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Volume 5
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Mathematics, Volume 5, Issue 4 (December 2017) – 38 articles

Cover Story (view full-size image): We investigate the solution of the integral equation describing the evolution of the intensity of the free electron laser. The method we propose employs a broader definition of the memory kernel derivative and is obtained by a generalization of the Volterra iteration procedure. The figures report the normalized intensity vs. the normalized frequency. View this paper
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2755 KiB  
Article
Extending the Characteristic Polynomial for Characterization of C20 Fullerene Congeners
by Dan-Marian Joiţa and Lorentz Jäntschi
Mathematics 2017, 5(4), 84; https://doi.org/10.3390/math5040084 - 19 Dec 2017
Cited by 25 | Viewed by 4942
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Applied and Computational Statistics)
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246 KiB  
Article
Isomorphic Classification of Reflexive Müntz Spaces
by Sergey V. Ludkowski
Mathematics 2017, 5(4), 83; https://doi.org/10.3390/math5040083 - 18 Dec 2017
Viewed by 2193
Abstract
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 [...] Read more.
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. Full article
407 KiB  
Article
Multiplicative Structure and Hecke Rings of Generator Matrices for Codes over Quotient Rings of Euclidean Domains
by Hajime Matsui
Mathematics 2017, 5(4), 82; https://doi.org/10.3390/math5040082 - 15 Dec 2017
Cited by 3 | Viewed by 2802
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Geometry of Numbers)
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252 KiB  
Article
Hyperfuzzy Ideals in BCK/BCI-Algebras
by Seok-Zun Song, Seon Jeong Kim and Young Bae Jun
Mathematics 2017, 5(4), 81; https://doi.org/10.3390/math5040081 - 14 Dec 2017
Cited by 4 | Viewed by 2519
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Fuzzy Mathematics)
1022 KiB  
Article
Global Analysis and Optimal Control of a Periodic Visceral Leishmaniasis Model
by Ibrahim M. ELmojtaba, Santanu Biswas and Joydev Chattopadhyay
Mathematics 2017, 5(4), 80; https://doi.org/10.3390/math5040080 - 14 Dec 2017
Cited by 3 | Viewed by 2977
Abstract
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 [...] Read more.
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. Full article
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237 KiB  
Article
Convertible Subspaces of Hessenberg-Type Matrices
by Henrique F. Da Cruz, Ilda Inácio Rodrigues, Rogério Serôdio, Alberto Simões and José Velhinho
Mathematics 2017, 5(4), 79; https://doi.org/10.3390/math5040079 - 13 Dec 2017
Cited by 1 | Viewed by 2991
Abstract
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 [...] Read more.
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. Full article
235 KiB  
Article
A Fixed Point Approach to the Stability of a Mean Value Type Functional Equation
by Soon-Mo Jung and Yang-Hi Lee
Mathematics 2017, 5(4), 78; https://doi.org/10.3390/math5040078 - 13 Dec 2017
Cited by 4 | Viewed by 2621
Abstract
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. [...] Read more.
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. Full article
(This article belongs to the Special Issue Fixed Point Theory)
265 KiB  
Article
Solving the Lane–Emden Equation within a Reproducing Kernel Method and Group Preserving Scheme
by Mir Sajjad Hashemi, Ali Akgül, Mustafa Inc, Idrees Sedeeq Mustafa and Dumitru Baleanu
Mathematics 2017, 5(4), 77; https://doi.org/10.3390/math5040077 - 12 Dec 2017
Cited by 16 | Viewed by 3203
Abstract
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. Full article
(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications)
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275 KiB  
Article
On Some New Properties of the Fundamental Solution to the Multi-Dimensional Space- and Time-Fractional Diffusion-Wave Equation
by Yuri Luchko
Mathematics 2017, 5(4), 76; https://doi.org/10.3390/math5040076 - 08 Dec 2017
Cited by 26 | Viewed by 3619
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Fractional Calculus: Theory and Applications)
398 KiB  
Article
Acting Semicircular Elements Induced by Orthogonal Projections on Von-Neumann-Algebras
by Ilwoo Cho
Mathematics 2017, 5(4), 74; https://doi.org/10.3390/math5040074 - 06 Dec 2017
Cited by 8 | Viewed by 2776
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Mathematical Physics and Quantum Information)
314 KiB  
Article
Geometric Structure of the Classical Lagrange-d’Alambert Principle and Its Application to Integrable Nonlinear Dynamical Systems
by Anatolij K. Prykarpatski, Oksana E. Hentosh and Yarema A. Prykarpatsky
Mathematics 2017, 5(4), 75; https://doi.org/10.3390/math5040075 - 05 Dec 2017
Cited by 5 | Viewed by 2832
Abstract
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 [...] Read more.
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. Full article
1128 KiB  
Article
Fractional Derivatives, Memory Kernels and Solution of a Free Electron Laser Volterra Type Equation
by Marcello Artioli, Giuseppe Dattoli, Silvia Licciardi and Simonetta Pagnutti
Mathematics 2017, 5(4), 73; https://doi.org/10.3390/math5040073 - 04 Dec 2017
Cited by 8 | Viewed by 3032
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Fractional Calculus: Theory and Applications)
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1394 KiB  
Article
Wavelet Neural Network Model for Yield Spread Forecasting
by Firdous Ahmad Shah and Lokenath Debnath
Mathematics 2017, 5(4), 72; https://doi.org/10.3390/math5040072 - 27 Nov 2017
Cited by 8 | Viewed by 3959
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Recent Developments in Wavelet Transforms and Their Applications)
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772 KiB  
Article
Some Types of Subsemigroups Characterized in Terms of Inequalities of Generalized Bipolar Fuzzy Subsemigroups
by Pannawit Khamrot and Manoj Siripitukdet
Mathematics 2017, 5(4), 71; https://doi.org/10.3390/math5040071 - 27 Nov 2017
Cited by 3 | Viewed by 2907
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Fuzzy Mathematics)
2263 KiB  
Article
Controlling Chaos—Forced van der Pol Equation
by Matthew Cooper, Peter Heidlauf and Timothy Sands
Mathematics 2017, 5(4), 70; https://doi.org/10.3390/math5040070 - 24 Nov 2017
Cited by 33 | Viewed by 9725
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Mathematics on Automation Control Systems)
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695 KiB  
Article
Impact of Parameter Variability and Environmental Noise on the Klausmeier Model of Vegetation Pattern Formation
by Merlin C. Köhnke and Horst Malchow
Mathematics 2017, 5(4), 69; https://doi.org/10.3390/math5040069 - 23 Nov 2017
Cited by 4 | Viewed by 4075
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Progress in Mathematical Ecology)
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358 KiB  
Article
Channel Engineering for Nanotransistors in a Semiempirical Quantum Transport Model
by Ulrich Wulf, Jan Kučera, Hans Richter, Manfred Horstmann, Maciej Wiatr and Jan Höntschel
Mathematics 2017, 5(4), 68; https://doi.org/10.3390/math5040068 - 22 Nov 2017
Cited by 3 | Viewed by 4988
Abstract
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 [...] Read more.
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. Full article
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745 KiB  
Article
On Edge Irregular Reflexive Labellings for the Generalized Friendship Graphs
by Martin Bača, Muhammad Irfan, Joe Ryan, Andrea Semaničová-Feňovčíková and Dushyant Tanna
Mathematics 2017, 5(4), 67; https://doi.org/10.3390/math5040067 - 21 Nov 2017
Cited by 20 | Viewed by 3239
Abstract
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 [...] Read more.
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. Full article
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434 KiB  
Article
Picard’s Iterative Method for Caputo Fractional Differential Equations with Numerical Results
by Rainey Lyons, Aghalaya S. Vatsala and Ross A. Chiquet
Mathematics 2017, 5(4), 65; https://doi.org/10.3390/math5040065 - 21 Nov 2017
Cited by 23 | Viewed by 7737
Abstract
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 [...] Read more.
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. Full article
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341 KiB  
Article
Generalized Langevin Equation and the Prabhakar Derivative
by Trifce Sandev
Mathematics 2017, 5(4), 66; https://doi.org/10.3390/math5040066 - 20 Nov 2017
Cited by 58 | Viewed by 5801
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Fractional Calculus: Theory and Applications)
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242 KiB  
Article
On the Inception of Financial Representative Bubbles
by Massimiliano Ferrara, Bruno A. Pansera and Francesco Strati
Mathematics 2017, 5(4), 64; https://doi.org/10.3390/math5040064 - 17 Nov 2017
Cited by 5 | Viewed by 3450
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Financial Mathematics)
565 KiB  
Article
Krylov Implicit Integration Factor Methods for Semilinear Fourth-Order Equations
by Michael Machen and Yong-Tao Zhang
Mathematics 2017, 5(4), 63; https://doi.org/10.3390/math5040063 - 16 Nov 2017
Cited by 2 | Viewed by 3441
Abstract
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 Physics, 253 (2013) 368–388], IIF methods are designed to efficiently solve stiff nonlinear advection–diffusion–reaction (ADR) equations. The methods can [...] Read more.
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 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. Full article
(This article belongs to the Special Issue Numerical Methods for Partial Differential Equations)
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312 KiB  
Article
Solution of Inhomogeneous Differential Equations with Polynomial Coefficients in Terms of the Green’s Function
by Tohru Morita and Ken-ichi Sato
Mathematics 2017, 5(4), 62; https://doi.org/10.3390/math5040062 - 10 Nov 2017
Cited by 4 | Viewed by 3358
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Operators of Fractional Calculus and Their Applications)
740 KiB  
Article
Mixed Order Fractional Differential Equations
by Michal Fečkan and JinRong Wang
Mathematics 2017, 5(4), 61; https://doi.org/10.3390/math5040061 - 07 Nov 2017
Cited by 4 | Viewed by 3187
Abstract
This paper studies fractional differential equations (FDEs) with mixed fractional derivatives. Existence, uniqueness, stability, and asymptotic results are derived. Full article
(This article belongs to the Special Issue Operators of Fractional Calculus and Their Applications)
950 KiB  
Article
Graph Structures in Bipolar Neutrosophic Environment
by Muhammad Akram, Muzzamal Sitara and Florentin Smarandache
Mathematics 2017, 5(4), 60; https://doi.org/10.3390/math5040060 - 06 Nov 2017
Cited by 4 | Viewed by 3416
Abstract
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 [...] Read more.
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. Full article
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353 KiB  
Article
Invariant Solutions for a Class of Perturbed Nonlinear Wave Equations
by Waheed A. Ahmed, F. D. Zaman and Khairul Saleh
Mathematics 2017, 5(4), 59; https://doi.org/10.3390/math5040059 - 01 Nov 2017
Cited by 2 | Viewed by 3211
Abstract
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. Full article
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496 KiB  
Article
Dynamics of Amoebiasis Transmission: Stability and Sensitivity Analysis
by Fidele Hategekimana, Snehanshu Saha and Anita Chaturvedi
Mathematics 2017, 5(4), 58; https://doi.org/10.3390/math5040058 - 01 Nov 2017
Cited by 10 | Viewed by 4729
Abstract
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 [...] Read more.
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. Full article
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716 KiB  
Article
The Theory of Connections: Connecting Points
by Daniele Mortari
Mathematics 2017, 5(4), 57; https://doi.org/10.3390/math5040057 - 01 Nov 2017
Cited by 82 | Viewed by 7600
Abstract
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 [...] Read more.
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. Full article
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340 KiB  
Article
A Constructive Method for Standard Borel Fixed Submodules with Given Extremal Betti Numbers
by Marilena Crupi
Mathematics 2017, 5(4), 56; https://doi.org/10.3390/math5040056 - 01 Nov 2017
Cited by 3 | Viewed by 2438
Abstract
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 [...] Read more.
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. Full article
530 KiB  
Article
On the Achievable Stabilization Delay Margin for Linear Plants with Time-Varying Delays
by Jing Zhu
Mathematics 2017, 5(4), 55; https://doi.org/10.3390/math5040055 - 25 Oct 2017
Viewed by 2688
Abstract
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 [...] Read more.
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. Full article
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