Dynamical convexity of the Euler problem of two fixed centers

We give thorough analysis for the rotation functions of the critical orbits from which one can understand bifurcations of periodic orbits. Moreover, we give explicit formulas of the Conley-Zehnder indices of the interior and exterior collision orbits and show that the universal cover of the regularized energy hypersurface of the Euler problem is dynamically convex for energies below the critical Jacobi energy.Comment: Final version, title changed, to appear in Math. Proc. Cambridge Philos. So

Equivariant wrapped Floer homology and symmetric periodic Reeb orbits

The aim of this article is to apply a Floer theory to study symmetric periodic Reeb orbits. We define positive equivariant wrapped Floer homology using a (anti-)symplectic involution on a Liouville domain and investigate its algebraic properties. By a careful analysis of index iterations, we obtain a non-trivial lower bound on the minimal number of geometrically distinct symmetric periodic Reeb orbits on a certain class of real contact manifolds. This includes non-degenerate real dynamically convex starshaped hypersurfaces in $\mathbb{R}^{2n}$ which are invariant under complex conjugation. As a result, we give a partial answer to the Seifert conjecture on brake orbits in the contact setting.Comment: 42 pages, 4 figures, final version, to appear in Ergodic Theory and Dynamical System

An Analysis of International Energy Conservation Code (IECC)-Compliant Single-Family Residential Energy Use

In 2001, the Texas State Senate passed Senate Bill 5 to reduce ozone levels by encouraging the reduction of emissions of NOx that were not regulated by the Texas Natural Resource Conservation Commission, including point sources (power plants), area sources (such as residential emissions), road mobile sources, and non-road mobile sources. For the building energy section, the Texas State Legislature adopted the 2000/2001 International Energy Conservation Code, as modified by the 2001 Supplement, as the state's building energy code. The 2000/2001 IECC is a comprehensive energy conservation code that establishes a standard for the insulation levels, glazing and cooling and heating system efficiencies through the use of prescriptive and performance-based provisions. Therefore, the purpose of this research is to improve the accuracy of a 2000/2001 IECCcompliant performance simulation using the DOE-2.1e simulation program to investigate the energy performance of a typical single-family house. To achieve this purpose, several objectives had to be accomplished, including: 1) the development of an IECC-compliant simulation model, 2) the development and testing of specific improvements to the existing code-traceable model, 3) the calibration and installation of sensors in a case-study house, 4) the validation of the improved simulation model with measured data from the case-study house, and 5) use the validated model to simulate the energy-conserving features of single-family residences that cannot be simulated with existing versions of the DOE-2.1e program. In order to create the code-traceable IECC-compliant simulation model, a base-case house simulation was created and the results calibrated with measured energy and environmental data from the case-study house. This was done in order to obtain an improved simulation model that would more accurately represent the case-study building. The calibrated model was then used to verify the accuracy of the improved simulation methods against previous models and measured data. After validation of the new simulation methodologies, the IECC simulation model was used to simulate different energy-conserving features for a single-family residence that could not be simulated with the previous version of the DOE-2 input file. Finally, areas for future work were identified in an effort to continue to improve the model

Genus zero transverse foliations for weakly convex Reeb flows on the tight 33-sphere

A contact form on the tight $3$-sphere $(S^3,\xi_0)$ is called weakly convex if the Conley-Zehnder index of every Reeb orbit is at least $2$. In this article, we study Reeb flows of weakly convex contact forms on $(S^3,\xi_0)$ admitting a prescribed finite set of index-$2$ Reeb orbits, which are all hyperbolic and mutually unlinked. We present conditions so that these index-$2$ orbits are binding orbits of a genus zero transverse foliation whose additional binding orbits have index $3$. In addition, we show in the real-analytic case that the topological entropy of the Reeb flow is positive if the branches of the stable/unstable manifolds of the index-$2$ orbits are mutually non-coincident.Comment: 47 pages; The main results remain unchange