Mathematics, Volume 8, Issue 12 (December 2020) – 163 articles

Evaluation of images of special functions under operators of fractional calculus has become a hot topic with hundreds of recently published papers. These are growing daily and we are able to comment here only on a few of them, including also some of the latest of 2019–2020, just for the purpose of illustrating our unified approach. Many authors are producing a flood of results for various operators of fractional order integration and differentiation and their generalizations of different special (and elementary) functions. This effect is natural because there are great varieties of special functions, respectively, of operators of (classical and generalized) fractional calculus, and thus, their combinations amount to a large number. As examples, we mentioned only two such operators from thousands of results found by a Google search. Most of the mentioned works use the same formal and standard procedures. Furthermore, in such results, often the originals and the images are special functions of different kinds, or the images are not recognized as known special functions, and thus are not easy to use. In this survey we present a unified approach to fulfill the mentioned task at once in a general setting and in a well visible form: for the operators of generalized fractional calculus (including also the classical operators of fractional calculus); and for all generalized hypergeometric functions such as ${}_p{\mathsf{\Psi }}_q$ and ${}_p{F}_q$, Fox H- and Meijer G-functions, thus incorporating wide classes of special functions. In this way, a great part of the results in the mentioned publications are well predicted and appear as very special cases of ours. The proposed general scheme is based on a few basic classical results (from the Bateman Project and works by Askey, Lavoie–Osler–Tremblay, etc.) combined with ideas and developments from more than 30 years of author’s research, and reflected in the cited recent works. The main idea is as follows: From one side, the operators considered by other authors are cases of generalized fractional calculus and so, are shown to be (m-times) compositions of weighted Riemann–Lioville, i.e., Erdélyi–Kober operators. On the other side, from each generalized hypergeometric function ${}_p{\mathsf{\Psi }}_q$ or ${}_p{F}_q$ ($p\le q$ or $p=q+1$) we can reach, from the final number of applications of such operators, one of the simplest cases where the classical results are known, for example: to ${}_0{F}_{q-p}$ (hyper-Bessel functions, in particular trigonometric functions of order $(q-p)$), ${}_0{F}_0$ (exponential function), or ${}_1{F}_0$ (beta-distribution of form ${(1-z)}^{\alpha }{z}^{\beta })$. The final result, written explicitly, is that any GFC operator (of multiplicity $m\ge 1$) transforms a generalized hypergeometric function into the same kind of special function with indices p and q increased by m. Full article

To increase the level of safety and prevent significant accidents, it is essential to prioritize risk factors and assess railway infrastructure. The key question is how to identify unsafe railway infrastructure so authorities can undertake safety improvement projects on time. The paper aims to introduce a picture fuzzy group multi-criteria decision-making approach for risk assessment of railway infrastructure. Firstly, picture fuzzy sets are employed for representing and handling risk-related information. Secondly, a picture fuzzy hybrid method based on the direct rating, and Tsallis–Havrda–Charvát entropy is provided to prioritize risk factors. Thirdly, a picture fuzzy measurement of alternatives and ranking according to compromise solution method is developed to rank railway infrastructures. Lastly, the formulated approach is implemented in the Czech Republic context. Two sensitivity analyses verified the high robustness of the formulated approach. The comparative analysis with five state-of-the-art picture fuzzy approaches approved its high reliability. Compared to the state-of-the-art picture fuzzy approaches, the provided three-parametric approach has superior flexibility. Full article

Enhancement of Cultural Heritage such as historical images is very crucial to safeguard the diversity of cultures. Automated colorization of black and white images has been subject to extensive research through computer vision and machine learning techniques. Our research addresses the problem of generating a plausible colored photograph of ancient, historically black, and white images of Nepal using deep learning techniques without direct human intervention. Motivated by the recent success of deep learning techniques in image processing, a feed-forward, deep Convolutional Neural Network (CNN) in combination with Inception- ResnetV2 is being trained by sets of sample images using back-propagation to recognize the pattern in RGB and grayscale values. The trained neural network is then used to predict two a* and b* chroma channels given grayscale, L channel of test images. CNN vividly colorizes images with the help of the fusion layer accounting for local features as well as global features. Two objective functions, namely, Mean Squared Error (MSE) and Peak Signal-to-Noise Ratio (PSNR), are employed for objective quality assessment between the estimated color image and its ground truth. The model is trained on the dataset created by ourselves with 1.2 K historical images comprised of old and ancient photographs of Nepal, each having 256 × 256 resolution. The loss i.e., MSE, PSNR, and accuracy of the model are found to be 6.08%, 34.65 dB, and 75.23%, respectively. Other than presenting the training results, the public acceptance or subjective validation of the generated images is assessed by means of a user study where the model shows 41.71% of naturalness while evaluating colorization results. Full article

Algebraic thinking, combinatorial thinking and reasoning skills are considered as playing central roles within teaching and learning in the field of mathematics, particularly in solving complex open-ended mathematical problems Specific relations between these three abilities, manifested in the solving of an open-ended ill-structured problem aimed at mathematical modeling, were investigated. We analyzed solutions received from 33 groups totaling 131 students, who solved a complex assignment within the mathematical contest Mathematics B-day 2018. Such relations were more obvious when solving a complex problem, compared to more structured closed subtasks. Algebraic generalization is an important prerequisite to prove mathematically and to solve combinatorial problem at higher levels, i.e., using expressions and formulas, therefore a special focus should be put on this ability in upper-secondary mathematics education. Full article

The existence, uniqueness and uniformly $L^p$ estimates for solutions of a high-order abstract Navier–Stokes problem on half space are derived. The equation involves an abstract operator in a Banach space E and small parameters. Since the Banach space E is arbitrary and A is a possible linear operator, by choosing spaces E and operators A, the existence, uniqueness and $L^p$ estimates of solutions for numerous classes of Navier–Stokes type problems are obtained. In application, the existence, uniqueness and uniformly $L^{\mathbf{p}}$ estimates for the solution of the Wentzell–Robin-type mixed problem for the Navier–Stokes equation and mixed problem for degenerate Navier–Stokes equations are established. Full article

This paper examines the role of non-cash flow factors over correlation jumps in financial markets. Utilizing time-varying risk aversion measure as a proxy for investor sentiment and the cross-quantilogram method applied to intraday data, we show that risk aversion captures significant predictive power over realized stock-bond correlation jumps at different quantiles and lags. The predictive relation between correlation jumps and time-varying risk aversion is found to be asymmetric, as we detect a heterogeneous dependence pattern across different quantiles and lag orders. Our findings underline the importance of non-cash flow factors over correlation jumps, highlighting the role of behavioral factors in optimal portfolio allocations and the effectiveness of diversification strategies. Full article

This paper presents an adaptive filtering-based maximum likelihood multi-innovation extended stochastic gradient algorithm to identify multivariable equation-error systems with colored noises. The data filtering and model decomposition techniques are used to simplify the structure of the considered system, in which a predefined filter is utilized to filter the observed data, and the multivariable system is turned into several subsystems whose parameters appear in the vectors. By introducing the multi-innovation identification theory to the stochastic gradient method, this study produces improved performances. The simulation numerical results indicate that the proposed algorithm can generate more accurate parameter estimates than the filtering-based maximum likelihood recursive extended stochastic gradient algorithm. Full article

The aim of this paper is to extend, from group theory to hypergroup theory, the class equation and the concept of commutativity degree. Both of them are studied in depth for complete hypergroups because we want to stress the similarities and the differences with respect to group theory, and the representation theorem of complete hypergroups helps us in this direction. We also find conditions under which the commutativity degree can be expressed by using the class equation. Full article

In matrix population modeling the multi-year monitoring of a population structure results in a set of annual population projection matrices (PPMs), which gives rise to the stochastic growth rate λ_S, a quantitative measure of long-term population viability. This measure is usually found in the paradigm of population growth in a variable environment. The environment is represented by the set of PPMs, and λ_S ensues from a long sequence of PPMs chosen at random from the given set. because the known rules of random choice, such as the iid (independent and identically distributed) matrices, are generally artificial, the challenge is to find a more realistic rule. We achieve this with the a following a Markov chain that models, in a certain sense, the real variations in the environment. We develop a novel method to construct the ruling Markov chain from long-term weather data and to simulate, in a Monte Carlo mode, the long sequences of PPMs resulting in the estimates of λ_S. The stochastic nature of sequences causes the estimates to vary within some range, and we compare the range obtained by the “realistic choice” from 10 PPMs for a local population of a Red-Book species to those using the iid choice. As noted in the title of this paper, this realistic choice contracts the range of λ_S estimates, thus improving the estimation and confirming the Red-Book status of the species. Full article

This research extended the model developed by Hull and White by integrating Taylor-series expansion into the model for deriving approximate analytical solutions for stochastic volatility forward-starting Asian options. Numerical experiments were performed to compare the proposed model with the Monte Carlo model over numerous simulations and demonstrated that the developed model has a pricing accuracy greater than 99%. Furthermore, the computation time was approximately 10⁻⁵ s for each simulation. The model’s outstanding computational performance demonstrates its capability to address the challenges of high-frequency trading. Full article

We study Riemann-Lebesgue integrability for interval-valued multifunctions relative to an interval-valued set multifunction. Some classic properties of the $RL$ integral, such as monotonicity, order continuity, bounded variation, convergence are obtained. An application of interval-valued multifunctions to image processing is given for the purpose of illustration; an example is given in case of fractal image coding for image compression, and for edge detection algorithm. In these contexts, the image modelization as an interval valued multifunction is crucial since allows to take into account the presence of quantization errors (such as the so-called round-off error) in the discretization process of a real world analogue visual signal into a digital discrete one. Full article

This study is devoted to the description of the asymptotic expansions of solutions of linear ordinary differential equations with holomorphic coefficients in the neighborhood of an infinitely distant singular point. This is a classical problem of analytical theory of differential equations and an important particular case of the general Poincare problem on constructing the asymptotics of solutions of linear ordinary differential equations with holomorphic coefficients in the neighborhoods of irregular singular points. In this study we consider such equations for which the principal symbol of the differential operator has multiple roots. The asymptotics of a solution for the case of equations with simple roots of the principal symbol were constructed earlier. Full article

In this paper we obtain a novel implementation for irregular clouds of nodes of the meshless method called Generalized Finite Difference Method for solving the complex Ginzburg–Landau equation. We derive the explicit formulae for the spatial derivative and an explicit scheme by splitting the equation into a system of two parabolic PDEs. We prove the conditional convergence of the numerical scheme towards the continuous solution under certain assumptions. We obtain a second order approximation as it is clear from the numerical results. Finally, we provide several examples of its application over irregular domains in order to test the accuracy of the explicit scheme, as well as comparison with other numerical methods. Full article

The fractional calculus is useful in describing the natural phenomena with memory effect. This paper addresses the fractional form of Ambartsumian equation with a delay parameter. It may be a challenge to obtain accurate approximate solution of such kinds of fractional delay equations. In the literature, several attempts have been conducted to analyze the fractional Ambartsumian equation. However, the previous approaches in the literature led to approximate power series solutions which converge in subdomains. Such difficulties are solved in this paper via the Homotopy Perturbation Method (HPM). The present approximations are expressed in terms of the Mittag-Leffler functions which converge in the whole domain of the studied model. The convergence issue is also addressed. Several comparisons with the previous published results are discussed. In particular, while the computed solution in the literature is physical in short domains, with our approach it is physical in the whole domain. The results reveal that the HPM is an effective tool to analyzing the fractional Ambartsumian equation. Full article

In this paper, we recall a more generalized integral transform, a generalized convolution product and a generalized first variation on function space. The Gaussian process and the bounded linear operators on function space are used to define them. We then establish the existence and various relationships between the generalized integral transform and the generalized convolution product. Furthermore, we obtain some relationships between the generalized integral transform and the generalized first variation with the generalized Cameron–Storvick theorem. Finally, some applications are demonstrated as examples. Full article

A necessity in the design of a path planning algorithm is to account for the environment. If the movement of the mobile robot is through a dynamic environment, the algorithm needs to include the main constraint: real-time collision avoidance. This kind of problem has been studied by different researchers suggesting different techniques to solve the problem of how to design a trajectory of a mobile robot avoiding collisions with dynamic obstacles. One of these algorithms is the artificial potential field (APF), proposed by O. Khatib in 1986, where a set of an artificial potential field is generated to attract the mobile robot to the goal and to repel the obstacles. This is one of the best options to obtain the trajectory of a mobile robot in real-time (RT). However, the main disadvantage is the presence of deadlocks. The mobile robot can be trapped in one of the local minima. In 1988, J.F. Canny suggested an alternative solution using harmonic functions satisfying the Laplace partial differential equation. When this article appeared, it was nearly impossible to apply this algorithm to RT applications. Years later a novel technique called proper generalized decomposition (PGD) appeared to solve partial differential equations, including parameters, the main appeal being that the solution is obtained once in life, including all the possible parameters. Our previous work, published in 2018, was the first approach to study the possibility of applying the PGD to designing a path planning alternative to the algorithms that nowadays exist. The target of this work is to improve our first approach while including dynamic obstacles as extra parameters. Full article

Hypothesis testing has been pointed out as one of the statistical topics in which students present more misconceptions. In this article, an approach based on the divide-and-conquer methodology is proposed to facilitate its learning. The proposed strategy is designed to sequentially explain and evaluate the different concepts involved in hypothesis testing, ensuring that a new concept is not presented until the previous one has been fully assimilated. The proposed approach, which contains several gamification elements (i.e., points or a leader-board), is implemented into an application via a modern game engine. The usefulness of the proposed approach was assessed in an experiment in which 89 first-year students enrolled in the Statistics course within the Industrial Engineering degree participated. Based on the results of a test aimed at evaluating the acquired knowledge, it was observed that students who used the developed application based on the proposed approach obtained statistically significant higher scores than those that attended a traditional class (p-value < 0.001), regardless of whether they used the learning tool before or after the traditional class. In addition, the responses provided by the students who participated in the study to a test of satisfaction showed their high satisfaction with the application and their interest in the promotion of these tools. However, despite the good results, they also considered that these learning tools should be considered as a complement to the master class rather than a replacement. Full article

The aim of the paper is to generalize results by Sikorska on some functional equations for set-valued functions. In the paper, a tool is described for solving a generalized type of an integral-functional equation for a set-valued function $F:X\to cc\left(Y\right)$, where X is a real vector space and Y is a locally convex real linear metric space with an invariant metric. Most general results are described in the case of a compact topological group G equipped with the right-invariant Haar measure acting on X. Further results are found if the group G is finite or Y is Asplund space. The main results are applied to an example where $X={\mathbb{R}}^2$ and $Y={\mathbb{R}}^n$, $n\in \mathbb{N}$, and G is the unitary group $U\left(1\right)$. Full article

Computational cardiology is rapidly becoming the gold standard for innovative medical treatments and device development. Despite a worldwide effort in mathematical and computational modeling research, the complexity and intrinsic multiscale nature of the heart still limit our predictability power raising the question of the optimal modeling choice for large-scale whole-heart numerical investigations. We propose an extended numerical analysis among two different electrophysiological modeling approaches: a simplified phenomenological one and a detailed biophysical one. To achieve this, we considered three-dimensional healthy and infarcted swine heart geometries. Heterogeneous electrophysiological properties, fine-tuned DT-MRI -based anisotropy features, and non-conductive ischemic regions were included in a custom-built finite element code. We provide a quantitative comparison of the electrical behaviors during steady pacing and sustained ventricular fibrillation for healthy and diseased cases analyzing cardiac arrhythmias dynamics. Action potential duration (APD) restitution distributions, vortex filament counting, and pseudo-electrocardiography (ECG) signals were numerically quantified, introducing a novel statistical description of restitution patterns and ventricular fibrillation sustainability. Computational cost and scalability associated with the two modeling choices suggests that ventricular fibrillation signatures are mainly controlled by anatomy and structural parameters, rather than by regional restitution properties. Finally, we discuss limitations and translational perspectives of the different modeling approaches in view of large-scale whole-heart in silico studies. Full article

We study properties of generalized solutions of the Dirichlet–Robin problem for an elasticity system in the exterior of a compact, as well as the asymptotic behavior of solutions of this mixed problem at infinity, with the condition that the energy integral with the weight ${\left|x\right|}^a$ is finite. Depending on the value of the parameter a, we have proved uniqueness (or non-uniqueness) theorems for the mixed Dirichlet–Robin problem, and also given exact formulas for the dimension of the space of solutions. The main method for studying the problem under consideration is the variational principle, which assumes the minimization of the corresponding functional in the class of admissible functions. Full article

Simheuristics combine metaheuristics with simulation in order to solve the optimization problems with stochastic elements. This paper introduces the concept of fuzzy simheuristics, which extends the simheuristics approach by making use of fuzzy techniques, thus allowing us to tackle optimization problems under a more general scenario, which includes uncertainty elements of both stochastic and non-stochastic nature. After reviewing the related work, the paper discusses, in detail, how the optimization, simulation, and fuzzy components can be efficiently integrated. In order to illustrate the potential of fuzzy simheuristics, we consider the team orienteering problem (TOP) under an uncertainty scenario, and perform a series of computational experiments. The obtained results show that our proposed approach is not only able to generate competitive solutions for the deterministic version of the TOP, but, more importantly, it can effectively solve more realistic TOP versions, including stochastic and other uncertainty elements. Full article

In this paper, we give an affirmative answer to an open question posed recently by Mlaiki et al. As a consequence of our results, we get some known results in the literature. We also give an application of our results to the existence of a solution of nonlinear fractional differential equations. Full article

We study some topological properties of the class of supermodular n-quasi-copulas and check that the topological size of the Dedekind–MacNeille completion of the set of n-copulas is small, in terms of the Baire category, in the Dedekind–MacNeille completion of the set of the supermodular n-quasi-copulas, and in turn, this set and the set of n-copulas are small in the set of n-quasi-copulas. Full article

This paper presents an easy-to-apply methodology that allows obtaining the permeability index and the initial hydraulic conductivity of clayey soils, basic constitutive parameters in non-linear models of consolidation, based on the laboratory oedometer test. For this, the data of the void ratio, compressibility index and characteristic consolidation time are taken from the test and, as an inverse problem, the constitutive permeability parameters sought are determined by applying the universal solutions of the characteristic time for a general non-linear consolidation model with constitutive relations void ratio-effective soil stress and hydraulic conductivity-void ratio of logarithmic type. The application protocol of the inverse problem is described in detail and illustrated by a series of applications carried out on real laboratory data belonging to two different soils. The influence that errors in laboratory parameter measurements can have on the final values of the permeability index and initial hydraulic conductivity is studied, showing the maximum deviations that may appear and, by last, the precision of the results obtained. Full article

The human lymphatic system (HLS) is a complex network of lymphatic organs linked through the lymphatic vessels. We present a graph theory-based approach to model and analyze the human lymphatic network. Two different methods of building a graph are considered: the method using anatomical data directly and the method based on a system of rules derived from structural analysis of HLS. A simple anatomical data-based graph is converted to an oriented graph by quantifying the steady-state fluid balance in the lymphatic network with the use of the Poiseuille equation in vessels and the mass conservation at vessel junctions. A computational algorithm for the generation of the rule-based random graph is developed and implemented. Some fundamental characteristics of the two types of HLS graph models are analyzed using different metrics such as graph energy, clustering, robustness, etc. Full article

The social adoption of change is usually hard because in reality, forces opposing the social adoption of change manifest. This situation of organizational conflict corresponds to the case where two competing groups of influential agents (“promoters” versus “adversaries” of change) operate concurrently within the same organizational network. We model and explore the co-evolution of interpersonal ties and attitudes in the presence of conflict, taking into account explicitly the microscopic “agent-to-agent” interactions. In this perspective, we propose a new ties-attitudes co-evolution model where the diffusion of attitudes depends on the weights and the evolution of weights is formulated as a “learning mechanism” (weight updates depend on the previous values of both weights and attitudes). As a result, the co-evolution is intrinsic/endogenous. We simulate representative scenarios of conflict in 4 real organizational networks. In order to formulate structural balance in directed networks, we extended Heider’s definition of balance considering directed triangles. The evolution of balance involves two stages: first, negative links pop up disorderly and destroy balance, but after some time, as new negative links are formed, a “new” balance is re-established. This “new” balance is emerging concurrently with the polarization of attitudes or domination of one attitude. Moreover, same-minded agents are positively linked and different-minded agents are negatively-linked. This macroscopic self-organization of the system is due only to agent-to-agent interactions, involving feedbacks on weight updates at the local microscopic level. Full article

In this paper we study flag codes on ${\mathbb{F}}_q^n$, being ${\mathbb{F}}_q$ the finite field with q elements. Special attention is given to the connection between the parameters and properties of a flag code and the ones of a family of constant dimension codes naturally associated to it (the projected codes). More precisely, we focus on consistent flag codes, that is, flag codes whose distance and size are completely determined by their projected codes. We explore some aspects of this family of codes and present examples of them by generalizing the concepts of equidistant and sunflower subspace code to the flag codes setting. Finally, we present a decoding algorithm for consistent flag codes that fully exploits the consistency condition. Full article

Monitoring the distribution of vehicles across the city is of great importance for urban traffic control. In particular, information on the number of vehicles entering and leaving a city, or moving between urban areas, gives a valuable estimate on potential bottlenecks and congestions. The possibility of predicting such flows in advance is even more beneficial, allowing for timely traffic management strategies and targeted congestion warnings. Our work is inserted in the context of short-term forecasting, aiming to predict rapid changes and sudden variations in the traffic volume, beyond the general trend. Moreover, it concurrently targets multiple locations in the city, providing an instant prediction outcome comprising the future distribution of vehicles across several urban locations. Specifically, we propose a multi-target deep learning regressor for simultaneous predictions of traffic volumes, in multiple entry and exit points among city neighborhoods. The experiment focuses on an hourly forecasting of the amount of vehicles accessing and moving between New York City neighborhoods through the Metropolitan Transportation Authority (MTA) bridges and tunnels. By leveraging a single training process for all location points, and an instant one-step volume inference for every location at each time update, our sequential modeling approach is able to grasp rapid variations in the time series and process the collective information of all entry and exit points, whose distinct predicted values are outputted at once. The multi-target model, based on long short-term memory (LSTM) recurrent neural network layers, was tested on a real-world dataset, achieving an average prediction error of 7% and demonstrating its feasibility for short-term spatially-distributed urban traffic forecasting. Full article

In the paper, we consider a multi-server retrial queueing system with setup time which is motivated by applications in power-saving data centers with the ON-OFF policy, where an idle server is immediately turned off and an off server is set up upon arrival of a customer. Customers that find all the servers busy join the orbit and retry for service after an exponentially distributed time. For this model, we derive the stability condition which depends on the setup time and turns out to be more strict than that of the corresponding model with an infinite buffer which is independent of the setup time. We propose asymptotic methods to analyze the system under the condition that the delay in the orbit is extremely long. We show that the scaled-number of customers in the orbit converges to a diffusion process. Using this diffusion limit, we obtain approximations for the steady-state probability distribution of the number of busy servers and that of the number of customers in the orbit. We verify the accuracy of the approximations by simulations and numerical analysis. Numerical results show that the retrial system under the limiting condition consumes more energy than that with an infinite buffer in front of the servers. Full article

In this paper, we consider a nonlinear fractional differential equation. This equation takes the form of the Bernoulli differential equation, where we use the Caputo fractional derivative of non-integer order instead of the first-order derivative. The paper proposes an exact solution for this equation, in which coefficients are power law functions. We also give conditions for the existence of the exact solution for this non-linear fractional differential equation. The exact solution of the fractional logistic differential equation with power law coefficients is also proposed as a special case of the proposed solution for the Bernoulli fractional differential equation. Some applications of the Bernoulli fractional differential equation to describe dynamic processes with power law memory in physics and economics are suggested. Full article