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)