Lifting connections to the r-jet prolongation of the cotangent bundle

We show that the problem of finding all Mfm -natural operators C : Q "M QJ r T ∗ lifting classical linear connections ∇ on m-manifolds M into classical linear connections CM (∇) on the r-jet prolongation J r T ∗M of the cotangent bundle T ∗M p of M can be reduced to that of finding all Mfm -natural operators D : Q "M® T ⊗ q ® T ∗ sending classical linear connections ∇ on M into tensor fields DM (∇) of type (p, q) on M

Government and Online Data: Creation, Access, Preservation

From the 1790 Census to the present, the US government has been a major user, producer, and distributor of data. Through its agencies and departments, it creates data; through funding, grants, and data sharing mandates, it makes research data accessible; through the FDLP and various agency platforms, it circulates and stores data online. This presentation discusses the implications of Government’s role in data creation, online access, and preservation. What are the potential strengths and weaknesses of this relationship, and how can librarians prepare for the future of data creation, preservation, and access? Special attention given to: Federal Data Mandates; Online Sources of Government Data; Challenges to Preservation and Access

Natural operators lifting functions to cotangent bundles of linear higher order tangent bundles

summary:The author studies the problem how a map L:M\to\bbfR on an $n$-dimensional manifold $M$ can induce canonically a map A_M(L):T^* T^{(r)}M\to \bbfR for $r$ a fixed natural number. He proves the following result: ``Let $A: T^{(0,0)}\to T^{(0,0)}(T^* T^{(r)})$ be a natural operator for $n$-manifolds. If $n\ge 3$ then there exists a uniquely determined smooth map H: \bbfR^{S(r)}\times \bbfR\to\bbfR such that $A= A^{(H)}$.''\par The conclusion is that all natural functions on $T^* T^{(r)}$ for $n$-manifolds $(n\ge 3)$ are of the form $\{H\circ(\lambda^{\langle 0,1\rangle}_M,\dots, \lambda^{\langle r,0\rangle}_M)\}$, where H\in C^\infty(\bbfR^r) is a function of $r$ variables

Natural liftings of foliations to the rr-tangent bunde

summary:Let $F$ be a $p$-dimensional foliation on an $n$-manifold $M$, and $T^r M$ the $r$-tangent bundle of $M$. The purpose of this paper is to present some reltionship between the foliation $F$ and a natural lifting of $F$ to the bundle $T^r M$. Let $L^r_q (F)$ $(q=0, 1, \dots, r)$ be a foliation on $T^r M$ projectable onto $F$ and $L^r_q= \{L^r_q (F)\}$ a natural lifting of foliations to $T^r M$. The author proves the following theorem: Any natural lifting of foliations to the $r$-tangent bundle is equal to one of the liftings $L^r_0, L^r_1, \dots, L^r_n$. \par The exposition is clear and well organized

Product preserving bundle functors on fibered manifolds

summary:The complete description of all product preserving bundle functors on fibered manifolds in terms of natural transformations between product preserving bundle functors on manifolds is given

The natural affinors on (JrT∗)∗(J^rT^*)^*

summary:For natural numbers $r$ and $n\ge 2$ a complete classification of natural affinors on the natural bundle $(J^rT^*)^*$ dual to $r$-jet prolongation $J^rT^*$ of the cotangent bundle over $n$-manifolds is given

Canonical 1-forms on higher order adapted frame bundles

summary:Let $(M,\mathcal{F})$ be a foliated $m+n$-dimensional manifold $M$ with $n$-dimensional foliation $\mathcal{F}$. Let $V$ be a finite dimensional vector space over $\mathbf{R}$. We describe all canonical ({\mathcal{F}}\mbox {\it ol}_{m,n}-invariant) $V$-valued $1$-forms $\Theta \colon TP^r(M,{\mathcal{F}}) \rightarrow V$ on the $r$-th order adapted frame bundle $P^r(M,\mathcal{F})$ of $(M,\mathcal{F})$

The natural affinors on some fiber product preserving gauge bundle functors of vector bundles

summary:We classify all natural affinors on vertical fiber product preserving gauge bundle functors $F$ on vector bundles. We explain this result for some more known such $F$. We present some applications. We remark a similar classification of all natural affinors on the gauge bundle functor $F^*$ dual to $F$ as above. We study also a similar problem for some (not all) not vertical fiber product preserving gauge bundle functors on vector bundles