Polynomial primitive

To understand the criteria for basis set choice, then, we need consider only the behavior of the primitive integrals. The primitive integrals over the basis functions can be expressed in terms of Hermite polynomials... 

Section III introduces the concept of nonmonotonic planning and outlines its basic features. It is shown that the tractability of nonmonotonic planning is directly related to the form of the operators employed simple propositional operators lead to polynomial-time algorithms, whereas conditional and functional operators lead to NP-hard formulations. In addition, three specific subsections establish the theoretical foundation for the conversion of operational constraints on the plans into temporal orderings of primitive operations. The three classes of constraints considered are (1) temporal ordering of abstract operations, (2) avoidable mixtures of chemical species, and (3) quantitative bounding constraints on the state of processing systems. 

In this construction the first row of the matrix is taken to be one period s worth of the output from a shift-register whose feedback polynomial is a primitive polynomial of degree n. The exact definition of these polynomials need not concern us here, since Table 1 gives examples of degrees 1 through 20, enough to construct S-matrices of order up to 2 0 - 1 = 1048575. 

Hosoya (1986a) observed a resemblance between the polynomial in (1) and the characteristic polynomial for benzene, viz. a — 6x4 gx — 4. He writes "The reason for this mystic coincidence, however, is not known." On the other hand, a substantial amount of work has been done in the area of combinatorial K formulas in the form of polynomials for primitive coronoids in general, and especially for those of regular hexagonal symmetry as in the case above... 

Quantum dynamics simulations of the UV absorption spectrum and of the electronic state population dynamics of the molecule excited by a short laser pulse resonant with the transition to the bright B2u t t ) state, based on the models described in Sect. 5.3 were performed using the MCTDH method in the multi-set formalism (see Sect. 4.2.5 in Chap. 4). For the representation of the Hamiltonian and the wave function, a Hermite polynomial DVR scheme [60] was used for all the degrees of freedom. The number of SPF and primitive basis functions used in the calculations are listed in Table 5.4. Test calculations with both larger primitive and SPF bases have been... 

Shor PW (1997) Polynomial-time algorithms for primitive factorization and discrete logarithms on a quantum computer. SIAM J Comput 26 1484—1509... 

---