Multi-Lagrangians for Integrable Systems

We propose a general scheme to construct multiple Lagrangians for completely integrable non-linear evolution equations that admit multi- Hamiltonian structure. The recursion operator plays a fundamental role in this construction. We use a conserved quantity higher/lower than the Hamiltonian in the potential part of the new Lagrangian and determine the corresponding kinetic terms by generating the appropriate momentum map. This leads to some remarkable new developments. We show that nonlinear evolutionary systems that admit $N$-fold first order local Hamiltonian structure can be cast into variational form with $2N-1$ Lagrangians which will be local functionals of Clebsch potentials. This number increases to $3N-2$ when the Miura transformation is invertible. Furthermore we construct a new Lagrangian for polytropic gas dynamics in $1+1$ dimensions which is a {\it local} functional of the physical field variables, namely density and velocity, thus dispensing with the necessity of introducing Clebsch potentials entirely. This is a consequence of bi-Hamiltonian structure with a compatible pair of first and third order Hamiltonian operators derived from Sheftel's recursion operator.Comment: typos corrected and a reference adde

A geometric study of the dispersionless Boussinesq type equation

We discuss the dispersionless Boussinesq type equation, which is equivalent to the Benney-Lax equation, being a system of equations of hydrodynamical type. This equation was discussed in . The results include: a description of local and nonlocal Hamiltonian and symplectic structures, hierarchies of symmetries, hierarchies of conservation laws, recursion operators for symmetries and generating functions of conservation laws (cosymmetries). Highly interesting are the appearances of operators that send conservation laws and symmetries to each other but are neither Hamiltonian, nor symplectic. These operators give rise to a noncommutative infinite-dimensional algebra of recursion operators

Poisson Structures for Aristotelian Model of Three Body Motion

We present explicitly Poisson structures, for both time-dependent and time-independent Hamiltonians, of a dynamical system with three degrees of freedom introduced and studied by Calogero et al [2005]. For the time-independent case, new constant of motion includes all parameters of the system. This extends the result of Calogero et al [2009] for semi-symmetrical motion. We also discuss the case of three bodies two of which are not interacting with each other but are coupled with the interaction of third one

Quantization with maximally degenerate Poisson brackets: The harmonic oscillator!

Nambu's construction of multi-linear brackets for super-integrable systems can be thought of as degenerate Poisson brackets with a maximal set of Casimirs in their kernel. By introducing privileged coordinates in phase space these degenerate Poisson brackets are brought to the form of Heisenberg's equations. We propose a definition for constructing quantum operators for classical functions which enables us to turn the maximally degenerate Poisson brackets into operators. They pose a set of eigenvalue problems for a new state vector. The requirement of the single valuedness of this eigenfunction leads to quantization. The example of the harmonic oscillator is used to illustrate this general procedure for quantizing a class of maximally super-integrable systems

Multi-Hamiltonian structure of equations of hydrodynamic type

We complete the discussion of the Hamiltonian structure of 2-component equations of hydrodynamic type by presenting the Hamiltonian operators for Euler's equation governing the motion of plane sound waves of nite amplitude and another quasi-linear second order wave equation. There exists a doubly in- nite family of conserved Hamiltonians for the equations of gas dynamics which degenerate into one, namely the Benney sequence, for shallow water waves. We present further in nite sequences of conserved quantities for these equations. In the case of multi-component equations of hydrodynamic type, we show that Kodama's generalization of the shallow water equations admits bi-Hamiltonian structure

Virulence markers of opportunistic black yeast in Exophiala

PubMedID: 26857806The black yeast genus Exophiala is known to cause a wide variety of diseases in severely ill individuals but can also affect immunocompetent individuals. Virulence markers and other physiological parameters were tested in eight clinical and 218 environmental strains, with a specific focus on human-dominated habitats for the latter. Urease and catalase were consistently present in all samples; four strains expressed proteinase and three strains expressed DNase, whereas none of the strains showed phospholipase, haemolysis, or co-haemolysis activities. Biofilm formation was identified in 30 (13.8%) of the environmental isolates, particularly in strains from dishwashers, and was noted in only two (25%) of the clinical strains. These results indicate that virulence factors are inconsistently present in the investigated Exophiala species, suggesting opportunism rather than pathogenicity. © 2016 Blackwell Verlag GmbH