Non-existence of exchange transformations of iterated jet functors

Miroslav Doupovec; Włodzimierz M. Mikulski

Displaying similar documents to “Non-existence of exchange transformations of iterated jet functors”

Natural transformations of the composition of Weil and cotangent functors

Miroslav Doupovec (2001)

Annales Polonici Mathematici

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We study geometrical properties of natural transformations $T^{A} T * \to T * T^{A}$ depending on a linear function defined on the Weil algebra A. We show that for many particular cases of A, all natural transformations $T^{A} T * \to T * T^{A}$ can be described in a uniform way by means of a simple geometrical construction.

Fiber product preserving bundle functors as modified vertical Weil functors

Włodzimierz M. Mikulski (2015)

Czechoslovak Mathematical Journal

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We introduce the concept of modified vertical Weil functors on the category $ℱ ℳ_{m}$ of fibred manifolds with $m$ -dimensional bases and their fibred maps with embeddings as base maps. Then we describe all fiber product preserving bundle functors on $ℱ ℳ_{m}$ in terms of modified vertical Weil functors. The construction of modified vertical Weil functors is an (almost direct) generalization of the usual vertical Weil functor. Namely, in the construction of the usual vertical Weil functors, we replace...

Convergence acceleration of shifted $L R$ transformations for totally nonnegative Hessenberg matrices

Akiko Fukuda, Yusaku Yamamoto, Masashi Iwasaki, Emiko Ishiwata, Yoshimasa Nakamura (2020)

Applications of Mathematics

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We design shifted $L R$ transformations based on the integrable discrete hungry Toda equation to compute eigenvalues of totally nonnegative matrices of the banded Hessenberg form. The shifted $L R$ transformation can be regarded as an extension of the extension employed in the well-known dqds algorithm for the symmetric tridiagonal eigenvalue problem. In this paper, we propose a new and effective shift strategy for the sequence of shifted $L R$ transformations by considering the concept of the Newton...

Direct summands of systems of continuous linear transformations

Uri Fixman, Frank A. Zorzitto

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CONTENTSIntroduction............................................................................................................ 51. The category of $C^{N}$ -systems........................................................................... 82. The problem of split monomorphisms................................................................ 103. Internal hom and tensor product........................................................................... 134. Characterizations of split monomorphisms..........................................................

Linear liftings of affinors to Weil bundles

Jacek Dębecki (2003)

Colloquium Mathematicae

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We give a classification of all linear natural operators transforming affinors on each n-dimensional manifold M into affinors on $T^{A} M$ , where $T^{A}$ is the product preserving bundle functor given by a Weil algebra A, under the condition that n ≥ 2.

Special Lagrangian linear subspaces in product symplectic space

Małgorzata Mikosz (2004)

Banach Center Publications

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The notes consist of a study of special Lagrangian linear subspaces. We will give a condition for the graph of a linear symplectomorphism $f : (ℝ^{2 n}, σ = \sum_{i = 1}^{n} d x_{i} \land d y_{i}) \to (ℝ^{2 n}, σ)$ to be a special Lagrangian linear subspace in $(ℝ^{2 n} \times ℝ^{2 n}, ω = π * ₂ σ - π * ₁ σ)$ . This way a special symplectic subset in the symplectic group is introduced. A stratification of special Lagrangian Grassmannian $S Λ_{2 n} ≃ S U (2 n) / S O (2 n)$ is defined.

Connected sequences of stable derived functors and their applications

Daniel Simson, Andrzej Tyc

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CONTENTS1. Introduction........................................................................................................................................................................................................ 52. Category of complexes.................................................................................................................................................................................... 73. Left stable derived functors of covariant functors..........................................................................................................................................

Lifting right-invariant vector fields and prolongation of connections

W. M. Mikulski (2009)

Annales Polonici Mathematici

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We describe all $_{m} (G)$ -gauge-natural operators lifting right-invariant vector fields X on principal G-bundles P → M with m-dimensional bases into vector fields (X) on the rth order principal prolongation $W^{r} P = P^{r} M \times_{M} J^{r} P$ of P → M. In other words, we classify all $_{m} (G)$ -natural transformations $J^{r} L P \times_{M} W^{r} P \to T W^{r} P = L W^{r} P \times_{M} W^{r} P$ covering the identity of $W^{r} P$ , where $J^{r} L P$ is the r-jet prolongation of the Lie algebroid LP=TP/G of P, i.e. we find all $_{m} (G)$ -natural transformations which are similar to the Kumpera-Spencer isomorphism $J^{r} L P = L W^{r} P$ . We formulate axioms which...

Liftings of 1-forms to $(J^{r} T ) $

Włodzimierz M. Mikulski (2002)

Colloquium Mathematicae

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Let $J^{r} T * M$ be the r-jet prolongation of the cotangent bundle of an n-dimensional manifold M and let $(J^{r} T * M) *$ be the dual vector bundle. For natural numbers r and n, a complete classification of all linear natural operators lifting 1-forms from M to 1-forms on $(J^{r} T * M) *$ is given.

Digital shapes, digital boundaries and rigid transformations: A topological discussion

Yukiko Kenmochi, Phuc Ngo, Nicolas Passat, Hugues Talbot (2013)

Actes des rencontres du CIRM

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Curvature is a continuous and infinitesimal notion. These properties induce geometrical difficulties in digital frameworks, and the following question is naturally asked: “How to define and compute curvatures of digital shapes?” In fact, not only geometrical but also topological difficulties are also induced in digital frameworks. The – deeper – question thus arises: “Can we still define and compute curvatures?” This latter question, that is relevant in the context of digitization, ,...

The natural linear operators $T * ⇝ T T^{(r)}$

J. Kurek, W. M. Mikulski (2003)

Colloquium Mathematicae

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For natural numbers n ≥ 3 and r a complete description of all natural bilinear operators $T * \times_{ℳ f ₙ} T^{(0, 0)} ⇝ T^{(0, 0)} T^{(r)}$ is presented. Next for natural numbers r and n ≥ 3 a full classification of all natural linear operators $T *_{| ℳ f ₙ} ⇝ T T^{(r)}$ is obtained.

Compatibility of the theta correspondence with the Whittaker functors

Vincent Lafforgue, Sergey Lysenko (2011)

Bulletin de la Société Mathématique de France

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We prove that the global geometric theta-lifting functor for the dual pair $(H, G)$ is compatible with the Whittaker functors, where $(H, G)$ is one of the pairs $({S 𝕆}_{2 n}, {𝕊 p}_{2 n})$ , $({𝕊 p}_{2 n}, {S 𝕆}_{2 n + 2})$ or $({𝔾 L}_{n}, {𝔾 L}_{n + 1})$ . That is, the composition of the theta-lifting functor from $H$ to $G$ with the Whittaker functor for $G$ is isomorphic to the Whittaker functor for $H$ .

The natural operators lifting 1-forms to some vector bundle functors

J. Kurek, W. M. Mikulski (2002)

Colloquium Mathematicae

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Let F:ℳ f→ ℬ be a vector bundle functor. First we classify all natural operators $T_{| ℳ f ₙ} ⇝ T^{(0, 0)} (F_{| ℳ f ₙ}) *$ transforming vector fields to functions on the dual bundle functor $(F_{| ℳ f ₙ}) *$ . Next, we study the natural operators $T *_{| ℳ f ₙ} ⇝ T * (F_{| ℳ f ₙ}) *$ lifting 1-forms to $(F_{| ℳ f ₙ}) *$ . As an application we classify the natural operators $T *_{| ℳ f ₙ} ⇝ T * (F_{| ℳ f ₙ}) *$ for some well known vector bundle functors F.

Characterizations of $H_{x} 1$ -sufficiency of jets

Le Tien Tam (1986)

Annales Polonici Mathematici

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Non-existence of some natural operators on connections

W. M. Mikulski (2003)

Annales Polonici Mathematici

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Let n,r,k be natural numbers such that n ≥ k+1. Non-existence of natural operators $C^{r} ₀ ⟿ Q (r e g T_{k}^{r} \to K_{k}^{r})$ and $C^{r} ₀ ⟿ Q (r e g T_{k}^{r *} \to K_{k}^{r *})$ over n-manifolds is proved. Some generalizations are obtained.

Natural transformations of separated jets

Miroslav Doupovec, Ivan Kolář (2000)

Archivum Mathematicum

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Given a map of a product of two manifolds into a third one, one can define its jets of separated orders $r$ and $s$ . We study the functor $J$ of separated $(r; s)$ -jets. We determine all natural transformations of $J$ into itself and we characterize the canonical exchange $J \to J^{s; r}$ from the naturality point of view.

Natural operators lifting linear vector fields from a vector bundle to its r-jet prolongations

Włodzimierz M. Mikulski (2003)

Annales Polonici Mathematici

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All natural operators A transforming a linear vector field X on a vector bundle E into a vector field A(X) on the r-jet prolongation $J^{r} E$ of E are given. Similar results are deduced for the r-jet prolongations $J_{v}^{r} E$ and $J^{[r]} E$ in place of $J^{r} E$ .

On natural vector bundle morphisms $T_{A} \circ ⨂_{s}^{q} \to ⨂_{s}^{q} \circ T_{A}$ over $i d_{T_{A}}$

A. Ntyam, A. Mba (2009)

Annales Polonici Mathematici

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Some properties and applications of natural vector bundle morphisms $T_{A} \circ ⨂_{s}^{q} \to ⨂_{s}^{q} \circ T_{A}$ over $i d_{T_{A}}$ are presented.