Schwartz equation

In clinical practice, glomerular filtration rate (GFR) can be estimated from prediction equations such as the Schwartz formula, which takes into account serum creatinine concentration and the patient s height, gender, and age. The advantage of estimating GFR using the Schwartz equation is rapid determination and the avoidance of a cumbersome 24-hour urine collection." " The following formula is used to estimate GFR ... 

Schwartz, H. M., Phys. Rev. 103, 110, "Ground state solution of the non-relativistic equation for He." More rapid convergence in the Ritz variational method by inclusion of half-integral powers in the Hylleraas function. 

Proof.—If k-6 = 0, k°6° = k-. Applying Schwartz inequality to this last equation, we obtain... 

Amotf was the first to develop a set of equations for acetone to simultaneously calculate chlorophyll a and chlorophyll b in 1949. Several authors later proposed different new equations based on more adjusted and accurate extinction coefficients due to the development of higher resolution spectrophotometers adapted to each special condition. Moreover, besides 80% acetone, coefficients for diethyl ether and ethanol were also established and their respective equations developed, as reviewed by Schwartz and Lorenzo and Eder. Solvents chosen should be those for which specific absorbance coefficients have been published to derive equations and updates should be carefully tracked for new values. 

Stochastic or probabilistic techniques can be applied to either the moisture module, or the solution of equation (3) — or for example the models of Schwartz Crowe (13) and Tang et al. (16), or can lead to new conceptual model developments as for example the work of Jury (17). Stochastic or probabilistic modeling is mainly aimed at describing breakthrough times of overall concentration threshold levels, rather than individual processes or concentrations in individual soil compartments. Coefficients or response functions and these models have to be calibrated to field data since major processes are studied via a black-box or response function approach and not individually. Other modeling concepts may be related to soil models for solid waste sites and specialized pollutant leachate issues (18). 

Alkenylzirconium reagents generated from alkynes and the Schwartz reagent react with aldimines under mild conditions in the presence of [RhCl(COD)]2 to give homoallylic amine derivatives (Equation (63)).418... 

More than twenty indole derivatives have been identified from bufonid skin extracts. The indolylalkylamines bufotenidine, bufotenine, de-hydrobufotenine, bufo-tionine and serotonin (5-hydroxytryptamine) (Fig. 39.2b) have been identified in skin secretions of Bufo marinus, while the latter four have been detected in parotoid gland secretions (Erspamer 1994 Maciel, Schwartz, Pires Jr, Sebben, Castro, Sousa, Fontes and Schwartz 2003). The concentration of serotonin in the dried secretion of B. marinus was found to equate to approximately 0.1% of the total composition and primarily acts as a vasoconstrictor (Gregerman 1952 Toledo and Jared 1995). 

The concentration data obtained from each sample analysis were expressed as fractional parts and normalized to sum to 100. The normalized data were statistically analyzed, and three principal components (A=3, Equation [1]) were calculated. The PCB constituents (varlbles) are numbered sequentially and correspond to peak 1, peak 2,. .. to peak 105. The structure and retention index of each constituent in the mixture were reported by Schwartz et al. (9). The tabular listing of the data is availedile from the present authors. 

The anions of primary nitramines, like other nucleophiles, can undergo Michael 1,4-addition reactions with a range of a,-unsaturated substrates to form secondary nitramines of varying molecular complexity (Equation 5.18). Kissinger and Schwartz prepared a number of secondary nitramines from the condensation of primary nitramines with a,/3-unsaturated ketones, esters, amides and cyanides. In a standard experiment a solution of the primary nitramine and... 

Fahien and Smith (F2) and Dorweiler and Fahien (D20) have considered the variation of Dr in packed beds, using a separation-of-vari-ables technique to solve Eq. (63). The Z-dependent part was solved analytically, and a set of difference equations was used to solve the /2-dependent part. Details are given in (FI). The velocity profile data of Schwartz and Smith (Sll) was used to calculate values of Dr R) in the packed column, typical results from Fahien and Smith (F2) being shown in Fig. 14. Dorweiler and Fahien s (D20) data, for a lower Rey-... 

We wish here to obtain the thermodynamic equations defining the liquidus surface of a solid solution, (At BB)2, ). It is assumed that the A and atoms occupy the sites of one sublattice of the structure and the C atoms the sites of a second sublattice. For the specific systems considered here Sb and play the role of C in the general formula above. It is also assumed that the composition variable is confined to values near unity so that the site fractions of atomic point defects is always small compared to unity. This apparently is the case for the solid solutions in the two systems considered. Then it can be shown theoretically (Brebrick, 1979), as well as experimentally for (Hgj CdJ2-yTe)l(s) (Schwartz et al, 1981 Tung et al., 1981b), that the sum of the chemical potentials of A and C and that of and C in the solid are independent of the composition variable y ... 

For the very small systems in Table 7.1, it is possible to approach the exact solution of the Schrodinger equation, but, as a rule, convergence of the correlation energy is depressingly slow. Mathematically, this derives from the poor ability of products of one-electron basis functions, which is what Slater determinants are, to describe the cusps in two-electron densities that characterize electronic structure. For the MP2 level of theory, Schwartz (1962)... 

In many cases ordinary differential equations (ODEs) provide adequate models of chemical reactors. When partial differential equations become necessary, their discretization will again lead to large systems of ODEs. Numerical methods for the location, continuation and stability analysis of periodic and quasi-periodic trajectories of systems of coupled nonlinear ODEs (both autonomous and nonautonomous) are extensively used in this work. We are not concerned with the numerical description of deterministic chaotic trajectories where they occur, we have merely inferred them from bifurcation sequences known to lead to deterministic chaos. Extensive literature, as well as a wide choice of algorithms, is available for the numerical analysis of periodic trajectories (Keller, 1976,1977 Curry, 1979 Doedel, 1981 Seydel, 1981 Schwartz, 1983 Kubicek and Hlavacek, 1983 Aluko and Chang, 1984). 

Schwartz and coworkers, in their development of hydrozirconation, studied the stereochemistry at carbon during the alkoxide forming reaction (equation 35).114 115... 

Several cyclofunctionalization reactions of alkynic alcohols are synthetically useful. Metal ion-promoted cyclofunctionalization of ris-2-propargylcyclopentanol systems proceeds by the 5-exo mode (equation 77 and Table 23).197 Protiodemetallation or reductive demetallation provides the cyclic enol ether in high yields. This method has been used by Noyori in the synthesis of prostacyclin (PGh).197b,197c Reactions with catalytic amounts of mercury(II) or palladium(II) salts gave the endocyclic enol ether as the major product.197 -198 A related cyclization with Ag2C03 has been reported by Chuche.191 Schwartz... 

Jackson and Mott23 obtained a solution of equation (10) in terms of Bessel functions, from which T was derived in exponential functions. Following a simplification similar to that noted by Schwartz, Slawsky and Herzfeld26 (ssh theory)... 

Various techniques have been devised for coping with Schwartz singularities. They may either be ignored, or they can be avoided by using a modified form of the Kohn variational method in which the asymptotic form of the trial function has an amplitude different from that of equation (3.42) ... 

Probably the most accurate positron-hydrogen s-wave phase shifts are those obtained by Bhatia et al. (4974), who avoided the possibility of Schwartz singularities by using a bounded variational method based on the optical potential formalism described previously. These authors chose their basis functions spanning the closed-channel Q-space, see equation (3.44), to be of essentially the same Hylleraas form as those used in the Kohn trial function, equation (3.42), and their most accurate results were obtained with 84 such terms. By extrapolating to infinite u in a somewhat similar way to that described in equation (3.54), they obtained phase shifts which are believed to be accurate to within 0.0002 rad. They also established that there are no Feshbach resonances below the positronium formation threshold. 

According to Schwartz, Slawksy, and Herzfeld, when v approaches vj and the energy transferred from translation approaches zero, the transition probability of Jackson and Mott [see equation (25)], for an exponential repulsive potential goes over to the form... 

Differentiation of equation (10.1) to obtain a value of dl/dt, U, permits the Reynolds number to be rewritten as equal to (ffLvCos0Ypd2/(6/72l)). Assuming the workpieces are completely wetted, substitution of Table 10.1 data yields Reynolds numbers for a typical braze joint 0.1 mm wide and 10 mm long, (Schwartz 1995), ranging from 49.5 for A1 to 117 for Au and even smaller numbers are obtained if the workpieces are not completely wetted so that cos()Y is less than 1. These values are much smaller than that needed to induce a transition in flow behaviour, and... 

Since the development of the equation, it has been tried to derive further information from it. Rees and Rue [129] determined the area under the Heckel plot. Duberg and Nystrom [137] used the nonlinear part for characterization of particle fracture. Paronen [138] deduced elastic deformation from the appearance of the Heckel plot during decompression. Morris and Schwartz [139] analyzed different phases of the Heckel plot. Imbert et al. [134] used, in analogy to Leuenberger and Ineichen [14], percolation theory for the compression process as described by the Heckel equation. Based on the Heckel equation, Kuentz and Leuenberger [135,140] developed a new derived equation for the pressure sensitivity of tablets. 

This bimetallic approach is based on hydride-transfer from M-H species. Schwartz (16 showed tliat CO in the presence of Cp2ZrCIj can be reduced with t-BuiAlH to C)—C linear alcohols. Complexes, as shown in Equation (4), are postulated as intermediates. 

---