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VI - Symmetric Hamiltonian Bifurcations (J.A. Montaldi)
- Edited by James Montaldi, University of Manchester Institute of Science and Technology, Tudor Ratiu, École Polytechnique Fédérale de Lausanne
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- Book:
- Geometric Mechanics and Symmetry
- Published online:
- 19 October 2009
- Print publication:
- 05 May 2005, pp 357-402
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Summary
Introduction
The purpose of these notes is to give a brief survey of bifurcation theory of Hamiltonian systems with symmetry; they are a slightly extended version of the five lectures given by JM on Hamiltonian Bifurcations with Symmetry. We focus our attention on bifurcation theory near equilibrium solutions and relative equilibria. The notes are composed of two parts. In the first, we review results on nonlinear normal modes in equivariant Hamiltonian systems, generic movement of eigenvalues in equivariant Hamiltonian matrices, one and two parameter bifurcation of equilibria and the Hamiltonian-Hopf Theorems with symmetry. The second part is about local dynamics near relative equilibria. Particular topics discussed are the existence, stability and persistence of relative equilibria, bifurcations from zero momentum relative equilibria and examples.
We begin with some basic facts on Lie group actions on symplectic manifolds and Hamiltonian systems with symmetry. The reader should refer to Ratiu's lectures for more details and examples.
Semisymplectic actions A Lie group G acts semisymplectically on a symplectic manifold (P, ω) if g*ω = ±ω. In this case the choice of sign determines a homomorphism X : G → Z2 called the temporal character, such that g*ω = X(g)ω. We denote the kernel of X by G+; it consists of those elements acting symplectically, and if G does contain antisymplectic elements then G+ is a subgroup of G of index 2. Some details on semisymplectic actions can be found in [MR00].
Controlled production of Camembert-type cheeses. Part I: Microbiological and physicochemical evolutions
- Marie-Noëlle Leclercq-Perlat, Frédéric Buono, Denis Lambert, Eric Latrille, Henry-Eric Spinnler, Georges Corrieu
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- Journal:
- Journal of Dairy Research / Volume 71 / Issue 3 / August 2004
- Published online by Cambridge University Press:
- 23 July 2004, pp. 346-354
- Print publication:
- August 2004
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A holistic approach of a mould cheese ripening is presented. The objective was to establish relationships between the different microbiological and biochemical changes during cheese ripening. Model cheeses were prepared from pasteurized milk inoculated with Kluyveromyces lactis, Geotrichum candidum, Penicillium camemberti and Brevibacterium linens under aseptic conditions. Two cheese-making trials with efficient control of environmental parameters were carried out and showed similar ripening characteristics. K. lactis grew rapidly between days 1 and 6 (generation time around 48 h). G. candidum grew exponentially between days 4 and 10 (generation time around 4·6 d). Brevi. linens also grew exponentially but after day 6 when Pen. camemberti mycelium began developing and the pH of the rind was close to 7. Its exponential growth presented 3 phases in relation to carbon and nitrogen substrate availability. Concentrations of Pen. camemberti mycelium were not followed by viable cell count but they were evaluated visually. The viable microorganism concentrations were well correlated with the carbon substrate concentrations in the core and in the rind. The lactose concentrations were negligible after 10 d ripening, and changes in lactate quantities were correlated with fungi flora. The pH of the inner part depended on NH3. Surface pH was significantly related to NH3 concentration and to fungi growth. The acid-soluble nitrogen (ASN) and non-protein nitrogen (NPN) indexes and NH3 concentrations of the rind were low until day 6, and then increased rapidly to follow the fungi concentrations until day 45. The ASN and NPN indexes and NH3 concentrations in the core were lower than in the rind and they showed the same evolution. G. candidum and Pen. camemberti populations have a major effect on proteolysis; nevertheless, K. lactis and Brevi. linens cell lysis also had an impact on proteolysis. Viable cell counts of K. lactis, G. candidum, Pen. camemberti and Brevi. linens were correlated with the environmental conditions, with proteolytic products and with carbon substrate assimilation. NH3 diffusion from surface to the cheese core during ripening was highly suspected. Interaction phenomena between microorganisms are discussed.