Two significant directions in the development of jet calculus are showed. First, jets are generalized to so-called quasijets. Second, jets of foliated and multifoliated manifold morphisms are presented. Although the paper has mainly a survey character, it also includes new results: jets modulo multifoliations are introduced and their relation to (R,S,Q)-jets is demonstrated.
It is proved that generalized polynomials with rational exponents over a commutative field form an elementary divisor ring; an algorithm for computing the Smith normal form is derived and implemented.
Let be a Weil algebra. The bijection between all natural operators lifting vector fields from -manifolds to the bundle functor of Weil contact elements and the subalgebra of fixed elements of the Weil algebra is determined and the bijection between all natural affinors on and is deduced. Furthermore, the rigidity of the functor is proved. Requisite results about the structure of are obtained by a purely algebraic approach, namely the existence of nontrivial is discussed.
Quaternion algebras are investigated and isomorphisms between them are described. Furthermore, the orders of these algebras are presented and the uniqueness of the discrete norm for such orders is proved.
The author considers the problem to give explicit descriptions for several types of bundles on smooth manifolds, naturally related with the bundle of -dimensional velocities, or -jets. In fact, this kind of bundles are very natural objects in differential geometry, mechanics and Lagrangian dynamics. For this the author considers Weil bundles that arose from Weil algebras. If a suitable combinatorial data is provided by a simplicial coloured structure, then the author describes the corresponding...
The torsions of a general connection on the th-order tangent bundle of a manifold are defined as the Frölicher-Nijenhuis bracket of with the natural affinors. The author deduces the basic properties of these torsions. Then he compares them with the classical torsion of a principal connection on the th-order frame bundle of .
The result of the distributed computing projectWieferich@Home is presented: the binary periodic numbers of bit pseudo-length j ≤ 3500 obtained by replication of a bit string of bit pseudo-length k ≤ 24 and increased by one are Wieferich primes only for the cases of 1092 or 3510.
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