INDEX polynomial

This generating function is easily obtained from another polynomial, called cycle index polynomial. It displays - in the exponents of its summands the cycle structure of the permutation g induced by e G on X ... 

Thus, the generating function for the enumeration of symmetry classes of mappings is obtained from the cycle index polynomial by simply replacing the indeterminate Zj by the power sum symmetric function (a polynomial is called symmetric if it is invariant under permutations of the indeterminates) Xj y], using... 

The polynomial (1.5) which I called cycle index is, if H is the symmetric group, equal to the principal character of H in representation theory. Professor Schur informed me that the cycle index of an arbitrary permutation group being really a subgroup of a symmetric group is of importance for the representation of this symmetric group. We will, however, not expand on the relationship between representation theory and our subject. 

One example of a structure (8) is the space of polynomials, where the ladder of subspaces corresponds to polynomials of increasing degree. As the index / of Sj increases, the subspaces become increasingly more complex where complexity is referred to the number of basis functions spanning each subspace. Since we seek the solution at the lowest index space, we express our bias toward simpler solutions. This is not, however, enough in guaranteeing smoothness for the approximating function. Additional restrictions will have to be imposed on the structure to accommodate better the notion of smoothness and that, in turn, depends on our ability to relate this intuitive requirement to mathematical descriptions. 

K -I- l)th-order (degree < K + 1) and Kth-order polynomials, respectively. (Here the notation k = 0,1 refers to the index k starting at zero but not equal... 

In the three-branch horseshoe, the periodic oibit 0 is hyperbolic with reflection and has a Maslov index equal to no = 3 while the off-diagonal orbits 1 and 2 are hyperbolic without reflection with the Maslov index n = 2 [10]. Fitting of numerical actions, stability eigenvalues, and rotation numbers to polynomial functions in E can then be used to reproduce the analytical dependence on E. The resonance spectrum is obtained in terms of the zeros of (4.16) in the complex energy surface. 

The shape of the vibration-rotation bands in infrared absorption and Raman scattering experiments on diatomic molecules dissolved in a host fluid have been used to determine2,15 the autocorrelation functions unit vector pointing along the molecular axis and P2(x) is the Legendre polynomial of index 2. These correlation functions measure the rate of rotational reorientation of the molecule in the host fluid. The observed temperature- and density-dependence of these functions yields a great deal of information about reorientation in solids, liquids, and gases. These correlation functions have been successfully evaluated on the basis of molecular models.15... 

The following covariates individually made no substantial changes in the excess risk fractions for men or women race, fat in diet, body mass index, use of Tagamet, use of Valium, Librium, or Tylenol, ever had kidney disease, ever had bronchitis, marital status, and ever had cancer. Age was represented by a cubic polynomial. 

Randic [21] and Aihara [22] independently proposed an idea of the index of local aromaticity, ILA, and overall index of aromaticity, OIA, based on the counting of the Kekule patterns. However, these concepts were found to be closely related to the sextet polynomial and its derivative as [9]... 

Find a polynomial regression applet on the web to fit the PDB structure data in Figure 9.2. (A good one is http // www.arachnoid.com/polysolve/index.html.) Select a high enough order to give a reasonable fit (r2 s 0.99). Based on the best-fit equation, how many structures would you predict to be in the PDB in 2015 2020 ... 

A proof for this statement is constructed in accordance with the fact that the latter inequality accounts for the sign of the coefficient in the polynomial P(/.) at kn T, which in turn is associated with the index of a steady-state point for the vector field (151) [60]. If this coefficient is positive at any point of the positive orthant R z, zt > 0, i = 1, 2,. . ., n, then the steady-state point is unique. 

However, the linear response of a dielectric to an applied field is an approximation the actual response is non-linear and is of the form indicated in Fig. 8.6. The electro-optic effect has its origins in this non-linearity, and the very large electric fields associated with high-intensity laser light lead to the non-linear optics technology discussed briefly in Section 8.1.4. Clearly the permittivity measured for small increments in field depends on the biasing field E0, from which it follows that the refractive index also depends on E0. The dependence can be expressed by the following polynomial ... 

THE FUNCTIONS OF MATHEMATICAL PHYSICS, Harry Hochstadt. Comprehensive treatment of orthogonal polynomials, hypergeometric functions. Hill s equation, much more. Bibliography. Index. S22pp. 55 x 854. 65214-9 Pa. 8.95... 

In addition to these, only a limited number of other topological indices of benzenoid molecules have been studied. With a few not too important exceptions, generally valid mathematical results were obtained only for one of them — namely for the Wiener index. Therefore the remaining part of this section is devoted to the Wiener index of benzenoid systems. (Further graph invariants worth mentioning in connection with benzenoids, especially unbranched catacondensed systems, are the Hosoya index [119-121], the Merrifield — Simmons index [122, 123], the modified Hosoya index [38] and the polynomials associated with them.)... 

Eigenvectors for different characters are linearly independent, since p(v) = v Xv and we know by (2.2) that the different Xv are independent Hence there are only finitely many different jf > and the subgroup H = g xtv — Xv bas finite index in S. But for each n in the equality Xv n) = Xv(ff ifW) is a polynomial equation in g, and. thus H is closed. A connected S cannot have a proper closed subgroup of finite index, since by (S.2) the cosets would disconnect S. Thus H = S, and S acts on all gv by the same character. 

Upon choosing an active orbital index pair i,j) and a rotation sine, variation of the energy through a EJR can be obtained from the preceding section. The net result is a fourth order sine polynomial... 

ASE for thiophene (20.2-22.4kcalmoP ) is calculated using molecular descriptors such as magnetic susceptibility exaltation ( ), NICSs, and electrotopological indexes (Els) via linear, quadratic, and cubic fitting polynomials. Theoretical estimations compare fairly well with experimental data when three variable multilinear regression equations are employed <2004MI145>. 

---