On solvability of linear differential equations in ${\Bbb R}^{\Bbb N}$

We construct $x^0$ in ${\Bbb R}^{\Bbb N}$ and a row-finite matrix $T=\{T_{i,j}(t)\}_{i,j\in\N}$ of polynomials of one real variable $t$ such that the Cauchy problem $\dot x(t)=T_tx(t)$, $x(0)=x^0$ in the Fr\'echet space $\R^\N$ has no solutions. We also construct a row-finite matrix $A=\{A_{i,j}(t)\}_{i,j\in\N}$ of $C^\infty(\R)$ functions such that the Cauchy problem $\dot x(t)=A_tx(t)$, $x(0)=x^0$ in ${\Bbb R}^{\Bbb N}$ has no solutions for any $x^0\in{\Bbb R}^{\Bbb N}\setminus\{0\}$. We provide some sufficient condition of solvability and of unique solvability for linear ordinary differential equations $\dot x(t)=T_tx(t)$ with matrix elements $T_{i,j}(t)$ analytically dependent on $t$.

Differential Scanning Calorimetry Study and Computer Modeling of b a Phase Transformation in Ti-6Al-4V Alloy

The relationship between heat-treatment parameters and microstructure in titanium alloys has so far been mainly studied empirically, using characterization techniques such as microscopy. Calculation and modeling of the kinetics of phase transformation have not yet been widely used for these alloys. Differential scanning calorimetry (DSC) has been widely used for the study of a variety of phase transformations. There has been much work done on the calculation and modeling of the kinetics of phase transformations for different systems based on the results from DSC study. In the present work, the kinetics of the transformation in a Ti-6Al-4V titanium alloy were studied using DSC, at continuous cooling conditions with constant cooling rates of 5 °C, 10 °C, 20 °C, 30 °C, 40 °C, and 50 °C/min. The results from calorimetry were then used to trace and model the transformation kinetics in continuous cooling conditions. Based on suitably interpreted DSC results, continuous cooling–transformation (CCT) diagrams were calculated with lines of isotransformed fraction. The kinetics of transformation were modeled using the Johnson–Mehl–Avrami (JMA) theory and by applying the "concept of additivity." The JMA kinetic parameters were derived. Good agreement between the calculated and experimental transformed fractions is demonstrated. Using the derived kinetic parameters, the transformation in a Ti-6Al-4V alloy can be described for any cooling path and condition. An interpretation of the results from the point of view of activation energy for nucleation is also presented.

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 $\Gamma$.

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.

Analytical approximations to numerical solutions of theoretical emission measure distributions

Emission line fluxes from cool stars are widely used to establish an apparent emission measure distribution, EmdApp(Te), between temperatures characteristic of the low transition region and the low corona. The true emission measure distribution, EmdTrue(Te), is determined by the energy balance and geometry adopted and, with a numerical model, can be used to predict EmdApp(Te), to guide further modelling. The scaling laws that exist between coronal parameters arise from the dimensions of the terms in the energy balance equation. Here, analytical approximations to numerical solutions for EmdTrue(Te) are presented, which show how the constants in the coronal scaling laws are determined. The apparent emission measure distributions show a minimum value at some T0 and a maximum at the mean coronal temperature Tc (although in some stars, emission from active regions can contribute). It is shown that, for the energy balance and geometry adopted, the analytical values of the emission measure and electron pressure at T0 and Tc depend on only three parameters: the stellar surface gravity and the values of T0 and Tc. The results are tested against full numerical solutions for e Eri (K2 V) and are applied to Procyon (a CMi, F5 IV/V). The analytical approximations can be used to restrict the required range of full numerical solutions, to check the assumed geometry and to show where the adopted energy balance may not be appropriate. © 2011 The Authors Monthly Notices of the Royal Astronomical Society © 2011 RAS.

GPU Acceleration of Partial Differential Equation Solvers

Differential equations are often directly solvable by analytical means only in their one dimensional version. Partial differential equations are generally not solvable by analytical means in two and three dimensions, with the exception of few special cases. In all other cases, numerical approximation methods need to be utilized. One of the most popular methods is the finite element method. The main areas of focus, here, are the Poisson heat equation and the plate bending equation. The purpose of this paper is to provide a quick walkthrough of the various approaches that the authors followed in pursuit of creating optimal solvers, accelerated with the use of graphical processing units, and comparing them in terms of accuracy and time efficiency with existing or self-made non-accelerated solvers.

Solutions of fourth-order parabolic equation modeling thin film growth

In this paper we study the well-posedness for a fourth-order parabolic equation modeling epitaxial thin film growth. Using Kato's Method [1], [2] and [3] we establish existence, uniqueness and regularity of the solution to the model, in suitable spaces, namelyC⁰([0,T];L^p(Ω)) where  with 1<α<2, n∈N and n≥2. We also show the global existence solution to the nonlinear parabolic equations for small initial data. Our main tools are L_p–L_q-estimates, regularization property of the linear part of e^−tΔ2 and successive approximations. Furthermore, we illustrate the qualitative behavior of the approximate solution through some numerical simulations. The approximate solutions exhibit some favorable absorption properties of the model, which highlight the stabilizing effect of our specific formulation of the source term associated with the upward hopping of atoms. Consequently, the solutions describe well some experimentally observed phenomena, which characterize the growth of thin film such as grain coarsening, island formation and thickness growth.

Segmentation of optic disc in retinal images using an improved gradient vector flow algorithm

Image segmentation plays an important role in the analysis of retinal images as the extraction of the optic disk provides important cues for accurate diagnosis of various retinopathic diseases. In recent years, gradient vector flow (GVF) based algorithms have been used successfully to successfully segment a variety of medical imagery. However, due to the compromise of internal and external energy forces within the resulting partial differential equations, these methods can lead to less accurate segmentation results in certain cases. In this paper, we propose the use of a new mean shift-based GVF segmentation algorithm that drives the internal/external energies towards the correct direction. The proposed method incorporates a mean shift operation within the standard GVF cost function to arrive at a more accurate segmentation. Experimental results on a large dataset of retinal images demonstrate that the presented method optimally detects the border of the optic disc.

STEADY-STATE SOLUTIONS IN NONLINEAR DIFFUSIVE SHOCK ACCELERATION

Stationary solutions to the equations of nonlinear diffusive shock acceleration play a fundamental role in the theory of cosmic-ray acceleration. Their existence usually requires that a fraction of the accelerated particles be allowed to escape from the system. Because the scattering mean free path is thought to be an increasing function of energy, this condition is conventionally implemented as an upper cutoff in energy space-particles are then permitted to escape from any part of the system, once their energy exceeds this limit. However, because accelerated particles are responsible for the substantial amplification of the ambient magnetic field in a region upstream of the shock front, we examine an alternative approach in which particles escape over a spatial boundary. We use a simple iterative scheme that constructs stationary numerical solutions to the coupled kinetic and hydrodynamic equations. For parameters appropriate for supernova remnants, we find stationary solutions with efficient acceleration when the escape boundary is placed at the point where growth and advection of strongly driven nonresonant waves are in balance. We also present the energy dependence of the distribution function close to the energy where it cuts off-a diagnostic that is in principle accessible to observation.

Mean shift based gradient vector flow for image segmentation

In recent years, gradient vector flow (GVF) based algorithms have been successfully used to segment a variety of 2-D and 3-D imagery. However, due to the compromise of internal and external energy forces within the resulting partial differential equations, these methods may lead to biased segmentation results. In this paper, we propose MSGVF, a mean shift based GVF segmentation algorithm that can successfully locate the correct borders. MSGVF is developed so that when the contour reaches equilibrium, the various forces resulting from the different energy terms are balanced. In addition, the smoothness constraint of image pixels is kept so that over- or under-segmentation can be reduced. Experimental results on publicly accessible datasets of dermoscopic and optic disc images demonstrate that the proposed method effectively detects the borders of the objects of interest.

Dynamic Modeling and Interaction of Hybrid Natural Gas and Electricity Supply System in Microgrid

Natural gas (NG) network and electric network are becoming tightly integrated by microturbines in the microgrid. Interactions between these two networks are not well captured by the traditional microturbine (MT) models. To address this issue, two improved models for single-shaft MT and split-shaft MT are proposed in this paper. In addition, dynamic models of the hybrid natural gas and electricity system (HGES) are developed for the analysis of their interactions. Dynamic behaviors of natural gas in pipes are described by partial differential equations (PDEs), while the electric network is described by differential algebraic equations (DAEs). So the overall network is a typical two-time scale dynamic system. Numerical studies indicate that the two-time scale algorithm is faster and can capture the interactions between the two networks. The results also show the HGES with a single-shaft MT is a weakly coupled system in which disturbances in the two networks mainly influence the dc link voltage of the MT, while the split-shaft MT is a strongly coupled system where the impact of an event will affect both networks.

A Simplified Method for Ductility Calculation in RC Jacketed Columns

Reinforced concrete (RC) jacketing is a common method to retrofit existing columns with poor structural performance. It can be applied in two different ways: if the continuity of the jacket is ensured, the axial load of the column can be transferred to the jacket, which will be directly loaded; conversely, if no continuity is provided, the jacket induces only confinement action. In both cases the strength and ductility evaluation is rather complex, due to the different physical phenomena included, such as confinement, composite action core-jacket, preload, buckling of longitudinal bars.
Although different theoretical studies have been carried out to calculate the confinement effects, a practical approach to evaluate the flexural capacity and ductility is still missing. The calculation of these quantities is often related to the use of commercial computer programs, taking advantage of numerical methods such as fiber method or finite element method.
This paper presents a simplified approach to calculate the flexural strength and ductility of square RC jacketed sections subjected to axial load and bending moment. In particular the proposed approach is based on the calibration of the stress-block parameters including the confinement effect. Equilibrium equations are determined and buckling of longitudinal bars is modeled with a suitable stress-strain law. Moment-curvature curves are derived with simple calculations. Finally, comparisons are made with numerical analyses carried out with the code OpenSees and with experimental data available in the literature, showing good agreement.