Pub Date : 2023-01-20DOI: 10.7494/opmath.2023.43.3.275
D. Alpay, Ilwoo Cho
In this paper, we consider a family ({ mathbb{H}_{t}}_{tinmathbb{R}}) of rings of hypercomplex numbers, indexed by the real numbers, which contain both the quaternions and the split-quaternions. We consider natural Hilbert-space representations ({(mathbb{C}^{2},pi_{t})}_{tinmathbb{R}}) of the hypercomplex system ({ mathbb{H}_{t}}_{tinmathbb{R}}), and study the realizations (pi_{t}(h)) of hypercomplex numbers (h in mathbb{H}_{t}), as ((2times 2))-matrices acting on (mathbb{C}^{2}), for an arbitrarily fixed scale (tinmathbb{R}). Algebraic, operator-theoretic, spectral-analytic, and free-probabilistic properties of them are considered.
{"title":"Operators induced by certain hypercomplex systems","authors":"D. Alpay, Ilwoo Cho","doi":"10.7494/opmath.2023.43.3.275","DOIUrl":"https://doi.org/10.7494/opmath.2023.43.3.275","url":null,"abstract":"In this paper, we consider a family ({ mathbb{H}_{t}}_{tinmathbb{R}}) of rings of hypercomplex numbers, indexed by the real numbers, which contain both the quaternions and the split-quaternions. We consider natural Hilbert-space representations ({(mathbb{C}^{2},pi_{t})}_{tinmathbb{R}}) of the hypercomplex system ({ mathbb{H}_{t}}_{tinmathbb{R}}), and study the realizations (pi_{t}(h)) of hypercomplex numbers (h in mathbb{H}_{t}), as ((2times 2))-matrices acting on (mathbb{C}^{2}), for an arbitrarily fixed scale (tinmathbb{R}). Algebraic, operator-theoretic, spectral-analytic, and free-probabilistic properties of them are considered.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49532333","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.7494/opmath.2023.43.4.475
Tom Cuchta, R. Ferreira
We present the use of a Fourier transform on time scales to solve a dynamic heat IVP. This is done by inverting a certain exponential function via contour integral. We include some specific examples and directions for further study.
{"title":"The heat equation on time scales","authors":"Tom Cuchta, R. Ferreira","doi":"10.7494/opmath.2023.43.4.475","DOIUrl":"https://doi.org/10.7494/opmath.2023.43.4.475","url":null,"abstract":"We present the use of a Fourier transform on time scales to solve a dynamic heat IVP. This is done by inverting a certain exponential function via contour integral. We include some specific examples and directions for further study.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342541","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.7494/opmath.2023.43.1.47
J. Graef, Doudja Hebboul, T. Moussaoui
In this paper the authors study the existence of positive radial solutions to the Kirchhoff type problem involving the (p)-Laplacian [-Big(a+bint_{Omega_e}|nabla u|^p dxBig)Delta_p u=lambda fleft(|x|,uright), xin Omega_e,quad u=0 text{on} partialOmega_e,] where (lambda gt 0) is a parameter, (Omega_e = lbrace xinmathbb{R}^N : |x|gt r_0rbrace), (r_0gt 0), (N gt p gt 1), (Delta_p) is the (p)-Laplacian operator, and (fin C(left[ r_0, +inftyright)timesleft[0,+inftyright),mathbb{R})) is a non-decreasing function with respect to its second variable. By using the Mountain Pass Theorem, they prove the existence of positive radial solutions for small values of (lambda).
{"title":"Existence of positive radial solutions to a p-Laplacian Kirchhoff type problem on the exterior of a ball","authors":"J. Graef, Doudja Hebboul, T. Moussaoui","doi":"10.7494/opmath.2023.43.1.47","DOIUrl":"https://doi.org/10.7494/opmath.2023.43.1.47","url":null,"abstract":"In this paper the authors study the existence of positive radial solutions to the Kirchhoff type problem involving the (p)-Laplacian [-Big(a+bint_{Omega_e}|nabla u|^p dxBig)Delta_p u=lambda fleft(|x|,uright), xin Omega_e,quad u=0 text{on} partialOmega_e,] where (lambda gt 0) is a parameter, (Omega_e = lbrace xinmathbb{R}^N : |x|gt r_0rbrace), (r_0gt 0), (N gt p gt 1), (Delta_p) is the (p)-Laplacian operator, and (fin C(left[ r_0, +inftyright)timesleft[0,+inftyright),mathbb{R})) is a non-decreasing function with respect to its second variable. By using the Mountain Pass Theorem, they prove the existence of positive radial solutions for small values of (lambda).","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342629","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.7494/opmath.2023.43.1.67
Kazuki Ishibashi
The proportional derivative (PD) controller of a differential operator is commonly referred to as the conformable derivative. In this paper, we derive a nonoscillation theorem for damped linear differential equations with a differential operator using the conformable derivative of control theory. The proof of the nonoscillation theorem utilizes the Riccati inequality corresponding to the equation considered. The provided nonoscillation theorem gives the nonoscillatory condition for a damped Euler-type differential equation with a PD controller. Moreover, the nonoscillation of the equation with a PD controller that can generalize Whittaker-Hill-type equations is also considered in this paper. The Whittaker-Hill-type equation considered in this study also includes the Mathieu-type equation. As a subtopic of this work, we consider the nonoscillation of Mathieu-type equations with a PD controller while making full use of numerical simulations.
{"title":"Nonoscillation of damped linear differential equations with a proportional derivative controller and its application to Whittaker-Hill-type and Mathieu-type equations","authors":"Kazuki Ishibashi","doi":"10.7494/opmath.2023.43.1.67","DOIUrl":"https://doi.org/10.7494/opmath.2023.43.1.67","url":null,"abstract":"The proportional derivative (PD) controller of a differential operator is commonly referred to as the conformable derivative. In this paper, we derive a nonoscillation theorem for damped linear differential equations with a differential operator using the conformable derivative of control theory. The proof of the nonoscillation theorem utilizes the Riccati inequality corresponding to the equation considered. The provided nonoscillation theorem gives the nonoscillatory condition for a damped Euler-type differential equation with a PD controller. Moreover, the nonoscillation of the equation with a PD controller that can generalize Whittaker-Hill-type equations is also considered in this paper. The Whittaker-Hill-type equation considered in this study also includes the Mathieu-type equation. As a subtopic of this work, we consider the nonoscillation of Mathieu-type equations with a PD controller while making full use of numerical simulations.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342680","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.7494/opmath.2023.43.1.81
S. Kozerenko, Andrii Serdiuk
An edge imbalance provides a local measure of how irregular a given graph is. In this paper, we study graphs with graphic imbalance sequences. We give a new proof of imbalance graphicness for trees and use the new idea to prove that the same holds for unicyclic graphs. We then show that antiregular graphs are imbalance graphic and consider the join operation on graphs as well as the double graph operation. Our main results are concerning imbalance graphicness of three classes of block graphs: block graphs having all cut vertices in a single block; block graphs in which the subgraph induced by the cut vertices is either a star or a path. In the end, we discuss open questions and conjectures regarding imbalance graphic graphs.
{"title":"New results on imbalance graphic graphs","authors":"S. Kozerenko, Andrii Serdiuk","doi":"10.7494/opmath.2023.43.1.81","DOIUrl":"https://doi.org/10.7494/opmath.2023.43.1.81","url":null,"abstract":"An edge imbalance provides a local measure of how irregular a given graph is. In this paper, we study graphs with graphic imbalance sequences. We give a new proof of imbalance graphicness for trees and use the new idea to prove that the same holds for unicyclic graphs. We then show that antiregular graphs are imbalance graphic and consider the join operation on graphs as well as the double graph operation. Our main results are concerning imbalance graphicness of three classes of block graphs: block graphs having all cut vertices in a single block; block graphs in which the subgraph induced by the cut vertices is either a star or a path. In the end, we discuss open questions and conjectures regarding imbalance graphic graphs.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342719","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.7494/opmath.2023.43.2.185
G. Ivanova, El�bieta Wagner-Bojakowska, W�adys�aw Wilczy�ski
Using the new method of the construction of lower density operator introduced in the earlier paper of the first two authors, we study how much the new operator can be different from the classical one. The aim of this paper is to show that if (f) is a good adjusted measure-preserving bijection then the lower density operator generated by (f) can be really different from the classical density operator.
{"title":"Lower density operators. ?_{f} versus ?_{d}","authors":"G. Ivanova, El�bieta Wagner-Bojakowska, W�adys�aw Wilczy�ski","doi":"10.7494/opmath.2023.43.2.185","DOIUrl":"https://doi.org/10.7494/opmath.2023.43.2.185","url":null,"abstract":"Using the new method of the construction of lower density operator introduced in the earlier paper of the first two authors, we study how much the new operator can be different from the classical one. The aim of this paper is to show that if (f) is a good adjusted measure-preserving bijection then the lower density operator generated by (f) can be really different from the classical density operator.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342804","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.7494/opmath.2023.43.4.559
Ahmed Sanhaji, A. Dakkak, M. Moussaoui
This article is intended to prove the existence and uniqueness of the first eigencurve, for a homogeneous Neumann problem with singular weights associated with the equation [-Delta_{p} u=alpha m_{1}|u|^{p-2}u+beta m_{2}|u|^{p-2}u] in a bounded domain (Omega subset mathbb{R}^{N}). We then establish many properties of this eigencurve, particularly the continuity, variational characterization, asymptotic behavior, concavity and the differentiability.
{"title":"The first eigencurve for a Neumann boundary problem involving p-Laplacian with essentially bounded weights","authors":"Ahmed Sanhaji, A. Dakkak, M. Moussaoui","doi":"10.7494/opmath.2023.43.4.559","DOIUrl":"https://doi.org/10.7494/opmath.2023.43.4.559","url":null,"abstract":"This article is intended to prove the existence and uniqueness of the first eigencurve, for a homogeneous Neumann problem with singular weights associated with the equation [-Delta_{p} u=alpha m_{1}|u|^{p-2}u+beta m_{2}|u|^{p-2}u] in a bounded domain (Omega subset mathbb{R}^{N}). We then establish many properties of this eigencurve, particularly the continuity, variational characterization, asymptotic behavior, concavity and the differentiability.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71343004","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.7494/opmath.2023.43.6.789
B. El-Matary, H. El-Morshedy, V. Benekas, I. Stavroulakis
A technique is developed to establish a new oscillation criterion for a first-order linear difference equation with several delays and non-negative coefficients. Our result improves recent oscillation criteria and covers the cases of monotone and non-monotone delays. Moreover, the paper is concluded with an illustrative example to show the applicability and strength of our result.
{"title":"Oscillation conditions for difference equations with several variable delays","authors":"B. El-Matary, H. El-Morshedy, V. Benekas, I. Stavroulakis","doi":"10.7494/opmath.2023.43.6.789","DOIUrl":"https://doi.org/10.7494/opmath.2023.43.6.789","url":null,"abstract":"A technique is developed to establish a new oscillation criterion for a first-order linear difference equation with several delays and non-negative coefficients. Our result improves recent oscillation criteria and covers the cases of monotone and non-monotone delays. Moreover, the paper is concluded with an illustrative example to show the applicability and strength of our result.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71343462","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.7494/opmath.2023.43.5.633
Lian Hu, Xiaosong Liu
A family of Dirichlet-Morrey spaces (mathcal{D}_{lambda,K}) of functions analytic in the open unit disk (mathbb{D}) are defined in this paper. We completely characterize the boundedness of the Volterra integral operators (T_g), (I_g) and the multiplication operator (M_g) on the space (mathcal{D}_{lambda,K}). In addition, the compactness and essential norm of the operators (T_g) and (I_g) on (mathcal{D}_{lambda,K}) are also investigated.
{"title":"Volterra integral operators on a family of Dirichlet-Morrey spaces","authors":"Lian Hu, Xiaosong Liu","doi":"10.7494/opmath.2023.43.5.633","DOIUrl":"https://doi.org/10.7494/opmath.2023.43.5.633","url":null,"abstract":"A family of Dirichlet-Morrey spaces (mathcal{D}_{lambda,K}) of functions analytic in the open unit disk (mathbb{D}) are defined in this paper. We completely characterize the boundedness of the Volterra integral operators (T_g), (I_g) and the multiplication operator (M_g) on the space (mathcal{D}_{lambda,K}). In addition, the compactness and essential norm of the operators (T_g) and (I_g) on (mathcal{D}_{lambda,K}) are also investigated.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71343163","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.7494/opmath.2023.43.5.663
L. N. Kamble, C. Deshpande, B. Athawale
An almost self-complementary 3-uniform hypergraph on (n) vertices exists if and only if (n) is congruent to 3 modulo 4 A hypergraph (H) with vertex set (V) and edge set (E) is called bipartite if (V) can be partitioned into two subsets (V_1) and (V_2) such that (ecap V_1neq emptyset) and (ecap V_2neq emptyset) for any (ein E). A bipartite self-complementary 3-uniform hypergraph (H) with partition ((V_1, V_2)) of the vertex set (V) such that (|V_1|=m) and (|V_2|=n) exists if and only if either (i) (m=n) or (ii) (mneq n) and either (m) or (n) is congruent to 0 modulo 4 or (iii) (mneq n) and both (m) and (n) are congruent to 1 or 2 modulo 4. In this paper we define a bipartite almost self-complementary 3-uniform hypergraph (H) with partition ((V_1, V_2)) of a vertex set (V) such that (|V_1|=m) and (|V_2|=n) and find the conditions on (m) and (n) for a bipartite 3-uniform hypergraph (H) to be almost self-complementary. We also prove the existence of bi-regular bipartite almost self-complementary 3-uniform hypergraphs.
{"title":"The existence of bipartite almost self-complementary 3-uniform hypergraphs","authors":"L. N. Kamble, C. Deshpande, B. Athawale","doi":"10.7494/opmath.2023.43.5.663","DOIUrl":"https://doi.org/10.7494/opmath.2023.43.5.663","url":null,"abstract":"An almost self-complementary 3-uniform hypergraph on (n) vertices exists if and only if (n) is congruent to 3 modulo 4 A hypergraph (H) with vertex set (V) and edge set (E) is called bipartite if (V) can be partitioned into two subsets (V_1) and (V_2) such that (ecap V_1neq emptyset) and (ecap V_2neq emptyset) for any (ein E). A bipartite self-complementary 3-uniform hypergraph (H) with partition ((V_1, V_2)) of the vertex set (V) such that (|V_1|=m) and (|V_2|=n) exists if and only if either (i) (m=n) or (ii) (mneq n) and either (m) or (n) is congruent to 0 modulo 4 or (iii) (mneq n) and both (m) and (n) are congruent to 1 or 2 modulo 4. In this paper we define a bipartite almost self-complementary 3-uniform hypergraph (H) with partition ((V_1, V_2)) of a vertex set (V) such that (|V_1|=m) and (|V_2|=n) and find the conditions on (m) and (n) for a bipartite 3-uniform hypergraph (H) to be almost self-complementary. We also prove the existence of bi-regular bipartite almost self-complementary 3-uniform hypergraphs.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71343217","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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