Two-step fourth order methods for linear ODEs of the second order

2008 ◽  
Vol 51 (4) ◽  
pp. 449-460 ◽  
Author(s):  
Mikhail V. Bulatov ◽  
Guido Vanden Berghe
Keyword(s):  
Fourth Order ◽  
Second Order ◽  
Linear Odes

scholarly journals Integrity basis for a second-order and a fourth-order tensor

1982 ◽  
Vol 5 (1) ◽  
pp. 87-96 ◽  
Author(s):  
Josef Betten
Keyword(s):  
Fourth Order ◽  
Second Order ◽  
Tensor Function ◽  
Order Tensor ◽  
Isotropic Tensor ◽  
Anisotropic Properties ◽  
Isotropic Tensor Function ◽  
Fourth Order Tensor

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.

scholarly journals Higher order Hamiltonian Monte Carlo sampling for cosmological large-scale structure analysis

2021 ◽  
Vol 502 (3) ◽  
pp. 3976-3992
Author(s):  
Mónica Hernández-Sánchez ◽  
Francisco-Shu Kitaura ◽  
Metin Ata ◽  
Claudio Dalla Vecchia
Keyword(s):  
Fourth Order ◽  
Large Scale ◽  
Equations Of Motion ◽  
Large Scale Structure ◽  
Real Space ◽  
Higher Order ◽  
Second Order ◽  
Scale Structure ◽  
Integration Schemes ◽  
Reconstruction Methods

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.

Bases for Second Order Linear ODEs

2020 ◽  
Vol 127 (9) ◽  
pp. 849-849
Author(s):  
Peter McGrath
Keyword(s):  
Second Order ◽  
Order Linear ◽  
Linear Odes

scholarly journals Second order splitting of a class of fourth order PDEs with point constraints

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

scholarly journals Relaxin induces rapid, transient vasodilation in the microcirculation of hamster skeletal muscle

2013 ◽  
Vol 218 (2) ◽  
pp. 179-191 ◽  
Author(s):  
Jordan M Willcox ◽  
Alastair J S Summerlee ◽  
Coral L Murrant
Keyword(s):  
Skeletal Muscle ◽  
Gap Junction ◽  
Fourth Order ◽  
Peripheral Resistance ◽  
Second Order ◽  
Systemic Resistance ◽  
Transient Nature ◽  
Channel Dependent ◽  
Synthase Inhibitor

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.

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