Nonlocal symmetries of Riccati and Abel chains and their similarity reductions

We study nonlocal symmetries and their similarity reductions of Riccati and Abel chains. Our results show that all the equations in Riccati chain share the same form of nonlocal symmetry. The similarity reduced $N^{th}$ order ordinary differential equation (ODE), $N=2, 3,4,...$, in this chain yields $(N-1)^{th}$ order ODE in the same chain. All the equations in the Abel chain also share the same form of nonlocal symmetry (which is different from the one that exist in Riccati chain) but the similarity reduced $N^{th}$ order ODE, $N=2, 3,4,$, in the Abel chain always ends at the $(N-1)^{th}$ order ODE in the Riccati chain. We describe the method of finding general solution of all the equations that appear in these chains from the nonlocal symmetry.Comment: Accepted for publication in J. Math. Phy

Transformations of Heun's equation and its integral relations

We find transformations of variables which preserve the form of the equation for the kernels of integral relations among solutions of the Heun equation. These transformations lead to new kernels for the Heun equation, given by single hypergeometric functions (Lambe-Ward-type kernels) and by products of two hypergeometric functions (Erd\'elyi-type). Such kernels, by a limiting process, also afford new kernels for the confluent Heun equation.Comment: This version was published in J. Phys. A: Math. Theor. 44 (2011) 07520

Exact partial information decompositions for Gaussian systems based on dependency constraints

The Partial Information Decomposition (PID) [arXiv:1004.2515] provides a theoretical framework to characterize and quantify the structure of multivariate information sharing. A new method (Idep) has recently been proposed for computing a two-predictor PID over discrete spaces. [arXiv:1709.06653] A lattice of maximum entropy probability models is constructed based on marginal dependency constraints, and the unique information that a particular predictor has about the target is defined as the minimum increase in joint predictor-target mutual information when that particular predictor-target marginal dependency is constrained. Here, we apply the Idep approach to Gaussian systems, for which the marginally constrained maximum entropy models are Gaussian graphical models. Closed form solutions for the Idep PID are derived for both univariate and multivariate Gaussian systems. Numerical and graphical illustrations are provided, together with practical and theoretical comparisons of the Idep PID with the minimum mutual information PID (Immi). [arXiv:1411.2832] In particular, it is proved that the Immi method generally produces larger estimates of redundancy and synergy than does the Idep method. In discussion of the practical examples, the PIDs are complemented by the use of deviance tests for the comparison of Gaussian graphical models.Comment: 39 pages, 9 figures, 9 table

A wave driver theory for vortical waves propagating across junctions with application to those between rigid and compliant walls

A theory is described for propagation of vortical waves across alternate rigid and compliant panels. The structure in the fluid side at the junction of panels is a highly vortical narrow viscous structure which is idealized as a wave driver. The wave driver is modelled as a ‘half source cum half sink’. The incoming wave terminates into this structure and the outgoing wave emanates from it. The model is described by half Fourier–Laplace transforms respectively for the upstream and downstream sides of the junction. The cases below cutoff and above cutoff frequencies are studied. The theory completely reproduces the direct numerical simulation results of Davies & Carpenter (J. Fluid Mech., vol. 335, 1997, p. 361). Particularly, the jumps across the junction in the kinetic energy integral, the vorticity integral and other related quantities as obtained in the work of Davies & Carpenter are completely reproduced. Also, some important new concepts emerge, notable amongst which is the concept of the pseudo group velocity

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

A systematic review of ongoing clinical trials in optic pathway gliomas

INTRODUCTION: Optic pathway gliomas (OPGs), also known as Visual Pathway Gliomas, are insidious, debilitating tumours. They are most commonly WHO grade 1 pilocytic astrocytomas and frequently occur in patients with neurofibromatosis type 1. The location of OPGs within the optic pathway typically precludes complete resection or optimal radiation dosing, hence outcomes remain poor compared to many other low-grade gliomas. The aim of this systematic review was to formulate a comprehensive list of all current ongoing clinical trials that are specifically looking at clinical care of OPGs in order to identify trends in current research and provide an overview to guide future research efforts. METHODS: This systematic review was conducted in line with the Preferred Reporting Items for Systematic Reviews and Meta-Analysis (PRISMA) guidelines. The Cochrane Controlled Register of Trials (CENTRAL) and ClinicalTrials.gov were searched. Inclusion and exclusion criteria were applied and final results were reviewed. RESULTS: 501 clinical trials were identified with the search strategy. All were screened and eligible studies extracted and reviewed. This yielded 36 ongoing clinical trials, 27 of which were pharmacological agents in phase I-III. The remaining trials were a mixture of biological agents, radiation optimisation, diagnostic imaging, surgical intervention, and a social function analysis. CONCLUSION: OPG is a complex multifaceted disease, and advances in care require ongoing research efforts across a spectrum of different research fields. This review provides an update on the current state of research in OPG and summarises ongoing trials

On the Causality and Stability of the Relativistic Diffusion Equation

This paper examines the mathematical properties of the relativistic diffusion equation. The peculiar solution which Hiscock and Lindblom identified as an instability is shown to emerge from an ill-posed initial value problem. These do not meet the mathematical conditions required for realistic physical problems and can not serve as an argument against the relativistic hydrodynamics of Landau and Lifshitz.Comment: 6 page

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

The vulnerable microcirculation in the critically ill pediatric patient

In neonates, cardiovascular system development does not stop after the transition from intra-uterine to extra-uterine life and is not limited to the macrocirculation. The microcirculation (MC), which is essential for oxygen, nutrient, and drug delivery to tissues and cells, also develops. Developmental changes in the microcirculatory structure continue to occur during the initial weeks of life in healthy neonates. The physiologic hallmarks of neonates and developing children make them particularly vulnerable during critical illness; however, the cardiovascular monitoring possibilities are limited compared with critically ill adult patients. Therefore, the development of non-invasive methods for monitoring the MC is necessary in pediatric critical care for early identification of impending deterioration and to enable the initiation and titration of therapy to ensure cell survival. To date, the MC may be non-invasively monitored at the bedside using hand-held videomicroscopy, which provides useful information regarding the microcirculation. There is an increasing number of studies on the MC in neonates and pediatric patients; however, additional steps are necessary to transition MC monitoring from bench to bedside. The recently introduced concept of hemodynamic coherence describes the relationship between changes in the MC and macrocirculation. The loss of hemodynamic coherence may result in a depressed MC despite an improvement in the macrocirculation, which represents a condition associated with adverse outcomes. In the pediatric intensive care unit, the concept of hemodynamic coherence may function as a framework to develop microcirculatory measurements towards implementation in daily clinical practice

Partial and Entropic Information Decompositions of a Neuronal Modulatory Interaction

Information processing within neural systems often depends upon selective amplification of relevant signals and suppression of irrelevant signals. This has been shown many times by studies of contextual effects but there is as yet no consensus on how to interpret such studies. Some researchers interpret the effects of context as contributing to the selective receptive field (RF) input about which neurons transmit information. Others interpret context effects as affecting transmission of information about RF input without becoming part of the RF information transmitted. Here we use partial information decomposition (PID) and entropic information decomposition (EID) to study the properties of a form of modulation previously used in neurobiologically plausible neural nets. PID shows that this form of modulation can affect transmission of information in the RF input without the binary output transmitting any information unique to the modulator. EID produces similar decompositions, except that information unique to the modulator and the mechanistic shared component can be negative when modulating and modulated signals are correlated. Synergistic and source shared components were never negative in the conditions studied. Thus, both PID and EID show that modulatory inputs to a local processor can affect the transmission of information from other inputs. Contrary to what was previously assumed, this transmission can occur without the modulatory inputs becoming part of the information transmitted, as shown by the use of PID with the model we consider. Decompositions of psychophysical data from a visual contrast detection task with surrounding context suggest that a similar form of modulation may also occur in real neural systems