Grassmann product

J. E. Harriman, Grassmann products, cumulants, and two-electron reduced density matrices. Phys. Rev. A 65, 052507 (2002). 

VIII. A Look Ahead Acknowledgments Appendix Grassmann Products References... 

J. M. Herbert and J. E. Harriman, Contraction relations for Grassmann products of reduced density matrices and implications for density matrix reconstruction. J. Chem. Phys. 117, 7464 (2002). 

Here A denotes an antisymmetiized product Grassmann product [47-56])... 

Particle number can be increased either by replacing A with A +i, or by incorporating an additional Grassmann product with Ai. As an example, consider generating D3 (Eq. (25c)) from D2 (Eq. (25b)). Given the first term in Eq. (25b), Ai A Ai, one can increase particle number in three ways, and from these one obtains A2 A Ai -f Ai A A2 + Ai A Ai A Ai. The second term in Eq. (25b), A2, affords A3 + A2 A Aj upon increase in particle number. Together, these terms afford Eq. (25c). 

Grassmann product scales asymptotically (A oo) like for some > 1. For example, the Grassmann product... 

These expressions are highly compact compared to brute-force expansions of the Grassmann products and For example, A ostensibly contains... 

Including permutations, this expression for contains 68 terms, a significant reduction as compared to the unsimplified Grassmann products. 

In the contraction of any wedge product the position of the upper and lower indices generates two types of terms in Grassmann algebra [26] (i) pure contraction terms where the upper and lower contraction indices appear on the same component of the wedge product, and (ii) transvection terms, where the upper and lower contraction indices appear on different components of the wedge product. To illustrate, we consider the contraction of the wedge product between A and A ... 

The condition that is a positive semidefinite operator ensures that belongs to the Kummer cone. Unfortunately, the Grassmann wedge product of two operators is not explained explicitly in Ref. [1] but the section, p.79 may be sufficient. Here are two other references [4, 5] that may be helpful. Be sure that you understand Exercise 4 on p.73 of Ref. [1]. 

Intramolecular dipolar cycloaddition may also result in products that do not correspond to experience with related intermolecular cyclizations. An example is the intramolecular reaction of 6-diazohex-l-ene (6.32), which was investigated by Kirmse and Grassmann (1966). This reaction leads to 2,3-diaza-bicyclo[3.3.0]oct-2-ene (6.34), i. e., with a regioselectivity that corresponds to the formation of a 4-alkyl-l-pyrazoline. 

In contrast to the Munsell system, the CIE system does not use color samples. The CIE system is based on the Grassmann laws, which state that a color valency is completely described by the sum of the vector products of three color values and their color value proportions ... 

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