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

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

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

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

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

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

Topology of the spaces of Morse functions on surfaces

Let $M$ be a smooth closed orientable surface, and let $F$ be the space of Morse functions on $M$ such that at least $\chi(M)+1$ critical points of each function of $F$ are labeled by different labels (enumerated). Endow the space $F$ with $C^\infty$-topology. We prove the homotopy equivalence $F\sim R\times{\widetilde{\cal M}}$ where $R$ is one of the manifolds ${\mathbb R}P^3$, $S^1\times S^1$ and the point in dependence on the sign of $\chi(M)$, and ${\widetilde{\cal M}}$ is the universal moduli space of framed Morse functions, which is a smooth stratified manifold. Morse inequalities for the Betti numbers of the space $F$ are obtained.Comment: 15 pages, in Russia

Topology of energy surfaces and existence of transversal Poincar\'e sections

Two questions on the topology of compact energy surfaces of natural two degrees of freedom Hamiltonian systems in a magnetic field are discussed. We show that the topology of this 3-manifold (if it is not a unit tangent bundle) is uniquely determined by the Euler characteristic of the accessible region in configuration space. In this class of 3-manifolds for most cases there does not exist a transverse and complete Poincar\'e section. We show that there are topological obstacles for its existence such that only in the cases of $S^1\times S^2$ and $T^3$ such a Poincar\'e section can exist.Comment: 10 pages, LaTe

Homology and K--Theory Methods for Classes of Branes Wrapping Nontrivial Cycles

We apply some methods of homology and K-theory to special classes of branes wrapping homologically nontrivial cycles. We treat the classification of four-geometries in terms of compact stabilizers (by analogy with Thurston's classification of three-geometries) and derive the K-amenability of Lie groups associated with locally symmetric spaces listed in this case. More complicated examples of T-duality and topology change from fluxes are also considered. We analyse D-branes and fluxes in type II string theory on ${\mathbb C}P^3\times \Sigma_g \times {\mathbb T}^2$ with torsion $H-$flux and demonstrate in details the conjectured T-duality to ${\mathbb R}P^7\times X^3$ with no flux. In the simple case of $X^3 = {\mathbb T}^3$, T-dualizing the circles reduces to duality between ${\mathbb C}P^3\times {\mathbb T}^2 \times {\mathbb T}^2$ with $H-$flux and ${\mathbb R}P^7\times {\mathbb T}^3$ with no flux.Comment: 27 pages, tex file, no figure