Mathematics

270 KiB

Open AccessArticle

Exponential Energy Decay of Solutions for a Transmission Problem With Viscoelastic Term and Delay

by Danhua Wang, Gang Li and Biqing Zhu

Mathematics 2016, 4(2), 42; https://doi.org/10.3390/math4020042 - 9 Jun 2016

Cited by 5 | Viewed by 4479

In our previous work (Journal of Nonlinear Science and Applications 9: 1202–1215, 2016), we studied the well-posedness and general decay rate for a transmission problem in a bounded domain with a viscoelastic term and a delay term. In this paper, we continue to [...] Read more.

In our previous work (Journal of Nonlinear Science and Applications 9: 1202–1215, 2016), we studied the well-posedness and general decay rate for a transmission problem in a bounded domain with a viscoelastic term and a delay term. In this paper, we continue to study the similar problem but without the frictional damping term. The main difficulty arises since we have no frictional damping term to control the delay term in the estimate of the energy decay. By introducing suitable energy and Lyapunov functionals, we establish an exponential decay result for the energy. Full article

(This article belongs to the Special Issue Applied Mathematics and Mechanics)

595 KiB

Open AccessFeature PaperArticle

Entropic Uncertainty Relations for Successive Generalized Measurements

by Kyunghyun Baek and Wonmin Son

Mathematics 2016, 4(2), 41; https://doi.org/10.3390/math4020041 - 7 Jun 2016

Cited by 8 | Viewed by 5769

Abstract

We derive entropic uncertainty relations for successive generalized measurements by using general descriptions of quantum measurement within two distinctive operational scenarios. In the first scenario, by merging two successive measurements into one we consider successive measurement scheme as a method to perform an [...] Read more.

We derive entropic uncertainty relations for successive generalized measurements by using general descriptions of quantum measurement within two distinctive operational scenarios. In the first scenario, by merging two successive measurements into one we consider successive measurement scheme as a method to perform an overall composite measurement. In the second scenario, on the other hand, we consider it as a method to measure a pair of jointly measurable observables by marginalizing over the distribution obtained in this scheme. In the course of this work, we identify that limits on one’s ability to measure with low uncertainty via this scheme come from intrinsic unsharpness of observables obtained in each scenario. In particular, for the Lüders instrument, disturbance caused by the first measurement to the second one gives rise to the unsharpness at least as much as incompatibility of the observables composing successive measurement. Full article

(This article belongs to the Special Issue Mathematics of Quantum Uncertainty)

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538 KiB

Open AccessArticle

Morphisms and Order Ideals of Toric Posets

by Matthew Macauley

Mathematics 2016, 4(2), 39; https://doi.org/10.3390/math4020039 - 4 Jun 2016

Viewed by 4359

Abstract

Toric posets are in some sense a natural “cyclic” version of finite posets in that they capture the fundamental features of a partial order but without the notion of minimal or maximal elements. They can be thought of combinatorially as equivalence classes of [...] Read more.

Toric posets are in some sense a natural “cyclic” version of finite posets in that they capture the fundamental features of a partial order but without the notion of minimal or maximal elements. They can be thought of combinatorially as equivalence classes of acyclic orientations under the equivalence relation generated by converting sources into sinks, or geometrically as chambers of toric graphic hyperplane arrangements. In this paper, we define toric intervals and toric order-preserving maps, which lead to toric analogues of poset morphisms and order ideals. We develop this theory, discuss some fundamental differences between the toric and ordinary cases, and outline some areas for future research. Additionally, we provide a connection to cyclic reducibility and conjugacy in Coxeter groups. Full article

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253 KiB

Open AccessFeature PaperReview

Uncertainty Relations and Possible Experience

by Gregg Jaeger

Mathematics 2016, 4(2), 40; https://doi.org/10.3390/math4020040 - 3 Jun 2016

Cited by 5 | Viewed by 4467

Abstract

The uncertainty principle can be understood as a condition of joint indeterminacy of classes of properties in quantum theory. The mathematical expressions most closely associated with this principle have been the uncertainty relations, various inequalities exemplified by the well known expression regarding position [...] Read more.

The uncertainty principle can be understood as a condition of joint indeterminacy of classes of properties in quantum theory. The mathematical expressions most closely associated with this principle have been the uncertainty relations, various inequalities exemplified by the well known expression regarding position and momentum introduced by Heisenberg. Here, recent work involving a new sort of “logical” indeterminacy principle and associated relations introduced by Pitowsky, expressable directly in terms of probabilities of outcomes of measurements of sharp quantum observables, is reviewed and its quantum nature is discussed. These novel relations are derivable from Boolean “conditions of possible experience” of the quantum realm and have been considered both as fundamentally logical and as fundamentally geometrical. This work focuses on the relationship of indeterminacy to the propositions regarding the values of discrete, sharp observables of quantum systems. Here, reasons for favoring each of these two positions are considered. Finally, with an eye toward future research related to indeterminacy relations, further novel approaches grounded in category theory and intended to capture and reconceptualize the complementarity characteristics of quantum propositions are discussed in relation to the former. Full article

(This article belongs to the Special Issue Mathematics of Quantum Uncertainty)

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887 KiB

Open AccessFeature PaperArticle

Measurement Uncertainty for Finite Quantum Observables

by René Schwonnek, David Reeb and Reinhard F. Werner

Mathematics 2016, 4(2), 38; https://doi.org/10.3390/math4020038 - 2 Jun 2016

Cited by 13 | Viewed by 5896

Abstract

Measurement uncertainty relations are lower bounds on the errors of any approximate joint measurement of two or more quantum observables. The aim of this paper is to provide methods to compute optimal bounds of this type. The basic method is semidefinite programming, which [...] Read more.

Measurement uncertainty relations are lower bounds on the errors of any approximate joint measurement of two or more quantum observables. The aim of this paper is to provide methods to compute optimal bounds of this type. The basic method is semidefinite programming, which we apply to arbitrary finite collections of projective observables on a finite dimensional Hilbert space. The quantification of errors is based on an arbitrary cost function, which assigns a penalty to getting result x rather than y, for any pair

(x, y)

. This induces a notion of optimal transport cost for a pair of probability distributions, and we include an Appendix with a short summary of optimal transport theory as needed in our context. There are then different ways to form an overall figure of merit from the comparison of distributions. We consider three, which are related to different physical testing scenarios. The most thorough test compares the transport distances between the marginals of a joint measurement and the reference observables for every input state. Less demanding is a test just on the states for which a “true value” is known in the sense that the reference observable yields a definite outcome. Finally, we can measure a deviation as a single expectation value by comparing the two observables on the two parts of a maximally-entangled state. All three error quantities have the property that they vanish if and only if the tested observable is equal to the reference. The theory is illustrated with some characteristic examples. Full article

(This article belongs to the Special Issue Mathematics of Quantum Uncertainty)

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213 KiB

Open AccessArticle

Smoothness in Binomial Edge Ideals

by Hamid Damadi and Farhad Rahmati

Mathematics 2016, 4(2), 37; https://doi.org/10.3390/math4020037 - 1 Jun 2016

Viewed by 3564

Abstract

In this paper we study some geometric properties of the algebraic set associated to the binomial edge ideal of a graph. We study the singularity and smoothness of the algebraic set associated to the binomial edge ideal of a graph. Some of these [...] Read more.

In this paper we study some geometric properties of the algebraic set associated to the binomial edge ideal of a graph. We study the singularity and smoothness of the algebraic set associated to the binomial edge ideal of a graph. Some of these algebraic sets are irreducible and some of them are reducible. If every irreducible component of the algebraic set is smooth we call the graph an edge smooth graph, otherwise it is called an edge singular graph. We show that complete graphs are edge smooth and introduce two conditions such that the graph G is edge singular if and only if it satisfies these conditions. Then, it is shown that cycles and most of trees are edge singular. In addition, it is proved that complete bipartite graphs are edge smooth. Full article

337 KiB

Open AccessFeature PaperArticle

SIC-POVMs and Compatibility among Quantum States

by Blake C. Stacey

Mathematics 2016, 4(2), 36; https://doi.org/10.3390/math4020036 - 1 Jun 2016

Cited by 19 | Viewed by 5512

Abstract

An unexpected connection exists between compatibility criteria for quantum states and Symmetric Informationally Complete quantum measurements (SIC-POVMs). Beginning with Caves, Fuchs and Schack’s "Conditions for compatibility of quantum state assignments", I show that a qutrit SIC-POVM studied in other contexts enjoys additional interesting [...] Read more.

An unexpected connection exists between compatibility criteria for quantum states and Symmetric Informationally Complete quantum measurements (SIC-POVMs). Beginning with Caves, Fuchs and Schack’s "Conditions for compatibility of quantum state assignments", I show that a qutrit SIC-POVM studied in other contexts enjoys additional interesting properties. Compatibility criteria provide a new way to understand the relationship between SIC-POVMs and mutually unbiased bases, as calculations in the SIC representation of quantum states make clear. This, in turn, illuminates the resources necessary for magic-state quantum computation, and why hidden-variable models fail to capture the vitality of quantum mechanics. Full article

(This article belongs to the Special Issue Mathematics of Quantum Uncertainty)

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223 KiB

Open AccessArticle

Three Identities of the Catalan-Qi Numbers

by Mansour Mahmoud and Feng Qi

Mathematics 2016, 4(2), 35; https://doi.org/10.3390/math4020035 - 26 May 2016

Cited by 23 | Viewed by 6904

Abstract

In the paper, the authors find three new identities of the Catalan-Qi numbers and provide alternative proofs of two identities of the Catalan numbers. The three identities of the Catalan-Qi numbers generalize three identities of the Catalan numbers. Full article

794 KiB

Open AccessArticle

Lie Symmetries of (1+2) Nonautonomous Evolution Equations in Financial Mathematics

by Andronikos Paliathanasis, Richard M. Morris and Peter G. L. Leach

Mathematics 2016, 4(2), 34; https://doi.org/10.3390/math4020034 - 13 May 2016

Cited by 4 | Viewed by 3922

Abstract

We analyse two classes of

(1 + 2)

evolution equations which are of special interest in Financial Mathematics, namely the Two-dimensional Black-Scholes Equation and the equation for the Two-factor Commodities Problem. Our approach is that of Lie Symmetry Analysis. We study [...] Read more.

We analyse two classes of

(1 + 2)

evolution equations which are of special interest in Financial Mathematics, namely the Two-dimensional Black-Scholes Equation and the equation for the Two-factor Commodities Problem. Our approach is that of Lie Symmetry Analysis. We study these equations for the case in which they are autonomous and for the case in which the parameters of the equations are unspecified functions of time. For the autonomous Black-Scholes Equation we find that the symmetry is maximal and so the equation is reducible to the

(1 + 2)

Classical Heat Equation. This is not the case for the nonautonomous equation for which the number of symmetries is submaximal. In the case of the two-factor equation the number of symmetries is submaximal in both autonomous and nonautonomous cases. When the solution symmetries are used to reduce each equation to a

(1 + 1)

equation, the resulting equation is of maximal symmetry and so equivalent to the

(1 + 1)

Classical Heat Equation. Full article

(This article belongs to the Special Issue Mathematical Finance)

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2435 KiB

Open AccessArticle

Chaos Control in Three Dimensional Cancer Model by State Space Exact Linearization Based on Lie Algebra

by Mohammad Shahzad

Mathematics 2016, 4(2), 33; https://doi.org/10.3390/math4020033 - 10 May 2016

Cited by 13 | Viewed by 4965

Abstract

This study deals with the control of chaotic dynamics of tumor cells, healthy host cells, and effector immune cells in a chaotic Three Dimensional Cancer Model (TDCM) by State Space Exact Linearization (SSEL) technique based on Lie algebra. A non-linear feedback control law [...] Read more.

This study deals with the control of chaotic dynamics of tumor cells, healthy host cells, and effector immune cells in a chaotic Three Dimensional Cancer Model (TDCM) by State Space Exact Linearization (SSEL) technique based on Lie algebra. A non-linear feedback control law is designed which induces a coordinate transformation thereby changing the original chaotic TDCM system into a controlled one linear system. Numerical simulation has been carried using Mathematica that witness the robustness of the technique implemented on the chosen chaotic system. Full article

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234 KiB

Open AccessArticle

On the Dimension of Algebraic-Geometric Trace Codes

by Phong Le and Sunil Chetty

Mathematics 2016, 4(2), 32; https://doi.org/10.3390/math4020032 - 7 May 2016

Cited by 1 | Viewed by 3923

Abstract

We study trace codes induced from codes defined by an algebraic curve X. We determine conditions on X which admit a formula for the dimension of such a trace code. Central to our work are several dimension reducing methods for the underlying [...] Read more.

We study trace codes induced from codes defined by an algebraic curve X. We determine conditions on X which admit a formula for the dimension of such a trace code. Central to our work are several dimension reducing methods for the underlying functions spaces associated to X. Full article

415 KiB

Open AccessFeature PaperArticle

Fractional Schrödinger Equation in the Presence of the Linear Potential

by André Liemert and Alwin Kienle

Mathematics 2016, 4(2), 31; https://doi.org/10.3390/math4020031 - 4 May 2016

Cited by 53 | Viewed by 10734

Abstract

In this paper, we consider the time-dependent Schrödinger equation: [...] Read more.

In this paper, we consider the time-dependent Schrödinger equation:

i \frac{\partial ψ (x, t)}{\partial t} = \frac{1}{2} {(- Δ)}^{\frac{α}{2}} ψ (x, t) + V (x) ψ (x, t),

x \in R, t > 0

with the Riesz space-fractional derivative of order

0 < α \leq 2

in the presence of the linear potential

V (x) = β x

. The wave function to the one-dimensional Schrödinger equation in momentum space is given in closed form allowing the determination of other measurable quantities such as the mean square displacement. Analytical solutions are derived for the relevant case of

α = 1

, which are useable for studying the propagation of wave packets that undergo spreading and splitting. We furthermore address the two-dimensional space-fractional Schrödinger equation under consideration of the potential

V (ρ) = F \cdot ρ

including the free particle case. The derived equations are illustrated in different ways and verified by comparisons with a recently proposed numerical approach. Full article

(This article belongs to the Special Issue Fractional Differential and Difference Equations)

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2125 KiB

Open AccessArticle

New Approach for Fractional Order Derivatives: Fundamentals and Analytic Properties

by Ali Karcı

Mathematics 2016, 4(2), 30; https://doi.org/10.3390/math4020030 - 4 May 2016

Cited by 8 | Viewed by 4553

Abstract

The rate of change of any function versus its independent variables was defined as a derivative. The fundamentals of the derivative concept were constructed by Newton and l’Hôpital. The followers of Newton and l’Hôpital defined fractional order derivative concepts. We express the derivative [...] Read more.

The rate of change of any function versus its independent variables was defined as a derivative. The fundamentals of the derivative concept were constructed by Newton and l’Hôpital. The followers of Newton and l’Hôpital defined fractional order derivative concepts. We express the derivative defined by Newton and l’Hôpital as an ordinary derivative, and there are also fractional order derivatives. So, the derivative concept was handled in this paper, and a new definition for derivative based on indefinite limit and l’Hôpital’s rule was expressed. This new approach illustrated that a derivative operator may be non-linear. Based on this idea, the asymptotic behaviors of functions were analyzed and it was observed that the rates of changes of any function attain maximum value at inflection points in the positive direction and minimum value (negative) at inflection points in the negative direction. This case brought out the fact that the derivative operator does not have to be linear; it may be non-linear. Another important result of this paper is the relationships between complex numbers and derivative concepts, since both concepts have directions and magnitudes. Full article

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5596 KiB

Open AccessFeature PaperArticle

An Adaptive WENO Collocation Method for Differential Equations with Random Coefficients

by Wei Guo, Guang Lin, Andrew J. Christlieb and Jingmei Qiu

Mathematics 2016, 4(2), 29; https://doi.org/10.3390/math4020029 - 3 May 2016

Cited by 3 | Viewed by 4776

Abstract

The stochastic collocation method for solving differential equations with random inputs has gained lots of popularity in many applications, since such a scheme exhibits exponential convergence with smooth solutions in the random space. However, in some circumstance the solutions do not fulfill the [...] Read more.

The stochastic collocation method for solving differential equations with random inputs has gained lots of popularity in many applications, since such a scheme exhibits exponential convergence with smooth solutions in the random space. However, in some circumstance the solutions do not fulfill the smoothness requirement; thus a direct application of the method will cause poor performance and slow convergence rate due to the well known Gibbs phenomenon. To address the issue, we propose an adaptive high-order multi-element stochastic collocation scheme by incorporating a WENO (Weighted Essentially non-oscillatory) interpolation procedure and an adaptive mesh refinement (AMR) strategy. The proposed multi-element stochastic collocation scheme requires only repetitive runs of an existing deterministic solver at each interpolation point, similar to the Monte Carlo method. Furthermore, the scheme takes advantage of robustness and the high-order nature of the WENO interpolation procedure, and efficacy and efficiency of the AMR strategy. When the proposed scheme is applied to stochastic problems with non-smooth solutions, the Gibbs phenomenon is mitigated by the WENO methodology in the random space, and the errors around discontinuities in the stochastic space are significantly reduced by the AMR strategy. The numerical experiments for some benchmark stochastic problems, such as the Kraichnan-Orszag problem and Burgers’ equation with random initial conditions, demonstrate the reliability, efficiency and efficacy of the proposed scheme. Full article

(This article belongs to the Special Issue New Trends in Applications of Orthogonal Polynomials and Special Functions)

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832 KiB

Open AccessFeature PaperArticle

Lie Symmetry Analysis of the Black-Scholes-Merton Model for European Options with Stochastic Volatility

by Andronikos Paliathanasis, K. Krishnakumar, K.M. Tamizhmani and Peter G.L. Leach

Mathematics 2016, 4(2), 28; https://doi.org/10.3390/math4020028 - 3 May 2016

Cited by 15 | Viewed by 5792

Abstract

We perform a classification of the Lie point symmetries for the Black-Scholes-Merton Model for European options with stochastic volatility, σ, in which the last is defined by a stochastic differential equation with an Orstein-Uhlenbeck term. In this model, the value of the [...] Read more.

We perform a classification of the Lie point symmetries for the Black-Scholes-Merton Model for European options with stochastic volatility, σ, in which the last is defined by a stochastic differential equation with an Orstein-Uhlenbeck term. In this model, the value of the option is given by a linear (1 + 2) evolution partial differential equation in which the price of the option depends upon two independent variables, the value of the underlying asset, S, and a new variable, y. We find that for arbitrary functional form of the volatility,

σ (y)

, the (1 + 2) evolution equation always admits two Lie point symmetries in addition to the automatic linear symmetry and the infinite number of solution symmetries. However, when

σ (y) = σ_{0}

and as the price of the option depends upon the second Brownian motion in which the volatility is defined, the (1 + 2) evolution is not reduced to the Black-Scholes-Merton Equation, the model admits five Lie point symmetries in addition to the linear symmetry and the infinite number of solution symmetries. We apply the zeroth-order invariants of the Lie symmetries and we reduce the (1 + 2) evolution equation to a linear second-order ordinary differential equation. Finally, we study two models of special interest, the Heston model and the Stein-Stein model. Full article

(This article belongs to the Special Issue Mathematical Finance)

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1187 KiB

Open AccessArticle

Stagnation-Point Flow towards a Stretching Vertical Sheet with Slip Effects

by Khairy Zaimi and Anuar Ishak

Mathematics 2016, 4(2), 27; https://doi.org/10.3390/math4020027 - 21 Apr 2016

Cited by 25 | Viewed by 4551

Abstract

The effects of partial slip on stagnation-point flow and heat transfer due to a stretching vertical sheet is investigated. Using a similarity transformation, the governing partial differential equations are reduced into a system of nonlinear ordinary differential equations. The resulting equations are solved [...] Read more.

The effects of partial slip on stagnation-point flow and heat transfer due to a stretching vertical sheet is investigated. Using a similarity transformation, the governing partial differential equations are reduced into a system of nonlinear ordinary differential equations. The resulting equations are solved numerically using a shooting method. The effect of slip and buoyancy parameters on the velocity, temperature, skin friction coefficient and the local Nusselt number are graphically presented and discussed. It is found that dual solutions exist in a certain range of slip and buoyancy parameters. The skin friction coefficient decreases while the Nusselt number increases as the slip parameter increases. Full article

(This article belongs to the Special Issue Applied Mathematics and Mechanics)

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11129 KiB

Open AccessArticle

POD-Based Constrained Sensor Placement and Field Reconstruction from Noisy Wind Measurements: A Perturbation Study

by Zhongqiang Zhang, Xiu Yang and Guang Lin

Mathematics 2016, 4(2), 26; https://doi.org/10.3390/math4020026 - 14 Apr 2016

Cited by 15 | Viewed by 4842

Abstract

It is shown in literature that sensor placement at the extrema of Proper Orthogonal Decomposition (POD) modes is efficient and leads to accurate reconstruction of the field of quantity of interest (velocity, pressure, salinity, etc.) from a limited number of measurements in [...] Read more.

It is shown in literature that sensor placement at the extrema of Proper Orthogonal Decomposition (POD) modes is efficient and leads to accurate reconstruction of the field of quantity of interest (velocity, pressure, salinity, etc.) from a limited number of measurements in the oceanography study. In this paper, we extend this approach of sensor placement and take into account measurement errors and detect possible malfunctioning sensors. We use the 24 hourly spatial wind field simulation data sets simulated using the Weather Research and Forecasting (WRF) model applied to the Maine Bay to evaluate the performances of our methods. Specifically, we use an exclusion disk strategy to distribute sensors when the extrema of POD modes are close. We demonstrate that this strategy can improve the accuracy of the reconstruction of the velocity field. It is also capable of reducing the standard deviation of the reconstruction from noisy measurements. Moreover, by a cross-validation technique, we successfully locate the malfunctioning sensors. Full article

(This article belongs to the Special Issue New Trends in Applications of Orthogonal Polynomials and Special Functions)

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225 KiB

Open AccessArticle

Recurrence Relations for Orthogonal Polynomials on Triangular Domains

by Abedallah Rababah

Mathematics 2016, 4(2), 25; https://doi.org/10.3390/math4020025 - 12 Apr 2016

Viewed by 4125

Abstract

In Farouki et al, 2003, Legendre-weighted orthogonal polynomials

P_{n, r} (u, v, w), r = 0, 1, \dots, n, n \geq 0

on the triangular domain [...] Read more.

In Farouki et al, 2003, Legendre-weighted orthogonal polynomials

P_{n, r} (u, v, w), r = 0, 1, \dots, n, n \geq 0

on the triangular domain

T = {(u, v, w) : u, v, w \geq 0, u + v + w = 1}

are constructed, where

u, v, w

are the barycentric coordinates. Unfortunately, evaluating the explicit formulas requires many operations and is not very practical from an algorithmic point of view. Hence, there is a need for a more efficient alternative. A very convenient method for computing orthogonal polynomials is based on recurrence relations. Such recurrence relations are described in this paper for the triangular orthogonal polynomials, providing a simple and fast algorithm for their evaluation. Full article

(This article belongs to the Special Issue New Trends in Applications of Orthogonal Polynomials and Special Functions)

266 KiB

Open AccessArticle

Qualitative Properties of Difference Equation of Order Six

by Abdul Khaliq and E.M. Elsayed

Mathematics 2016, 4(2), 24; https://doi.org/10.3390/math4020024 - 12 Apr 2016

Cited by 10 | Viewed by 4003

Abstract

In this paper we study the qualitative properties and the periodic nature of the solutions of the difference equation [...] Read more.

In this paper we study the qualitative properties and the periodic nature of the solutions of the difference equation

x_{n + 1} = α x_{n - 2} + \frac{β x_{n - 2}^{2}}{γ x_{n - 2} + δ x_{n - 5}}, n = 0, 1, . . .,

where the initial conditions

x_{- 5}, x_{- 4}, x_{- 3}, x_{- 2}, x_{- 1}, x_{0}

are arbitrary positive real numbers and

α, β, γ, δ

are positive constants. In addition, we derive the form of the solutions of some special cases of this equation. Full article

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270 KiB

Open AccessArticle

Existence of Semi Linear Impulsive Neutral Evolution Inclusions with Infinite Delay in Frechet Spaces

by Dimplekumar N. Chalishajar, Kulandhivel Karthikeyan and Annamalai Anguraj

Mathematics 2016, 4(2), 23; https://doi.org/10.3390/math4020023 - 6 Apr 2016

Cited by 4 | Viewed by 3692

Abstract

In this paper, sufficient conditions are given to investigate the existence of mild solutions on a semi-infinite interval for first order semi linear impulsive neutral functional differential evolution inclusions with infinite delay using a recently developed nonlinear alternative for contractive multivalued maps in [...] Read more.

In this paper, sufficient conditions are given to investigate the existence of mild solutions on a semi-infinite interval for first order semi linear impulsive neutral functional differential evolution inclusions with infinite delay using a recently developed nonlinear alternative for contractive multivalued maps in Frechet spaces due to Frigon combined with semigroup theory. The existence result has been proved without assumption of compactness of the semigroup. We introduced a new phase space for impulsive system with infinite delay and claim that the phase space considered by different authors are not correct. Full article

650 KiB

Open AccessArticle

Higher Order Methods for Nonlinear Equations and Their Basins of Attraction

by Kalyanasundaram Madhu and Jayakumar Jayaraman

Mathematics 2016, 4(2), 22; https://doi.org/10.3390/math4020022 - 1 Apr 2016

Cited by 11 | Viewed by 4363

Abstract

In this paper, we have presented a family of fourth order iterative methods, which uses weight functions. This new family requires three function evaluations to get fourth order accuracy. By the Kung–Traub hypothesis this family of methods is optimal and has an efficiency [...] Read more.

In this paper, we have presented a family of fourth order iterative methods, which uses weight functions. This new family requires three function evaluations to get fourth order accuracy. By the Kung–Traub hypothesis this family of methods is optimal and has an efficiency index of 1.587. Furthermore, we have extended one of the methods to sixth and twelfth order methods whose efficiency indices are 1.565 and 1.644, respectively. Some numerical examples are tested to demonstrate the performance of the proposed methods, which verifies the theoretical results. Further, we discuss the extraneous fixed points and basins of attraction for a few existing methods, such as Newton’s method and the proposed family of fourth order methods. An application problem arising from Planck’s radiation law has been verified using our methods. Full article

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666 KiB

Open AccessArticle

Optimal Control and Treatment of Infectious Diseases. The Case of Huge Treatment Costs

by Andrea Di Liddo

Mathematics 2016, 4(2), 21; https://doi.org/10.3390/math4020021 - 1 Apr 2016

Cited by 14 | Viewed by 4418

Abstract

The representation of the cost of a therapy is a key element in the formulation of the optimal control problem for the treatment of infectious diseases. The cost of the treatment is usually modeled by a function of the price and quantity of [...] Read more.

The representation of the cost of a therapy is a key element in the formulation of the optimal control problem for the treatment of infectious diseases. The cost of the treatment is usually modeled by a function of the price and quantity of drugs administered; this function should be the cost as subjectively perceived by the decision-maker. Nevertheless, in literature, the choice of the cost function is often simply done to make the problem more tractable. A specific problem is also given by very expensive therapies in the presence of a very high number of patients to be treated. Firstly, we investigate the optimal treatment of infectious diseases in the simplest case of a two-class population (susceptible and infectious people) and compare the results coming from five different shapes of cost functions. Finally, a model for the treatment of the HCV virus using the blowing-up cost function is investigated. Some numerical simulations are also given. Full article

(This article belongs to the Special Issue Optimal Control and Management of Infectious Diseases)

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