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

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

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

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

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

History of Karras Colony and Nearby Settlements of Europeans in 19th — First Half of 20th Centuries at Present Stage of Study

The issues of the current stage of studying the history of the Karras colony and nearby European settlements in the 19th and first half of the 20th centuries are considered. A review and analysis of new sources and historiography from 2000 to 2020 has been carried out. The relevance of the study is due to the poorly studied and fragmentary coverage of the history of European settlements in the central part of the North Caucasus in the 19th — first half of the 20th centuries in Russian historiography. The authors dwell on terminology issues. It is emphasized that the terms-cliches ‘mountaineers’ and ‘Tatars’ are characteristic of the historical literature of the 19th century and are inaccurately used by some authors today. The novelty of the research is seen in the fact that in this work the history of the Karras colony and neighboring settlements of Europeans in the 19th — first half of the 20th centuries is considered based on publications of 2000—2020. It is concluded that there is a possibility and a need for an independent review of the history of the Scottish mission, the center of which was originally located in Karras. The authors proceed from the fact that the history of the settlements of the colonists has a broader chronological framework and the main task of the colonists was not always missionary activity

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

Three natural mechanical systems on Stiefel varieties

We consider integrable generalizations of the spherical pendulum system to the Stiefel variety $V(n,r)=SO(n)/SO(n-r)$ for a certain metric. For the case of V(n,2) an alternative integrable model of the pendulum is presented. We also describe a system on the Stiefel variety with a four-degree potential. The latter has invariant relations on $T^*V(n,r)$ which provide the complete integrability of the flow reduced on the oriented Grassmannian variety $G^+(n,r)=SO(n)/SO(r)\times SO(n-r)$.Comment: 14 page

A New High Energy Photon Tagger for the H1 - Detector at HERA

The H1 detector at HERA has been upgraded by the addition of a new electromagnetic calorimeter. This is installed in the HERA tunnel close to the electron beam line at a position 8m from the interaction point in the electron beam direction. The new calorimeter extends the acceptance for tagged photoproduction events to the high y range, 0.85 < y < 0.95, and thus significantly improves the capability of H1 to study high energy gamma-p processes. The calorimeter design, performance and first results obtained during the 1996-1999 HERA running are described.Comment: 17 pages, 16 figure

The Maslov index and nondegenerate singularities of integrable systems

We consider integrable Hamiltonian systems in R^{2n} with integrals of motion F = (F_1,...,F_n) in involution. Nondegenerate singularities are critical points of F where rank dF = n-1 and which have definite linear stability. The set of nondegenerate singularities is a codimension-two symplectic submanifold invariant under the flow. We show that the Maslov index of a closed curve is a sum of contributions +/- 2 from the nondegenerate singularities it is encloses, the sign depending on the local orientation and stability at the singularities. For one-freedom systems this corresponds to the well-known formula for the Poincar\'e index of a closed curve as the oriented difference between the number of elliptic and hyperbolic fixed points enclosed. We also obtain a formula for the Liapunov exponent of invariant (n-1)-dimensional tori in the nondegenerate singular set. Examples include rotationally symmetric n-freedom Hamiltonians, while an application to the periodic Toda chain is described in a companion paper.Comment: 27 pages, 1 figure; published versio