scholarly journals Geometric properties of the scattering map of a normally hyperbolic invariant manifold

2008 ◽  
Vol 217 (3) ◽  
pp. 1096-1153 ◽  
Author(s):  
Amadeu Delshams ◽  
Rafael de la Llave ◽  
Tere M. Seara
Keyword(s):  
Invariant Manifold ◽  
Geometric Properties ◽  
Normally Hyperbolic Invariant Manifold ◽  
Scattering Map

Dynamics Associated with the Normally Hyperbolic Invariant Manifold that Governs the Ionization of Hydrogen in a Circularly Polarized Electric Field

2020 ◽  
Vol 30 (08) ◽  
pp. 2030022 ◽  
Author(s):  
Christof Jung ◽  
Holger Waalkens
Keyword(s):  
Electric Field ◽  
Transition State ◽  
Invariant Manifold ◽  
Numerical Procedure ◽  
Homoclinic Orbits ◽  
Stable And Unstable Manifolds ◽  
Circularly Polarized ◽  
High Energies ◽  
Normally Hyperbolic Invariant Manifold ◽  
Unstable Manifolds

The ionization of hydrogen in a circularly polarized electric field consists of the electron escaping over a rotating Stark saddle. The escape is mediated by the stable and unstable manifolds of a normally hyperbolic invariant manifold associated with the saddle. General principles guarantee the existence of this normally hyperbolic invariant manifold which is referred to as a transition state for energies close to the saddle. We use a numerical procedure to continue the transition state to high energies and study the dynamics restricted to the transition state. We find that for the strength of the electric field under consideration, the transition state persists to very high energies and the restricted dynamics remains amazingly regular. Our analysis moreover includes a study of homoclinic orbits connecting the transition state to itself.

Some Properties of Para-Sasakian Manifolds

2017 ◽  
Vol 5 (2) ◽  
pp. 73-78
Author(s):  
Jay Prakash Singh ◽  
Keyword(s):  
Vector Field ◽  
Killing Vector ◽  
Sasakian Manifold ◽  
Subject Classification ◽  
Killing Vector Field ◽  
Sasakian Manifolds ◽  
Geometric Properties ◽  
Paper Author

In this paper author present an investigation of some differential geometric properties of Para-Sasakian manifolds. Condition for a vector field to be Killing vector field in Para-Sasakian manifold is obtained. Mathematics Subject Classification (2010). 53B20, 53C15.

Effect of 12 months of creatine supplementation and whole-body resistance training on measures of bone, muscle and strength in older males

2020 ◽  
pp. 026010602097524
Author(s):  
Darren G Candow ◽  
Philip D Chilibeck ◽  
Julianne Gordon ◽  
Emelie Vogt ◽  
Tim Landeryou ◽  
...  
Keyword(s):  
Bone Mineral Density ◽  
Resistance Training ◽  
Bone Mineral ◽  
Creatine Supplementation ◽  
Whole Body ◽  
Mineral Density ◽  
Geometric Properties ◽  
Section Modulus ◽  
Body Resistance ◽  
Older Males

Background: The combination of creatine supplementation and resistance training (10–12 weeks) has been shown to increase bone mineral content and reduce a urinary indicator of bone resorption in older males compared with placebo. However, the longer-term effects (12 months) of creatine and resistance training on bone mineral density and bone geometric properties in older males is unknown. Aim: To assess the effects of 12 months of creatine supplementation and supervised, whole-body resistance training on bone mineral density, bone geometric properties, muscle accretion, and strength in older males. Methods: Participants were randomized to supplement with creatine ( n = 18, 49–69 years, 0.1 g·kg-1·d-1) or placebo ( n = 20, 49–67 years, 0.1 g·kg-1·d-1) during 12 months of supervised, whole-body resistance training. Results: After 12 months of training, both groups experienced similar changes in bone mineral density and geometry, bone speed of sound, lean tissue and fat mass, muscle thickness, and muscle strength. There was a trend ( p = 0.061) for creatine to increase the section modulus of the narrow part of the femoral neck, an indicator of bone bending strength, compared with placebo. Adverse events did not differ between creatine and placebo. Conclusions: Twelve months of creatine supplementation and supervised, whole-body resistance training had no greater effect on measures of bone, muscle, or strength in older males compared with placebo.

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