Iterated Differential Forms IV: C-Spectral Sequence

For the multiple differential algebra of iterated differential forms (see math.DG/0605113 and math.DG/0609287) on a diffiety (O,C) an analogue of C-spectral sequence is constructed. The first term of it is naturally interpreted as the algebra of secondary iterated differential forms on (O,C). This allows to develop secondary tensor analysis on generic diffieties, some simplest elements of which are sketched here. The presented here general theory will be specified to infinite jet spaces and infinitely prolonged PDEs in subsequent notes.Comment: 8 pages, submitted to Math. Dok

Domains in Infinite Jets: C-Spectral Sequence

Domains in infinite jets present the simplest class of diffieties with boundary. In this note some basic elements of geometry of these domains are introduced and an analogue of the C-spectral sequence in this context is studied. This, in particular, allows cohomological interpretation and analysis of initial data, boundary conditions, etc, for general partial differential equations and of transversality conditions in calculus of variations. This kind applications and extensions to arbitrary diffieties will be considered in subsequent publications.Comment: 7 pages; no proofs give

Iterated Differential Forms I: Tensors

We interpret tensors on a smooth manifold M as differential forms over a graded commutative algebra called the algebra of iterated differential forms over M. This allows us to put standard tensor calculus in a new differentially closed context and, in particular, enriches it with new natural operations. Applications will be considered in subsequent notes.Comment: 9 pages, extended version of the published note Dokl. Math. 73, n. 2 (2006) 16

Iterated Differential Forms II: Riemannian Geometry Revisited

A natural extension of Riemannian geometry to a much wider context is presented on the basis of the iterated differential form formalism developed in math.DG/0605113 and an application to general relativity is given.Comment: 12 pages, extended version of the published note Dokl. Math. 73, n. 2 (2006) 18

Differentiation of the matter of the moon

The following facts were uncovered in comparing the basaltic surface rocks of the moon with terrestrial tholeiitic basalts and ordinary chondrites: (1) there is an excess of the so-called refractory chemical elements, including the group of truly refractory elements, the rare earths, U, and Th, in comparison with their content in primitive terrestrial basalts and chondrites; (2) the so-called siderophilic elements have lower contents in the lunar surface rocks than in terrestrial rocks; (3) the low alkali content (Na, K, Rb) in lunar rocks is established; (4) there is a low content of H2O and the ordinary gases CO2, halides, etc.; (5) the low content of metals with high vapor pressure, (In, Tl, etc.) has been established. It is proposed that U and Th were carried from the internal areas to the peripheral rocks of the moon during magmatic activity, i.e., up to 3 billion years ago. This redistribution of U and Th lead to their concentration in surface layers of the moon, and the heat which they generated was lost into surrounding space. The conclusion is then reached that in order to understand processes on the moon, the chondritic model cannot be used

Iterated Differential Forms III: Integral Calculus

Basic elements of integral calculus over algebras of iterated differential forms, are presented. In particular, defining complexes for modules of integral forms are described and the corresponding berezinians and complexes of integral forms are computed. Various applications and the integral calculus over the algebra $\Lambda_{\infty}$ will be discussed in subsequent notes.Comment: 7 pages, submitted to Math. Dok

Iterated Differential Forms VI: Differential Equations

We describe the first term of the $\Lambda_{k-1}\mathcal{C}$--spectral sequence (see math.DG/0610917) of the diffiety (E,C), E being the infinite prolongation of an l-normal system of partial differential equations, and C the Cartan distribution on it.Comment: 8 pages, to appear in Dokl. Mat

Special functions from quantum canonical transformations

Quantum canonical transformations are used to derive the integral representations and Kummer solutions of the confluent hypergeometric and hypergeometric equations. Integral representations of the solutions of the non-periodic three body Toda equation are also found. The derivation of these representations motivate the form of a two-dimensional generalized hypergeometric equation which contains the non-periodic Toda equation as a special case and whose solutions may be obtained by quantum canonical transformation.Comment: LaTeX, 24 pp., Imperial-TP-93-94-5 (revision: two sections added on the three-body Toda problem and a two-dimensional generalization of the hypergeometric equation

Iterated Differential Forms V: C-Spectral Sequence on Infinite Jet Spaces

In the preceding note math.DG/0610917 the $\Lambda_{k-1}\mathcal{C}$--spectral sequence, whose first term is composed of \emph{secondary iterated differential forms}, was constructed for a generic diffiety. In this note the zero and first terms of this spectral sequence are explicitly computed for infinite jet spaces. In particular, this gives an explicit description of secondary covariant tensors on these spaces and some basic operations with them. On the basis of these results a description of the $\Lambda_{k-1}\mathcal{C}$--spectral sequence for infinitely prolonged PDE's will be given in the subsequent note.Comment: 9 pages, to appear in Math. Dok

Analytical Representation of the Longitudinal Hadronic Shower Development

The analytical representation of the longitudinal hadronic shower development from the face of a calorimeter is presented and compared with experimental data. The suggested formula is particularly useful at designing, testing and calibration of huge calorimeter complex like in ATLAS at LHC.Comment: 5 pages, 1 figur