770 research outputs found
Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics
In this paper we apply dynamical systems techniques to the problem of heteroclinic connections and resonance transitions in the planar circular restricted three-body problem. These related phenomena have been of concern for some time in topics such as the capture of comets and asteroids and with the design of trajectories for space missions such as the Genesis Discovery Mission. The main new technical result in this paper is the numerical demonstration of the existence of a heteroclinic connection between pairs of periodic orbits: one around the libration point L1 and the other around L2, with the two periodic orbits having the same energy. This result is applied to the resonance transition problem and to the explicit numerical construction of interesting orbits with prescribed itineraries. The point of view developed in this paper is that the invariant manifold structures associated to L1 and L2 as well as the aforementioned heteroclinic connection are fundamental tools that can aid in understanding dynamical channels throughout the solar system as well as transport between the "interior" and "exterior" Hill's regions and other resonant phenomena
A 1-parameter family of spherical CR uniformizations of the figure eight knot complement
We describe a simple fundamental domain for the holonomy group of the
boundary unipotent spherical CR uniformization of the figure eight knot
complement, and deduce that small deformations of that holonomy group (such
that the boundary holonomy remains parabolic) also give a uniformization of the
figure eight knot complement. Finally, we construct an explicit 1-parameter
family of deformations of the boundary unipotent holonomy group such that the
boundary holonomy is twist-parabolic. For small values of the twist of these
parabolic elements, this produces a 1-parameter family of pairwise
non-conjugate spherical CR uniformizations of the figure eight knot complement
Preparation and Measurement Uncertainty in Quantum Mechanics
This thesis addresses two forms of quantum uncertainty. In part I, we focus on preparation uncertainty, an expression of the fact that there are sets of observables for which the induced probability distributions are not simultaneously sharp in any state. We exactly characterise the preparation uncertainty regions for several finite dimensional case studies, including a new derivation of the preparation uncertainty region for the Pauli observables of qubits, and two qutrit case studies which have not previously been addressed in the literature. We also consider the variance based preparation uncertainty for position and momentum observables for the well known βparticle in a boxβ system. We see that the appropriate momentum observable is not given by the spectral measure of a self-adjoint operator, although the position observable is. The box system lacks the phase-space symmetry used to determine the free particle and particle on a ring systems so determining the box uncertainty region is rather more difficult than in these cases. We give upper and lower bounds on the boundary of the uncertainty region, and show that our upper bound is exact in an interval.
In part II we turn our attention to measurement uncertainty, exploring the space of compatible joint approximations to incompatible target observables. We prove a general theorem, which shows that, for a broad class of figures of merit, the optimal compatible approximations to covariant targets are themselves covariant. This substantially simplifies the problem of determining measurement uncertainty regions for covariant observables, since the space of covariant compatible approximations is smaller than the space of all compatible approximations. We employ this theorem to derive measurement uncertainty regions for three mutually orthogonal Pauli observables, and for the quantum Fourier pair acting in any finite dimension
Clock and Category; IS QUANTUM GRAVITY ALGEBRAIC
We investigate the possibility that the quantum theory of gravity could be
constructed discretely using algebraic methods. The algebraic tools are similar
to ones used in constructing topological quantum field theories.The algebraic
tools are related to ideas about the reinterpretation of quantum mechanics in a
general relativistic context.Comment: To appear in special issue of JMP. Latex documen
Upper and lower bounds for an eigenvalue associated with a positive eigenvector
When an eigenvector of a semi-bounded operator is positive, we show that a
remarkably simple argument allows to obtain upper and lower bounds for its
associated eigenvalue. This theorem is a substantial generalization of
Barta-like inequalities and can be applied to non-necessarily purely quadratic
Hamiltonians. An application for a magnetic Hamiltonian is given and the case
of a discrete Schrodinger operator is also discussed. It is shown how this
approach leads to some explicit bounds on the ground-state energy of a system
made of an arbitrary number of attractive Coulombian particles
ΠΠΎΠΊΠ°Π»ΠΈΠ·Π°ΡΠΈΡ ΠΈΠ½Π²Π°ΡΠΈΠ°Π½ΡΠ½ΡΡ ΠΊΠΎΠΌΠΏΠ°ΠΊΡΠΎΠ² ΡΠΈΡΡΠ΅ΠΌΡ Lorenz-84
Localization of compact invariant sets of a dynamical system is one way to conduct a qualitative analysis of dynamical system. The localization task is aimed at evaluating the location of invariant compact sets of systems, which are equilibrium, periodic trajectories, attractors and repellers, and invariant tori. Such sets and their properties largely determine the structure of the phase portrait of the system. For this purpose, one can use a localization set, i.e. a set in the phase space of the system that contains all invariant compact sets.This article considers the problem of localization of invariant compact sets of an Autonomous version of the Lorenz-84 system. The system represents a simple model of the General circulation of the atmosphere in middle latitudes. The model was used in various climatological studies. To build localization set of the system the so-called functional localization method is applied. The article describes the main provisions of this method, lists the main properties of the localization sets. The simplest version of the Lorenz-84 system when there are no thermal loads is analyzed, and a common variant of the Autonomous Lorenz-84 system, in which for some values of system parameters chaotic dynamics occurs is investigated. In the first case it is shown that the only invariant compact set of the system is its equilibrium position, and localization function turned out to be a Lyapunov function of the system. For the General version of the system a family of localization sets is built and the intersection of this family is described. Graphical illustration for the localization set at fixed values of the parameters is shown. The result of the study partially overlaps with the result of K.E. Starkov on the subject, but provides additional information.The theme of localization of invariant compact sets is discussed quite actively in the literature. Research focuses both on the development of the method and its application to dynamical systems of other classes, and on the investigation of specific dynamical systems.DOI: 10.7463/mathm.0415.0812317ΠΠ΄ΠΈΠ½ ΠΈΠ· ΠΌΠ΅ΡΠΎΠ΄ΠΎΠ² ΠΏΠΎΡΡΡΠΎΠ΅Π½ΠΈΡ Π»ΠΎΠΊΠ°Π»ΠΈΠ·ΠΈΡΡΡΡΠΈΡ
ΠΌΠ½ΠΎΠΆΠ΅ΡΡΠ² ΠΎΡΠ½ΠΎΠ²Π°Π½ Π½Π° ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°Π½ΠΈΠΈ ΡΡΠ½ΠΊΡΠΈΠΉ, ΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½Π½ΡΡ
Π½Π° ΡΠ°Π·ΠΎΠ²ΠΎΠΌ ΠΏΡΠΎΡΡΡΠ°Π½ΡΡΠ²Π΅ ΡΠΈΡΡΠ΅ΠΌΡ - ΡΠ°ΠΊ Π½Π°Π·ΡΠ²Π°Π΅ΠΌΡΠΉ ΡΡΠ½ΠΊΡΠΈΠΎΠ½Π°Π»ΡΠ½ΡΠΉ ΠΌΠ΅ΡΠΎΠ΄ Π»ΠΎΠΊΠ°Π»ΠΈΠ·Π°ΡΠΈΠΈ. Π ΡΡΠ°ΡΡΠ΅ Ρ ΠΏΠΎΠΌΠΎΡΡΡ Π΄Π°Π½Π½ΠΎΠ³ΠΎ ΠΌΠ΅ΡΠΎΠ΄Π° ΠΏΠΎΠ»ΡΡΠ΅Π½Ρ ΠΎΡΠ΅Π½ΠΊΠΈ ΠΏΠΎΠ»ΠΎΠΆΠ΅Π½ΠΈΡ ΠΈΠ½Π²Π°ΡΠΈΠ°Π½ΡΠ½ΡΡ
ΠΊΠΎΠΌΠΏΠ°ΠΊΡΠ½ΡΡ
ΠΌΠ½ΠΎΠΆΠ΅ΡΡΠ² Π°Π²ΡΠΎΠ½ΠΎΠΌΠ½ΠΎΠΉ ΡΠΈΡΡΠ΅ΠΌΡ Lorenz-84, ΠΈΡΠΏΠΎΠ»ΡΠ·ΡΠ΅ΠΌΠΎΠΉ ΠΏΡΠΈ ΠΏΠΎΡΡΡΠΎΠ΅Π½ΠΈΠΈ Π½Π΅ΠΊΠΎΡΠΎΡΡΡ
ΠΌΠ΅ΡΠ΅ΠΎΡΠΎΠ»ΠΎΠ³ΠΈΡΠ΅ΡΠΊΠΈΡ
ΠΌΠΎΠ΄Π΅Π»Π΅ΠΉ. Π Π°ΡΡΠΌΠΎΡΡΠ΅Π½ ΠΏΡΠΎΡΡΠ΅ΠΉΡΠΈΠΉ Π²Π°ΡΠΈΠ°Π½Ρ ΡΠΈΡΡΠ΅ΠΌΡ Ρ ΠΎΡΡΡΡΡΡΠ²ΡΡΡΠΈΠΌΠΈ ΡΠ΅ΡΠΌΠΈΡΠ΅ΡΠΊΠΈΠΌΠΈ ΠΏΠ΅ΡΠ΅Π³ΡΡΠ·ΠΊΠ°ΠΌΠΈ, ΠΈ ΠΎΠ±ΡΠΈΠΉ Π²Π°ΡΠΈΠ°Π½Ρ ΡΠΈΡΡΠ΅ΠΌΡ, Π² ΠΊΠΎΡΠΎΡΠΎΠΉ ΠΏΡΠΈ Π½Π΅ΠΊΠΎΡΠΎΡΡΡ
Π·Π½Π°ΡΠ΅Π½ΠΈΡΡ
ΠΏΠ°ΡΠ°ΠΌΠ΅ΡΡΠΎΠ² Π²ΠΎΠ·Π½ΠΈΠΊΠ°Π΅Ρ Ρ
Π°ΠΎΡΠΈΡΠ΅ΡΠΊΠ°Ρ Π΄ΠΈΠ½Π°ΠΌΠΈΠΊΠ°. ΠΡΠΈ Π²ΡΠ΅Ρ
Π·Π½Π°ΡΠ΅Π½ΠΈΡΡ
ΠΏΠ°ΡΠ°ΠΌΠ΅ΡΡΠΎΠ² ΡΠΈΡΡΠ΅ΠΌΡ ΠΎΠΏΠΈΡΠ°Π½ΠΎ ΠΊΠΎΠΌΠΏΠ°ΠΊΡΠ½ΠΎΠ΅ Π»ΠΎΠΊΠ°Π»ΠΈΠ·ΠΈΡΡΡΡΠ΅Π΅ ΠΌΠ½ΠΎΠΆΠ΅ΡΡΠ²ΠΎ.DOI: 10.7463/mathm.0415.081231
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