Stable mappings of discriminant varieties

J. W. Bruce

doi:10.1017/s030500410006463x

Stable mappings of discriminant varieties

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500410006463x ◽

1988 ◽

Vol 103 (1) ◽

pp. 69-82

Author(s):

J. W. Bruce

Keyword(s):

Wave Front ◽

Smooth Surfaces ◽

Isolated Singularity ◽

Isolated Singularities ◽

Low Dimensions

Smooth mappings defined on discriminant varieties of -versal unfoldings of isolated singularities arise in many interesting geometrical contexts, for example when classifying outlines of smooth surfaces in ℝ3 and their duals, or wave-front evolution [1, 2, 5]. In three previous papers we have classified various stable mappings on discriminants. When the isolated singularity is weighted homogeneous the discriminant is not a local smooth product, and this makes the classification of stable germs considerably easier than in general. Moreover, discriminants arising from weighted homogeneous singularities predominate in low dimensions, so such classifications are very useful for applications.

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Classification of Filiform Leibniz Algebras in Low Dimensions

Leibniz Algebras ◽

10.1201/9780429344336-6 ◽

2019 ◽

pp. 223-249

Author(s):

Shavkat Ayupov ◽

Bakhrom Omirov ◽

Isamiddin Rakhimov

Keyword(s):

Low Dimensions ◽

Leibniz Algebras

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Isolated Singularities of Holomorphic Functions at ∞ (Continued). Orders of Poles at ∞. Casorati-Weierstrass’ Theorem for an Isolated Singularity at ∞. Residues. Cauchy’s Residue Theorem. Computing Residues

Springer Undergraduate Mathematics Series - Twenty-One Lectures on Complex Analysis ◽

10.1007/978-3-319-68170-2_15 ◽

2017 ◽

pp. 127-135

Author(s):

Alexander Isaev

Keyword(s):

Holomorphic Functions ◽

Isolated Singularity ◽

Residue Theorem ◽

Weierstrass Theorem ◽

Isolated Singularities ◽

Cauchy’S Residue Theorem

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Ternary and n-ary f-distributive structures

Open Mathematics ◽

10.1515/math-2018-0006 ◽

2018 ◽

Vol 16 (1) ◽

pp. 32-45 ◽

Cited By ~ 1

Author(s):

Indu R. U. Churchill ◽

M. Elhamdadi ◽

M. Green ◽

A. Makhlouf

Keyword(s):

Cohomology Theory ◽

Extension Theory ◽

Low Dimensions

AbstractWe introduce and study ternary f-distributive structures, Ternary f-quandles and more generally their higher n-ary analogues. A classification of ternary f-quandles is provided in low dimensions. Moreover, we study extension theory and introduce a cohomology theory for ternary, and more generally n-ary, f-quandles. Furthermore, we give some computational examples.

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NULL HELICES IN LORENTZIAN SPACE FORMS

International Journal of Modern Physics A ◽

10.1142/s0217751x01005821 ◽

2001 ◽

Vol 16 (30) ◽

pp. 4845-4863 ◽

Cited By ~ 55

Author(s):

ANGEL FERRÁNDEZ ◽

ANGEL GIMÉNEZ ◽

PASCUAL LUCAS

Keyword(s):

Minkowski Space ◽

Complete Classification ◽

Space Forms ◽

Lorentzian Space ◽

Low Dimensions ◽

Minkowski Space Time ◽

Minimum Number ◽

Null Curves ◽

Lorentzian Space Forms

In this paper we introduce a reference along a null curve in an n-dimensional Lorentzian space with the minimum number of curvatures. That reference generalizes the reference of Bonnor for null curves in Minkowski space–time and it is called the Cartan frame of the curve. The associated curvature functions are called the Cartan curvatures of the curve. We characterize the null helices (that is, null curves with constant Cartan curvatures) in n-dimensional Lorentzian space forms and we obtain a complete classification of them in low dimensions.

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The classification of Novikov algebras in low dimensions: invariant bilinear forms

Journal of Physics A General Physics ◽

10.1088/0305-4470/34/39/401 ◽

2001 ◽

Vol 34 (39) ◽

pp. 8193-8197 ◽

Cited By ~ 17

Author(s):

Chengming Bai ◽

Daoji Meng

Keyword(s):

Bilinear Forms ◽

Low Dimensions ◽

Novikov Algebras

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A Classification of Isolated Singularities of Elliptic Monge-Ampère Equations in Dimension Two

Communications on Pure and Applied Mathematics ◽

10.1002/cpa.21551 ◽

2014 ◽

pp. n/a-n/a

Author(s):

José A. Gálvez ◽

Asun Jiménez ◽

Pablo Mira

Keyword(s):

Isolated Singularities ◽

Monge Ampere

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Castelnuovo-Mumford regularity bound for smooth threefolds in ℙ5 and extremal examples

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crll.1999.509.21 ◽

1999 ◽

Vol 1999 (509) ◽

pp. 21-34

Author(s):

Si-Jong Kwak

Keyword(s):

Complete Intersection ◽

Number Field ◽

Complex Number ◽

Smooth Surfaces ◽

Veronese Surface ◽

Optimal Bound ◽

Integral Curves ◽

Complex Number Field

Abstract Let X be a nondegenerate integral subscheme of dimension n and degree d in ℙN defined over the complex number field ℂ. X is said to be k-regular if Hi(ℙN, ℐX (k – i)) = 0 for all i ≧ 1, where ℐX is the sheaf of ideals of ℐℙN and Castelnuovo-Mumford regularity reg(X) of X is defined as the least such k. There is a well-known conjecture concerning k-regularity: reg(X) ≦ deg(X) – codim(X) + 1. This regularity conjecture including the classification of borderline examples was verified for integral curves (Castelnuovo, Gruson, Lazarsfeld and Peskine), and an optimal bound was also obtained for smooth surfaces (Pinkham, Lazarsfeld). It will be shown here that reg(X) ≦ deg(X) – 1 for smooth threefolds X in ℙ5 and that the only extremal cases are the rational cubic scroll and the complete intersection of two quadrics. Furthermore, every smooth threefold X in ℙ5 is k-normal for all k ≧ deg(X) – 4, which is the optimal bound as the Palatini 3-fold of degree 7 shows. The same bound also holds for smooth regular surfaces in ℙ4 other than for the Veronese surface.

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