Construction of classical superintegrable systems with higher order integrals of motion from ladder operators

We construct integrals of motion for multidimensional classical systems from ladder operators of one-dimensional systems. This method can be used to obtain new systems with higher order integrals. We show how these integrals generate a polynomial Poisson algebra. We consider a one-dimensional system with third order ladders operators and found a family of superintegrable systems with higher order integrals of motion. We obtain also the polynomial algebra generated by these integrals. We calculate numerically the trajectories and show that all bounded trajectories are closed.Comment: 10 pages, 4 figures, to appear in j.math.phys

Differentiation of dementia with Lewy bodies from Alzheimer's disease using a dopaminergic presynaptic ligand

Background: Dementia with Lewy bodies (DLB) is one of the main differential diagnoses of Alzheimer's disease (AD). Key pathological features of patients with DLB are not only the presence of cerebral cortical neuronal loss, with Lewy bodies in surviving neurones, but also loss of nigrostriatal dopaminergic neurones, similar to that of Parkinson's disease (PD). In DLB there is 40-70% loss of striatal dopamine.Objective: To determine if detection of this dopaminergic degeneration can help to distinguish DLB from AD during life.Methods: The integrity of the nigrostriatal metabolism in 27 patients with DLB, 17 with AD, 19 drug naive patients with PD, and 16 controls was assessed using a dopaminergic presynaptic ligand, I-123-labelled 2beta-carbomethoxy-3beta-(4-iodophenyl)-N-(3-fluoropropyl)nortropane (FP-CIT), and single photon emission tomography (SPET). A SPET scan was carried out with a single slice, brain dedicated tomograph (SME 810) 3.5 hours after intravenous injection of 185 MBq FP-CIT. With occipital cortex used as a radioactivity uptake reference, ratios for the caudate nucleus and the anterior and posterior putamen of both hemispheres were calculated. All scans were also rated by a simple visual method.Results: Both DLB and PD patients had significantly lower uptake of radioactivity than patients with (p<0.01) and controls (p<0.001) in the caudate nucleus and the anterior and posterior Putamen.Conclusion: FP-CIT SPET provides a means of distinguishing DLB from AD during life

Exact Solutions of a (2+1)-Dimensional Nonlinear Klein-Gordon Equation

The purpose of this paper is to present a class of particular solutions of a C(2,1) conformally invariant nonlinear Klein-Gordon equation by symmetry reduction. Using the subgroups of similitude group reduced ordinary differential equations of second order and their solutions by a singularity analysis are classified. In particular, it has been shown that whenever they have the Painlev\'e property, they can be transformed to standard forms by Moebius transformations of dependent variable and arbitrary smooth transformations of independent variable whose solutions, depending on the values of parameters, are expressible in terms of either elementary functions or Jacobi elliptic functions.Comment: 16 pages, no figures, revised versio

Microcirculation: Physiology, pathophysiology, and clinical application

This paper briefly reviews the physiological components of the microcirculation, focusing on its function in homeostasis and its central function in the realization of oxygen transport to tissue cells. Its pivotal role in the understanding of circulatory compromise in states of shock and renal compromise is discussed. Our introduction of hand-held vital microscopes (HVM) to clinical medicine has revealed the importance of the microcirculation as a central target organ in states of critical illness and inadequate response to therapy. Technical and methodological developments have been made in hardware and in software including our recent introduction and validation of automatic analysis software called MicroTools, which now allows point-of-care use of HVM imaging at the bedside for instant availability of functional microcirculatory parameters needed for microcirculatory targeted resuscitation procedures to be a reality

Movable algebraic singularities of second-order ordinary differential equations

Any nonlinear equation of the form y''=\sum_{n=0}^N a_n(z)y^n has a (generally branched) solution with leading order behaviour proportional to (z-z_0)^{-2/(N-1)} about a point z_0, where the coefficients a_n are analytic at z_0 and a_N(z_0)\ne 0. We consider the subclass of equations for which each possible leading order term of this form corresponds to a one-parameter family of solutions represented near z_0 by a Laurent series in fractional powers of z-z_0. For this class of equations we show that the only movable singularities that can be reached by analytic continuation along finite-length curves are of the algebraic type just described. This work generalizes previous results of S. Shimomura. The only other possible kind of movable singularity that might occur is an accumulation point of algebraic singularities that can be reached by analytic continuation along infinitely long paths ending at a finite point in the complex plane. This behaviour cannot occur for constant coefficient equations in the class considered. However, an example of R. A. Smith shows that such singularities do occur in solutions of a simple autonomous second-order differential equation outside the class we consider here

Analytic Behaviour of Competition among Three Species

We analyse the classical model of competition between three species studied by May and Leonard ({\it SIAM J Appl Math} \textbf{29} (1975) 243-256) with the approaches of singularity analysis and symmetry analysis to identify values of the parameters for which the system is integrable. We observe some striking relations between critical values arising from the approach of dynamical systems and the singularity and symmetry analyses.Comment: 14 pages, to appear in Journal of Nonlinear Mathematical Physic

Thermodynamic large fluctuations from uniformized dynamics

Large fluctuations have received considerable attention as they encode information on the fine-scale dynamics. Large deviation relations known as fluctuation theorems also capture crucial nonequilibrium thermodynamical properties. Here we report that, using the technique of uniformization, the thermodynamic large deviation functions of continuous-time Markov processes can be obtained from Markov chains evolving in discrete time. This formulation offers new theoretical and numerical approaches to explore large deviation properties. In particular, the time evolution of autonomous and non-autonomous processes can be expressed in terms of a single Poisson rate. In this way the uniformization procedure leads to a simple and efficient way to simulate stochastic trajectories that reproduce the exact fluxes statistics. We illustrate the formalism for the current fluctuations in a stochastic pump model

On Approximation of the Eigenvalues of Perturbed Periodic Schrodinger Operators

This paper addresses the problem of computing the eigenvalues lying in the gaps of the essential spectrum of a periodic Schrodinger operator perturbed by a fast decreasing potential. We use a recently developed technique, the so called quadratic projection method, in order to achieve convergence free from spectral pollution. We describe the theoretical foundations of the method in detail, and illustrate its effectiveness by several examples.Comment: 17 pages, 2 tables and 2 figure

Nonlocal aspects of λ\lambda-symmetries and ODEs reduction

A reduction method of ODEs not possessing Lie point symmetries makes use of the so called $\lambda$-symmetries (C. Muriel and J. L. Romero, \emph{IMA J. Appl. Math.} \textbf{66}, 111-125, 2001). The notion of covering for an ODE $\mathcal{Y}$ is used here to recover $\lambda$-symmetries of $\mathcal{Y}$ as nonlocal symmetries. In this framework, by embedding $\mathcal{Y}$ into a suitable system $\mathcal{Y}^{\prime}$ determined by the function $\lambda$, any $\lambda$-symmetry of $\mathcal{Y}$ can be recovered by a local symmetry of $\mathcal{Y}^{\prime}$. As a consequence, the reduction method of Muriel and Romero follows from the standard method of reduction by differential invariants applied to $\mathcal{Y}^{\prime}$.Comment: 13 page

Solutions for certain classes of Riccati differential equation

We derive some analytic closed-form solutions for a class of Riccati equation y'(x)-\lambda_0(x)y(x)\pm y^2(x)=\pm s_0(x), where \lambda_0(x), s_0(x) are C^{\infty}-functions. We show that if \delta_n=\lambda_n s_{n-1}-\lambda_{n-1}s_n=0, where \lambda_{n}= \lambda_{n-1}^\prime+s_{n-1}+\lambda_0\lambda_{n-1} and s_{n}=s_{n-1}^\prime+s_0\lambda_{k-1}, n=1,2,..., then The Riccati equation has a solution given by y(x)=\mp s_{n-1}(x)/\lambda_{n-1}(x). Extension to the generalized Riccati equation y'(x)+P(x)y(x)+Q(x)y^2(x)=R(x) is also investigated.Comment: 10 page