Tau functions for the Dirac operator on the cylinder

The goal of the present paper is to calculate the determinant of the Dirac operator with a mass in the cylindrical geometry. The domain of this operator consists of functions that realize a unitary one-dimensional representation of the fundamental group of the cylinder with $n$ marked points. The determinant represents a version of the isomonodromic $\tau$-function, itroduced by M. Sato, T. Miwa and M. Jimbo. It is calculated by comparison of two sections of the $\mathrm{det}^*$-bundle over an infinite-dimensional grassmannian. The latter is composed of the spaces of boundary values of some local solutions to Dirac equation. The principal ingredients of the computation are the formulae for the Green function of the singular Dirac operator and for the so-called canonical basis of global solutions on the 1-punctured cylinder. We also derive a set of deformation equations satisfied by the expansion coefficients of the canonical basis in the general case and find a more explicit expression for the $\tau$-function in the simplest case $n=2$.Comment: 32 pages, 5 figure

On Painleve VI transcendents related to the Dirac operator on the hyperbolic disk

Dirac hamiltonian on the Poincare disk in the presence of an Aharonov-Bohm flux and a uniform magnetic field admits a one-parameter family of self-adjoint extensions. We determine the spectrum and calculate the resolvent for each element of this family. Explicit expressions for Green functions are then used to find Fredholm determinant representations for the tau function of the Dirac operator with two branch points on the Poincare disk. Isomonodromic deformation theory for the Dirac equation relates this tau function to a one-parameter class of solutions of the Painleve VI equation with $\gamma=0$. We analyze long distance behaviour of the tau function, as well as the asymptotics of the corresponding Painleve VI transcendents as $s\to 1$. Considering the limit of flat space, we also obtain a class of solutions of the Painleve V equation with $\beta=0$.Comment: 38 pages, 5 figure

DC-10 composite vertical stabilizer ground test program

A review of the structural configuration and ground test program is presented. Particular emphasis is placed on the testing of a full-scale stub box test subcomponent and full span ground test unit. The stub box subcomponent was tested in an environmental chamber under ambient, cold/wet, and hot/wet conditions. The test program included design limit static loads, fatigue spectrum loading to approximately two service lifetimes (with and without damage), design limit damage tolerance tests, and a final residual strength test to a structural failure. The first full-scale ground test unit was tested under ambient conditions. The test unit was to have undergone static, fatigue, and damage tolerance tests but a premature structural failure occurred at design limit load during the third limit load test. A failure theory was developed which explains the similarity in types of failure and the large load discrepancy at failure between the two test articles. The theory attributes both failures to high stress concentrations at the edge of the lower rear spar access opening. A second full-scale ground test unit has been modified to incorporate the various changes resulting from the premature failure. The article has been assembled and is active in the test program

Thermal Management of Satellite Electronics via Gallium Phase Change Heat Sink Devices

The purpose of this research was to determine the effectiveness and feasibility of additively manufactured heat sinks using gallium as a phase change material in the thermal management of satellite electronics. A design was created based on the footprint of an Astronautical Development, LLC Lithium 1 UHF radio and six heat sinks were additively manufactured; two each of stainless steel 316, Inconel 718, and ULTEM 9085. Each heat sink was filled with gallium for testing purposes. Models were created to simulate the behavior of the heat transfer and phase change processes occurring within the heat sink. Additionally, laboratory data was gathered on the actual processes occurring. Testing was carried out in a thermal vacuum chamber with the use of film heaters that were attached to the heat sink to simulate a radio in transmitting mode while a satellite is in contact with a ground station. Finally, temperature profiles of the laboratory data were created to gain insight into the characteristics of the phase change process and its effectiveness in thermal management of satellite electronics