Abstract We give some versions of Hahn-Banach, sandwich, duality, Moreau--Rockafellar-type theorems, optimality conditions and a formula for the subdifferential of composite functions for order continuous vector lattice-valued operators, invariant or equivariant with respect to a fixed group G of homomorphisms. As applications to optimization problems with both convex and linear constraints, we present some Farkas and Kuhn-Tucker-type results.
{"title":"Hahn-Banach-Type Theorems and Subdifferentials for Invariant and Equivariant Order Continuous Vector Lattice-Valued Operators with Applications to Optimization","authors":"A. Boccuto","doi":"10.2478/tmmp-2021-0010","DOIUrl":"https://doi.org/10.2478/tmmp-2021-0010","url":null,"abstract":"Abstract We give some versions of Hahn-Banach, sandwich, duality, Moreau--Rockafellar-type theorems, optimality conditions and a formula for the subdifferential of composite functions for order continuous vector lattice-valued operators, invariant or equivariant with respect to a fixed group G of homomorphisms. As applications to optimization problems with both convex and linear constraints, we present some Farkas and Kuhn-Tucker-type results.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"78 1","pages":"139 - 156"},"PeriodicalIF":0.0,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46125574","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 measurable fuzzy sets carrying the multivalued Łukasiewicz logic lead to a natural generalization of the classical Kolmogorovian probability theory. The transition from Boolean logic to Łukasiewicz logic has a categorical background and the resulting divisible probability theory possesses both fuzzy and quantum qualities. Observables of the divisible probability theory play an analogous role as classical random variables: to convey stochastic information from one system to another one. Observables preserving the Łukasiewicz logic are called conservative and characterize the “classical core” of divisible probability theory. They send crisp random events to crisp random events and Dirac probability measures to Dirac probability measures. The nonconservative observables send some crisp random events to genuine fuzzy events and some Dirac probability measures to nondegenerated probability measures. They constitute the added value of transition from classical to divisible probability theory.
{"title":"Łukasiewicz Logic and the Divisible Extension of Probability Theory","authors":"R. Fric","doi":"10.2478/tmmp-2021-0008","DOIUrl":"https://doi.org/10.2478/tmmp-2021-0008","url":null,"abstract":"Abstract We show that measurable fuzzy sets carrying the multivalued Łukasiewicz logic lead to a natural generalization of the classical Kolmogorovian probability theory. The transition from Boolean logic to Łukasiewicz logic has a categorical background and the resulting divisible probability theory possesses both fuzzy and quantum qualities. Observables of the divisible probability theory play an analogous role as classical random variables: to convey stochastic information from one system to another one. Observables preserving the Łukasiewicz logic are called conservative and characterize the “classical core” of divisible probability theory. They send crisp random events to crisp random events and Dirac probability measures to Dirac probability measures. The nonconservative observables send some crisp random events to genuine fuzzy events and some Dirac probability measures to nondegenerated probability measures. They constitute the added value of transition from classical to divisible probability theory.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"78 1","pages":"119 - 128"},"PeriodicalIF":0.0,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43526544","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 space X is said to have the star-K-I-Hurewicz property (SKIH) [Tyagi, B. K.—Singh, S.—Bhardwaj, M. Ideal analogues of some variants of Hurewicz property, Filomat 33 (2019), no. 9, 2725–2734] if for each sequence (Un : n ∈ ℕ) of open covers of X there is a sequence (Kn : n ∈ ℕ) of compact subsets of X such that for each x ∈ X, {n ∈ ℕ : x ∉ St(Kn, Un)} ∈ I, where I is the proper admissible ideal of ℕ. In this paper, we continue to investigate the relationship between the SKIH property and other related properties and study the topological properties of the SKIH property.
{"title":"On Star-K-I-Hurewicz Property","authors":"Sumit Singh, Harsh V. S. Chauhan, Vikesh Kumar","doi":"10.2478/tmmp-2021-0011","DOIUrl":"https://doi.org/10.2478/tmmp-2021-0011","url":null,"abstract":"Abstract A space X is said to have the star-K-I-Hurewicz property (SKIH) [Tyagi, B. K.—Singh, S.—Bhardwaj, M. Ideal analogues of some variants of Hurewicz property, Filomat 33 (2019), no. 9, 2725–2734] if for each sequence (Un : n ∈ ℕ) of open covers of X there is a sequence (Kn : n ∈ ℕ) of compact subsets of X such that for each x ∈ X, {n ∈ ℕ : x ∉ St(Kn, Un)} ∈ I, where I is the proper admissible ideal of ℕ. In this paper, we continue to investigate the relationship between the SKIH property and other related properties and study the topological properties of the SKIH property.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"78 1","pages":"157 - 166"},"PeriodicalIF":0.0,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44193686","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 pexiderized functional equations for studying their Hyers-Ulam-Rassias stability. This stability has been studied for a variety of mathematical structures. Our framework of discussion is a modular space. We adopt a fixed-point approach to the problem in which we use a generalized contraction mapping principle in modular spaces. The result is illustrated with an example.
{"title":"A Fixed Point Approach to the Hyers-Ulam-Rassias Stability Problem of Pexiderized Functional Equation in Modular Spaces","authors":"P. Saha, P. Mondal, Binayak S. Chqudhury","doi":"10.2478/tmmp-2021-0005","DOIUrl":"https://doi.org/10.2478/tmmp-2021-0005","url":null,"abstract":"Abstract In this paper, we consider pexiderized functional equations for studying their Hyers-Ulam-Rassias stability. This stability has been studied for a variety of mathematical structures. Our framework of discussion is a modular space. We adopt a fixed-point approach to the problem in which we use a generalized contraction mapping principle in modular spaces. The result is illustrated with an example.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"78 1","pages":"59 - 72"},"PeriodicalIF":0.0,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48700053","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}
B. Venkateswarlu, P. Reddy, S. Sridevi, Vaishnavy Sujatha
Abstract In this paper, we introduce a new subclass of analytic functions with negative coefficients defined by Gegenbauer polynomials. We obtain coefficient bounds, growth and distortion properties, extreme points and radii of starlikeness, convexity and close-to-convexity for functions belonging to the class TSλm(γ,e,k,v)TS_lambda ^m(gamma ,e,k,v). Furthermore, we obtained the Fekete-Szego problem for this class.
{"title":"A Certain Subclass of Analytic Functions with Negative Coefficients Defined by Gegenbauer Polynomials","authors":"B. Venkateswarlu, P. Reddy, S. Sridevi, Vaishnavy Sujatha","doi":"10.2478/tmmp-2021-0006","DOIUrl":"https://doi.org/10.2478/tmmp-2021-0006","url":null,"abstract":"Abstract In this paper, we introduce a new subclass of analytic functions with negative coefficients defined by Gegenbauer polynomials. We obtain coefficient bounds, growth and distortion properties, extreme points and radii of starlikeness, convexity and close-to-convexity for functions belonging to the class TSλm(γ,e,k,v)TS_lambda ^m(gamma ,e,k,v). Furthermore, we obtained the Fekete-Szego problem for this class.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"78 1","pages":"73 - 84"},"PeriodicalIF":0.0,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41867738","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 As in the recent article of M. Balcerzak, T. Filipczak and P. Nowakowski, we identify the family CS of central Cantor subsets of [0, 1] with the Polish space X : = (0, 1)ℕ equipped with the probability product measure µ. We investigate the size of the family P0 of sets in CS with packing dimension zero. We show that P0 is meager and of µ measure zero while it is treated as the corresponding subset of X. We also check possible inclusions between P0 and other subfamilies CS consisting of small sets.
在M. Balcerzak, T. Filipczak和P. Nowakowski最近的一篇文章中，我们用概率积测度µ在波兰空间X: = (0,1) n上识别了[0,1]的中心Cantor子集CS族。研究了CS中集族P0的大小。我们证明了P0是微测度零的，而它被看作x的相应子集。我们还检查了P0和其他由小集合组成的子族CS之间可能存在的包含。
{"title":"The Family of Central Cantor Sets with Packing Dimension Zero","authors":"P. Nowakowski","doi":"10.2478/tmmp-2021-0001","DOIUrl":"https://doi.org/10.2478/tmmp-2021-0001","url":null,"abstract":"Abstract As in the recent article of M. Balcerzak, T. Filipczak and P. Nowakowski, we identify the family CS of central Cantor subsets of [0, 1] with the Polish space X : = (0, 1)ℕ equipped with the probability product measure µ. We investigate the size of the family P0 of sets in CS with packing dimension zero. We show that P0 is meager and of µ measure zero while it is treated as the corresponding subset of X. We also check possible inclusions between P0 and other subfamilies CS consisting of small sets.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"78 1","pages":"1 - 8"},"PeriodicalIF":0.0,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44545999","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}
Dlmplekumar N. Chalishajar, K. Karthikeyan, Dhachinamoorthi Tamizharasan
Abstract This paper is concerned with the controllability of impulsive differential equations with nonlocal conditions. First, we establish a property of measure of noncompactness in the space of piecewise continuous functions. Then, by using this property and Darbo-Sadovskii’s Fixed Point Theorem, we get the controllability of nonlocal impulsive differential equations under compactness conditions, Lipschitz conditions and mixed-type conditions, respectively.
{"title":"Controllability of Nonlocal Impulsive Functional Differential Equations with Measure of Noncompactness in Banach Spaces","authors":"Dlmplekumar N. Chalishajar, K. Karthikeyan, Dhachinamoorthi Tamizharasan","doi":"10.22436/JNSA.014.06.03","DOIUrl":"https://doi.org/10.22436/JNSA.014.06.03","url":null,"abstract":"Abstract This paper is concerned with the controllability of impulsive differential equations with nonlocal conditions. First, we establish a property of measure of noncompactness in the space of piecewise continuous functions. Then, by using this property and Darbo-Sadovskii’s Fixed Point Theorem, we get the controllability of nonlocal impulsive differential equations under compactness conditions, Lipschitz conditions and mixed-type conditions, respectively.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"79 1","pages":"59 - 80"},"PeriodicalIF":0.0,"publicationDate":"2021-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47098599","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 introduce a novel method for map registration and apply it to transformation of the river Ister from Strabo’s map of the World to the current map in the World Geodetic System. This transformation leads to the surprising but convincing result that Strabo’s river Ister best coincides with the nowadays Tauernbach-Isel-Drava-Danube course and not with the Danube river what is commonly assumed. Such a result is supported by carefully designed mathematical measurements and it resolves all related controversies otherwise appearing in understanding and translation of Strabo’s original text. Based on this result, we also show that Strabo’s Suevi in the Hercynian Forest corresponds to the Slavic people in the Carpathian-Alpine basin and thus that the compact Slavic settlement was there already at the beginning of the first millennium AD.
{"title":"What was the River Ister in the Time of Strabo? A Mathematical Approach","authors":"K. Mikula, M. Ambroz, Renáta Mokošová","doi":"10.2478/tmmp-2021-0032","DOIUrl":"https://doi.org/10.2478/tmmp-2021-0032","url":null,"abstract":"Abstract We introduce a novel method for map registration and apply it to transformation of the river Ister from Strabo’s map of the World to the current map in the World Geodetic System. This transformation leads to the surprising but convincing result that Strabo’s river Ister best coincides with the nowadays Tauernbach-Isel-Drava-Danube course and not with the Danube river what is commonly assumed. Such a result is supported by carefully designed mathematical measurements and it resolves all related controversies otherwise appearing in understanding and translation of Strabo’s original text. Based on this result, we also show that Strabo’s Suevi in the Hercynian Forest corresponds to the Slavic people in the Carpathian-Alpine basin and thus that the compact Slavic settlement was there already at the beginning of the first millennium AD.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"80 1","pages":"71 - 118"},"PeriodicalIF":0.0,"publicationDate":"2021-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47549094","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 paper, we continue the research of Borsík and Doboš on functions which allow us to introduce a metric to the product of metric spaces. We extend their scope to a broader class of spaces which usually fail to satisfy the triangle inequality, albeit they tend to satisfy some weaker form of this axiom. In particular, we examine the behavior of functions preserving b-metric inequality. We provide analogues of the results of Borsík and Doboš adjusted to the new broader setting. The results we obtained are illustrated with multitude of examples. Furthermore, the connections of newly introduced families of functions with the ones already known from the literature are investigated.
{"title":"On Functions Preserving Products of Certain Classes of Semimetric Spaces","authors":"Mateusz Lichman, P. Nowakowski, Filip Tcroboś","doi":"10.2478/tmmp-2021-0013","DOIUrl":"https://doi.org/10.2478/tmmp-2021-0013","url":null,"abstract":"Abstract In the paper, we continue the research of Borsík and Doboš on functions which allow us to introduce a metric to the product of metric spaces. We extend their scope to a broader class of spaces which usually fail to satisfy the triangle inequality, albeit they tend to satisfy some weaker form of this axiom. In particular, we examine the behavior of functions preserving b-metric inequality. We provide analogues of the results of Borsík and Doboš adjusted to the new broader setting. The results we obtained are illustrated with multitude of examples. Furthermore, the connections of newly introduced families of functions with the ones already known from the literature are investigated.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"78 1","pages":"175 - 198"},"PeriodicalIF":0.0,"publicationDate":"2020-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47691531","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 a generalization of Jacobsthal hybrid numbers – J(r, n)-Jacobsthal hybrid numbers. We give some of their properties: character, Binet’s formula, a summation formula and a generating function.
{"title":"On J(r, n)-Jacobsthal Hybrid Numbers","authors":"D. Bród, A. Szynal-Liana","doi":"10.2478/tmmp-2020-0028","DOIUrl":"https://doi.org/10.2478/tmmp-2020-0028","url":null,"abstract":"Abstract In this paper we introduce a generalization of Jacobsthal hybrid numbers – J(r, n)-Jacobsthal hybrid numbers. We give some of their properties: character, Binet’s formula, a summation formula and a generating function.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"77 1","pages":"13 - 26"},"PeriodicalIF":0.0,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41417106","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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