Authors:Sundas Shahzadi, Muhammad Akram First page: 20 Abstract: In this research article, we present a novel framework for handling intuitionistic fuzzy soft information by combining the theory of intuitionistic fuzzy soft sets with graphs. We introduce the notion of certain types of intuitionistic fuzzy soft graphs including neighbourly edge regular intuitionistic fuzzy soft graphs and strongyl edge irregular intuitionistic fuzzy soft graphs. We illustrate these novel concepts by several examples, and investigate some of their related properties. We present an application of intuitionistic fuzzy soft graph in a decision-making problem and also present our methods as an algorithm that is used in this application. Citation: Axioms PubDate: 2018-03-23 DOI: 10.3390/axioms7020020 Issue No:Vol. 7, No. 2 (2018)

Authors:Surapati Pramanik, Partha Pratim Dey, Florentin Smarandache, Jun Ye First page: 21 Abstract: The bipolar neutrosophic set is an important extension of the bipolar fuzzy set. The bipolar neutrosophic set is a hybridization of the bipolar fuzzy set and neutrosophic set. Every element of a bipolar neutrosophic set consists of three independent positive membership functions and three independent negative membership functions. In this paper, we develop cross entropy measures of bipolar neutrosophic sets and prove their basic properties. We also define cross entropy measures of interval bipolar neutrosophic sets and prove their basic properties. Thereafter, we develop two novel multi-attribute decision-making strategies based on the proposed cross entropy measures. In the decision-making framework, we calculate the weighted cross entropy measures between each alternative and the ideal alternative to rank the alternatives and choose the best one. We solve two illustrative examples of multi-attribute decision-making problems and compare the obtained result with the results of other existing strategies to show the applicability and effectiveness of the developed strategies. At the end, the main conclusion and future scope of research are summarized. Citation: Axioms PubDate: 2018-03-24 DOI: 10.3390/axioms7020021 Issue No:Vol. 7, No. 2 (2018)

Authors:Yilmaz Simsek First page: 22 Abstract: In this paper, by applying umbral calculus methods to generating functions for the combinatorial numbers and the Apostol type polynomials and numbers of order k, we derive some identities and relations including the combinatorial numbers, the Apostol-Bernoulli polynomials and numbers of order k and the Apostol-Euler polynomials and numbers of order k. Moreover, by using p-adic integral technique, we also derive some combinatorial sums including the Bernoulli numbers, the Euler numbers, the Apostol-Euler numbers and the numbers y 1 n , k ; λ . Finally, we make some remarks and observations regarding these identities and relations. Citation: Axioms PubDate: 2018-04-01 DOI: 10.3390/axioms7020022 Issue No:Vol. 7, No. 2 (2018)

Authors:Young Jun, Seon Kim, Florentin Smarandache First page: 23 Abstract: For i , j , k , l , m , n ∈ { 1 , 2 , 3 , 4 } , the notion of ( T ( i , j ) , I ( k , l ) , F ( m , n ) ) -interval neutrosophic subalgebra in B C K / B C I -algebra is introduced, and their properties and relations are investigated. The notion of interval neutrosophic length of an interval neutrosophic set is also introduced, and related properties are investigated. Citation: Axioms PubDate: 2018-04-09 DOI: 10.3390/axioms7020023 Issue No:Vol. 7, No. 2 (2018)

Authors:Ömer Kişi, Hafize Gümüş, Ekrem Savas First page: 24 Abstract: In this paper, using an infinite matrix of complex numbers, a modulus function and a lacunary sequence, we generalize the concept of I -statistical convergence, which is a recently introduced summability method. The names of our new methods are A I -lacunary statistical convergence and strongly A I -lacunary convergence with respect to a sequence of modulus functions. These spaces are denoted by S θ A I , F and N θ A I , F , respectively. We give some inclusion relations between S A I , F , S θ A I , F and N θ A I , F . We also investigate Cesáro summability for A I and we obtain some basic results between A I -Cesáro summability, strongly A I -Cesáro summability and the spaces mentioned above. Citation: Axioms PubDate: 2018-04-13 DOI: 10.3390/axioms7020024 Issue No:Vol. 7, No. 2 (2018)

Authors:Kevin Burrage, Pamela Burrage, Ian Turner, Fanhai Zeng First page: 25 Abstract: In this paper, we study the class of mixed-index time fractional differential equations in which different components of the problem have different time fractional derivatives on the left-hand side. We prove a theorem on the solution of the linear system of equations, which collapses to the well-known Mittag–Leffler solution in the case that the indices are the same and also generalises the solution of the so-called linear sequential class of time fractional problems. We also investigate the asymptotic stability properties of this class of problems using Laplace transforms and show how Laplace transforms can be used to write solutions as linear combinations of generalised Mittag–Leffler functions in some cases. Finally, we illustrate our results with some numerical simulations. Citation: Axioms PubDate: 2018-04-17 DOI: 10.3390/axioms7020025 Issue No:Vol. 7, No. 2 (2018)

Authors:Seok-Zun Song, Hashem Bordbar, Young Bae Jun First page: 26 Abstract: Relations between I-quasi-valuation maps and ideals in B C K / B C I -algebras are investigated. Using the notion of an I-quasi-valuation map of a B C K / B C I -algebra, the quasi-metric space is induced, and several properties are investigated. Relations between the I-quasi-valuation map and the I-valuation map are considered, and conditions for an I-quasi-valuation map to be an I-valuation map are provided. A congruence relation is introduced by using the I-valuation map, and then the quotient structures are established and related properties are investigated. Isomorphic quotient B C K / B C I -algebras are discussed. Citation: Axioms PubDate: 2018-04-23 DOI: 10.3390/axioms7020026 Issue No:Vol. 7, No. 2 (2018)

Authors:Hanaa M. Zayed, Mohamed Kamal Aouf, Adela O. Mostafa First page: 27 Abstract: Using of the principle of subordination, we investigate some subordination and convolution properties for classes of multivalent functions under certain assumptions on the parameters involved, which are defined by a generalized fractional differintegral operator under certain assumptions on the parameters involved. Citation: Axioms PubDate: 2018-04-24 DOI: 10.3390/axioms7020027 Issue No:Vol. 7, No. 2 (2018)

Authors:Antti Rasila, Tommi Sottinen First page: 28 Abstract: This paper continues our earlier investigation, where a walk-on-spheres (WOS) algorithm for Monte Carlo simulation of the solutions of the Yukawa and the Helmholtz partial differential equations (PDEs) was developed by using the Duffin correspondence. In this paper, we investigate the foundations behind the algorithm for the case of the Yukawa PDE. We study the panharmonic measure, which is a generalization of the harmonic measure for the Yukawa PDE. We show that there are natural stochastic definitions for the panharmonic measure in terms of the Brownian motion and that the harmonic and the panharmonic measures are all mutually equivalent. Furthermore, we calculate their Radon–Nikodym derivatives explicitly for some balls, which is a key result behind the WOS algorithm. Citation: Axioms PubDate: 2018-05-01 DOI: 10.3390/axioms7020028 Issue No:Vol. 7, No. 2 (2018)

Authors:María-Jesús Campión, Cristina Gómez-Polo, Esteban Induráin, Armajac Raventós-Pujol First page: 29 Abstract: Different abstract versions of entropy, encountered in science, are interpreted in the light of numerical representations of several ordered structures, as total-preorders, interval-orders and semiorders. Intransitivities, other aspects of entropy as competitive systems, additivity, etc., are also viewed in terms of representability of algebraic structures endowed with some compatible ordering. A particular attention is paid to the problem of the construction of an entropy function or their mathematical equivalents. Multidisciplinary comparisons to other similar frameworks are also discussed, pointing out the mathematical foundations. Citation: Axioms PubDate: 2018-05-08 DOI: 10.3390/axioms7020029 Issue No:Vol. 7, No. 2 (2018)

Authors:Ravi Agarwal, Snezhana Hristova, Donal O’Regan First page: 30 Abstract: One of the main properties of solutions of nonlinear Caputo fractional neural networks is stability and often the direct Lyapunov method is used to study stability properties (usually these Lyapunov functions do not depend on the time variable). In connection with the Lyapunov fractional method we present a brief overview of the most popular fractional order derivatives of Lyapunov functions among Caputo fractional delay differential equations. These derivatives are applied to various types of neural networks with variable coefficients and time-varying delays. We show that quadratic Lyapunov functions and their Caputo fractional derivatives are not applicable in some cases when one studies stability properties. Some sufficient conditions for stability of equilibrium of nonlinear Caputo fractional neural networks with time dependent transmission delays, time varying self-regulating parameters of all units and time varying functions of the connection between two neurons in the network are obtained. The cases of time varying Lipschitz coefficients as well as nonLipschitz activation functions are studied. We illustrate our theory on particular nonlinear Caputo fractional neural networks. Citation: Axioms PubDate: 2018-05-09 DOI: 10.3390/axioms7020030 Issue No:Vol. 7, No. 2 (2018)

Authors:Ann-Eva Christensen, Jon Johnsen First page: 31 Abstract: This article concerns the basic understanding of parabolic final value problems, and a large class of such problems is proved to be well posed. The clarification is obtained via explicit Hilbert spaces that characterise the possible data, giving existence, uniqueness and stability of the corresponding solutions. The data space is given as the graph normed domain of an unbounded operator occurring naturally in the theory. It induces a new compatibility condition, which relies on the fact, shown here, that analytic semigroups always are invertible in the class of closed operators. The general set-up is evolution equations for Lax–Milgram operators in spaces of vector distributions. As a main example, the final value problem of the heat equation on a smooth open set is treated, and non-zero Dirichlet data are shown to require a non-trivial extension of the compatibility condition by addition of an improper Bochner integral. Citation: Axioms PubDate: 2018-05-09 DOI: 10.3390/axioms7020031 Issue No:Vol. 7, No. 2 (2018)

Authors: Axioms Editorial Office First page: 2 Abstract: Peer review is an essential part in the publication process, ensuring that Axioms maintains high quality standards for its published papers.[...] Citation: Axioms PubDate: 2018-01-11 DOI: 10.3390/axioms7010002 Issue No:Vol. 7, No. 1 (2018)

Authors:Young Jun, Florentin Smarandache, Seok-Zun Song, Madad Khan First page: 3 Abstract: The notion of a neutrosophic positive implicative N -ideal in B C K -algebras is introduced, and several properties are investigated. Relations between a neutrosophic N -ideal and a neutrosophic positive implicative N -ideal are discussed. Characterizations of a neutrosophic positive implicative N -ideal are considered. Conditions for a neutrosophic N -ideal to be a neutrosophic positive implicative N -ideal are provided. An extension property of a neutrosophic positive implicative N -ideal based on the negative indeterminacy membership function is discussed. Citation: Axioms PubDate: 2018-01-15 DOI: 10.3390/axioms7010003 Issue No:Vol. 7, No. 1 (2018)

Authors:Teresa González-Arteaga, Rocio de Andrés Calle, Luis Martínez First page: 4 Abstract: The need for organizations to evaluate their environmental practices has been recently increasing. This fact has led to the development of many approaches to appraise such practices. In this paper, a novel decision model to evaluate company’s environmental practices is proposed to improve traditional evaluation process in different facets. Firstly, different reviewers’ collectives related to the company’s activity are taken into account in the process to increase company internal efficiency and external legitimacy. Secondly, following the standard ISO 14031, two general categories of environmental performance indicators, management and operational, are considered. Thirdly, since the assumption of independence among environmental indicators is rarely verified in environmental context, an aggregation operator to bear in mind the relationship among such indicators in the evaluation results is proposed. Finally, this new model integrates quantitative and qualitative information with different scales using a multi-granular linguistic model that allows to adapt diverse evaluation scales according to appraisers’ knowledge. Citation: Axioms PubDate: 2018-01-16 DOI: 10.3390/axioms7010004 Issue No:Vol. 7, No. 1 (2018)

Authors:Sidra Sayed, Nabeela Ishfaq, Muhammad Akram, Florentin Smarandache First page: 5 Abstract: A rough neutrosophic set model is a hybrid model which deals with vagueness by using the lower and upper approximation spaces. In this research paper, we apply the concept of rough neutrosophic sets to graphs. We introduce rough neutrosophic digraphs and describe methods of their construction. Moreover, we present the concept of self complementary rough neutrosophic digraphs. Finally, we consider an application of rough neutrosophic digraphs in decision-making. Citation: Axioms PubDate: 2018-01-18 DOI: 10.3390/axioms7010005 Issue No:Vol. 7, No. 1 (2018)

Authors:Tahsin Oner, Tugce Katican First page: 6 Abstract: In this work, we introduce Wajsberg algebras which are equivalent structures to MV-algebras in their implicational version, and then we define new notions and give new solutions to the set-theoretical Yang-Baxter equation by using Wajsberg algebras. Citation: Axioms PubDate: 2018-01-20 DOI: 10.3390/axioms7010006 Issue No:Vol. 7, No. 1 (2018)

Authors:Young Jun, Seok-Zun Song, Seon Kim First page: 7 Abstract: As a new extension of a cubic set, the notion of a cubic interval-valued intuitionistic fuzzy set is introduced, and its application in B C K / B C I -algebra is considered. The notions of α -internal, β -internal, α -external and β -external cubic IVIF set are introduced, and the P-union, P-intersection, R-union and R-intersection of α -internal and α -external cubic IVIF sets are discussed. The concepts of cubic IVIF subalgebra and ideal in B C K / B C I -algebra are introduced, and related properties are investigated. Relations between cubic IVIF subalgebra and cubic IVIF ideal are considered, and characterizations of cubic IVIF subalgebra and cubic IVIF ideal are discussed. Citation: Axioms PubDate: 2018-01-23 DOI: 10.3390/axioms7010007 Issue No:Vol. 7, No. 1 (2018)

Authors:Christian Servin, Gerardo Muela, Vladik Kreinovich First page: 8 Abstract: In modern mathematics, many concepts and ideas are described in terms of category theory. From this viewpoint, it is desirable to analyze what can be determined if, instead of the basic category of sets, we consider a similar category of fuzzy sets. In this paper, we describe a natural fuzzy analog of the category of sets and functions, and we show that, in this category, fuzzy relations (a natural fuzzy analogue of functions) can be determined in category terms—of course, modulo 1-1 mapping of the corresponding universe of discourse and 1-1 re-scaling of fuzzy degrees. Citation: Axioms PubDate: 2018-01-23 DOI: 10.3390/axioms7010008 Issue No:Vol. 7, No. 1 (2018)

Authors:Carlton-James Osakwe First page: 9 Abstract: In this paper, we examine the real options approach to capital budgeting decision making in the presence of managerial adverse incentives. We show that real options have the potential to be value enhancing or value destroying depending on the managerial incentives that may result from having objectives different from firm value maximization. We further examine the possibility of using a generic residual income based rule of managerial compensation to induce the proper investment incentives and we seek to determine the cost-of-capital that must be employed in such a rule. Using numerical examples it is demonstrated that in general, a range of incentive compatible costs-of-capital exists across all managerial investment horizons but not across all managerial hurdle rates. Citation: Axioms PubDate: 2018-02-07 DOI: 10.3390/axioms7010009 Issue No:Vol. 7, No. 1 (2018)

Authors:Jun Jiang, Yuqiang Feng, Shougui Li First page: 10 Abstract: In this paper, the solvability of nonlinear fractional partial differential equations (FPDEs) with mixed partial derivatives is considered. The invariant subspace method is generalized and is then used to derive exact solutions to the nonlinear FPDEs. Some examples are solved to illustrate the effectiveness and applicability of the method. Citation: Axioms PubDate: 2018-02-11 DOI: 10.3390/axioms7010010 Issue No:Vol. 7, No. 1 (2018)

Authors:Gerardo Febres First page: 11 Abstract: When considering perceptions, the observation scale and resolution are closely related properties. There is consensus on considering resolution as the density of the elementary pieces of information in a specified information space. On the other hand, with the concept of scale, several conceptions compete for a consistent meaning. Scale is typically regarded as a way to indicate the degree of detail in which an observation is performed. Surprisingly, there is not a unified definition of scale as a description’s property. This paper offers a precise definition of scale and a method to quantify it as a property associated with the interpretation of a description. To complete the parameters needed to describe the perception of a description, the concepts of scope and resolution are also revealed with an exact meaning. A model describing a recursive process of interpretation, based on evolving steps of scale, scope and resolution, is introduced. The model relies on the conception of observation scale and its association to the selection of symbols. Five experiments illustrate the application of these concepts, showing that resolution, scale and scope integrate the set of properties to define any point of view from which an observation is performed and interpreted. The results obtained for descriptions expressed in one and two dimensions, are the basis for a comparison of the perceivable symbolic information from different interpretations of the same descriptions. In conclusion, this study provides a framework for building models of our interpretation process and suggests ways to understand some mechanisms in the formation of information from initially meaningless symbols. Citation: Axioms PubDate: 2018-02-13 DOI: 10.3390/axioms7010011 Issue No:Vol. 7, No. 1 (2018)

Authors:Kalyan Mondal, Surapati Pramanik, Bibhas C. Giri, Florentin Smarandache First page: 12 Abstract: A neutrosophic number (a + bI) is a significant mathematical tool to deal with indeterminate and incomplete information which exists generally in real-world problems, where a and bI denote the determinate component and indeterminate component, respectively. We define score functions and accuracy functions for ranking neutrosophic numbers. We then define a cosine function to determine the unknown weight of the criteria. We define the neutrosophic number harmonic mean operators and prove their basic properties. Then, we develop two novel multi-criteria group decision-making (MCGDM) strategies using the proposed aggregation operators. We solve a numerical example to demonstrate the feasibility, applicability, and effectiveness of the two proposed strategies. Sensitivity analysis with the variation of “I” on neutrosophic numbers is performed to demonstrate how the preference ranking order of alternatives is sensitive to the change of “I”. The efficiency of the developed strategies is ascertained by comparing the results obtained from the proposed strategies with the results obtained from the existing strategies in the literature. Citation: Axioms PubDate: 2018-02-23 DOI: 10.3390/axioms7010012 Issue No:Vol. 7, No. 1 (2018)

Authors:Jun Ye, Wenhua Cui, Zhikang Lu First page: 13 Abstract: In practical situations, we often have to handle programming problems involving indeterminate information. Building on the concepts of indeterminacy I and neutrosophic number (NN) (z = p + qI for p, q ∈ ℝ), this paper introduces some basic operations of NNs and concepts of NN nonlinear functions and inequalities. These functions and/or inequalities contain indeterminacy I and naturally lead to a formulation of NN nonlinear programming (NN-NP). These techniques include NN nonlinear optimization models for unconstrained and constrained problems and their general solution methods. Additionally, numerical examples are provided to show the effectiveness of the proposed NN-NP methods. It is obvious that the NN-NP problems usually yield NN optimal solutions, but not always. The possible optimal ranges of the decision variables and NN objective function are indicated when the indeterminacy I is considered for possible interval ranges in real situations. Citation: Axioms PubDate: 2018-02-24 DOI: 10.3390/axioms7010013 Issue No:Vol. 7, No. 1 (2018)

Authors:Muhammad Akram, Hafsa M. Malik, Sundas Shahzadi, Florentin Smarandache First page: 14 Abstract: Neutrosophic sets (NSs) handle uncertain information while fuzzy sets (FSs) and intuitionistic fuzzy sets (IFs) fail to handle indeterminate information. Soft set theory, neutrosophic set theory, and rough set theory are different mathematical models for handling uncertainties and they are mutually related. The neutrosophic soft rough set (NSRS) model is a hybrid model by combining neutrosophic soft sets with rough sets. We apply neutrosophic soft rough sets to graphs. In this research paper, we introduce the idea of neutrosophic soft rough graphs (NSRGs) and describe different methods of their construction. We consider the application of NSRG in decision-making problems. In particular, we develop efficient algorithms to solve decision-making problems. Citation: Axioms PubDate: 2018-02-26 DOI: 10.3390/axioms7010014 Issue No:Vol. 7, No. 1 (2018)

Authors:Solomon Marcus, Florin Nichita First page: 15 Abstract: Bringing toghether mathematical and philosophical ideas related to transcendental numbers, this paper begins with a survey on transcendence and transcendental numbers, it then presents several properties of the transcendental numbers e and π , and then it gives the proof of a new inequality for transcendental numbers. Also, in relationship with these topics, we study solutions to the Yang-Baxter equation from hyperbolic functions and from logical implication. Citation: Axioms PubDate: 2018-03-01 DOI: 10.3390/axioms7010015 Issue No:Vol. 7, No. 1 (2018)

Authors:Krzysztof Piasecki First page: 16 Abstract: Ordered fuzzy numbers are defined by Kosiński. In this way, he was going to supplement a fuzzy number by orientation. A significant drawback of Kosiński’s theory is that there exist such ordered fuzzy numbers which, in fact, are not fuzzy numbers. For this reason, a fully formalized correct definition of ordered fuzzy numbers is proposed here. Also, the arithmetic proposed by Kosiński has a significant disadvantage. The space of ordered fuzzy numbers is not closed under Kosiński’s addition. On the other hand, many mathematical applications require the considered space be closed under used arithmetic operations. Therefore, the Kosinski’s theory is modified in this way that the space of ordered fuzzy numbers is closed under revised arithmetic operations. In addition, it is shown that the multiple revised sum of finite sequence of ordered fuzzy numbers depends on its summands ordering. Citation: Axioms PubDate: 2018-03-02 DOI: 10.3390/axioms7010016 Issue No:Vol. 7, No. 1 (2018)

Authors:Juan Candeal First page: 17 Abstract: A general characterization result of projective aggregation functions is shown, the proof of which makes use of the celebrated Arrow’s theorem, thus providing a link between aggregation functions theory and social choice theory. The result can be viewed as a generalization of a theorem obtained by Kim (1990) for real-valued aggregation functions defined on the n-dimensional Euclidean space in the context of measurement theory. In addition, two applications of the core theorem of the article are shown. The first is a simple extension of the main result to the context of multi-valued aggregation functions. The second offers a new characterization of projective bijection aggregators, thus connecting aggregation operators theory with social choice. Citation: Axioms PubDate: 2018-03-20 DOI: 10.3390/axioms7010017 Issue No:Vol. 7, No. 1 (2018)

Authors:Manseob Lee First page: 18 Abstract: We show that if a C 1 generic diffeomorphism of a closed smooth two-dimensional manifold has the average shadowing property or the asymptotic average shadowing property, then it is Anosov. Moreover, if a C 1 generic vector field of a closed smooth three-dimensional manifold has the average shadowing property or the asymptotic average shadowing property, then it satisfies singular Axiom A without cycles. Citation: Axioms PubDate: 2018-03-20 DOI: 10.3390/axioms7010018 Issue No:Vol. 7, No. 1 (2018)

Authors:Muhammad Akram, Sundas Shahzadi, Florentin Smarandache First page: 19 Abstract: Soft sets (SSs), neutrosophic sets (NSs), and rough sets (RSs) are different mathematical models for handling uncertainties, but they are mutually related. In this research paper, we introduce the notions of soft rough neutrosophic sets (SRNSs) and neutrosophic soft rough sets (NSRSs) as hybrid models for soft computing. We describe a mathematical approach to handle decision-making problems in view of NSRSs. We also present an efficient algorithm of our proposed hybrid model to solve decision-making problems. Citation: Axioms PubDate: 2018-03-20 DOI: 10.3390/axioms7010019 Issue No:Vol. 7, No. 1 (2018)

Authors:Nor Jaini, Sergey Utyuzhnikov First page: 1 Abstract: The aim of this paper is to present a trade-off ranking method in a fuzzy multi-criteria decision-making environment. The triangular fuzzy numbers are used to represent the imprecise numerical quantities in the criteria values of each alternative and the weight of each criterion. A fuzzy trade-off ranking method is developed to rank alternatives in the fuzzy multi-criteria decision-making problem with conflicting criteria. The trade-off ranking method tackles this type of multi-criteria problems by giving the least compromise solution as the best option. The proposed method for the fuzzy decision-making problems is compared against two other fuzzy decision-making approaches, fuzzy Technique for Order Preference by Similarity to the Ideal Solution (TOPSIS) and fuzzy VlseKriterijuska Optimizacija I Komoromisno Resenje (VIKOR), used for ranking alternatives. Citation: Axioms PubDate: 2017-12-26 DOI: 10.3390/axioms7010001 Issue No:Vol. 7, No. 1 (2017)

Authors:George Willis First page: 27 Abstract: The scale of an endomorphism of a totally disconnected, locally compact group G is defined and an example is presented which shows that the scale function is not always continuous with respect to the Braconnier topology on the automorphism group of G. Methods for computing the scale, which is a positive integer, are surveyed and illustrated by applying them in diverse cases, including when G is compact; an automorphism group of a tree; Neretin’s group of almost automorphisms of a tree; and a p-adic Lie group. The information required to compute the scale is reviewed from the perspective of the, as yet incomplete, general theory of totally disconnected, locally compact groups. Citation: Axioms PubDate: 2017-10-20 DOI: 10.3390/axioms6040027 Issue No:Vol. 6, No. 4 (2017)

Authors:Ol’ga Sipacheva First page: 28 Abstract: Various notions of large sets in groups, including the classical notions of thick, syndetic, and piecewise syndetic sets and the new notion of vast sets in groups, are studied with emphasis on the interplay between such sets in Boolean groups. Natural topologies closely related to vast sets are considered; as a byproduct, interesting relations between vast sets and ultrafilters are revealed. Citation: Axioms PubDate: 2017-10-24 DOI: 10.3390/axioms6040028 Issue No:Vol. 6, No. 4 (2017)

Authors:Paolo Bevilacqua, Gianni Bosi, Magalì Zuanon First page: 29 Abstract: Looking at decisiveness as crucial, we discuss the existence of an order-preserving function for the nontotal crisp preference relation naturally associated to a nontotal fuzzy preference relation. We further present conditions for the existence of an upper semicontinuous order-preserving function for a fuzzy binary relation on a crisp topological space. Citation: Axioms PubDate: 2017-10-28 DOI: 10.3390/axioms6040029 Issue No:Vol. 6, No. 4 (2017)

Authors:Jin Liang, Yunyi Mu First page: 30 Abstract: In this paper, we present new existence theorems of mild solutions to Cauchy problem for some fractional differential equations with delay. Our main tools to obtain our results are the theory of analytic semigroups and compact semigroups, the Kuratowski measure of non-compactness, and fixed point theorems, with the help of some estimations. Examples are also given to illustrate the applicability of our results. Citation: Axioms PubDate: 2017-11-20 DOI: 10.3390/axioms6040030 Issue No:Vol. 6, No. 4 (2017)

Authors:Mehmet Şahin, Rızvan Erol First page: 31 Abstract: This study proposes a mathematical model of dynamic pricing for soccer game tickets. The logic behind the dynamic ticket pricing model is price change based on multipliers which reflect the effects of time and inventory. Functions are formed for the time and inventory multipliers. The optimization algorithm attempts to find optimal values of these multipliers in order to maximize revenue. By multiplying the mean season ticket price (used as the reference price) by the multipliers, dynamic ticket prices are obtained. Demand rates at different prices are needed for the model, and they are provided by a unique fuzzy logic model. The results of this model are compared with real data to test the model’s effectiveness. According to the results of the dynamic pricing model, the total revenue generated is increased by 8.95% and 0.76% compared with the static pricing strategy in the first and second cases, respectively. The results of the fuzzy logic model are also found to be competitive and effective. This is the first time a fuzzy logic model has been designed to forecast the attendance of soccer games. It is also the first time this type of mathematical model of dynamic pricing for soccer game tickets has been designed. Citation: Axioms PubDate: 2017-11-19 DOI: 10.3390/axioms6040031 Issue No:Vol. 6, No. 4 (2017)

Authors:Simon Lentner, Andreas Lochmann First page: 32 Abstract: A ubiquitous observation for finite-dimensional Nichols algebras is that as a graded algebra the Hilbert series factorizes into cyclotomic polynomials. For Nichols algebras of diagonal type (e.g., Borel parts of quantum groups), this is a consequence of the existence of a root system and a Poincare-Birkhoff-Witt (PBW) basis basis, but, for nondiagonal examples (e.g., Fomin–Kirillov algebras), this is an ongoing surprise. In this article, we discuss this phenomenon and observe that it continues to hold for the graded character of the involved group and for automorphisms. First, we discuss thoroughly the diagonal case. Then, we prove factorization for a large class of nondiagonal Nichols algebras obtained by the folding construction. We conclude empirically by listing all remaining examples, which were in size accessible to the computer algebra system GAP and find that again all graded characters factorize. Citation: Axioms PubDate: 2017-12-04 DOI: 10.3390/axioms6040032 Issue No:Vol. 6, No. 4 (2017)

Authors:Vsevolod Gubarev First page: 33 Abstract: Universal enveloping commutative Rota–Baxter algebras of pre- and post-commutative algebras are constructed. The pair of varieties (RBλCom, postCom) is proved to be a Poincaré–Birkhoff–Witt-pair (PBW)-pair and the pair (RBCom, preCom) is proven not to be. Citation: Axioms PubDate: 2017-12-07 DOI: 10.3390/axioms6040033 Issue No:Vol. 6, No. 4 (2017)

Authors:Juan-José Miñana, Oscar Valero First page: 34 Abstract: The notion of indistinguishability operator was introduced by Trillas, E. in 1982, with the aim of fuzzifying the crisp notion of equivalence relation. Such operators allow for measuring the similarity between objects when there is a limitation on the accuracy of the performed measurement or a certain degree of similarity can be only determined between the objects being compared. Since Trillas introduced such kind of operators, many authors have studied their properties and applications. In particular, an intensive research line is focused on the metric behavior of indistinguishability operators. Specifically, the existence of a duality between metrics and indistinguishability operators has been explored. In this direction, a technique to generate metrics from indistinguishability operators, and vice versa, has been developed by several authors in the literature. Nowadays, such a measurement of similarity is provided by the so-called fuzzy metrics when the degree of similarity between objects is measured relative to a parameter. The main purpose of this paper is to extend the notion of indistinguishability operator in such a way that the measurements of similarity are relative to a parameter and, thus, classical indistinguishability operators and fuzzy metrics can be retrieved as a particular case. Moreover, we discuss the relationship between the new operators and metrics. Concretely, we prove the existence of a duality between them and the so-called modular metrics, which provide a dissimilarity measurement between objects relative to a parameter. The new duality relationship allows us, on the one hand, to introduce a technique for generating the new indistinguishability operators from modular metrics and vice versa and, on the other hand, to derive, as a consequence, a technique for generating fuzzy metrics from modular metrics and vice versa. Furthermore, we yield examples that illustrate the new results. Citation: Axioms PubDate: 2017-12-15 DOI: 10.3390/axioms6040034 Issue No:Vol. 6, No. 4 (2017)

Authors:Ümit Budak, Yanhui Guo, Abdulkadir Şengür, Florentin Smarandache First page: 35 Abstract: Hough transform (HT) is a useful tool for both pattern recognition and image processing communities. In the view of pattern recognition, it can extract unique features for description of various shapes, such as lines, circles, ellipses, and etc. In the view of image processing, a dozen of applications can be handled with HT, such as lane detection for autonomous cars, blood cell detection in microscope images, and so on. As HT is a straight forward shape detector in a given image, its shape detection ability is low in noisy images. To alleviate its weakness on noisy images and improve its shape detection performance, in this paper, we proposed neutrosophic Hough transform (NHT). As it was proved earlier, neutrosophy theory based image processing applications were successful in noisy environments. To this end, the Hough space is initially transferred into the NS domain by calculating the NS membership triples (T, I, and F). An indeterminacy filtering is constructed where the neighborhood information is used in order to remove the indeterminacy in the spatial neighborhood of neutrosophic Hough space. The potential peaks are detected based on thresholding on the neutrosophic Hough space, and these peak locations are then used to detect the lines in the image domain. Extensive experiments on noisy and noise-free images are performed in order to show the efficiency of the proposed NHT algorithm. We also compared our proposed NHT with traditional HT and fuzzy HT methods on variety of images. The obtained results showed the efficiency of the proposed NHT on noisy images. Citation: Axioms PubDate: 2017-12-18 DOI: 10.3390/axioms6040035 Issue No:Vol. 6, No. 4 (2017)

Authors:Tahsin Oner, Ibrahim Senturk, Gulsah Oner First page: 17 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 the Yang–Baxter equation by using MV-algebras. Citation: Axioms PubDate: 2017-06-22 DOI: 10.3390/axioms6030017 Issue No:Vol. 6, No. 3 (2017)

Authors:Ram Saxena, Rakesh Parmar First page: 18 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 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. Citation: Axioms PubDate: 2017-06-27 DOI: 10.3390/axioms6030018 Issue No:Vol. 6, No. 3 (2017)

Authors:María Campión, Edurne Falcó, José García-Lapresta, Esteban Induráin First page: 19 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 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. Citation: Axioms PubDate: 2017-07-06 DOI: 10.3390/axioms6030019 Issue No:Vol. 6, No. 3 (2017)

Authors:Xiaosheng Zhuang First page: 20 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 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. Citation: Axioms PubDate: 2017-07-06 DOI: 10.3390/axioms6030020 Issue No:Vol. 6, No. 3 (2017)

Authors:Christopher Fuchs, Michael Hoang, Blake Stacey First page: 21 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 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. Citation: Axioms PubDate: 2017-07-18 DOI: 10.3390/axioms6030021 Issue No:Vol. 6, No. 3 (2017)

Authors:Pablo Hernández, Susana Cubillo, Carmen Torres-Blanc, José Guerrero First page: 22 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 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. Citation: Axioms PubDate: 2017-07-28 DOI: 10.3390/axioms6030022 Issue No:Vol. 6, No. 3 (2017)

Authors:Taras Banakh First page: 23 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. Citation: Axioms PubDate: 2017-07-30 DOI: 10.3390/axioms6030023 Issue No:Vol. 6, No. 3 (2017)

Authors:Paul Alsing, Howard Blair, Matthew Corne, Gordon Jones, Warner Miller, Konstantin Mischaikow, Vidit Nanda First page: 24 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 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. Citation: Axioms PubDate: 2017-08-01 DOI: 10.3390/axioms6030024 Issue No:Vol. 6, No. 3 (2017)

Authors:Leire Legarreta, Inmaculada Lizasoain, Iraide Mardones-Pérez First page: 25 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 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. Citation: Axioms PubDate: 2017-09-20 DOI: 10.3390/axioms6030025 Issue No:Vol. 6, No. 3 (2017)

Authors:Eli Appleboim First page: 26 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 triangulated orientable Riemannian surface. Experimental results of the two dimensional case are given as well. Citation: Axioms PubDate: 2017-09-20 DOI: 10.3390/axioms6030026 Issue No:Vol. 6, No. 3 (2017)

Authors:John Herr, Eric Weber First page: 7 Abstract: Using the Kaczmarz algorithm, we prove that for any singular Borel probability measure μ on [ 0 , 1 ) , every f ∈ L 2 ( μ ) possesses a Fourier series of the form f ( x ) = ∑ n = 0 ∞ c n e 2 π i n x . We show that the coefficients c n can be computed in terms of the quantities f ^ ( n ) = ∫ 0 1 f ( x ) e − 2 π i n x d μ ( x ) . We also demonstrate a Shannon-type sampling theorem for functions that are in a sense μ -bandlimited. Citation: Axioms PubDate: 2017-03-28 DOI: 10.3390/axioms6020007 Issue No:Vol. 6, No. 2 (2017)

Authors:David Galas, Gregory Dewey, James Kunert-Graf, Nikita Sakhanenko First page: 8 Abstract: Inferring and comparing complex, multivariable probability density functions is fundamental to problems in several fields, including probabilistic learning, network theory, and data analysis. Classification and prediction are the two faces of this class of problem. This study takes an approach that simplifies many aspects of these problems by presenting a structured, series expansion of the Kullback-Leibler divergence—a function central to information theory—and devise a distance metric based on this divergence. Using the Möbius inversion duality between multivariable entropies and multivariable interaction information, we express the divergence as an additive series in the number of interacting variables, which provides a restricted and simplified set of distributions to use as approximation and with which to model data. Truncations of this series yield approximations based on the number of interacting variables. The first few terms of the expansion-truncation are illustrated and shown to lead naturally to familiar approximations, including the well-known Kirkwood superposition approximation. Truncation can also induce a simple relation between the multi-information and the interaction information. A measure of distance between distributions, based on Kullback-Leibler divergence, is then described and shown to be a true metric if properly restricted. The expansion is shown to generate a hierarchy of metrics and connects this work to information geometry formalisms. An example of the application of these metrics to a graph comparison problem is given that shows that the formalism can be applied to a wide range of network problems and provides a general approach for systematic approximations in numbers of interactions or connections, as well as a related quantitative metric. Citation: Axioms PubDate: 2017-04-01 DOI: 10.3390/axioms6020008 Issue No:Vol. 6, No. 2 (2017)

Authors:Jianzhong Wang First page: 9 Abstract: For a given pair of s-dimensional real Laurent polynomials ( a → ( z ) , b → ( z ) ) , which has a certain type of symmetry and satisfies the dual condition b → ( z ) T a → ( z ) = 1 , an s × s Laurent polynomial matrix A ( z ) (together with its inverse A - 1 ( z ) ) is called a symmetric Laurent polynomial matrix extension of the dual pair ( a → ( z ) , b → ( z ) ) if A ( z ) has similar symmetry, the inverse A - 1 ( Z ) also is a Laurent polynomial matrix, the first column of A ( z ) is a → ( z ) and the first row of A - 1 ( z ) is ( b → ( z ) ) T . In this paper, we introduce the Euclidean symmetric division and the symmetric elementary matrices in the Laurent polynomial ring and reveal their relation. Based on the Euclidean symmetric division algorithm in the Laurent polynomial ring, we develop a novel and effective algorithm for symmetric Laurent polynomial matrix extension. We also apply the algorithm in the construction of multi-band symmetric perfect reconstruction filter banks. Citation: Axioms PubDate: 2017-04-20 DOI: 10.3390/axioms6020009 Issue No:Vol. 6, No. 2 (2017)

Authors:Evgenii Proutorov, Hiroshi Koibuchi First page: 10 Abstract: We study triangulated surface models with nontrivial surface metrices for membranes. The surface model is defined by a mapping r from a two-dimensional parameter space M to the three-dimensional Euclidean space R 3 . The metric variable g a b , which is always fixed to the Euclidean metric δ a b , can be extended to a more general non-Euclidean metric on M in the continuous model. The problem we focus on in this paper is whether such an extension is well defined or not in the discrete model. We find that a discrete surface model with a nontrivial metric becomes well defined if it is treated in the context of Finsler geometry (FG) modeling, where triangle edge length in M depends on the direction. It is also shown that the discrete FG model is orientation asymmetric on invertible surfaces in general, and for this reason, the FG model has a potential advantage for describing real physical membranes, which are expected to have some asymmetries for orientation-changing transformations. Citation: Axioms PubDate: 2017-04-25 DOI: 10.3390/axioms6020010 Issue No:Vol. 6, No. 2 (2017)

Authors:Dhannya Joseph First page: 11 Abstract: In this paper, I consider multivariate analogues of the extended gamma density, which will provide multivariate extensions to Tsallis statistics and superstatistics. By making use of the pathway parameter β , multivariate generalized gamma density can be obtained from the model considered here. Some of its special cases and limiting cases are also mentioned. Conditional density, best predictor function, regression theory, etc., connected with this model are also introduced. Citation: Axioms PubDate: 2017-04-24 DOI: 10.3390/axioms6020011 Issue No:Vol. 6, No. 2 (2017)

Authors:Zengqiang Chen, Matthias Dehmer, Frank Emmert-Streib, Abbe Mowshowitz, Yongtang Shi First page: 12 Abstract: In this exploratory paper, we discuss quantitative graph-theoretical measures of network aesthetics. Related work in this area has typically focused on geometrical features (e.g., line crossings or edge bendiness) of drawings or visual representations of graphs which purportedly affect an observer’s perception. Here we take a very different approach, abandoning reliance on geometrical properties, and apply information-theoretic measures to abstract graphs and networks directly (rather than to their visual representaions) as a means of capturing classical appreciation of structural symmetry. Examples are used solely to motivate the approach to measurement, and to elucidate our symmetry-based mathematical theory of network aesthetics. Citation: Axioms PubDate: 2017-05-06 DOI: 10.3390/axioms6020012 Issue No:Vol. 6, No. 2 (2017)

Authors:Gianluca Paolini, Saharon Shelah First page: 13 Abstract: We prove that if G is a Polish group and A a group admitting a system of generators whose associated length function satisfies: (i) if 0 < k < ω , then l g ( x ) ≤ l g ( x k ) ; (ii) if l g ( y ) < k < ω and x k = y , then x = e , then there exists a subgroup G * of G of size b (the bounding number) such that G * is not embeddable in A. In particular, we prove that the automorphism group of a countable structure cannot be an uncountable right-angled Artin group. This generalizes analogous results for free and free abelian uncountable groups. Citation: Axioms PubDate: 2017-05-11 DOI: 10.3390/axioms6020013 Issue No:Vol. 6, No. 2 (2017)

Authors:Sonja Jäckle, Karsten Keller First page: 14 Abstract: The Tsallis entropy given for a positive parameter α can be considered as a generalization of the classical Shannon entropy. For the latter, corresponding to α = 1 , there exist many axiomatic characterizations. One of them based on the well-known Khinchin-Shannon axioms has been simplified several times and adapted to Tsallis entropy, where the axiom of (generalized) Shannon additivity is playing a central role. The main aim of this paper is to discuss this axiom in the context of Tsallis entropy. We show that it is sufficient for characterizing Tsallis entropy, with the exceptions of cases α = 1 , 2 discussed separately. Citation: Axioms PubDate: 2017-06-14 DOI: 10.3390/axioms6020014 Issue No:Vol. 6, No. 2 (2017)

Authors:Miao Jin, Su Xia, Hongyi Wu, Xianfeng Gu First page: 15 Abstract: This work proposes a novel connectivity-based localization algorithm, well suitable for large-scale sensor networks with complex shapes and a non-uniform nodal distribution. In contrast to current state-of-the-art connectivity-based localization methods, the proposed algorithm is highly scalable with linear computation and communication costs with respect to the size of the network; and fully distributed where each node only needs the information of its neighbors without cumbersome partitioning and merging process. The algorithm is theoretically guaranteed and numerically stable. Moreover, the algorithm can be readily extended to the localization of networks with a one-hop transmission range distance measurement, and the propagation of the measurement error at one sensor node is limited within a small area of the network around the node. Extensive simulations and comparison with other methods under various representative network settings are carried out, showing the superior performance of the proposed algorithm. Citation: Axioms PubDate: 2017-06-15 DOI: 10.3390/axioms6020015 Issue No:Vol. 6, No. 2 (2017)

Authors:Kai Liu, YangQuan Chen, Xi Zhang First page: 16 Abstract: Strong coupling between values at different times that exhibit properties of long range dependence, non-stationary, spiky signals cannot be processed by the conventional time series analysis. The autoregressive fractional integral moving average (ARFIMA) model, a fractional order signal processing technique, is the generalization of the conventional integer order models—autoregressive integral moving average (ARIMA) and autoregressive moving average (ARMA) model. Therefore, it has much wider applications since it could capture both short-range dependence and long range dependence. For now, several software programs have been developed to deal with ARFIMA processes. However, it is unfortunate to see that using different numerical tools for time series analysis usually gives quite different and sometimes radically different results. Users are often puzzled about which tool is suitable for a specific application. We performed a comprehensive survey and evaluation of available ARFIMA tools in the literature in the hope of benefiting researchers with different academic backgrounds. In this paper, four aspects of ARFIMA programs concerning simulation, fractional order difference filter, estimation and forecast are compared and evaluated, respectively, in various software platforms. Our informative comments can serve as useful selection guidelines. Citation: Axioms PubDate: 2017-06-17 DOI: 10.3390/axioms6020016 Issue No:Vol. 6, No. 2 (2017)

Authors:Dan Kučerovský First page: 1 Abstract: The classical Cuntz semigroup has an important role in the study of C*-algebras, being one of the main invariants used to classify recalcitrant C*-algebras up to isomorphism. We consider C*-algebras that have Hopf algebra structure, and find additional structure in their Cuntz semigroups. We show that in many cases, isomorphisms of Cuntz semigroups that respect this additional structure can be lifted to Hopf algebra (bi)isomorphisms, up to a possible flip of the co-product. This shows that the Cuntz semigroup provides an interesting invariant of C*-algebraic quantum groups. Citation: Axioms PubDate: 2017-01-04 DOI: 10.3390/axioms6010001 Issue No:Vol. 6, No. 1 (2017)

Authors:M. Khokulan, K. Thirulogasanthar, S. Srisatkunarajah First page: 3 Abstract: An introductory theory of frames on finite dimensional left quaternion Hilbert spaces is demonstrated along the lines of their complex counterpart. Citation: Axioms PubDate: 2017-02-21 DOI: 10.3390/axioms6010003 Issue No:Vol. 6, No. 1 (2017)

Authors:Dana Černá, Václav Finĕk First page: 4 Abstract: We propose a construction of a Hermite cubic spline-wavelet basis on the interval and hypercube. The basis is adapted to homogeneous Dirichlet boundary conditions. The wavelets are orthogonal to piecewise polynomials of degree at most seven on a uniform grid. Therefore, the wavelets have eight vanishing moments, and the matrices arising from discretization of differential equations with coefﬁcients that are piecewise polynomials of degree at most four on uniform grids are sparse. Numerical examples demonstrate the efﬁciency of an adaptive wavelet method with the constructed wavelet basis for solving the one-dimensional elliptic equation and the two-dimensional Black–Scholes equation with a quadratic volatility. Citation: Axioms PubDate: 2017-02-22 DOI: 10.3390/axioms6010004 Issue No:Vol. 6, No. 1 (2017)

Authors:Dagmar Markechová First page: 5 Abstract: The main aim of this contribution is to define the notions of Kullback-Leibler divergence and conditional mutual information in fuzzy probability spaces and to derive the basic properties of the suggested measures. In particular, chain rules for mutual information of fuzzy partitions and for Kullback-Leibler divergence with respect to fuzzy P-measures are established. In addition, a convexity of Kullback-Leibler divergence and mutual information with respect to fuzzy P-measures is studied. Citation: Axioms PubDate: 2017-03-03 DOI: 10.3390/axioms6010005 Issue No:Vol. 6, No. 1 (2017)

Authors:Peter Casazza, Dorsa Ghoreishi, Shani Jose, Janet Tremain First page: 6 Abstract: We make a detailed study of norm retrieval. We give several classification theorems for norm retrieval and give a large number of examples to go with the theory. One consequence is a new result about Parseval frames: If a Parseval frame is divided into two subsets with spans W 1 , W 2 and W 1 ∩ W 2 = { 0 } , then W 1 ⊥ W 2 . Citation: Axioms PubDate: 2017-03-04 DOI: 10.3390/axioms6010006 Issue No:Vol. 6, No. 1 (2017)