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Axioms, Volume 5, Issue 3 (September 2016) – 6 articles

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287 KiB  
Article
On the q-Laplace Transform and Related Special Functions
by Shanoja R. Naik and Hans J. Haubold
Axioms 2016, 5(3), 24; https://doi.org/10.3390/axioms5030024 - 6 Sep 2016
Cited by 6 | Viewed by 4983
Abstract
Motivated by statistical mechanics contexts, we study the properties of the q-Laplace transform, which is an extension of the well-known Laplace transform. In many circumstances, the kernel function to evaluate certain integral forms has been studied. In this article, we establish relationships [...] Read more.
Motivated by statistical mechanics contexts, we study the properties of the q-Laplace transform, which is an extension of the well-known Laplace transform. In many circumstances, the kernel function to evaluate certain integral forms has been studied. In this article, we establish relationships between q-exponential and other well-known functional forms, such as Mittag–Leffler functions, hypergeometric and H-function, by means of the kernel function of the integral. Traditionally, we have been applying the Laplace transform method to solve differential equations and boundary value problems. Here, we propose an alternative, the q-Laplace transform method, to solve differential equations, such as as the fractional space-time diffusion equation, the generalized kinetic equation and the time fractional heat equation. Full article
266 KiB  
Article
The Universe in Leśniewski’s Mereology: Some Comments on Sobociński’s Reflections
by Marcin Łyczak, Marek Porwolik and Kordula Świętorzecka
Axioms 2016, 5(3), 23; https://doi.org/10.3390/axioms5030023 - 6 Sep 2016
Cited by 1 | Viewed by 4655
Abstract
Stanisław Leśniewski’s mereology was originally conceived as a theory of foundations of mathematics and it is also for this reason that it has philosophical connotations. The ‘philosophical significance’ of mereology was upheld by Bolesław Sobociński who expressed the view in his correspondence with [...] Read more.
Stanisław Leśniewski’s mereology was originally conceived as a theory of foundations of mathematics and it is also for this reason that it has philosophical connotations. The ‘philosophical significance’ of mereology was upheld by Bolesław Sobociński who expressed the view in his correspondence with J.M. Bocheński. As he wrote to Bocheński in 1948: “[...] it is interesting that, being such a simple deductive theory, mereology may prove a number of very general theses reminiscent of metaphysical ontology”. The theses which Sobociński had in mind were related to the mereological notion of “the Universe”. Sobociński listed them in the letter adding his philosophical commentary but he did not give proofs for them and did not specify precisely the theory lying behind them. This is what we want to supply in the first part of our paper. We indicate some connections between the notion of the universe and other specific mereological notions. Motivated by Sobociński’s informal suggestions showing his preference for mereology over the axiomatic set theory in application to philosophy we propose to consider Sobociński’s formalism in a new frame which is the ZFM theory—an extension of Zermelo-Fraenkel set theory by mereological axioms, developed by A. Pietruszczak. In this systematic part we investigate reasons of ’philosophical hopes’ mentioned by Sobociński, pinned on the mereological concept of “the Universe”. Full article
(This article belongs to the Special Issue Lvov—Warsaw School)
245 KiB  
Review
A Method for Ordering of LR-Type Fuzzy Numbers: An Important Decision Criteria
by José A. González Campos and Ronald A. Manríquez Peñafiel
Axioms 2016, 5(3), 22; https://doi.org/10.3390/axioms5030022 - 31 Aug 2016
Cited by 2 | Viewed by 3775
Abstract
Methods for ordering fuzzy numbers play an important role as decision criteria, with applications in areas such as optimization and data mining, among others. Although there are several proposals for ordering methods in the fuzzy literature, many of them are difficult to apply [...] Read more.
Methods for ordering fuzzy numbers play an important role as decision criteria, with applications in areas such as optimization and data mining, among others. Although there are several proposals for ordering methods in the fuzzy literature, many of them are difficult to apply and present some problems with ranking computation. For that reason, this work proposes an ordering method for fuzzy numbers based on a simple application of a polynomial function. We study some properties of our new method, comparing our results with those generated by other methods previously discussed in literature. Full article
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184 KiB  
Article
Is Kazimierz Ajdukiewicz’s Concept of a Real Definition Still Important?
by Robert Kublikowski
Axioms 2016, 5(3), 21; https://doi.org/10.3390/axioms5030021 - 17 Aug 2016
Viewed by 3941
Abstract
The concept of a real definition worked out by Kazimierz Ajdukiewicz is still important in the theory of definition and can be developed by applying Hilary Putnam’s theory of reference of natural kind terms and Karl Popper’s fallibilism. On the one hand, the [...] Read more.
The concept of a real definition worked out by Kazimierz Ajdukiewicz is still important in the theory of definition and can be developed by applying Hilary Putnam’s theory of reference of natural kind terms and Karl Popper’s fallibilism. On the one hand, the definiendum of a real definition refers to a natural kind of things and, on the other hand, the definiens of such a definition expresses actual, empirical, fallible knowledge which can be revised and changed. Full article
(This article belongs to the Special Issue Lvov—Warsaw School)
1155 KiB  
Review
Approach of Complexity in Nature: Entropic Nonuniqueness
by Constantino Tsallis
Axioms 2016, 5(3), 20; https://doi.org/10.3390/axioms5030020 - 12 Aug 2016
Cited by 18 | Viewed by 4663
Abstract
Boltzmann introduced in the 1870s a logarithmic measure for the connection between the thermodynamical entropy and the probabilities of the microscopic configurations of the system. His celebrated entropic functional for classical systems was then extended by Gibbs to the entire phase space of [...] Read more.
Boltzmann introduced in the 1870s a logarithmic measure for the connection between the thermodynamical entropy and the probabilities of the microscopic configurations of the system. His celebrated entropic functional for classical systems was then extended by Gibbs to the entire phase space of a many-body system and by von Neumann in order to cover quantum systems, as well. Finally, it was used by Shannon within the theory of information. The simplest expression of this functional corresponds to a discrete set of W microscopic possibilities and is given by S B G = k i = 1 W p i ln p i (k is a positive universal constant; BG stands for Boltzmann–Gibbs). This relation enables the construction of BGstatistical mechanics, which, together with the Maxwell equations and classical, quantum and relativistic mechanics, constitutes one of the pillars of contemporary physics. The BG theory has provided uncountable important applications in physics, chemistry, computational sciences, economics, biology, networks and others. As argued in the textbooks, its application in physical systems is legitimate whenever the hypothesis of ergodicity is satisfied, i.e., when ensemble and time averages coincide. However, what can we do when ergodicity and similar simple hypotheses are violated, which indeed happens in very many natural, artificial and social complex systems. The possibility of generalizing BG statistical mechanics through a family of non-additive entropies was advanced in 1988, namely S q = k 1 i = 1 W p i q q 1 , which recovers the additive S B G entropy in the q→ 1 limit. The index q is to be determined from mechanical first principles, corresponding to complexity universality classes. Along three decades, this idea intensively evolved world-wide (see the Bibliography in http://tsallis.cat.cbpf.br/biblio.htm) and led to a plethora of predictions, verifications and applications in physical systems and elsewhere. As expected, whenever a paradigm shift is explored, some controversy naturally emerged, as well, in the community. The present status of the general picture is here described, starting from its dynamical and thermodynamical foundations and ending with its most recent physical applications. Full article
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219 KiB  
Article
A Logical Analysis of Existential Dependence and Some Other Ontological Concepts—A Comment to Some Ideas of Eugenia Ginsberg-Blaustein
by Marek Magdziak
Axioms 2016, 5(3), 19; https://doi.org/10.3390/axioms5030019 - 15 Jul 2016
Cited by 2 | Viewed by 3750
Abstract
This paper deals with several problems concerning notion of existential dependence and ontological notions of existence, necessity and fusion. Following some ideas of Eugenia Ginsberg-Blaustein, the notions are treated in reference to objects, in relation to the concepts of state of affairs and [...] Read more.
This paper deals with several problems concerning notion of existential dependence and ontological notions of existence, necessity and fusion. Following some ideas of Eugenia Ginsberg-Blaustein, the notions are treated in reference to objects, in relation to the concepts of state of affairs and subject of state of affairs. It provides an axiomatic characterization of these concepts within the framework of a multi-modal propositional logic and then presents a semantic analysis of these concepts. The semantics are a slight modification to the standard relational semantics for normal modal propositional logic. Full article
(This article belongs to the Special Issue Lvov—Warsaw School)
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