CUMULATIVE PART

In this paper, we examine the electron correlation of one-dimensional and quasi-one-dimensional Hubbard models with two sets of approximate iV-representability conditions. While recent RDM calculations have examined linear [20] as well as 4 x 4 and 6x6 Hubbard lattices [2, 57], there has not been an exploration of ROMs on quasi-one-dimensional Hubbard lattices with a comparison to the one-dimensional Hubbard lattices. How does the electron correlation change as we move from a one-dimensional to a quasi-one-dimensional Hubbard model How are these changes in correlation reflected in the required A -repre-sentability conditions on the 2-RDM One- and two-par-ticle correlation functions are used to compare the electronic structure of the half-filled states of the 1 x 10 and 2x10 lattices with periodic boundary conditions. The degree of correlation captured by approximate A -repre-sentability conditions is probed by examining the one-particle occupations around the Fermi surfaces of both lattices and measuring the entanglement with a size-extensive correlation metric, the Frobenius norm squared of the cumulant part of the 2-RDM [23]. 

Fig. 6 Frobenius norm squared of the cumulant part of the 2-RDM is shown for the (a) 2 x 10 and (b) 1x10 lattices where the 2-RDMs are computed with variational 2-RDM calculations with DQG and DQGT conditions...

Recall a typical cumulative probability curve of reserves for an exploration prospect in which the probability of success (POS) is 30%. The success part of the probability axis can be divided into three equal bands, and the average reserves for each band is calculated to provide a low, medium and high estimate of reserves, //there are hydrocarbons present. 

Critical micelle concentration (Section 19 5) Concentration above which substances such as salts of fatty acids aggre gate to form micelles in aqueous solution Crown ether (Section 16 4) A cyclic polyether that via lon-dipole attractive forces forms stable complexes with metal 10ns Such complexes along with their accompany mg anion are soluble in nonpolar solvents C terminus (Section 27 7) The amino acid at the end of a pep tide or protein chain that has its carboxyl group intact—that IS in which the carboxyl group is not part of a peptide bond Cumulated diene (Section 10 5) Diene of the type C=C=C in which a single carbon atom participates in double bonds with two others... 

If necessary, the fit can be improved by increasing the order of the polynomial part of Eq. (9-89), so that this approach provides a veiy flexible method of simulation of a cumulative-frequency distribution. The method can even be extended to J-shaped cui ves, which are characterized by a maximum frequency at x = 0 and decreasing frequency for increasing values of x, by considering the reflexion of the cui ve in the y axis to exist. The resulting single maximum cui ve can then be sampled correctly by Monte Carlo methods if the vertical scale is halved and only absolute values of x are considered. 

In order to reinforce such conclusion for this part of the phase boundary we also studied the cumulants as functions of the rotational constant at constant temperature in the range T < 0.2424. The fourth-order cumulant plots show that the intersection occurs in this low temperature range at a... 

This work was supported in part by NIH Grant GM-26898. The author particularly wishes to thank present and past members of the Laboratory of Molecular Biophysics whose cumulative efforts of more than a decade have made this review possible. 

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