Non-atomicity and the Darboux property for fuzzy and non-fuzzy Borel/Baire multivalued set functions

Gavriluţ, Alina; Merches, Ioan; Agop, Maricel

doi:10.1007/978-3-030-29593-6_4

Cited by 2 publications

(8 citation statements)

References 10 publications

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“…We firstly recall the following notions (Gavrilut [7,8], Gavriluţ and Croitoru [5,6])). Definition 2.…”

Section: Minimal Atomsmentioning

confidence: 99%

“…Precisely, interference allows the union of sets of measure zero to have nonzero measure. Definition 3 (Gavrilut [7,8], Gavriluţ and Croitoru [5,6]). (I) A set multifunction : C → P ( ), with (0) = {0}, is said to be (i) null-additive if ( ∪ ) = ( ), ∀ , ∈ C, with ( ) = {0};…”

Section: Minimal Atomsmentioning

confidence: 99%

“…(iii) (Gavrilut [7,8], Gavriluţ and Croitoru [5,6]) a pseudo-atom of if ( ) ⊋ {0} and ∀ ∈ C, ⊆ , one has ( ) = {0} or ( ) = ( ).…”

Section: Minimal Atomsmentioning

confidence: 99%

“…Moreover, conversely, if ∈ C is a minimal atom of , then it is also an atom of , so ( ) ≥ | ( )|, whence is a minimal atom of . In consequence, if, moreover, is a multisubmeasure (in the sense of Gavrilut [8]) of finite variation, then is finitely additive, so ( ) = ∑ =1 | ( )| (Dinculeanu [45]).…”

Section: Minimal Atomsmentioning

confidence: 99%

“…(Non)atomic measures and purely atomic measures have been investigated in different variants (Chiţescu [1,2], Cavaliere and Ventriglia [3], Gavriluţ and Agop [4], Gavriluţ and Croitoru [5,6], Gavrilut [7,8], Khare and Singh [9], Li et al [10,11], Pap [12][13][14], Pap et al [15], Rao and Rao [16], Suzuki [17], and Wu and Bo [18]).…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Extended Minimal Atomicity through Nondifferentiability: A Mathematical-Physical Approach

Gavriluț

Gavriluţ

Agop

2019

Advances in Mathematical Physics

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The mathematical concept of minimal atomicity is extended to fractal minimal atomicity, based on the nondifferentiability of the motion curves of physical system entities on a fractal manifold. For this purpose, firstly, different results concerning minimal atomicity from the mathematical procedure of the Quantum Measure Theory and also several physical implications are obtained. Further, an inverse method with respect to the common developments concerning the minimal atomicity concept has been used, showing that Quantum Mechanics is identified as a particular case of Fractal Mechanics at a given scale resolution. More precisely, for fractality through Markov type stochastic processes, i.e., fractalization through stochasticization, the standard Schrödinger equation is identified with the geodesics of a fractal space for motions of the physical system entities on nondifferentiable curves on fractal dimension two at Compton scale resolution. In the one-dimensional stationary case of the fractal Schrödinger type geodesics, a special symmetry induced by the homographic group in Barbilian’s form “makes possible the synchronicity” of all entities of a given physical system. The integral and differential properties of this group under the restriction of defining a parallelism of directions in Levi-Civita’s sense impose correspondences with the “dynamics” of the hyperbolic plane so that harmonic mappings between the ordinary flat space and the hyperbolic one generate (by means of a variational principle) a priori probabilities in Jaynes’ sense. The explicitation of such situation specifies the fact that the hydrodynamical variant of a Fractal Mechanics is more easily approached and, from this, the fact that Quantum Measure Theory can be a particular case of a possible Fractal Measure Theory.

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“…We firstly recall the following notions (Gavrilut [7,8], Gavriluţ and Croitoru [5,6])). Definition 2.…”

Section: Minimal Atomsmentioning

confidence: 99%

Section: Minimal Atomsmentioning

confidence: 99%

“…(iii) (Gavrilut [7,8], Gavriluţ and Croitoru [5,6]) a pseudo-atom of if ( ) ⊋ {0} and ∀ ∈ C, ⊆ , one has ( ) = {0} or ( ) = ( ).…”

Section: Minimal Atomsmentioning

confidence: 99%

Section: Minimal Atomsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Extended Minimal Atomicity through Nondifferentiability: A Mathematical-Physical Approach

Gavriluț

Gavriluţ

Agop

2019

Advances in Mathematical Physics

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Fractal Atomicity, a Fundamental Concept in the Dynamics of Complex Systems

Agop

Gavriluţ

Eva

et al. 2021

13th Chaotic Modeling and Simulation International Conference

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Applying a fractal method of analyzing the dynamics of the structural units of any complex system, a mathematical concept is built, namely that of fractal atomicity. The construction of such a concept involves defining dynamic variables in the form of fractal functions, defining scale resolutions, defining a principle of scale covariance as a fundamental principle of motion, equations of evolution, etc. Finally, some specific mathematical properties of the fractal atom are also established. IntroductionThe notion of non-atomicity for set functions plays a key role in Measure Theory and its applications and extensions. For classical measures taking values in finite dimensional Banach spaces, it guarantees the connectedness of range. Even just replacing σ-additivity with finite additivity for measures requires some stronger non-atomicity property for the same conclusion to hold.Because of their multiple applications in game theory or mathematical economics, the study concerning atoms and non-atomicity for additive, respectively, non-additive set functions has developed. Particularly, (non)atomic (purely) measures have been studied in different forms due to their special form and their special properties

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Non-atomicity and the Darboux property for fuzzy and non-fuzzy Borel/Baire multivalued set functions

Cited by 2 publications

References 10 publications

Extended Minimal Atomicity through Nondifferentiability: A Mathematical-Physical Approach

Extended Minimal Atomicity through Nondifferentiability: A Mathematical-Physical Approach

Fractal Atomicity, a Fundamental Concept in the Dynamics of Complex Systems

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