scholarly journals Computer Assisted Proof of Transverse Saddle-to-Saddle Connecting Orbits for First Order Vector Fields

2014 ◽  
Vol 26 (2) ◽  
pp. 267-313 ◽  
Author(s):  
Jean-Philippe Lessard ◽  
Jason D. Mireles James ◽  
Christian Reinhardt
Keyword(s):  
Vector Fields ◽  
Computer Assisted ◽  
Connecting Orbits ◽  
First Order ◽  
Computer Assisted Proof

A computer-assisted proof method of the invertibility to elliptic operators

2014 ◽  
Vol 1 ◽  
pp. 816-819
Author(s):  
Akitoshi Takayasu ◽  
Shin'ichi Oishi
Keyword(s):  
Elliptic Operators ◽  
Computer Assisted ◽  
Computer Assisted Proof

scholarly journals Off-Shell Noether Currents and Potentials for First-Order General Relativity

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 348
Author(s):  
Merced Montesinos ◽  
Diego Gonzalez ◽  
Rodrigo Romero ◽  
Mariano Celada
Keyword(s):  
General Relativity ◽  
Vector Fields ◽  
Killing Vector ◽  
Spherically Symmetric ◽  
Killing Vector Fields ◽  
Arbitrary Vector ◽  
Four Dimensions ◽  
First Order ◽  
Direct Form ◽  
Definition Of

We report off-shell Noether currents obtained from off-shell Noether potentials for first-order general relativity described by n-dimensional Palatini and Holst Lagrangians including the cosmological constant. These off-shell currents and potentials are achieved by using the corresponding Lagrangian and the off-shell Noether identities satisfied by diffeomorphisms generated by arbitrary vector fields, local SO(n) or SO(n−1,1) transformations, ‘improved diffeomorphisms’, and the ‘generalization of local translations’ of the orthonormal frame and the connection. A remarkable aspect of our approach is that we do not use Noether’s theorem in its direct form. By construction, the currents are off-shell conserved and lead naturally to the definition of off-shell Noether charges. We also study what we call the ‘half off-shell’ case for both Palatini and Holst Lagrangians. In particular, we find that the resulting diffeomorphism and local SO(3,1) or SO(4) off-shell Noether currents and potentials for the Holst Lagrangian generically depend on the Immirzi parameter, which holds even in the ‘half off-shell’ and on-shell cases. We also study Killing vector fields in the ‘half off-shell’ and on-shell cases. The current theoretical framework is illustrated for the ‘half off-shell’ case in static spherically symmetric and Friedmann–Lemaitre–Robertson–Walker spacetimes in four dimensions.

scholarly journals On the Computation of Degenerate Hopf Bifurcations forn-Dimensional Multiparameter Vector Fields

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Michail P. Markakis ◽  
Panagiotis S. Douris
Keyword(s):  
Vector Fields ◽  
Center Manifold ◽  
Normal Forms ◽  
Computer Assisted ◽  
Hopf Bifurcations ◽  
Symbolic Form ◽  
Parametric System ◽  
Analytical Expressions

The restriction of ann-dimensional nonlinear parametric system on the center manifold is treated via a new proper symbolic form and analytical expressions of the involved quantities are obtained as functions of the parameters by lengthy algebraic manipulations combined with computer assisted calculations. Normal forms regarding degenerate Hopf bifurcations up to codimension 3, as well as the corresponding Lyapunov coefficients and bifurcation portraits, can be easily computed for any system under consideration.

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