Surfaces Whose Asymptotic Curves are Twisted Cubics

1938 ◽  
Vol 60 (2) ◽  
pp. 337 ◽  
Author(s):  
E. P. Lane ◽  
M. L. MacQueen
Keyword(s):  
Asymptotic Curves ◽  
Twisted Cubics

scholarly journals The Geometry of Marked Contact Engel Structures

2020 ◽  
Author(s):  
Gianni Manno ◽  
Paweł Nurowski ◽  
Katja Sagerschnig
Keyword(s):  
Local Geometry ◽  
Dimensional Manifold ◽  
Twisted Cubic ◽  
Homogeneous Models ◽  
Local Invariants ◽  
Twisted Cubics ◽  
Cubic Structure ◽  
Contact Distribution ◽  
Engel Structures ◽  
Engel Structure

AbstractA contact twisted cubic structure$$({\mathcal M},\mathcal {C},{\varvec{\upgamma }})$$ ( M , C , γ ) is a 5-dimensional manifold $${\mathcal M}$$ M together with a contact distribution $$\mathcal {C}$$ C and a bundle of twisted cubics $${\varvec{\upgamma }}\subset \mathbb {P}(\mathcal {C})$$ γ ⊂ P ( C ) compatible with the conformal symplectic form on $$\mathcal {C}$$ C . The simplest contact twisted cubic structure is referred to as the contact Engel structure; its symmetry group is the exceptional group $$\mathrm {G}_2$$ G 2 . In the present paper we equip the contact Engel structure with a smooth section $$\sigma : {\mathcal M}\rightarrow {\varvec{\upgamma }}$$ σ : M → γ , which “marks” a point in each fibre $${\varvec{\upgamma }}_x$$ γ x . We study the local geometry of the resulting structures $$({\mathcal M},\mathcal {C},{\varvec{\upgamma }}, \sigma )$$ ( M , C , γ , σ ) , which we call marked contact Engel structures. Equivalently, our study can be viewed as a study of foliations of $${\mathcal M}$$ M by curves whose tangent directions are everywhere contained in $${\varvec{\upgamma }}$$ γ . We provide a complete set of local invariants of marked contact Engel structures, we classify all homogeneous models with symmetry groups of dimension $$\ge 6$$ ≥ 6 up to local equivalence, and we prove an analogue of the classical Kerr theorem from Relativity.

ASYMPTOTIC CURVES ON SURFACES IN ℝ5

2008 ◽  
Vol 10 (03) ◽  
pp. 309-335 ◽  
Author(s):  
M. C. ROMERO-FUSTER ◽  
M. A. S. RUAS ◽  
F. TARI
Keyword(s):  
Fundamental Form ◽  
Second Fundamental Form ◽  
Immersed Surfaces ◽  
The Second Fundamental Form ◽  
Asymptotic Curves ◽  
Curves On Surfaces

We study asymptotic curves on generically immersed surfaces in ℝ5. We characterize asymptotic directions via the contact of the surface with flat objects (k-planes, k = 1 - 4), give the equation of the asymptotic curves in terms of the coefficients of the second fundamental form and study their generic local configurations.

II.—The Irreducible System of Concomitants of Two Double Binary (2, 1) Forms

1926 ◽  
Vol 45 (1) ◽  
pp. 3-13
Author(s):  
W. Saddler
Keyword(s):  
Quadric Surface ◽  
Binary Forms ◽  
Irreducible System ◽  
Simultaneous System ◽  
Twisted Cubics

Little is known of the details of systems of concomitants belonging to double binary forms. The cases of the single ground form of orders (1, 1), (2, 1), (2, 2) respectively, together with the simultaneous system of any number of (1, 1) forms, are the only four cases, which have been published. The following pages establish the simultaneous system of two (2, 1) forms.This system is fundamental for the geometrical treatment of two twisted cubics lying upon a quadric surface and having four common points.

scholarly journals Generalized twisted cubics on a cubic fourfold as a moduli space of stable objects

2018 ◽  
Vol 114 ◽  
pp. 85-117 ◽  
Author(s):  
Martí Lahoz ◽  
Manfred Lehn ◽  
Emanuele Macrì ◽  
Paolo Stellari
Keyword(s):  
Moduli Space ◽  
Twisted Cubics ◽  
Cubic Fourfold

scholarly journals STICKINESS IN CHAOS

2008 ◽  
Vol 18 (10) ◽  
pp. 2929-2949 ◽  
Author(s):  
G. CONTOPOULOS ◽  
M. HARSOULA
Keyword(s):  
Degrees Of Freedom ◽  
Outer Boundary ◽  
Characteristic Number ◽  
Heteroclinic Orbits ◽  
Escape Time ◽  
Unstable Periodic Orbits ◽  
Time Direction ◽  
Asymptotic Curves ◽  
Long Time ◽  
Stickiness Effects

We distinguish two types of stickiness in systems of two degrees of freedom: (a) stickiness around an island of stability, and (b) stickiness in chaos, along the unstable asymptotic curves of unstable periodic orbits. In fact, there are asymptotic curves of unstable orbits near the outer boundary of an island that remain close to the island for some time, and then extend to large distances into the surrounding chaotic sea. But later the asymptotic curves return close to the island and contribute to the overall stickiness that produces dark regions around the islands and dark lines extending far from the islands. We have studied these effects in the standard map with a rather large nonlinearity K = 5, and we emphasized the role of the asymptotic curves U , S from the central orbit O (x = 0.5, y = 0), that surround two large islands O 1 and O ′1, and the asymptotic curves U + U - S + S - from the simplest unstable orbit around the island O 1. This is the orbit 4/9 that has 9 points around the island O 1 and 9 more points around the symmetric island O ′1. The asymptotic curves produce stickiness in the positive time direction ( U , U +, U -) and in the negative time direction ( S , S +, S -). The asymptotic curves U +, S + are closer to the island O 1 and make many oscillations before reaching the chaotic sea. The curves U -, S - are further away from the island O 1 and escape faster. Nevertheless all curves return many times close to O 1 and contribute to the stickiness near this island. The overall stickiness effects of U +, U - are very similar and the stickiness effects along S +, S - are also very similar. However, the stickiness in the forward time direction, along U +, U -, is very different from the stickiness in the opposite time direction along S +, S -. We calculated the finite time LCN (Lyapunov characteristic number) χ( t ), which is initially smaller for U +, S + than for U -, S -. However, after a long time all the values of χ( t ) in the chaotic zone approach the same final value LCN = lim t → ∞ χ(t). The stretching number (LCN for one iteration only) varies along an asymptotic curve going through minima at the turning points of the asymptotic curve. We calculated the escape times (initial stickiness times) for many initial points outside but close to the island O 1. The lines that separate the regions of the fast from the slow escape time follow the shape of the asymptotic curves S +, S -. We explained this phenomenon by noting that lines close to S + on its inner side (closer to O 1) approach a point of the orbit 4/9, say P 1, and then follow the oscillations of the asymptotic curve U +, and escape after a rather long time, while the curves outside S + after their approach to P 1 follow the shape of the asymptotic curves U - and escape fast into the chaotic sea. All these curves return near the original arcs of U +, U - and contribute to the overall stickiness close to U +, U -. The isodensity curves follow the shape of the curves U +, U - and the maxima of density are along U +, U -. For a rather long time, the stickiness effects along U +, U - are very pronounced. However, after much longer times (about 1000 iterations) the overall stickiness effects are reduced and the distribution of points in the chaotic sea outside the islands tends to be uniform. The stickiness along the asymptotic curve U of the orbit O is very similar to the stickiness along the asymptotic curves U +, U - of the orbit 4/9. This is related to the fact that the asymptotic curves of O and 4/9 are connected by heteroclinic orbits. However, the main reason for this similarity is the fact that the asymptotic curves U , U +, U - cannot intersect but follow each other.

Asymptotic curves of submanifolds

1992 ◽  
Vol 52 (5) ◽  
pp. 1081-1087 ◽  
Author(s):  
Yu. A. Aminov
Keyword(s):  
Asymptotic Curves

scholarly journals Leucine and isoleucine requirements of the kitten

1984 ◽  
Vol 52 (3) ◽  
pp. 595-605 ◽  
Author(s):  
Diane M. Hargrove ◽  
Quinton R. Rogers ◽  
James G. Morris
Keyword(s):  
Plasma Concentrations ◽  
Latin Square ◽  
Nitrogen Retention ◽  
Separate Experiment ◽  
Maximal Response ◽  
Maximal Growth ◽  
Response Relationship ◽  
Asymptotic Curves ◽  
Dietary Requirements ◽  
Plasma Leucine

1. In separate experiments the isoleucine and leucine requirements of the kitten were determined on the basis of growth and nitrogen retention. The dietary concentrations of isoleucine tested were (g/kg diet) 1.4, 2.2, 3.0, 3.8, 4.6 and 9.0 with adequate (12.0 g/kg diet) leucine. The levels of leucine tested were (g/kg diet) 5.0, 7.5, 9.0, 10.5, 12.0 and 20.0 in diets containing adequate (9.0 g/kg diet) isoleucine. In both experiments six male and six female kittens received each dietary level of isoleucine or leucine for periods of 10 d in a balanced 6 x 6 Latin-square experimental design.2. Asymptotic curves were fitted to the response relationships and the minimal dietary requirements for maximal response were estimated from the values at 0.95 of the asymptote. On this basis, the requirements for maximal growth were 6.2 g isoleucine/kg and 7.8 g leucine/kg diet. The requirements for maximal N retention were higher; 8.4 g isoleucine and 10.6 g leucine/kg diet. The isoleucine requirements suggested by this method are probably overestimations and might be slightly above 4.6 g/kg diet.3. Plasma isoleucine and leucine concentrations were not useful in estimating the requirements. Plasma leucine increased rectilinearly with increasing dietary leucine while the response of plasma isoleucine to increasing dietary isoleucine was non-rectilinear. Neither response relationship exhibited a breakpoint at the level of requirement. Below the suggested minimal requirement for leucine there were significant increases in the concentrations of isoleucine and valine in the plasma. Dietary isoleucine below the level of requirement had no effect on plasma valine and leucine. Dietary leucine had no effect on the plasma concentrations of methionine, phenylalanine and threonine, suggesting that the effect of decreasing dietary leucine on plasma isoleucine and valine is a result of decreased oxidation rather than decreased protein anabolism.4. In a separate experiment six kittens, presented a diet containing 2.2 g isoleucine/kg, developed crusty exudates around their eyes within 27 d and six kittens, presented diets containing 3.8 g isoleucine/kg, showed this clinical sign but with less severity within 47 d. Cultures of conjunctival swabs taken from the most severely affected kittens showed the presence of staphylococcal species, suggesting that in isoleucine-deficient kittens there was impaired resistance to these dermal microbes.

Chords of Twisted Cubics

1923 ◽  
Vol s2-21 (1) ◽  
pp. 98-113 ◽  
Author(s):  
E. K. Wakeford
Keyword(s):  
Twisted Cubics

scholarly journals FAMILIES OF SURFACES IN ℝ4

2002 ◽  
Vol 45 (1) ◽  
pp. 181-203 ◽  
Author(s):  
J. W. Bruce ◽  
F. Tari
Keyword(s):  
Subject Classification ◽  
Secondary 53A05 ◽  
Asymptotic Curves ◽  
Primary 58C27

AbstractWe study the geometry of surfaces in $\mathbb{R}^4$ associated to contact with hyperplanes. We list all possible transitions that occur on the parabolic and so-called $A_3$-set, and analyse the configurations of the asymptotic curves and their bifurcations in generic one-parameter families.AMS 2000 Mathematics subject classification: Primary 58C27. Secondary 53A05

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