ALBERT

Description: Mathematics, Vol. 5, Pages 43: On Minimal Covolume Hyperbolic Lattices Mathematics doi: 10.3390/math5030043 Authors: Ruth Kellerhals We study lattices with a non-compact fundamental domain of small volume in hyperbolic space H n . First, we identify the arithmetic lattices in Isom + H n of minimal covolume for even n up to 18. Then, we discuss the related problem in higher odd dimensions and provide solutions for n = 11 and n = 13 in terms of the rotation subgroup of certain Coxeter pyramid groups found by Tumarkin. The results depend on the work of Belolipetsky and Emery, as well as on the Euler characteristic computation for hyperbolic Coxeter polyhedra with few facets by means of the program CoxIter developed by Guglielmetti. This work complements the survey about hyperbolic orbifolds of minimal volume.

Description: Mathematics, Vol. 5, Pages 45: Fusion Estimation from Multisensor Observations with Multiplicative Noises and Correlated Random Delays in Transmission Mathematics doi: 10.3390/math5030045 Authors: Raquel Caballero-Águila Aurora Hermoso-Carazo Josefa Linares-Pérez In this paper, the information fusion estimation problem is investigated for a class of multisensor linear systems affected by different kinds of stochastic uncertainties, using both the distributed and the centralized fusion methodologies. It is assumed that the measured outputs are perturbed by one-step autocorrelated and cross-correlated additive noises, and also stochastic uncertainties caused by multiplicative noises and randomly missing measurements in the sensor outputs are considered. At each sampling time, every sensor output is sent to a local processor and, due to some kind of transmission failures, one-step correlated random delays may occur. Using only covariance information, without requiring the evolution model of the signal process, a local least-squares (LS) filter based on the measurements received from each sensor is designed by an innovation approach. All these local filters are then fused to generate an optimal distributed fusion filter by a matrix-weighted linear combination, using the LS optimality criterion. Moreover, a recursive algorithm for the centralized fusion filter is also proposed and the accuracy of the proposed estimators, which is measured by the estimation error covariances, is analyzed by a simulation example.

Description: Mathematics, Vol. 5, Pages 41: On the Duality of Regular and Local Functions Mathematics doi: 10.3390/math5030041 Authors: Jens Fischer In this paper, we relate Poisson’s summation formula to Heisenberg’s uncertainty principle. They both express Fourier dualities within the space of tempered distributions and these dualities are also inverse of each other. While Poisson’s summation formula expresses a duality between discretization and periodization, Heisenberg’s uncertainty principle expresses a duality between regularization and localization. We define regularization and localization on generalized functions and show that the Fourier transform of regular functions are local functions and, vice versa, the Fourier transform of local functions are regular functions.

Description: In this article, Müntz spaces M Λ , C of continuous functions supplied with the absolute maximum norm are considered. An existence of Schauder bases in Müntz spaces M Λ , C is investigated. Moreover, Fourier series approximation of functions in Müntz spaces M Λ , C is studied.

Description: Mathematics, Vol. 5, Pages 46: The Catastrophe of Electric Vehicle Sales Mathematics doi: 10.3390/math5030046 Authors: Timothy Sands Electric vehicles have undergone a recent faddy trend in the United States and Europe, and several recent publications trumpet the continued rise of electric vehicles citing steadily-climbing monthly vehicle sales. The broad purpose of this study is to examine this optimism with some degree of mathematical rigor. Specifically, the methodology will use catastrophe theory to explore the possibility of a sudden, seemingly-unexplainable crash in vehicle sales. The study begins by defining optimal system equations that well-model the available sales data. Next, these optimal models are used to investigate the potential response to a slow dynamic acting on the relatively faster dynamic of the optimal system equations. Catastrophe theory indicates a potential sudden crash in sales when a slow dynamic is at-work. It is noteworthy that the prediction can be made even while sales are increasing.

Description: Mathematics, Vol. 5, Pages 42: On the Uniqueness Results and Value Distribution of Meromorphic Mappings Mathematics doi: 10.3390/math5030042 Authors: Rahman Ullah Xiao-Min Li Faiz Faizullah Hong-Xun Yi Riaz Khan This research concentrates on the analysis of meromorphic mappings. We derived several important results for value distribution of specific difference polynomials of meromorphic mappings, which generalize the work of Laine and Yang. In addition, we proved uniqueness theorems of meromorphic mappings. The difference polynomials of these functions have the same fixed points or share a nonzero value. This extends the research work of Qi, Yang and Liu, where they used the finite ordered meromorphic mappings.

Description: Mathematics, Vol. 5, Pages 44: Topics of Measure Theory on Infinite Dimensional Spaces Mathematics doi: 10.3390/math5030044 Authors: José Velhinho This short review is devoted to measures on infinite dimensional spaces. We start by discussing product measures and projective techniques. Special attention is paid to measures on linear spaces, and in particular to Gaussian measures. Transformation properties of measures are considered, as well as fundamental results concerning the support of the measure.

Description: Finite trigonometric sums occur in various branches of physics, mathematics, and their applications. These sums may contain various powers of one or more trigonometric functions. Sums with one trigonometric function are known; however, sums with products of trigonometric functions can become complicated, and may not have a simple expression in a number of cases. Some of these sums have interesting properties, and can have amazingly simple values. However, only some of them are available in the literature. We obtain a number of such sums using the method of residues.

Description: A hypergraph is the most developed tool for modeling various practical problems in different fields, including computer sciences, biological sciences, social networks and psychology. Sometimes, given data in a network model are based on bipolar information rather than one sided. To deal with such types of problems, we use mathematical models that are based on bipolar fuzzy (BF) sets. In this research paper, we introduce the concept of BF directed hypergraphs. We describe certain operations on BF directed hypergraphs, including addition, multiplication, vertex-wise multiplication and structural subtraction. We introduce the concept of B = ( m + , m − ) -tempered BF directed hypergraphs and investigate some of their properties. We also present an algorithm to compute the minimum arc length of a BF directed hyperpath.

Description: In this paper, we have proved a metrization theorem that gives the sufficient conditions for a uniform IP-loop X to be metrizable by a left-invariant metric. It is shown that by consideration of topological IP-loop dual to X we obtain an analogical theorem for the case of the right-invariant metric.

Description: The asset flow differential equation (AFDE) is the mathematical model that plays an essential role for planning to predict the financial behavior in the market. In this paper, we introduce the fractional asset flow differential equations (FAFDEs) based on the Liouville-Caputo derivative. We prove the existence and uniqueness of a solution for the FAFDEs. Furthermore, the stability analysis of the model is investigated and the numerical simulation is accordingly performed to support the proposed model.

Description: Lie symmetries and their Lie group transformations for a class of Timoshenko systems are presented. The class considered is the class of nonlinear Timoshenko systems of partial differential equations (PDEs). An optimal system of one-dimensional sub-algebras of the corresponding Lie algebra is derived. All possible invariant variables of the optimal system are obtained. The corresponding reduced systems of ordinary differential equations (ODEs) are also provided. All possible non-similar invariant conditions prescribed on invariant surfaces under symmetry transformations are given. As an application, explicit solutions of the system are given where the beam is hinged at one end and free at the other end.

Description: In this article, we show how to use the first and second Minkowski Theorems and some Diophantine geometry to bound explicitly the height of the points of rank N - 1 on transverse curves in E N , where E is an elliptic curve without Complex Multiplication (CM). We then apply our result to give a method for finding the rational points on such curves, when E has Q -rank ≤ N - 1 . We also give some explicit examples. This result generalises from rank 1 to rank N - 1 previous results of S. Checcoli, F. Veneziano and the author.

Description: The contribution of this article is quadruple. It (1) unifies various schemes of premodels/models including situations such as presheaves/sheaves, sheaves/flabby sheaves, prespectra/ Ω -spectra, simplicial topological spaces/(complete) Segal spaces, pre-localised rings/localised rings, functors in categories/strong stacks and, to some extent, functors from a limit sketch to a model category versus the homotopical models for the limit sketch; (2) provides a general construction from the premodels to the models; (3) proposes technics that allow one to assess the nature of the universal properties associated with this construction; (4) shows that the obtained localisation admits a particular presentation, which organises the structural and relational information into bundles of data. This presentation is obtained via a process called an elimination of quotients and its aim is to facilitate the handling of the relational information appearing in the construction of higher dimensional objects such as weak ( ω , n ) -categories, weak ω -groupoids and higher moduli stacks.

Description: Mathematics, Vol. 5, Pages 39: Confidence Intervals for Mean and Difference between Means of Normal Distributions with Unknown Coefficients of Variation Mathematics doi: 10.3390/math5030039 Authors: Warisa Thangjai Suparat Niwitpong Sa-Aat Niwitpong This paper proposes confidence intervals for a single mean and difference of two means of normal distributions with unknown coefficients of variation (CVs). The generalized confidence interval (GCI) approach and large sample (LS) approach were proposed to construct confidence intervals for the single normal mean with unknown CV. These confidence intervals were compared with existing confidence interval for the single normal mean based on the Student’s t-distribution (small sample size case) and the z-distribution (large sample size case). Furthermore, the confidence intervals for the difference between two normal means with unknown CVs were constructed based on the GCI approach, the method of variance estimates recovery (MOVER) approach and the LS approach and then compared with the Welch–Satterthwaite (WS) approach. The coverage probability and average length of the proposed confidence intervals were evaluated via Monte Carlo simulation. The results indicated that the GCIs for the single normal mean and the difference of two normal means with unknown CVs are better than the other confidence intervals. Finally, three datasets are given to illustrate the proposed confidence intervals.

Description: Mathematics, Vol. 5, Pages 40: Integral Representations of the Catalan Numbers and Their Applications Mathematics doi: 10.3390/math5030040 Authors: Feng Qi Bai-Ni Guo In the paper, the authors survey integral representations of the Catalan numbers and the Catalan–Qi function, discuss equivalent relations between these integral representations, supply alternative and new proofs of several integral representations, collect applications of some integral representations, and present sums of several power series whose coefficients involve the Catalan numbers.

Description: Mathematics, Vol. 5, Pages 52: Solutions Modulo p of Gauss–Manin Differential Equations for Multidimensional Hypergeometric Integrals and Associated Bethe Ansatz Mathematics doi: 10.3390/math5040052 Authors: Alexander Varchenko We consider the Gauss–Manin differential equations for hypergeometric integrals associated with a family of weighted arrangements of hyperplanes moving parallel to themselves. We reduce these equations modulo a prime integer p and construct polynomial solutions of the new differential equations as p-analogs of the initial hypergeometric integrals. In some cases, we interpret the p-analogs of the hypergeometric integrals as sums over points of hypersurfaces defined over the finite field Fp. This interpretation is similar to the classical interpretation by Yu. I. Manin of the number of points on an elliptic curve depending on a parameter as a solution of a Gauss hypergeometric differential equation. We discuss the associated Bethe ansatz.

Description: Mathematics, Vol. 5, Pages 48: Least-Squares Solution of Linear Differential Equations Mathematics doi: 10.3390/math5040048 Authors: Daniele Mortari This study shows how to obtain least-squares solutions to initial value problems (IVPs), boundary value problems (BVPs), and multi-value problems (MVPs) for nonhomogeneous linear differential equations (DEs) with nonconstant coefficients of any order. However, without loss of generality, the approach has been applied to second-order DEs. The proposed method has two steps. The first step consists of writing a constrained expression, that has the DE constraints embedded. These kind of expressions are given in terms of a new unknown function, g ( t ) , and they satisfy the constraints, no matter what g ( t ) is. The second step consists of expressing g ( t ) as a linear combination of m independent known basis functions. Specifically, orthogonal polynomials are adopted for the basis functions. This choice requires rewriting the DE and the constraints in terms of a new independent variable, x ∈ [ − 1 , + 1 ] . The procedure leads to a set of linear equations in terms of the unknown coefficients of the basis functions that are then computed by least-squares. Numerical examples are provided to quantify the solutions’ accuracy for IVPs, BVPs and MVPs. In all the examples provided, the least-squares solution is obtained with machine error accuracy.

Description: Mathematics, Vol. 5, Pages 49: An Optimal Control Approach for the Treatment of Solid Tumors with Angiogenesis Inhibitors Mathematics doi: 10.3390/math5040049 Authors: Adam Glick Antonio Mastroberardino Cancer is a disease of unregulated cell growth that is estimated to kill over 600,000 people in the United States in 2017 according to the National Institute of Health. While there are several therapies to treat cancer, tumor resistance to these therapies is a concern. Drug therapies have been developed that attack proliferating endothelial cells instead of the tumor in an attempt to create a therapy that is resistant to resistance in contrast to other forms of treatment such as chemotherapy and radiation therapy. In this study, a two-compartment model in terms of differential equations is presented in order to determine the optimal protocol for the delivery of anti-angiogenesis therapy. Optimal control theory is applied to the model with a range of anti-angiogenesis doses to determine optimal doses to minimize tumor volume at the end of a two week treatment and minimize drug toxicity to the patient. Applying a continuous optimal control protocol to our model of angiogenesis and tumor cell growth shows promising results for tumor control while minimizing the toxicity to the patients. By investigating a variety of doses, we determine that the optimal angiogenesis inhibitor dose is in the range of 10–20 mg/kg. In this clinically useful range of doses, good tumor control is achieved for a two week treatment period. This work shows that varying the toxicity of the treatment to the patient will change the optimal dosing scheme but tumor control can still be achieved.

Description: Mathematics, Vol. 5, Pages 50: The Stability of Parabolic Problems with Nonstandard p(x, t)-Growth Mathematics doi: 10.3390/math5040050 Authors: André Erhardt In this paper, we study weak solutions to the following nonlinear parabolic partial differential equation ∂ t u − div a ( x , t , ∇ u ) + λ ( | u | p ( x , t ) − 2 u ) = 0 in Ω T , where λ ≥ 0 and ∂ t u denote the partial derivative of u with respect to the time variable t, while ∇ u denotes the one with respect to the space variable x. Moreover, the vector-field a ( x , t , · ) satisfies certain nonstandard p ( x , t ) -growth and monotonicity conditions. In this manuscript, we establish the existence of a unique weak solution to the corresponding Dirichlet problem. Furthermore, we prove the stability of this solution, i.e., we show that two weak solutions with different initial values are controlled by these initial values.

Description: Mathematics, Vol. 5, Pages 47: New Analytical Technique for Solving a System of Nonlinear Fractional Partial Differential Equations Mathematics doi: 10.3390/math5040047 Authors: Hayman Thabet Subhash Kendre Dimplekumar Chalishajar This paper introduces a new analytical technique (NAT) for solving a system of nonlinear fractional partial differential equations (NFPDEs) in full general set. Moreover, the convergence and error analysis of the proposed technique is shown. The approximate solutions for a system of NFPDEs are easily obtained by means of Caputo fractional partial derivatives based on the properties of fractional calculus. However, analytical and numerical traveling wave solutions for some systems of nonlinear wave equations are successfully obtained to confirm the accuracy and efficiency of the proposed technique. Several numerical results are presented in the format of tables and graphs to make a comparison with results previously obtained by other well-known methods.

Description: Mathematics, Vol. 5, Pages 51: Euclidean Submanifolds via Tangential Components of Their Position Vector Fields Mathematics doi: 10.3390/math5040051 Authors: Bang-Yen Chen The position vector field is the most elementary and natural geometric object on a Euclidean submanifold. The position vector field plays important roles in physics, in particular in mechanics. For instance, in any equation of motion, the position vector x (t) is usually the most sought-after quantity because the position vector field defines the motion of a particle (i.e., a point mass): its location relative to a given coordinate system at some time variable t. This article is a survey article. The purpose of this article is to survey recent results of Euclidean submanifolds associated with the tangential components of their position vector fields. In the last section, we present some interactions between torqued vector fields and Ricci solitons.

Description: Mathematics, Vol. 5, Pages 66: Generalized Langevin Equation and the Prabhakar Derivative Mathematics doi: 10.3390/math5040066 Authors: Trifce Sandev We consider a generalized Langevin equation with regularized Prabhakar derivative operator. We analyze the mean square displacement, time-dependent diffusion coefficient and velocity autocorrelation function. We further introduce the so-called tempered regularized Prabhakar derivative and analyze the corresponding generalized Langevin equation with friction term represented through the tempered derivative. Various diffusive behaviors are observed. We show the importance of the three parameter Mittag-Leffler function in the description of anomalous diffusion in complex media. We also give analytical results related to the generalized Langevin equation for a harmonic oscillator with generalized friction. The normalized displacement correlation function shows different behaviors, such as monotonic and non-monotonic decay without zero-crossings, oscillation-like behavior without zero-crossings, critical behavior, and oscillation-like behavior with zero-crossings. These various behaviors appear due to the friction of the complex environment represented by the Mittag-Leffler and tempered Mittag-Leffler memory kernels. Depending on the values of the friction parameters in the system, either diffusion or oscillations dominate.

Description: Mathematics, Vol. 5, Pages 69: Impact of Parameter Variability and Environmental Noise on the Klausmeier Model of Vegetation Pattern Formation Mathematics doi: 10.3390/math5040069 Authors: Merlin Köhnke Horst Malchow Semi-arid ecosystems made up of patterned vegetation, for instance, are thought to be highly sensitive. This highlights the importance of understanding the dynamics of the formation of vegetation patterns. The most renowned mathematical model describing such pattern formation consists of two partial differential equations and is often referred to as the Klausmeier model. This paper provides analytical and numerical investigations regarding the influence of different parameters, including the so-far not contemplated evaporation, on the long-term model results. Another focus is set on the influence of different initial conditions and on environmental noise, which has been added to the model. It is shown that patterning is beneficial for semi-arid ecosystems, that is, vegetation is present for a broader parameter range. Both parameter variability and environmental noise have only minor impacts on the model results. Increasing mortality has a high, nonlinear impact underlining the importance of further studies in order to gain a sufficient understanding allowing for suitable management strategies of this natural phenomenon.

Description: Mathematics, Vol. 5, Pages 71: Some Types of Subsemigroups Characterized in Terms of Inequalities of Generalized Bipolar Fuzzy Subsemigroups Mathematics doi: 10.3390/math5040071 Authors: Pannawit Khamrot Manoj Siripitukdet In this paper, we introduce a generalization of a bipolar fuzzy (BF) subsemigroup, namely, a ( α 1 , α 2 ; β 1 , β 2 ) -BF subsemigroup. The notions of ( α 1 , α 2 ; β 1 , β 2 ) -BF quasi(generalized bi-, bi-) ideals are discussed. Some inequalities of ( α 1 , α 2 ; β 1 , β 2 ) -BF quasi(generalized bi-, bi-) ideals are obtained. Furthermore, any regular semigroup is characterized in terms of generalized BF semigroups.

Description: Mathematics, Vol. 5, Pages 72: Wavelet Neural Network Model for Yield Spread Forecasting Mathematics doi: 10.3390/math5040072 Authors: Firdous Shah Lokenath Debnath In this study, a hybrid method based on coupling discrete wavelet transforms (DWTs) and artificial neural network (ANN) for yield spread forecasting is proposed. The discrete wavelet transform (DWT) using five different wavelet families is applied to decompose the five different yield spreads constructed at shorter end, longer end, and policy relevant area of the yield curve to eliminate noise from them. The wavelet coefficients are then used as inputs into Levenberg-Marquardt (LM) ANN models to forecast the predictive power of each of these spreads for output growth. We find that the yield spreads constructed at the shorter end and policy relevant areas of the yield curve have a better predictive power to forecast the output growth, whereas the yield spreads, which are constructed at the longer end of the yield curve do not seem to have predictive information for output growth. These results provide the robustness to the earlier results.

Description: Mathematics, Vol. 5, Pages 63: Krylov Implicit Integration Factor Methods for Semilinear Fourth-Order Equations Mathematics doi: 10.3390/math5040063 Authors: Michael Machen Yong-Tao Zhang Implicit integration factor (IIF) methods were developed for solving time-dependent stiff partial differential equations (PDEs) in literature. In [Jiang and Zhang, Journal of Computational [-15]Physics, 253 (2013) 368-388], IIF methods are designed to efficiently solve stiff nonlinear advection–diffusion–reaction (ADR) equations. The methods can be designed for an arbitrary order of accuracy. The stiffness of the system is resolved well, and large-time-step-size computations are achieved. To efficiently calculate large matrix exponentials, a Krylov subspace approximation is directly applied to the IIF methods. In this paper, we develop Krylov IIF methods for solving semilinear fourth-order PDEs. As a result of the stiff fourth-order spatial derivative operators, the fourth-order PDEs have much stricter constraints in time-step sizes than the second-order ADR equations. We analyze the truncation errors of the fully discretized schemes. Numerical examples of both scalar equations and systems in one and higher spatial dimensions are shown to demonstrate the accuracy, efficiency and stability of the methods. Large time-step sizes that are of the same order as the spatial grid sizes have been achieved in the simulations of the fourth-order PDEs.

Description: Mathematics, Vol. 5, Pages 64: On the Inception of Financial Representative Bubbles Mathematics doi: 10.3390/math5040064 Authors: Massimiliano Ferrara Bruno Pansera Francesco Strati In this work, we aim to formalize the inception of representative bubbles giving the condition under which they may arise. We will find that representative bubbles may start at any time, depending on the definition of a behavioral component. This result is at odds with the theory of classic rational bubbles, which are those models that rely on the fulfillment of the transversality condition by which a bubble in a financial asset can arise just at its first trade. This means that a classic rational bubble (differently from our model) cannot follow a cycle since if a bubble exists, it will burst by definition and never arise again.

Description: Mathematics, Vol. 5, Pages 65: Picard’s Iterative Method for Caputo Fractional Differential Equations with Numerical Results Mathematics doi: 10.3390/math5040065 Authors: Rainey Lyons Aghalaya Vatsala Ross Chiquet With fractional differential equations (FDEs) rising in popularity and methods for solving them still being developed, approximations to solutions of fractional initial value problems (IVPs) have great applications in related fields. This paper proves an extension of Picard’s Iterative Existence and Uniqueness Theorem to Caputo fractional ordinary differential equations, when the nonhomogeneous term satisfies the usual Lipschitz’s condition. As an application of our method, we have provided several numerical examples.

Description: Mathematics, Vol. 5, Pages 67: On Edge Irregular Reflexive Labellings for the Generalized Friendship Graphs Mathematics doi: 10.3390/math5040067 Authors: Martin Bača Muhammad Irfan Joe Ryan Andrea Semaničová-Feňovčíková Dushyant Tanna We study an edge irregular reflexive k-labelling for the generalized friendship graphs, also known as flowers (a symmetric collection of cycles meeting at a common vertex), and determine the exact value of the reflexive edge strength for several subfamilies of the generalized friendship graphs.

Description: Mathematics, Vol. 5, Pages 70: Controlling Chaos—Forced van der Pol Equation Mathematics doi: 10.3390/math5040070 Authors: Matthew Cooper Peter Heidlauf Timothy Sands Nonlinear systems are typically linearized to permit linear feedback control design, but, in some systems, the nonlinearities are so strong that their performance is called chaotic, and linear control designs can be rendered ineffective. One famous example is the van der Pol equation of oscillatory circuits. This study investigates the control design for the forced van der Pol equation using simulations of various control designs for iterated initial conditions. The results of the study highlight that even optimal linear, time-invariant (LTI) control is unable to control the nonlinear van der Pol equation, but idealized nonlinear feedforward control performs quite well after an initial transient effect of the initial conditions. Perhaps the greatest strength of ideal nonlinear control is shown to be the simplicity of analysis. Merely equate coefficients order-of-differentiation insures trajectory tracking in steady-state (following dissipation of transient effects of initial conditions), meanwhile the solution of the time-invariant linear-quadratic optimal control problem with infinite time horizon is needed to reveal constant control gains for a linear-quadratic regulator. Since analytical development is so easy for ideal nonlinear control, this article focuses on numerical demonstrations of trajectory tracking error.

Description: Mathematics, Vol. 5, Pages 68: Channel Engineering for Nanotransistors in a Semiempirical Quantum Transport Model Mathematics doi: 10.3390/math5040068 Authors: Ulrich Wulf Jan Kučera Hans Richter Manfred Horstmann Maciej Wiatr Jan Höntschel One major concern of channel engineering in nanotransistors is the coupling of the conduction channel to the source/drain contacts. In a number of previous publications, we have developed a semiempirical quantum model in quantitative agreement with three series of experimental transistors. On the basis of this model, an overlap parameter 0 ≤ C ≤ 1 can be defined as a criterion for the quality of the contact-to-channel coupling: A high level of C means good matching between the wave functions in the source/drain and in the conduction channel associated with a low contact-to-channel reflection. We show that a high level of C leads to a high saturation current in the ON-state and a large slope of the transfer characteristic in the OFF-state. Furthermore, relevant for future device miniaturization, we analyze the contribution of the tunneling current to the total drain current. It is seen for a device with a gate length of 26 nm that for all gate voltages, the share of the tunneling current becomes small for small drain voltages. With increasing drain voltage, the contribution of the tunneling current grows considerably showing Fowler–Nordheim oscillations. In the ON-state, the classically allowed current remains dominant for large drain voltages. In the OFF-state, the tunneling current becomes dominant.

Description: Mathematics, Vol. 5, Pages 73: Fractional Derivatives, Memory Kernels and Solution of a Free Electron Laser Volterra Type Equation Mathematics doi: 10.3390/math5040073 Authors: Marcello Artioli Giuseppe Dattoli Silvia Licciardi Simonetta Pagnutti The high gain free electron laser (FEL) equation is a Volterra type integro-differential equation amenable for analytical solutions in a limited number of cases. In this note, a novel technique, based on an expansion employing a family of two variable Hermite polynomials, is shown to provide straightforward analytical solutions for cases hardly solvable with conventional means. The possibility of extending the method by the use of expansion using different polynomials (two variable Legendre like) expansion is also discussed.

Description: Mathematics, Vol. 5, Pages 75: Geometric Structure of the Classical Lagrange-d’Alambert Principle and its Application to Integrable Nonlinear Dynamical Systems Mathematics doi: 10.3390/math5040075 Authors: Anatolij Prykarpatski Oksana Hentosh Yarema Prykarpatsky The classical Lagrange-d’Alembert principle had a decisive influence on formation of modern analytical mechanics which culminated in modern Hamilton and Poisson mechanics. Being mainly interested in the geometric interpretation of this principle, we devoted our review to its deep relationships to modern Lie-algebraic aspects of the integrability theory of nonlinear heavenly type dynamical systems and its so called Lax-Sato counterpart. We have also analyzed old and recent investigations of the classical M. A. Buhl problem of describing compatible linear vector field equations, its general M.G. Pfeiffer and modern Lax-Sato type special solutions. Especially we analyzed the related Lie-algebraic structures and integrability properties of a very interesting class of nonlinear dynamical systems called the dispersionless heavenly type equations, which were initiated by Plebański and later analyzed in a series of articles. As effective tools the AKS-algebraic and related R -structure schemes are used to study the orbits of the corresponding co-adjoint actions, which are intimately related to the classical Lie-Poisson structures on them. It is demonstrated that their compatibility condition coincides with the corresponding heavenly type equations under consideration. It is also shown that all these equations originate in this way and can be represented as a Lax-Sato compatibility condition for specially constructed loop vector fields on the torus. Typical examples of such heavenly type equations, demonstrating in detail their integrability via the scheme devised herein, are presented.

Description: Mathematics, Vol. 5, Pages 74: Acting Semicircular Elements Induced by Orthogonal Projections on Von-Neumann-Algebras Mathematics doi: 10.3390/math5040074 Authors: Ilwoo Cho In this paper, we construct a free semicircular family induced by Z -many mutually-orthogonal projections, and construct Banach *-probability spaces containing the family, called the free filterizations. By acting a free filterization on fixed von Neumann algebras, we construct the corresponding Banach *-probability spaces, called affiliated free filterizations. We study free-probabilistic properties on such new structures, determined by both semicircularity and free-distributional data on von Neumann algebras. In particular, we study how the freeness on free filterizations, and embedded freeness conditions on fixed von Neumann algebras affect free-distributional data on affiliated free filterizations.

Description: Mathematics, Vol. 5, Pages 55: On the Achievable Stabilization Delay Margin for Linear Plants with Time-Varying Delays Mathematics doi: 10.3390/math5040055 Authors: Jing Zhu The paper contributes to stabilization problems of linear systems subject to time-varying delays. Drawing upon small gain criteria and robust analysis techniques, upper and lower bounds on the largest allowable time-varying delay are developed by using bilinear transformation and rational approximates. The results achieved are not only computationally efficient but also conceptually appealing. Furthermore, analytical expressions of the upper and lower bounds are derived for specific situations that demonstrate the dependence of those bounds on the unstable poles and nonminumum phase zeros of systems.

Description: Mathematics, Vol. 5, Pages 60: Graph Structures in Bipolar Neutrosophic Environment Mathematics doi: 10.3390/math5040060 Authors: Muhammad Akram Muzzamal Sitara Florentin Smarandache A bipolar single-valued neutrosophic (BSVN) graph structure is a generalization of a bipolar fuzzy graph. In this research paper, we present certain concepts of BSVN graph structures. We describe some operations on BSVN graph structures and elaborate on these with examples. Moreover, we investigate some related properties of these operations.

Description: Mathematics, Vol. 5, Pages 54: An Investigation of Radial Basis Function-Finite Difference (RBF-FD) Method for Numerical Solution of Elliptic Partial Differential Equations Mathematics doi: 10.3390/math5040054 Authors: Suranon Yensiri Ruth Skulkhu The Radial Basis Function (RBF) method has been considered an important meshfree tool for numerical solutions of Partial Differential Equations (PDEs). For various situations, RBF with infinitely differentiable functions can provide accurate results and more flexibility in the geometry of computation domains than traditional methods such as finite difference and finite element methods. However, RBF does not suit large scale problems, and, therefore, a combination of RBF and the finite difference (RBF-FD) method was proposed because of its own strengths not only on feasibility and computational cost, but also on solution accuracy. In this study, we try the RBF-FD method on elliptic PDEs and study the effect of it on such equations with different shape parameters. Most importantly, we study the solution accuracy after additional ghost node strategy, preconditioning strategy, regularization strategy, and floating point arithmetic strategy. We have found more satisfactory accurate solutions in most situations than those from global RBF, except in the preconditioning and regularization strategies.

Description: Mathematics, Vol. 5, Pages 57: The Theory of Connections: Connecting Points Mathematics doi: 10.3390/math5040057 Authors: Daniele Mortari This study introduces a procedure to obtain all interpolating functions, y = f ( x ) , subject to linear constraints on the function and its derivatives defined at specified values. The paper first shows how to express these interpolating functions passing through a single point in three distinct ways: linear, additive, and rational. Then, using the additive formalism, interpolating functions with linear constraints on one, two, and n points are introduced as well as those satisfying relative constraints. In particular, for expressions passing through n points, a generalization of the Waring’s interpolation form is introduced. An alternative approach to derive additive constraint interpolating expressions is introduced requiring the inversion of a matrix with dimensions equally the number of constraints. Finally, continuous and discontinuous interpolating periodic functions passing through a set of points with specified periods are provided. This theory has already been applied to obtain least-squares solutions of initial and boundary value problems applied to nonhomogeneous linear differential equations with nonconstant coefficients.

Description: Mathematics, Vol. 5, Pages 59: Invariant Solutions for a Class of Perturbed Nonlinear Wave Equations Mathematics doi: 10.3390/math5040059 Authors: Waheed Ahmed F. Zaman Khairul Saleh Approximate symmetries of a class of perturbed nonlinear wave equations are computed using two newly-developed methods. Invariant solutions associated with the approximate symmetries are constructed for both methods. Symmetries and solutions are compared through discussing the advantages and disadvantages of each method.

Description: Mathematics, Vol. 5, Pages 84: Extending the Characteristic Polynomial for Characterization of C20 Fullerene Congeners Mathematics doi: 10.3390/math5040084 Authors: Dan-Marian Joiţa Lorentz Jäntschi The characteristic polynomial (ChP) has found its use in the characterization of chemical compounds since Hückel’s method of molecular orbitals. In order to discriminate the atoms of different elements and different bonds, an extension of the classical definition is required. The extending characteristic polynomial (EChP) family of structural descriptors is introduced in this article. Distinguishable atoms and bonds in the context of chemical structures are considered in the creation of the family of descriptors. The extension finds its uses in problems requiring discrimination among same-patterned graph representations of molecules as well as in problems involving relations between the structure and the properties of chemical compounds. The ability of the EChP to explain two properties, namely, area and volume, is analyzed on a sample of C20 fullerene congeners. The results have shown that the EChP-selected descriptors well explain the properties.

Description: Mathematics, Vol. 6, Pages 3: Letnikov vs. Marchaud: A Survey on Two Prominent Constructions of Fractional Derivatives Mathematics doi: 10.3390/math6010003 Authors: Sergei Rogosin Maryna Dubatovskaya In this survey paper, we analyze two constructions of fractional derivatives proposed by Aleksey Letnikov (1837–1888) and by André Marchaud (1887–1973), respectively. These derivatives play very important roles in Fractional Calculus and its applications.

Description: Mathematics, Vol. 5, Pages 83: Isomorphic Classification of Reflexive Müntz Spaces Mathematics doi: 10.3390/math5040083 Authors: Sergey Ludkowski The article is devoted to reflexive Müntz spaces M Λ , p of L p functions with 1 < p < ∞ . The Stieltjes transform and a potential transform are studied for these spaces. Isomorphisms of the reflexive Müntz spaces fulfilling the gap and Müntz conditions are investigated.

Description: Mathematics, Vol. 6, Pages 1: A Numerical Solution of Fractional Lienard’s Equation by Using the Residual Power Series Method Mathematics doi: 10.3390/math6010001 Authors: Muhammed Syam In this paper, we investigate a numerical solution of Lienard’s equation. The residual power series (RPS) method is implemented to find an approximate solution to this problem. The proposed method is a combination of the fractional Taylor series and the residual functions. Numerical and theoretical results are presented.

Description: Mathematics, Vol. 6, Pages 2: An Iterative Method for Solving a Class of Fractional Functional Differential Equations with “Maxima” Mathematics doi: 10.3390/math6010002 Authors: Khadidja Nisse Lamine Nisse In the present work, we deal with nonlinear fractional differential equations with “maxima” and deviating arguments. The nonlinear part of the problem under consideration depends on the maximum values of the unknown function taken in time-dependent intervals. Proceeding by an iterative approach, we obtain the existence and uniqueness of the solution, in a context that does not fit within the framework of fixed point theory methods for the self-mappings, frequently used in the study of such problems. An example illustrating our main result is also given.

Description: Mathematics, Vol. 5, Pages 58: Dynamics of Amoebiasis Transmission: Stability and Sensitivity Analysis Mathematics doi: 10.3390/math5040058 Authors: Fidele Hategekimana Snehanshu Saha Anita Chaturvedi Compartmental epidemic models are intriguing in the sense that the generic model may explain different kinds of infectious diseases with minor modifications. However, there may exist some ailments that may not fit the generic capsule. Amoebiasis is one such example where transmission through the population demands a more detailed and sophisticated approach, both mathematical and numerical. The manuscript engages in a deep analytical study of the compartmental epidemic model; susceptible-exposed-infectious-carrier-recovered-susceptible (SEICRS), formulated for Amoebiasis. We have shown that the model allows the single disease-free equilibrium (DFE) state if R 0 , the basic reproduction number, is less than unity and the unique endemic equilibrium (EE) state if R 0 is greater than unity. Furthermore, the basic reproduction number depends uniquely on the input parameters and constitutes a key threshold indicator to portray the general trends of the dynamics of Amoebiasis transmission. We have also shown that R 0 is highly sensitive to the changes in values of the direct transmission rate in contrast to the change in values of the rate of transfer from latent infection to the infectious state. Using the Routh–Hurwitz criterion and Lyapunov direct method, we have proven the conditions for the disease-free equilibrium and the endemic equilibrium states to be locally and globally asymptotically stable. In other words, the conditions for Amoebiasis “die-out” and “infection propagation” are presented.

Description: Mathematics, Vol. 5, Pages 56: A Constructive Method for Standard Borel Fixed Submodules with Given Extremal Betti Numbers Mathematics doi: 10.3390/math5040056 Authors: Marilena Crupi Let S be a polynomial ring in n variables over a field K of any characteristic. Given a strongly stable submodule M of a finitely generated graded free S-module F, we propose a method for constructing a standard Borel-fixed submodule M ˜ of F so that the extremal Betti numbers of M, values as well as positions, are preserved by passing from M to M ˜ . As a result, we obtain a numerical characterization of all possible extremal Betti numbers of any standard Borel-fixed submodule of a finitely generated graded free S-module F.

Description: In this paper, we introduced a new generalization method to solve fractional convection–diffusion equations based on the well-known variational iteration method (VIM) improved by an auxiliary parameter. The suggested method was highly effective in controlling the convergence region of the approximate solution. By solving some fractional convection–diffusion equations with a propounded method and comparing it with standard VIM, it was concluded that complete reliability, efficiency, and accuracy of this method are guaranteed. Additionally, we studied and investigated the convergence of the proposed method, namely the VIM with an auxiliary parameter. We also offered the optimal choice of the auxiliary parameter in the proposed method. It was noticed that the approach could be applied to other models of physics.

Description: Wire coating process is a continuous extrusion process for primary insulation of conducting wires with molten polymers for mechanical strength and protection in aggressive environments. Nylon, polysulfide, low/high density polyethylene (LDPE/HDPE) and plastic polyvinyl chloride (PVC) are the common and important plastic resin used for wire coating. In the current study, wire coating is performed using viscoelastic third grade fluid in the presence of applied magnetic field and porous medium. The governing equations are first modeled and then solved analytically by utilizing the homotopy analysis method (HAM). The convergence of the series solution is established. A numerical technique called ND-solve method is used for comparison and found good agreement. The effect of pertinent parameters on the velocity field and temperature profile is shown with the help of graphs. It is observed that the velocity profiles increase as the value of viscoelastic third grade parameter β increase and decrease as the magnetic parameter M and permeability parameter K increase. It is also observed that the temperature profiles increases as the Brinkman number B r , permeability parameter K , magnetic parameter M and viscoelastic third grade parameter (non-Newtonian parameter) β increase.

Description: A numerical method is proposed for estimating piecewise-constant solutions for Fredholm integral equations of the first kind. Two functionals, namely the weighted total variation (WTV) functional and the simplified Modica-Mortola (MM) functional, are introduced. The solution procedure consists of two stages. In the first stage, the WTV functional is minimized to obtain an approximate solution f TV * . In the second stage, the simplified MM functional is minimized to obtain the final result by using the damped Newton (DN) method with f TV * as the initial guess. The numerical implementation is given in detail, and numerical results of two examples are presented to illustrate the efficiency of the proposed approach.

Description: The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. The additively weighted Harary index H A ( G ) is a modification of the Harary index in which the contributions of vertex pairs are weighted by the sum of their degrees. This new invariant was introduced in (Alizadeh, Iranmanesh and Došlić. Additively weighted Harary index of some composite graphs, Discrete Math, 2013) and they posed the following question: What is the behavior of H A ( G ) when G is a composite graph resulting for example by: splice, link, corona and rooted product? We investigate the additively weighted Harary index for these standard graph products. Then we obtain lower and upper bounds for some of them.

Description: We shall discuss the numerical solution of the Cauchy problem for the fully fractional Fokker-Planck (fFP) equation in connection with Sinc convolution methods. The numerical approximation is based on Caputo and Riesz-Feller fractional derivatives. The use of the transfer function in Laplace and Fourier spaces in connection with Sinc convolutions allow to find exponentially converging computing schemes. Examples using different initial conditions demonstrate the effective computations with a small number of grid points on an infinite spatial domain.

Description: In this paper, we introduce an iterative algorithm for solving the split common fixed point problem for a family of multi-valued quasinonexpansive mappings and totally asymptotically strictly pseudocontractive mappings, as well as for a family of totally quasi-ϕ-asymptotically nonexpansive mappings and k-quasi-strictly pseudocontractive mappings in the setting of Banach spaces. Our results improve and extend the results of Tang et al., Takahashi, Moudafi, Censor et al., and Byrne et al.

Description: In this manuscript, we implement Bohnenblust–Karlin’s fixed point theorem to demonstrate the existence of mild solutions for a class of impulsive fractional integro-differential inclusions (IFIDI) with state-dependent delay (SDD) in Banach spaces. An example is provided to illustrate the obtained abstract results.

Description: Müntz spaces satisfying the Müntz and gap conditions are considered. A Fourier approximation of functions in the Müntz spaces MΛ,p of Lp functions is studied, where 1 〈 p 〈 ∞. It is proven that up to an isomorphism and a change of variables, these spaces are contained in Weil–Nagy’s class. Moreover, the existence of Schauder bases in the Müntz spaces MΛ,p is investigated.

Description: Mathematics, Vol. 5, Pages 82: Multiplicative Structure and Hecke Rings of Generator Matrices for Codes over Quotient Rings of Euclidean Domains Mathematics doi: 10.3390/math5040082 Authors: Hajime Matsui In this study, we consider codes over Euclidean domains modulo their ideals. In the first half of the study, we deal with arbitrary Euclidean domains. We show that the product of generator matrices of codes over the rings mod a and mod b produces generator matrices of all codes over the ring mod a b , i.e., this correspondence is onto. Moreover, we show that if a and b are coprime, then this correspondence is one-to-one, i.e., there exist unique codes over the rings mod a and mod b that produce any given code over the ring mod a b through the product of their generator matrices. In the second half of the study, we focus on the typical Euclidean domains such as the rational integer ring, one-variable polynomial rings, rings of Gaussian and Eisenstein integers, p-adic integer rings and rings of one-variable formal power series. We define the reduced generator matrices of codes over Euclidean domains modulo their ideals and show their uniqueness. Finally, we apply our theory of reduced generator matrices to the Hecke rings of matrices over these Euclidean domains.

Description: A formulation and solution of the discrete-time fractional optimal control problem in terms of the Caputo fractional derivative is presented in this paper. The performance index (PI) is considered in a quadratic form. The necessary and transversality conditions are obtained using a Hamiltonian approach. Both the free and fixed final state cases have been considered. Numerical examples are taken up and their solution technique is presented. Results are produced for different values of α .

Description: In the present article, we introduce the new concept of start point in a directed graph and provide the characterizations required for a directed graph to have a start point. We also define the notion of a self path set valued map and establish its relation with start point in the setting of a metric space endowed with a directed graph. Further, some fixed point theorems for set valued maps have been proven in this context. A version of the Knaster–Tarski theorem has also been established using our results.

Description: In a certain class of graphs, a graph is called minimizing if the least eigenvalue of its adjacency matrix attains the minimum. A connected graph containing two or three cycles is called a bicyclic graph if its number of edges is equal to its number of vertices plus one. Let G 1 , n c and G 2 , n c be the classes of the connected graphs of order n whose complements are bicyclic with exactly two and three cycles, respectively. In this paper, we characterize the unique minimizing graph among all the graphs which belong to G n c = G 1 , n c ∪ G 2 , n c , a class of the connected graphs of order n whose complements are bicyclic.

Description: In the present paper, we consider the large scale Stein matrix equation with a low-rank constant term A X B − X + E F T = 0 . These matrix equations appear in many applications in discrete-time control problems, filtering and image restoration and others. The proposed methods are based on projection onto the extended block Krylov subspace with a Galerkin approach (GA) or with the minimization of the norm of the residual. We give some results on the residual and error norms and report some numerical experiments.

Description: The main objective of this paper is to deal with some properties of interest in two types of fuzzy ordered proximal contractions of cyclic self-mappings T integrated in a pair ( g , T ) of mappings. In particular, g is a non-contractive fuzzy self-mapping, in the framework of non-Archimedean ordered fuzzy complete metric spaces and T is a p -cyclic proximal contraction. Two types of such contractions (so called of type I and of type II) are dealt with. In particular, the existence, uniqueness and limit properties for sequences to optimal fuzzy best proximity coincidence points are investigated for such pairs of mappings.

Description: In this work, we deal with the autonomy issue in the perturbation expansion for the eigenfunctions of a compact Hilbert–Schmidt integral operator. Here, the autonomy points to the perturbation expansion coefficients of the relevant eigenfunction not depending on the perturbation parameter explicitly, but the dependence on this parameter arises from the coordinate change at the zero interval limit. Moreover, the related half interval length is utilized as the perturbation parameter in the perturbative analyses. Thus, the zero interval limit perturbation for solving the eigenproblem under consideration is developed. The aim of this work is to show that the autonomy imposition brings an important restriction on the kernel of the corresponding integral operator, and the constructed perturbation series is not capable of expressing the exact solution approximately unless a specific type of kernel is considered. The general structure for the encountered constraints is revealed, and the specific class of kernels is identified to this end. Error analysis of the developed method is given. These analyses are also supported by certain illustrative implementations involving the kernels, which are and are not in the specific class addressed above. Thus, the efficiency of the developed method is shown, and the relevant analyses are confirmed.

Description: Every finite simple group P can be generated by two of its elements. Pairs of generators for P are available in the Atlas of finite group representations as (not necessarily minimal) permutation representations P . It is unusual, but significant to recognize that a P is a Grothendieck’s “dessin d’enfant” D and that a wealth of standard graphs and finite geometries G —such as near polygons and their generalizations—are stabilized by a D . In our paper, tripods P − D − G of rank larger than two, corresponding to simple groups, are organized into classes, e.g., symplectic, unitary, sporadic, etc. (as in the Atlas). An exhaustive search and characterization of non-trivial point-line configurations defined from small index representations of simple groups is performed, with the goal to recognize their quantum physical significance. All of the defined geometries G ′ s have a contextuality parameter close to its maximal value of one.

Description: The subject known as Programming Contests in the Bachelor’s Degree in Computer Engineering course focuses on solving programming problems frequently met within contests such as the Southwest Europe Regional Contest (SWERC). In order to solve these problems one first needs to model the problem correctly, find the ideal solution, and then be able to program it without making any mistakes in a very short period of time. Leading multinationals such as Google, Apple, IBM, Facebook and Microsoft place a very high value on these abilities when selecting candidates for posts in their companies. In this communication we present some preliminary results of an analysis of the interaction between two optional subjects in the Computer Science Degree course: Programming Contests (PC) and Graphs, Models and Applications (GMA). The results of this analysis enabled us to make changes to some of the contents in GMA in order to better prepare the students to deal with the challenges they have to face in programming contests.

Description: There are two probabilistic algebras: one for classical probability and the other for quantum mechanics. Naturally, it is the relation to the object that decides, as in the case of logic, which algebra is to be used. From a paraconsistent multivalued logic therefore, one can derive a probability theory, adding the correspondence between truth value and fortuity.

Description: In this paper we introduce a new iterative algorithm for approximating fixed points of totally asymptotically quasi-nonexpansive mappings on CAT(0) spaces. We prove a strong convergence theorem under suitable conditions. The result we obtain improves and extends several recent results stated by many others; they also complement many known recent results in the literature. We then provide some numerical examples to illustrate our main result and to display the efficiency of the proposed algorithm.

Description: Mathematics, Vol. 5, Pages 62: Solution of Inhomogeneous Differential Equations with Polynomial Coefficients in Terms of the Green’s Function Mathematics doi: 10.3390/math5040062 Authors: Tohru Morita Ken-ichi Sato The particular solutions of inhomogeneous differential equations with polynomial coefficients in terms of the Green’s function are obtained in the framework of distribution theory. In particular, discussions are given on Kummer’s and the hypergeometric differential equation. Related discussions are given on the particular solution of differential equations with constant coefficients, by the Laplace transform.

Description: Mathematics, Vol. 5, Pages 77: Solving the Lane–Emden Equation within a Reproducing Kernel Method and Group Preserving Scheme Mathematics doi: 10.3390/math5040077 Authors: Mir Hashemi Ali Akgül Mustafa Inc Idrees Mustafa Dumitru Baleanu We apply the reproducing kernel method and group preserving scheme for investigating the Lane–Emden equation. The reproducing kernel method is implemented by the useful reproducing kernel functions and the numerical approximations are given. These approximations demonstrate the preciseness of the investigated techniques.

Description: Mathematics, Vol. 5, Pages 76: On Some New Properties of the Fundamental Solution to the Multi-Dimensional Space- and Time-Fractional Diffusion-Wave Equation Mathematics doi: 10.3390/math5040076 Authors: Yuri Luchko In this paper, some new properties of the fundamental solution to the multi-dimensional space- and time-fractional diffusion-wave equation are deduced. We start with the Mellin-Barnes representation of the fundamental solution that was derived in the previous publications of the author. The Mellin-Barnes integral is used to obtain two new representations of the fundamental solution in the form of the Mellin convolution of the special functions of the Wright type. Moreover, some new closed-form formulas for particular cases of the fundamental solution are derived. In particular, we solve the open problem of the representation of the fundamental solution to the two-dimensional neutral-fractional diffusion-wave equation in terms of the known special functions.

Description: Mathematics, Vol. 5, Pages 78: A Fixed Point Approach to the Stability of a Mean Value Type Functional Equation Mathematics doi: 10.3390/math5040078 Authors: Soon-Mo Jung Yang-Hi Lee We prove the generalized Hyers–Ulam stability of a mean value type functional equation f ( x ) − g ( y ) = ( x − y ) h ( x + y ) by applying a method originated from fixed point theory.

Description: Mathematics, Vol. 5, Pages 79: Convertible Subspaces of Hessenberg-Type Matrices Mathematics doi: 10.3390/math5040079 Authors: Henrique da Cruz Ilda Rodrigues Rogério Serôdio Alberto Simões José Velhinho We describe subspaces of generalized Hessenberg matrices where the determinant is convertible into the permanent by affixing ± signs. An explicit characterization of convertible Hessenberg-type matrices is presented. We conclude that convertible matrices with the maximum number of nonzero entries can be reduced to a basic set.

Description: Mathematics, Vol. 5, Pages 81: Hyperfuzzy Ideals in BCK/BCI-Algebras Mathematics doi: 10.3390/math5040081 Authors: Seok-Zun Song Seon Kim Young Jun The notions of hyperfuzzy ideals in B C K / B C I -algebras are introduced, and related properties are investigated. Characterizations of hyperfuzzy ideals are established. Relations between hyperfuzzy ideals and hyperfuzzy subalgebras are discussed. Conditions for hyperfuzzy subalgebras to be hyperfuzzy ideals are provided.

Description: Mathematics, Vol. 5, Pages 80: Global Analysis and Optimal Control of a Periodic Visceral Leishmaniasis Model Mathematics doi: 10.3390/math5040080 Authors: Ibrahim ELmojtaba Santanu Biswas Joydev Chattopadhyay In this paper, we propose and analyze a mathematical model for the dynamics of visceral leishmaniasis with seasonality. Our results show that the disease-free equilibrium is globally asymptotically stable under certain conditions when R 0 , the basic reproduction number, is less than unity. When R 0 > 1 and under some conditions, then our system has a unique positive ω -periodic solution that is globally asymptotically stable. Applying two controls, vaccination and treatment, to our model forces the system to be non-periodic, and all fractions of infected populations settle on a very low level.

Description: Mathematics, Vol. 5, Pages 38: Variable Shape Parameter Strategy in Local Radial Basis Functions Collocation Method for Solving the 2D Nonlinear Coupled Burgers’ Equations Mathematics doi: 10.3390/math5030038 Authors: Hananeh Nojavan Saeid Abbasbandy Tofigh Allahviranloo This study aimed at investigating a local radial basis function collocation method (LRBFCM) in the reproducing kernel Hilbert space. This method was, in fact, a meshless one which applied the local sub-clusters of domain nodes for the approximation of the arbitrary field. For time-dependent partial differential equations (PDEs), it would be changed to a system of ordinary differential equations (ODEs). Here, we intended to decrease the error through utilizing variable shape parameter (VSP) strategies. This method was an appropriate way to solve the two-dimensional nonlinear coupled Burgers’ equations comprised of Dirichlet and mixed boundary conditions. Numerical examples indicated that the variable shape parameter strategies were more efficient than constant ones for various values of the Reynolds number.

Description: In this paper, some properties of F -harmonic and conformal F -harmonic maps between doubly warped product manifolds are studied and new examples of non-harmonic F -harmonic maps are constructed.

Description: In this paper, inspired by the concept of b-metric space, we introduce the concept of extended b-metric space. We also establish some fixed point theorems for self-mappings defined on such spaces. Our results extend/generalize many pre-existing results in literature.

Description: In this paper, we introduce the new notion of Suzuki-type ( α , β , θ , γ ) -contractive mapping and investigate the existence and uniqueness of the best proximity point for such mappings in non-Archimedean modular metric space using the weak P λ -property. Meanwhile, we present an illustrative example to emphasize the realized improvements. These obtained results extend and improve certain well-known results in the literature.

Description: In this article the coincidence points of a self map and a sequence of multivalued maps are found in the settings of complete metric space endowed with a graph. A novel result of Asrifa and Vetrivel is generalized and as an application we obtain an existence theorem for a special type of fractional integral equation. Moreover, we establish a result on the convergence of successive approximation of a system of Bernstein operators on a Banach space.

Description: We present the emergence of a root system in six dimensions from the tetrahedra of an icosahedral core known as the 20-group (20G) within the framework of Clifford’s geometric algebra. Consequently, we establish a connection between a three-dimensional icosahedral seed, a six-dimensional (6D) Dirichlet quantized host and a higher dimensional lattice structure. The 20G, owing to its icosahedral symmetry, bears the signature of a 6D lattice that manifests in the Dirichlet integer representation. We present an interpretation whereby the three-dimensional 20G can be regarded as the core substratum from which the higher dimensional lattices emerge. This emergent geometry is based on an induction principle supported by the Clifford multi-vector formalism of three-dimensional (3D) Euclidean space. This lays a geometric framework for understanding several physics theories related to S U ( 5 ) , E 6 , E 8 Lie algebras and their composition with the algebra associated with the even unimodular lattice in R 3 , 1 . The construction presented here is inspired by Penrose’s three world mode.

Description: In this paper, some nonlinear Gronwall–Bellman type inequalities are established. Then, the obtained results are applied to study the Hyers–Ulam stability of a fractional differential equation and the boundedness of solutions to an integral equation, respectively.

Description: Mathematics, Vol. 5, Pages 61: Mixed Order Fractional Differential Equations Mathematics doi: 10.3390/math5040061 Authors: Michal Fečkan JinRong Wang This paper studies fractional differential equations (FDEs) with mixed fractional derivatives. Existence, uniqueness, stability, and asymptotic results are derived.

Description: Mathematics, Vol. 5, Pages 53: Stability of a Monomial Functional Equation on a Restricted Domain Mathematics doi: 10.3390/math5040053 Authors: Yang-Hi Lee In this paper, we prove the stability of the following functional equation ∑ i = 0 n n C i ( − 1 ) n − i f ( i x + y ) − n ! f ( x ) = 0 on a restricted domain by employing the direct method in the sense of Hyers.

Description: Data clustering is a vital tool for data analysis. This work shows that some existing useful methods in data clustering are actually based on quantum mechanics and can be assembled into a powerful and accurate data clustering method where the efficiency of computational quantum chemistry eigenvalue methods is therefore applicable. These methods can be applied to scientific data, engineering data and even text.

Description: In this paper, the author proposes a new SEIRS model that generalizes several classical deterministic epidemic models (e.g., SIR and SIS and SEIR and SEIRS) involving the relationships between the susceptible S, exposed E, infected I, and recovered R individuals for understanding the proliferation of infectious diseases. As a way to incorporate the most important features of the previous models under the assumption of homogeneous mixing (mass-action principle) of the individuals in the population N, the SEIRS model utilizes vital dynamics with unequal birth and death rates, vaccinations for newborns and non-newborns, and temporary immunity. In order to determine the equilibrium points, namely the disease-free and endemic equilibrium points, and study their local stability behaviors, the SEIRS model is rescaled with the total time-varying population and analyzed according to its epidemic condition R0 for two cases of no epidemic (R0 ≤ 1) and epidemic (R0 > 1) using the time-series and phase portraits of the susceptible s, exposed e, infected i, and recovered r individuals. Based on the experimental results using a set of arbitrarily-defined parameters for horizontal transmission of the infectious diseases, the proportional population of the SEIRS model consisted primarily of the recovered r (0.7–0.9) individuals and susceptible s (0.0–0.1) individuals (epidemic) and recovered r (0.9) individuals with only a small proportional population for the susceptible s (0.1) individuals (no epidemic). Overall, the initial conditions for the susceptible s, exposed e, infected i, and recovered r individuals reached the corresponding equilibrium point for local stability: no epidemic (DFE X ¯ D F E ) and epidemic (EE X ¯ E E ).

Description: The generalized elastic model encompasses several linear stochastic models describing the dynamics of polymers, membranes, rough surfaces, and fluctuating interfaces. While usually defined in the overdamped case, in this paper we formally include the inertial term to account for the initial diffusive stages of the stochastic dynamics. We derive the generalized Langevin equation for a probe particle and we show that this equation reduces to the usual Langevin equation for Brownian motion, and to the fractional Langevin equation on the long-time limit.

Description: In the paper by Riečan and Markechová (Fuzzy Sets Syst. 96, 1998), some fuzzy modifications of Shannon’s and Kolmogorov-Sinai’s entropy were studied and the general scheme involving the presented models was introduced. Our aim in this contribution is to provide analogies of these results for the case of the logical entropy. We define the logical entropy and logical mutual information of finite partitions on the appropriate algebraic structure and prove basic properties of these measures. It is shown that, as a special case, we obtain the logical entropy of fuzzy partitions studied by Markechová and Riečan (Entropy 18, 2016). Finally, using the suggested concept of entropy of partitions we define the logical entropy of a dynamical system and prove that it is the same for two dynamical systems that are isomorphic.