Mathematics — Open Access Journal

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