10,694 research outputs found
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
Algebraic theories of brackets and related (co)homologies
A general theory of the Frolicher-Nijenhuis and Schouten-Nijenhuis brackets
in the category of modules over a commutative algebra is described. Some
related structures and (co)homology invariants are discussed, as well as
applications to geometry.Comment: 14 pages; v2: minor correction
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 will be discussed in subsequent notes.Comment: 7 pages, submitted to Math. Dok
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 VI: Differential Equations
We describe the first term of the --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
Iterated Differential Forms V: C-Spectral Sequence on Infinite Jet Spaces
In the preceding note math.DG/0610917 the
--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
--spectral sequence for infinitely prolonged PDE's
will be given in the subsequent note.Comment: 9 pages, to appear in Math. Dok
- …