Sequentially continuous non-linear fundamental systems of solutions of affine equations in locally convex spaces

According to the Mickael's selection theorem any surjective continuous linear operator from one Fr\'echet space onto another has a continuous (not necessarily linear) right inverse. Using this theorem Herzog and Lemmert proved that if $E$ is a Fr\'echet space and $T:E\to E$ is a continuous linear operator such that the Cauchy problem $\dot x=Tx$, $x(0)=x_0$ is solvable in $[0,1]$ for any $x_0\in E$, then for any $f\in C([0,1],E)$, there exists a continuos map $S:[0,1]\times E\to E$, $(t,x)\mapsto S_tx$ such that for any $x_0\in E$, the function $x(t)=S_tx_0$ is a solution of the Cauchy problem $\dot x(t)=Tx(t)+f(t)$, $x(0)=x_0$ (they call $S$ a fundamental system of solutions of the equation $\dot x=Tx+f$). We prove the same theorem, replacing "continuous" by "sequentially continuous" for locally convex spaces from a class which contains strict inductive limits of Fr\'echet spaces and strong duals of Fr\'echet--Schwarz spaces and is closed with respect to finite products and sequentially closed subspaces. The key-point of the proof is an extension of the theorem on existence of a sequentially continuous right inverse of any surjective sequentially continuous linear operator to some class of non-metrizable locally convex spaces.

On the solvability of Hamilton's equations in Hilbert spaces

We construct a bounded function $H : l_2\times l_2 \to R$ with continuous Frechet derivative such that for any $q_0\in l_2$ the Cauchy problem $\dot p= - {\partial H\over\partial q}$, $\dot q={\partial H\over\partial p}$, $p(0) = 0$, q(0) = q_0$ has no solutions in any neighborhood of zero in R.

Integral completeness of locally convex spaces

A locally convex space X is said to be integrally complete if each continuous mapping f: [0, 1] --> X is Riemann integrable. A criterion for integral completeness is established. Readily verifiable sufficient conditions of integral completeness are proved.

Some results on solvability of ordinary linear differential equations in locally convex spaces

Let $\Gamma$ be the class of sequentially complete locally convex spaces such that an existence theorem holds for the linear Cauchy problem $\dot x = Ax$, $x(0) = x_0$ with respect to functions $x: R\to E$. It is proved that if $E\in \Gamma$, then $E\times R^A$ is-an-element-of $\Gamma$ for an arbitrary set $A$. It is also proved that a topological product of infinitely many infinite-dimensional Frechet spaces, each not isomorphic to $\omega$, does not belong to $\Gamma$.

Resting energy expenditure in non-ventilated, non-sedated patients recovering from serious traumatic brain injury: Comparison of prediction equations with indirect calorimetry values

Background & aims: Little is known about energy requirements in brain injured (TBI) patients, despite evidence suggesting adequate nutritional support can improve clinical outcomes. The study aim was to compare predicted energy requirements with measured resting energy expenditure (REE) values, in patients recovering from TBI.

Methods: Indirect calorimetry (IC) was used to measure REE in 45 patients with TBI. Predicted energy requirements were determined using FAO/WHO/UNU and Harris–Benedict (HB) equations. Bland– Altman and regression analysis were used for analysis.

Results: One-hundred and sixty-seven successful measurements were recorded in patients with TBI. At an individual level, both equations predicted REE poorly. The mean of the differences of standardised areas of measured REE and FAO/WHO/UNU was near zero (9 kcal) but the variation in both directions was substantial (range 591 to þ573 kcal). Similarly, the differences of areas of measured REE and HB demonstrated a mean of 1.9 kcal and range 568 to þ571 kcal. Glasgow coma score, patient status, weight and body temperature were signi?cant predictors of measured REE (p < 0.001; R²⁼ 0.47).

Conclusions: Clinical equations are poor predictors of measured REE in patients with TBI. The variability in REE is substantial. Clinicians should be aware of the limitations of prediction equations when estimating energy requirements in TBI patients.

Modeling Multivariate Interest Rates Using Time-Varying Copulas and Reducible Nonlinear Stochastic Differential Equations

We propose a new approach for modeling nonlinear multivariate interest rate processes based on time-varying copulas and reducible stochastic differential equations (SDEs). In the modeling of the marginal processes, we consider a class of nonlinear SDEs that are reducible to Ornstein--Uhlenbeck (OU) process or Cox, Ingersoll, and Ross (1985) (CIR) process. The reducibility is achieved via a nonlinear transformation function. The main advantage of this approach is that these SDEs can account for nonlinear features, observed in short-term interest rate series, while at the same time leading to exact discretization and closed-form likelihood functions. Although a rich set of specifications may be entertained, our exposition focuses on a couple of nonlinear constant elasticity volatility (CEV) processes, denoted as OU-CEV and CIR-CEV, respectively. These two processes encompass a number of existing models that have closed-form likelihood functions. The transition density, the conditional distribution function, and the steady-state density function are derived in closed form as well as the conditional and unconditional moments for both processes. In order to obtain a more flexible functional form over time, we allow the transformation function to be time varying. Results from our study of U.S. and UK short-term interest rates suggest that the new models outperform existing parametric models with closed-form likelihood functions. We also find the time-varying effects in the transformation functions statistically significant. To examine the joint behavior of interest rate series, we propose flexible nonlinear multivariate models by joining univariate nonlinear processes via appropriate copulas. We study the conditional dependence structure of the two rates using Patton (2006a) time-varying symmetrized Joe--Clayton copula. We find evidence of asymmetric dependence between the two rates, and that the level of dependence is positively related to the level of the two rates. (JEL: C13, C32, G12) Copyright The Author 2010. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org, Oxford University Press.

Discretization and solution of the inhomogeneous Bogoliubov-de Gennes equations

In the presence of inhomogeneities, defects and currents, the equations describing a Bose-condensed ensemble of alkali atoms have to be solved numerically. By combining both linear and nonlinear equations within a Discrete Variable Representation framework, we describe a computational scheme for the solution of the coupled Bogoliubov-de Gennes (BdG) and nonlinear Schrodinger (NLS) equations for fields in a 3D spheroidal potential. We use the method to calculate the collective excitation spectrum and quasiparticle mode densities for excitations of a Bose condensed gas in a spheroidal trap. The method is compared against finite-difference and spectral methods, and we find the DVR computational scheme to be superior in accuracy and efficiency for the cases we consider. (C) 2004 Elsevier B.V. All rights reserved.

A review of the equations used to predict the velocity distribution within a ship’s propeller jet

Predicting the velocity within the ship’s propeller jet is the initial step to investigate the scouring made by the propeller jet. Albertson et al. (1950) suggested the investigation of a submerged jet can be undertaken through observation of the plain water jet from an orifice. The plain water jet investigation of Albertson et al. (1950) was based on the axial momentum theory. This has been the basis of all subsequent work with propeller jets. In reality, the velocity characteristic of a ship’s propeller jet is more complicated than a plain water jet. Fuehrer and Römisch (1977), Blaauw and van de Kaa (1978), Berger et al. (1981), Verhey (1983) and Hamill (1987) have carried out investigations using physical model. This paper reviews the state-of-art of the equations used to predict the time-averaged axial, tangential and radial components of velocity within the zone of flow establishment and the zone of established flow of a ship’s propeller jet.

Critical assessment of two-qubit post-Markovian master equations

A post-Markovian master equation has been recently proposed as a tool to describe the evolution of a system coupled to a memory-keeping environment [A. Shabani and D. A. Lidar, Phys. Rev. A 71, 020101 ( R) ( 2005)]. For a single qubit affected by appropriately chosen environmental conditions, the corresponding dynamics is always legitimate and physical. Here we extend such a situation to the case of two qubits, only one of which experiences the environmental effects. We show how, despite the innocence of such an extension, the introduction of the second qubit should be done cum grano salis to avoid consequences such as the breaking of the positivity of the associated dynamical map. This hints at the necessity of using care when adopting phenomenologically derived models for evolutions occurring outside the Markovian framework.