Axioms

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From Normal Surfaces to Normal Curves to Geodesics on Surfaces

by Eli Appleboim

Axioms 2017, 6(3), 26; https://doi.org/10.3390/axioms6030026 - 20 Sep 2017

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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 [...] Read more.

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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261 KiB

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Orness For Idempotent Aggregation Functions

by Leire Legarreta, Inmaculada Lizasoain and Iraide Mardones-Pérez

Axioms 2017, 6(3), 25; https://doi.org/10.3390/axioms6030025 - 20 Sep 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 [...] Read more.

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)

3431 KiB

Open AccessArticle

Topological Signals of Singularities in Ricci Flow

by Paul M. Alsing, Howard A. Blair, Matthew Corne, Gordon Jones, Warner A. Miller, Konstantin Mischaikow and Vidit Nanda

Axioms 2017, 6(3), 24; https://doi.org/10.3390/axioms6030024 - 1 Aug 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 [...] Read more.

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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433 KiB

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Categorically Closed Topological Groups

by Taras Banakh

Axioms 2017, 6(3), 23; https://doi.org/10.3390/axioms6030023 - 30 Jul 2017

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Abstract

Let \({\overset{\rightarrow}{\mathcal{C}} }\) 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 \({\overset{\rightarrow}{\mathcal{C}} }\) is called \({\overset{\rightarrow}{\mathcal{C}} }\)-closed if for each morphism \({\Phi\subset X\times Y}\) in [...] Read more.

Let \({\overset{\rightarrow}{\mathcal{C}} }\) 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 \({\overset{\rightarrow}{\mathcal{C}} }\) is called \({\overset{\rightarrow}{\mathcal{C}} }\)-closed if for each morphism \({\Phi\subset X\times Y}\) in the category \({\overset{\rightarrow}{\mathcal{C}} }\) the image \({\Phi(X)=\{y\in Y:\exists x\in X\;(x,y)\in\Phi\}}\) is closed in Y. In the paper we survey existing and new results on topological groups, which are \({\overset{\rightarrow}{\mathcal{C}} }\)-closed for various categories \({\overset{\rightarrow}{\mathcal{C}} }\) of topologized semigroups. Full article

(This article belongs to the Collection Topological Groups)

1403 KiB

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New Order on Type 2 Fuzzy Numbers

by Pablo Hernández, Susana Cubillo, Carmen Torres-Blanc and José A. Guerrero

Axioms 2017, 6(3), 22; https://doi.org/10.3390/axioms6030022 - 28 Jul 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 [...] Read more.

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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344 KiB

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The SIC Question: History and State of Play

by Christopher A. Fuchs, Michael C. Hoang and Blake C. Stacey

Axioms 2017, 6(3), 21; https://doi.org/10.3390/axioms6030021 - 18 Jul 2017

Cited by 145 | Viewed by 10414

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 [...] Read more.

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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805 KiB

Open AccessArticle

Quincunx Fundamental Refinable Functions in Arbitrary Dimensions

by Xiaosheng Zhuang

Axioms 2017, 6(3), 20; https://doi.org/10.3390/axioms6030020 - 6 Jul 2017

Cited by 2 | Viewed by 3339

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 [...] Read more.

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)

284 KiB

Open AccessArticle

Assigning Numerical Scores to Linguistic Expressions

by María Jesús Campión, Edurne Falcó, José Luis García-Lapresta and Esteban Induráin

Axioms 2017, 6(3), 19; https://doi.org/10.3390/axioms6030019 - 6 Jul 2017

Cited by 2 | Viewed by 4471

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 [...] Read more.

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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243 KiB

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Fractional Integration and Differentiation of the Generalized Mathieu Series

by Ram K. Saxena and Rakesh K. Parmar

Axioms 2017, 6(3), 18; https://doi.org/10.3390/axioms6030018 - 27 Jun 2017

Cited by 6 | Viewed by 4250

Abstract

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 [...] Read more.

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

(This article belongs to the Special Issue Special Functions: Fractional Calculus and the Pathway for Entropy Dedicated to Professor Dr. A.M. Mathai on the occasion of his 80th Birthday)

707 KiB

Open AccessArticle

An Independent Set of Axioms of MV-Algebras and Solutions of the Set-Theoretical Yang–Baxter Equation

by Tahsin Oner, Ibrahim Senturk and Gulsah Oner

Axioms 2017, 6(3), 17; https://doi.org/10.3390/axioms6030017 - 22 Jun 2017

Cited by 16 | Viewed by 5193

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 [...] Read more.

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)

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Axioms, Volume 6, Issue 3 (September 2017) – 10 articles

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