scholarly journals Almost existence from the feral perspective and some questions

2021 ◽  
pp. 1-43
Author(s):  
JOEL W. FISH ◽  
HELMUT H. W. HOFER
Keyword(s):  
Periodic Orbit ◽  
Energy Level ◽  
Existence Result ◽  
Symplectic Manifolds ◽  
Extended Version ◽  
Pseudoholomorphic Curves ◽  
Energy Surfaces

Abstract We use feral pseudoholomorphic curves and adiabatic degeneration to prove an extended version of the so-called ‘almost existence result’ for regular compact Hamiltonian energy surfaces. That is, that for a variety of symplectic manifolds equipped with a Hamiltonian, almost every (non-empty) compact energy level has a periodic orbit.

scholarly journals Non-contractible periodic orbits in Hamiltonian dynamics on closed symplectic manifolds

2016 ◽  
Vol 152 (9) ◽  
pp. 1777-1799 ◽  
Author(s):  
Viktor L. Ginzburg ◽  
Başak Z. Gürel
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Symplectic Manifold ◽  
Floer Homology ◽  
Hamiltonian Dynamics ◽  
Symplectic Manifolds ◽  
Homotopy Classes ◽  
Hamiltonian Diffeomorphisms ◽  
Favorable Conditions ◽  
Hamiltonian Diffeomorphism

We study Hamiltonian diffeomorphisms of closed symplectic manifolds with non-contractible periodic orbits. In a variety of settings, we show that the presence of one non-contractible periodic orbit of a Hamiltonian diffeomorphism of a closed toroidally monotone or toroidally negative monotone symplectic manifold implies the existence of infinitely many non-contractible periodic orbits in a specific collection of free homotopy classes. The main new ingredient in the proofs of these results is a filtration of Floer homology by the so-called augmented action. This action is independent of capping and, under favorable conditions, the augmented action filtration for toroidally (negative) monotone manifolds can play the same role as the ordinary action filtration for atoroidal manifolds.

scholarly journals A polyfold proof of the Arnold conjecture

2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Benjamin Filippenko ◽  
Katrin Wehrheim
Keyword(s):  
Field Theory ◽  
Moduli Spaces ◽  
Symplectic Manifolds ◽  
Joint Work ◽  
Pseudoholomorphic Curves ◽  
Countable Collection ◽  
Arnold Conjecture ◽  
Detailed Proof ◽  
Symplectic Field Theory

AbstractWe give a detailed proof of the homological Arnold conjecture for nondegenerate periodic Hamiltonians on general closed symplectic manifolds M via a direct Piunikhin–Salamon–Schwarz morphism. Our constructions are based on a coherent polyfold description for moduli spaces of pseudoholomorphic curves in a family of symplectic manifolds degenerating from $${{\mathbb {C}}{\mathbb {P}}}^1\times M$$ C P 1 × M to $${{\mathbb {C}}}^+ \times M$$ C + × M and $${{\mathbb {C}}}^-\times M$$ C - × M , as developed by Fish–Hofer–Wysocki–Zehnder as part of the Symplectic Field Theory package. To make the paper self-contained we include all polyfold assumptions, describe the coherent perturbation iteration in detail, and prove an abstract regularization theorem for moduli spaces with evaluation maps relative to a countable collection of submanifolds. The 2011 sketch of this proof was joint work with Peter Albers, Joel Fish.

Particle Number Dependence of Collective Properties of Nuclei in the Rare Earth Region

1974 ◽  
Vol 29 (8) ◽  
pp. 1131-1137
Author(s):  
H. Rafelski ◽  
B. Fink
Keyword(s):  
Experimental Data ◽  
Perturbation Theory ◽  
Rare Earth ◽  
Kinetic Energy ◽  
Particle Number ◽  
Interaction Model ◽  
Quantitative Agreement ◽  
Spin Model ◽  
Extended Version ◽  
Energy Surfaces

Within the framework of the pairing plus quadrupole interaction model and by using the technique of quasi spin formalism it is possible to determine the collective potential and kinetic energy surfaces as analytic functions of the particle number in the limit that single particle splittings are neglected. Pushing the quasi spin model in an extended version up to the 4th order in perturbation theory the stiffness and mass parameters of harmonic and anharmonic terms for Dy, Er, Yb, and Hf-isotopes have been calculated. The theoretical particle dependence of collective quantities shows a good qualitative and even quantitative agreement with experimental data and former calculations.

Convex Hamiltonians without conjugate points

1999 ◽  
Vol 19 (4) ◽  
pp. 901-952 ◽  
Author(s):  
GONZALO CONTRERAS ◽  
RENATO ITURRIAGA
Keyword(s):  
Periodic Orbit ◽  
Energy Level ◽  
Conjugate Points ◽  
Exponential Map ◽  
Metric Entropy ◽  
Hamiltonian Flow ◽  
Local Diffeomorphism ◽  
Index Form ◽  
Exponential Maps ◽  
Contact Flows

We construct the Green bundles for an energy level without conjugate points of a convex Hamiltonian. In this case we give a formula for the metric entropy of the Liouville measure and prove that the exponential map is a local diffeomorphism. We prove that the Hamiltonian flow is Anosov if and only if the Green bundles are transversal. Using the Clebsch transformation of the index form we prove that if the unique minimizing measure of a generic Lagrangian is supported on a periodic orbit, then it is a hyperbolic periodic orbit.We also show some examples of differences with the behaviour of a geodesic flow without conjugate points, namely: (non-contact) flows and periodic orbits without invariant transversal bundles, segments without conjugate points but with crossing solutions and non-surjective exponential maps.

The orbit of Pluto over a long interval of time

1966 ◽  
Vol 25 ◽  
pp. 227-229 ◽  
Author(s):  
D. Brouwer
Keyword(s):  
Periodic Orbit ◽  
Numerical Integration ◽  
Restricted Problem ◽  
Three Dimensional ◽  
The Third ◽  
Circular Restricted Problem ◽  
Resonance Relation

The paper presents a summary of the results obtained by C. J. Cohen and E. C. Hubbard, who established by numerical integration that a resonance relation exists between the orbits of Neptune and Pluto. The problem may be explored further by approximating the motion of Pluto by that of a particle with negligible mass in the three-dimensional (circular) restricted problem. The mass of Pluto and the eccentricity of Neptune's orbit are ignored in this approximation. Significant features of the problem appear to be the presence of two critical arguments and the possibility that the orbit may be related to a periodic orbit of the third kind.

Use of fractals to describe microstructures of porous thick-film platinum electrodes

1992 ◽  
Vol 50 (1) ◽  
pp. 314-315
Author(s):  
Steven D. Toteda
Keyword(s):  
Fractal Dimension ◽  
Power Plants ◽  
Electrical Response ◽  
Cubic Phase ◽  
Platinum Electrodes ◽  
Fine Grained ◽  
Impedance Response ◽  
Platinum Particles ◽  
Energy Surfaces ◽  
Highly Porous

Zirconia oxygen sensors, in such applications as power plants and automobiles, generally utilize platinum electrodes for the catalytic reaction of dissociating O2 at the surface. The microstructure of the platinum electrode defines the resulting electrical response. The electrode must be porous enough to allow the oxygen to reach the zirconia surface while still remaining electrically continuous. At low sintering temperatures, the platinum is highly porous and fine grained. The platinum particles sinter together as the firing temperatures are increased. As the sintering temperatures are raised even further, the surface of the platinum begins to facet with lower energy surfaces. These microstructural changes can be seen in Figures 1 and 2, but the goal of the work is to characterize the microstructure by its fractal dimension and then relate the fractal dimension to the electrical response. The sensors were fabricated from zirconia powder stabilized in the cubic phase with 8 mol% percent yttria. Each substrate was sintered for 14 hours at 1200°C. The resulting zirconia pellets, 13mm in diameter and 2mm in thickness, were roughly 97 to 98 percent of theoretical density. The Engelhard #6082 platinum paste was applied to the zirconia disks after they were mechanically polished ( diamond). The electrodes were then sintered at temperatures ranging from 600°C to 1000°C. Each sensor was tested to determine the impedance response from 1Hz to 5,000Hz. These frequencies correspond to the electrode at the test temperature of 600°C.

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