PARABOLIC COMPACTIFICATION OF HOMOGENEOUS SPACES

In this article, we study compactifications of homogeneous spaces coming from equivariant, open embeddings into a generalized flag manifold $G/P$ . The key to this approach is that in each case $G/P$ is the homogeneous model for a parabolic geometry; the theory of such geometries provides a large supply of geometric tools and invariant differential operators that can be used for this study. A classical theorem of Wolf shows that any involutive automorphism of a semisimple Lie group $G$ with fixed point group $H$ gives rise to a large family of such compactifications of homogeneous spaces of $H$ . Most examples of (classical) Riemannian symmetric spaces as well as many non-symmetric examples arise in this way. A specific feature of the approach is that any compactification of that type comes with the notion of ‘curved analog’ to which the tools we develop also apply. The model example of this is a general Poincaré–Einstein manifold forming the curved analog of the conformal compactification of hyperbolic space. In the first part of the article, we derive general tools for the analysis of such compactifications. In the second part, we analyze two families of examples in detail, which in particular contain compactifications of the symmetric spaces $\mathit{SL}(n,\mathbb{R})/\mathit{SO}(p,n-p)$ and $\mathit{SO}(n,\mathbb{C})/\mathit{SO}(n)$ . We describe the decomposition of the compactification into orbits, show how orbit closures can be described as the zero sets of smooth solutions to certain invariant differential operators and prove a local slice theorem around each orbit in these examples.

Development of a new high-current triode extraction system for helicon ion source: design and simulation

Three triode extraction systems are simulated by IBSimu ion optical code for Amirkabir Helicon Ion Source (AHIS). The optimized pierce and suggested parabolic electrodes are introduced for the first time in this paper. The obtained N+ beam for parabolic geometry designed for ion implantation has 66 keV energy, and 10.4 mA current. Ion beam emittance and Twiss parameters of the emittance ellipse as the function of x term index are calculated for parabolic electrode equation. The simulated triode extraction systems have been evaluated by using of optimized parameters such as the extraction voltage, gap distance, plasma electrode (PE) aperture, and ion temperature. The extraction voltage, gap distance, PE aperture, and ion temperature have been changed in the range of 58–70 kV, 35–39 mm, 4–6 mm, and 0.5–4.4 eV in the simulations, respectively.

Indoor Light Performance of Coil Type Cylindrical Dye Sensitized Solar Cells

A very good performance under low/diffused light intensities is one of the application areas in which dye-sensitized solar cells (DSSCs) can be utilized effectively compared to their inorganic silicon solar cell counterparts. In this article, we have investigated the 1 SUN and low intensity fluorescent light performance of Titanium (Ti)-coil based cylindrical DSSC (C-DSSC) using ruthenium based N719 dye and organic dyes such as D205 and Y123. Electrochemical impedance spectroscopic results were analyzed for variable solar cell performances. Reflecting mirror with parabolic geometry as concentrator was also utilized to tap diffused light for indoor applications. Fluorescent light at relatively lower illumination intensities (0.2 mW/cm2 to 0.5 mW/cm2) were used for the investigation of TCO-less C-DSSC performance with and without reflector geometry. Furthermore, the DSSC performances were analyzed and compared with the commercially available amorphous silicon based solar cell for indoor applications.

Submaximally symmetric c-projective structures

[Formula: see text]-projective structures are analogues of projective structures in the almost complex setting. The maximal dimension of the Lie algebra of [Formula: see text]-projective symmetries of a complex connection on an almost complex manifold of [Formula: see text]-dimension [Formula: see text] is classically known to be [Formula: see text]. We prove that the submaximal dimension is equal to [Formula: see text]. If the complex connection is minimal (encoded as a normal parabolic geometry), the harmonic curvature of the [Formula: see text]-projective structure has three components and we specify the submaximal symmetry dimensions and the corresponding geometric models for each of these three pure curvature types. If the connection is non-minimal, we introduce a modified normalization condition on the parabolic geometry and use this to resolve the symmetry gap problem. We prove that the submaximal symmetry dimension in the class of Levi-Civita connections for pseudo-Kähler metrics is [Formula: see text], and specializing to the Kähler case, we obtain [Formula: see text]. This resolves the symmetry gap problem for metrizable [Formula: see text]-projective structures.

Graphical Proof of the Main Theorem of Non-Euclidean Geometry

The tragedy for the pioneers of non-Euclidean geometry (N. Lobachevsky and J. Boyai) was their quarrel with the scientific tradition. Figuratively speaking, in the judgment of the scientific world they could not provide proof of their views, and substantive law of science was not on their side despite the efforts of such an influential advocate as Karl Friedrich Gauss. They lost the civil process to the scientific layman, who sincerely believed that the earth is flat. Traditionally mathematical logic considers a new idea proven, if it is derived by inference from already proven ones, or recognized as obvious, or recognized without proof (postulates). Yet the founders of non-Euclidean geometry could not imagine such traditional evidence at all desire, because it had not yet been developed, and most importantly respective starting points (axioms, postulates, and theorems) had not been recognized by mathematicians. The paper outlines the original concept of non-Euclidean geometries. Hyperbolic geometry of Lobachevsky is considered based on viewing the sphere as a surface of zero curvature. In this case, the plane will have a real curvature properties of hyperboloid or a pseudosphere depending on the absolute and space anisotropy index, which replaces the concept of curvature of space; i.e. the notion of the curvature of the surface is converted to purely analytical attributes. Parabolic geometry of Euclid with degenerate absolute becomes a special case of geometries with non-degenerate absolute. The geometry of Riemann having the absolute of imaginary surface with negative Gaussian curvature at all points is declared not real but imaginary, since, according to the authors, it is impossible for plotting. References to textbooks of mechanics and mathematics departments of universities.

Asymptotic Formulas for the Reflection/Transmission of Long Water Waves Propagating in a Tapered and Slender Harbor

We obtain asymptotic formulas for the reflection/transmission coefficients of linear long water waves, propagating in a harbor composed of a tapered and slender region connected to uniform inlet and outlet regions. The region with variable character obeys a power-law. The governing equations are presented in dimensionless form. The reflection/transmission coefficients are obtained for the limit of the parameterκ2≪1, which corresponds to a wavelength shorter than the characteristic horizontal length of the harbor. The asymptotic formulas consider those cases when the geometry of the harbor can be variable in width and depth: linear or parabolic among other transitions or a combination of these geometries. For harbors with nonlinear transitions, the parabolic geometry is less reflective than the other cases. The reflection coefficient for linear transitions just presents an oscillatory behavior. We can infer that the deducted formulas provide as first approximation a practical reference to the analysis of wave reflection/transmission in harbors.

NORMAL BGG SOLUTIONS AND POLYNOMIALS

First BGG operators are a large class of overdetermined linear differential operators intrinsically associated to a parabolic geometry on a manifold. The corresponding equations include those controlling infinitesimal automorphisms, higher symmetries and many other widely studied PDE of geometric origin. The machinery of BGG sequences also singles out a subclass of solutions called normal solutions. These correspond to parallel tractor fields and hence to (certain) holonomy reductions of the canonical normal Cartan connection. Using the normal Cartan connection, we define a special class of local frames for any natural vector bundle associated to a parabolic geometry. We then prove that the coefficient functions of any normal solution of a first BGG operator with respect to such a frame are polynomials in the normal coordinates of the parabolic geometry. A bound on the degree of these polynomials in terms of representation theory data is derived. For geometries locally isomorphic to the homogeneous model of the geometry we explicitly compute the local frames mentioned above. Together with the fact that on such structures all solutions are normal, we obtain a complete description of all first BGG solutions in this case. Finally, we prove that in the general case the polynomial system coming from a normal solution is the pull-back of a polynomial system that solves the corresponding problem on the homogeneous model. Thus we can derive a complete list of potential normal solutions on curved geometries. Moreover, questions concerning the zero locus of solutions, as well as related finer geometric and smooth data, are reduced to a study of polynomial systems and real algebraic sets.