The author considers the problem to give explicit descriptions for several types of bundles on smooth manifolds, naturally related with the bundle of $k$-dimensional velocities, or $k$-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 $\Gamma $ on the $r$th-order tangent bundle of a manifold $M$ are defined as the Frölicher-Nijenhuis bracket of $\Gamma $ 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 $r$th-order frame bundle of $M$.

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.

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 $A$ be a Weil algebra. The bijection between all natural operators lifting vector fields from $m$-manifolds to the bundle functor ${K}^{A}$ of Weil contact elements and the subalgebra of fixed elements $SA$ of the Weil algebra $A$ is determined and the bijection between all natural affinors on ${K}^{A}$ and $SA$ is deduced. Furthermore, the rigidity of the functor ${K}^{A}$ is proved. Requisite results about the structure of $SA$ are obtained by a purely algebraic approach, namely the existence of nontrivial $SA$ is discussed.

Quaternion algebras $\left(\frac{-1,b}{\mathbb{Q}}\right)$ 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.

Download Results (CSV)