Natural transformations of the composition of Weil and cotangent functors

Miroslav Doupovec

Displaying similar documents to “Natural transformations of the composition of Weil and cotangent functors”

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

Similarity:

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...

Fiber product preserving bundle functors as modified vertical Weil functors

Włodzimierz M. Mikulski (2015)

Czechoslovak Mathematical Journal

Similarity:

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...

Compatibility of the theta correspondence with the Whittaker functors

Vincent Lafforgue, Sergey Lysenko (2011)

Bulletin de la Société Mathématique de France

Similarity:

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 $H$ .

Non-existence of exchange transformations of iterated jet functors

Miroslav Doupovec, Włodzimierz M. Mikulski (2007)

Banach Center Publications

Similarity:

We study the problem of the non-existence of natural transformations $J^{r} J^{s} Y \to J^{s} J^{r} Y$ of iterated jet functors depending on some geometric object on the base of Y.

Lifting right-invariant vector fields and prolongation of connections

W. M. Mikulski (2009)

Annales Polonici Mathematici

Similarity:

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...

Connected sequences of stable derived functors and their applications

Daniel Simson, Andrzej Tyc

Similarity:

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

The general rigidity result for bundles of $A$ -covelocities and $A$ -jets

Jiří M. Tomáš (2017)

Czechoslovak Mathematical Journal

Similarity:

Let $M$ be an $m$ -dimensional manifold and $A = 𝔻_{k}^{r} / I = ℝ \oplus N_{A}$ a Weil algebra of height $r$ . We prove that any $A$ -covelocity $T_{x}^{A} f \in T_{x}^{A *} M$ , $x \in M$ is determined by its values over arbitrary $max {width A, m}$ regular and under the first jet projection linearly independent elements of $T_{x}^{A} M$ . Further, we prove the rigidity of the so-called universally reparametrizable Weil algebras. Applying essentially those partial results we give the proof of the general rigidity result $T^{A *} M ≃ T^{r *} M$ without coordinate computations, which improves and generalizes the partial...

On the Schröder equation

M. Kuczma

Similarity:

CONTENTSPART IIntroduction............................................................................................... 31. General solution.................................................................................. 42. Preliminaries and notation................................................................ 53. $C^{p}$ solutions in $ℭ$ *................................................ 74. Change of variables..............................................................................

Explicit construction of normal lattice configurations

Mordechay B. Levin, Meir Smorodinsky (2005)

Colloquium Mathematicae

Similarity:

We extend Champernowne’s construction of normal numbers to base b to the $ℤ^{d}$ case and obtain an explicit construction of a generic point of the $ℤ^{d}$ shift transformation of the set ${0, 1, . . ., b - 1}^{ℤ^{d}}$ .

Lipschitz constants for a hyperbolic type metric under Möbius transformations

Yinping Wu, Gendi Wang, Gaili Jia, Xiaohui Zhang (2024)

Czechoslovak Mathematical Journal

Similarity:

Let $D$ be a nonempty open set in a metric space $(X, d)$ with $\partial D \neq \emptyset$ . Define $h_{D, c} (x, y) = log (1 + c \frac{d (x, y)}{\sqrt{d_{D} (x) d_{D} (y)}}),$ where $d_{D} (x) = d (x, \partial D)$ is the distance from $x$ to the boundary of $D$ . For every $c \geq 2$ , $h_{D, c}$ is a metric. We study the sharp Lipschitz constants for the metric $h_{D, c}$ under Möbius transformations of the unit ball, the upper half space, and the punctured unit ball.

On lifting of connections to Weil bundles

Jan Kurek, Włodzimierz M. Mikulski (2012)

Annales Polonici Mathematici

Similarity:

We prove that the problem of finding all $ℳ f_{m}$ -natural operators $B : Q ⟿ Q T^{A}$ lifting classical linear connections ∇ on m-manifolds M to classical linear connections $B_{M} (\nabla)$ on the Weil bundle $T^{A} M$ corresponding to a p-dimensional (over ℝ) Weil algebra A is equivalent to the one of finding all $ℳ f_{m}$ -natural operators $C : Q ⟿ (T ¹_{p - 1}, T * \otimes T * \otimes T)$ transforming classical linear connections ∇ on m-manifolds M into base-preserving fibred maps $C_{M} (\nabla) : T ¹_{p - 1} M = ⨁_{M}^{p - 1} T M \to T * M \otimes T * M \otimes T M$ .

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

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

Colloquium Mathematicae

Similarity:

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.

Taylor towers of symmetric and exterior powers

Brenda Johnson, Randy McCarthy (2008)

Fundamenta Mathematicae

Similarity:

We study the Taylor towers of the nth symmetric and exterior power functors, Spⁿ and Λⁿ. We obtain a description of the layers of the Taylor towers, $D_{k} S p ⁿ$ and $D_{k} Λ ⁿ$ , in terms of the first terms in the Taylor towers of $S p^{t}$ and $Λ^{t}$ for t < n. The homology of these first terms is related to the stable derived functors (in the sense of Dold and Puppe) of $S p^{t}$ and $Λ^{t}$ . We use stable derived functor calculations of Dold and Puppe to determine the lowest nontrivial homology groups for $D_{k} S p ⁿ$ and $D_{k} Λ ⁿ$ .

Direct summands of systems of continuous linear transformations

Uri Fixman, Frank A. Zorzitto

Similarity:

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..........................................................

Geometric theta-lifting for the dual pair ${𝕊𝕆}_{2 m}, 𝕊 p_{2 n}$

Sergey Lysenko (2011)

Annales scientifiques de l'École Normale Supérieure

Similarity:

Let $X$ be a smooth projective curve over an algebraically closed field of characteristic $> 2$ . Consider the dual pair $H = {SO}_{2 m}, G = {Sp}_{2 n}$ over $X$ with $H$ split. Write ${Bun}_{G}$ and ${Bun}_{H}$ for the stacks of $G$ -torsors and $H$ -torsors on $X$ . The theta-kernel ${Aut}_{G, H}$ on ${Bun}_{G} \times {Bun}_{H}$ yields theta-lifting functors $F_{G} : D ({Bun}_{H}) \to D ({Bun}_{G})$ and $F_{H} : D ({Bun}_{G}) \to D ({Bun}_{H})$ between the corresponding derived categories. We describe the relation of these functors with Hecke operators. In two particular cases these functors realize the geometric Langlands functoriality for the above pair (in the non...

On almost complex structures from classical linear connections

Jan Kurek, Włodzimierz M. Mikulski (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

Similarity:

Let $ℳ f_{m}$ be the category of $m$ -dimensional manifolds and local diffeomorphisms and let $T$ be the tangent functor on $ℳ f_{m}$ . Let $𝒱$ be the category of real vector spaces and linear maps and let $𝒱_{m}$ be the category of $m$ -dimensional real vector spaces and linear isomorphisms. We characterize all regular covariant functors $F : 𝒱_{m} \to 𝒱$ admitting $ℳ f_{m}$ -natural operators $\tilde{J}$ transforming classical linear connections $\nabla$ on $m$ -dimensional manifolds $M$ into almost complex structures $\tilde{J} (\nabla)$ on $F (T) M = ⋃_{x \in M} F (T_{x} M)$ .

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

Le Tien Tam (1986)

Annales Polonici Mathematici

Similarity:

On maps preserving connectedness and/or compactness

István Juhász, Jan van Mill (2018)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We call a function $f : X \to Y$ P-preserving if, for every subspace $A \subset X$ with property P, its image $f (A)$ also has property P. Of course, all continuous maps are both compactness- and connectedness-preserving and the natural question about when the converse of this holds, i.e. under what conditions such a map is continuous, has a long history. Our main result is that any nontrivial product function, i.e. one having at least two nonconstant factors, that has connected domain, $T_{1}$ range, and is connectedness-preserving...

The natural operators $T^{(0, 0)} ⇝ T^{(1, 1)} T^{(r)}$

Włodzimierz M. Mikulski (2003)

Colloquium Mathematicae

Similarity:

We study the problem of how a map f:M → ℝ on an n-manifold M induces canonically an affinor $A (f) : T T^{(r)} M \to T T^{(r)} M$ on the vector r-tangent bundle $T^{(r)} M = (J^{r} (M, ℝ) ₀) *$ over M. This problem is reflected in the concept of natural operators $A : T_{| ℳ f ₙ}^{(0, 0)} ⇝ T^{(1, 1)} T^{(r)}$ . For integers r ≥ 1 and n ≥ 2 we prove that the space of all such operators is a free (r+1)²-dimensional module over $^{\infty} (T^{(r)} ℝ)$ and we construct explicitly a basis of this module.

Nilakantha's accelerated series for π

David Brink (2015)

Acta Arithmetica

Similarity:

We show how the idea behind a formula for π discovered by the Indian mathematician and astronomer Nilakantha (1445-1545) can be developed into a general series acceleration technique which, when applied to the Gregory-Leibniz series, gives the formula $π = \sum_{n = 0}^{\infty} ((5 n + 3) n! (2 n)!) / (2^{n - 1} (3 n + 2)!)$ with convergence as $13 . 5^{- n}$ , in much the same way as the Euler transformation gives $π = \sum_{n = 0}^{\infty} (2^{n + 1} n! n!) / (2 n + 1)!$ with convergence as $2^{- n}$ . Similar transformations lead to other accelerated series for π, including three “BBP-like” formulas, all of which are collected in...

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

Włodzimierz M. Mikulski (2002)

Colloquium Mathematicae

Similarity:

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.

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

Włodzimierz M. Mikulski (2003)

Annales Polonici Mathematici

Similarity:

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 $J^{r} E$ .

The Roquette category of finite $p$ -groups

Serge Bouc (2015)

Journal of the European Mathematical Society

Similarity:

Let $p$ be a prime number. This paper introduces the Roquette category $ℛ_{p}$ of finite $p$ -groups, which is an additive tensor category containing all finite $p$ -groups among its objects. In $ℛ_{p}$ , every finite $p$ -group $P$ admits a canonical direct summand $\partial P$ , called the edge of $P$ . Moreover $P$ splits uniquely as a direct sum of edges of Roquette $p$ -groups, and the tensor structure of $ℛ_{p}$ can be described in terms of such edges. The main motivation for considering this category is that the additive functors...

The natural transformations between r-tangent and r-cotangent bundles over Riemannian manifolds

Jan Kurek, Włodzimierz Mikulski (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

Similarity:

If $(M, g)$ is a Riemannian manifold, we have the well-known base preserving vector bundle isomorphism $T M \tilde{=} T^{*} M$ given by $v \to g (v, -)$ between the tangent $T M$ and the cotangent $T^{*} M$ bundles of $M$ . In the present note, we generalize this isomorphism to the one $T^{(r)} M \tilde{=} T^{r *} M$ between the $r$ -th order vector tangent $T^{(r)} M = {(J^{r} {(M, R)}_{0})}^{*}$ and the $r$ -th order cotangent $T^{r *} M = J^{r} {(M, R)}_{0}$ bundles of $M$ . Next, we describe all base preserving vector bundle maps $C_{M} (g) : T^{(r)} M \to T^{r *} M$ depending on a Riemannian metric $g$ in terms of natural (in $g$ ) tensor fields on $M$ .

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

Similarity:

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.