Stability analysis of some integrable Euler equations for SO(n)

A family of special cases of the integrable Euler equations on $so(n)$ introduced by Manakov in 1976 is considered. The equilibrium points are found and their stability is studied. Heteroclinic orbits are constructed that connect unstable equilibria and are given by the orbits of certain 1-parameter subgroups of SO(n). The results are complete in the case $n=4$ and incomplete for $n>4$.Comment: 15 pages, LaTeX, minor stylistic changes in v

Force dipoles and stable local defects on fluid vesicles

An exact description is provided of an almost spherical fluid vesicle with a fixed area and a fixed enclosed volume locally deformed by external normal forces bringing two nearby points on the surface together symmetrically. The conformal invariance of the two-dimensional bending energy is used to identify the distribution of energy as well as the stress established in the vesicle. While these states are local minima of the energy, this energy is degenerate; there is a zero mode in the energy fluctuation spectrum, associated with area and volume preserving conformal transformations, which breaks the symmetry between the two points. The volume constraint fixes the distance $S$, measured along the surface, between the two points; if it is relaxed, a second zero mode appears, reflecting the independence of the energy on $S$; in the absence of this constraint a pathway opens for the membrane to slip out of the defect. Logarithmic curvature singularities in the surface geometry at the points of contact signal the presence of external forces. The magnitude of these forces varies inversely with $S$ and so diverges as the points merge; the corresponding torques vanish in these defects. The geometry behaves near each of the singularities as a biharmonic monopole, in the region between them as a surface of constant mean curvature, and in distant regions as a biharmonic quadrupole. Comparison of the distribution of stress with the quadratic approximation in the height functions points to shortcomings of the latter representation. Radial tension is accompanied by lateral compression, both near the singularities and far away, with a crossover from tension to compression occurring in the region between them.Comment: 26 pages, 10 figure

Invariant metrics and Hamiltonian Systems

Via a non degenerate symmetric bilinear form we identify the coadjoint representation with a new representation and so we induce on the orbits a simplectic form. By considering Hamiltonian systems on the orbits we study some features of them and finally find commuting functions under the corresponding Lie-Poisson bracketComment: 16 pages corrected typos, changed contents (Prop. 3.4 and Theorem in Section 3

An electrooptical muscle contraction sensor

An electrooptical sensor for the detection of muscle contraction is described. Infrared light is injected into the muscle, the backscattering is observed, and the contraction is detected by measuring the change, that occurs during muscle contraction, between the light scattered in the direction parallel and perpendicular to the muscle cells. With respect to electromyography and to optical absorption-based sensors, our device has the advantage of lower invasiveness, of lower sensitivity to electromagnetic noise and to movement artifacts, and of being able to distinguish between isometric and isotonic contractions

Giant magnetoresistance in semiconductor / granular film heterostructures with cobalt nanoparticles

We have studied the electron transport in SiO${}_2$(Co)/GaAs and SiO${}_2$(Co)/Si heterostructures, where the SiO${}_2$(Co) structure is the granular SiO${}_2$ film with Co nanoparticles. In SiO${}_2$(Co)/GaAs heterostructures giant magnetoresistance effect is observed. The effect has positive values, is expressed, when electrons are injected from the granular film into the GaAs semiconductor, and has the temperature-peak type character. The temperature location of the effect depends on the Co concentration and can be shifted by the applied electrical field. For the SiO${}_2$(Co)/GaAs heterostructure with 71 at.% Co the magnetoresistance reaches 1000 ($10^5$ %) at room temperature. On the contrary, for SiO${}_2$(Co)/Si heterostructures magnetoresistance values are very small (4%) and for SiO${}_2$(Co) films the magnetoresistance has an opposite value. High values of the magnetoresistance effect in SiO${}_2$(Co)/GaAs heterostructures have been explained by magnetic-field-controlled process of impact ionization in the vicinity of the spin-dependent potential barrier formed in the semiconductor near the interface. Kinetic energy of electrons, which pass through the barrier and trigger the avalanche process, is reduced by the applied magnetic field. This electron energy suppression postpones the onset of the impact ionization to higher electric fields and results in the giant magnetoresistance. The spin-dependent potential barrier is due to the exchange interaction between electrons in the accumulation electron layer in the semiconductor and $d$-electrons of Co.Comment: 25 pages, 16 figure

Fomenko-Mischenko Theory, Hessenberg Varieties, and Polarizations

The symmetric algebra g (denoted S(\g)) over a Lie algebra \g (frak g) has the structure of a Poisson algebra. Assume \g is complex semi-simple. Then results of Fomenko- Mischenko (translation of invariants) and A.Tarasev construct a polynomial subalgebra \cal H = \bf C[q_1,...,q_b] of S(\g) which is maximally Poisson commutative. Here b is the dimension of a Borel subalgebra of \g. Let G be the adjoint group of \g and let \ell = rank \g. Identify \g with its dual so that any G-orbit O in \g has the structure (KKS) of a symplectic manifold and S(\g) can be identified with the affine algebra of \g. An element x \in \g is strongly regular if \{(dq_i)_x\}, i=1,...,b, are linearly independent. Then the set \g^{sreg} of all strongly regular elements is Zariski open and dense in \g, and also \g^{sreg \subset \g^{reg} where \g^{reg} is the set of all regular elements in \g. A Hessenberg variety is the b-dimensional affine plane in \g, obtained by translating a Borel subalgebra by a suitable principal nilpotent element. This variety was introduced in [K2]. Defining Hess to be a particular Hessenberg variety, Tarasev has shown that Hess \subset \g^sreg. Let R be the set of all regular G-orbits in \g. Thus if O \in R, then O is a symplectic manifold of dim 2n where n= b-\ell. For any O\in R let O^{sreg} = \g^{sreg}\cap O. We show that O^{sreg} is Zariski open and dense in O so that O^{sreg} is again a symplectic manifold of dim 2n. For any O \in R let Hess (O) = Hess \cap O. We prove that Hess(O) is a Lagrangian submanifold of O^{sreg} and Hess =\sqcup_{O \in R} Hess(O). The main result here shows that there exists, simultaneously over all O \in R, an explicit polarization (i.e., a "fibration" by Lagrangian submanifolds) of O^{sreg} which makes O^{sreg} simulate, in some sense, the cotangent bundle of Hess(O).Comment: 36 pages, plain te

Spinor representation of surfaces and complex stresses on membranes and interfaces

Variational principles are developed within the framework of a spinor representation of the surface geometry to examine the equilibrium properties of a membrane or interface. This is a far-reaching generalization of the Weierstrass-Enneper representation for minimal surfaces, introduced by mathematicians in the nineties, permitting the relaxation of the vanishing mean curvature constraint. In this representation the surface geometry is described by a spinor field, satisfying a two-dimensional Dirac equation, coupled through a potential associated with the mean curvature. As an application, the mesoscopic model for a fluid membrane as a surface described by the Canham-Helfrich energy quadratic in the mean curvature is examined. An explicit construction is provided of the conserved complex-valued stress tensor characterizing this surface.Comment: 17 page

Tagging High Energy Photons in the H1 Detector at HERA

Measures taken to extend the acceptance of the H1 detector at HERA for photoproduction events are described. These will enable the measurement of electrons scattered in events in the high y range 0.85 < y < 0.95 in the 1998 and 1999 HERA run period. The improvement is achieved by the installation of an electromagnetic calorimeter, the ET8, in the HERA tunnel close to the electron beam line 8 m downstream of the H1 interaction point in the electron direction. The ET8 will allow the study of tagged gamma p interactions at centre-of-mass energies significantly higher than those previously attainable. The calorimeter design and expected performance are discussed, as are results obtained using a prototype placed as close as possible to the position of the ET8 during the 1996 and 1997 HERA running.Comment: 13 pages, 13 figure

On the Cartan Model of the Canonical Vector Bundles over Grassmannians

We give a representation of canonical vector bundles over Grassmannian manifolds as non-compact affine symmetric spaces as well as their Cartan model in the group of the Euclidean motions.Comment: 6 page