Abstract In this paper, we present a Jacobi spectral collocation method to solve nonlinear Volterra-Fredholm integral equations with smooth kernels. The main idea in this approach is to convert the original problem into an equivalent one through appropriate variable transformations so that the resulting equation can be accurately solved using spectral collocation at the Jacobi-Gauss points. The convergence and error analysis are discussed for both L∞ and weighted L2 norms. We confirm the theoretical prediction of the exponential rate of convergence by the numerical results which are compared with well-known methods.
{"title":"Solving Nonlinear Volterra-Fredholm Integral Equations using an Accurate Spectral Collocation Method","authors":"Fatima Hamani, A. Rahmoune","doi":"10.2478/tmmp-2021-0030","DOIUrl":"https://doi.org/10.2478/tmmp-2021-0030","url":null,"abstract":"Abstract In this paper, we present a Jacobi spectral collocation method to solve nonlinear Volterra-Fredholm integral equations with smooth kernels. The main idea in this approach is to convert the original problem into an equivalent one through appropriate variable transformations so that the resulting equation can be accurately solved using spectral collocation at the Jacobi-Gauss points. The convergence and error analysis are discussed for both L∞ and weighted L2 norms. We confirm the theoretical prediction of the exponential rate of convergence by the numerical results which are compared with well-known methods.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"80 1","pages":"35 - 52"},"PeriodicalIF":0.0,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49150794","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 the presented paper, the main attention is given to fractal sets whose elements have certain restrictions on using digits or combinations of digits in their own nega-P-representation. Topological, metric, and fractal properties of images of certain self-similar fractals under the action of some singular distributions, are investigated.
{"title":"Certain Singular Distributions and Fractals","authors":"Serbenyuk Symon","doi":"10.2478/tmmp-2021-0026","DOIUrl":"https://doi.org/10.2478/tmmp-2021-0026","url":null,"abstract":"Abstract In the presented paper, the main attention is given to fractal sets whose elements have certain restrictions on using digits or combinations of digits in their own nega-P-representation. Topological, metric, and fractal properties of images of certain self-similar fractals under the action of some singular distributions, are investigated.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"31 1","pages":"163 - 198"},"PeriodicalIF":0.0,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77712960","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 consider the issues of local entropy for a finite family of generators (that generates the semigroup). Our main aim is to show that any continuous function can be approximated by s-chaotic family of generators.
{"title":"Local Properties of Entropy for Finite Family of Functions","authors":"R. Pawlak","doi":"10.2478/tmmp-2021-0004","DOIUrl":"https://doi.org/10.2478/tmmp-2021-0004","url":null,"abstract":"Abstract In this paper we consider the issues of local entropy for a finite family of generators (that generates the semigroup). Our main aim is to show that any continuous function can be approximated by s-chaotic family of generators.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"78 1","pages":"43 - 58"},"PeriodicalIF":0.0,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41446315","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 prove compactness of the operator MhCg on a subspace of the space of 2π-periodic functions of Riesz bounded variation on [−π, π], for appropriate functions g and h. Here Mh denotes multiplication by h and Cg convolution by g.
{"title":"Compactness of Multiplication Operators on Riesz Bounded Variation Spaces","authors":"Martha Guzmán-Partida","doi":"10.2478/tmmp-2021-0012","DOIUrl":"https://doi.org/10.2478/tmmp-2021-0012","url":null,"abstract":"Abstract We prove compactness of the operator MhCg on a subspace of the space of 2π-periodic functions of Riesz bounded variation on [−π, π], for appropriate functions g and h. Here Mh denotes multiplication by h and Cg convolution by g.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"78 1","pages":"167 - 174"},"PeriodicalIF":0.0,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43063003","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 The paper tries to survey the recent results about relationships between covering properties of a topological space X and the space USC(X) of upper semicontinuous functions on X with the topology of pointwise convergence. Dealing with properties of continuous functions C(X), we need shrinkable covers. The results are extended for A-measurable and upper A-semimeasurable functions where A is a family of subsets of X. Similar results for covers respecting a bornology and spaces USC(X) or C(X) endowed by a topology defined by using the bornology are presented. Some of them seem to be new.
{"title":"Real Functions, Covers and Bornologies","authors":"L. Bukovský","doi":"10.2478/tmmp-2021-0014","DOIUrl":"https://doi.org/10.2478/tmmp-2021-0014","url":null,"abstract":"Abstract The paper tries to survey the recent results about relationships between covering properties of a topological space X and the space USC(X) of upper semicontinuous functions on X with the topology of pointwise convergence. Dealing with properties of continuous functions C(X), we need shrinkable covers. The results are extended for A-measurable and upper A-semimeasurable functions where A is a family of subsets of X. Similar results for covers respecting a bornology and spaces USC(X) or C(X) endowed by a topology defined by using the bornology are presented. Some of them seem to be new.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"78 1","pages":"199 - 214"},"PeriodicalIF":0.0,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42587175","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 2014, J. Borsík and J. Holos defined porouscontinuous functions. Using the notion of density in O’Malley sense, we introduce new definitions of porouscontinuity, namely MOr and SOr-continuity. Some relevant properties of these classes of functions are discussed.
{"title":"On O’Malley Porouscontinuous Functions","authors":"Irena Domnik, S. Kowalczyk, M. Turowska","doi":"10.2478/tmmp-2021-0002","DOIUrl":"https://doi.org/10.2478/tmmp-2021-0002","url":null,"abstract":"Abstract In 2014, J. Borsík and J. Holos defined porouscontinuous functions. Using the notion of density in O’Malley sense, we introduce new definitions of porouscontinuity, namely MOr and SOr-continuity. Some relevant properties of these classes of functions are discussed.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"78 1","pages":"9 - 24"},"PeriodicalIF":0.0,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45550198","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 examine some generalized densities (called (ψ, n)-densities) obtained as a result of strengthening the Lebesgue Density Theorem. It turns out that these notions are the generalizations of superdensity, enhanced density and m-density, and have some applications in the theory of sets of finite perimeter and in Sobolev spaces.
{"title":"Generalized Densities on ℝn and their Applications","authors":"M. Filipczak, Małgorzata Terepeta","doi":"10.2478/tmmp-2021-0003","DOIUrl":"https://doi.org/10.2478/tmmp-2021-0003","url":null,"abstract":"Abstract We examine some generalized densities (called (ψ, n)-densities) obtained as a result of strengthening the Lebesgue Density Theorem. It turns out that these notions are the generalizations of superdensity, enhanced density and m-density, and have some applications in the theory of sets of finite perimeter and in Sobolev spaces.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"78 1","pages":"25 - 42"},"PeriodicalIF":0.0,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47400194","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 show there are strong relations between the algebraic properties of a graded commutative ring R and topological properties of open subsets of Zariski topology on the graded prime spectrum of R. We examine some algebraic conditions for open subsets of Zariski topology to become quasi-compact, dense, and irreducible. We also present a characterization for the radical of a graded ideal in R by using topological properties.
{"title":"Zariski Topologies on Graded Ideals","authors":"M. Bataineh, A. Alshehry, R. Abu-Dawwas","doi":"10.2478/tmmp-2021-0015","DOIUrl":"https://doi.org/10.2478/tmmp-2021-0015","url":null,"abstract":"Abstract In this paper, we show there are strong relations between the algebraic properties of a graded commutative ring R and topological properties of open subsets of Zariski topology on the graded prime spectrum of R. We examine some algebraic conditions for open subsets of Zariski topology to become quasi-compact, dense, and irreducible. We also present a characterization for the radical of a graded ideal in R by using topological properties.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"78 1","pages":"215 - 224"},"PeriodicalIF":0.0,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42353859","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 If R is a relation on X to Y, U is a relation on P (X) to Y, and V is a relation on P (X) to P (Y), then we say that R is an ordinary relation, U is a super relation, and V is a hyper relation on X to Y. Motivated by an ingenious idea of Emilia Przemska on a unified treatment of open- and closed-like sets, we shall introduce and investigate here four reasonable notions of product relations for super relations. In particular, for any two super relations U and V on X, we define two super relations U * V and U * V, and two hyper relations U ★ V and U * V on X such that : (U*V)(A)=(A∪U(A))∩V(A),(U*V)(A)=(A∩U(A))∪U(A) begin{array}{*{20}{l}} {(U*V)(A) = (Amathop cup nolimits^ U(A))mathop cap nolimits^ V(A),} {(U*V)(A) = (Amathop cap nolimits^ U(A))mathop cup nolimits^ U(A)} end{array} and (U★V)(A)={B⊆X: (U*V)(A)⊆B⊆(U*V)(A)},(U*V)(A)={B⊆X: (U∩V)(A)⊆B⊆(U∪V)(A)}begin{array}{*{20}{l}} {(UV)(A) = { B subseteq X:,(U*V)(A) subseteq B subseteq (U*V)(A)} ,} {(U*V)(A) = { B subseteq X:,(Umathop cap nolimits^ V)(A) subseteq B subseteq (Umathop cup nolimits^ V)(A)} } end{array} for all A ⊆ X. By using the distributivity of the operation ∩ over ∪, we can at once see that U * V ⊆ U * V. Moreover, if U ⊆ V, then we can also see that U * V = U * V. The most simple case is when U is an interior relation on X and V is the associated closure relation defined such that V (A) = U (Ac)c for all A ⊆ X.
如果R是X到Y上的关系，U是P (X)到Y上的关系，V是P (X)到P (Y)上的关系，那么我们就说R是普通关系，U是超关系，V是X到Y上的超关系。在Emilia Przemska关于开闭集统一处理的一个巧妙思想的启发下，我们将引入并研究超关系的四个合理的积关系概念。特别地，对于X上的任意两个超关系U和V，我们定义了两个超关系U * V和U * V，以及X上的两个超关系U★V和U * V，使得:(U*V)(A)=(A∪U(A))∩V(A)，(U*V)(A)=(A∩U(A))∪U(A) begin{array}{*{20}{l}} {(U*V)(A) = (Amathop cup nolimits^ U(A))mathop cap nolimits^ V(A),} {(U*V)(A) = (Amathop cap nolimits^ U(A))mathop cup nolimits^ U(A)} end{array}和(U★V)(A)= {b白日梦:(u * v)(a)}，(U*V)(A)= {b蔓蔓性:(u∩v)(a)贝蔓蔓性(u∩v)(a)}begin{array}{*{20}{l}} {(UV)(A) = { B subseteq X:,(U*V)(A) subseteq B subseteq (U*V)(A)} ,} {(U*V)(A) = { B subseteq X:,(Umathop cap nolimits^ V)(A) subseteq B subseteq (Umathop cup nolimits^ V)(A)} } end{array}对于所有的A (X)，利用运算∩在∪上的分布性，我们可以立即得到U*V≠U*V，如果U≠V，那么我们也可以得到U*V = U*V。最简单的情况是，U是X上的一个内关系，V是所有A (X)定义为V(A) = U(Ac)c的关联闭包关系。
{"title":"Super and Hyper Products of Super Relations","authors":"Á. Száz","doi":"10.2478/tmmp-2021-0007","DOIUrl":"https://doi.org/10.2478/tmmp-2021-0007","url":null,"abstract":"Abstract If R is a relation on X to Y, U is a relation on P (X) to Y, and V is a relation on P (X) to P (Y), then we say that R is an ordinary relation, U is a super relation, and V is a hyper relation on X to Y. Motivated by an ingenious idea of Emilia Przemska on a unified treatment of open- and closed-like sets, we shall introduce and investigate here four reasonable notions of product relations for super relations. In particular, for any two super relations U and V on X, we define two super relations U * V and U * V, and two hyper relations U ★ V and U * V on X such that : (U*V)(A)=(A∪U(A))∩V(A),(U*V)(A)=(A∩U(A))∪U(A) begin{array}{*{20}{l}} {(U*V)(A) = (Amathop cup nolimits^ U(A))mathop cap nolimits^ V(A),} {(U*V)(A) = (Amathop cap nolimits^ U(A))mathop cup nolimits^ U(A)} end{array} and (U★V)(A)={B⊆X: (U*V)(A)⊆B⊆(U*V)(A)},(U*V)(A)={B⊆X: (U∩V)(A)⊆B⊆(U∪V)(A)}begin{array}{*{20}{l}} {(UV)(A) = { B subseteq X:,(U*V)(A) subseteq B subseteq (U*V)(A)} ,} {(U*V)(A) = { B subseteq X:,(Umathop cap nolimits^ V)(A) subseteq B subseteq (Umathop cup nolimits^ V)(A)} } end{array} for all A ⊆ X. By using the distributivity of the operation ∩ over ∪, we can at once see that U * V ⊆ U * V. Moreover, if U ⊆ V, then we can also see that U * V = U * V. The most simple case is when U is an interior relation on X and V is the associated closure relation defined such that V (A) = U (Ac)c for all A ⊆ X.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"78 1","pages":"85 - 118"},"PeriodicalIF":0.0,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42870182","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 Egoroff’s classical theorem shows that from a pointwise convergence we can get a uniform convergence outside the set of an arbitrary small measure. Taylor’s theorem shows the possibility of controlling the convergence of the sequences of functions on the set of the full measure. Namely, for every sequence of real-valued measurable factions |fn}n∈ℕ pointwise converging to a function f on a measurable set E, there exist a decreasing sequence |δn}n∈ℕ of positive reals converging to 0 and a set A ⊆ E such that E A is a nullset and limn→+∞|fn(x)−f(x)|δn=0 for all x∈A. Let J(A, {fn}) {lim _{n to + infty }}frac{{|{f_n}(x) - f(x)|}}{{{delta _n}}} = 0,{rm{for}},{rm{all}},x in A.,{rm{Let}},J(A,,{ {f_n}} ) denote the set of all such sequences |δn}n∈ℕ. The main results of the paper concern basic properties of sets of all such sequences for a given set A and a given sequence of functions. A relationship between pointwise convergence, uniform convergence and the Taylor’s type of convergence is considered.
Egoroff经典定理证明了从点向收敛可以得到任意小测度集合外的一致收敛。泰勒定理证明了在全测度集合上控制函数序列收敛的可能性。即，对于在可测集合E上点收敛于函数f的每一个实值可测组序列|fn｝n∈＿1，存在一个收敛于0的正实数递减序列|δn｝n∈＿1，且存在一个集a≥≥a，使得E≥a为空集，且对于所有x∈a, limn→+∞|fn(x)−f(x)|δn=0。设J(a, {fn}) {lim ＿n{to + infty}}frac{{|{f_n}(x) - f(x)|}}{{{delta _n}}} =0 {rm{for}}{rm{all}}，，，x in a ，{rm{Let}}，J(a，，｛{f＿n}｝)表示所有这样的序列|δn｝n∈＿1的集合。本文的主要结果是关于给定集合a和给定函数序列的所有这类序列的集合的基本性质。考虑了点向收敛、一致收敛和泰勒型收敛之间的关系。
{"title":"Around Taylor’s Theorem on the Convergence of Sequences of Functions","authors":"G. Horbaczewska, Patrycja Rychlewicz","doi":"10.2478/tmmp-2021-0009","DOIUrl":"https://doi.org/10.2478/tmmp-2021-0009","url":null,"abstract":"Abstract Egoroff’s classical theorem shows that from a pointwise convergence we can get a uniform convergence outside the set of an arbitrary small measure. Taylor’s theorem shows the possibility of controlling the convergence of the sequences of functions on the set of the full measure. Namely, for every sequence of real-valued measurable factions |fn}n∈ℕ pointwise converging to a function f on a measurable set E, there exist a decreasing sequence |δn}n∈ℕ of positive reals converging to 0 and a set A ⊆ E such that E A is a nullset and limn→+∞|fn(x)−f(x)|δn=0 for all x∈A. Let J(A, {fn}) {lim _{n to + infty }}frac{{|{f_n}(x) - f(x)|}}{{{delta _n}}} = 0,{rm{for}},{rm{all}},x in A.,{rm{Let}},J(A,,{ {f_n}} ) denote the set of all such sequences |δn}n∈ℕ. The main results of the paper concern basic properties of sets of all such sequences for a given set A and a given sequence of functions. A relationship between pointwise convergence, uniform convergence and the Taylor’s type of convergence is considered.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"78 1","pages":"129 - 138"},"PeriodicalIF":0.0,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68923056","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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