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The index theorem
Published in Peter B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, 2018
be a homogeneous invariant polynomial and let P∼ be the complete polarization. Let Ωg be the curvature of the Levi-Civita connection. Then the associated characteristic form is given by: () P(∇g):=P∼(Ωg,…,Ωg).
Discrete Linear Systems
Published in Janos J. Gertler, Fault Detection and Diagnosis in Engineering Systems, 2017
Zeroes. The zeroes of the multiple-input multiple-output system are defined as the roots of the invariant (numerator) polynomial φ1 (ϕ)φ2(ϕ)…φρ (ϕ) of the Smith-McMillan matrix. Again, multiplicities are possible, if and as shown by the invariant polynomial. Note that a multiple-input multiple-output system may have a pole and a zero at the same value (the individual φi, ηi pairs are relative prime but their products may not be). It can be shown (MacFarlane and Karcanias, 1976) that the zeroes defined above are the same as the invariant zeroes defined in terms of the system matrix Γ+(ϕ) (see Subsection 2.5.5) of the minimum (observable and controllable) realizations. If the realization contains unobservable and/or uncontrollable modes, these result in extra zeroes and poles, but these are invisible in the transfer function due to cancellation (see Subsections 2.6.4 and 2.6.5).
Limit Cycles and Centres: An Example
Published in K. D. Elworthy, W. Norrie Evenitt, E. Bruce Lee, Differential equations, dynamical systems, and control science, 2017
where q{x) = x(x2 – 1)(a2x2+ 1). The transformed system is of Liénard form and the origin is immediately seen to be a centre by symmetry in the Y-axis. Note that (1.2) has an invariant polynomial if and only if (3.9) does. Reverting to coordinates (x,y) let
CH4·F− revisited: full-dimensional ab initio potential energy surface and variational vibrational states
Published in Molecular Physics, 2023
Dóra Papp, Viktor Tajti, Gustavo Avila, Edit Mátyus, Gábor Czakó
The PES is fitted using the permutationally invariant polynomial method [43], based on polynomials of Morse-like variables of the internuclear distances with a = 3.0 bohr. The full-dimensional analytical function, representing the PES, which is invariant under the permutation of like atoms, has a highest total order of the polyanomials of 7. We note that a 7th-order polynomial of yij can accurately describe the asymptotic region of the PES, even better than a single-parameter (σ) –σ/rijn function, as demonstrated for the CH4·Ar complex, where n = 6, in Ref. [32]. The PES is determined using 9355 coefficients in a weighted least-squares fit on the energy points, with a weight of (E0/(E0 + E)) × (E1/(E1 + E)) with E0 = 0.05 hartree and E1 = 0.5 hartree applied on a given energy E relative to the global minimum of the data set.
The state-to-state dynamics of the N + NH(3Σ−) → N2(X1 Σg +) + H reaction: based on a new global potential energy surface
Published in Molecular Physics, 2023
Ziliang Zhu, Yinghua Feng, Wentao Li
As discussed above, the PESs of the ground state of the HN2 system were mainly reported by Varandas et al. [14,15,17] before 2009 using the DMBE method. In addition, the rate constants based on these PESs are larger than those of the experimental data. Therefore, to further understand the reaction, we constructed a new global PES with a permutation invariant polynomial neural network (PIP-NN) [29,30] method. As far as we know, there are no quantum state-resolved dynamic studies for the N + NH(3Σ−) → N2(X1) + H reaction until now. The aims of this work are to construct a more accurate PES of the HN2 system and presented dynamic results at a state-to-state level of theory. This paper is organised as follows: The theory introduction is shown in Section 2; The topographic features of PES are presented in Section 3; The dynamical results of the title reaction are discussed in Section 4; The conclusions are displayed in Section 5.
An analytic global potential energy surface of the CsH2 system and the dynamic calculations of the H + CsH reaction
Published in Molecular Physics, 2023
The permutation invariant polynomial neural network (PIP-NN) method [48,49] was used to construct PES, which is widely employed in PES construction such as LiH2 [6], BrH2 [50], KH2 [51] and so on. Herein, we only briefly introduce the PIP-NN method, more details can be found in the literature [52]. First, the three bond lengths of the CsH2 system were transformed into Q1 = exp(−0.2·RCsH), Q2 = exp(−0.2·RHH), and Q3 = exp(−0.2·RCsH). To avoid the discontinuities of the derivatives, the PIP method was adopted. Then, the input term Gi (i = 1, 2, 3) was expressed as