract> <title style='display:none'>Abstract</title> <p>The aim of this paper is to present background information in relation with some fractional-order type operators in the complex plane, which is designed by the fractional-order derivative operator(s). Next we state various implications of that operator and then we show some interesting-special results of those applications.</p> </abstract>ARTICLEtrue a certain characterisation of the semigroup of positive natural numbers with multiplication<abstract> <title style='display:none'>Abstract</title> <p>In this paper we continue our investigation concerning the concept of a <italic>liken</italic>. This notion has been defined as a sequence of non-negative real numbers, tending to infinity and closed with respect to addition in ℝ. The most important examples of likens are clearly the set of natural numbers ℕ with addition and the set of positive natural numbers ℕ* with multiplication, represented by the sequence <inline-formula> <alternatives> <inline-graphic xmlns:xlink="" xlink:href="graphic/j_aupcsm-2022-0007_eq_001.png"/> <mml:math xmlns:mml="" display="inline"><mml:mrow><mml:msubsup><mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo>ln</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mo>∞</mml:mo></mml:msubsup></mml:mrow></mml:math> <tex-math>\left( {\ln \left( {n + 1} \right)} \right)_{n = 0}^\infty</tex-math> </alternatives> </inline-formula>. The set of all likens can be parameterized by the points of some infinite dimensional, complete metric space. In this <italic>space of likens</italic> we consider elements up to isomorphism and define <italic>properties of likens</italic> as such that are isomorphism invariant. The main result of this paper is a theorem characterizing the liken ℕ* of natural numbers with multiplication in the space of all likens.</p> </abstract>ARTICLEtrue simpler method to get only the true solutions of cubic and quartic equations using Tschirnhaus transformation<abstract> <title style='display:none'>Abstract</title> <p>The classic method of solving the cubic and the quartic equations using Tschirnhaus transformation yields true as well as false solutions. Recently some papers on this topic are published, in which methods are given to get only the true solutions of cubic and quartic equations. However these methods have some limitations. In this paper the author presents a method of solving cubic and quartic equations using Tschirnhaus transformation, which yields only the true solutions. The proposed method is much simpler than the methods published earlier.</p> </abstract>ARTICLEtrue of quadruple band matrix operator and geometric properties<abstract> <title style='display:none'>Abstract</title> <p>In this work, we characterize the class of compact matrix operators from <italic>c</italic><sub>0</sub>(<italic>Q</italic>), <italic>c</italic>(<italic>Q</italic>) and <italic>ℓ</italic><sub>∞</sub> (<italic>Q</italic>) into <italic>c</italic><sub>0</sub>, <italic>c</italic> and <italic>ℓ</italic><sub>∞</sub>, respectively, with the notion of the Hausdorff measure of noncompactness. Moreover, we determine some geometric properties of the sequence space <italic>ℓ<sub>p</sub></italic>(<italic>Q</italic>).</p> </abstract>ARTICLEtrue to algebraic equations of degree 4 and the fundamental theorem of algebra by Carl Friedrich Gauss<abstract> <title style='display:none'>Abstract</title> <p>Since <italic>Geronimo Cardano</italic>, algebraic equations of degree 4 have been solved analytically. Frequently, the solution algorithm is given in its entirety. We discovered two algorithms that lead to the same <italic>resolvente</italic>, each with two solutions; therefore, six formal solutions appear to solve an algebraic equation of degree four. Given that a square was utilized to derive the solution in both instances, it is critical to verify each solution. This check reveals that the four Cardanic solutions are the only four solutions to an algebraic equation of degree four. This demonstrates that <italic>Carl Friedrich Gauss</italic>’ (1799) <italic>fundamental theorem of algebra</italic> is not simple, despite the fact that it is a fundamental theorem. This seems to be a novel insight.</p> </abstract>ARTICLEtrueℋ-compactness in GTS<abstract> <title style='display:none'>Abstract</title> <p>Using the notion of operations on a generalized topological space (<italic>X, µ</italic>) and a hereditary class we have introduced the notion of <italic><sub>γµ</sub></italic>-compactness modulo a hereditary class ℋ termed as <italic><sub>γµ</sub></italic>ℋ -compactness. We have studied <italic><sub>γµ</sub></italic>ℋ-compact spaces and <italic><sub>γµ</sub></italic>ℋ-compact sets relative to <italic>µ</italic>.</p> </abstract>ARTICLEtrue systems of pseudo-M algebras<abstract> <title style='display:none'>Abstract</title> <p>The class of pseudo-M algebras contains pseudo-BCK, pseudo-BCI, pseudo-BCH, pseudo-BE, pseudo-CI algebras and many other algebras of logic. In this paper, the notion of deductive system in a pseudo-M algebra is introduced and its elementary properties are investigated. Closed deductive systems are defined and studied. The homomorphic properties of (closed) deductive systems are provided. The concepts of translation deductive systems and R-congruences in pseudo-M algebras are introduced and investigated. It is shown that there is a bijection between closed translation deductive systems and R-congruences. Finally, the construction of quotient algebra 𝒜/<italic>D</italic> of a pseudo-M algebra 𝒜 via a translation deductive system <italic>D</italic> of 𝒜 is given.</p> </abstract>ARTICLEtrue existence and blow-up of generalized self-similar solutions for a space-fractional diffusion equation with mixed conditions<abstract> <title style='display:none'>Abstract</title> <p>This paper investigates the problem of the existence and uniqueness of solutions under the generalized self-similar forms to the space-fractional diffusion equation. Therefore, through applying the properties of Schauder’s and Banach’s fixed point theorems; we establish several results on the global existence and blow-up of generalized self-similar solutions to this equation.</p> </abstract>ARTICLEtrue