Asymptotic curves of submanifolds

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

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.

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.

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 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.

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

scholarly journals Generalized normal ruled surface of a curve in the Euclidean 3-space

2021 ◽  
Vol 13 (1) ◽  
pp. 217-238
Author(s):  
Onur Kaya ◽  
Mehmet Önder
Keyword(s):  
Ruled Surface ◽  
Ruled Surfaces ◽  
Asymptotic Curves ◽  
Lines Of Curvature

Abstract In this study, we define the generalized normal ruled surface of a curve in the Euclidean 3-space E3. We study the geometry of such surfaces by calculating the Gaussian and mean curvatures to determine when the surface is flat or minimal (equivalently, helicoid). We examine the conditions for the curves lying on this surface to be asymptotic curves, geodesics or lines of curvature. Finally, we obtain the Frenet vectors of generalized normal ruled surface and get some relations with helices and slant ruled surfaces and we give some examples for the obtained results.

Isophote curves on spacelike surfaces in Lorentz–Minkowski space

2021 ◽  
pp. 2150180
Author(s):  
Fatih Doğan ◽  
Yusuf Yaylı
Keyword(s):  
Minkowski Space ◽  
Spacelike Surface ◽  
Constant Angle ◽  
Asymptotic Curves ◽  
Lines Of Curvature ◽  
Darboux Frame ◽  
Curves On Surfaces ◽  
Normal Vectors ◽  
Spacelike Surfaces

An isophote curve consists of a locus of surface points whose normal vectors make a constant angle with a fixed vector (the axis). In this paper, we define an isophote curve on a spacelike surface in Lorentz–Minkowski space [Formula: see text] and then find its axis as timelike and spacelike vectors via the Darboux frame. Besides, we give some relations between isophote curves and special curves on surfaces such as geodesic curves, asymptotic curves or lines of curvature.

Cantori, Islands and Asymptotic Curves in the Stickiness Region

1999 ◽  
pp. 221-230 ◽  
Author(s):  
C. Efthymiopoulos ◽  
G. Contopoulos ◽  
N. Voglis
Keyword(s):  
Asymptotic Curves

The effect of herbage allowance on herbage intake and performance of lambs grazing perennial ryegrass and red clover swards

1976 ◽  
Vol 86 (2) ◽  
pp. 355-365 ◽  
Author(s):  
M. J. Gibb ◽  
T. T. Treacher
Keyword(s):  
Growth Rate ◽  
Organic Matter ◽  
Perennial Ryegrass ◽  
Nitrogen Fertilizer ◽  
Daily Intake ◽  
Live Weight ◽  
Red Clover ◽  
Asymptotic Curves ◽  
Digestible Organic Matter ◽  
The Relationship

SummaryThe effect of daily herbage allowance on herbage intakes and growth rates lambs grazing perennial ryegrass and red clover was investigated in two experiments. Herbage allowances defined as g herbage D.M./kg live weight (LW)/day were controlled by varying the areas of plots grazed for 2 days by groups of six lambs.In the first experiment five herbage allowances in the range 20–120 g D.M./kg LW/day were offered on two areas of a perennial ryegrass (cv. S. 23) sward that received nitrogen fertilizer applications of 39 or 78 kg N/ha/28 days. In the second experiment five herbage allowances in the range 30–160 g D.M./kg LW/day were offered on perennial ryegrass (cv. S. 23) and red clover (cv. Hungaropoly) swards.Asymptotic curves were fitted to describe the relationship between herbage allowance and daily intake of herbage. In Expt 1 nitrogen fertilizer rates did not affect the yield of herbage or animal performance. In Expt 2 intakes were higher on the clover sward than on the ryegrass sward at the higher herbage allowances.The asymptotic curves to describe the relationship between herbage allowance and growth rate of lambs differed widely between periods. Growth rate of the lambs increased linearly with increase in digestible organic matter intake. Live-weight gain per unit of digestible organic matter intake was higher on the red clover than on the ryegrass.The conclusion is drawn that if the herbage present to ground level is not more than three times the daily intake of the animals, intake of herbage of the animals may bo restricted.

scholarly journals INSTABILITIES AND STICKINESS IN A 3D ROTATING GALACTIC POTENTIAL

2013 ◽  
Vol 23 (02) ◽  
pp. 1330005 ◽  
Author(s):  
M. KATSANIKAS ◽  
P. A. PATSIS ◽  
G. CONTOPOULOS
Keyword(s):  
Periodic Orbits ◽  
Invariant Manifolds ◽  
Rotational Axis ◽  
Unstable Periodic Orbits ◽  
Chaotic Orbits ◽  
Asymptotic Curves ◽  
Axis Orbit ◽  
Diffusion Speed ◽  
Transition Points ◽  
Sticky Chaotic Orbits

We study the dynamics in the neighborhood of simple and double unstable periodic orbits in a rotating 3D autonomous Hamiltonian system of galactic type. In order to visualize the four-dimensional spaces of section, we use the method of color and rotation. We investigate the structure of the invariant manifolds that we found in the neighborhood of simple and double unstable periodic orbits in 4D spaces of section. We consider orbits in the neighborhood of the families x1v2, belonging to the x1 tree, and the z-axis (the rotational axis of our system). Close to the transition points from stability to simple instability, in the neighborhood of the bifurcated simple unstable x1v2 periodic orbits, we encounter the phenomenon of stickiness as the asymptotic curves of the unstable manifold surround regions of the phase space occupied by rotational tori existing in the region. For larger energies, away from the bifurcating point, the consequents of the chaotic orbits form clouds of points with mixing of color in their 4D representations. In the case of double instability, close to x1v2 orbits, we find clouds of points in the four-dimensional spaces of section. However, in some cases of double unstable periodic orbits belonging to the z-axis family we can visualize the associated unstable eigensurface. Chaotic orbits close to the periodic orbit remain sticky to this surface for long times (of the order of a Hubble time or more). Among the orbits we studied, we found those close to the double unstable orbits of the x1v2 family having the largest diffusion speed. The sticky chaotic orbits close to the bifurcation point of the simple unstable x1v2 orbit and close to the double unstable z-axis orbit that we have examined, have comparable diffusion speeds. These speeds are much slower than of the orbits in the neighborhood of x1v2 simple unstable periodic orbits away from the bifurcating point, or of the double unstable orbits of the same family having very different eigenvalues along the corresponding unstable eigendirections.

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