Modelling and performance analysis of four and eight element TCAS

This semi-annual report describes the work performed during the period September 1989 through March 1990. The first section presents a description of the effect of the engines of the Boeing 737-200 on the performance of a bottom mounted eight-element traffic alert and collision avoidance system (TCAS). The second section deals exclusively with a four element TCAS antenna. The model obtained to simulate the four element TCAS and new algorithms developed for studying its performance are described. The effect of location on its performance when mounted on top of a Boeing 737-200 operating at 1060 MHz is discussed. It was found that the four element TCAS generally does not perform as well as the eight element TCAS III

Class preserving automorphisms of unitriangular groups

Let $\textrm{UT}_n (K)$ be a unitriangular group over a field $K$ and $\Gamma_{n,k} := \textrm{UT}_n (K)/ \gamma_k(\textrm{UT}_n (K))$, where $\gamma_k (\mathrm{UT}_n(K))$ denotes the $k$-th term of the lower central series of $\mathrm{UT}_n (K)$, $2 \le k \le n$. We prove that the group of all class preserving automorphisms of $\Gamma_{n,k}$ is equal to \Inn(\Gamma_{n,k}) if and only if $K$ is a prime field. Let $G_n^{(m)} := \mathrm{UT}_n (\mathbb{F}_{p^m}) / \gamma_3 (\mathrm{UT}_n(\mathbb{F}_{p^m}))$. We calculate the group of all class preserving automorphisms and class preserving outer automorphisms of $G_n^{(m)}$.Comment: 19 pages, accepted for publication in International Journal of Algebra and Computatio

Space Shuttle Communications Coverage Analysis for Thermal Tile Inspection

The space shuttle ultra-high frequency Space-to-Space Communication System has to provide adequate communication coverage for astronauts who are performing thermal tile inspection and repair on the underside of the space shuttle orbiter (SSO). Careful planning and quantitative assessment are necessary to ensure successful system operations and mission safety in this work environment. This study assesses communication systems performance for astronauts who are working in the underside, non-line-of-sight shadow region on the space shuttle. All of the space shuttle and International Space Station (ISS) transmitting antennas are blocked by the SSO structure. To ensure communication coverage at planned inspection worksites, the signal strength and link margin between the SSO/ISS antennas and the extravehicular activity astronauts, whose line-of-sight is blocked by vehicle structure, was analyzed. Investigations were performed using rigorous computational electromagnetic modeling techniques. Signal strength was obtained by computing the reflected and diffracted fields along the signal propagation paths between transmitting and receiving antennas. Radio frequency (RF) coverage was determined for thermal tile inspection and repair missions using the results of this computation. Analysis results from this paper are important in formulating the limits on reliable communication range and RF coverage at planned underside inspection and repair worksites

On uniformization of Burnside's curve y2=x5−xy^2=x^5-x

Main objects of uniformization of the curve $y^2=x^5-x$ are studied: its Burnside's parametrization, corresponding Schwarz's equation, and accessory parameters. As a result we obtain the first examples of solvable Fuchsian equations on torus and exhibit number-theoretic integer $q$-series for uniformizing functions, relevant modular forms, and analytic series for holomorphic Abelian integrals. A conjecture of Whittaker for hyperelliptic curves and its hypergeometric reducibility are discussed. We also consider the conversion between Burnside's and Whittaker's uniformizations.Comment: Final version. LaTeX, 23 pages, 1 figure. The handbook for elliptic functions has been moved to arXiv:0808.348

Performance of traffic-alert collision avoidance (TCAS) antennas in the presence of scatterers

The performance of two TCAS systems is studied in the presence of electromagnetic scatterers. TCAS is an aircraft mounted angle of arrival (AOA) system, which estimates the bearing of a signal transmitted from a mode-S transponder on another nearby aircraft (intruder). Two systems are studied: (1) Comparison of Relative Amplitude system (CRA), and (2) Spiral Phase Antenna (SPA). The CRA antenna receives the reply via four switched beams. The bearing is estimated by comparing the amplitudes of the received signal. The SPA is based on the phase interferometer, which utilizes the received phase via sum and difference beams. The AOA is computed by comparing the reply with similar values on a calibration table, which is generated by modeling the TCAS antenna on the bare fuselage of a Boeing 727-200. The antenna patterns for the TCAS are found via high frequency methods based on the Uniform Geometric theory of Diffraction (UTD). By minimizing the standard deviation of the bearing error in a specified angular sector, optimal locations for top and bottom mounted TCAS antennas are found on the Boeing 727-200, 737-300 and 747-200 airframes. It will be shown that the overall bearing errors of the amplitude system are consistently smaller than the spiral phase TCAS. The effect of two types of nearby scatterers--antennas, and engine inlets--is studied. The AT741 L-band blade, DMC60-1 VHF Communication antenna were chosen as being representative antenna interference examples. Models are derived for the blades via a moment method analysis followed by a least squares procedure to synthesize the scattering patterns. Studies were conducted to estimate the minimum separation between the two antennas for acceptable operation. It will be shown that the spiral phase TCAS is adversely affected by the presence of a blade antenna. The amplitude system does not suffer from this limitation, especially for the forward look angles which are of most interest here. A model to represent the inlet scattering is based on the multiple scattering method and UTD. The engine on top of the B727-200 fuselage is modeled by a terminated circular waveguide. Then, the effect of moving the antenna forward on the fuselage is studied. It is again shown that the performance of the amplitude system is superior

Large Deviation Approach to the Randomly Forced Navier-Stokes Equation

The random forced Navier-Stokes equation can be obtained as a variational problem of a proper action. By virtue of incompressibility, the integration over transverse components of the fields allows to cast the action in the form of a large deviation functional. Since the hydrodynamic operator is nonlinear, the functional integral yielding the statistics of fluctuations can be practically computed by linearizing around a physical solution of the hydrodynamic equation. We show that this procedure yields the dimensional scaling predicted by K41 theory at the lowest perturbative order, where the perturbation parameter is the inverse Reynolds number. Moreover, an explicit expression of the prefactor of the scaling law is obtained.Comment: 24 page

Classical and Quantum Gravitational Collapse in d-dim AdS Spacetime I. Classical Solutions

We study the collapse of a spherically symmetric dust distribution in $d$-dimensional AdS spacetime. We investigate the role of dimensionality, and the presence of a negative cosmological constant, in determining the formation of trapped surfaces and the end state of gravitational collapse. We obtain the self-similar solution for the case of zero cosmological constant, and show that one cannot construct a self-similar solution when a cosmological constant is included.Comment: 19 pages, sections on surface gravity removed, section on self similar solution expanded, appendix removed, references added. To appear in Phys. Rev.

On the trace of the antipode and higher indicators

We introduce two kinds of gauge invariants for any finite-dimensional Hopf algebra H. When H is semisimple over C, these invariants are respectively, the trace of the map induced by the antipode on the endomorphism ring of a self-dual simple module, and the higher Frobenius-Schur indicators of the regular representation. We further study the values of these higher indicators in the context of complex semisimple quasi-Hopf algebras H. We prove that these indicators are non-negative provided the module category over H is modular, and that for a prime p, the p-th indicator is equal to 1 if, and only if, p is a factor of dim H. As an application, we show the existence of a non-trivial self-dual simple H-module with bounded dimension which is determined by the value of the second indicator.Comment: additional references, fixed some typos, minor additions including a questions and some remark