Graphene as a quantum surface with curvature-strain preserving dynamics

We discuss how the curvature and the strain density of the atomic lattice generate the quantization of graphene sheets as well as the dynamics of geometric quasiparticles propagating along the constant curvature/strain levels. The internal kinetic momentum of Riemannian oriented surface (a vector field preserving the Gaussian curvature and the area) is determined.Comment: 13p, minor correction

New Bryokhutuliinia species (bryophyta) with sporophytes from the upper jurassic of Transbaikalia

A new species of the moss genus Bryokhutuliinia, B. crassimarginata is described from the Upper Jurassic deposits from the Olov, Transbaikal Area of South Siberia. Its excellent preservation demonstra- tes that the leaves were not only complanate, but truly distichous. In addition to anatomically pre- served gametophytes, sporophytes on short lateral branches were found, although carbonized and not exhibiting structural details. Possible relationships with pleurocarpous mosses and with Fissidentaceae are discussedyesBelgorod State National Research Universit

Cotangent bundle quantization: Entangling of metric and magnetic field

For manifolds $\cal M$ of noncompact type endowed with an affine connection (for example, the Levi-Civita connection) and a closed 2-form (magnetic field) we define a Hilbert algebra structure in the space $L^2(T^*\cal M)$ and construct an irreducible representation of this algebra in $L^2(\cal M)$. This algebra is automatically extended to polynomial in momenta functions and distributions. Under some natural conditions this algebra is unique. The non-commutative product over $T^*\cal M$ is given by an explicit integral formula. This product is exact (not formal) and is expressed in invariant geometrical terms. Our analysis reveals this product has a front, which is described in terms of geodesic triangles in $\cal M$. The quantization of $\delta$-functions induces a family of symplectic reflections in $T^*\cal M$ and generates a magneto-geodesic connection $\Gamma$ on $T^*\cal M$. This symplectic connection entangles, on the phase space level, the original affine structure on $\cal M$ and the magnetic field. In the classical approximation, the $\hbar^2$-part of the quantum product contains the Ricci curvature of $\Gamma$ and a magneto-geodesic coupling tensor.Comment: Latex, 38 pages, 5 figures, minor correction

Analogues of the central point theorem for families with dd-intersection property in Rd\mathbb R^d

In this paper we consider families of compact convex sets in $\mathbb R^d$ such that any subfamily of size at most $d$ has a nonempty intersection. We prove some analogues of the central point theorem and Tverberg's theorem for such families

Global action-angle coordinates for completely integrable systems with noncompact invariant submanifolds

The obstruction to the existence of global action-angle coordinates of Abelian and noncommutative (non-Abelian) completely integrable systems with compact invariant submanifolds has been studied. We extend this analysis to the case of noncompact invariant submanifolds.Comment: 13 pages, to be published in J. Math. Phys. (2007

Cohomologies of the Poisson superalgebra

Cohomology spaces of the Poisson superalgebra realized on smooth Grassmann-valued functions with compact support on $R^{2n}$ ($C^{2n}) are investigated under suitable continuity restrictions on cochains. The first and second cohomology spaces in the trivial representation and the zeroth and first cohomology spaces in the adjoint representation of the Poisson superalgebra are found for the case of a constant nondegenerate Poisson superbracket for arbitrary n>0. The third cohomology space in the trivial representation and the second cohomology space in the adjoint representation of this superalgebra are found for arbitrary n>1.Comment: Comments: 40 pages, the text to appear in Theor. Math. Phys. supplemented by computation of the 3-rd trivial cohomolog

Fractional Derivative as Fractional Power of Derivative

Definitions of fractional derivatives as fractional powers of derivative operators are suggested. The Taylor series and Fourier series are used to define fractional power of self-adjoint derivative operator. The Fourier integrals and Weyl quantization procedure are applied to derive the definition of fractional derivative operator. Fractional generalization of concept of stability is considered.Comment: 20 pages, LaTe

Heisenberg Evolution WKB and Symplectic Area Phases

The Schrodinger and Heisenberg evolution operators are represented in quantum phase space by their Weyl symbols. Their semiclassical approximations are constructed in the short and long time regimes. For both evolution problems, the WKB representation is purely geometrical: the amplitudes are functions of a Poisson bracket and the phase is the symplectic area of a region in phase space bounded by trajectories and chords. A unified approach to the Schrodinger and Heisenberg semiclassical evolutions is developed by introducing an extended phase space. In this setting Maslov's pseudodifferential operator version of WKB analysis applies and represents these two problems via a common higher dimensional Schrodinger evolution, but with different extended Hamiltonians. The evolution of a Lagrangian manifold in the extended phase space, defined by initial data, controls the phase, amplitude and caustic behavior. The symplectic area phases arise as a solution of a boundary condition problem. Various applications and examples are considered.Comment: 32 pages, 7 figure