Mathematics — Open Access Journal

In this paper, we report on recent findings in the numerical solution of Hamiltonian Partial Differential Equations (PDEs) by using energy-conserving line integral methods in the Hamiltonian Boundary Value Methods (HBVMs) class. In particular, we consider the semilinear wave equation, the nonlinear Schrödinger equation, and the Korteweg–de Vries equation, to illustrate the main features of this novel approach. Full article

Multivariate copulas have been widely used to handle risk in the financial market. This paper aimed to adopt two novel multivariate copulas, Vine copulas and Factor copulas, to measure and compare the financial risks of the emerging economy, developed economy, and global economy. In this paper, we used data from three groups (BRICS, which stands for emerging markets, specifically, those of Brazil, Russia, India, China, and South Africa; G7, which refers to developed countries; and G20, which represents the global market), separated into three periods (pre-crisis, crisis, and post-crisis) and weighed Value at Risk (VaR) and Expected Shortfall (ES) (based on their market capitalization) to compare among three copulas, C-Vine, D-Vine, and Factor copulas. Also, real financial data demonstrated that Factor copulas have stronger stability and perform better than the other two copulas in high-dimensional data. Moreover, we showed that BRICS has the highest risk and G20 has the lowest risk of the three groups. Full article

With the advent of the artificial intelligence (AI) era, the beauty camera is widely used, and makeup transfer has attracted increasing attention. In this paper, we propose an adaptive makeup transfer based on the bat algorithm to solve the problem that only a single makeup effect can be transferred. According to the characteristics of makeup style, the algorithm optimizes the weight value to get the appropriate makeup lightness by using the adaptive method. The improved algorithm can not only help to get the optimal weight values in the process of transferring the same makeup style to different targets, but also to transfer different makeup styles to the same target. Moreover, this algorithm can choose the most suitable makeup style and also the most appropriate lightness for a certain person. Experimental results show that the algorithm proposed in the paper has a better effect than the existing algorithm of makeup transfer, and the algorithm can provide users with a suitable makeup style and appropriate lightness. Full article

The analysis presented in this paper regards the security of a present value given as an ordered fuzzy number. The present value was estimated in an imprecise manner and supplemented by the forecast of its coming changes. A discount factor of such security is an ordered fuzzy number of the orientation identical to the oriented present value that determines it. All classical methods of portfolio analysis are based on the definition of the return rate. In the case of securities with a fuzzy present value, a discount factor is a better tool for portfolio analysis than the return rate, which implies the chosen methods of management of securities should be revised and transformed to equivalent methods based on a discount factor. This would enable the use of those methods in the case of a financial instrument of the oriented fuzzy present value. This paper presents example results of the realization of such a postulate. The main aim of the paper is to generalize Sharpe’s ratio to a case of investment recommendations management formulated for a security characterized by an oriented discount factor. A five-degree rating scale was used. The whole deliberation is illustrated by broad numerical examples. Full article

A graph is said to be a regular graph if all its vertices have the same degree, otherwise, it is irregular. Irregularity indices are usually used for quantitative characterization of the topological structure of non-regular graphs. In numerous applications and problems in material engineering and chemistry, it is useful to be aware that how irregular a molecular structure is? Furthermore, evaluations of the irregularity of underline molecular graphs could be valuable for QSAR/QSPR studies, and for the expressive determines of chemical and physical properties, such as enthalpy of vaporization, toxicity, resistance, Entropy, melting and boiling points. In this paper, we think over the following four irregularity measures: the irregularity index by Albertson, $\sigma$ irregularity index, the total irregularity index and the variance of vertex degrees. By way of graph structural estimation and derivations, we determine these irregularity measures of the molecular graphs of different classes of dendrimers. Full article

This paper deals with the dynamics of a delayed cooperative system without quasimonotonicity. Using the contracting rectangles, we obtain a sufficient condition on the stability of the unique positive steady state of the functional differential system. When the spatial domain is whole $\mathbb{R},$ the existence and nonexistence of traveling wave solutions are investigated, during which the asymptotic behavior is investigated by the contracting rectangles. Full article

In this paper, some new properties of Abel Grassmann‘s Neutrosophic Extended Triplet Loop (AG-NET-Loop) were further studied. The following important results were proved: (1) an AG-NET-Loop is weakly commutative if, and only if, it is a commutative neutrosophic extended triplet (NETG); (2) every AG-NET-Loop is the disjoint union of its maximal subgroups. At the same time, the new notion of Abel Grassmann’s (l, l)-Loop (AG-(l, l)-Loop), which is the Abel-Grassmann’s groupoid with the local left identity and local left inverse, were introduced. The strong AG-(l, l)-Loops were systematically analyzed, and the following decomposition theorem was proved: every strong AG-(l, l)-Loop is the disjoint union of its maximal sub-AG-groups. Full article

The super connectivity ${\kappa }^{\prime }\left(G\right)$ of a graph G is the minimum cardinality of vertices, if any, whose deletion results in a disconnected graph that contains no isolated vertex. G is said to be r-super connected if ${\kappa }^{\prime }\left(G\right)\ge r$. In this note, we establish some asymptotic almost sure results on r-super connectedness for classical Erdős–Rényi random graphs as the number of nodes tends to infinity. The known results for r-connectedness are extended to r-super connectedness by pairing off vertices and estimating the probability of disconnecting the graph that one gets by identifying the two vertices of each pair. Full article

A class of generalized $(\psi ,\alpha ,\beta )$—weak contraction is introduced and some fixed-point theorems in a framework of partially ordered metric spaces are proved. The main result of this paper is applied to a first-order ordinary differential equation to find its solution. Full article

The q-homotopy analysis transform method (q-HATM) is employed to find the solution for the fractional Kolmogorov–Petrovskii–Piskunov (FKPP) equation in the present frame work. To ensure the applicability and efficiency of the proposed algorithm, we consider three distinct initial conditions with two of them having Jacobi elliptic functions. The numerical simulations have been conducted to verify that the proposed scheme is reliable and accurate. Moreover, the uniqueness and convergence analysis for the projected problem is also presented. The obtained results elucidate that the proposed technique is easy to implement and very effective to analyze the complex problems arising in science and technology. Full article

In this article, Optimal Homotopy Asymptotic Method (OHAM) is used to approximate results of time-fractional order Fokker-Planck equations. In this work, 3rd order results obtained through OHAM are compared with the exact solutions. It was observed that results from OHAM have better convergence rate for time-fractional order Fokker-Planck equations. The solutions are plotted and the relative errors are tabulated. Full article

Given a multivariate complex centered Gaussian vector $Z=(Z_1,\cdots ,Z_p)$ with non-singular covariance matrix $\mathsf{\Sigma }$, we derive sufficient conditions on the nullity of the complex moments and we give a closed-form expression for the non-null complex moments. We present conditions for the factorisation of the complex moments. Computational consequences of these results are discussed. Full article

It is well known that feed-forward neural networks can be used for approximation to functions based on an appropriate activation function. In this paper, employing a new sigmoidal function with a parameter for an activation function, we consider a constructive feed-forward neural network approximation on a closed interval. The developed approximation method takes a simple form of a superposition of the parametric sigmoidal function. It is shown that the proposed method is very effective in approximation of discontinuous functions as well as continuous ones. For some examples, the availability of the presented method is demonstrated by comparing its numerical results with those of an existing neural network approximation method. Furthermore, the efficiency of the method in extended application to the multivariate function is also illustrated. Full article

Networks have an important role in our daily lives. The effectiveness of the network decreases with the breaking down of some vertices or links. Therefore, a less vulnerable communication network is required for greater stability. Vulnerability is the measure of resistance of the network after failure of communication links. In this article, a graph has been taken for modeling a network and integrity as a measure of vulnerability. The approach is to estimate the integrity or upper bound of integrity of at least one connected graph or network constructed from the given graphic integer sequence. Experiments have been done with random graphs, complex networks and also a comparison between two parameters, namely the vertex connectivity and graph integrity as a measure of the network vulnerability have been carried out by removing vertices randomly from various complex networks. A comparison with the existing method shows that the algorithm proposed in this article provides a much better integrity measurement. Full article

The concept of q-rung orthopair fuzzy sets generalizes the notions of intuitionistic fuzzy sets and Pythagorean fuzzy sets to describe complicated uncertain information more effectively. Their most dominant attribute is that the sum of the $q^{\mathrm{th}}$ power of the truth-membership and the $q^{\mathrm{th}}$ power of the falsity-membership must be equal to or less than one, so they can broaden the space of uncertain data. This set can adjust the range of indication of decision data by changing the parameter q, $q\ge 1$. In this research study, we design a new framework for handling uncertain data by means of the combinative theory of q-rung orthopair fuzzy sets and hypergraphs. We define q-rung orthopair fuzzy hypergraphs to achieve the advantages of both theories. Further, we propose certain novel concepts, including adjacent levels of q-rung orthopair fuzzy hypergraphs, $(\alpha ,\beta )$-level hypergraphs, transversals, and minimal transversals of q-rung orthopair fuzzy hypergraphs. We present a brief comparison of our proposed model with other existing theories. Moreover, we implement some interesting concepts of q-rung orthopair fuzzy hypergraphs for decision-making to prove the effectiveness of our proposed model. Full article

In this article, our main purpose is to introduce a new and generalized quadratic Gauss sum. By using analytic methods, the properties of classical Gauss sums, and character sums, we consider the calculating problem of its fourth power mean and give two interesting computational formulae for it. Full article

Fuzzy information in venture capital can be well expressed by neutrosophic numbers, and TODIM method is an effective tool for multi-attribute decision-making. The distance measure is an essential step in TODIM method. The keystone of this paper is to define several new distance measures, in particular the improved interval neutrosophic Euclidean distance, and these measures are applied in the TODIM method for multi-attribute decision-making. Firstly, the normalized generalized interval neutrosophic Hausdorff distance is defined and proved to be valid in this paper. Secondly, we define a weighted parameter interval neutrosophic distance and discuss whether different weight parameters affect the decision result based on TODIM method. Thirdly, considering the preference perspective of decision-makers in behavioral economics, we define the improved interval neutrosophic Euclidean distance with the known parameter of risk preference. Finally, an application example is given to compare the effects of different parameters on the result and discuss the feasibility of these two distance measures in TODIM method. Full article