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Axioms, Volume 6, Issue 3 (September 2017)

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Research

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Open AccessArticle An Independent Set of Axioms of MV-Algebras and Solutions of the Set-Theoretical Yang–Baxter Equation
Axioms 2017, 6(3), 17; doi:10.3390/axioms6030017
Received: 3 May 2017 / Revised: 18 June 2017 / Accepted: 20 June 2017 / Published: 22 June 2017
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Abstract
The aim of this paper is to give a new equivalent set of axioms for MV-algebras, and to show that the axioms are independent. In addition to this, we handle Yang–Baxter equation problem. In conclusion, we construct a new set-theoretical solution for
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The aim of this paper is to give a new equivalent set of axioms for MV-algebras, and to show that the axioms are independent. In addition to this, we handle Yang–Baxter equation problem. In conclusion, we construct a new set-theoretical solution for the Yang–Baxter equation by using MV-algebras. Full article
(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang–Baxter Equations 2017)
Open AccessArticle Fractional Integration and Differentiation of the Generalized Mathieu Series
Axioms 2017, 6(3), 18; doi:10.3390/axioms6030018
Received: 27 April 2017 / Revised: 15 June 2017 / Accepted: 22 June 2017 / Published: 27 June 2017
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Abstract
We aim to present some formulas for the Saigo hypergeometric fractional integral and differential operators involving the generalized Mathieu series Sμ(r), which are expressed in terms of the Hadamard product of the generalized Mathieu series Sμ(
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We aim to present some formulas for the Saigo hypergeometric fractional integral and differential operators involving the generalized Mathieu series S μ ( r ) , which are expressed in terms of the Hadamard product of the generalized Mathieu series S μ ( r ) and the Fox–Wright function p Ψ q ( z ) . Corresponding assertions for the classical Riemann–Liouville and Erdélyi–Kober fractional integral and differential operators are deduced. Further, it is emphasized that the results presented here, which are for a seemingly complicated series, can reveal their involved properties via the series of the two known functions. Full article
Open AccessArticle Assigning Numerical Scores to Linguistic Expressions
Axioms 2017, 6(3), 19; doi:10.3390/axioms6030019
Received: 7 June 2017 / Revised: 29 June 2017 / Accepted: 30 June 2017 / Published: 6 July 2017
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Abstract
In this paper, we study different methods of scoring linguistic expressions defined on a finite set, in the search for a linear order that ranks all those possible expressions. Among them, particular attention is paid to the canonical extension, and its representability through
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In this paper, we study different methods of scoring linguistic expressions defined on a finite set, in the search for a linear order that ranks all those possible expressions. Among them, particular attention is paid to the canonical extension, and its representability through distances in a graph plus some suitable penalization of imprecision. The relationship between this setting and the classical problems of numerical representability of orderings, as well as extension of orderings from a set to a superset is also explored. Finally, aggregation procedures of qualitative rankings and scorings are also analyzed. Full article
(This article belongs to the Special Issue New Trends in Fuzzy Set Theory and Related Items)
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Open AccessArticle Quincunx Fundamental Refinable Functions in Arbitrary Dimensions
Axioms 2017, 6(3), 20; doi:10.3390/axioms6030020
Received: 16 June 2017 / Revised: 3 July 2017 / Accepted: 4 July 2017 / Published: 6 July 2017
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Abstract
In this paper, we generalize the family of Deslauriers–Dubuc’s interpolatory masks from dimension one to arbitrary dimensions with respect to the quincunx dilation matrices, thereby providing a family of quincunx fundamental refinable functions in arbitrary dimensions. We show that a family of unique
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In this paper, we generalize the family of Deslauriers–Dubuc’s interpolatory masks from dimension one to arbitrary dimensions with respect to the quincunx dilation matrices, thereby providing a family of quincunx fundamental refinable functions in arbitrary dimensions. We show that a family of unique quincunx interpolatory masks exists and such a family of masks is of real value and has the full-axis symmetry property. In dimension d = 2 , we give the explicit form of such unique quincunx interpolatory masks, which implies the nonnegativity property of such a family of masks. Full article
(This article belongs to the Special Issue Wavelet and Frame Constructions, with Applications)
Open AccessArticle New Order on Type 2 Fuzzy Numbers
Axioms 2017, 6(3), 22; doi:10.3390/axioms6030022
Received: 5 June 2017 / Revised: 14 July 2017 / Accepted: 24 July 2017 / Published: 28 July 2017
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Abstract
Since Lotfi A. Zadeh introduced the concept of fuzzy sets in 1965, many authors have devoted their efforts to the study of these new sets, both from a theoretical and applied point of view. Fuzzy sets were later extended in order to get
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Since Lotfi A. Zadeh introduced the concept of fuzzy sets in 1965, many authors have devoted their efforts to the study of these new sets, both from a theoretical and applied point of view. Fuzzy sets were later extended in order to get more adequate and flexible models of inference processes, where uncertainty, imprecision or vagueness is present. Type 2 fuzzy sets comprise one of such extensions. In this paper, we introduce and study an extension of the fuzzy numbers (type 1), the type 2 generalized fuzzy numbers and type 2 fuzzy numbers. Moreover, we also define a partial order on these sets, which extends into these sets the usual order on real numbers, which undoubtedly becomes a new option to be taken into account in the existing total preorders for ranking interval type 2 fuzzy numbers, which are a subset of type 2 generalized fuzzy numbers. Full article
(This article belongs to the Special Issue New Trends in Fuzzy Set Theory and Related Items)
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Open AccessArticle Categorically Closed Topological Groups
Axioms 2017, 6(3), 23; doi:10.3390/axioms6030023
Received: 30 June 2017 / Revised: 26 July 2017 / Accepted: 27 July 2017 / Published: 30 July 2017
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Abstract
Let C be a category whose objects are semigroups with topology and morphisms are closed semigroup relations, in particular, continuous homomorphisms. An object X of the category C is called C -closed if for each morphism Φ X × Y in the category C the image Φ ( X ) = { y Y : x X ( x , y ) Φ } is closed in Y. In the paper we survey existing and new results on topological groups, which are C -closed for various categories C of topologized semigroups. Full article
(This article belongs to the collection Topological Groups)
Open AccessArticle Topological Signals of Singularities in Ricci Flow
Axioms 2017, 6(3), 24; doi:10.3390/axioms6030024
Received: 13 June 2017 / Revised: 25 July 2017 / Accepted: 27 July 2017 / Published: 1 August 2017
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Abstract
We implement methods from computational homology to obtain a topological signal of singularity formation in a selection of geometries evolved numerically by Ricci flow. Our approach, based on persistent homology, produces precise, quantitative measures describing the behavior of an entire collection of data
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We implement methods from computational homology to obtain a topological signal of singularity formation in a selection of geometries evolved numerically by Ricci flow. Our approach, based on persistent homology, produces precise, quantitative measures describing the behavior of an entire collection of data across a discrete sample of times. We analyze the topological signals of geometric criticality obtained numerically from the application of persistent homology to models manifesting singularities under Ricci flow. The results we obtain for these numerical models suggest that the topological signals distinguish global singularity formation (collapse to a round point) from local singularity formation (neckpinch). Finally, we discuss the interpretation and implication of these results and future applications. Full article
(This article belongs to the Special Issue Discrete Geometry and its Applications)
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Open AccessArticle Orness For Idempotent Aggregation Functions
Axioms 2017, 6(3), 25; doi:10.3390/axioms6030025
Received: 23 August 2017 / Revised: 15 September 2017 / Accepted: 17 September 2017 / Published: 20 September 2017
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Abstract
Aggregation functions are mathematical operators that merge given data in order to obtain a global value that preserves the information given by the data as much as possible. In most practical applications, this value is expected to be between the infimum and the
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Aggregation functions are mathematical operators that merge given data in order to obtain a global value that preserves the information given by the data as much as possible. In most practical applications, this value is expected to be between the infimum and the supremum of the given data, which is guaranteed only when the aggregation functions are idempotent. Ordered weighted averaging (OWA) operators are particular cases of this kind of function, with the particularity that the obtained global value depends on neither the source nor the expert that provides each datum, but only on the set of values. They have been classified by means of the orness—a measurement of the proximity of an OWA operator to the OR-operator. In this paper, the concept of orness is extended to the framework of idempotent aggregation functions defined both on the real unit interval and on a complete lattice with a local finiteness condition. Full article
(This article belongs to the Special Issue New Trends in Fuzzy Set Theory and Related Items)
Open AccessArticle From Normal Surfaces to Normal Curves to Geodesics on Surfaces
Axioms 2017, 6(3), 26; doi:10.3390/axioms6030026
Received: 23 March 2017 / Revised: 7 August 2017 / Accepted: 25 August 2017 / Published: 20 September 2017
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Abstract
This paper gives a study of a two dimensional version of the theory of normal surfaces; namely, a study o normal curves and their relations with respect to geodesic curves. This study results with a nice discrete approximation of geodesics embedded in a
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This paper gives a study of a two dimensional version of the theory of normal surfaces; namely, a study o normal curves and their relations with respect to geodesic curves. This study results with a nice discrete approximation of geodesics embedded in a triangulated orientable Riemannian surface. Experimental results of the two dimensional case are given as well. Full article
(This article belongs to the Special Issue Discrete Geometry and its Applications)
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Review

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Open AccessReview The SIC Question: History and State of Play
Axioms 2017, 6(3), 21; doi:10.3390/axioms6030021
Received: 30 June 2017 / Revised: 14 July 2017 / Accepted: 15 July 2017 / Published: 18 July 2017
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Abstract
Recent years have seen significant advances in the study of symmetric informationally complete (SIC) quantum measurements, also known as maximal sets of complex equiangular lines. Previously, the published record contained solutions up to dimension 67, and was with high confidence complete up through
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Recent years have seen significant advances in the study of symmetric informationally complete (SIC) quantum measurements, also known as maximal sets of complex equiangular lines. Previously, the published record contained solutions up to dimension 67, and was with high confidence complete up through dimension 50. Computer calculations have now furnished solutions in all dimensions up to 151, and in several cases beyond that, as large as dimension 844. These new solutions exhibit an additional type of symmetry beyond the basic definition of a SIC, and so verify a conjecture of Zauner in many new cases. The solutions in dimensions 68 through 121 were obtained by Andrew Scott, and his catalogue of distinct solutions is, with high confidence, complete up to dimension 90. Additional results in dimensions 122 through 151 were calculated by the authors using Scott’s code. We recap the history of the problem, outline how the numerical searches were done, and pose some conjectures on how the search technique could be improved. In order to facilitate communication across disciplinary boundaries, we also present a comprehensive bibliography of SIC research. Full article
(This article belongs to the Special Issue Wavelet and Frame Constructions, with Applications)
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