scholarly journals Cosmology in generalized Horndeski theories with second-order equations of motion

2014 ◽  
Vol 90 (4) ◽  
Author(s):  
Ryotaro Kase ◽  
Shinji Tsujikawa
Keyword(s):  
Equations Of Motion ◽  
Second Order ◽  
Second Order Equations

scholarly journals Modified Gravity Models Admitting Second Order Equations of Motion

Entropy ◽  
2015 ◽  
Vol 17 (12) ◽  
pp. 6643-6662 ◽  
Author(s):  
Aimeric Colléaux ◽  
Sergio Zerbini
Keyword(s):  
Modified Gravity ◽  
Equations Of Motion ◽  
Second Order ◽  
Gravity Models ◽  
Second Order Equations ◽  
Modified Gravity Models

scholarly journals Applying the method of normal forms to second-order nonlinear vibration problems

2010 ◽  
Vol 467 (2128) ◽  
pp. 1141-1163 ◽  
Author(s):  
Simon A. Neild ◽  
David J. Wagg
Keyword(s):  
Normal Form ◽  
Nonlinear Vibration ◽  
Normal Forms ◽  
Equations Of Motion ◽  
Second Order ◽  
Form Analysis ◽  
First Order ◽  
Invariance Properties ◽  
Vibration Problems ◽  
Second Order Equations

Vibration problems are naturally formulated with second-order equations of motion. When the vibration problem is nonlinear in nature, using normal form analysis currently requires that the second-order equations of motion be put into first-order form. In this paper, we demonstrate that normal form analysis can be carried out on the second-order equations of motion. In addition, for forced, damped, nonlinear vibration problems, we show that the invariance properties of the first- and second-order transforms differ. As a result, using the second-order approach leads to a simplified formulation for forced, damped, nonlinear vibration problems.

scholarly journals E11, Romans theory and higher level duality relations

2017 ◽  
Vol 32 (05) ◽  
pp. 1750023 ◽  
Author(s):  
Alexander G. Tumanov ◽  
Peter West
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Second Order ◽  
Duality Relation ◽  
First Order ◽  
Duality Relations ◽  
Second Order Equations

From the underlying nonlinear realisation, we compute the complete E[Formula: see text] invariant equations of motion in eleven dimensions, at the linearised level, up to and including level four in the fields. Thus, we include the metric, the three and six forms, the dual graviton and three fields at level four. The fields are linked by a set of duality equations, which are first-order in derivatives and transform into each other under the E[Formula: see text] symmetries. From these duality relations, we deduce second-order equations of motion, including those for the usual supergravity fields. As a result the on-shell degrees of freedom are those of the eleven-dimensional supergravity. We also show that the level four fields provide an eleven-dimensional origin of Romans theory and lead to a novel duality relation.

scholarly journals A Note on the Strong Maximum Principle for Fully Nonlinear Equations on Riemannian Manifolds

2021 ◽  
Author(s):  
Alessandro Goffi ◽  
Francesco Pediconi
Keyword(s):  
Maximum Principle ◽  
Riemannian Manifolds ◽  
Mean Curvature ◽  
Strong Maximum Principle ◽  
Second Order ◽  
Fully Nonlinear Equations ◽  
Nonlinear Operators ◽  
Fully Nonlinear ◽  
Second Order Equations ◽  
Curvature Type

AbstractWe investigate strong maximum (and minimum) principles for fully nonlinear second-order equations on Riemannian manifolds that are non-totally degenerate and satisfy appropriate scaling conditions. Our results apply to a large class of nonlinear operators, among which Pucci’s extremal operators, some singular operators such as those modeled on the p- and $$\infty $$ ∞ -Laplacian, and mean curvature-type problems. As a byproduct, we establish new strong comparison principles for some second-order uniformly elliptic problems when the manifold has nonnegative sectional curvature.

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.

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