A method is presented for solving the Bloch equation for Morse potential, based on the application of Fourier transformation of the density matrix. The results obtained agree completely with the results obtained by other methods, however its advantage lies in its elegance and intuitivity of the approach. q 2000 Elsevier Science Ltd. All rights reserved.

Motivated by numerical examples in solving semilinear elliptic PDEs for multiple solutions, some properties of Newton homotopy continuation method, such as its continuation on symmetries, the Morse index and certain functional structures, are established. Those results provide useful information on selecting initial points for the method to find desired solutions. As an application, a bifurcati...

We show that the self-shuffle of Thue-Morse given by Charlier et al. is optimal/canonical in the sense that among self-shuffles of Thue-Morse, it has the lexicographically least directive sequence starting with 1.

The present paper contains an interpretation and generalization of Novikov’s theory for Morse type inequalities for closed 1-forms in terms of concepts from Conley’s theory for dynamical systems. We introduce the concept of a flow carrying a cocycle α, (generalized) α-flow for short, where α is a cocycle in bounded Alexander-Spanier cohomology theory. Gradient-like flows can then be characteriz...

We investigate how syntactic properties of Thue-Morse words are related to special type of automata/graphs. The directed acyclic subword graph (dawg, in short) is a useful deterministic automaton accepting all su xes of the word. Its compacted version (resulted by compressing chains of states) is denoted by cdawg. The cdawgs of Thue-Morse words have regular and very simple structure, in particu...

3 Morse-Bott theory 30 3.1 Framed moduli space . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 Gradient flow lines . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3 Relative Morse Index . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.4 Decay estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.5 Transversality of M(Oa, Ob) . . . . . . . . . . . . . . . ...

This paper uses Morse-theoretic techniques to compute the equivariant Betti numbers of the space of semistable rank two degree zero Higgs bundles over a compact Riemann surface, a method in the spirit of Atiyah and Bott’s original approach for semistable holomorphic bundles. This leads to a natural proof that the hyperkähler Kirwan map is surjective for the non-fixed determinant case. CONTENTS

We introduce graded, enriched characteristic cycles as a method for encoding Morse modules of strata with respect to a constructible complex of sheaves. Using this new device, we obtain results for arbitrary complex analytic functions on arbitrarily singular complex analytic spaces. §

The Variational Method is applied within the context of Supersymmetric Quantum Mechanics to provide information about the energy and eigenfunction of the lowest levels of a Hamiltonian. The approach is illustrated by the case of the Morse potential applied to several diatomic molecules and the results are compared with stablished results.

Egorov's theorem on the classical propagation of quantum observables is related to prominent quasi-classical descriptions of quantum molecular dynamics as the linearized semiclassical initial value representation, the Wigner phase space method, or the statistical quasiclassical method. The error estimates show that different accuracies are achievable for the computation of expectation values an...