Mathematics, Volume 9, Issue 3 (February-1 2021) – 93 articles

For a simple graph $G=(V,E)$ with no isolated vertices, a total Roman {3}-dominating function(TR3DF) on G is a function $f:V\left(G\right)\to \{0,1,2,3\}$ having the property that (i) ${\sum }_{w\in N\left(v\right)}f\left(w\right)\ge 3$ if $f\left(v\right)=0$; (ii) ${\sum }_{w\in N\left(v\right)}f\left(w\right)\ge 2$ if $f\left(v\right)=1$; and (iii) every vertex v with $f\left(v\right)\ne 0$ has a neighbor u with $f\left(u\right)\ne 0$ for every vertex $v\in V\left(G\right)$. The weight of a TR3DF f is the sum $f\left(V\right)={\sum }_{v\in V\left(G\right)}f\left(v\right)$ and the minimum weight of a total Roman {3}-dominating function on G is called the total Roman {3}-domination number denoted by ${\gamma }_{t\{R3\}}\left(G\right)$. In this paper, we show that the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs. Finally, we present a linear-time algorithm to compute the value of ${\gamma }_{t\{R3\}}$ for trees. Full article

The exact solution of the movement equation of the Coriolis vibratory gyroscope (CVG) with a linear law of variation of the angular rate of rotation of the base is given. The solution is expressed in terms of the Weber functions (the parabolic cylinder functions) and their asymptotic representations. On the basis of the obtained solution, an analytical solution to the equation of the ring dynamics in the case of piecewise linear approximation of an arbitrary angular velocity profile on a time grid is derived. The piecewise linear solution is compared with the more rough piecewise constant solution and the dependence of the error of such approximations on the sampling step in time is estimated numerically. The results obtained make it possible to significantly reduce the number of operations when it is necessary to study long-range dynamics of oscillations of the system, as well as quantitatively and qualitatively control the convergence of finite-difference schemes for solving the movement equations of the Coriolis vibratory gyroscope. Full article

A novel semi-supervised learning method is proposed to better utilize labeled and unlabeled samples to improve classification performance. However, there is exists the limitation that Laplace regularization in a semi-supervised extreme learning machine (SSELM) tends to lead to poor generalization ability and it ignores the role of labeled information. To solve the above problems, a Joint Regularized Semi-Supervised Extreme Learning Machine (JRSSELM) is proposed, which uses Hessian regularization instead of Laplace regularization and adds supervised information regularization. In order to solve the problem of slow convergence speed and the easy to fall into local optimum of marine predator algorithm (MPA), a multi-strategy marine predator algorithm (MSMPA) is proposed, which first uses a chaotic opposition learning strategy to generate high-quality initial population, then uses adaptive inertia weights and adaptive step control factor to improve the exploration, utilization, and convergence speed, and then uses neighborhood dimensional learning strategy to maintain population diversity. The parameters in JRSSELM are then optimized using MSMPA. The MSMPA-JRSSELM is applied to logging oil formation identification. The experimental results show that MSMPA shows obvious superiority and strong competitiveness in terms of convergence accuracy and convergence speed. Also, the classification performance of MSMPA-JRSSELM is better than other classification methods, and the practical application is remarkable. Full article

Given a digital image (or digital object) $(X,k),X\subset {\mathbb{Z}}^n$, this paper initially establishes a group structure of the set of self-k-isomorphisms of $(X,k)$ with the function composition, denoted by $Is{o}_k(X)$ or $Au{t}_k(X)$. In particular, let $C_k^{n,l}$ be a simple closed k-curve with l elements in ${\mathbb{Z}}^n$. Then, the group $Is{o}_k(C_k^{n,l})$ is proved to be isomorphic to the standard dihedral group $D_l$ with order l. The calculation of this quantity $Is{o}_k(C_k^{n,l})$ is a key step for obtaining many new results. Indeed, it is essential for exploring many features of $Is{o}_k(X)$. Furthermore, this quantity is proved to be a digital topological invariant. After proceeding with an $Is{o}_k(X)$-action on $(X,k)$, we investigate some properties of fixed point sets by this action. Finally, we explore various features of fixed point sets by this action from the viewpoint of digital k-curve theory. This paper only deals with k-connected digital images $(X,k)$ whose cardinality is equal to or greater than 2. Besides, we discuss some errors that have appeared in the lilterature. Full article

Few studies analyze the endogenous emergence of price competition in a new product market. This paper analyzes two differentiated products, an existing product and a newly introduced substitutable product, and investigates conditions under which a price competition endogenously emerges in a new product market in the context of a choice between engaging in price competition and holding price leadership. We demonstrate that Bertrand price competition emerges when the setup cost for the new product is high enough. This result implies that government policies reducing setup costs such as subsidies could change the type of competition to price leadership in a new product market. Full article

In this paper, we propose an approach to take into account, in a robust way, part-worth uncertainty in a share-of-choice (SOC) model. More precisely, we extend the method proposed by Wang and Curry by endogenously including competition. Indeed in their approach, competition is described exogenously and the model cannot take into account part-worth uncertainty for the competition’s products. Our extension permits us to take into account all effects of part-worth uncertainty, even those relative to the competition, and therefore improve substantially Wang and Curry’s approach. Full article

Computer vision has shown great accomplishments in a wide variety of classification, segmentation and object recognition tasks, but tends to encounter more difficulties when tasks require more contextual assessment. Measuring the engagement of students is an example of such a complex task, as it requires a strong interpretative component. This research describes a methodology to measure students’ engagement, taking both an individual (student-level) and a collective (classroom) approach. Results show that students’ individual behaviour, such as note-taking or hand-raising, is challenging to recognise, and does not correlate with students’ self-reported engagement. Interestingly, students’ collective behaviour can be quantified in a more generic way using measures for students’ symmetry, reaction times and eye-gaze intersections. Nonetheless, the evidence for a connection between these collective measures and engagement is rather weak. Although this study does not succeed in providing a proxy of students’ self-reported engagement, our approach sheds light on the needs for future research. More concretely, we suggest that not only the behavioural, but also the emotional and cognitive component of engagement should be captured. Full article

In this paper, we apply the pseudospectral method based on the Chebyshev cardinal function to solve the parabolic partial integro-differential equations (PIDEs). Since these equations play a key role in mathematics, physics, and engineering, finding an appropriate solution is important. We use an efficient method to solve PIDEs, especially for the integral part. Unlike when using Chebyshev functions, when using Chebyshev cardinal functions it is no longer necessary to integrate to find expansion coefficients of a given function. This reduces the computation. The convergence analysis is investigated and some numerical examples guarantee our theoretical results. We compare the presented method with others. The results confirm the efficiency and accuracy of the method. Full article

This communication provides a discussion of a scheme originally proposed by Falcón in a paper entitled “Latin squares associated to principal autotopisms of long cycles. Applications in cryptography”. Falcón outlines the protocol for a cryptographical scheme that uses the $\mathfrak{F}$-critical sets associated with a particular Latin square to generate access levels for participants of the scheme. Accompanying the scheme is an example, which applies the protocol to a particular Latin square of order six. Exploration of the example itself, revealed some interesting observations about both the structure of the Latin square itself and the autotopisms associated with the Latin square. These observations give rise to necessary conditions for the generation of the $\mathfrak{F}$-critical sets associated with certain autotopisms of the given Latin square. The communication culminates with a table which outlines the various access levels for the given Latin square in accordance with the scheme detailed by Falcón. Full article

Time series forecasting is an important research topic with many practical applications. As shown earlier, the problems of lossless data compression and prediction are very similar mathematically. In this article, we propose several forecasting methods based on real-world data compressors. We consider predicting univariate and multivariate data, describe how multiple data compressors can be combined into one forecasting method with automatic selection of the best algorithm for the input data. The developed forecasting techniques are not inferior to the known ones. We also propose a way to reduce the computation time of the combined method by using the so-called time-universal codes. To test the proposed techniques, we make predictions for real-world data such as sunspot numbers and some social indicators of Novosibirsk region, Russia. The results of our computations show that the described methods find non-trivial regularities in data, and time universal codes can reduce the computation time without losing accuracy. Full article

This paper introduces a new approach for the fabrication of generalized developable cubic trigonometric Bézier (GDCT-Bézier) surfaces with shape parameters to address the fundamental issue of local surface shape adjustment. The GDCT-Bézier surfaces are made by means of GDCT-Bézier-basis-function-based control planes and alter their shape by modifying the shape parameter value. The GDCT-Bézier surfaces are designed by maintaining the classic Bézier surface characteristics when the shape parameters take on different values. In addition, the terms are defined for creating a geodesic interpolating surface for the GDCT-Bézier surface. The conditions appropriate and suitable for $G^1$, Farin–Boehm $G^2$, and $G^2$ Beta continuity in two adjacent GDCT-Bézier surfaces are also created. Finally, a few important aspects of the newly formed surfaces and the influence of the shape parameters are discussed. The modeling example shows that the proposed approach succeeds and can also significantly improve the capability of solving problems in design engineering. Full article

In this paper, we review overdispersed Bayesian generalized spatial conditional count data models. Their usefulness is illustrated with their application to infant mortality rates from Colombian regions and by comparing them with the widely used Besag–York–Mollié (BYM) models. These overdispersed models assume that excess of dispersion in the data may be partially caused from the possible spatial dependence existing among the different spatial units. Thus, specific regression structures are then proposed both for the conditional mean and for the dispersion parameter in the models, including covariates, as well as an assumed spatial neighborhood structure. We focus on the case of response variables following a Poisson distribution, specifically concentrating on the spatial generalized conditional normal overdispersion Poisson model. Models were fitted by making use of the Markov Chain Monte Carlo (MCMC) and Integrated Nested Laplace Approximation (INLA) algorithms in the specific context of Bayesian estimation methods. Full article

In the present work, a new extension of the two-variable Fubini polynomials is introduced by means of the polyexponential function, which is called the two-variable type 2 poly-Fubini polynomials. Then, some useful relations including the Stirling numbers of the second and the first kinds, the usual Fubini polynomials, and the higher-order Bernoulli polynomials are derived. Also, some summation formulas and an integral representation for type 2 poly-Fubini polynomials are investigated. Moreover, two-variable unipoly-Fubini polynomials are introduced utilizing the unipoly function, and diverse properties involving integral and derivative properties are attained. Furthermore, some relationships covering the two-variable unipoly-Fubini polynomials, the Stirling numbers of the second and the first kinds, and the Daehee polynomials are acquired. Full article

We consider a generalization of Awad–Shannon entropy, namely Awad–Varma entropy, introduce a stochastic order on Awad–Varma residual entropy and study some properties of this order, like closure, reversed closure and preservation in some stochastic models (the proportional hazard rate model, the proportional reversed hazard rate model, the proportional odds model and the record values model). Full article

We study a class of nonlinear operators that can be written as the composition of a linear operator and a nonlinear map. We obtain results on fixed point index based on parameters that are related to the definitions of nonlinear spectra. As a particular case, existence of positive solutions for a second-order differential equation with separated boundary conditions is proved. The result also provides a spectral interval for the corresponding Hammerstein integral operator. Full article

In recent years, many advanced techniques have been applied to financial problems; however, very few scholars have used the Lie theory. The purpose of this study was to examine the options for a trade account through Lie symmetry analysis. According to our results, it is effective for determining analytical solutions for pricing issues and solving other partial differential equations. The proposed solution can be used by further researchers or practitioners in option pricing problems for better performance compared with the classical Black–Scholes model. Full article

F-contractions have inspired a branch of metric fixed point theory committed to the generalization of the classical Banach contraction principle. The study of these contractions and $\alpha$-fuzzy mappings in b-metric spaces was attempted timidly and was not successful. In this article, the main objective is to obtain common $\alpha$-fuzzy fixed point results for F-contractions in b-metric spaces. Some multivalued fixed point results in the literature are derived as consequences of our main results. We also provide a non-trivial example to show the validity of our results. As applications, we investigate the solution for fuzzy initial value problems in the context of a generalized Hukuhara derivative. Our results generalize, improve and complement several developments from the existing literature. Full article

The mathematical modeling of unsteady flow of micropolar Cu–Al₂O₃/water nanofluid driven by a deformable sheet in stagnation region with thermal radiation effect has been explored numerically. To achieve the system of nonlinear ordinary differential equations (ODEs), we have employed some appropriate transformations and solved it numerically using MATLAB software (built-in solver called bvp4c). Influences of relevant parameters on fluid flow and heat transfer characteristic are discussed and presented in graphs. The findings expose that double solutions appear in shrinking sheet case in which eventually contributes to the analysis of stability. The stability analysis therefore confirms that merely the first solution is a stable solution. Addition of nanometer-sized particle (Cu) has been found to significantly strengthen the heat transfer rate of micropolar nanofluid. When the copper nanoparticle volume fraction increased from 0 to 0.01 (1%) in micropolar nanofluid, the heat transfer rate increased roughly to an average of 17.725%. The result also revealed that an upsurge in the unsteady and radiation parameters have been noticed to enhance the local Nusselt number of micropolar hybrid nanofluid. Meanwhile, the occurrence of material parameter conclusively decreases it. Full article

This paper deals with an optimal control problem for a nonlocal model of the steady-state flow of a differential type fluid of complexity 2 with variable viscosity. We assume that the fluid occupies a bounded three-dimensional (or two-dimensional) domain with the impermeable boundary. The control parameter is the external force. We discuss both strong and weak solutions. Using one result on the solvability of nonlinear operator equations with weak-to-weak and weak-to-strong continuous mappings in Sobolev spaces, we construct a weak solution that minimizes a given cost functional subject to natural conditions on the model data. Moreover, a necessary condition for the existence of strong solutions is derived. Simultaneously, we introduce the concept of the marginal function and study its properties. In particular, it is shown that the marginal function of this control system is lower semicontinuous with respect to the directed Hausdorff distance. Full article

A series of computational models and simulations were conducted for determining the dynamic responses of a solid metal projectile impacting a target under a prescribed high strain rate loading scenario in three-dimensional space. The focus of this study was placed on two different modeling techniques within finite element analysis available in the Abaqus software suite. The first analysis technique relied heavily on more traditional Lagrangian analysis methods utilizing a fixed mesh, while still taking advantage of the finite difference integration present under the explicit analysis approach. A symmetry reduced model using the Lagrangian coordinate system was also developed for comparison in physical and computational performance. The second analysis technique relied on a mixed model that still made use of some Lagrangian modeling, but included smoothed particle hydrodynamics techniques as well, which are mesh free. The inclusion of the smoothed particle hydrodynamics was intended to address some of the known issues in Lagrangian analysis under high displacement and deformation. A comparison of the models was first performed against experimental results as a validation of the models, then the models were compared against each other based on closeness to experimentation and computational performance. Full article

Currently, much research attention has focused on generalizations of known mathematical objects in order to obtain adequate models describing real phenomena. An important role in the applied theory of probability and mathematical statistics is the gamma class of distributions, which has proven to be a convenient and effective tool for modeling many real processes. The gamma class is quite wide and includes distributions that have useful properties such as, for example, infinite divisibility and stability, which makes it possible to use distributions from this class as asymptotic approximations in various limit theorems. One of the most important tasks of applied statistics is to obtain estimates of the parameters of the model distribution from the available real data. In this paper, we consider the gamma-exponential distribution, which is a generalization of the distributions from the gamma class. Estimators for some parameters of this distribution are given, and the asymptotic normality of these estimators is proven. When obtaining the estimates, a modified method of moments was used, based on logarithmic moments calculated on the basis of the Mellin transform for the generalized gamma distribution. On the basis of the results obtained, asymptotic confidence intervals for the estimated parameters are constructed. The results of this work can be used in the study of probabilistic models based on continuous distributions with an unbounded non-negative support. Full article

The coalgebraic method is of great significance to research in process algebra, modal logic, object-oriented design and component-based software engineering. In recent years, fuzzy control has been widely used in many fields, such as handwriting recognition and the control of robots or air conditioners. It is then an interesting topic to analyze the behavior of fuzzy automata from a coalgebraic point of view. This paper models different types of fuzzy automata as coalgebras with a monad structure capturing fuzzy behavior. Based on the coalgebraic models, we can define a notion of fuzzy language and consider several versions of bisimulation for fuzzy automata. A group of combinators is defined to compose fuzzy automata of two branches: state transition and output function. A case study illustrates the coalgebraic models proposed and their composition. Full article

We study power-mixture type functional equations in terms of Laplace–Stieltjes transforms of probability distributions on the right half-line $[0,\infty ).$ These equations arise when studying distributional equations of the type $Z\stackrel{\mathrm{d}}{=}X+TZ$, where the random variable $T\ge 0$ has known distribution, while the distribution of the random variable $Z\ge 0$ is a transformation of that of $X\ge 0$, and we want to find the distribution of X. We provide necessary and sufficient conditions for such functional equations to have unique solutions. The uniqueness is equivalent to a characterization property of a probability distribution. We present results that are either new or extend and improve previous results about functional equations of compound-exponential and compound-Poisson types. In particular, we give another affirmative answer to a question posed by J. Pitman and M. Yor in 2003. We provide explicit illustrative examples and deal with related topics. Full article

In this research, a new approach for tackling the permutation flow shop scheduling problem (PFSSP) is proposed. This algorithm is based on the steps of the elitism continuous genetic algorithm improved by two strategies and used the largest rank value (LRV) rule to transform the continuous values into discrete ones for enabling of solving the combinatorial PFSSP. The first strategy is combining the arithmetic crossover with the uniform crossover to give the algorithm a high capability on exploitation in addition to reducing stuck into local minima. The second one is re-initializing an individual selected randomly from the population to increase the exploration for avoiding stuck into local minima. Afterward, those two strategies are combined with the proposed algorithm to produce an improved one known as the improved efficient genetic algorithm (IEGA). To increase the exploitation capability of the IEGA, it is hybridized a local search strategy in a version abbreviated as HIEGA. HIEGA and IEGA are validated on three common benchmarks and compared with a number of well-known robust evolutionary and meta-heuristic algorithms to check their efficacy. The experimental results show that HIEGA and IEGA are competitive with others for the datasets incorporated in the comparison, such as Carlier, Reeves, and Heller. Full article

Asphalt production plants play an important role in the field of civil engineering, but also in the entire economic system since the construction of roads enables uninterrupted functioning within it. In this paper, the ranking of asphalt production plants on the territory of the Autonomous Province of Vojvodina has been performed. The modern economy needs contemporary models and methods to solve complicated MCDM problems and, for these purposes, it has been developed an original Interval Rough Number (IRN) Multi-criteria decision-making (MCDM) model that implies an extension of two methods belonging to the field with interval rough numbers. After forming a list of eight most significant criteria for assessing the efficiency of asphalt production plants, the Interval Rough Number PIvot Pairwise RElative Criteria Importance Assessment (IRN PIPRECIA) method was developed to determine the significance of the criteria. A total of 21 locations with asphalt mixture installation were considered. For that purpose, seven asphalt production plants were included, and for their ranking, the IRN EDAS (Evaluation based on Distance from Average Solution) method was created. The aim of this paper is to develop a novel interval rough model that can be useful for determining the efficiency of asphalt production plants. Averaging in group decision-making (GDM) for both methods was performed using an IRN Dombi weighted geometric averaging (IRNDWGA) aggregator. The obtained results show that (A15) Ruma (SP)–Mačvanska Mitrovica–Zasavica has the best characteristics out of the set of locations considered in this study. However, Alternatives A6 and A19 are also variants with remarkably good characteristics since there is very little difference in values compared to the first-ranked alternative. Also, the obtained results have shown that the developed model is applicable, which is proven through a comparative analysis. Full article

This note is concerned with two new methods for the solution of a Cauchy problem. The first method is based on homotopy-perturbation approach which leads to solving a series of well-posed boundary value problems. No regularization is needed in this method. Laplace and Helmholtz equations are considered in an annular region. It is also proved that the homotopy solution for the Laplace operator converges to the actual exact solution. The second method is also non-iterative. It is based on the application of the Green’s second identity which leads to a moment problem for the unknown boundary condition. Tikhonov regularization is used to obtain a stable and close approximation of the missing boundary condition. A number of examples are used to study the applicability of the methods with the presence of noise. Full article

Since the launch of Bitcoin, there has been a lot of controversy surrounding what asset class it is. Several authors recognize the potential of cryptocurrencies but also certain deviations with respect to the functions of a conventional currency. Instead, Bitcoin’s diversifying factor and its high return potential have generated the attention of portfolio managers. In this context, understanding how its volatility is explained is a critical element of investor decision-making. By modeling the volatility of classic assets, nonlinear models such as Generalized Autoregressive Conditional Heteroskedasticity (GARCH) offer suitable results. Therefore, taking GARCH(1,1) as a reference point, the main aim of this study is to model and assess the relationship between the Bitcoin volatility and key financial environment variables through a Conditional Correlation (CC) Multivariate GARCH (MGARCH) approach. For this, several commodities, exchange rates, stock market indices, and company stocks linked to cryptocurrencies have been tested. The results obtained show certain heterogeneity in the fit of the different variables, highlighting the uncorrelation with respect to traditional safe haven assets such as gold and oil. Focusing on the CC-MGARCH model, a better behavior of the dynamic conditional correlation is found compared to the constant. Full article

A new class of differential variational inequalities (DVIs), governed by a variational inequality and an evolution equation formulated in infinite-dimensional spaces, is investigated in this paper. More precisely, based on Browder’s result, optimal control theory, measurability of set-valued mappings and the theory of semigroups, we establish that the solution set of DVI is nonempty and compact. In addition, the theoretical developments are accompanied by an application to differential Nash games. Full article

The paper is devoted to an emerging trend in control—a machine learning control. Despite the popularity of the idea of machine learning, there are various interpretations of this concept, and there is an urgent need for its strict mathematical formalization. An attempt to formalize the concept of machine learning is presented in this paper. The concepts of an unknown function, work area, training set are introduced, and a mathematical formulation of the machine learning problem is presented. Based on the presented formulation, the concept of machine learning control is considered. One of the problems of machine learning control is the general synthesis of control. It implies finding a control function that depends on the state of the object, which ensures the achievement of the control goal with the optimal value of the quality criterion from any initial state of some admissible region. Supervised and unsupervised approaches to solving a problem based on symbolic regression methods are considered. As a computational example, a problem of general synthesis of optimal control for a spacecraft landing on the surface of the Moon is considered as supervised machine learning control with a training set. Full article

Markov-type inequalities are often used in numerical solutions of differential equations, and their constants improve error bounds. In this paper, the upper approximation of the constant in a Markov-type inequality on a simplex is considered. To determine the constant, the minimal polynomial and pluripotential theories were employed. They include a complex equilibrium measure that solves the extreme problem by minimizing the energy integral. Consequently, examples of polynomials of the second degree are introduced. Then, a challenging bilevel optimization problem that uses the polynomials for the approximation was formulated. Finally, three popular meta-heuristics were applied to the problem, and their results were investigated. Full article