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Real Functions, Dynamical Systems and Applications

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Volumen 78 (2021): Heft 1 (October 2021)

Volumen 77 (2020): Heft 1 (December 2020)

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Real Functions, Dynamical Systems and their Applications

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Real Functons, Ideals, Measurable Functions, Functional Equations

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Real Functions '15 — Measure Theory, Real Functions, General Topology. Editors: J. Borsík, 2016.

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Special Heft Title: Applied Mathematics ‘14

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Special Heft Title: Number Theory ‘14

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Real Functions ‘13 Real Functions, Topology, Real and Functional Analysis, Locally Convex Spaces

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PROBASTAT ‘11

Volumen 50 (2011): Heft 1 (December 2011)
Applied Mathematics and Informatics

Volumen 49 (2011): Heft 1 (August 2011)
Real Functions ‘10

Volumen 48 (2011): Heft 1 (April 2011)
Differential and Difference Equations and Applications 2010

Volumen 47 (2010): Heft 1 (December 2010)
CCEC ‘09

Volumen 46 (2010): Heft 1 (August 2010)
Real Functions ‘09

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NILCRYPT ‘10

Volumen 44 (2009): Heft 1 (December 2009)
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Differential and Difference Equations and Applications 2008

Volumen 42 (2009): Heft 1 (April 2009)
Real Function ‘07

Zeitschriftendaten
Format
Zeitschrift
eISSN
1338-9750
Erstveröffentlichung
12 Nov 2012
Erscheinungsweise
3 Hefte pro Jahr
Sprachen
Englisch

Suche

Volumen 78 (2021): Heft 1 (October 2021)

Zeitschriftendaten
Format
Zeitschrift
eISSN
1338-9750
Erstveröffentlichung
12 Nov 2012
Erscheinungsweise
3 Hefte pro Jahr
Sprachen
Englisch

Suche

15 Artikel
Open Access

The Family of Central Cantor Sets with Packing Dimension Zero

Online veröffentlicht: 01 Jan 2022
Seitenbereich: 1 - 8

Zusammenfassung

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) 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.

Schlüsselwörter

  • central Cantor sets
  • Baire category
  • product measure
  • sets of packing dimension zero
Open Access

On O’Malley Porouscontinuous Functions

Online veröffentlicht: 01 Jan 2022
Seitenbereich: 9 - 24

Zusammenfassung

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.

Schlüsselwörter

  • porosity
  • porouscontinuity
  • porouscontinuity in O’Malley sense
Open Access

Generalized Densities on ℝn and their Applications

Online veröffentlicht: 01 Jan 2022
Seitenbereich: 25 - 42

Zusammenfassung

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.

Schlüsselwörter

  • - density point
  • - density topology
  • similarity of topologies
  • differentiation basis
  • base operator
  • set of finite perimeter
  • Sobolev space
Open Access

Local Properties of Entropy for Finite Family of Functions

Online veröffentlicht: 01 Jan 2022
Seitenbereich: 43 - 58

Zusammenfassung

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.

Schlüsselwörter

  • entropy
  • semigroup
  • set of generators
  • entropy of I, II, III type
  • (periodic) dynamical system
  • -invariant set
  • -entropy point ( ∈ {I, II, III})
  • s-chaotic set of generators
Open Access

A Fixed Point Approach to the Hyers-Ulam-Rassias Stability Problem of Pexiderized Functional Equation in Modular Spaces

Online veröffentlicht: 01 Jan 2022
Seitenbereich: 59 - 72

Zusammenfassung

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.

Schlüsselwörter

  • Hyers-Ulam-Rassias stability
  • pexiderized functional equation
  • modular spaces
  • fixed point method
Open Access

A Certain Subclass of Analytic Functions with Negative Coefficients Defined by Gegenbauer Polynomials

Online veröffentlicht: 01 Jan 2022
Seitenbereich: 73 - 84

Zusammenfassung

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.

Schlüsselwörter

  • analytic
  • coefficient bounds
  • extreme points
  • convolution
  • polynomial
Open Access

Super and Hyper Products of Super Relations

Online veröffentlicht: 01 Jan 2022
Seitenbereich: 85 - 118

Zusammenfassung

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 UV and U * V on X such that : (U*V)(A)=(AU(A))V(A),(U*V)(A)=(AU(A))U(A) \begin{array}{*{20}{l}} {(U*V)(A) = (A\mathop \cup \nolimits^ U(A))\mathop \cap \nolimits^ V(A),}\\ {(U*V)(A) = (A\mathop \cap \nolimits^ U(A))\mathop \cup \nolimits^ U(A)} \end{array} and (UV)(A)={BX:(U*V)(A)B(U*V)(A)},(U*V)(A)={BX:(UV)(A)B(UV)(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:\,(U\mathop \cap \nolimits^ V)(A) \subseteq B \subseteq (U\mathop \cup \nolimits^ V)(A)\} } \end{array} for all AX.

By using the distributivity of the operation ∩ over ∪, we can at once see that U * VU * V. Moreover, if UV, 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 AX.

Schlüsselwörter

  • relations on ordinary and power sets
  • unary and binary operations for relations
  • Galois connections
  • generalized uniformities
  • generalized open sets
Open Access

Łukasiewicz Logic and the Divisible Extension of Probability Theory

Online veröffentlicht: 01 Jan 2022
Seitenbereich: 119 - 128

Zusammenfassung

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.

Schlüsselwörter

  • divisible extension of probability theory
  • Boolean logic
  • random variable
  • observable
  • statistical map
  • stochastic channel
  • Łukasiewicz logic
  • full Łukasiewicz tribe
  • conservative observable
Open Access

Around Taylor’s Theorem on the Convergence of Sequences of Functions

Online veröffentlicht: 01 Jan 2022
Seitenbereich: 129 - 138

Zusammenfassung

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 AE such that E \ A is a nullset and limn+|fn(x)f(x)|δn=0forallxA.LetJ(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.

Schlüsselwörter

  • sequences of real-valued functions
  • convergence
Open Access

Hahn-Banach-Type Theorems and Subdifferentials for Invariant and Equivariant Order Continuous Vector Lattice-Valued Operators with Applications to Optimization

Online veröffentlicht: 01 Jan 2022
Seitenbereich: 139 - 156

Zusammenfassung

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.

Schlüsselwörter

  • vector lattice
  • order bounded functional
  • order continuous functional
  • amenability
  • Hahn-Banach theorem
  • sandwich theorem
  • Fenchel duality theorem
  • subgradient
  • subdifferential of composite functions
  • optimality condition
  • Moreau-Rockafellar formula
  • Farkas theorem
  • Kuhn-Tucker theorem
Open Access

On Star-K-I-Hurewicz Property

Online veröffentlicht: 01 Jan 2022
Seitenbereich: 157 - 166

Zusammenfassung

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 xX, {n ∈ ℕ : xSt(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.

Schlüsselwörter

  • Hurewicz
  • star-K--Hurewicz
  • star-K-Hurewicz
  • ideal
  • covering
  • star-covering
  • topological space
Open Access

Compactness of Multiplication Operators on Riesz Bounded Variation Spaces

Online veröffentlicht: 01 Jan 2022
Seitenbereich: 167 - 174

Zusammenfassung

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.

Schlüsselwörter

  • Riesz bounded variation
  • multiplication operator
  • compactness
Open Access

On Functions Preserving Products of Certain Classes of Semimetric Spaces

Online veröffentlicht: 01 Jan 2022
Seitenbereich: 175 - 198

Zusammenfassung

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.

Schlüsselwörter

  • semimetrics
  • metric-preserving functions
  • -metrics
  • subadditive functions
Open Access

Real Functions, Covers and Bornologies

Online veröffentlicht: 01 Jan 2022
Seitenbereich: 199 - 214

Zusammenfassung

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.

Schlüsselwörter

  • cover
  • upper semicontinuous function
  • measurable function
  • upper semimeasurable function
  • bornology
Open Access

Zariski Topologies on Graded Ideals

Online veröffentlicht: 01 Jan 2022
Seitenbereich: 215 - 224

Zusammenfassung

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.

Schlüsselwörter

  • graded prime ideal
  • graded radical ideal
  • Zariski topology
15 Artikel
Open Access

The Family of Central Cantor Sets with Packing Dimension Zero

Online veröffentlicht: 01 Jan 2022
Seitenbereich: 1 - 8

Zusammenfassung

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) 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.

Schlüsselwörter

  • central Cantor sets
  • Baire category
  • product measure
  • sets of packing dimension zero
Open Access

On O’Malley Porouscontinuous Functions

Online veröffentlicht: 01 Jan 2022
Seitenbereich: 9 - 24

Zusammenfassung

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.

Schlüsselwörter

  • porosity
  • porouscontinuity
  • porouscontinuity in O’Malley sense
Open Access

Generalized Densities on ℝn and their Applications

Online veröffentlicht: 01 Jan 2022
Seitenbereich: 25 - 42

Zusammenfassung

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.

Schlüsselwörter

  • - density point
  • - density topology
  • similarity of topologies
  • differentiation basis
  • base operator
  • set of finite perimeter
  • Sobolev space
Open Access

Local Properties of Entropy for Finite Family of Functions

Online veröffentlicht: 01 Jan 2022
Seitenbereich: 43 - 58

Zusammenfassung

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.

Schlüsselwörter

  • entropy
  • semigroup
  • set of generators
  • entropy of I, II, III type
  • (periodic) dynamical system
  • -invariant set
  • -entropy point ( ∈ {I, II, III})
  • s-chaotic set of generators
Open Access

A Fixed Point Approach to the Hyers-Ulam-Rassias Stability Problem of Pexiderized Functional Equation in Modular Spaces

Online veröffentlicht: 01 Jan 2022
Seitenbereich: 59 - 72

Zusammenfassung

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.

Schlüsselwörter

  • Hyers-Ulam-Rassias stability
  • pexiderized functional equation
  • modular spaces
  • fixed point method
Open Access

A Certain Subclass of Analytic Functions with Negative Coefficients Defined by Gegenbauer Polynomials

Online veröffentlicht: 01 Jan 2022
Seitenbereich: 73 - 84

Zusammenfassung

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.

Schlüsselwörter

  • analytic
  • coefficient bounds
  • extreme points
  • convolution
  • polynomial
Open Access

Super and Hyper Products of Super Relations

Online veröffentlicht: 01 Jan 2022
Seitenbereich: 85 - 118

Zusammenfassung

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 UV and U * V on X such that : (U*V)(A)=(AU(A))V(A),(U*V)(A)=(AU(A))U(A) \begin{array}{*{20}{l}} {(U*V)(A) = (A\mathop \cup \nolimits^ U(A))\mathop \cap \nolimits^ V(A),}\\ {(U*V)(A) = (A\mathop \cap \nolimits^ U(A))\mathop \cup \nolimits^ U(A)} \end{array} and (UV)(A)={BX:(U*V)(A)B(U*V)(A)},(U*V)(A)={BX:(UV)(A)B(UV)(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:\,(U\mathop \cap \nolimits^ V)(A) \subseteq B \subseteq (U\mathop \cup \nolimits^ V)(A)\} } \end{array} for all AX.

By using the distributivity of the operation ∩ over ∪, we can at once see that U * VU * V. Moreover, if UV, 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 AX.

Schlüsselwörter

  • relations on ordinary and power sets
  • unary and binary operations for relations
  • Galois connections
  • generalized uniformities
  • generalized open sets
Open Access

Łukasiewicz Logic and the Divisible Extension of Probability Theory

Online veröffentlicht: 01 Jan 2022
Seitenbereich: 119 - 128

Zusammenfassung

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.

Schlüsselwörter

  • divisible extension of probability theory
  • Boolean logic
  • random variable
  • observable
  • statistical map
  • stochastic channel
  • Łukasiewicz logic
  • full Łukasiewicz tribe
  • conservative observable
Open Access

Around Taylor’s Theorem on the Convergence of Sequences of Functions

Online veröffentlicht: 01 Jan 2022
Seitenbereich: 129 - 138

Zusammenfassung

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 AE such that E \ A is a nullset and limn+|fn(x)f(x)|δn=0forallxA.LetJ(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.

Schlüsselwörter

  • sequences of real-valued functions
  • convergence
Open Access

Hahn-Banach-Type Theorems and Subdifferentials for Invariant and Equivariant Order Continuous Vector Lattice-Valued Operators with Applications to Optimization

Online veröffentlicht: 01 Jan 2022
Seitenbereich: 139 - 156

Zusammenfassung

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.

Schlüsselwörter

  • vector lattice
  • order bounded functional
  • order continuous functional
  • amenability
  • Hahn-Banach theorem
  • sandwich theorem
  • Fenchel duality theorem
  • subgradient
  • subdifferential of composite functions
  • optimality condition
  • Moreau-Rockafellar formula
  • Farkas theorem
  • Kuhn-Tucker theorem
Open Access

On Star-K-I-Hurewicz Property

Online veröffentlicht: 01 Jan 2022
Seitenbereich: 157 - 166

Zusammenfassung

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 xX, {n ∈ ℕ : xSt(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.

Schlüsselwörter

  • Hurewicz
  • star-K--Hurewicz
  • star-K-Hurewicz
  • ideal
  • covering
  • star-covering
  • topological space
Open Access

Compactness of Multiplication Operators on Riesz Bounded Variation Spaces

Online veröffentlicht: 01 Jan 2022
Seitenbereich: 167 - 174

Zusammenfassung

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.

Schlüsselwörter

  • Riesz bounded variation
  • multiplication operator
  • compactness
Open Access

On Functions Preserving Products of Certain Classes of Semimetric Spaces

Online veröffentlicht: 01 Jan 2022
Seitenbereich: 175 - 198

Zusammenfassung

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.

Schlüsselwörter

  • semimetrics
  • metric-preserving functions
  • -metrics
  • subadditive functions
Open Access

Real Functions, Covers and Bornologies

Online veröffentlicht: 01 Jan 2022
Seitenbereich: 199 - 214

Zusammenfassung

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.

Schlüsselwörter

  • cover
  • upper semicontinuous function
  • measurable function
  • upper semimeasurable function
  • bornology
Open Access

Zariski Topologies on Graded Ideals

Online veröffentlicht: 01 Jan 2022
Seitenbereich: 215 - 224

Zusammenfassung

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.

Schlüsselwörter

  • graded prime ideal
  • graded radical ideal
  • Zariski topology

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