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Mathematics, Volume 5, Issue 4 (December 2017)

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Cover Story (view full-size image) We investigate the solution of the integral equation describing the evolution of the intensity of [...] Read more.
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Open AccessFeature PaperArticle Extending the Characteristic Polynomial for Characterization of C20 Fullerene Congeners
Mathematics 2017, 5(4), 84; https://doi.org/10.3390/math5040084
Received: 28 November 2017 / Revised: 28 November 2017 / Accepted: 13 December 2017 / Published: 19 December 2017
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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
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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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Open AccessArticle Isomorphic Classification of Reflexive Müntz Spaces
Mathematics 2017, 5(4), 83; https://doi.org/10.3390/math5040083
Received: 6 November 2017 / Revised: 1 December 2017 / Accepted: 5 December 2017 / Published: 18 December 2017
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Abstract
The article is devoted to reflexive Müntz spaces MΛ,p of Lp functions with 1<p<. The Stieltjes transform and a potential transform are studied for these spaces. Isomorphisms of the reflexive Müntz spaces fulfilling the
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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
Open AccessArticle Multiplicative Structure and Hecke Rings of Generator Matrices for Codes over Quotient Rings of Euclidean Domains
Mathematics 2017, 5(4), 82; https://doi.org/10.3390/math5040082
Received: 30 June 2017 / Revised: 20 November 2017 / Accepted: 8 December 2017 / Published: 15 December 2017
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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
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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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Open AccessArticle Hyperfuzzy Ideals in BCK/BCI-Algebras
Mathematics 2017, 5(4), 81; https://doi.org/10.3390/math5040081
Received: 17 November 2017 / Revised: 10 December 2017 / Accepted: 10 December 2017 / Published: 14 December 2017
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Abstract
The notions of hyperfuzzy ideals in BCK/BCI-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
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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)
Open AccessArticle Global Analysis and Optimal Control of a Periodic Visceral Leishmaniasis Model
Mathematics 2017, 5(4), 80; https://doi.org/10.3390/math5040080
Received: 2 October 2017 / Revised: 6 November 2017 / Accepted: 12 November 2017 / Published: 14 December 2017
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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 R0, the basic reproduction number, is less than
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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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Open AccessFeature PaperArticle Convertible Subspaces of Hessenberg-Type Matrices
Mathematics 2017, 5(4), 79; https://doi.org/10.3390/math5040079
Received: 16 November 2017 / Revised: 9 December 2017 / Accepted: 10 December 2017 / Published: 13 December 2017
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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
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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
Open AccessArticle A Fixed Point Approach to the Stability of a Mean Value Type Functional Equation
Mathematics 2017, 5(4), 78; https://doi.org/10.3390/math5040078
Received: 3 November 2017 / Revised: 30 November 2017 / Accepted: 5 December 2017 / Published: 13 December 2017
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Abstract
We prove the generalized Hyers–Ulam stability of a mean value type functional equation f(x)g(y)=(xy)h(x+y) by applying a method originated from fixed point theory.
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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)
Open AccessArticle Solving the Lane–Emden Equation within a Reproducing Kernel Method and Group Preserving Scheme
Mathematics 2017, 5(4), 77; https://doi.org/10.3390/math5040077
Received: 17 October 2017 / Revised: 26 November 2017 / Accepted: 5 December 2017 / Published: 12 December 2017
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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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Open AccessArticle On Some New Properties of the Fundamental Solution to the Multi-Dimensional Space- and Time-Fractional Diffusion-Wave Equation
Mathematics 2017, 5(4), 76; https://doi.org/10.3390/math5040076
Received: 11 November 2017 / Revised: 1 December 2017 / Accepted: 4 December 2017 / Published: 8 December 2017
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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
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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)
Open AccessFeature PaperArticle Acting Semicircular Elements Induced by Orthogonal Projections on Von-Neumann-Algebras
Mathematics 2017, 5(4), 74; https://doi.org/10.3390/math5040074
Received: 18 September 2017 / Revised: 20 November 2017 / Accepted: 21 November 2017 / Published: 6 December 2017
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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
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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)
Open AccessFeature PaperArticle Geometric Structure of the Classical Lagrange-d’Alambert Principle and Its Application to Integrable Nonlinear Dynamical Systems
Mathematics 2017, 5(4), 75; https://doi.org/10.3390/math5040075
Received: 14 September 2017 / Revised: 11 November 2017 / Accepted: 25 November 2017 / Published: 5 December 2017
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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
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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
Open AccessFeature PaperArticle Fractional Derivatives, Memory Kernels and Solution of a Free Electron Laser Volterra Type Equation
Mathematics 2017, 5(4), 73; https://doi.org/10.3390/math5040073
Received: 17 October 2017 / Revised: 20 November 2017 / Accepted: 24 November 2017 / Published: 4 December 2017
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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
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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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Open AccessFeature PaperArticle Wavelet Neural Network Model for Yield Spread Forecasting
Mathematics 2017, 5(4), 72; https://doi.org/10.3390/math5040072
Received: 28 September 2017 / Revised: 14 November 2017 / Accepted: 20 November 2017 / Published: 27 November 2017
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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
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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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Open AccessArticle Some Types of Subsemigroups Characterized in Terms of Inequalities of Generalized Bipolar Fuzzy Subsemigroups
Mathematics 2017, 5(4), 71; https://doi.org/10.3390/math5040071
Received: 14 November 2017 / Revised: 21 November 2017 / Accepted: 23 November 2017 / Published: 27 November 2017
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Abstract
In this paper, we introduce a generalization of a bipolar fuzzy (BF) subsemigroup, namely, a (α1,α2;β1,β2)-BF subsemigroup. The notions of (α1,α2;β1,
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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)
Open AccessArticle Controlling Chaos—Forced van der Pol Equation
Mathematics 2017, 5(4), 70; https://doi.org/10.3390/math5040070
Received: 6 October 2017 / Revised: 14 November 2017 / Accepted: 17 November 2017 / Published: 24 November 2017
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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
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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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