Mathematics, Volume 7, Issue 10 (October 2019) – 132 articles

In this paper, approximation properties in Felbin fuzzy normed spaces are considered. These approximation properties are new concepts in Felbin fuzzy normed spaces. Definitions and examples of such properties are given and we make a comparative study among approximation properties in Bag and Samanta fuzzy normed spaces and Felbin fuzzy normed spaces. We develop the representation of finite rank bounded operators in our context. By using this representation, characterizations of approximation properties are established in Felbin fuzzy normed spaces. Full article

In this paper, we introduce and study a new three-parameter lifetime distribution constructed from the so-called type I half-logistic-G family and the inverted Kumaraswamy distribution, naturally called the type I half-logistic inverted Kumaraswamy distribution. The main feature of this new distribution is to add a new tuning parameter to the inverted Kumaraswamy (according to the type I half-logistic structure), with the aim to increase the flexibility of the related inverted Kumaraswamy model and thus offering more precise diagnostics in data analyses. The new distribution is discussed in detail, exhibiting various mathematical and statistical properties, with related graphics and numerical results. An exhaustive simulation was conducted to investigate the estimation of the model parameters via several well-established methods, including the method of maximum likelihood estimation, methods of least squares and weighted least squares estimation, and method of Cramer-von Mises minimum distance estimation, showing their numerical efficiency. Finally, by considering the method of maximum likelihood estimation, we apply the new model to fit two practical data sets. In this regards, it is proved to be better than recent models, also derived to the inverted Kumaraswamy distribution. Full article

The Boerdijk–Coxeter helix is a helical structure of tetrahedra which possesses no non-trivial translational or rotational symmetries. In this document, we develop a procedure by which this structure is modified to obtain both translational and rotational (upon projection) symmetries along/about its central axis. We show by construction that a helix can be obtained whose shortest period is any whole number of tetrahedra greater than one except six, while a period of six necessarily entails a shorter period. We give explicit examples of two particular forms related to the pentagonal and icosahedral aggregates of tetrahedra as well as Buckminster Fuller’s “jitterbug transformation”. Full article

As a generalization of several fuzzy tools, picture fuzzy sets (PFSs) hold a special ability to perfectly portray inherent uncertain and vague decision preferences. The intention of this paper is to present a Pearson’s picture fuzzy correlation-based model for multi-attribute decision-making (MADM) analysis. To this end, we develop a new correlation coefficient for picture fuzzy sets, based on which a Pearson’s picture fuzzy closeness index is introduced to simultaneously calculate the relative proximity to the positive ideal point and the relative distance from the negative ideal point. On the basis of the presented concepts, a Pearson’s correlation-based model is further presented to address picture fuzzy MADM problems. Finally, an illustrative example is provided to examine the usefulness and feasibility of the proposed methodology. Full article

Governments often support their preferences for decentralised (centralised) bureaucracies on the grounds of efficiency considerations (production side). Here, we consider the demand side, i.e., whether the government perception of citizens’ demand for differentiated goods/services might increase efficiency by simply reshuffling bureaucratic production activities. We represent the budgetary process—between an incumbent governing party and n-competing bureaus producing differentiated goods/services—as a simultaneous Nash-compliance game with complete information. On these grounds, we analyse—in terms of public production, players’ rents and payoffs—the effects of increasing competition (as for the number of bureaus) in the political–bureaucratic market. Moreover, we evaluate, ceteris paribus, the effects of bureaucratic reshuffling from the point of view of society, assumed to prefer those policies that approximate social efficiency by minimising bureaucratic and political rents. Full article

Let $G=(V,E)$ be a graph and $f:V⟶\{0,1,2\}$ be a function. Given a vertex u with $f\left(u\right)=0$ , if all neighbors of u have zero weights, then u is called undefended with respect to f. Furthermore, if every vertex u with $f\left(u\right)=0$ has a neighbor v with $f\left(v\right)>0$ and the function $f^{\prime }:V⟶\{0,1,2\}$ with $f^{\prime }\left(u\right)=1$ , $f^{\prime }\left(v\right)=f\left(v\right)-1$ , $f^{\prime }\left(w\right)=f\left(w\right)$ if $w\in V\setminus \{u,v\}$ has no undefended vertex, then f is called a weak Roman dominating function. Also, the function f is a perfect Roman dominating function if every vertex u with $f\left(u\right)=0$ is adjacent to exactly one vertex v for which $f\left(v\right)=2$ . Let the weight of f be $w\left(f\right)={\sum }_{v\in V}f\left(v\right)$ . The weak (resp., perfect) Roman domination number, denoted by ${\gamma }_r\left(G\right)$ (resp., ${\gamma }_R^p\left(G\right)$ ), is the minimum weight of the weak (resp., perfect) Roman dominating function in G. In this paper, we characterize those trees where the perfect Roman domination number strongly equals the weak Roman domination number, in the sense that each weak Roman dominating function of minimum weight is, at the same time, perfect Roman dominating. Full article

The aim of this article is to propose the 2-tuple picture fuzzy linguistic aggregation operators and a decision-making model to deal with uncertainties in the form of 2-tuple picture fuzzy linguistic sets; 2-tuple picture fuzzy linguistic operators have more flexibility than general fuzzy set. We proposed a number of aggregation operators, namely, 2-TPFLWA, 2-TPFLOWA, 2-TPFLHA, 2-TPFLWG, 2-TPFLOWG, and 2-TPFLHG operators. The distinguished feature of the developed operators are studied. At that point, we used these operators to design a model to deal with multiple attribute decision-making issues under the 2-tuple picture fuzzy linguistic information. Then, a practical application of robot selection by manufacturing unit is given to prove the introduced technique and to show its practicability and effectiveness. Besides this, a systematic comparison analysis with other existent approaches is conducted to reveal the advantage of our developed method. Results indicate that the proposed method is suitable and effective for decision-making problems. Full article

In this article, we define an extended version of the Pochhammer symbol and then introduce the corresponding extension of the τ-Gauss hypergeometric function. The basic properties of the extended τ-Gauss hypergeometric function, including integral and derivative formulas involving the Mellin transform and the operators of fractional calculus, are derived. We also consider some new and known results as consequences of our proposed extension of the τ-Gauss hypergeometric function. Full article

Suppose that G is a simple undirected connected graph. Denote by $D\left(G\right)$ the distance matrix of G and by $Tr\left(G\right)$ the diagonal matrix of the vertex transmissions in G, and let $\alpha \in [0,1]$ . The generalized distance matrix $D_{\alpha }\left(G\right)$ is defined as $D_{\alpha }\left(G\right)=\alpha Tr\left(G\right)+(1-\alpha )D\left(G\right)$ , where $0\le \alpha \le 1$ . If ${\partial }_1\ge {\partial }_2\ge \dots \ge {\partial }_n$ are the eigenvalues of $D_{\alpha }\left(G\right)$ ; we define the generalized distance Estrada index of the graph G as $D_{\alpha }E\left(G\right)={\sum }_{i=1}^n{e}^{\left({\partial }_i-\frac{2\alpha W\left(G\right)}{n}\right)},$ where $W\left(G\right)$ denotes for the Wiener index of G. It is clear from the definition that $D_{0}E\left(G\right)=DEE\left(G\right)$ and $2D_{\frac{1}{2}}E\left(G\right)=D^{Q}EE\left(G\right)$ , where $DEE\left(G\right)$ denotes the distance Estrada index of G and $D^{Q}EE\left(G\right)$ denotes the distance signless Laplacian Estrada index of G. This shows that the concept of generalized distance Estrada index of a graph G merges the theories of distance Estrada index and the distance signless Laplacian Estrada index. In this paper, we obtain some lower and upper bounds for the generalized distance Estrada index, in terms of various graph parameters associated with the structure of the graph G, and characterize the extremal graphs attaining these bounds. We also highlight relationship between the generalized distance Estrada index and the other graph-spectrum-based invariants, including generalized distance energy. Moreover, we have worked out some expressions for $D_{\alpha }E\left(G\right)$ of some special classes of graphs. Full article

In this manuscript, we consider some hybrid contractions that merge linear and nonlinear contractions in the abstract spaces induced by the Branciari distance and the Branciari b-distance. More precisely, we introduce the notion of a $(p,c)$ -weight type $\psi$ -contraction in the setting of Branciari distance spaces and the concept of a $(p,c)$ -weight type contraction in Branciari b-distance spaces. We investigate the existence of a fixed point of such operators in Branciari type distance spaces and illustrate some examples to show that the presented results are genuine in the literature. Full article

Contraflow technique has gained a considerable focus in evacuation planning research over the past several years. In this work, we design efficient algorithms to solve the maximum, lex-maximum, earliest arrival, and quickest dynamic flow problems having constant attributes and their generalizations with partial contraflow reconfiguration in the context of evacuation planning. The partial static contraflow problems, that are foundations to the dynamic flows, are also studied. Moreover, the contraflow model with inflow-dependent transit time on arcs is introduced. A strongly polynomial time algorithm to compute approximate solution of the quickest partial contraflow problem on two terminal networks is presented, which is substantiated by numerical computations considering Kathmandu road network as an evacuation network. Our results show that the quickest time to evacuate a flow of value 100,000 units is reduced by more than 42% using the partial contraflow technique, and the difference is more with the increase in the flow value. Moreover, the technique keeps the record of the portions of the road network not used by the evacuees. Full article

This article concerns the expressive power of depth in neural nets with ReLU activations and a bounded width. We are particularly interested in the following questions: What is the minimal width $w_{\min }\left(d\right)$ so that ReLU nets of width $w_{\min }\left(d\right)$ (and arbitrary depth) can approximate any continuous function on the unit cube ${[0,1]}^d$ arbitrarily well? For ReLU nets near this minimal width, what can one say about the depth necessary to approximate a given function? We obtain an essentially complete answer to these questions for convex functions. Our approach is based on the observation that, due to the convexity of the ReLU activation, ReLU nets are particularly well suited to represent convex functions. In particular, we prove that ReLU nets with width $d+1$ can approximate any continuous convex function of d variables arbitrarily well. These results then give quantitative depth estimates for the rate of approximation of any continuous scalar function on the d-dimensional cube ${[0,1]}^d$ by ReLU nets with width $d+3$ . Full article

We investigate the main statistical parameters of the integral over time of the fractional Brownian motion and of a kind of pseudo-fractional Gaussian process, obtained as a classical Gauss–Markov process from Doob representation by replacing Brownian motion with fractional Brownian motion. Possible applications in the context of neuronal models are highlighted. A fractional Ornstein–Uhlenbeck process is considered and relations with the integral of the pseudo-fractional Gaussian process are provided. Full article

In this paper, noise removing of the well test data is considered. We use the Legendre expansion to approximate well test data and a truncated strategy has been employed to reduce noise. The parameter of the truncation will be chosen by a discrepancy principle and a corresponding convergence result has been obtained. The theoretical analysis shows that a well numerical approximation can be obtained by the new method. Moreover, we can directly obtain the stable numerical derivatives of the pressure data in this method. Finally, we give some numerical tests to show the effectiveness of the method. Full article

Understanding ecosystem response to drier climates calls for modeling the dynamics of dryland plant populations, which are crucial determinants of ecosystem function, as they constitute the basal level of whole food webs. Two modeling approaches are widely used in population dynamics, individual (agent)-based models and continuum partial-differential-equation (PDE) models. The latter are advantageous in lending themselves to powerful methodologies of mathematical analysis, but the question of whether they are suitable to describe small discrete plant populations, as is often found in dryland ecosystems, has remained largely unaddressed. In this paper, we first draw attention to two aspects of plants that distinguish them from most other organisms—high phenotypic plasticity and dispersal of stress-tolerant seeds—and argue in favor of PDE modeling, where the state variables that describe population sizes are not discrete number densities, but rather continuous biomass densities. We then discuss a few examples that demonstrate the utility of PDE models in providing deep insights into landscape-scale behaviors, such as the onset of pattern forming instabilities, multiplicity of stable ecosystem states, regular and irregular, and the possible roles of front instabilities in reversing desertification. We briefly mention a few additional examples, and conclude by outlining the nature of the information we should and should not expect to gain from PDE model studies. Full article

The purpose of the present work is to determine a bound for the functional $H_2\left(2\right)=a_2{a}_4-a_3^2$ for functions belonging to the class ${\mathcal{C}}_{\Sigma }$ of bi-close-to-convex functions. The main result presented here provides much improved estimation compared with the previous result by means of different proof methods than those used by others. Full article

In this study, we propose a new flexible two-parameter continuous distribution with support on the unit interval. It can be identified as a special member of the so-called type I half-logistic-G family of distributions, defined with the Topp–Leone distribution as baseline. Among its features, the corresponding probability density function can be left skewed, right-skewed, approximately symmetric, J-shaped, as well as reverse J-shaped, making it suitable for modeling a wide variety of data sets. It thus provides an alternative to the so-called beta and Kumaraswamy distributions. The mathematical properties of the new distribution are determined, deriving the asymptotes, shapes, quantile function, skewness, kurtosis, some power series expansions, ordinary moments, incomplete moments, moment-generating function, stress strength parameter, and order statistics. Then, a statistical treatment of the related model is proposed. The estimation of the unknown parameters is performed by a simulation study exploring seven methods, all described in detail. Two practical data sets are analyzed, showing the usefulness of the new proposed model. Full article

This paper presents a new procedure for designing a fractional order unknown input observer (FOUIO) for nonlinear systems represented by a fractional-order Takagi–Sugeno (FOTS) model with unmeasurable premise variables (UPV). Most of the current research on fractional order systems considers models using measurable premise variables (MPV) and therefore cannot be utilized when premise variables are not measurable. The concept of the proposed is to model the FOTS with UPV into an uncertain FOTS model by presenting the estimated state in the model. First, the fractional-order extension of Lyapunov theory is used to investigate the convergence conditions of the FOUIO, and the linear matrix inequalities (LMIs) provide the stability condition. Secondly, performances of the proposed FOUIO are improved by the reduction of bounded external disturbances. Finally, an example is provided to clarify the proposed method. The obtained results show that a good convergence of the outputs and the state estimation errors were observed using the new proposed FOUIO. Full article

Based on the natural vector addition and scalar multiplication, the set of all bounded and closed intervals in $\mathbb{R}$ cannot form a vector space. This is mainly because the zero element does not exist. In this paper, we endow a norm to the interval space in which the axioms are almost the same as the axioms of conventional norm by involving the concept of null set. Under this consideration, we shall propose two different concepts of open balls. Based on the open balls, we shall also propose the different types of open sets, which can generate many different topologies. Full article

The prevalence of business-to-business (B2B) has made the relationship among firms more closer than ever. Whether in simple arm-length transactions or business cooperation, many firms, in order to reduce costs and achieve efficiency, have shifted their day-to-day operations from the tradition of relying on manpower to the use of information technology in handling tasks such as inventory, procurement, production planning, distribution, etc. As a result, the need of a business process information system is imminent for firms to coordinate with partners in the supply chain and to be sustainable in the competitive market. This study thus proposes a hybrid multi-criteria decision-making approach for evaluating business process information systems. First, the factors that should be taken into account in selecting an appropriate system are explored. The Decision-Making Trial and Evaluation Laboratory (DEMATEL) is adopted next to understand the interrelationships among the criteria. Based on the results from the DEMATEL, the Fuzzy Analytic Network Process (FANP) is applied to calculate the importance of the factors. Fuzzy Techniques for Order of Preference by Similarity to Ideal Solution (FTOPSIS) is used to rank the business process information systems. The interrelationship among the factors should be considered in the decision-making; thus, the FANP can be a recommended methodology. However, the FANP questionnaire is usually very lengthy and cumbersome. The use of DEMATEL in advance can shorten the questionnaire substantially. FTOPSIS is used to rank the alternatives so that the pairwise comparisons of the alternatives required in the FANP can be avoided. Fuzzy set theory is incorporated in the study so that the uncertainty and ambiguity present in decision-making can be considered. The proposed approach can provide references for decision makers for making relevant decisions and can be revised and adopted in similar problems. Full article

To be able to compete in the domestic plastic industry, small and medium-sized enterprises producing plastic need to proactively find the supply of raw materials, avoiding shortages like in the previous years. Purchasing is extremely important and will create a competitive advantage with competitors in the market, so finding suppliers will determine the success in the later stages of the production chain. With the development of the current information system, selection and evaluation have become important in order to achieve effective decision-making through optimal options. In this study, the authors provide a new approach for decision-makers in evaluating and selecting suppliers, which is formulated based on the supply chain operation reference (SCOR) model, fuzzy analytic network process (FANP), and VIseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR). The contribution of this research is to propose a multicriteria decision-making model (MCDM) for raw material supplier selection in the plastic industry. This research also provided a useful guideline for supplier selection in other industry. Full article

We show that if a differentiable map f of a compact smooth Riemannian manifold M is $C^1$ robustly positive continuum-wise expansive, then f is expanding. Moreover, $C^1$ -generically, if a differentiable map f of a compact smooth Riemannian manifold M is positively continuum-wise expansive, then f is expanding. Full article

Owing to its high accuracy, the radial basis function (RBF) is gaining popularity in function interpolation and for solving partial differential equations (PDEs). The implementation of RBF methods is independent of the locations of the points and the dimensionality of the problems. However, the stability and accuracy of RBF methods depend significantly on the shape parameter, which is mainly affected by the basis function and the node distribution. If the shape parameter has a small value, then the RBF becomes accurate but unstable. Several approaches have been proposed in the literature to overcome the instability issue. Changing or expanding the radial basis function is one of the most commonly used approaches because it addresses the stability problem directly. However, the main issue with most of those approaches is that they require the optimization of additional parameters, such as the truncation order of the expansion, to obtain the desired accuracy. In this work, the Hermite polynomial is used to expand the RBF with respect to the shape parameter to determine a stable basis, even when the shape parameter approaches zero, and the approach does not require the optimization of any parameters. Furthermore, the Hermite polynomial properties enable the RBF to be evaluated stably even when the shape parameter equals zero. The proposed approach was benchmarked to test its reliability, and the obtained results indicate that the accuracy is independent of or weakly dependent on the shape parameter. However, the convergence depends on the order of the truncation of the expansion. Additionally, it is observed that the new approach improves accuracy and yields the accurate interpolation, derivative approximation, and PDE solution. Full article

The first part of the article contains integral expressions for products of two Bessel functions of the first kind having either different integer orders or different arguments. A similar question for a product of modified Bessel functions of the first kind is solved next, when the input functions are of different integer orders and have different arguments. Full article

The purpose of this paper is to introduce the new notion of a specific point in the space of the bounded real-valued functions on a given non-empty set and present a result based on the existence and uniqueness of such points. As a consequence of our results, we discuss the existence of a unique common solution to coupled systems of functional equations arising in dynamic programming. Full article

In this paper, we propose a separable reversible data hiding method in encrypted image (RDHEI) based on two-dimensional permutation and exploiting modification direction (EMD). The content owner uses two-dimensional permutation to encrypt original image through encryption key, which provides confidentiality for the original image. Then the data hider divides the encrypted image into a series of non-overlapping blocks and constructs histogram of adjacent encrypted pixel errors. Secret bits are embedded into a series of peak points of the histogram through EMD. Direct decryption, data extraction and image recovery can be performed separately by the receiver according to the availability of encryption key and data-hiding key. Different from some state-of-the-art RDHEI methods, visual quality of the directly decrypted image can be further improved by the receiver holding the encryption key. Experimental results demonstrate that the proposed method outperforms some state-of-the-art methods in embedding capacity and visual quality. Full article

Managing invasive species in rivers can be assisted by appropriate adjustment of flow rates. Using a partial differential equation (PDE) model representing an invasive population in a river, we investigate controlling the water discharge rate as a management strategy. Our goal is to see how controlling the water discharge rate will affect the invasive population, and more specifically how water discharges may force the invasive population downstream. We complete the analysis of a flow control problem, which seeks to minimize the invasive population upstream while minimizing the cost of this management. Using an optimality system, consisting of our population PDE, an adjoint PDE, and corresponding optimal control characterization, we illustrate some numerical simulations in which parameters are varied to determine how far upstream the invasive population reaches. We also change the river’s cross-sectional area to investigate its impact on the optimal control. Full article

A convex combined expectations regularized gap function with uncertain variable is presented to deal with uncertain nonlinear variational inequality problems (UNVIP). The UNVIP is transformed into a minimization problem through an uncertain weighted expected residual function. Moreover, the convergence of the global optimal solutions of the uncertain weighted expected residual minimization model is given through the integration by parts method under the compact space of the uncertain event. The limiting behaviors of the transformed model are analyzed. Furthermore, a compact approximation method is proposed in the unbounded uncertain event space. Through analysis of the convergence of UWERM model and reasonable hypothesis, the compact approximation method is verified under the circumstance of Holder continuity. Full article