HOME > Journal Current TOC
Mathematics [1 followers] Follow Open Access journal ISSN (Online) 2227-7390 Published by MDPI [146 journals] |
- Mathematics, Vol. 4, Pages 43: Cohen Macaulayness and Arithmetical Rank of
Generalized Theta Graphs
Authors: Seyyede Seyyedi, Farhad Rahmati
First page: 43
Abstract: In this paper, we study some algebraic invariants of the edge ideal of generalized theta graphs, such as arithmetical rank, big height and height. We give an upper bound for the difference between the arithmetical rank and big height. Moreover, all Cohen-Macaulay (and unmixed) graphs of this type will be characterized.
PubDate: 2016-06-29
DOI: 10.3390/math4030043
Issue No: Vol. 4, No. 3 (2016)
- Mathematics, Vol. 4, Pages 44: Exact Discrete Analogs of Canonical
Commutation and Uncertainty Relations
Authors: Vasily Tarasov
First page: 44
Abstract: An exact discretization of the canonical commutation and corresponding uncertainty relations are suggested. We prove that the canonical commutation relations of discrete quantum mechanics, which is based on standard finite difference, holds for constant wave functions only. In this paper, we use the recently proposed exact discretization of derivatives, which is based on differences that are represented by infinite series. This new mathematical tool allows us to build sensible discrete quantum mechanics based on the suggested differences and includes the correct canonical commutation and uncertainty relations.
PubDate: 2016-06-28
DOI: 10.3390/math4030044
Issue No: Vol. 4, No. 3 (2016)
- Mathematics, Vol. 4, Pages 45: Fourier Spectral Methods for Some Linear
Stochastic Space-Fractional Partial Differential Equations
Authors: Yanmei Liu, Monzorul Khan, Yubin Yan
First page: 45
Abstract: Fourier spectral methods for solving some linear stochastic space-fractional partial differential equations perturbed by space-time white noises in the one-dimensional case are introduced and analysed. The space-fractional derivative is defined by using the eigenvalues and eigenfunctions of the Laplacian subject to some boundary conditions. We approximate the space-time white noise by using piecewise constant functions and obtain the approximated stochastic space-fractional partial differential equations. The approximated stochastic space-fractional partial differential equations are then solved by using Fourier spectral methods. Error estimates in the L 2 -norm are obtained, and numerical examples are given.
PubDate: 2016-07-01
DOI: 10.3390/math4030045
Issue No: Vol. 4, No. 3 (2016)
- Mathematics, Vol. 4, Pages 46: Geometrical Inverse Preconditioning for
Symmetric Positive Definite Matrices
Authors: Jean-Paul Chehab, Marcos Raydan
First page: 46
Abstract: We focus on inverse preconditioners based on minimizing F ( X ) = 1 − cos ( X A , I ) , where X A is the preconditioned matrix and A is symmetric and positive definite. We present and analyze gradient-type methods to minimize F ( X ) on a suitable compact set. For this, we use the geometrical properties of the non-polyhedral cone of symmetric and positive definite matrices, and also the special properties of F ( X ) on the feasible set. Preliminary and encouraging numerical results are also presented in which dense and sparse approximations are included.
PubDate: 2016-07-09
DOI: 10.3390/math4030046
Issue No: Vol. 4, No. 3 (2016)
- Mathematics, Vol. 4, Pages 47: Uncertainty Relations for Quantum Coherence
Authors: Uttam Singh, Arun Pati, Manabendra Bera
First page: 47
Abstract: Coherence of a quantum state intrinsically depends on the choice of the reference basis. A natural question to ask is the following: if we use two or more incompatible reference bases, can there be some trade-off relation between the coherence measures in different reference bases? We show that the quantum coherence of a state as quantified by the relative entropy of coherence in two or more noncommuting reference bases respects uncertainty like relations for a given state of single and bipartite quantum systems. In the case of bipartite systems, we find that the presence of entanglement may tighten the above relation. Further, we find an upper bound on the sum of the relative entropies of coherence of bipartite quantum states in two noncommuting reference bases. Moreover, we provide an upper bound on the absolute value of the difference of the relative entropies of coherence calculated with respect to two incompatible bases.
PubDate: 2016-07-16
DOI: 10.3390/math4030047
Issue No: Vol. 4, No. 3 (2016)
- Mathematics, Vol. 4, Pages 48: Sharing of Nonlocality of a Single Member
of an Entangled Pair of Qubits Is Not Possible by More than Two Unbiased
Observers on the Other Wing
Authors: Shiladitya Mal, Archan Majumdar, Dipankar Home
First page: 48
Abstract: We address the recently posed question as to whether the nonlocality of a single member of an entangled pair of spin 1 / 2 particles can be shared among multiple observers on the other wing who act sequentially and independently of each other. We first show that the optimality condition for the trade-off between information gain and disturbance in the context of weak or non-ideal measurements emerges naturally when one employs a one-parameter class of positive operator valued measures (POVMs). Using this formalism we then prove analytically that it is impossible to obtain violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality by more than two Bobs in one of the two wings using unbiased input settings with an Alice in the other wing.
PubDate: 2016-07-16
DOI: 10.3390/math4030048
Issue No: Vol. 4, No. 3 (2016)
- Mathematics, Vol. 4, Pages 49: Preparational Uncertainty Relations for N
Continuous Variables
Authors: Spiros Kechrimparis, Stefan Weigert
First page: 49
Abstract: A smooth function of the second moments of N continuous variables gives rise to an uncertainty relation if it is bounded from below. We present a method to systematically derive such bounds by generalizing an approach applied previously to a single continuous variable. New uncertainty relations are obtained for multi-partite systems that allow one to distinguish entangled from separable states. We also investigate the geometry of the “uncertainty region” in the N ( 2 N + 1 ) -dimensional space of moments. It is shown to be a convex set, and the points on its boundary are found to be in one-to-one correspondence with pure Gaussian states of minimal uncertainty. For a single degree of freedom, the boundary can be visualized as one sheet of a “Lorentz-invariant” hyperboloid in the three-dimensional space of second moments.
PubDate: 2016-07-19
DOI: 10.3390/math4030049
Issue No: Vol. 4, No. 3 (2016)
- Mathematics, Vol. 4, Pages 50: Complete Classification of Cylindrically
Symmetric Static Spacetimes and the Corresponding Conservation Laws
Authors: Farhad Ali, Tooba Feroze
First page: 50
Abstract: In this paper we find the Noether symmetries of the Lagrangian of cylindrically symmetric static spacetimes. Using this approach we recover all cylindrically symmetric static spacetimes appeared in the classification by isometries and homotheties. We give different classes of cylindrically symmetric static spacetimes along with the Noether symmetries of the corresponding Lagrangians and conservation laws.
PubDate: 2016-08-08
DOI: 10.3390/math4030050
Issue No: Vol. 4, No. 3 (2016)
- Mathematics, Vol. 4, Pages 51: A New Approach to Study Fixed Point of
Multivalued Mappings in Modular Metric Spaces and Applications
Authors: Dilip Jain, Anantachai Padcharoen, Poom Kumam, Dhananjay Gopal
First page: 51
Abstract: The purpose of this paper is to present a new approach to study the existence of fixed points for multivalued F-contraction in the setting of modular metric spaces. In establishing this connection, we introduce the notion of multivalued F-contraction and prove corresponding fixed point theorems in complete modular metric space with some specific assumption on the modular. Then we apply our results to establish the existence of solutions for a certain type of non-linear integral equations.
PubDate: 2016-08-08
DOI: 10.3390/math4030051
Issue No: Vol. 4, No. 3 (2016)
- Mathematics, Vol. 4, Pages 52: Role of Measurement Incompatibility and
Uncertainty in Determining Nonlocality
Authors: Guruprasad Kar, Sibasish Ghosh, Sujit Choudhary, Manik Banik
First page: 52
Abstract: It has been recently shown that measurement incompatibility and fine grained uncertainty—a particular form of preparation uncertainty relation—are deeply related to the nonlocal feature of quantum mechanics. In particular, the degree of measurement incompatibility in a no-signaling theory determines the bound on the violation of Bell-CHSH inequality, and a similar role is also played by (fine-grained) uncertainty along with steering, a subtle non-local phenomenon. We review these connections, along with comments on the difference in the roles played by measurement incompatibility and uncertainty. We also discuss why the toy model of Spekkens (Phys. Rev. A 75, 032110 (2007)) shows no nonlocal feature even though steering is present in this theory.
PubDate: 2016-08-15
DOI: 10.3390/math4030052
Issue No: Vol. 4, No. 3 (2016)
- Mathematics, Vol. 4, Pages 53: Solution for Rational Systems of Difference
Equations of Order Three
Authors: Mohamed El-Dessoky
First page: 53
Abstract: In this paper, we consider the solution and periodicity of the following systems of difference equations: x n + 1 = y n − 2 − 1 + y n − 2 x n − 1 y n , y n + 1 = x n − 2 ± 1 ± x n − 2 y n − 1 x n , with initial conditions x − 2 , x − 1 , x 0 , y − 2 , y − 1 , and y 0 are nonzero real numbers.
PubDate: 2016-09-03
DOI: 10.3390/math4030053
Issue No: Vol. 4, No. 3 (2016)
- Mathematics, Vol. 4, Pages 54: Quantum Incompatibility in Collective
Measurements
Authors: Claudio Carmeli, Teiko Heinosaari, Daniel Reitzner, Jussi Schultz, Alessandro Toigo
First page: 54
Abstract: We study the compatibility (or joint measurability) of quantum observables in a setting where the experimenter has access to multiple copies of a given quantum system, rather than performing the experiments on each individual copy separately. We introduce the index of incompatibility as a quantifier of incompatibility in this multi-copy setting, as well as the notion of the compatibility stack representing various compatibility relations present in a given set of observables. We then prove a general structure theorem for multi-copy joint observables and use it to prove that all abstract compatibility stacks with three vertices have realizations in terms of quantum observables.
PubDate: 2016-09-10
DOI: 10.3390/math4030054
Issue No: Vol. 4, No. 3 (2016)
- Mathematics, Vol. 4, Pages 55: Amenability Modulo an Ideal of Second Duals
of Semigroup Algebras
Authors: Hamidreza Rahimi, Khalil Nabizadeh
First page: 55
Abstract: The aim of this paper is to investigate the amenability modulo, an ideal of Banach algebras with emphasis on applications to homological algebras. In doing so, we show that amenability modulo, an ideal of A * * implies amenability modulo, an ideal of A. Finally, for a large class of semigroups, we prove that l 1 ( S ) * * is amenable modulo I σ * * if and only if an appropriate group homomorphic image of S is finite, where I σ is the closed ideal induced by the least group congruence σ .
PubDate: 2016-09-13
DOI: 10.3390/math4030055
Issue No: Vol. 4, No. 3 (2016)
- Mathematics, Vol. 4, Pages 56: Quantum Measurements, Stochastic Networks,
the Uncertainty Principle, and the Not So Strange “Weak Values”
Authors: Dmitri Sokolovski
First page: 56
Abstract: Suppose we make a series of measurements on a chosen quantum system. The outcomes of the measurements form a sequence of random events, which occur in a particular order. The system, together with a meter or meters, can be seen as following the paths of a stochastic network connecting all possible outcomes. The paths are shaped from the virtual paths of the system, and the corresponding probabilities are determined by the measuring devices employed. If the measurements are highly accurate, the virtual paths become “real”, and the mean values of a quantity (a functional) are directly related to the frequencies with which the paths are traveled. If the measurements are highly inaccurate, the mean (weak) values are expressed in terms of the relative probabilities’ amplitudes. For pre- and post-selected systems they are bound to take arbitrary values, depending on the chosen transition. This is a direct consequence of the uncertainty principle, which forbids one from distinguishing between interfering alternatives, while leaving the interference between them intact.
PubDate: 2016-09-15
DOI: 10.3390/math4030056
Issue No: Vol. 4, No. 3 (2016)
- Mathematics, Vol. 4, Pages 21: Optimal Control and Treatment of Infectious
Diseases. The Case of Huge Treatment Costs
Authors: Andrea Di Liddo
First page: 21
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 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.
PubDate: 2016-04-01
DOI: 10.3390/math4020021
Issue No: Vol. 4, No. 2 (2016)
- Mathematics, Vol. 4, Pages 22: Higher Order Methods for Nonlinear
Equations and Their Basins of Attraction
Authors: Kalyanasundaram Madhu, Jayakumar Jayaraman
First page: 22
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 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.
PubDate: 2016-04-01
DOI: 10.3390/math4020022
Issue No: Vol. 4, No. 2 (2016)
- Mathematics, Vol. 4, Pages 23: Existence of Semi Linear Impulsive Neutral
Evolution Inclusions with Infinite Delay in Frechet Spaces
Authors: Dimplekumar Chalishajar, Kulandhivel Karthikeyan, Annamalai Anguraj
First page: 23
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 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.
PubDate: 2016-04-06
DOI: 10.3390/math4020023
Issue No: Vol. 4, No. 2 (2016)
- Mathematics, Vol. 4, Pages 24: Qualitative Properties of Difference
Equation of Order Six
Authors: Abdul Khaliq, E.M. Elsayed
First page: 24
Abstract: 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 + 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.
PubDate: 2016-04-12
DOI: 10.3390/math4020024
Issue No: Vol. 4, No. 2 (2016)
- Mathematics, Vol. 4, Pages 25: Recurrence Relations for Orthogonal
Polynomials on Triangular Domains
Authors: Abedallah Rababah
First page: 25
Abstract: In Farouki et al, 2003, Legendre-weighted orthogonal polynomials P n , r ( u , v , w ) , r = 0 , 1 , … , n , n ≥ 0 on the triangular domain T = { ( u , v , w ) : u , v , w ≥ 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.
PubDate: 2016-04-12
DOI: 10.3390/math4020025
Issue No: Vol. 4, No. 2 (2016)
- Mathematics, Vol. 4, Pages 26: POD-Based Constrained Sensor Placement and
Field Reconstruction from Noisy Wind Measurements: A Perturbation Study
Authors: Zhongqiang Zhang, Xiu Yang, Guang Lin
First page: 26
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 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.
PubDate: 2016-04-14
DOI: 10.3390/math4020026
Issue No: Vol. 4, No. 2 (2016)
- Mathematics, Vol. 4, Pages 27: Stagnation-Point Flow towards a Stretching
Vertical Sheet with Slip Effects
Authors: Khairy Zaimi, Anuar Ishak
First page: 27
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 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.
PubDate: 2016-04-21
DOI: 10.3390/math4020027
Issue No: Vol. 4, No. 2 (2016)
- Mathematics, Vol. 4, Pages 28: Lie Symmetry Analysis of the
Black-Scholes-Merton Model for European Options with Stochastic Volatility
Authors: Andronikos Paliathanasis, K. Krishnakumar, K.M. Tamizhmani, Peter Leach
First page: 28
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 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.
PubDate: 2016-05-03
DOI: 10.3390/math4020028
Issue No: Vol. 4, No. 2 (2016)
- Mathematics, Vol. 4, Pages 29: An Adaptive WENO Collocation Method for
Differential Equations with Random Coefficients
Authors: Wei Guo, Guang Lin, Andrew Christlieb, Jingmei Qiu
First page: 29
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 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.
PubDate: 2016-05-03
DOI: 10.3390/math4020029
Issue No: Vol. 4, No. 2 (2016)
- Mathematics, Vol. 4, Pages 30: New Approach for Fractional Order
Derivatives: Fundamentals and Analytic Properties
First page: 30
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 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.
PubDate: 2016-05-04
DOI: 10.3390/math4020030
Issue No: Vol. 4, No. 2 (2016)
- Mathematics, Vol. 4, Pages 31: Fractional Schrödinger Equation in the
Presence of the Linear Potential
First page: 31
Abstract: In this paper, we consider the time-dependent Schrödinger equation: i ∂ ψ ( x , t ) ∂ t = 1 2 ( − Δ ) α 2 ψ ( x , t ) + V ( x ) ψ ( x , t ) , x ∈ R , t > 0 with the Riesz space-fractional derivative of order 0 < α ≤ 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 · ρ including the free particle case. The derived equations are illustrated in different ways and verified by comparisons with a recently proposed numerical approach.
PubDate: 2016-05-04
DOI: 10.3390/math4020031
Issue No: Vol. 4, No. 2 (2016)
- Mathematics, Vol. 4, Pages 32: On the Dimension of Algebraic-Geometric
Trace Codes
Authors: Phong Le, Sunil Chetty
First page: 32
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 functions spaces associated to X.
PubDate: 2016-05-07
DOI: 10.3390/math4020032
Issue No: Vol. 4, No. 2 (2016)
- Mathematics, Vol. 4, Pages 33: Chaos Control in Three Dimensional Cancer
Model by State Space Exact Linearization Based on Lie Algebra
Authors: Mohammad Shahzad
First page: 33
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 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.
PubDate: 2016-05-10
DOI: 10.3390/math4020033
Issue No: Vol. 4, No. 2 (2016)
- Mathematics, Vol. 4, Pages 34: Lie Symmetries of (1+2) Nonautonomous
Evolution Equations in Financial Mathematics
Authors: Andronikos Paliathanasis, Richard Morris, Peter Leach
First page: 34
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 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.
PubDate: 2016-05-13
DOI: 10.3390/math4020034
Issue No: Vol. 4, No. 2 (2016)
- Mathematics, Vol. 4, Pages 35: Three Identities of the Catalan-Qi Numbers
Authors: Mansour Mahmoud, Feng Qi
First page: 35
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.
PubDate: 2016-05-26
DOI: 10.3390/math4020035
Issue No: Vol. 4, No. 2 (2016)
- Mathematics, Vol. 4, Pages 36: SIC-POVMs and Compatibility among Quantum
States
Authors: Blake Stacey
First page: 36
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 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.
PubDate: 2016-06-01
DOI: 10.3390/math4020036
Issue No: Vol. 4, No. 2 (2016)
- Mathematics, Vol. 4, Pages 37: Smoothness in Binomial Edge Ideals
Authors: Hamid Damadi, Farhad Rahmati
First page: 37
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 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.
PubDate: 2016-06-01
DOI: 10.3390/math4020037
Issue No: Vol. 4, No. 2 (2016)
- Mathematics, Vol. 4, Pages 38: Measurement Uncertainty for Finite Quantum
Observables
First page: 38
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 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.
PubDate: 2016-06-02
DOI: 10.3390/math4020038
Issue No: Vol. 4, No. 2 (2016)
- Mathematics, Vol. 4, Pages 39: Morphisms and Order Ideals of Toric Posets
Authors: Matthew Macauley
First page: 39
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 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.
PubDate: 2016-06-04
DOI: 10.3390/math4020039
Issue No: Vol. 4, No. 2 (2016)
- Mathematics, Vol. 4, Pages 40: Uncertainty Relations and Possible
Experience
Authors: Gregg Jaeger
First page: 40
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 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.
PubDate: 2016-06-03
DOI: 10.3390/math4020040
Issue No: Vol. 4, No. 2 (2016)
- Mathematics, Vol. 4, Pages 41: Entropic Uncertainty Relations for
Successive Generalized Measurements
Authors: Kyunghyun Baek, Wonmin Son
First page: 41
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 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.
PubDate: 2016-06-07
DOI: 10.3390/math4020041
Issue No: Vol. 4, No. 2 (2016)
- Mathematics, Vol. 4, Pages 42: Exponential Energy Decay of Solutions for a
Transmission Problem With Viscoelastic Term and Delay
Authors: Danhua Wang, Gang Li, Biqing Zhu
First page: 42
Abstract: 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.
PubDate: 2016-06-09
DOI: 10.3390/math4020042
Issue No: Vol. 4, No. 2 (2016)
- Mathematics, Vol. 4, Pages 2: Barrier Option Under Lévy Model : A PIDE
and Mellin Transform Approach
Authors: Sudip Chandra, Diganta Mukherjee
First page: 2
Abstract: We propose a stochastic model to develop a partial integro-differential equation (PIDE) for pricing and pricing expression for fixed type single Barrier options based on the Itô-Lévy calculus with the help of Mellin transform. The stock price is driven by a class of infinite activity Lévy processes leading to the market inherently incomplete, and dynamic hedging is no longer risk free. We first develop a PIDE for fixed type Barrier options, and apply the Mellin transform to derive a pricing expression. Our main contribution is to develop a PIDE with its closed form pricing expression for the contract. The procedure is easy to implement for all class of Lévy processes numerically. Finally, the algorithm for computing numerically is presented with results for a set of Lévy processes.
PubDate: 2016-01-04
DOI: 10.3390/math4010002
Issue No: Vol. 4, No. 1 (2016)
- Mathematics, Vol. 4, Pages 3: Multiplicative Expression for the
Coefficient in Fermionic 3–3 Relation
Authors: Igor Korepanov
First page: 3
Abstract: Recently, a family of fermionic relations were discovered corresponding to Pachner move 3–3 and parameterized by complex-valued 2-cocycles, where the weight of a pentachoron (4-simplex) is a Grassmann–Gaussian exponent. Here, the proportionality coefficient between Berezin integrals in the l.h.s. and r.h.s. of such relations is written in a form multiplicative over simplices.
PubDate: 2016-01-20
DOI: 10.3390/math4010003
Issue No: Vol. 4, No. 1 (2016)
- Mathematics, Vol. 4, Pages 4: Acknowledgement to Reviewers of Mathematics
in 2015
Authors: Mathematics Editorial Office
First page: 4
Abstract: The editors of Mathematics would like to express their sincere gratitude to the following reviewers for assessing manuscripts in 2015. [...]
PubDate: 2016-01-25
DOI: 10.3390/math4010004
Issue No: Vol. 4, No. 1 (2016)
- Mathematics, Vol. 4, Pages 5: Modular Forms and Weierstrass Mock Modular
Forms
Authors: Amanda Clemm
First page: 5
Abstract: Alfes, Griffin, Ono, and Rolen have shown that the harmonic Maass forms arising from Weierstrass ζ-functions associated to modular elliptic curves “encode” the vanishing and nonvanishing for central values and derivatives of twisted Hasse-Weil L-functions for elliptic curves. Previously, Martin and Ono proved that there are exactly five weight 2 newforms with complex multiplication that are eta-quotients. In this paper, we construct a canonical harmonic Maass form for these five curves with complex multiplication. The holomorphic part of this harmonic Maass form arises from the Weierstrass ζ-function and is referred to as the Weierstrass mock modular form. We prove that the Weierstrass mock modular form for these five curves is itself an eta-quotient or a twist of one. Using this construction, we also obtain p-adic formulas for the corresponding weight 2 newform using Atkin’s U-operator.
PubDate: 2016-02-02
DOI: 10.3390/math4010005
Issue No: Vol. 4, No. 1 (2016)
- Mathematics, Vol. 4, Pages 6: Microtubules Nonlinear Models Dynamics
Authors: Nur Alam, Fethi Belgacem
First page: 6
Abstract: In this research article, we present exact solutions with parameters for two nonlinear model partial differential equations(PDEs) describing microtubules, by implementing the exp(−Φ(ξ))-Expansion Method. The considered models, describing highly nonlinear dynamics of microtubules, can be reduced to nonlinear ordinary differential equations. While the first PDE describes the longitudinal model of nonlinear dynamics of microtubules, the second one describes the nonlinear model of dynamics of radial dislocations in microtubules. The acquired solutions are then graphically presented, and their distinct properties are enumerated in respect to the corresponding dynamic behavior of the microtubules they model. Various patterns, including but not limited to regular, singular kink-like, as well as periodicity exhibiting ones, are detected. Being the method of choice herein, the exp(−Φ(ξ))-Expansion Method not disappointing in the least, is found and declared highly efficient.
PubDate: 2016-02-04
DOI: 10.3390/math4010006
Issue No: Vol. 4, No. 1 (2016)
- Mathematics, Vol. 4, Pages 7: Nevanlinna’s Five Values Theorem on
Annuli
Authors: Hong-Yan Xu, Hua Wang
First page: 7
Abstract: By using the second main theorem of the meromorphic function on annuli, we investigate the problem on two meromorphic functions partially sharing five or more values and obtain some theorems that improve and generalize the previous results given by Cao and Yi.
PubDate: 2016-02-18
DOI: 10.3390/math4010007
Issue No: Vol. 4, No. 1 (2016)
- Mathematics, Vol. 4, Pages 8: Tight State-Independent Uncertainty
Relations for Qubits
Authors: Alastair Abbott, Pierre-Louis Alzieu, Michael Hall, Cyril Branciard
First page: 8
Abstract: The well-known Robertson–Schrödinger uncertainty relations have state-dependent lower bounds, which are trivial for certain states. We present a general approach to deriving tight state-independent uncertainty relations for qubit measurements that completely characterise the obtainable uncertainty values. This approach can give such relations for any number of observables, and we do so explicitly for arbitrary pairs and triples of qubit measurements. We show how these relations can be transformed into equivalent tight entropic uncertainty relations. More generally, they can be expressed in terms of any measure of uncertainty that can be written as a function of the expectation value of the observable for a given state.
PubDate: 2016-02-24
DOI: 10.3390/math4010008
Issue No: Vol. 4, No. 1 (2016)
- Mathematics, Vol. 4, Pages 9: Coefficient Inequalities of Second Hankel
Determinants for Some Classes of Bi-Univalent Functions
Authors: Rayaprolu Bharavi Sharma, Kalikota Rajya Laxmi
First page: 9
Abstract: In this paper, we investigate two sub-classes S∗ (θ, β) and K∗ (θ, β) of bi-univalent functions in the open unit disc Δ that are subordinate to certain analytic functions. For functions belonging to these classes, we obtain an upper bound for the second Hankel determinant H2 (2).
PubDate: 2016-02-25
DOI: 10.3390/math4010009
Issue No: Vol. 4, No. 1 (2016)
- Mathematics, Vol. 4, Pages 10: A Note on Burg’s Modified Entropy in
Statistical Mechanics
Authors: Amritansu Ray, S. Majumder
First page: 10
Abstract: Burg’s entropy plays an important role in this age of information euphoria, particularly in understanding the emergent behavior of a complex system such as statistical mechanics. For discrete or continuous variable, maximization of Burg’s Entropy subject to its only natural and mean constraint always provide us a positive density function though the Entropy is always negative. On the other hand, Burg’s modified entropy is a better measure than the standard Burg’s entropy measure since this is always positive and there is no computational problem for small probabilistic values. Moreover, the maximum value of Burg’s modified entropy increases with the number of possible outcomes. In this paper, a premium has been put on the fact that if Burg’s modified entropy is used instead of conventional Burg’s entropy in a maximum entropy probability density (MEPD) function, the result yields a better approximation of the probability distribution. An important lemma in basic algebra and a suitable example with tables and graphs in statistical mechanics have been given to illustrate the whole idea appropriately.
PubDate: 2016-02-27
DOI: 10.3390/math4010010
Issue No: Vol. 4, No. 1 (2016)
- Mathematics, Vol. 4, Pages 11: Solution of Excited Non-Linear Oscillators
under Damping Effects Using the Modified Differential Transform Method
Authors: H. Abdelhafez
First page: 11
Abstract: The modified differential transform method (MDTM), Laplace transform and Padé approximants are used to investigate a semi-analytic form of solutions of nonlinear oscillators in a large time domain. Forced Duffing and forced van der Pol oscillators under damping effect are studied to investigate semi-analytic forms of solutions. Moreover, solutions of the suggested nonlinear oscillators are obtained using the fourth-order Runge-Kutta numerical solution method. A comparison of the result by the numerical Runge-Kutta fourth-order accuracy method is compared with the result by the MDTM and plotted in a long time domain.
PubDate: 2016-03-02
DOI: 10.3390/math4010011
Issue No: Vol. 4, No. 1 (2016)
- Mathematics, Vol. 4, Pages 12: Inverse Eigenvalue Problems for Two Special
Acyclic Matrices
Authors: Debashish Sharma, Mausumi Sen
First page: 12
Abstract: In this paper, we study two inverse eigenvalue problems (IEPs) of constructing two special acyclic matrices. The first problem involves the reconstruction of matrices whose graph is a path, from given information on one eigenvector of the required matrix and one eigenvalue of each of its leading principal submatrices. The second problem involves reconstruction of matrices whose graph is a broom, the eigen data being the maximum and minimum eigenvalues of each of the leading principal submatrices of the required matrix. In order to solve the problems, we use the recurrence relations among leading principal minors and the property of simplicity of the extremal eigenvalues of acyclic matrices.
PubDate: 2016-03-03
DOI: 10.3390/math4010012
Issue No: Vol. 4, No. 1 (2016)
- Mathematics, Vol. 4, Pages 13: Existence Results for a New Class of
Boundary Value Problems of Nonlinear Fractional Differential Equations
Authors: Meysam Alvan, Rahmat Darzi, Amin Mahmoodi
First page: 13
Abstract: In this article, we study the following fractional boundary value problem \(_{}^{c}D_{0^{+}}^{\alpha}u\left( t \right) + 2r\ _{}^{c}D_{0^{+}}^{\alpha - 1}u\left( t \right) \)\( + r^{2}\ _{}^{c}D_{0^{+}}^{\alpha - 2}u\left( t \right) = f\left( {t,u\left( t \right)} \right),\quad r > 0,\quad 0 < t < 1, \;\; u\left( 0 \right) = u\left( 1 \right),\quad u^{\prime}\left( 0 \right) = u^{\prime}\left( 1 \right),\quad u^{\prime}\left( \xi \right) + ru\left( \xi \right) = \eta,\)\(\quad\xi \in \left( {0,1} \right)\) Where \(2 \leq \alpha < 3,\ _{}^{c}D_{0^{+}}^{\alpha - i}\left( {i = 0,1,2} \right) \) are the standard Caputo derivative and \(\eta \) is a positive real number. Some new existence results are obtained by means of the contraction mapping principle and Schauder fixed point theorem. Some illustrative examples are also presented.
PubDate: 2016-03-04
DOI: 10.3390/math4010013
Issue No: Vol. 4, No. 1 (2016)
- Mathematics, Vol. 4, Pages 14: Cost Effectiveness Analysis of Optimal
Malaria Control Strategies in Kenya
Authors: Gabriel Otieno, Joseph Koske, John Mutiso
First page: 14
Abstract: Malaria remains a leading cause of mortality and morbidity among the children under five and pregnant women in sub-Saharan Africa, but it is preventable and controllable provided current recommended interventions are properly implemented. Better utilization of malaria intervention strategies will ensure the gain for the value for money and producing health improvements in the most cost effective way. The purpose of the value for money drive is to develop a better understanding (and better articulation) of costs and results so that more informed, evidence-based choices could be made. Cost effectiveness analysis is carried out to inform decision makers on how to determine where to allocate resources for malaria interventions. This study carries out cost effective analysis of one or all possible combinations of the optimal malaria control strategies (Insecticide Treated Bednets—ITNs, Treatment, Indoor Residual Spray—IRS and Intermittent Preventive Treatment for Pregnant Women—IPTp) for the four different transmission settings in order to assess the extent to which the intervention strategies are beneficial and cost effective. For the four different transmission settings in Kenya the optimal solution for the 15 strategies and their associated effectiveness are computed. Cost-effective analysis using Incremental Cost Effectiveness Ratio (ICER) was done after ranking the strategies in order of the increasing effectiveness (total infections averted). The findings shows that for the endemic regions the combination of ITNs, IRS, and IPTp was the most cost-effective of all the combined strategies developed in this study for malaria disease control and prevention; for the epidemic prone areas is the combination of the treatment and IRS; for seasonal areas is the use of ITNs plus treatment; and for the low risk areas is the use of treatment only. Malaria transmission in Kenya can be minimized through tailor-made intervention strategies for malaria control which produces health improvements in the most cost effective way for different epidemiological zones. This offers the good value for money for the public health programs and can guide in the allocation of malaria control resources for the post-2015 malaria eradication strategies and the achievement of the Sustainable Development Goals.
PubDate: 2016-03-09
DOI: 10.3390/math4010014
Issue No: Vol. 4, No. 1 (2016)
- Mathematics, Vol. 4, Pages 15: Conformal Maps, Biharmonic Maps, and the
Warped Product
Authors: Seddik Ouakkas, Djelloul Djebbouri
First page: 15
Abstract: In this paper we study some properties of conformal maps between equidimensional manifolds, we construct new example of non-harmonic biharmonic maps and we characterize the biharmonicity of some maps on the warped product manifolds.
PubDate: 2016-03-08
DOI: 10.3390/math4010015
Issue No: Vol. 4, No. 1 (2016)
- Mathematics, Vol. 4, Pages 16: New Method of Randomized Forecasting Using
Authors: Yuri Popkov, Yuri Dubnov, Alexey Popkov
First page: 16
Abstract: We propose a new method of randomized forecasting (RF-method), which operates with models described by systems of linear ordinary differential equations with random parameters. The RF-method is based on entropy-robust estimation of the probability density functions (PDFs) of model parameters and measurement noises. The entropy-optimal estimator uses a limited amount of data. The method of randomized forecasting is applied to World population prediction. Ensembles of entropy-optimal prognostic trajectories of World population and their probability characteristics are generated. We show potential preferences of the proposed method in comparison with existing methods.
PubDate: 2016-03-11
DOI: 10.3390/math4010016
Issue No: Vol. 4, No. 1 (2016)
- Mathematics, Vol. 4, Pages 17: Skew Continuous Morphisms of Ordered
Lattice Ringoids
Authors: Sergey Ludkowski
First page: 17
Abstract: Skew continuous morphisms of ordered lattice semirings and ringoids are studied. Different associative semirings and non-associative ringoids are considered. Theorems about properties of skew morphisms are proved. Examples are given. One of the main similarities between them is related to cones in algebras of non locally compact groups.
PubDate: 2016-03-16
DOI: 10.3390/math4010017
Issue No: Vol. 4, No. 1 (2016)
- Mathematics, Vol. 4, Pages 18: Dynamics and the Cohomology of Measured
Laminations
First page: 18
Abstract: In this paper, the interconnection between the cohomology of measured group actions and the cohomology of measured laminations is explored, the latter being a generalization of the former for the case of discrete group actions and cocycles evaluated on abelian groups. This relation gives a rich interplay between these concepts. Several results can be adapted to this setting—for instance, Zimmer’s reduction of the coefficient group of bounded cocycles or Fustenberg’s cohomological obstruction for extending the ergodicity \(\mathbb{Z}\)-action to a skew product relative to an \(S^{1}\) evaluated cocycle. Another way to think about foliated cocycles is also shown, and a particular application is the characterization of the existence of certain classes of invariant measures for smooth foliations in terms of the \(L^{\infty}\)-cohomology class of the infinitesimal holonomy.
PubDate: 2016-03-15
DOI: 10.3390/math4010018
Issue No: Vol. 4, No. 1 (2016)
- Mathematics, Vol. 4, Pages 19: Solution of Differential Equations with
Polynomial Coefficients with the Aid of an Analytic Continuation of
Laplace Transform
Authors: Tohru Morita, Ken-ichi Sato
First page: 19
Abstract: In a series of papers, we discussed the solution of Laplace’s differential equation (DE) by using fractional calculus, operational calculus in the framework of distribution theory, and Laplace transform. The solutions of Kummer’s DE, which are expressed by the confluent hypergeometric functions, are obtained with the aid of the analytic continuation (AC) of Riemann–Liouville fractional derivative (fD) and the distribution theory in the space D′R or the AC of Laplace transform. We now obtain the solutions of the hypergeometric DE, which are expressed by the hypergeometric functions, with the aid of the AC of Riemann–Liouville fD, and the distribution theory in the space D′r,R, which is introduced in this paper, or by the term-by-term inverse Laplace transform of AC of Laplace transform of the solution expressed by a series.
PubDate: 2016-03-17
DOI: 10.3390/math4010019
Issue No: Vol. 4, No. 1 (2016)
- Mathematics, Vol. 4, Pages 20: Birkhoff Normal Forms, KAM Theory and Time
Reversal Symmetry for Certain Rational Map
First page: 20
Abstract: By using the KAM(Kolmogorov-Arnold-Moser) theory and time reversal symmetries, we investigate the stability of the equilibrium solutions of the system: x n + 1 = 1 y n , y n + 1 = β x n 1 + y n , n = 0 , 1 , 2 , … , where the parameter β > 0 , and initial conditions x 0 and y 0 are positive numbers. We obtain the Birkhoff normal form for this system and prove the existence of periodic points with arbitrarily large periods in every neighborhood of the unique positive equilibrium. We use invariants to find a Lyapunov function and Morse’s lemma to prove closedness of invariants. We also use the time reversal symmetry method to effectively find some feasible periods and the corresponding periodic orbits.
PubDate: 2016-03-18
DOI: 10.3390/math4010020
Issue No: Vol. 4, No. 1 (2016)