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The Degenerate Parametric Oscillator and Ince's Equation  [PDF]
Ricardo Cordero-Soto,Sergei K. Suslov
Physics , 2010, DOI: 10.1088/1751-8113/44/1/015101
Abstract: We construct Green's function for the quantum degenerate parametric oscillator in terms of standard solutions of Ince's equation in a framework of a general approach to harmonic oscillators. Exact time-dependent wave functions and their connections with dynamical invariants and SU(1,1) group are also discussed.
Parameter estimation of ODE's via nonparametric estimators  [PDF]
Nicolas J-B. Brunel
Mathematics , 2007, DOI: 10.1214/07-EJS132
Abstract: Ordinary differential equations (ODE's) are widespread models in physics, chemistry and biology. In particular, this mathematical formalism is used for describing the evolution of complex systems and it might consist of high-dimensional sets of coupled nonlinear differential equations. In this setting, we propose a general method for estimating the parameters indexing ODE's from times series. Our method is able to alleviate the computational difficulties encountered by the classical parametric methods. These difficulties are due to the implicit definition of the model. We propose the use of a nonparametric estimator of regression functions as a first-step in the construction of an M-estimator, and we show the consistency of the derived estimator under general conditions. In the case of spline estimators, we prove asymptotic normality, and that the rate of convergence is the usual $\sqrt{n}$-rate for parametric estimators. Some perspectives of refinements of this new family of parametric estimators are given.
Construction of Exact Parametric or Closed Form Solutions of Some Unsolvable Classes of Nonlinear ODEs (Abel's Nonlinear ODEs of the First Kind and Relative Degenerate Equations)  [PDF]
Dimitrios E. Panayotounakos,Theodoros I. Zarmpoutis
International Journal of Mathematics and Mathematical Sciences , 2011, DOI: 10.1155/2011/387429
Abstract: We provide a new mathematical technique leading to the construction of the exact parametric or closed form solutions of the classes of Abel's nonlinear differential equations (ODEs) of the first kind. These solutions are given implicitly in terms of Bessel functions of the first and the second kind (Neumann functions), as well as of the free member of the considered ODE; the parameter being introduced furnishes the order of the above Bessel functions and defines also the desired solutions of the considered ODE as one-parameter family of surfaces. The nonlinear initial or boundary value problems are also investigated. Finally, introducing a relative mathematical methodology, we construct the exact parametric or closed form solutions for several degenerate Abel's equation of the first kind. 1. Introduction One of the most known nonlinear differential equations (ODEs) is the Abel equation of the first kind. There are admissible functional transformations that can reduce the general first kind nonlinear Abel (ODE) to an Abel equation of the first kind of the normal form . None of these types of equations admits general solutions in terms of known (tabulated) functions, since only very special cases (depending on the form of the free member or on ) can be solved in parametric form [1–3]. We underline here that adding a cubic term in a Riccati (ODE), an Abel nonlinear (ODE) of the first kind arises. Our motivation of studying this problem was to provide a mathematical methodology through which the construction of exact parametric or closed form solutions of an Abel first kind nonlinear ODE form is possible. According to this method, separating properly the terms of the considered ODE, one leads to an equivalent system of two equations. The first one is a Riccati ODE, and the second one is a cubic equation that must be simultaneously hold true. In other words, any solution which satisfies these two equations (the Riccati ODE and the cubic one) satisfies also the Abel equation of the first kind of the normal form. Using basic results given by Kamke [1], the above Riccati equation is transformed to a second-order linear ODE of variable and determinable coefficients. We succeed in defining the coefficients of the above second-order linear ODE, so that the prescribed cubic equation hold true, and in the sequel in solving exactly this second-order linear ODE. This introducing procedure guides to the construction of the above-mentioned exact parametric or closed form solutions of the initial Abel equation. Here, we underline that all the above introduced functional
Studies on the equations of Ince's table  [PDF]
Yusuke Sasano
Mathematics , 2008,
Abstract: We study the phase space of the equations of Ince's table from the viewpoint of its accessible singularities and local index.
Parametric Bayesian Estimation of Differential Entropy and Relative Entropy  [PDF]
Maya Gupta,Santosh Srivastava
Entropy , 2010, DOI: 10.3390/e12040818
Abstract: Given iid samples drawn from a distribution with known parametric form, we propose the minimization of expected Bregman divergence to form Bayesian estimates of differential entropy and relative entropy, and derive such estimators for the uniform, Gaussian, Wishart, and inverse Wishart distributions. Additionally, formulas are given for a log gamma Bregman divergence and the differential entropy and relative entropy for the Wishart and inverse Wishart. The results, as always with Bayesian estimates, depend on the accuracy of the prior parameters, but example simulations show that the performance can be substantially improved compared to maximum likelihood or state-of-the-art nonparametric estimators.
Ince's limits for confluent and double-confluent Heun equations  [PDF]
B. D. Bonorino Figueiredo
Mathematics , 2005, DOI: 10.1063/1.2104267
Abstract: We find pairs of solutions to a differential equation which is obtained as a special limit of a generalized spheroidal wave equation (this is also known as confluent Heun equation). One solution in each pair is given by a series of hypergeometric functions and converges for any finite value of the independent variable $z$, while the other is given by a series of modified Bessel functions and converges for $|z|>|z_{0}|$, where $z_{0}$ denotes a regular singularity. For short, the preceding limit is called Ince's limit after Ince who have used the same procedure to get the Mathieu equations from the Whittaker-Hill ones. We find as well that, when $z_{0}$ tends to zero, the Ince limit of the generalized spheroidal wave equation turns out to be the Ince limit of a double-confluent Heun equation, for which solutions are provided. Finally, we show that the Schr\"odinger equation for inverse fourth and sixth-power potentials reduces to peculiar cases of the double-confluent Heun equation and its Ince's limit, respectively.
Parametric Integration by Magic Point Empirical Interpolation  [PDF]
Maximilian Ga?,Kathrin Glau
Mathematics , 2015,
Abstract: We derive analyticity criteria for explicit error bounds and an exponential rate of convergence of the magic point empirical interpolation method introduced by Barrault et al. (2004). Furthermore, we investigate its application to parametric integration. We find that the method is well-suited to Fourier transforms and has a wide range of applications in such diverse fields as probability and statistics, signal and image processing, physics, chemistry and mathematical finance. To illustrate the method, we apply it to the evaluation of probability densities by parametric Fourier inversion. Our numerical experiments display convergence of exponential order, even in cases where the theoretical results do not apply.
Entangled Singularity patterns of Photons in Ince-Gauss modes  [PDF]
Mario Krenn,Robert Fickler,Marcus Huber,Radek Lapkiewicz,William Plick,Sven Ramelow,Anton Zeilinger
Physics , 2012, DOI: 10.1103/PhysRevA.87.012326
Abstract: Photons with complex spatial mode structures open up possibilities for new fundamental high-dimensional quantum experiments and for novel quantum information tasks. Here we show for the first time entanglement of photons with complex vortex and singularity patterns called Ince-Gauss modes. In these modes, the position and number of singularities vary depending on the mode parameters. We verify 2-dimensional and 3-dimensional entanglement of Ince-Gauss modes. By measuring one photon and thereby defining its singularity pattern, we non-locally steer the singularity structure of its entangled partner, while the initial singularity structure of the photons is undefined. In addition we measure an Ince-Gauss specific quantum-correlation function with possible use in future quantum communication protocols.
Analytical solution in the Ince-Gaussian form of the beam propagating in the strong nonlocal media
强非局域非线性介质中光束传输的Ince-Gauss解

Zhang Xia-Ping,Liu You-Wen,
张霞萍
,刘友文

物理学报 , 2009,
Abstract: 利用强非局域非线性介质中傍轴光束传输的线性模型(Snyder-Mitchell模型)讨论了椭圆坐标系下光束传输过程,通过设立Ince多项式对Gauss函数的调制解得到了强非局域非线性介质中光束稳定传输的Ince-Gauss解.当Ince-Gauss光束的入射功率为临界功率时,光束保持孤子形式传输,否则传输光束的束宽呈现周期性波动,即为呼吸子形式.同时还数值模拟了呼吸子的传输过程.Ince-Gauss光在一定条件下可以连续转换为Hermite-Gauss光或Laguerre-Gauss光,图示展现了几个低阶Ince型光孤子及其转换情况.
Global attractor and stabilization for a coupled PDE-ODE system  [PDF]
Messoud Efendiev,Sergey Zelik
Mathematics , 2011,
Abstract: We study the asymptotic behavior of solutions of one coupled PDE-ODE system arising in mathematical biology as a model for the development of a forest ecosystem. In the case where the ODE-component of the system is monotone, we establish the existence of a smooth global attractor of finite Hausdorff and fractal dimension. The case of the non-monotone ODE-component is much more complicated. In particular, the set of equilibria becomes non-compact here and contains a huge number of essentially discontinuous solutions. Nevertheless, we prove the stabilization of any trajectory to a single equilibrium if the coupling constant is small enough.
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