Shepard function

B(x). Here, superscript k denotes the order of the polynomial basis. The lowest order form of the moving least square, i.e., for = 0, is called the Shepard function... 

Shepard R (1987) Geometrical energy derivative evaluation with MRCI wave functions. Int J Quantum Chem 31 33... 

Shepard R (1995) The analytic gradient method for configuration interaction wave functions. Yarkony DR (ed) In Modern electronic structure theory part I, World Scientific, Singapore, p 345... 

This method was used for optimization of electronic wave functions by A. Baneijee and F. Grein, Int. J. Quantum Chem. 10, 123 (1976) and D. R. Yarkony, Chem. Phys. Lett. 77, 634 (1981). It was applied to surface studies by A. Baneijee, N. Adams, J. Simons, and R. Shepard, J. Phys. Chem. 89, 52 (1985). It has, to the author s knowledge, not been discussed in textbooks on optimization. 

Plyley MJ, Shepard RJ, Davis GM, Goode RC. Sleep deprivation and cardiorespiratory function—influence of intermittent submaximal exercise. Eur J Appl Physiol 1987 56 338-344. 

Sarup, J. C., Johnson, R. M., King, K. L., Fendly, B. M., Lipari, M. T., Napier, M. A., Ullrich, A., and Shepard, H. M. (1991). Characterization of an anti-pl85HER2 monoclonal antibody that stimulates receptor function and inhibits tumor cell growth. Growth Regul. 1, 72-82. 

Shepard interpolation has been applied to several systems. Wu et al. have used it to construct the ground potential energy surface for the reaction CH4 -I- H -> CH3 -I- H2 [63]. Neural networks can be described as general, non-linear fitting functions that do not require any assumptions about the functional form of the... 

Shepard, R., Discussion of some multiconfiguration wave function optimization methods , 183rd ACS National Meeting, March 1982. 

Shepard, J.S. 1993. Using a fractal model to compute the hydraulic conductivity function. Soil Sci. Soc. Am. J. 57 300-306. 

Bracken MB, Shepard MJ, Hellenbrand KG, et al. Methylprednisolone and neurological function 1 year after spinal cord injury. Results of the National Acute Spinal Cord Injury Study. J Neurosurg 1985 63(5) 704-713. 

The reviews of the MCSCF method by Shepard [129] and Werner [130] are too technical to summarize briefly. We shall content ourselves with pointing out that the structure of our MCDHF(B) equations is consistent with the nonrela-tivistic theory, so that much of the nonrelativistic formalism can be adopted with little change. The nonrelativistic energy expression is of the form of (231) and can be regarded as a function of the CSF coefficients Cj and the MO expansion coefficients where... 

A. Banerjee, R. Shepard and J. Simons, One-particle Green s function with multiconfiguration reference states, Int. J. Quantum Chem. Quantum Chem. Symp., 12 (1978) 389. 

Wettability Studies of Bartelland Shepard. In Figures 4 and 5 we have plotted the contact angle data of Bartell and Shepard [2, 3,11] as a function... 

Patton, J.S., Shepard, H.M., Wilking, H., Lewis, G., Aggarwal, B.B., Eessalu, T.E., Gavin, L.A., and Grunfeld, C. 1986. Interferons and tumor necrosis factors have similar catabolic effects on 3T3-L1 cells. Proc. Natl. Acad. Sci. USA. 83 8313-8317. Uysal, K.T., Wiesbrock, S.M., Marino, M.W., and Hotamisligil, G.S. 1997. Protection from obesity-induced insulin resistance in mice lacking TNF-alpha function. Nature. 389 610-614. 

Shepard, R. N. The analysis of proximities Multidimensional scaling with an unknown distance function. II. Psychometrika, 1962, 27, 219-246. 

The optimum MCSCF orbitals can be found by an iterative process somewhat similar to the iterative process used to find SCF wave functions see A. C Wahl and G. Das, Chapter 3 in Schaefer, Methods of Electronic Structure Theory. By optimizing the orbitals, one can get good results with inclusion of relatively few CSFs. Because the orbitals are varied, the amount of calculation required in the MCSCF procedure is great, but advances in methods of computing MCSCF wave functions [R. Shepard, Adv. Chem. Phys., 69,63 (1987)] have led to wide use of the MCSCF and related methods... 

An alternative approach to estimating the metabolic capabilities of chloroplasts entails measurement of the light-dependent metabolism of radioactive tracers. Using isolated pea chloroplasts. Mills and Wilson (1978a) found that lysine, methionine, threonine, and isoleucine were synthesized from [ C]aspartate. Further evidence that aspartate was being metabolized via the anticipated pathways was provided by the demonstration that the synthesis of homoserine was inhibited by lysine and threonine (Lea et al., 1979). These results, combined with those relating to enzyme localization, lead to the concept that chloroplasts contain a complete functional sequence of enzymes which can facilitate the synthesis of the aspartate family and at least some of the branched-chain amino acids. This is consistent with the importance of chloroplasts in ammonia assimilation (Miflin and Lea, this volume. Chapter 4) and with the evidence that protein can be synthesized from CO2 in isolated plastids (Shepard and Leven, 1972 Huberer al., 1977). The actual fraction of [ ]02 which is utilized for amino acid biosynthesis in isolated plastids is usually quite small. Thus, reactions which normally occur outside of chloroplasts are considered to be of major importance in the synthesis of carbon skeletons such as oxaloacetate or pyruvate (Kirk and Leech, 1972 Leech and Murphy, 1976). 

R. Shepard, I. Shavitt, and J. Simons, Comparison of Convergence Characteristics of Some Iterative Wave Function Optimization Methods, J. Chem. Phys. 76, 543-557 (1982). 

Shepard R, Gidofalvi G, Brozell SR (2014) The multifacet graphically contracted function method I. Formulation and implementation. J Chem Phys (in press)... 

Gradient methods discussed above use a quadratic function (energy, gradient and approximate Hessian) to model the energy surface near the transition state. Distance-weighted interpolants provide a more flexible functional form that can interpolate arbitrarily spaced points with a smooth differentiable function. For a gradient-based optimization, the Shepard interpolation functions seem appropriate... 

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