K-loops and quasidirect products in 2-dlmensional Linear Groups over a Pythagorean Field

For a group G = (G, ·), we define the (internal) quasidirect product f · U = F × U of a certain K-loop (F,+) with F ? G and a suitable subgroup il of G (cf. (3.1)). Let K be a commutative pythagorean field and let L = K(i) be the quadratic extension of K with i2 = ～-1. Then the future cone H:= A ∈ GL(2,L) ¦ A = A*, det A ∈ K⁺, Tr A ∈ K⁺ is a K-loop with respect to the binary operation $A?ggsquaredplus B:=sqrt{AB^{2}A},{? where}sqrt{A}=({? Tr}A+2sqrt {{? det}A})^{1?er 2}(sqrt {? det}AE+A)$} (cf. (2.4)), and the (internal) quasidirect product $H^{}</Emphasis>{\mathop \times\limits_{Q}}Q_{1}$ of the K-loop (H},+) and the group Q₁:= {X ∈ GL(2,L) ¦ X*X = E) is a subgroup of GL(2,L) (cf. (3.2)). Moreover, S L(2,1) = $H^{1+}{\mathop \times\limits_{Q}}Q^{1}$ , where H¹+ = SL(2,L)∩ H ≤} (H},+), Q¹ = S L(2, L) ∩ Q₁ (cf. (3.4)), and if K is euclidean, then (cf. (3.6)).     

Pulsed-laser photolysis of chromium and molybdenum hexacarbonyls at 308 nm: Simultaneous monitoring of optical and acoustic emissions

An acoustic technique developed earlier was applied to measure the molar number of fragments produced in the XeCl-laser-induced gas-phase photolysis of Cr(CO)₆ and Mo(CO)₆. The bluish-green Cr and Mo emissions were also monitored and correlated with the acoustic signal. For laser fluence ranging up to about 25 mJ/cm² (38 mJ/cm²), the Cr (Mo) optical signal varied with laser fluence asy =dx ³, in agreement with the model that highly excited metal atoms were produced by direct three-photon processes. The acoustic signal varied with fluence asy = ax + bx ², consistent with the model that most fragments were formed via sequential and direct two-photon processes. Quite expectedly, all coefficientsa,b andd vary linearly with carbonyl partial pressure. Both optical and acoustic signals showed onset of saturation at 25 mJ/cm² (38 mJ/cm²) for the case of chromium (molybdenum) hexacarbonyl, suggesting that all photolysis channels were equally affected by the depletion of parent molecules. By assuming a common depletion mechanism, upper limits on the branching ratios of the various photolysis channels were estimated.     

Fastiterative methods for least squares estimations

Least squares estimations have been used extensively in many applications, e.g. system identification and signal prediction. When the stochastic process is stationary, the least squares estimators can be found by solving a Toeplitz or near-Toeplitz matrix system depending on the knowledge of the data statistics. In this paper, we employ the preconditioned conjugate gradient method with circulant preconditioners to solve such systems. Our proposed circulant preconditioners are derived from the spectral property of the given stationary process. In the case where the spectral density functions() of the process is known, we prove that ifs() is a positive continuous function, then the spectrum of the preconditioned system will be clustered around 1 and the method converges superlinearly. However, if the statistics of the process is unknown, then we prove that with probability 1, the spectrum of the preconditioned system is still clustered around 1 provided that large data samples are taken. For finite impulse response (FIR) system identification problems, our numerical results show that annth order least squares estimator can usually be obtained inO(n logn) operations whenO(n) data samples are used. Finally, we remark that our algorithm can be modified to suit the applications of recursive least squares computations with the proper use of sliding window method arising in signal processing applications.Research supported in part by HKRGC grant no. 221600070, ONR contract no. N00014-90-J-1695 and DOE grant no. DE-FG03-87ER25037.