Matrix efficiency

The interpretation of the density (and efficiency) matrices is rather obvious the diagonal elements represent the probabilities of finding a certain magnetic quantum number of the ensemble, and the non-diagonal elements contain the phase information of the system for the corresponding different magnetic quantum numbers. However, in calculations with these matrices one still has to work out the cumbersome summations over these quantum numbers. It is more convenient to replace the density (and efficiency) matrices by statistical and efficiency tensors, also called state multipoles. 

Substitution of the density and efficiency matrices in the expression for the transition rate P in equ. (8.77a) then leads to the result... 

The most efficient matrix for retention of actinides and fission products is the uraninite mineral. However, it has been shown that other matricies such as apatite, clay minerals, zirconium silicates, and oxides (Fe, Mn) may also be important in the retention of fission products and actinides. For example, Pu was stored in apatite (Bros et al. 1996) and chlorite (Bros et al. 1993) in the core of the reactor 10. In the core of the reactors, between uraninite grains, 20-200 (j.m-sized metallic aggregates containing fissiogenic Ru, Rh, and Te associated with As, Pb, and S were found. These aggregates also exist in spent fuels of water-pressured type reactor plants, suggesting their analogy with spent fuels. 

Q. Explain why matrix modification may lead to more efficient matrix removal. 

In the final state the property of the detector is important. Since the relevant density matrix depends upon this detection efficiency, it is called the efficiency matrix < f>. In the present example the detection efficiency is independent of Mf, and one gets... 

Second, the transition rate P follows from the trace of the matrix built from the density and the efficiency matrix of the final state ... 

Shirahata N, Hozumi A, Miura Y, Kobayashi K, Sakka Y, Yonezawa T. An efficient matrix that resists the nonspecific adsorption of protein to fabricate carbohydrate arrays on silicon. 

Peric, M. (1987), Efficient semi-implicit solving algorithm for nine-diagonal co-efficient matrix. Numerical Heat Transfer, 11, 251-279. 

Note that all expressions for matrix elements of H that are shown in this paper have the same general form. Each matrix element of H is a product of at most two matrices (typically, a product of the previously generated intermediate and a cluster amplitude). This is to make sure that the resulting equations have a fully vectorizable form (matrix products can always be evaluated using efficient matrix multiplication routines cf., e.g., Ref. 92). It can be easily verified that none of the matrix elements h ( 1,2) and hp ( 1,2) listed above requires steps that are more expensive than... 

A common feature of both the gradient vector and Hessian matrix element construction is that the elimination of the formula file in favor of efficient matrix and vector operations must be accompanied by a particular arrangement of the integrals and density matrix elements. This sometimes requires an expansion of these quantities, with some elements stored redundantly, in order to simplify the memory references within these vector operations. Care must be taken during the computational algorithm design that this overhead does not become a significant fraction of the total effort. 

Formulations involving the use of the mole fractions as independent variables have been proposed by Bruno et al.4 and Ishii and Otto.15 Bruno et al. formulated the problem in terms of N(c + 1) independent variables the V, 7, and xj2 xj3 xjc. To solve an extractive distillation problem which involved nine plates and three components, Bruno et al. reported an execution time of 1.5 minutes on an IBM 360-50. Gallun6 solved the same problem (except for a minor difference in specifications) in nine seconds of IBM 360-50 execution time with six iterations. The difference in execution time of the two methods was attributed to the efficient matrix solving techniques used by Gallun. 

A second source of uncertainty Is associated with the quantities comprising the overall callhratlon factor A, such as recovery. Instrumental detection efficiency, matrix absorption or scattering, etc. If A Is determined as a random variable each time X (concentration) is estimated, then there is no problem its random error is automatically taken into account through error... 

The linear prediction equations can be solved by the autocorrelation or covariance methods. Each involves solving the equations by means of efficient matrix inversion. 

A two-step approach was used for the determination of method detection and quantification limits for the sulfur analytes, as described in Lee Aizawa (2003). The two step approach takes into consideration several factors that affect the analyte signal, including instrumental noise, variability in instrumental sensitivity, and variability in method efficiency, matrix effects and interference, and is simple to follow. Other methods, such as the Hubaux-Vos approach for the calculation of the detection limit can also be used, as reported in Fedrizzi et al. (2007). However, this later approach is complicated, time consuming and does not take either the variabiUty in method efficiency or the matrix effects into consideration (Lee Aizawa, 2003). A brief discussion on how to conduct the method vahdation using the two steps approach is mentioned in this section. 

The computational complexities of Methods I-IV are compared with those of four existing serial algorithms ([2, 24, 25, 42]) in Table 3.7. The numbo of required scalar opmtions (multiplications, additions) is compared for the case of an jV-link, soial, open-chain manipulator with simple revolute and prismatic joints only. The efficient matrix transformations and other simplifications described in Section 3.5 have been plied in each of Methods I-IV. The computations necessary for detomining individual link transformation matrices have also been included in the expressions for these four methods, while this may not be true of the others referenced. The numbers shown in parentheses indi-... 

Tables S.l and 5.2 list the computational requirements for the new dynamic simulation algorithm, using the most efficient algorithms known for each calculation for different values of N. The computations are tabulated in toms of the matrix and vector quantities which are found in the first three stq>s of the algorithm. The requited scalar opoations (multiplications, additions) are given for an AT-link, serial manipulator with simple revolute and prismatic joints only. The efficient matrix transformations and oth simplifications described in Chapter 3 have been applied in each stq>, and the computations necessary to determine the individual link transformation matrices have also been included.

Besides size consistency, pair approaches have always exploited special features of two-electron functions as mentioned at the beginning of this review. However, a remarkable convergence of methods has taken place in this respect. Efficient matrix-oriented direct Cl algorithms—which avoid logic in inner loops and are well suited for vector computers —were first... 

In fact, due to the same chemical nature of fibers and matrix, an efficient matrix-fiber stress transfer will probably be ensured by hydrogen bonds between the two components. 

Calibration Solutions (Matrix Free) Clean solutions of the analytical standard (and SIS if appropriate) used to calibrate instrument response use of this clean solutions can lead to failure to correct for variations in extraction efficiency, matrix effects (ionization suppression), instrument response etc. (see Chapter 8). 

A notable and perhaps unexpected observation (Figure 5.6) is that, when matrix suppression occurs, all matrix ions are suppressed, e.g., an analyte that appears in the spectrum as the protonated molecule can suppress matrix alkah ion adducts [Ma-F Na] as well as protonated matrix [Ma-FH]+. This observation has been interpreted (Knochenmuss 2000) in terms of quantitative considerations of secondary plume reactions among matrix ion species that can be interconverted by reactions with neutral matrix. These reactions are sufficienily close to isoenergetic that they should proceed rapidly under plume conditions, so that when a highly efficient matrix-analyte reaction depletes one matrix species (e.g. [Ma-FH]+), all others (e.g., [Ma-FNa]+) wiU also be efficiently depleted via this interconversion reaction channel. 

Recovery is a term often used to describe the extraction efficiency viewed as the percentage of the amount of analyte carried through the sample extraction and processing steps of the method into the sample extract (see the discussions of Fj in several portions of Section 8.4). The best measurements of F(j are made by comparing the response observed for the extract of a spiked QC sample to that for an extract of the same control matrix used to prepare the QCs that is spiked post-extraction with the same quantity of analyte. Clean solutions of pure analytical standard can be used instead of the post-extraction-spiked extract of control matrix, but such a procedure may confound the desired extraction efficiency with matrix effects (see Section 5.3.6a for a discussion of the inter-relationships (Matuszewski 2003) among extraction efficiency, matrix effects and process efficiency). Of course, if the control matrix is in short supply a clean solution of analytical standard must be used for this purpose. 

Li ZQ, Guo XL, Palmer AF, Das H, Guan JJ. High-efficiency matrix modulus-induced cardiac differentiation of human mesenchymal stem cells inside a thermosensitive hydrogel. 

The visual appearance of the matrix layer is a reliable indicator of its ionization/desorption efficiency potential. The most efficient matrix layers show a homogenous and shiny appearance whereas layers with a mat finish render lower quality spectral data. The laser spot size is estimated to be between 100 and 150 p,m. 

The five point algorithm presented above can be witten for all internal mesh nodes to form a complete, linear, non-homogeneous, algebraic system to define the discretized P-field. The system is blockwise tridiagonal, and can be readily solved by standard efficient matrix methods. 

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