Triangular determinant

LV Factorization of a Matrix To eveiy m X n matrix A there exists a permutation matrix P, a lower triangular matrix L with unit diagonal elements, and a.nm X n (upper triangular) echelon matrix U such that PA = LU. The Gauss elimination is in essence an algorithm to determine U, P, and L. The permutation matrix P may be needed since it may be necessaiy in carrying out the Gauss elimination to... 

Solution of Ax = b hy Using LU Factorization Suppose that the indicated system is compatible and that A = LU (the case PA = LU is similarly handled and amounts to rearranging the equations). Let z = Ux. Then Ax = LUx = h imphes that Lz = h. Thus to solve Ax = h we first solve Lz = h for z and then solve Ux = z for x. This procedure does not require that A be invertible and can be used to determine all solutions of a compatible system Ax = h. Note that the systems Lz = h and Ux = z are both in triangular forms and thus can be easily solved. 

The response-factor approach is based on a method in which the response factors represent the transfer functions of the wall due to unit impulse excitations. The real excitation is approximated by a superposition of such impulses (mostly of triangular shape), and the real response is determined by the superposition of the impulse responses (see Figs. 11.33 and 11.34). ... 

IFIGI3RE 11.34 Determination of the real evolution of the heat flux at the inside wall surface The temperature condition at the exterior surface is approximated by triangular pulses, and the heat flux response at the interior surface is determined by superposition of the heat flux responses of the individual pulses. 

A geodesic structure with all triangular faces is determinate. 

For example, clusters identified by IR spectra and extraction as Ir4(CO)i2 on y-Al203 were found by EXAFS spectroscopy to have an Ir-Ir coordination number of nearly 3, consistent with the tetrahedral structure of the metal frame EXAFS spectroscopy produces the equivalent result for sohd Ir4(CO)i2 [27]. EXAFS spectroscopy is the most appropriate method for determination of framework structures of supported clusters, but it is limited by the errors to clusters with at most about six metal atoms. Thus, it has been used to determine frameworks that are triangular (EXAFS first-shell metal-metal coordination number of 2), tetrahedral (EXAFS first-shell metal-metal coordination number of 3), and octahedral (EXAFS first-shell metal-metal... 

FIGURE 4.2 Stahl triangular scheme to determine proper chromatographic conditions. (Adapted from Hahn-Deinstrop, E., J. Planar Chromatogr., 5, 57-61, 1992. With permission.)... 

The number of stages required for a given separation can be determined from the triangular diagram using a method analogous to the McCabe-Thiele diagram used to determine the... 

Draw the liquid-liquid equilibrium data on triangular graph paper. Show sufficient tie-lines to enable the equilibrium compositions to be determined at each stage. 

It has been several decades since oxo-centered triruthenium-carboxylate complexes with triangular cluster frameworks of Ru3(p3-0)(p-00CR)6 (R = alkyl or aryl) were first isolated [1,2]. In the early 1970s, the first oxo-centered triruthenium complex was structurally characterized by Cotton through X-ray crystal structural determination [3]. Since then, oxo-centered trinuclear ruthenium-carboxylate cluster complexes with general formula [Ru30(00CR)6(L)2L ]n+ (R = aryl or alkyl, L and... 

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