A representation of the coalgebra of derivations for smooth spaces

Fischer, Gerald

Displaying similar documents to “A representation of the coalgebra of derivations for smooth spaces”

On smooth points of boundaries of open sets

S. Rolewicz (2009)

Studia Mathematica

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The notions of smooth points of the boundary of an open set and α(·) intrinsically paraconvex sets are introduced. It is shown that for an α(·) intrinsically paraconvex open set the set of smooth points is a dense $G_{δ}$ -set of the boundary.

On the Extension of Certain Maps with Values in Spheres

Carlos Biasi, Alice K. M. Libardi, Pedro L. Q. Pergher, Stanisław Spież (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let E be an oriented, smooth and closed m-dimensional manifold with m ≥ 2 and V ⊂ E an oriented, connected, smooth and closed (m-2)-dimensional submanifold which is homologous to zero in E. Let $S^{n - 2} \subset S ⁿ$ be the standard inclusion, where Sⁿ is the n-sphere and n ≥ 3. We prove the following extension result: if $h : V \to S^{n - 2}$ is a smooth map, then h extends to a smooth map g: E → Sⁿ transverse to $S^{n - 2}$ and with $g^{- 1} (S^{n - 2}) = V$ . Using this result, we give a new and simpler proof of a theorem of Carlos Biasi related to the ambiental...

On the range of the derivative of a real-valued function with bounded support

T. Gaspari (2002)

Studia Mathematica

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We study the set f’(X) = f’(x): x ∈ X when f:X → ℝ is a differentiable bump. We first prove that for any C²-smooth bump f: ℝ² → ℝ the range of the derivative of f must be the closure of its interior. Next we show that if X is an infinite-dimensional separable Banach space with a $C^{p}$ -smooth bump b:X → ℝ such that $| | b^{(p)} {| |}_{\infty}$ is finite, then any connected open subset of X* containing 0 is the range of the derivative of a $C^{p}$ -smooth bump. We also study the finite-dimensional case which is quite different....

Using boundaries to find smooth norms

Victor Bible (2014)

Studia Mathematica

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The aim of this paper is to present a tool used to show that certain Banach spaces can be endowed with $C^{k}$ smooth equivalent norms. The hypothesis uses particular countable decompositions of certain subsets of $B_{X *}$ , namely boundaries. Of interest is that the main result unifies two quite well known results. In the final section, some new corollaries are given.

A Lipschitz function which is $C^{\infty}$ on a.e. line need not be generically differentiable

Luděk Zajíček (2013)

Colloquium Mathematicae

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We construct a Lipschitz function f on X = ℝ ² such that, for each 0 ≠ v ∈ X, the function f is $C^{\infty}$ smooth on a.e. line parallel to v and f is Gâteaux non-differentiable at all points of X except a first category set. Consequently, the same holds if X (with dimX > 1) is an arbitrary Banach space and “a.e.” has any usual “measure sense”. This example gives an answer to a natural question concerning the author’s recent study of linearly essentially smooth functions (which generalize essentially...

Are All Smooth Solutions of Euclidean $S U (2)$ -Yang-Mills Equations Selfdual or Antiselfdual?

R. Flume (1979)

Recherche Coopérative sur Programme n°25

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A group law on smooth real quartics having at least $3$ real branches

Johan Huisman (2002)

Journal de théorie des nombres de Bordeaux

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Let $C$ be a smooth real quartic curve in $ℙ^{2}$ . Suppose that $C$ has at least $3$ real branches $B_{1}, B_{2}, B_{3}$ . Let $B = B_{1} \times B_{2} \times B_{3}$ and let $O \in B$ . Let $τ_{O}$ be the map from $B$ into the neutral component Jac $(C) {(ℝ)}^{0}$ of the set of real points of the jacobian of $C$ , defined by letting $τ_{O} (P)$ be the divisor class of the divisor $\sum P_{i} - O_{i}$ . Then, $τ_{O}$ is a bijection. We show that this allows an explicit geometric description of the group law on Jac $(C) {(ℝ)}^{0}$ . It generalizes the classical geometric description of the group law on the neutral component of the set of real...

Smooth Gevrey normal forms of vector fields near a fixed point

Laurent Stolovitch (2013)

Annales de l’institut Fourier

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We study germs of smooth vector fields in a neighborhood of a fixed point having an hyperbolic linear part at this point. It is well known that the “small divisors” are invisible either for the smooth linearization or normal form problem. We prove that this is completely different in the smooth Gevrey category. We prove that a germ of smooth $α$ -Gevrey vector field with an hyperbolic linear part admits a smooth $β$ -Gevrey transformation to a smooth $β$ -Gevrey normal form. The Gevrey order...

Real algebraic actions on projective spaces - A survey

Ted Petrie (1973)

Annales de l'institut Fourier

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Let $G$ be a compact lie group. We introduce the set $S_{G} (Y)$ for every smooth $G$ manifold $Y$ . It consists of equivalence classes of pair $(X, f)$ where $f : X \to Y$ is a $G$ map which defines a homotopy equivalence from $X$ to $Y$ . Two pairs $(X_{i}, f_{i})$ , for $i = 0, 1$ , are equivalent if there is a $G$ homotopy equivalence $φ : X_{0} \to X_{1}$ such that $f_{0}$ is $G$ homotopic to $f_{1} \circ φ$ . Properties of the set $S_{G} (Y)$ and related to the representation of $G$ on the tangent spaces of $X$ and $Y$ at the fixed points. For the case $G = S^{1}$ and $Y$ is the $S^{1}$ manifold defined by a “linear”...

Generators for algebras dense in $L^{p}$ -spaces

Alexander J. Izzo, Bo Li (2013)

Studia Mathematica

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For various $L^{p}$ -spaces (1 ≤ p < ∞) we investigate the minimum number of complex-valued functions needed to generate an algebra dense in the space. The results depend crucially on the regularity imposed on the generators. For μ a positive regular Borel measure on a compact metric space there always exists a single bounded measurable function that generates an algebra dense in $L^{p} (μ)$ . For M a Riemannian manifold-with-boundary of finite volume there always exists a single continuous function...

Minimal number of periodic points for smooth self-maps of S³

Grzegorz Graff, Jerzy Jezierski (2009)

Fundamenta Mathematicae

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Let f be a continuous self-map of a smooth compact connected and simply-connected manifold of dimension m ≥ 3 and r a fixed natural number. A topological invariant $D_{r}^{m} [f]$ , introduced by the authors [Forum Math. 21 (2009)], is equal to the minimal number of r-periodic points for all smooth maps homotopic to f. In this paper we calculate $D ³_{r} [f]$ for all self-maps of S³.

A characterization of graphs which can be approximated in area by smooth graphs

Domenico Mucci (2001)

Journal of the European Mathematical Society

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For vector valued maps, convergence in $W^{1, 1}$ and of all minors of the Jacobian matrix in $L^{1}$ is equivalent to convergence weakly in the sense of currents and in area for graphs. We show that maps defined on domains of dimension $n \geq 3$ can be approximated strongly in this sense by smooth maps if and only if the same property holds for the restriction to a.e. 2-dimensional plane intersecting the domain.

$Z ₂^{k}$ -actions with a special fixed point set

Pedro L. Q. Pergher, Rogério de Oliveira (2005)

Fundamenta Mathematicae

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Let Fⁿ be a connected, smooth and closed n-dimensional manifold satisfying the following property: if $N^{m}$ is any smooth and closed m-dimensional manifold with m > n and $T : N^{m} \to N^{m}$ is a smooth involution whose fixed point set is Fⁿ, then m = 2n. We describe the equivariant cobordism classification of smooth actions $(M^{m}; Φ)$ of the group $G = Z ₂^{k}$ on closed smooth m-dimensional manifolds $M^{m}$ for which the fixed point set of the action is a submanifold Fⁿ with the above property. This generalizes a result of F....

Quantum expanders and geometry of operator spaces

Gilles Pisier (2014)

Journal of the European Mathematical Society

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We show that there are well separated families of quantum expanders with asymptotically the maximal cardinality allowed by a known upper bound. This has applications to the “growth" of certain operator spaces: It implies asymptotically sharp estimates for the growth of the multiplicity of $M_{N}$ -spaces needed to represent (up to a constant $C > 1$ ) the $M_{N}$ -version of the $n$ -dimensional operator Hilbert space $O H_{n}$ as a direct sum of copies of $M_{N}$ . We show that, when $C$ is close to 1, this multiplicity grows...

Commuting involutions whose fixed point set consists of two special components

Pedro L. Q. Pergher, Rogério de Oliveira (2008)

Fundamenta Mathematicae

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Let Fⁿ be a connected, smooth and closed n-dimensional manifold. We call Fⁿ a manifold with property when it has the following property: if $N^{m}$ is any smooth closed m-dimensional manifold with m > n and $T : N^{m} \to N^{m}$ is a smooth involution whose fixed point set is Fⁿ, then m = 2n. Examples of manifolds with this property are: the real, complex and quaternionic even-dimensional projective spaces $R P^{2 n}$ , $C P^{2 n}$ and $H P^{2 n}$ , and the connected sum of $R P^{2 n}$ and any number of copies of Sⁿ × Sⁿ, where Sⁿ is the n-sphere...

Remarks on Sekine quantum groups

Jialei Chen, Shilin Yang (2022)

Czechoslovak Mathematical Journal

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We first describe the Sekine quantum groups $𝒜_{k}$ (the finite-dimensional Kac algebra of Kac-Paljutkin type) by generators and relations explicitly, which maybe convenient for further study. Then we classify all irreducible representations of $𝒜_{k}$ and describe their representation rings $r (𝒜_{k})$ . Finally, we compute the the Frobenius-Perron dimension of the Casimir element and the Casimir number of $r (𝒜_{k})$ .

Herman’s last geometric theorem

Bassam Fayad, Raphaël Krikorian (2009)

Annales scientifiques de l'École Normale Supérieure

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We present a proof of Herman’s Last Geometric Theorem asserting that if $F$ is a smooth diffeomorphism of the annulus having the intersection property, then any given $F$ -invariant smooth curve on which the rotation number of $F$ is Diophantine is accumulated by a positive measure set of smooth invariant curves on which $F$ is smoothly conjugated to rotation maps. This implies in particular that a Diophantine elliptic fixed point of an area preserving diffeomorphism of the plane is stable. The...