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Real Functons, Ideals, Measurable Functions, Functional Equations

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

Suche

Volumen 74 (2019): Heft 1 (December 2019)
Real Functons, Ideals, Measurable Functions, Functional Equations

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

Suche

13 Artikel
Uneingeschränkter Zugang

Ján Borsík (1955 – 2019)

Online veröffentlicht: 15 Nov 2019
Seitenbereich: 1 - 6

Zusammenfassung

Uneingeschränkter Zugang

The Zariski Topology on the Graded Primary Spectrum Over Graded Commutative Rings

Online veröffentlicht: 15 Nov 2019
Seitenbereich: 7 - 16

Zusammenfassung

Abstract

Let G be a group with identity e and let R be a G-graded ring. A proper graded ideal P of R is called a graded primary ideal if whenever rgsh∈P, we have rg∈ P or sh∈ Gr(P), where rg,sg∈ h(R). The graded primary spectrum p.Specg(R) is defined to be the set of all graded primary ideals of R.In this paper, we define a topology on p.Specg(R), called Zariski topology, which is analogous to that for Specg(R), and investigate several properties of the topology.

Schlüsselwörter

  • Zariski topology
  • graded primary spectrum
  • graded primary ideals

MSC 2010

  • 13A02
  • 16W50
Uneingeschränkter Zugang

Real Functions in Stochastic Dependence

Online veröffentlicht: 15 Nov 2019
Seitenbereich: 17 - 34

Zusammenfassung

Abstract

In a fuzzified probability theory, random events are modeled by measurable functions into [0,1] and probability measures are replaced with probability integrals. The transition from Boolean two-valued logic to Lukasiewicz multivalued logic results in an upgraded probability theory in which we define and study asymmetrical stochastic dependence/independence and conditional probability based on stochastic channels and joint experiments so that the classical constructions follow as particular cases. Elementary categorical methods enable us to put the two theories into a perspective.

Schlüsselwörter

  • measurable function
  • stochastic dependence
  • fuzzy random event
  • observable
  • probability measure
  • probability integral
  • state map
  • statistical map
  • joint experiment
  • asymmetrical independence

MSC 2010

  • 26E50
  • 28A35
  • 60A86
  • 60A05
Uneingeschränkter Zugang

I-Completeness in Function Spaces

Online veröffentlicht: 15 Nov 2019
Seitenbereich: 35 - 46

Zusammenfassung

Abstract

In this paper, we have studied the idea of ideal completeness of function spaces YX with respect to pointwise uniformity and uniformity of uniform convergence. Further, involving topological structure on X,wehaveobtained relationships between the uniformity of uniform convergence on compacta on YX and uniformity of uniform convergence on Y X in terms of I-Cauchy condition and I-convergence of a net. Also, using the notion of a k-space, we have given a sufficient condition for C(X, Y) to be ideal complete with respect to the uniformity of uniform convergence on compacta.

Schlüsselwörter

  • ideal
  • filter
  • uniform space
  • -Cauchy condition
  • -convergence
  • ideal completeness

MSC 2010

  • Primary 54A20
  • Secondary 40A35, 54E15
Uneingeschränkter Zugang

Product of Measurable Spaces and Applications

Online veröffentlicht: 15 Nov 2019
Seitenbereich: 47 - 56

Zusammenfassung

Abstract

We deal with products of measurable spaces and relationships between measures on products and asymmetrical stochastic dependence/independence of one extended probability space on another one.

Schlüsselwörter

  • Measurable space
  • measurable function
  • product of measurable spaces
  • probability measure on product
  • probability integral
  • observable
  • degenerated observable
  • statistical map
  • joint probability space
  • extended probability space
  • -joint extended probability space
  • stochastic dependence
  • asymmetrical stochastic independence
  • extended outcome
  • conditional expectation
  • conditional probability

MSC 2010

  • 28A35
  • 60A86
  • 60A05
  • 18A32
Uneingeschränkter Zugang

A Short Proof of Alienation of Additivity from Quadraticity

Online veröffentlicht: 15 Nov 2019
Seitenbereich: 57 - 62

Zusammenfassung

Abstract

Without the use of pexiderized versions of abstract polynomials theory, we show that on 2-divisible groups the functional equation

f(x+y)+g(x+y)+g(x-y)=f(x)+f(y)+2g(x)+2g(y)f\left( {x + y} \right) + g\left( {x + y} \right) + g\left( {x - y} \right) = f(x) + f(y) + 2g(x) + 2g(y)

forces the unknown functions f and g to be additive and quadratic, respectively, modulo a constant.

Motivated by the observation that the equation

f(x+y)+f(x2)=f(x)+f(y)+f(x2)f\left( {x + y} \right) + f({x^2}) = f(x) + f(y) + f({x^2})

implies both the additivity and multiplicativity of f, we deal also with the alienation phenomenon of equations in a single and several variables.

Schlüsselwörter

  • alienation phenomenon
  • additive and quadratic mappings
  • systems of functional equations

MSC 2010

  • 39B52
  • 39B72
Uneingeschränkter Zugang

A Three Dimensional Modification of the Gaussian Number Field

Online veröffentlicht: 15 Nov 2019
Seitenbereich: 63 - 76

Zusammenfassung

Abstract

For vectors in E3 we introduce an associative, commutative and distributive multiplication. We describe the related algebraic and geometrical properties, and hint some applications.

Based on properties of hyperbolic (Clifford) complex numbers, we prove that the resulting algebra 𝕋 is an associative algebra over a field and contains a subring isomorphic to hyperbolic complex numbers. Moreover, the algebra 𝕋 is isomorphic to direct product ℂ×ℝ, and so it contains a subalgebra isomorphic to the Gaussian complex plane.

Schlüsselwörter

  • Normed field
  • three dimensions
  • factor ring
  • generalized complex numbers

MSC 2010

  • 12J05
  • 12D99
  • 11R52
Uneingeschränkter Zugang

Generalized Derivative and Generalized Continuity

Online veröffentlicht: 15 Nov 2019
Seitenbereich: 77 - 84

Zusammenfassung

Abstract

The main objective of this article is to show that generalized differentiation can be understood as a process of comparing functions and their generalized continuity properties. We show it by working with generalized notions of derivative and continuity. The article covers wide range of types of generalized continuity.

Schlüsselwörter

  • generalized derivative
  • generalized continuity
  • -continuity
  • -continuity

MSC 2010

  • Primary 54C08, 26A24, 26A21
  • Secondary 26A06, 54C30, 26A99
Uneingeschränkter Zugang

How to Obtain Maximal and Minimal Subranges of Two-Dimensional Vector Measures

Online veröffentlicht: 15 Nov 2019
Seitenbereich: 85 - 90

Zusammenfassung

Abstract

Let (X, ℱ) be a measurable space with a nonatomic vector measure µ =(µ1, µ2). Denote by R(Y) the subrange R(Y)= (Z): Z ∈ ℱ, ZY }. For a given pµ(ℱ) consider a family of measurable subsets ℱp = {Z ∈ ℱ : µ(Z)= p}. Dai and Feinberg proved the existence of a maximal subset Z*Fp having the maximal subrange R(Z*) and also a minimal subset M* ∈ ℱp with the minimal subrange R(M*). We present a method of obtaining the maximal and the minimal subsets. Hence, we get simple proofs of the results of Dai and Feinberg.

Schlüsselwörter

  • Lyapunov convexity theorem
  • range of a vector measure
  • maximal subset
  • maximal subrange of a vector measure

MSC 2010

  • Primary 60A10
  • Secondary 28A10
Uneingeschränkter Zugang

On Functions of Bounded (φ, k)-Variation

Online veröffentlicht: 15 Nov 2019
Seitenbereich: 91 - 116

Zusammenfassung

Abstract

Given a φ-function φ and k ∈ ℕ, we introduce and study the concept of (φ, k)-variation in the sense of Riesz of a real function on a compact interval. We show that a function u :[a, b] ℝ has a bounded (φ, k)-variation if and only if u(k−1) is absolutely continuous on [a, b]and u(k) belongs to the Orlicz class L φ[a, b]. We also show that the space generated by this class of functions is a Banach space. Our approach simultaneously generalizes the concepts of the Riesz φ-variation, the de la Vallée Poussin second-variation and the Popoviciu kth variation.

Schlüsselwörter

  • Riesz -variation
  • De la Vallée Poussin second-variation
  • Popoviciu th variation
  • bounded ()-variation

MSC 2010

  • 93B05
  • 93C25
Uneingeschränkter Zugang

On One Application of Infinite Systems of Functional Equations in Function Theory

Online veröffentlicht: 15 Nov 2019
Seitenbereich: 117 - 144

Zusammenfassung

Abstract

The paper presents the investigation of applications of infinite systems of functional equations for modeling functions with complicated local structure that are defined in terms of the nega-˜Q-representation. The infinite systems of functional equations

f(φˆk(x))=β˜ik+1,k+1+p˜ik+1,k+1f(φˆk+1(x)),f\left( {{{\hat \varphi }^k}(x)} \right) = \tilde \beta {i_{k + 1}},k + 1 + \tilde p{i_{k + 1}},k + 1f\left( {{{\hat \varphi }^{k + 1}}(x)} \right),

where x=Δi1(x)i2(x)in(x)-Q˜x = \Delta _{{i_1}(x){i_2}(x) \ldots {i_n}(x) \ldots }^{ - \tilde Q}, and φ ̑ is the shift operator of the Q̃-expansion, are investigated. It is proved that the system has a unique solution in the class of determined and bounded on [0, 1] functions. Its analytical presentation is founded. The continuity of the solution is studied. Conditions of its monotonicity and nonmonotonicity, differential, and integral properties are studied. Conditions under which the solution of the system of functional equations is a distribution function of the random variable η=Δξ1ξ2ξnQ˜\eta = \Delta _{{\xi _1}\,\xi 2 \ldots {\xi _n} \ldots }^{\tilde Q} with independent Q̃-symbols are founded.

Schlüsselwörter

  • Function with complicated local structure
  • systems of functional equations
  • singular function
  • nowhere differentiable function
  • distribution function
  • nowhere monotone function
  • ̃-representation
  • nega-̃-representation
  • Lebesgue integral

MSC 2010

  • 39B72
  • 11K55
  • 26A27
  • 26A30
  • 26A42
Uneingeschränkter Zugang

On a Lindenbaum Composition Theorem

Online veröffentlicht: 15 Nov 2019
Seitenbereich: 145 - 158

Zusammenfassung

Abstract

We discuss several questions about Borel measurable functions on a topological space. We show that two Lindenbaum composition theorems [Lindenbaum, A. Sur les superpositions des fonctions représentables analytiquement, Fund. Math. 23 (1934), 15–37] proved for the real line hold in perfectly normal topological space as well. As an application, we extend a characterization of a certain class of topological spaces with hereditary Jayne-Rogers property for perfectly normal topological space. Finally, we pose an interesting question about lower and upper Δ02-measurable functions.

Schlüsselwörter

  • semicontinuous function
  • Young hierarchy
  • Δ-measurable function
  • composition
  • piecewise continuous

MSC 2010

  • Primary 26A21
  • Secondary 26A15, 54C50
Uneingeschränkter Zugang

Strictly Increasing Additive Generators of the Second Kind of Associative Binary Operations

Online veröffentlicht: 15 Nov 2019
Seitenbereich: 159 - 176

Zusammenfassung

Abstract

The class of strictly increasing additive generators of the second kind is defined and analyzed. Necessary and sufficient conditions for a binary operation generated by a strictly increasing additive generator of the second kind to be associative are introduced. The relation between the class of strictly increasing additive generators of the second kind of associative binary operations and the class of discrete upper additive generators of associative binary operations is revealed.

Schlüsselwörter

  • additive generator
  • aggregation function
  • associativity
  • discrete additive generator
  • t-conorm

MSC 2010

  • 03E72
  • 06F05
13 Artikel
Uneingeschränkter Zugang

Ján Borsík (1955 – 2019)

Online veröffentlicht: 15 Nov 2019
Seitenbereich: 1 - 6

Zusammenfassung

Uneingeschränkter Zugang

The Zariski Topology on the Graded Primary Spectrum Over Graded Commutative Rings

Online veröffentlicht: 15 Nov 2019
Seitenbereich: 7 - 16

Zusammenfassung

Abstract

Let G be a group with identity e and let R be a G-graded ring. A proper graded ideal P of R is called a graded primary ideal if whenever rgsh∈P, we have rg∈ P or sh∈ Gr(P), where rg,sg∈ h(R). The graded primary spectrum p.Specg(R) is defined to be the set of all graded primary ideals of R.In this paper, we define a topology on p.Specg(R), called Zariski topology, which is analogous to that for Specg(R), and investigate several properties of the topology.

Schlüsselwörter

  • Zariski topology
  • graded primary spectrum
  • graded primary ideals

MSC 2010

  • 13A02
  • 16W50
Uneingeschränkter Zugang

Real Functions in Stochastic Dependence

Online veröffentlicht: 15 Nov 2019
Seitenbereich: 17 - 34

Zusammenfassung

Abstract

In a fuzzified probability theory, random events are modeled by measurable functions into [0,1] and probability measures are replaced with probability integrals. The transition from Boolean two-valued logic to Lukasiewicz multivalued logic results in an upgraded probability theory in which we define and study asymmetrical stochastic dependence/independence and conditional probability based on stochastic channels and joint experiments so that the classical constructions follow as particular cases. Elementary categorical methods enable us to put the two theories into a perspective.

Schlüsselwörter

  • measurable function
  • stochastic dependence
  • fuzzy random event
  • observable
  • probability measure
  • probability integral
  • state map
  • statistical map
  • joint experiment
  • asymmetrical independence

MSC 2010

  • 26E50
  • 28A35
  • 60A86
  • 60A05
Uneingeschränkter Zugang

I-Completeness in Function Spaces

Online veröffentlicht: 15 Nov 2019
Seitenbereich: 35 - 46

Zusammenfassung

Abstract

In this paper, we have studied the idea of ideal completeness of function spaces YX with respect to pointwise uniformity and uniformity of uniform convergence. Further, involving topological structure on X,wehaveobtained relationships between the uniformity of uniform convergence on compacta on YX and uniformity of uniform convergence on Y X in terms of I-Cauchy condition and I-convergence of a net. Also, using the notion of a k-space, we have given a sufficient condition for C(X, Y) to be ideal complete with respect to the uniformity of uniform convergence on compacta.

Schlüsselwörter

  • ideal
  • filter
  • uniform space
  • -Cauchy condition
  • -convergence
  • ideal completeness

MSC 2010

  • Primary 54A20
  • Secondary 40A35, 54E15
Uneingeschränkter Zugang

Product of Measurable Spaces and Applications

Online veröffentlicht: 15 Nov 2019
Seitenbereich: 47 - 56

Zusammenfassung

Abstract

We deal with products of measurable spaces and relationships between measures on products and asymmetrical stochastic dependence/independence of one extended probability space on another one.

Schlüsselwörter

  • Measurable space
  • measurable function
  • product of measurable spaces
  • probability measure on product
  • probability integral
  • observable
  • degenerated observable
  • statistical map
  • joint probability space
  • extended probability space
  • -joint extended probability space
  • stochastic dependence
  • asymmetrical stochastic independence
  • extended outcome
  • conditional expectation
  • conditional probability

MSC 2010

  • 28A35
  • 60A86
  • 60A05
  • 18A32
Uneingeschränkter Zugang

A Short Proof of Alienation of Additivity from Quadraticity

Online veröffentlicht: 15 Nov 2019
Seitenbereich: 57 - 62

Zusammenfassung

Abstract

Without the use of pexiderized versions of abstract polynomials theory, we show that on 2-divisible groups the functional equation

f(x+y)+g(x+y)+g(x-y)=f(x)+f(y)+2g(x)+2g(y)f\left( {x + y} \right) + g\left( {x + y} \right) + g\left( {x - y} \right) = f(x) + f(y) + 2g(x) + 2g(y)

forces the unknown functions f and g to be additive and quadratic, respectively, modulo a constant.

Motivated by the observation that the equation

f(x+y)+f(x2)=f(x)+f(y)+f(x2)f\left( {x + y} \right) + f({x^2}) = f(x) + f(y) + f({x^2})

implies both the additivity and multiplicativity of f, we deal also with the alienation phenomenon of equations in a single and several variables.

Schlüsselwörter

  • alienation phenomenon
  • additive and quadratic mappings
  • systems of functional equations

MSC 2010

  • 39B52
  • 39B72
Uneingeschränkter Zugang

A Three Dimensional Modification of the Gaussian Number Field

Online veröffentlicht: 15 Nov 2019
Seitenbereich: 63 - 76

Zusammenfassung

Abstract

For vectors in E3 we introduce an associative, commutative and distributive multiplication. We describe the related algebraic and geometrical properties, and hint some applications.

Based on properties of hyperbolic (Clifford) complex numbers, we prove that the resulting algebra 𝕋 is an associative algebra over a field and contains a subring isomorphic to hyperbolic complex numbers. Moreover, the algebra 𝕋 is isomorphic to direct product ℂ×ℝ, and so it contains a subalgebra isomorphic to the Gaussian complex plane.

Schlüsselwörter

  • Normed field
  • three dimensions
  • factor ring
  • generalized complex numbers

MSC 2010

  • 12J05
  • 12D99
  • 11R52
Uneingeschränkter Zugang

Generalized Derivative and Generalized Continuity

Online veröffentlicht: 15 Nov 2019
Seitenbereich: 77 - 84

Zusammenfassung

Abstract

The main objective of this article is to show that generalized differentiation can be understood as a process of comparing functions and their generalized continuity properties. We show it by working with generalized notions of derivative and continuity. The article covers wide range of types of generalized continuity.

Schlüsselwörter

  • generalized derivative
  • generalized continuity
  • -continuity
  • -continuity

MSC 2010

  • Primary 54C08, 26A24, 26A21
  • Secondary 26A06, 54C30, 26A99
Uneingeschränkter Zugang

How to Obtain Maximal and Minimal Subranges of Two-Dimensional Vector Measures

Online veröffentlicht: 15 Nov 2019
Seitenbereich: 85 - 90

Zusammenfassung

Abstract

Let (X, ℱ) be a measurable space with a nonatomic vector measure µ =(µ1, µ2). Denote by R(Y) the subrange R(Y)= (Z): Z ∈ ℱ, ZY }. For a given pµ(ℱ) consider a family of measurable subsets ℱp = {Z ∈ ℱ : µ(Z)= p}. Dai and Feinberg proved the existence of a maximal subset Z*Fp having the maximal subrange R(Z*) and also a minimal subset M* ∈ ℱp with the minimal subrange R(M*). We present a method of obtaining the maximal and the minimal subsets. Hence, we get simple proofs of the results of Dai and Feinberg.

Schlüsselwörter

  • Lyapunov convexity theorem
  • range of a vector measure
  • maximal subset
  • maximal subrange of a vector measure

MSC 2010

  • Primary 60A10
  • Secondary 28A10
Uneingeschränkter Zugang

On Functions of Bounded (φ, k)-Variation

Online veröffentlicht: 15 Nov 2019
Seitenbereich: 91 - 116

Zusammenfassung

Abstract

Given a φ-function φ and k ∈ ℕ, we introduce and study the concept of (φ, k)-variation in the sense of Riesz of a real function on a compact interval. We show that a function u :[a, b] ℝ has a bounded (φ, k)-variation if and only if u(k−1) is absolutely continuous on [a, b]and u(k) belongs to the Orlicz class L φ[a, b]. We also show that the space generated by this class of functions is a Banach space. Our approach simultaneously generalizes the concepts of the Riesz φ-variation, the de la Vallée Poussin second-variation and the Popoviciu kth variation.

Schlüsselwörter

  • Riesz -variation
  • De la Vallée Poussin second-variation
  • Popoviciu th variation
  • bounded ()-variation

MSC 2010

  • 93B05
  • 93C25
Uneingeschränkter Zugang

On One Application of Infinite Systems of Functional Equations in Function Theory

Online veröffentlicht: 15 Nov 2019
Seitenbereich: 117 - 144

Zusammenfassung

Abstract

The paper presents the investigation of applications of infinite systems of functional equations for modeling functions with complicated local structure that are defined in terms of the nega-˜Q-representation. The infinite systems of functional equations

f(φˆk(x))=β˜ik+1,k+1+p˜ik+1,k+1f(φˆk+1(x)),f\left( {{{\hat \varphi }^k}(x)} \right) = \tilde \beta {i_{k + 1}},k + 1 + \tilde p{i_{k + 1}},k + 1f\left( {{{\hat \varphi }^{k + 1}}(x)} \right),

where x=Δi1(x)i2(x)in(x)-Q˜x = \Delta _{{i_1}(x){i_2}(x) \ldots {i_n}(x) \ldots }^{ - \tilde Q}, and φ ̑ is the shift operator of the Q̃-expansion, are investigated. It is proved that the system has a unique solution in the class of determined and bounded on [0, 1] functions. Its analytical presentation is founded. The continuity of the solution is studied. Conditions of its monotonicity and nonmonotonicity, differential, and integral properties are studied. Conditions under which the solution of the system of functional equations is a distribution function of the random variable η=Δξ1ξ2ξnQ˜\eta = \Delta _{{\xi _1}\,\xi 2 \ldots {\xi _n} \ldots }^{\tilde Q} with independent Q̃-symbols are founded.

Schlüsselwörter

  • Function with complicated local structure
  • systems of functional equations
  • singular function
  • nowhere differentiable function
  • distribution function
  • nowhere monotone function
  • ̃-representation
  • nega-̃-representation
  • Lebesgue integral

MSC 2010

  • 39B72
  • 11K55
  • 26A27
  • 26A30
  • 26A42
Uneingeschränkter Zugang

On a Lindenbaum Composition Theorem

Online veröffentlicht: 15 Nov 2019
Seitenbereich: 145 - 158

Zusammenfassung

Abstract

We discuss several questions about Borel measurable functions on a topological space. We show that two Lindenbaum composition theorems [Lindenbaum, A. Sur les superpositions des fonctions représentables analytiquement, Fund. Math. 23 (1934), 15–37] proved for the real line hold in perfectly normal topological space as well. As an application, we extend a characterization of a certain class of topological spaces with hereditary Jayne-Rogers property for perfectly normal topological space. Finally, we pose an interesting question about lower and upper Δ02-measurable functions.

Schlüsselwörter

  • semicontinuous function
  • Young hierarchy
  • Δ-measurable function
  • composition
  • piecewise continuous

MSC 2010

  • Primary 26A21
  • Secondary 26A15, 54C50
Uneingeschränkter Zugang

Strictly Increasing Additive Generators of the Second Kind of Associative Binary Operations

Online veröffentlicht: 15 Nov 2019
Seitenbereich: 159 - 176

Zusammenfassung

Abstract

The class of strictly increasing additive generators of the second kind is defined and analyzed. Necessary and sufficient conditions for a binary operation generated by a strictly increasing additive generator of the second kind to be associative are introduced. The relation between the class of strictly increasing additive generators of the second kind of associative binary operations and the class of discrete upper additive generators of associative binary operations is revealed.

Schlüsselwörter

  • additive generator
  • aggregation function
  • associativity
  • discrete additive generator
  • t-conorm

MSC 2010

  • 03E72
  • 06F05

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