Polynomial representations

The thermodynamic properties of single-component condensed phases are traditionally given in tabulated form in large data monographs. Separate tables are given for each solid phase as well as for the liquid and for the gas. In recent years analytical representations have been increasingly used to ease the implementation of the data in computations. These polynomial representations typically describe the thermodynamic properties above room temperature (or 200 K) only. 

Thermodynamic representation of transitions often represents a challenge. First-order phase transitions are more easily handled numerically than second-order transitions. The enthalpy and entropy of first-order phase transitions can be calculated at any temperature using the heat capacity of the two phases and the enthalpy and entropy of transition at the equilibrium transition temperature. Small pre-tran-sitional contributions to the heat capacity, often observed experimentally, are most often not included in the polynomial representations since the contribution to the... 

Analytical Continuation of the Polynomial Representation of the Full, Interacting Time-Independent Green Function. 

Energy-Separable Polynomial Representation of the Time-Independent Full Green Operator with Application to Time-Independent Wavepacket Forms of Schrodinger and Lippmann-Schwinger Equations. 

Energy-Separable Faber Polynomial Representation of Operator Functions Theory and Application in Quantum Scattering. 

As an alternative procedure to predict coefficients of a radial function p(x) for electric dipolar moment, one might attempt to convert the latter function from polynomial form, as in formula 91, which has unreliable properties beyond its range of validity from experimental data, into a rational function [13] that conforms to properties of electric dipolar moment as a function of intemuclear distance R towards limits of united and separate atoms. When such a rational function is constrained to yield the values of its derivatives the same as coefficients pj in a polynomial representation, that rational function becomes a Fade approximant. For CO an appropriate formula that conforms to properties described above would be... 

Huang Y., Kouri, D.J. and Hoffman, D.K. (1994) General, energy-separable Faber polynomial representation of operator-functions - theory and application in quantum scattering J. Chem. Phys. 101, 10493-10506. 

D. Levin Using Laurent polynomial representation for the analysis of non-uniform binary subdivision schemes. Adv. Comput. Math 11, pp41-54, 1999... 

Hollander-Kasteleyn expressions or from polynomial representations of numerical data on n) for lattices of various [A. d, v]. It was noted previously that when calculations are performed in which all symmetry-distinct sites, both internal and external in the geometries represented are considered, values of n) for the Menger sponge lie between the corresponding values for (n) ind — 2 and J = 3 for all N > 72. The data in Table III.l 1 show that this correlation holds even if one restricts consideration to surface-only locations of the reaction center, except for a few special site locations (viz., the corner sites in all cases, and the additional site 2 in I). 

State calculations. With the extensions provided, the method can be applied to the full Watson Hamiltonian [51] for the vibrational problem. The efficiency of the method depends greatly on the nature of the anharmonic potential that represents couphng between different vibrational modes. In favorable cases, the latter can be represented as a low-order polynomial in the normal-mode displacements. When this is not the case, the computational effort increases rapidly. The Cl-VSCF is expected to scale as or worse with the number N of vibrational modes. The most favorable situation is obtained when only pairs of normal modes are coupled in the terms of the polynomial representation of the potential. The VSCF-Cl method was implemented in MULTIMODE [47,52], a code for anharmonic vibrational spectra that has been used extensively. MULTIMODE has been successfully applied to relatively large molecules such as benzene [53]. Applications to much larger systems could be difficult in view of the unfavorable scalability trend mentioned above. 

Several Correlative Liquid Mixture Activity Coefficient Models 431 The simplest polynomial representation of G satisfying these criteria is... 

Y. Huang, D. J. Kouri, and D. K. Hoffman, A general, energy-separable polynomial representation of the time-independent full Green operator with application to time-independent wavepacket forms of Schrodinger and Lippmann-Schwinger equations, Chem. Phys. Lett. 225 31 (1994). 

This chapter is intended to present an integrated description of this general approach to quantum dynamics. Applications of the equations and strategies both to scattering and bound state problems will be discussed. In the next section, we begin with a detailed summary of the salient features of the DAFs as they are used to represent the Hamiltonian operator. Then in Sec. Ill, we discuss the TIWSE and some of the choices that can be made in solving for bound states and scattering information. Included in this is a discussion of the polynomial representations of various operators involved in the TIW form of quantum mechanics. Finally, in Sec. IV we briefly summarize some of the applications made to date of this overall approach. 

Vol. 830 J.A. Green, Polynomial Representations of GLn, with an Appendix on Schensted Correspondence and Littelmann Paths by K. Erdmann, J.A. Green and M. Schoker 1980 - 2nd corr. and augmented edition... 

CAD Model Representation The choice for a certain surface representation system will have far-reaching consequences for the capabilities and performance of the tool path generation algorithm. In general, there are three different surface representation techniques polynomial representation, faceted models, and Z-map models. For five-axis tool path generation, only the two first are the most important and described below. 

Knowing the mean concentration, determine the time-dependent coefficients of the assumed polynomial solution, and finally obtain the complete polynomial representation. 

Modification of a mesh can be best automated by the application of the novel parametric mesh. The polynomial representation of a mesh can be modified by changed parameters. Typical or task related values of parameters can be stored in databases. 

Used by Babcock and Wilcox—Polynomial representation of Lyon s JfcdT... 

As an example of this approach let us consider the constitutive equation arrived at (a) by adopting unchanged the field equations and boundary conditions of the linear theory, and (b) introducing cubic and higher order terms in the polynomial representation... ... 

Here A x, t) is usual normalized thickness of diffusive boundary. The simplest polynomial representation of c(x, y, t) which satisfies boundary conditions is... 

A careful restructuring of a quasiclassical trajectory code and the adoption of a polynomial representation of the potential energy has made it possible to take profit from both vector and parallel features of the IBM 3090/200 VF supercomputer to perform extended calculations of the reactivity of some alkali/alkaline earth -f- hydrogen halide systems. Thanks to the reduction of the needed cpu time it has thus been possible to examine in detail the dynamics of these reactions. In particular, it has been possible to improve ab initio potential... 

The goal is to construct and to evaluate a continuous representation without additional evaluations of the right hand side function /. The construction of a continuous representation is straight forward for integration methods based on a polynomial representation of the solution or its derivative, like Adams or BDF multistep methods or Runge-Kutta methods based on collocation. 

In all cases, the polynomial of lowest order which adequately fitted the data was chosen. A comparison of these polynomial representations and the corresponding wheelbase and track characteristics data is presented in figures 2 and 3. 

Using absolute entropy values from Table A.8 and heat-capacity values from Table A.6 or Table A.8, calculate values for the substances at the indicated temperatures. If no polynomial representations are available, assume constant heat capacities. 

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