Mapping the geometry of the F4 group

In this paper we present a construction of the compact form of the exceptional Lie group F4 by exponentiating the corresponding Lie algebra f4. We realize F4 as the automorphisms group of the exceptional Jordan algebra, whose elements are 3 x 3 hermitian matrices with octonionic entries. We use a parametrization which generalizes the Euler angles for SU(2) and is based on the fibration of F4 via a Spin(9) subgroup as a fiber. This technique allows us to determine an explicit expression for the Haar invariant measure on the F4 group manifold. Apart from shedding light on the structure of F4 and its coset manifold OP2=F4/Spin(9), the octonionic projective plane, these results are a prerequisite for the study of E6, of which F4 is a (maximal) subgroup.Comment: 50 pages; some typos correcte

E7 groups from octonionic magic square

In this paper we continue our program, started in [2], of building up explicit generalized Euler angle parameterizations for all exceptional compact Lie groups. Here we solve the problem for E7, by first providing explicit matrix realizations of the Tits construction of a Magic Square product between the exceptional octonionic algebra J and the quaternionic algebra H, both in the adjoint and the 56 dimensional representations. Then, we provide the Euler parametrization of E7 starting from its maximal subgroup U=(E6 x U(1))/Z3. Next, we give the constructions for all the other maximal compact subgroups.Comment: 23 pages, added sections with new construction

Heat exchanger with oscillating flow

Various heat exchange apparatuses are described in which an oscillating flow of primary coolant is used to dissipate an incident heat flux. The oscillating flow may be imparted by a reciprocating piston, a double action twin reciprocating piston, fluidic oscillators, or electromagnetic pumps. The oscillating fluid flows through at least one conduit in either an open loop or a closed loop. A secondary flow of coolant may be used to flow over the outer walls of at least one conduit to remove heat transferred from the primary coolant to the walls of the conduit

Automated detection of asteroids in real-time with the Spacewatch telescope

The Spacewatch telescope on Kitt Peak is being used to survey for near-earth asteroids using a Tektronix TK2048 CCD in scanning mode. We hope to identify suitable low delta v candidates amongst the near-earth asteroid population as possible exploration targets, to identify those objects which pose a danger to life on earth, and to study the physical properties of the objects in near-earth space. Between Sep. 1990 and Jun. 1991, 14 new earth-approaching asteroids including 1 Aten, 9 Apollo, and 4 Amor type asteroids were detected by automated software and discriminated by their angular rates from the rest of the detected asteroids in near-real time by the observer. The average of about 1.5 earth-approaching asteroids per month is comparable to the total number found by all other observatories combined. One other Apollo type asteroid was detected by the observer as a long trailed image. The positions of this last object were measured and the object was tracked by the observer in real time. This object was determined to be a 5-10 meter diameter object which passed within 170,000 kilometers of earth. Of the 14 automatically detected earth-approaching asteroids, 10 have been found at distances in excess of 0.5 AU from earth. An average of more than 2000 asteroids are detected each month. Positions, angular rates, and brightnesses are determined for each of these asteroids in real-time

Mapping the geometry of the E6 group

In this paper we present a construction for the compact form of the exceptional Lie group E6 by exponentiating the corresponding Lie algebra e6, which we realize as the the sum of f4, the derivations of the exceptional Jordan algebra J3 of dimension 3 with octonionic entries, and the right multiplication by the elements of J3 with vanishing trace. Our parametrization is a generalization of the Euler angles for SU(2) and it is based on the fibration of E6 via a F4 subgroup as the fiber. It makes use of a similar construction we have performed in a previous article for F4. An interesting first application of these results lies in the fact that we are able to determine an explicit expression for the Haar invariant measure on the E6 group manifold.Comment: 30 page