On meteor stream spatial structure theory

The classical spatial representation of meteor streams is an elliptical torus with variable cross section. The position of this torus in space is determined by the mean orbit elements that may be obtained directly from observations of individual meteor stream particles when crossed by the Earth. Since the orbits of individual particles of a stream differ from each other, the distance between them on a plane normal to the mean orbit of elliptical torus forms some area, i.e., a cross section. The size and form of these cross sections change with the change of the direction among the mean orbit and are completely defined by the dispersion values of the orbit elements in a stream. An attempt was made to create an analytical method that would permit description of the spatial and time parameters of meteor streams, i.e., the form and size of their cross section, density of incident flux and their variations along the mean orbit and in time. In this case, the stream is considered as a continuous flux rather than a set of individual particles

Higher dimensional uniformisation and W-geometry

We formulate the uniformisation problem underlying the geometry of W_n-gravity using the differential equation approach to W-algebras. We construct W_n-space (analogous to superspace in supersymmetry) as an (n-1) dimensional complex manifold using isomonodromic deformations of linear differential equations. The W_n-manifold is obtained by the quotient of a Fuchsian subgroup of PSL(n,R) which acts properly discontinuously on a simply connected domain in CP^{n-1}. The requirement that a deformation be isomonodromic furnishes relations which enable one to convert non-linear W-diffeomorphisms to (linear) diffeomorphisms on the W_n-manifold. We discuss how the Teichmuller spaces introduced by Hitchin can then be interpreted as the space of complex structures or the space of projective structures with real holonomy on the W_n-manifold. The projective structures are characterised by Halphen invariants which are appropriate generalisations of the Schwarzian. This construction will work for all ``generic'' W-algebras.Comment: LaTeX file; 25/13 pages in b/l mode ; version to appear in Nuc. Phys.

The Chazy XII Equation and Schwarz Triangle Functions

Dubrovin [Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 120-348] showed that the Chazy XII equation $y'''- 2yy''+3y'^2 = K(6y'-y^2)^2$, $K \in \mathbb{C}$, is equivalent to a projective-invariant equation for an affine connection on a one-dimensional complex manifold with projective structure. By exploiting this geometric connection it is shown that the Chazy XII solution, for certain values of $K$, can be expressed as $y=a_1w_1+a_2w_2+a_3w_3$ where $w_i$ solve the generalized Darboux-Halphen system. This relationship holds only for certain values of the coefficients $(a_1,a_2,a_3)$ and the Darboux-Halphen parameters $(\alpha, \beta, \gamma)$, which are enumerated in Table 2. Consequently, the Chazy XII solution $y(z)$ is parametrized by a particular class of Schwarz triangle functions $S(\alpha, \beta, \gamma; z)$ which are used to represent the solutions $w_i$ of the Darboux-Halphen system. The paper only considers the case where $\alpha+\beta+\gamma<1$. The associated triangle functions are related among themselves via rational maps that are derived from the classical algebraic transformations of hypergeometric functions. The Chazy XII equation is also shown to be equivalent to a Ramanujan-type differential system for a triple $(\hat{P}, \hat{Q},\hat{R})$

Self-dual SU(2) invariant Einstein metrics and modular dependence of theta-functions

We simplify Hitchin's description of SU(2)-invariant self-dual Einstein metrics, making use of the tau-function of related four-pole Schlesinger system.Comment: A wrong sign in the formula for W_1 is corrected; we thank Owen Dearricott who pointed out this mistake in the original version of the pape

A Proof of the Puiseux - Halphen Inequality in the Theory of Spherical Pendulum Based on an Interesting Identity of S. Ramanujan

The classical problem of finding bounds for the apsidal angle in the case of the spherical pendulum has been considered by Halphen in his classical treatise on 'Fonctions Elliptiques'. His proof is based on certain inequalities among the Elliptic-function constant. We prove these inequalities of Halphen and Puiseux, in a simple way, using an interesting identity of S. Ramanujan; besides, we obtain positive-term series for the quantities involved which may enable one to improve such bounds

On the genus of projective curves not contained in hypersurfaces of given degree

Fix integers r &gt;= 4 and i &gt;= 2 (for r = 4 assume i &gt;= 3). Assume that the rational number s defined by the equation ((i + 1)(2))s + (i + 1) = ((r + i) )(i)is( ) an integer. Fix an integer d &gt;= s. Divide d - 1 = ms + epsilon, 0 &lt;= epsilon &lt;= s - 1, and set G(r;d, i) := ((m)(2))s + m epsilon. As a number, 2 G(r; d, i) is nothing but the Castelnuovo's bound G(s + 1;d) for a curve of degree d in Ps+1. In the present paper we prove that G(r; d, i) is also an upper bound for the genus of a reduced and irreducible complex projective curve in P-r, of degree d &gt;&gt; max{ r,i}, not contained in hypersurfaces of degree &lt;= i. We prove that the bound G(r; d, i) is sharp if and only if there exists an integral surface S subset of P-r of degree s, not contained in hypersurfaces of degree &lt;= i. Such a surface, if existing, is necessarily the isomorphic projection of a rational normal scroll surface of degree s in Ps+1 The existence of such a surface S is known for r &gt;= 5, and 2 &lt;= i &lt;= 3. It follows that, when r &gt;= 5, and i = 2 or i = 3, the bound G(r; d, i) is sharp, and the extremal curves are isomorphic projection in P-r of Castelnuovo's curves of degree d in Ps+1. We do not know whether the bound G(r; d, i) is sharp for i &gt; 3