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The GCD Sequences of the Altered Lucas Sequences 修改后的卢卡斯序列的GCD序列
IF 0.4
Annales Mathematicae Silesianae Pub Date : 2020-09-01 DOI: 10.2478/AMSIL-2020-0005
Fikri Köken
{"title":"The GCD Sequences of the Altered Lucas Sequences","authors":"Fikri Köken","doi":"10.2478/AMSIL-2020-0005","DOIUrl":"https://doi.org/10.2478/AMSIL-2020-0005","url":null,"abstract":"","PeriodicalId":52359,"journal":{"name":"Annales Mathematicae Silesianae","volume":"34 1","pages":"222-240"},"PeriodicalIF":0.4,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69155159","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}
引用次数: 1
The Hybrid Numbers of Padovan and Some Identities 帕多万的杂交数和一些恒等式
IF 0.4
Annales Mathematicae Silesianae Pub Date : 2020-07-16 DOI: 10.2478/amsil-2020-0019
M. Mangueira, R. Vieira, F. R. Alves, P. Catarino
{"title":"The Hybrid Numbers of Padovan and Some Identities","authors":"M. Mangueira, R. Vieira, F. R. Alves, P. Catarino","doi":"10.2478/amsil-2020-0019","DOIUrl":"https://doi.org/10.2478/amsil-2020-0019","url":null,"abstract":"Abstract In this article, we will define Padovan’s hybrid numbers, based on the new noncommutative numbering system studied by Özdemir ([7]). Such a system that is a set involving complex, hyperbolic and dual numbers. In addition, Padovan’s hybrid numbers are created by combining this set, satisfying the relation ih = −hi = ɛ + i. Given this, some properties and identities are shown for these numbers, such as Binet’s formula, generating matrix, characteristic equation, norm, and generating function. In addition, these numbers are extended to the integer field and some identities are made.","PeriodicalId":52359,"journal":{"name":"Annales Mathematicae Silesianae","volume":"34 1","pages":"256 - 267"},"PeriodicalIF":0.4,"publicationDate":"2020-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43658499","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}
引用次数: 9
Numerical Comparison of FNVIM and FNHPM for Solving a Certain Type of Nonlinear Caputo Time-Fractional Partial Differential Equations 求解一类非线性卡普托时间-分数阶偏微分方程的FNVIM和FNHPM的数值比较
IF 0.4
Annales Mathematicae Silesianae Pub Date : 2020-07-09 DOI: 10.2478/amsil-2020-0008
Ali Khalouta, A. Kadem
{"title":"Numerical Comparison of FNVIM and FNHPM for Solving a Certain Type of Nonlinear Caputo Time-Fractional Partial Differential Equations","authors":"Ali Khalouta, A. Kadem","doi":"10.2478/amsil-2020-0008","DOIUrl":"https://doi.org/10.2478/amsil-2020-0008","url":null,"abstract":"Abstract This work presents a numerical comparison between two efficient methods namely the fractional natural variational iteration method (FNVIM) and the fractional natural homotopy perturbation method (FNHPM) to solve a certain type of nonlinear Caputo time-fractional partial differential equations in particular, nonlinear Caputo time-fractional wave-like equations with variable coefficients. These two methods provided an accurate and efficient tool for solving this type of equations. To show the efficiency and capability of the proposed methods we have solved some numerical examples. The results show that there is an excellent agreement between the series solutions obtained by these two methods. However, the FNVIM has an advantage over FNHPM because it takes less time to solve this type of nonlinear problems without using He’s polynomials. In addition, the FNVIM enables us to overcome the diffi-culties arising in identifying the general Lagrange multiplier and it may be considered as an added advantage of this technique compared to the FNHPM.","PeriodicalId":52359,"journal":{"name":"Annales Mathematicae Silesianae","volume":"34 1","pages":"203 - 221"},"PeriodicalIF":0.4,"publicationDate":"2020-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47631151","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}
引用次数: 2
Some Kinds of Sparseness on the Real Line and Ideals on ω 实线上的几种稀疏性及ω上的理想
IF 0.4
Annales Mathematicae Silesianae Pub Date : 2020-07-01 DOI: 10.2478/amsil-2020-0014
M. Filipczak, G. Horbaczewska
{"title":"Some Kinds of Sparseness on the Real Line and Ideals on ω","authors":"M. Filipczak, G. Horbaczewska","doi":"10.2478/amsil-2020-0014","DOIUrl":"https://doi.org/10.2478/amsil-2020-0014","url":null,"abstract":"Abstract We show that a large class of summable ideals can be defined using a certain kind of “sparseness” of subsets of the line near zero, but it is still an open question whether this gives a characterization of the whole class.","PeriodicalId":52359,"journal":{"name":"Annales Mathematicae Silesianae","volume":"34 1","pages":"45 - 50"},"PeriodicalIF":0.4,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48893384","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}
引用次数: 2
Zygfryd Kominek, a Mathematician, a Teacher, a Friend 齐格弗里德·科米内克，一位数学家，一位老师，一位朋友
IF 0.4
Annales Mathematicae Silesianae Pub Date : 2020-07-01 DOI: 10.2478/amsil-2020-0016
M. Sablik
{"title":"Zygfryd Kominek, a Mathematician, a Teacher, a Friend","authors":"M. Sablik","doi":"10.2478/amsil-2020-0016","DOIUrl":"https://doi.org/10.2478/amsil-2020-0016","url":null,"abstract":"For the first time I heard about a young mathematician of the name Zygfryd Kominek was in early 1970’s when I was a student of mathematics at the Silesian University in Katowice. I had friends studying physics at the same university, and one day I was told by one of them, that they had the calculus course with a guy who went in a military uniform. That was Zygfryd, who then performed military service, or rather a complement to it allowing him to be promoted to officer’s level. Not that he wanted it but that was in times of communist Poland where the military service was mandatory (students used to perform it during Summer holidays), and fresh PhD’s were proposed to make an additional service this was the case of Dr. Kominek. Later I met him personally, first indirectly (I used to have classes with Bożena Szymura","PeriodicalId":52359,"journal":{"name":"Annales Mathematicae Silesianae","volume":"34 1","pages":"1 - 26"},"PeriodicalIF":0.4,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48718679","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}
引用次数: 0
On Iteration of Bijective Functions with Discontinuities 关于不连续双射函数的迭代
IF 0.4
Annales Mathematicae Silesianae Pub Date : 2020-07-01 DOI: 10.2478/amsil-2020-0009
H. Fripertinger
{"title":"On Iteration of Bijective Functions with Discontinuities","authors":"H. Fripertinger","doi":"10.2478/amsil-2020-0009","DOIUrl":"https://doi.org/10.2478/amsil-2020-0009","url":null,"abstract":"Abstract We present three different types of bijective functions f : I → I on a compact interval I with finitely many discontinuities where certain iterates of these functions will be continuous. All these examples are strongly related to permutations, in particular to derangements in the first case, and permutations with a certain number of successions (or small ascents) in the second case. All functions of type III form a direct product of a symmetric group with a wreath product. It will be shown that any iterative root F : J → J of the identity of order k on a compact interval J with finitely many discontinuities is conjugate to a function f of type III, i.e., F = φ−1 ∘ f ∘ φ where φ is a continuous, bijective, and increasing mapping between J and [0, n] for some integer n.","PeriodicalId":52359,"journal":{"name":"Annales Mathematicae Silesianae","volume":"34 1","pages":"51 - 72"},"PeriodicalIF":0.4,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41691220","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}
引用次数: 0
On a Functional Equation Appearing on the Margins of a Mean Invariance Problem 关于一个出现在均值不变问题边值上的函数方程
IF 0.4
Annales Mathematicae Silesianae Pub Date : 2020-07-01 DOI: 10.2478/amsil-2020-0012
J. Jarczyk, W. Jarczyk
{"title":"On a Functional Equation Appearing on the Margins of a Mean Invariance Problem","authors":"J. Jarczyk, W. Jarczyk","doi":"10.2478/amsil-2020-0012","DOIUrl":"https://doi.org/10.2478/amsil-2020-0012","url":null,"abstract":"Abstract Given a continuous strictly monotonic real-valued function α, defined on an interval I, and a function ω : I → (0, +∞) we denote by Bαω the Bajraktarević mean generated by α and weighted by ω: Bωα(x,y)=α-1(ω(x)ω(x)+ω(y)α(x)+ω(y)ω(x)+ω(y)α(y)),   x,y∈I. B_omega ^alpha left({x,y} right) = {alpha ^{- 1}}left({{{omega left(x right)} over {omega left(x right) + omega left(y right)}}alpha left(x right) + {{omega left(y right)} over {omega left(x right) + omega left(y right)}}alpha left(y right)} right),,,,x,y in I. We find a necessary integral formula for all possible three times differentiable solutions (φ, ψ) of the functional equation r(x)Bsϕ(x,y)+r(y)Btψ(x,y)=r(x)x+r(y)y, rleft(x right)B_s^varphi left({x,y} right) + rleft(y right)B_t^psi left({x,y} right) = rleft(x right)x + rleft(y right)y, where r, s, t : I → (0, +∞) are three times differentiable functions and the first derivatives of φ, ψ and r do not vanish. However, we show that not every pair (φ, ψ) given by the found formula actually satisfies the above equation.","PeriodicalId":52359,"journal":{"name":"Annales Mathematicae Silesianae","volume":"34 1","pages":"103 - 96"},"PeriodicalIF":0.4,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43606086","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}
引用次数: 0
Generalization of the Harmonic Weighted Mean Via Pythagorean Invariance Identity and Application 调和加权均值的毕达哥拉斯不变性恒等式推广及其应用
IF 0.4
Annales Mathematicae Silesianae Pub Date : 2020-07-01 DOI: 10.2478/amsil-2020-0015
P. Kahlig, J. Matkowski
{"title":"Generalization of the Harmonic Weighted Mean Via Pythagorean Invariance Identity and Application","authors":"P. Kahlig, J. Matkowski","doi":"10.2478/amsil-2020-0015","DOIUrl":"https://doi.org/10.2478/amsil-2020-0015","url":null,"abstract":"Abstract Under some simple conditions on the real functions f and g defined on an interval I ⊂ (0, ∞), the two-place functions Af (x, y) = f (x) + y − f (y) and Gg(x,y)=g(x)g(y)y {G_g}left({x,y} right) = {{gleft(x right)} over {gleft(y right)}}y generalize, respectively, A and G, the classical weighted arithmetic and geometric means. In this note, basing on the invariance identity G ∘ (H, A) = G (equivalent to the Pythagorean harmony proportion), a suitable weighted extension Hf,g of the classical harmonic mean H is introduced. An open problem concerning the symmetry of Hf,g is proposed. As an application a method of effective solving of some functional equations involving means is presented.","PeriodicalId":52359,"journal":{"name":"Annales Mathematicae Silesianae","volume":"34 1","pages":"104 - 122"},"PeriodicalIF":0.4,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47947834","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}
引用次数: 1
On the Borel Classes of Set-Valued Maps of Two Variables 关于二元集值映射的Borel类
IF 0.4
Annales Mathematicae Silesianae Pub Date : 2020-07-01 DOI: 10.2478/amsil-2020-0018
L. Holá, G. Kwiecińska
{"title":"On the Borel Classes of Set-Valued Maps of Two Variables","authors":"L. Holá, G. Kwiecińska","doi":"10.2478/amsil-2020-0018","DOIUrl":"https://doi.org/10.2478/amsil-2020-0018","url":null,"abstract":"Abstract Using the Borel classification of set-valued maps, we present here some new results on set-valued maps which are similar to some of the well known theorems on functions due to Lebesgue and Kuratowski. We consider set-valued maps of two variables in perfectly normal topological spaces. It was proved in [11] that a set-valued map lower semicontinuous (i.e. of lower Borel class 0) in the first and upper semicontinuous (i.e. of upper Borel class 0) in the second variable is of upper Borel class 1 and also (with stronger assumptions) of lower Borel class 1. This result cannot be generalized into higher Borel classes. In this paper we show that a set-valued map of the upper (resp. lower) Borel class α in the first and lower semicontinuous and upper quasicontinuous (upper semicontinuous and lower quasicontinuous) in the second variable is of the lower (resp. upper) Borel class α + 1. Also other cases are considered.","PeriodicalId":52359,"journal":{"name":"Annales Mathematicae Silesianae","volume":"34 1","pages":"81 - 95"},"PeriodicalIF":0.4,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43656459","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}
引用次数: 0
Remarks Connected with the Weak Limit of Iterates of Some Random-Valued Functions and Iterative Functional Equations 关于一些随机值函数和迭代函数方程迭代次数弱极限的注记
IF 0.4
Annales Mathematicae Silesianae Pub Date : 2020-07-01 DOI: 10.2478/amsil-2019-0015
K. Baron
{"title":"Remarks Connected with the Weak Limit of Iterates of Some Random-Valued Functions and Iterative Functional Equations","authors":"K. Baron","doi":"10.2478/amsil-2019-0015","DOIUrl":"https://doi.org/10.2478/amsil-2019-0015","url":null,"abstract":"Abstract The paper consists of two parts. At first, assuming that (Ω, A, P) is a probability space and (X, ϱ) is a complete and separable metric space with the σ-algebra 𝒝 of all its Borel subsets we consider the set 𝒭c of all 𝒝 ⊗ 𝒜-measurable and contractive in mean functions f : X × Ω → X with finite integral ∫ Ω ϱ (f(x, ω), x) P (dω) for x ∈ X, the weak limit π f of the sequence of iterates of f ∈ 𝒭c, and investigate continuity-like property of the function f ↦ π f, f ∈ 𝒭c, and Lipschitz solutions φ that take values in a separable Banach space of the equation φ(x)=∫Ωφ(f(x,ω))P(dω)+F(x). varphi left( x right) = int_Omega {varphi left( {fleft( {x,omega } right)} right)Pleft( {domega } right)} + Fleft( x right). Next, assuming that X is a real separable Hilbert space, Λ: X → X is linear and continuous with ||Λ || < 1, and µ is a probability Borel measure on X with finite first moment we examine continuous at zero solutions φ : X → 𝔺 of the equation φ(x)=μ⌢(x)φ(Λx) varphi left( x right) = mathord{buildrel{lower3pthbox{$scriptscriptstylefrown$}}over mu } left( x right)varphi left( {Lambda x} right) which characterizes the limit distribution π f for some special f ∈ 𝒭c.","PeriodicalId":52359,"journal":{"name":"Annales Mathematicae Silesianae","volume":"34 1","pages":"36 - 44"},"PeriodicalIF":0.4,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42148420","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}
引用次数: 3
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