Zeszyty czasopisma

Tom 81 (2022): Zeszyt 1 (November 2022)
Real Functions, Dynamical Systems and Applications

Tom 80 (2021): Zeszyt 3 (December 2021)

Tom 79 (2021): Zeszyt 2 (December 2021)

Tom 78 (2021): Zeszyt 1 (October 2021)

Tom 77 (2020): Zeszyt 1 (December 2020)

Tom 76 (2020): Zeszyt 1 (December 2020)
Real Functions, Dynamical Systems and their Applications

Tom 75 (2020): Zeszyt 1 (April 2020)
Applied Mathematics'19

Tom 74 (2019): Zeszyt 1 (December 2019)
Real Functons, Ideals, Measurable Functions, Functional Equations

Tom 73 (2019): Zeszyt 1 (August 2019)
Number Theory, Algebra and Cryptology '18

Tom 72 (2018): Zeszyt 1 (December 2018)

Tom 71 (2018): Zeszyt 1 (December 2018)

Tom 70 (2017): Zeszyt 1 (September 2017)

Tom 69 (2017): Zeszyt 1 (June 2017)

Tom 68 (2017): Zeszyt 1 (March 2017)
Special Zeszyt: Real Functions ’16, Real Functions, Density Topologies, Porosity

Tom 67 (2016): Zeszyt 1 (September 2016)

Tom 66 (2016): Zeszyt 1 (June 2016)
Zeszyt title: Applied Mathematics ’16

Tom 65 (2016): Zeszyt 1 (March 2016)
Real Functions '15 — Measure Theory, Real Functions, General Topology. Editors: J. Borsík, 2016.

Tom 64 (2015): Zeszyt 1 (September 2015)
Number Theory and Cryptology ’15

Tom 62 (2015): Zeszyt 1 (March 2015)
Special Zeszyt Title: Real Functions ’14

Tom 61 (2014): Zeszyt 1 (December 2014)
Special Zeszyt Title: Applied Mathematics ‘14

Tom 60 (2014): Zeszyt 1 (September 2014)
Special Zeszyt Title: Cryptology ’14

Tom 59 (2014): Zeszyt 1 (June 2014)
Special Zeszyt Title: Number Theory ‘14

Tom 58 (2014): Zeszyt 1 (March 2014)
Real Functions ‘13 Real Functions, Topology, Real and Functional Analysis, Locally Convex Spaces

Tom 57 (2013): Zeszyt 1 (December 2013)
Cryptology

Tom 56 (2013): Zeszyt 1 (November 2013)
Number Theory

Tom 55 (2013): Zeszyt 1 (August 2013)

Tom 54 (2013): Zeszyt 1 (April 2013)
Differential and Difference Equations and Applications ‘2012

Tom 53 (2012): Zeszyt 1 (December 2012)
TATRACRYPT ‘12

Tom 52 (2012): Zeszyt 1 (August 2012)

Tom 51 (2012): Zeszyt 1 (April 2012)
PROBASTAT ‘11

Tom 50 (2011): Zeszyt 1 (December 2011)
Applied Mathematics and Informatics

Tom 49 (2011): Zeszyt 1 (August 2011)
Real Functions ‘10

Tom 48 (2011): Zeszyt 1 (April 2011)
Differential and Difference Equations and Applications 2010

Tom 47 (2010): Zeszyt 1 (December 2010)
CCEC ‘09

Tom 46 (2010): Zeszyt 1 (August 2010)
Real Functions ‘09

Tom 45 (2010): Zeszyt 1 (April 2010)
NILCRYPT ‘10

Tom 44 (2009): Zeszyt 1 (December 2009)
Real Function ’08 Functional Equation, Measures, Integration and Harmonic Analysis

Tom 43 (2009): Zeszyt 1 (August 2009)
Differential and Difference Equations and Applications 2008

Tom 42 (2009): Zeszyt 1 (April 2009)
Real Function ‘07

Informacje o czasopiśmie
Format
Czasopismo
eISSN
1338-9750
Pierwsze wydanie
12 Nov 2012
Częstotliwość wydawania
3 razy w roku
Języki
Angielski

Wyszukiwanie

Tom 78 (2021): Zeszyt 1 (October 2021)

Informacje o czasopiśmie
Format
Czasopismo
eISSN
1338-9750
Pierwsze wydanie
12 Nov 2012
Częstotliwość wydawania
3 razy w roku
Języki
Angielski

Wyszukiwanie

15 Artykułów
Otwarty dostęp

The Family of Central Cantor Sets with Packing Dimension Zero

Data publikacji: 01 Jan 2022
Zakres stron: 1 - 8

Abstrakt

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.

Słowa kluczowe

  • central Cantor sets
  • Baire category
  • product measure
  • sets of packing dimension zero
Otwarty dostęp

On O’Malley Porouscontinuous Functions

Data publikacji: 01 Jan 2022
Zakres stron: 9 - 24

Abstrakt

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.

Słowa kluczowe

  • porosity
  • porouscontinuity
  • porouscontinuity in O’Malley sense
Otwarty dostęp

Generalized Densities on ℝn and their Applications

Data publikacji: 01 Jan 2022
Zakres stron: 25 - 42

Abstrakt

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.

Słowa kluczowe

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

Local Properties of Entropy for Finite Family of Functions

Data publikacji: 01 Jan 2022
Zakres stron: 43 - 58

Abstrakt

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.

Słowa kluczowe

  • 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
Otwarty dostęp

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

Data publikacji: 01 Jan 2022
Zakres stron: 59 - 72

Abstrakt

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.

Słowa kluczowe

  • Hyers-Ulam-Rassias stability
  • pexiderized functional equation
  • modular spaces
  • fixed point method
Otwarty dostęp

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

Data publikacji: 01 Jan 2022
Zakres stron: 73 - 84

Abstrakt

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.

Słowa kluczowe

  • analytic
  • coefficient bounds
  • extreme points
  • convolution
  • polynomial
Otwarty dostęp

Super and Hyper Products of Super Relations

Data publikacji: 01 Jan 2022
Zakres stron: 85 - 118

Abstrakt

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.

Słowa kluczowe

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

Łukasiewicz Logic and the Divisible Extension of Probability Theory

Data publikacji: 01 Jan 2022
Zakres stron: 119 - 128

Abstrakt

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.

Słowa kluczowe

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

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

Data publikacji: 01 Jan 2022
Zakres stron: 129 - 138

Abstrakt

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.

Słowa kluczowe

  • sequences of real-valued functions
  • convergence
Otwarty dostęp

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

Data publikacji: 01 Jan 2022
Zakres stron: 139 - 156

Abstrakt

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.

Słowa kluczowe

  • 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
Otwarty dostęp

On Star-K-I-Hurewicz Property

Data publikacji: 01 Jan 2022
Zakres stron: 157 - 166

Abstrakt

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.

Słowa kluczowe

  • Hurewicz
  • star-K--Hurewicz
  • star-K-Hurewicz
  • ideal
  • covering
  • star-covering
  • topological space
Otwarty dostęp

Compactness of Multiplication Operators on Riesz Bounded Variation Spaces

Data publikacji: 01 Jan 2022
Zakres stron: 167 - 174

Abstrakt

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.

Słowa kluczowe

  • Riesz bounded variation
  • multiplication operator
  • compactness
Otwarty dostęp

On Functions Preserving Products of Certain Classes of Semimetric Spaces

Data publikacji: 01 Jan 2022
Zakres stron: 175 - 198

Abstrakt

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.

Słowa kluczowe

  • semimetrics
  • metric-preserving functions
  • -metrics
  • subadditive functions
Otwarty dostęp

Real Functions, Covers and Bornologies

Data publikacji: 01 Jan 2022
Zakres stron: 199 - 214

Abstrakt

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.

Słowa kluczowe

  • cover
  • upper semicontinuous function
  • measurable function
  • upper semimeasurable function
  • bornology
Otwarty dostęp

Zariski Topologies on Graded Ideals

Data publikacji: 01 Jan 2022
Zakres stron: 215 - 224

Abstrakt

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.

Słowa kluczowe

  • graded prime ideal
  • graded radical ideal
  • Zariski topology
15 Artykułów
Otwarty dostęp

The Family of Central Cantor Sets with Packing Dimension Zero

Data publikacji: 01 Jan 2022
Zakres stron: 1 - 8

Abstrakt

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.

Słowa kluczowe

  • central Cantor sets
  • Baire category
  • product measure
  • sets of packing dimension zero
Otwarty dostęp

On O’Malley Porouscontinuous Functions

Data publikacji: 01 Jan 2022
Zakres stron: 9 - 24

Abstrakt

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.

Słowa kluczowe

  • porosity
  • porouscontinuity
  • porouscontinuity in O’Malley sense
Otwarty dostęp

Generalized Densities on ℝn and their Applications

Data publikacji: 01 Jan 2022
Zakres stron: 25 - 42

Abstrakt

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.

Słowa kluczowe

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

Local Properties of Entropy for Finite Family of Functions

Data publikacji: 01 Jan 2022
Zakres stron: 43 - 58

Abstrakt

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.

Słowa kluczowe

  • 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
Otwarty dostęp

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

Data publikacji: 01 Jan 2022
Zakres stron: 59 - 72

Abstrakt

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.

Słowa kluczowe

  • Hyers-Ulam-Rassias stability
  • pexiderized functional equation
  • modular spaces
  • fixed point method
Otwarty dostęp

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

Data publikacji: 01 Jan 2022
Zakres stron: 73 - 84

Abstrakt

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.

Słowa kluczowe

  • analytic
  • coefficient bounds
  • extreme points
  • convolution
  • polynomial
Otwarty dostęp

Super and Hyper Products of Super Relations

Data publikacji: 01 Jan 2022
Zakres stron: 85 - 118

Abstrakt

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.

Słowa kluczowe

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

Łukasiewicz Logic and the Divisible Extension of Probability Theory

Data publikacji: 01 Jan 2022
Zakres stron: 119 - 128

Abstrakt

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.

Słowa kluczowe

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

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

Data publikacji: 01 Jan 2022
Zakres stron: 129 - 138

Abstrakt

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.

Słowa kluczowe

  • sequences of real-valued functions
  • convergence
Otwarty dostęp

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

Data publikacji: 01 Jan 2022
Zakres stron: 139 - 156

Abstrakt

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.

Słowa kluczowe

  • 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
Otwarty dostęp

On Star-K-I-Hurewicz Property

Data publikacji: 01 Jan 2022
Zakres stron: 157 - 166

Abstrakt

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.

Słowa kluczowe

  • Hurewicz
  • star-K--Hurewicz
  • star-K-Hurewicz
  • ideal
  • covering
  • star-covering
  • topological space
Otwarty dostęp

Compactness of Multiplication Operators on Riesz Bounded Variation Spaces

Data publikacji: 01 Jan 2022
Zakres stron: 167 - 174

Abstrakt

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.

Słowa kluczowe

  • Riesz bounded variation
  • multiplication operator
  • compactness
Otwarty dostęp

On Functions Preserving Products of Certain Classes of Semimetric Spaces

Data publikacji: 01 Jan 2022
Zakres stron: 175 - 198

Abstrakt

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.

Słowa kluczowe

  • semimetrics
  • metric-preserving functions
  • -metrics
  • subadditive functions
Otwarty dostęp

Real Functions, Covers and Bornologies

Data publikacji: 01 Jan 2022
Zakres stron: 199 - 214

Abstrakt

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.

Słowa kluczowe

  • cover
  • upper semicontinuous function
  • measurable function
  • upper semimeasurable function
  • bornology
Otwarty dostęp

Zariski Topologies on Graded Ideals

Data publikacji: 01 Jan 2022
Zakres stron: 215 - 224

Abstrakt

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.

Słowa kluczowe

  • graded prime ideal
  • graded radical ideal
  • Zariski topology

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