Abstract In this paper we define dual-complex numbers with generalized Jacobsthal coefficients. We introduce one-parameter generalization of dual-complex Jacobsthal numbers - dual-complex r-Jacobsthal numbers. We investigate some algebraic properties of introduced numbers, among others Binet type formula, Catalan, Cassini, d’Ocagne and Honsberger type identities. Moreover, we present the generating function, summation formula and matrix generator for these numbers. The results are generalization of the properties for the dual-complex Jacobsthal numbers.
{"title":"On a new one-parameter generalization of dual-complex Jacobsthal numbers","authors":"D. Bród, A. Szynal-Liana, I. Włoch","doi":"10.2478/ausm-2021-0007","DOIUrl":"https://doi.org/10.2478/ausm-2021-0007","url":null,"abstract":"Abstract In this paper we define dual-complex numbers with generalized Jacobsthal coefficients. We introduce one-parameter generalization of dual-complex Jacobsthal numbers - dual-complex r-Jacobsthal numbers. We investigate some algebraic properties of introduced numbers, among others Binet type formula, Catalan, Cassini, d’Ocagne and Honsberger type identities. Moreover, we present the generating function, summation formula and matrix generator for these numbers. The results are generalization of the properties for the dual-complex Jacobsthal numbers.","PeriodicalId":43054,"journal":{"name":"Acta Universitatis Sapientiae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74101032","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}
Abstract Here, we employ soft semi open sets to define new soft ordered maps, namely soft x-semi continuous, soft x-semi open, soft x-semi closed and soft x-semi homeomorphism maps, where x denotes the type of monotonicity. To show the relationships among them, we provide some illustrative examples. Then we give complete descriptions for each one of them. Also, we investigate “transmission” of these maps between soft and classical topological ordered spaces.
{"title":"Defining and investigating new soft ordered maps by using soft semi open sets","authors":"T. Al-shami","doi":"10.2478/ausm-2021-0008","DOIUrl":"https://doi.org/10.2478/ausm-2021-0008","url":null,"abstract":"Abstract Here, we employ soft semi open sets to define new soft ordered maps, namely soft x-semi continuous, soft x-semi open, soft x-semi closed and soft x-semi homeomorphism maps, where x denotes the type of monotonicity. To show the relationships among them, we provide some illustrative examples. Then we give complete descriptions for each one of them. Also, we investigate “transmission” of these maps between soft and classical topological ordered spaces.","PeriodicalId":43054,"journal":{"name":"Acta Universitatis Sapientiae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81857596","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}
Abstract A rectifying curve in the Euclidean 4-space 𝔼4 is defined as an arc length parametrized curve γ in 𝔼4 such that its position vector always lies in its rectifying space (i.e., the orthogonal complement Nγ ˔ of its principal normal vector field Nγ) in 𝔼4. In this paper, we introduce the notion of an f-rectifying curve in 𝔼4 as a curve γ in 𝔼4 parametrized by its arc length s such that its f-position vector γf, defined by γf (s) = ∫ f(s)dγ for all s, always lies in its rectifying space in 𝔼4, where f is a nowhere vanishing integrable function in parameter s of the curve γ. Also, we characterize and classify such curves in 𝔼4.
欧几里得4空间𝔼4中的校正曲线定义为𝔼4中的弧长参数化曲线γ，其位置向量始终位于其校正空间(即其主法向量场Nγ的正交补Nγ˔)𝔼4中。在本文中，我们引入了𝔼4中的f-校正曲线的概念，即𝔼4中的曲线γ被其弧长s参数化，使得它的f位置向量γf，定义为γf (s) =∫f(s)dγ，对于所有s，总是位于𝔼4中的校正空间，其中f是曲线γ在参数s中的无处消失的可积函数。此外，我们在𝔼4中对这些曲线进行了表征和分类。
{"title":"On f-rectifying curves in the Euclidean 4-space","authors":"Zafar Iqbal, J. Sengupta","doi":"10.2478/ausm-2021-0011","DOIUrl":"https://doi.org/10.2478/ausm-2021-0011","url":null,"abstract":"Abstract A rectifying curve in the Euclidean 4-space 𝔼4 is defined as an arc length parametrized curve γ in 𝔼4 such that its position vector always lies in its rectifying space (i.e., the orthogonal complement Nγ ˔ of its principal normal vector field Nγ) in 𝔼4. In this paper, we introduce the notion of an f-rectifying curve in 𝔼4 as a curve γ in 𝔼4 parametrized by its arc length s such that its f-position vector γf, defined by γf (s) = ∫ f(s)dγ for all s, always lies in its rectifying space in 𝔼4, where f is a nowhere vanishing integrable function in parameter s of the curve γ. Also, we characterize and classify such curves in 𝔼4.","PeriodicalId":43054,"journal":{"name":"Acta Universitatis Sapientiae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88727063","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}
Abstract In this paper, we introduce the notion of Walsh shift-invariant space and present a unified approach to the study of shift-invariant systems to be frames in L2(ℝ+). We obtain a necessary condition and three sufficient conditions under which the Walsh shift-invariant systems constitute frames for L2(ℝ+). Furthermore, we discuss applications of our main results to obtain some known conclusions about the Gabor frames and wavelet frames on positive half line.
{"title":"Frames associated with shift invariant spaces on positive half line","authors":"O. Ahmad, Mobin Ahmad, Neyaz Ahmad","doi":"10.2478/ausm-2021-0002","DOIUrl":"https://doi.org/10.2478/ausm-2021-0002","url":null,"abstract":"Abstract In this paper, we introduce the notion of Walsh shift-invariant space and present a unified approach to the study of shift-invariant systems to be frames in L2(ℝ+). We obtain a necessary condition and three sufficient conditions under which the Walsh shift-invariant systems constitute frames for L2(ℝ+). Furthermore, we discuss applications of our main results to obtain some known conclusions about the Gabor frames and wavelet frames on positive half line.","PeriodicalId":43054,"journal":{"name":"Acta Universitatis Sapientiae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87894642","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}
Abstract We show that the sum of two intervals in an ordered dense Abelian group is also an interval such that the endpoints of the sum are equal to the sums of the endpoints. We prove analogous statements concerning to the product of two intervals.
{"title":"Sums and products of intervals in ordered groups and fields","authors":"T. Glavosits, Zsolt Karácsony","doi":"10.2478/ausm-2021-0010","DOIUrl":"https://doi.org/10.2478/ausm-2021-0010","url":null,"abstract":"Abstract We show that the sum of two intervals in an ordered dense Abelian group is also an interval such that the endpoints of the sum are equal to the sums of the endpoints. We prove analogous statements concerning to the product of two intervals.","PeriodicalId":43054,"journal":{"name":"Acta Universitatis Sapientiae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78142625","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}
Abstract An induced star-triangle factor of a graph G is a spanning subgraph F of G such that each component of F is an induced subgraph on the vertex set of that component and each component of F is a star (here star means either K1,n, n ≥ 2 or K2) or a triangle (cycle of length 3) in G. In this paper, we establish that every graph without isolated vertices admits an induced star-triangle factor in which any two leaves from different stars K1,n (n ≥ 2) are non-adjacent.
文摘的诱导star-triangle因素图G是一个生成子图G的每个组件F是一个诱导子图的顶点集F是一个恒星的组件,每个组件(明星意味着要么K1, n, n≥2或K2)或一个三角形(周期长度3)G在本文中,我们建立,每个图没有孤立的顶点承认一个诱导star-triangle因素在其中任意两个不同的恒星K1, n (n≥2)是不相邻的。
{"title":"Induced star-triangle factors of graphs","authors":"S. P. S. Kainth, R. Kumar, S. Pirzada","doi":"10.2478/ausm-2021-0012","DOIUrl":"https://doi.org/10.2478/ausm-2021-0012","url":null,"abstract":"Abstract An induced star-triangle factor of a graph G is a spanning subgraph F of G such that each component of F is an induced subgraph on the vertex set of that component and each component of F is a star (here star means either K1,n, n ≥ 2 or K2) or a triangle (cycle of length 3) in G. In this paper, we establish that every graph without isolated vertices admits an induced star-triangle factor in which any two leaves from different stars K1,n (n ≥ 2) are non-adjacent.","PeriodicalId":43054,"journal":{"name":"Acta Universitatis Sapientiae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78197118","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}
Yogesh J. Bagul, M. Kostic, C. Chesneau, R. Dhaigude
Abstract In this paper, we establish several generalized Becker-Stark type inequalities for the tangent function. We present unified proofs of many inequalities in the existing literature. Graphical illustrations of some obtained results are also presented.
{"title":"On the generalized Becker-Stark type inequalities","authors":"Yogesh J. Bagul, M. Kostic, C. Chesneau, R. Dhaigude","doi":"10.2478/ausm-2021-0005","DOIUrl":"https://doi.org/10.2478/ausm-2021-0005","url":null,"abstract":"Abstract In this paper, we establish several generalized Becker-Stark type inequalities for the tangent function. We present unified proofs of many inequalities in the existing literature. Graphical illustrations of some obtained results are also presented.","PeriodicalId":43054,"journal":{"name":"Acta Universitatis Sapientiae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78173293","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}
Abstract Let G be a group with identity e. Let R be a graded ring, I a graded ideal of R and M be a G-graded R-module. Let ψ: Sgr(M) → S gr(M) ∪ {∅} be a function, where Sgr(M) denote the set of all graded submodules of M. In this article, we introduce and study the concepts of graded ψ -second submodules and graded I-second submodules of a graded R-module which are generalizations of graded second submodules of M and investigate some properties of this class of graded modules.
{"title":"Generalizations of graded second submodules","authors":"P. Ghiasvand, F. Farzalipour","doi":"10.2478/ausm-2021-0009","DOIUrl":"https://doi.org/10.2478/ausm-2021-0009","url":null,"abstract":"Abstract Let G be a group with identity e. Let R be a graded ring, I a graded ideal of R and M be a G-graded R-module. Let ψ: Sgr(M) → S gr(M) ∪ {∅} be a function, where Sgr(M) denote the set of all graded submodules of M. In this article, we introduce and study the concepts of graded ψ -second submodules and graded I-second submodules of a graded R-module which are generalizations of graded second submodules of M and investigate some properties of this class of graded modules.","PeriodicalId":43054,"journal":{"name":"Acta Universitatis Sapientiae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82407549","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}
Abstract In this article, we are going to look at the requirements regarding a monotone function f ∈ ℝ →ℝ ≥0, and regarding the sets of natural numbers (Ai)i=1∞⊆dmn(f) left( {{A_i}} right)_{i = 1}^infty subseteq dmnleft( f right) , which requirements are sufficient for the asymptotic ∑n∈ANP(n)≤Nθf(n)∼ρ(1/θ)∑n∈ANf(n) sumlimits_{matrix{{n in {A_N}} hfill cr {Pleft( n right) le {N^theta }} hfill cr } } {fleft( n right) sim rho left( {1/theta } right)sumlimits_{n in {A_N}} {fleft( n right)} } to hold, where N is a positive integer, θ ∈ (0, 1) is a constant, P(n) denotes the largest prime factor of n, and ρ is the Dickman function.
在本文中，我们将研究单调函数f∈∈∈→∈≥0，以及自然数(Ai)i=1∞≥≥f的集合(Ai)i=1∞≥f (f) left ({{A＿i}}right){＿i =1} ^ infty≥subseteq dmn left (f right)的要求。∑n∈ANP(n)≤n θf(n)∼ρ(1/θ)∑n∈ANf(n) sumlimits ＿ {matrix{{n in {A_N}} hfill cr {Pleft( n right) le {N^theta }} hfill cr } f }{left (n right) simrholeft ({1/theta}right) sumlimits ＿n{in A＿N{ f}}{left (n right)，}其中n为正整数，θ∈(0,1)为常数，P(n)表示n的最大素数因子，ρ是Dickman函数。}
{"title":"On sums of monotone functions over smooth numbers","authors":"G. Román","doi":"10.2478/ausm-2021-0016","DOIUrl":"https://doi.org/10.2478/ausm-2021-0016","url":null,"abstract":"Abstract In this article, we are going to look at the requirements regarding a monotone function f ∈ ℝ →ℝ ≥0, and regarding the sets of natural numbers (Ai)i=1∞⊆dmn(f) left( {{A_i}} right)_{i = 1}^infty subseteq dmnleft( f right) , which requirements are sufficient for the asymptotic ∑n∈ANP(n)≤Nθf(n)∼ρ(1/θ)∑n∈ANf(n) sumlimits_{matrix{{n in {A_N}} hfill cr {Pleft( n right) le {N^theta }} hfill cr } } {fleft( n right) sim rho left( {1/theta } right)sumlimits_{n in {A_N}} {fleft( n right)} } to hold, where N is a positive integer, θ ∈ (0, 1) is a constant, P(n) denotes the largest prime factor of n, and ρ is the Dickman function.","PeriodicalId":43054,"journal":{"name":"Acta Universitatis Sapientiae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85795287","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}
Abstract In this paper, we investigate the asymptotic properties of a nonparametric conditional quantile estimation in the single functional index model for dependent functional data and censored at random responses are observed. First of all, we establish asymptotic properties for a conditional distribution estimator from which we derive an central limit theorem (CLT) of the conditional quantile estimator. Simulation study is also presented to illustrate the validity and finite sample performance of the considered estimator. Finally, the estimation of the functional index via the pseudo-maximum likelihood method is discussed, but not tackled.
{"title":"CLT for single functional index quantile regression under dependence structure","authors":"Nadia Kadiri, A. Rabhi, S. Khardani, Fatima Akkal","doi":"10.2478/ausm-2021-0003","DOIUrl":"https://doi.org/10.2478/ausm-2021-0003","url":null,"abstract":"Abstract In this paper, we investigate the asymptotic properties of a nonparametric conditional quantile estimation in the single functional index model for dependent functional data and censored at random responses are observed. First of all, we establish asymptotic properties for a conditional distribution estimator from which we derive an central limit theorem (CLT) of the conditional quantile estimator. Simulation study is also presented to illustrate the validity and finite sample performance of the considered estimator. Finally, the estimation of the functional index via the pseudo-maximum likelihood method is discussed, but not tackled.","PeriodicalId":43054,"journal":{"name":"Acta Universitatis Sapientiae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84544157","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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