Bogoliubov method

Kolmogorov, A. N., 114,139,159 Konigs thorem applied to Bernoulli method, 81 Koopman, B., 307 Roster, G.F., 727,768 Kraft theorem, 201 Kronig-Penney problem, 726 antiferromagnetic, 747 Krylov-Bogoliubov method, 359 Krylov method, 73 Krylov, N., 322 Kuhn, W. H., 289,292,304 Kuratowski s theorem, 257... 

Appendix B. Approximate Solution of the Non-linear Second-Order Differential Equation by Small-Parameter (Krylov-Bogoliubov) Method [20]... 

The use of this theory in studies of nonlinear oscillations was suggested in 1929 (by Andronov). At a later date (1937) Krylov and Bogoliubov (K.B.) simplified somewhat the method of attack by a device resembling Lagrange s method of the variation of parameters, and in this form the method became useful for solving practical problems. Most of these early applications were to autonomous systems (mainly the self-excited oscillations), but later the method was extended to... 

In 1958 N. N. Bogoliubov and Y. A. Mitropolsky (B.M.) published a treatise entitled Asymptotic Methods in the Theory of Nonlinear Oscillations,18 which presents a considerable generalization of the early K.B. theory. Since a detailed account of this work is beyond the scope of this book, we give only a few of its salient points. 

Block relaxation, 61 Bogoliubov, N., 322,361 Boltzmann distribution, 471 Boltzmann equation Burnett method of solution, 25 Chapman-Enskog method of solution, 24... 

These bounds are the nonequilibrium equivalents of the Gibbs-Bogoliubov bounds discussed in Chap. 2. Having the free energy now bounded from above and below already demonstrates the power of using both forward and backward transformations. Moreover, as was shown by Crooks [18, 19], the distribution of work values from forward and backward paths satisfies a relation that is central to histogram methods in free energy calculations... 

It is important to stress that use of the generalised Bogoliubov transformatin provides an elegant physical interpretation of the Casimir effect as a consequence of the condensation in the vacuum of the fermion or the boson field. The method can be extended to other geometries such as spherical or cylindrical. 

We propose and demonstrate [Gershnabel 2003] an echo method of reducing the inhomogeneous broadening of Bogoliubov excitations in a harmonically-trapped BEC. Our proposal includes transfer of +q to —q momentum excitations, using a double two-photon Bragg process, in which a substantial... 

The high sensitivity of this method is related to the fact that we observe a matter-wave interference between the excitation and the condensate, i.e., a heterodyne measurement. Expansion in the inhomogeneous Bogoliubov projection basis confirm this picture [Tozzo 2004] We estimate that this improved sensitivity should give us access to the singly quantized excitation regime. 

N. N. Bogoliubov and Y. A. Mitropolsky, Asymptotic Methods in the Theory of Non-Linear Oscillations, Gordon Breach, New York, 1985. 

It should be noted that these equations are to be solved for each position of the centroid q. The frequency in Eq. (2.27) is the same as the effective frequency obtained for the optimized LHO reference system using the path-integral centroid density version of the Gibbs-Bogoliubov variational method [1, pp. 303-307 2, pp. 86-96], Correspondingly, Eqs. (2.27) and (2.28) are exactly the same as those in the quadratic effective potential theory [1,21-23], The derivation above does not make use of the variational principle but, instead, is the result of the vertex renormalization procedure. The diagrammatic analysis thus provides a method of systematic identification and evaluation of the corrections to the variational theory [3],... 

Bogoliubov, N.N. and Mitropolskii, Y.A., The Asymptotic Method in the Theory of Nonlinear Vibrations, Nauka, Moscow, 1974 (in Russian). 

Bogoliubov NN, MitropoTskiy YA (1974) Asymptotic methods in theory of non-linear oscillations. Nauka, Moscow (in Russian)... 

If there are many valence protons and neutrons present in the nucleus, traditional shell model calculations lead to insurmountable difficulties. Fortunately, the Bardeen-Cooper-Schrieffer (BCS) theory provides a good approximation method to the seniority-zero shell model, and allows to describe very complex nuclei, too. In the BCS quasiparticle calculations long chains of nuclei can be treated in a relatively simple way. The method was first applied in the theory of superconductivity by Bardeen et al. (1957), then used for nuclear physics by Bohr et al. (1958), Soloviev (1958), and Belyaev (1959). The quasiparticle concept was introduced into nuclear physics by Valatin (1958) and Bogoliubov (1958). The theory is explained in detail in several textbooks (Lawson 1980 Ring and Schuck 1980 Soloviev 1981 Heyde 1990 Nilsson and Ragnarsson 1995 Fenyes 2002). 

Auchmuty, J. F. G., Nicolis, G. (1975) Bifurcation analysis of nonlinear reaction-diffusion equations I. Evolution equations and the steady state solutions. Bull. Math. Biol. 37, 323 Auchmuty, J. F. G., Nicolis, G. (1976) Bifurcation analysis of nonlinear reaction-diffusion equations III. Chemical oscillations. Bull. Math. Biol. 38, 325 Bogoliubov, N. N., Mitropolskii, I. A. (1961) Asymptotic Methods in the Theory of Nonlinear Oscillations (Gordon and Breach, New York) [English transl.]... 

Bogoliubov NN, Mitropolsky YA (1961) Asymptotic methods in the theory of non-linear oscillations, 2nd edn. (translated from Russian). Hindustan, Delhi... 

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