On some second order and fourth order elliptic overdetermined problems

1995 ◽  
Vol 46 (2) ◽  
pp. 188-197 ◽  
Author(s):  
G. A. Philippin ◽  
L. Ragoub
Author(s):  
Josef Betten

In this paper a scalar-valued isotropic tensor function is considered, the variables of which are constitutive tensors of orders two and four, for instance, characterizing the anisotropic properties of a material. Therefore, the system of irreducible invariants of a fourth-order tensor is constructed. Furthermore, the joint or simultaneous invariants of a second-order and a fourth-order tensor are found. In a similar way one can construct an integrity basis for a tensor of order greater than four, as shown in the paper, for instance, for a tensor of order six.


2021 ◽  
Vol 502 (3) ◽  
pp. 3976-3992
Author(s):  
Mónica Hernández-Sánchez ◽  
Francisco-Shu Kitaura ◽  
Metin Ata ◽  
Claudio Dalla Vecchia

ABSTRACT We investigate higher order symplectic integration strategies within Bayesian cosmic density field reconstruction methods. In particular, we study the fourth-order discretization of Hamiltonian equations of motion (EoM). This is achieved by recursively applying the basic second-order leap-frog scheme (considering the single evaluation of the EoM) in a combination of even numbers of forward time integration steps with a single intermediate backward step. This largely reduces the number of evaluations and random gradient computations, as required in the usual second-order case for high-dimensional cases. We restrict this study to the lognormal-Poisson model, applied to a full volume halo catalogue in real space on a cubical mesh of 1250 h−1 Mpc side and 2563 cells. Hence, we neglect selection effects, redshift space distortions, and displacements. We note that those observational and cosmic evolution effects can be accounted for in subsequent Gibbs-sampling steps within the COSMIC BIRTH algorithm. We find that going from the usual second to fourth order in the leap-frog scheme shortens the burn-in phase by a factor of at least ∼30. This implies that 75–90 independent samples are obtained while the fastest second-order method converges. After convergence, the correlation lengths indicate an improvement factor of about 3.0 fewer gradient computations for meshes of 2563 cells. In the considered cosmological scenario, the traditional leap-frog scheme turns out to outperform higher order integration schemes only when considering lower dimensional problems, e.g. meshes with 643 cells. This gain in computational efficiency can help to go towards a full Bayesian analysis of the cosmological large-scale structure for upcoming galaxy surveys.


2020 ◽  
Vol 89 (326) ◽  
pp. 2613-2648
Author(s):  
Charles M. Elliott ◽  
Philip J. Herbert
Keyword(s):  

2008 ◽  
Vol 51 (4) ◽  
pp. 449-460 ◽  
Author(s):  
Mikhail V. Bulatov ◽  
Guido Vanden Berghe

2013 ◽  
Vol 218 (2) ◽  
pp. 179-191 ◽  
Author(s):  
Jordan M Willcox ◽  
Alastair J S Summerlee ◽  
Coral L Murrant

Relaxin produces a sustained decrease in total peripheral resistance, but the effects of relaxin on skeletal muscle arterioles, an important contributor to systemic resistance, are unknown. Using the intact, blood-perfused hamster cremaster muscle preparationin situ, we tested the effects of relaxin on skeletal muscle arteriolar microvasculature by applying 10−10 M relaxin to second-, third- and fourth-order arterioles and capillaries. The mechanisms responsible for relaxin-induced dilations were explored by applying 10−10 M relaxin to second-order arterioles in the presence of 10−5 M N(G)-nitro-l-arginine methyl ester (l-NAME, nitric oxide (NO) synthase inhibitor), 10−5 M glibenclamide (GLIB, ATP-dependent potassium (K+) channel inhibitor), 10−3 M tetraethylammonium (TEA) or 10−7 M iberiotoxin (IBTX, calcium-associated K+channel inhibitor). Relaxin caused second- (peak change in diameter: 8.3±1.7 μm) and third (4.5±1.1 μm)-order arterioles to vasodilate transiently while fourth-order arterioles did not (0.01±0.04 μm). Relaxin-induced vasodilations were significantly inhibited byl-NAME, GLIB, TEA and IBTX. Relaxin stimulated capillaries to induce a vasodilation in upstream fourth-order arterioles (2.1±0.3 μm), indicating that relaxin can induce conducted responses vasodilation that travels through blood vessel walls via gap junctions. We confirmed gap junction involvement by showing that gap junction uncouplers (18-β-glycyrrhetinic acid (40×10−6 M) or 0.07% halothane) inhibited upstream vasodilations to localised relaxin stimulation of second-order arterioles. Therefore, relaxin produces transient NO- and K+channel-dependent vasodilations in skeletal muscle arterioles and stimulates capillaries to initiate conducted responses. The transient nature of the arteriolar dilation brings into question the role of skeletal muscle vascular beds in generating the sustained systemic haemodynamic effects induced by relaxin.


1989 ◽  
Vol 79 (4) ◽  
pp. 1210-1230
Author(s):  
C. R. Daudt ◽  
L. W. Braile ◽  
R. L. Nowack ◽  
C. S. Chiang

Abstract The Fourier method, the second-order finite-difference method, and a fourth-order implicit finite-difference method have been tested using analytical phase and group velocity calculations, homogeneous velocity model calculations for disperson analysis, two-dimensional layered-interface calculations, comparisons with the Cagniard-de Hoop method, and calculations for a laterally heterogeneous model. Group velocity rather than phase velocity dispersion calculations are shown to be a more useful aid in predicting the frequency-dependent travel-time errors resulting from grid dispersion, and in establishing criteria for estimating equivalent accuracy between discrete grid methods. Comparison of the Fourier method with the Cagniard-de Hoop method showed that the Fourier method produced accurate seismic traces for a planar interface model even when a relatively coarse grid calculation was used. Computations using an IBM 3083 showed that Fourier method calculations using fourth-order time derivatives can be performed using as little as one-fourth the CPU time of an equivalent second-order finite-difference calculation. The Fourier method required a factor of 20 less computer storage than the equivalent second-order finite-difference calculation. The fourth-order finite-difference method required two-thirds the CPU time and a factor of 4 less computer storage than the second-order calculation. For comparison purposes, equivalent runs were determined by allowing a group velocity error tolerance of 2.5 per cent numerical dispersion for the maximum seismic frequency in each calculation. The Fourier method was also applied to a laterally heterogeneous model consisting of random velocity variations in the lower half-space. Seismograms for the random velocity model resulted in anticipated variations in amplitude with distance, particularly for refracted phases.


2021 ◽  
Vol 5 (4) ◽  
pp. 274
Author(s):  
Jinfeng Wang ◽  
Baoli Yin ◽  
Yang Liu ◽  
Hong Li ◽  
Zhichao Fang

In this article, a new mixed finite element (MFE) algorithm is presented and developed to find the numerical solution of a two-dimensional nonlinear fourth-order Riemann–Liouville fractional diffusion-wave equation. By introducing two auxiliary variables and using a particular technique, a new coupled system with three equations is constructed. Compared to the previous space–time high-order model, the derived system is a lower coupled equation with lower time derivatives and second-order space derivatives, which can be approximated by using many time discrete schemes. Here, the second-order Crank–Nicolson scheme with the modified L1-formula is used to approximate the time direction, while the space direction is approximated by the new MFE method. Analyses of the stability and optimal L2 error estimates are performed and the feasibility is validated by the calculated data.


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