General-affine invariants of plane curves and space curves

Shimpei Kobayashi; Takeshi Sasaki

Displaying similar documents to “General-affine invariants of plane curves and space curves”

A Remark on a Paper of Crachiola and Makar-Limanov

Robert Dryło (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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A. Crachiola and L. Makar-Limanov [J. Algebra 284 (2005)] showed the following: if X is an affine curve which is not isomorphic to the affine line $¹_{k}$ , then ML(X×Y) = k[X]⊗ ML(Y) for every affine variety Y, where k is an algebraically closed field. In this note we give a simple geometric proof of a more general fact that this property holds for every affine variety X whose set of regular points is not k-uniruled.

On the affine rectification of convex curves.

Leichtweiss, Kurt (1999)

Beiträge zur Algebra und Geometrie

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Bifurcations of affine invariants for one-parameter family of generic convex plane curves

Takashi Sano (1999)

Banach Center Publications

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We study affine invariants of plane curves from the view point of the singularity theory of smooth functions. We describe how affine vertices and affine inflexions are created and destroyed.

Restrictions of Fourier transforms to curves

S. W. Drury (1985)

Annales de l'institut Fourier

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Let $x (t) = (t, \frac{1}{2} t^{2}, \frac{1}{6} t^{3})$ a certain curve in $R^{3} .$ We investigate inequalities of the type ${\int | \hat{f} (x (t)) |^{b} d t}^{1 / b} \leq C ∥ f ∥_{a}$ for $f \in 𝒮 (R$ 3). Our results improve improve an earlier restriction theorem of Prestini. Various generalizations are also discussed.

Affine differential invariants for planar curves.

Nadjafikhah, Mehdi (2002)

Balkan Journal of Geometry and its Applications (BJGA)

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Curvature functionals for curves in the equi-affine plane

Steven Verpoort (2011)

Czechoslovak Mathematical Journal

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After having given the general variational formula for the functionals indicated in the title, the critical points of the integral of the equi-affine curvature under area constraint and the critical points of the full-affine arc-length are studied in greater detail. Notice. An extended version of this article is available on arXiv:0912.4075.

Remark on the equisingularity of families of affine plane curves

Hà Huy Vui, Phạkm Tien Son (1998)

Annales Polonici Mathematici

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We give some criteria for the equisingularity of families of affine plane curves.

Affine Dunkl processes of type ${\tilde{A}}_{1}$

François Chapon (2012)

Annales de l'I.H.P. Probabilités et statistiques

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We introduce the analogue of Dunkl processes in the case of an affine root system of type ${\tilde{A}}_{1}$ . The construction of the affine Dunkl process is achieved by a skew-product decomposition by means of its radial part and a jump process on the affine Weyl group, where the radial part of the affine Dunkl process is given by a Gaussian process on the ultraspherical hypergroup $[0, 1]$ . We prove that the affine Dunkl process is a càdlàg Markov process as well as a local martingale, study its jumps, and...

Affine Curves in Characteristic p.

Werner Lütkebohmert (1980)

Mathematische Zeitschrift

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The natural operators of general affine connections into general affine connections

Jan Kurek, Włodzimierz M. Mikulski (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We reduce the problem of describing all $ℳ f_{m}$ -natural operators transforming general affine connections on $m$ -manifolds into general affine ones to the known description of all $G L (𝐑^{m})$ -invariant maps $𝐑^{m *} \otimes 𝐑^{m} \to \otimes^{k} 𝐑^{m *} \otimes \otimes^{k} 𝐑^{m}$ for $k = 1, 3$ .

Hypersurfaces with almost complex structures in the real affine space

Mayuko Kon (2007)

Colloquium Mathematicae

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We study affine hypersurface immersions $f : M \to ℝ^{2 n + 1}$ , where M is an almost complex n-dimensional manifold. The main purpose is to give a condition for (M,J) to be a special Kähler manifold with respect to the Levi-Civita connection of an affine fundamental form.

On the Hausdorff dimension of certain self-affine sets

Abercrombie Alex G.., Nair R. (2002)

Studia Mathematica

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A subset E of ℝⁿ is called self-affine with respect to a collection ϕ₁,...,ϕₜ of affinities if E is the union of the sets ϕ₁(E),...,ϕₜ(E). For S ⊂ ℝⁿ let $Φ (S) = ⋃_{1 \leq j \leq t} ϕ_{j} (S)$ . If Φ(S) ⊂ S let $E_{Φ} (S)$ denote $⋂_{k \geq 0} Φ^{k} (S)$ . For given Φ consisting of contracting “pseudo-dilations” (affinities which preserve the directions of the coordinate axes) and subject to further mild technical restrictions we show that there exist self-affine sets $E_{Φ} (S)$ of each Hausdorff dimension between zero and a positive number depending on Φ. We also...

Affine braid group actions on derived categories of Springer resolutions

Roman Bezrukavnikov, Simon Riche (2012)

Annales scientifiques de l'École Normale Supérieure

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In this paper we construct and study an action of the affine braid group associated with a semi-simple algebraic group on derived categories of coherent sheaves on various varieties related to the Springer resolution of the nilpotent cone. In particular, we describe explicitly the action of the Artin braid group. This action is a “categorical version” of Kazhdan-Lusztig-Ginzburg’s construction of the affine Hecke algebra, and is used in particular by the first author and I. Mirković...

Sufficient conditions for the spectrality of self-affine measures with prime determinant

Jian-Lin Li (2014)

Studia Mathematica

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Let $μ_{M, D}$ be a self-affine measure associated with an expanding matrix M and a finite digit set D. We study the spectrality of $μ_{M, D}$ when |det(M)| = |D| = p is a prime. We obtain several new sufficient conditions on M and D for $μ_{M, D}$ to be a spectral measure with lattice spectrum. As an application, we present some properties of the digit sets of integral self-affine tiles, which are connected with a conjecture of Lagarias and Wang.