scholarly journals INITIAL DATA FOR NUMERICAL RELATIVITY: THE USE OF SPECTRAL METHODS

2011 ◽  
Vol 03 ◽  
pp. 417-427
Author(s):  
M. A. ALCOFORADO ◽  
H. P. DE OLIVEIRA ◽  
E. L. RODRIGUES

The determination of physical initial data is an important task in numerical relativity. In this direction we have applied the Galerkin and collocation methods to solve the Hamiltonian constraints resulting from the Cauchy formulation in the cases of spacetimes containing black holes as described by Ref. 1. We have shown that a considerable improvement in the accuracy is obtained if the basis functions are chosen such that the boundary conditions are satisfied. We have also introduced a new approach to solve numerically the constraint equations which consists in transforming them into parabolic equations after introducing fictitious diffusion terms. As a consequence, the application of Galerkin or collocation methods produces a dynamical system whose stationary solution corresponds to the initial data.

2019 ◽  
Vol 38 ◽  
pp. 11-25
Author(s):  
Hasib Uddin Molla ◽  
Goutam Saha

In this research work, Galerkin and collocation methods have been introduced for approximating the solution of FIE of 2nd kind using LH (product of Laguerre and Hermite) polynomials which are considered as basis functions. Also, a comparison has been done between the solutions of Galerkin and collocation method with the exact solution. Both of these methods show the outcome in terms of the approximate polynomial which is a linear combination of basis functions. Results reveal that performance of collocation method is better than Galerkin method. Moreover, five different polynomials such as Legendre, Laguerre, Hermite, Chebyshev 1st kind and Bernstein are also considered as a basis functions. And it is found that all these approximate solutions converge to same polynomial solution and then a comparison has been made with the exact solution. In addition, five different set of collocation points are also being considered and then the approximate results are compared with the exact analytical solution. It is observed that collocation method performed well compared to Galerkin method. GANIT J. Bangladesh Math. Soc.Vol. 38 (2018) 11-25


2017 ◽  
Vol 30 (1) ◽  
pp. 273-289
Author(s):  
Anmari Meerkotter

The Constitutional Court (CC) judgment of Lee v Minister of Correction Services 2013 2SA 144 (CC) is a recent contribution to transformative constitutional jurisprudence in the field of the law of delict. This matter turned on the issue of factual causation in the context of wrongful and negligent systemic omissions by the state. In this case note, I explore the law relating to this element of delictual liability with specific regard to the traditional test for factual causation – the conditio sine qua non (‘but-for’) test. In particular, I note the problems occasioned by formalistic adherence to this test in the context of systemic state omissions as evidenced by the SCA judgment in the same matter. I also consider the manner in which English courts have addressed this problem. Thereafter, I analyse the CC’s broader approach to the determination of factual causation as one based on common sense and justice. I argue that this approach endorses a break from a formalistic application of the test and constitutes a step towards an approach which resonates with the foundational constitutional values of freedom, dignity and equality. Furthermore, it presents an appropriate solution to the problems associated with factual causation where systemic omissions are concerned. I then consider the transformative impact of the Lee judgment. In particular, I argue that the broader enquiry favoured by the CC facilitates the realisation of constitutionally guaranteed state accountability, and amounts to an extension of the existing norm of accountability jurisprudence. Hence, I contend that the judgment presents a further effort by the Constitutional Court to effect wholesale the constitutionalisation of the law of delict, as well as a vindicatory tool to be used by litigants who have been adversely affected by systemic state omissions.


Author(s):  
Romain Desplats ◽  
Timothee Dargnies ◽  
Jean-Christophe Courrege ◽  
Philippe Perdu ◽  
Jean-Louis Noullet

Abstract Focused Ion Beam (FIB) tools are widely used for Integrated Circuit (IC) debug and repair. With the increasing density of recent semiconductor devices, FIB operations are increasingly challenged, requiring access through 4 or more metal layers to reach a metal line of interest. In some cases, accessibility from the front side, through these metal layers, is so limited that backside FIB operations appear to be the most appropriate approach. The questions to be resolved before starting frontside or backside FIB operations on a device are: 1. Is it do-able, are the metal lines accessible? 2. What is the optimal positioning (e.g. accessing a metal 2 line is much faster and easier than digging down to a metal 6 line)? (for the backside) 3. What risk, time and cost are involved in FIB operations? In this paper, we will present a new approach, which allows the FIB user or designer to calculate the optimal FIB operation for debug and IC repair. It automatically selects the fastest and easiest milling and deposition FIB operations.


Sensors ◽  
2021 ◽  
Vol 21 (9) ◽  
pp. 3070
Author(s):  
Sebastian Iwaszenko ◽  
Jakub Munk ◽  
Stefan Baron ◽  
Adam Smoliński

Modern dentistry commonly uses a variety of imaging methods to support diagnosis and treatment. Among them, cone beam computed tomography (CBCT) is particularly useful in presenting head structures, such as the temporomandibular joint (TMJ). The determination of the morphology of the joint is an important part of the diagnosis as well as the monitoring of the treatment results. It can be accomplished by measurement of the TMJ gap width at three selected places, taken at a specific cross-section. This study presents a new approach to these measurements. First, the CBCT images are denoised using curvilinear methods, and the volume of interest is determined. Then, the orientation of the vertical cross-section plane is computed based on segmented axial sections of the TMJ head. Finally, the cross-section plane is used to determine the standardized locations, at which the width of the gap between condyle and fossa is measured. The elaborated method was tested on selected TMJ CBCT scans with satisfactory results. The proposed solution lays the basis for the development of an autonomous method of TMJ index identification.


2020 ◽  
Vol 10 (1) ◽  
pp. 353-370 ◽  
Author(s):  
Hans-Christoph Grunau ◽  
Nobuhito Miyake ◽  
Shinya Okabe

Abstract This paper is concerned with the positivity of solutions to the Cauchy problem for linear and nonlinear parabolic equations with the biharmonic operator as fourth order elliptic principal part. Generally, Cauchy problems for parabolic equations of fourth order have no positivity preserving property due to the change of sign of the fundamental solution. One has eventual local positivity for positive initial data, but on short time scales, one will in general have also regions of negativity. The first goal of this paper is to find sufficient conditions on initial data which ensure the existence of solutions to the Cauchy problem for the linear biharmonic heat equation which are positive for all times and in the whole space. The second goal is to apply these results to show existence of globally positive solutions to the Cauchy problem for a semilinear biharmonic parabolic equation.


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