Hyperbolic Lines Research Articles

Lagrangian structures and stirring in the Earth’s mantle

In this paper we investigate three Lagrangian methods that have been recently proposed to quantify mixing in chaotic and time-aperiodic geophysical flows. The analytical method proposed by Haller [e.g., G. Haller, Chaos 10 (2000) 99–108] has a strong mathematical foundation and seeks to determine the location of stable (i.e., most attracting) and unstable (i.e., most repelling) material lines. The main hyperbolic lines describe the spatial organization of chaotic mixing. The finite-size Lyapunov exponents estimate the local mixing properties of the flow using the finite dispersion of particles, while the finite-time Lyapunov exponents have been often used to locate dynamically distinguished regions in geophysical flows. Mantle stirring is induced by the repeated action of stretching and folding, thus requiring to follow the strain history of a fluid element along a trajectory. We calculate the trajectories of more than half a million passive tracers forward and backward in time. The Eulerian velocity field is computed using a finite element code for solid state convection. We focus on a thermochemical model with a chemically denser layer at the base of the Earth’s mantle. This case allows us to test the ability of the Lagrangian techniques to detect the location of a dynamical barrier that inhibits mass exchanges and delimits domains characterized by different efficiency of stirring. We find that the Lagrangian techniques provide a satisfactory description of the main structures governing stirring, and enlighten different and complementary aspects: the methods based on the Lyapunov exponents provide a clear picture of mantle domains characterized by different strength of stirring, while the method proposed by Haller identifies the skeleton of the main structures around which stirring is organized. Our paper builds toward a more rigorous analysis of the stirring processes in the Earth’s mantle, which is required to understand the existence of geochemical reservoirs under a dynamical prospective.

Bailout embeddings and neutrally buoyant particles in three-dimensional flows.

We use the bailout embeddings of three-dimensional volume-preserving maps to study qualitatively the dynamics of small spherical neutrally buoyant impurities suspended in a time-periodic incompressible fluid flow. The accumulation of impurities in tubular vortical structures, the detachment of particles from fluid trajectories near hyperbolic invariant lines, and the formation of nontrivial three-dimensional structures in the distribution of particles are predicted.

Hyperbolic lines and the stratospheric polar vortex.

The necessary and sufficient conditions for Lagrangian hyperbolicity recently derived in the literature are reviewed in the light of older concepts of effective local rotation in strain coordinates. In particular, we introduce the simple interpretation of the necessary condition as a constraint on the local angular displacement in strain coordinates. These mathematically rigorous conditions are applied to the winter stratospheric circulation of the southern hemisphere, using analyzed wind data from the European Center for Medium-Range Weather Forecasts. Our results demonstrate that the sufficient condition is too strong and the necessary condition is too weak, so that both conditions fail to identify hyperbolic lines in the stratosphere. However a phenomenological, nonrigorous, criterion based on the necessary condition reveals the hyperbolic structure of the flow. Another (still nonrigorous) alternative is the finite-size Lyapunov exponent (FSLE) which is shown to produce good candidates for hyperbolic lines. In addition, we also tested the sufficient condition for Lagrangian ellipticity and found that it is too weak to detect elliptic coherent structures (ECS) in the stratosphere, of which the polar vortex is an obvious candidate. Yet, the FSLE method reveals a clear ECS-like barrier to mixing along the polar vortex edge. Further theoretical advancement is needed to explain the apparent success of nonrigorous methods, such as the FSLE approach, so as to achieve a sound kinematic understanding of chaotic mixing in the winter stratosphere and other geophysical flows. (c) 2002 American Institute of Physics.

Lagrangian structures and the rate of strain in a partition of two-dimensional turbulence

We derive analytic criteria for the existence of hyperbolic (attracting or repelling), elliptic, and parabolic material lines in two-dimensional turbulence. The criteria use a frame-independent Eulerian partition of the physical space that is based on the sign definiteness of the strain acceleration tensor over directions of zero strain. For Navier–Stokes flows, our hyperbolicity criterion can be reformulated in terms of strain, vorticity, pressure, viscous and body forces. The special material lines we identify allow us to locate different kinds of material structures that enhance or suppress finite-time turbulent mixing: stretching and folding lines, Lagrangian vortex cores, and shear jets. We illustrate the use of our criteria on simulations of two-dimensional barotropic turbulence.

Radial conformal motions in Minkowski space–time

A study of radial conformal Killing fields (RCKF) in Minkowski space–time is carried out, which leads to their classification into three disjointed classes. Their integral curves are straight or hyperbolic lines admitting orthogonal surfaces of constant curvature, whose sign is related to the causal character of the field. Otherwise, the kinematic properties of the timelike RCKF are given and their applications in kinematic cosmology is discussed.

Kennzeichnungen hyperbolischer Bewegungen durch Lineationen

LetIn be the set of points ofn-dimensional hyperbolic geometry,n≥2. THEOREM 1. Letf∶In→In be a surjection such that images of hyperbolic lines are contained in hyperbolic lines. Thenf is a hyperbolic motion. THEOREM 2. Iff∶In→In maps hyperbolic lines onto hyperbolic lines, then the image off is a hyperbolic line, orf is a hyperbolic motion. We note that Theorem 1 is a consequence of [3, Theorem 2.4].

Hyperbolic lines in generalized polygons

In this paper we develop the theory of hyperbolic lines in generalized polygons. In particular, we investigate the extremal situation where hyperbolic lines are long. We give restrictions on the existence of long hyperbolic lines and characterize the symplectic generalized quadrangles W (k) and generalized hexagons H(k) related to the groups G2(k) by the existence of certain classes of long hyperbolic lines.

Least squares range difference location

An array of n sensors at known locations receives the signal from an emitter whose location is desired. By measuring the time differences of arrival (TDOAs) between pairs of sensors, the range differences (RDs) are available and it becomes possible to compute the emitter location. Traditionally geometric solutions have been based on intersections of hyperbolic lines of position (LOPs). Each measured TDOA provides one hyperbolic LOP. In the absence of measurement noise, the RDs taken around any closed circuit of sensors add to zero. A bivector is introduced from exterior algebra such that when noise is present, the measured bivector of RDs is generally infeasible in that there does not correspond any actual emitter position exhibiting them. A circuital sum trivector is also introduced to represent the infeasibility; a null trivector implies a feasible RD bivector. A 2-step RD Emitter Location algorithm is proposed which exploits this implicit structure. Given the observed noisy RD bivector /spl Delta/, (1) calculate the nearest feasible RD bivector /spl Delta//spl circ/, and (2) calculate the nearest point to the (/sub 3//sup n/) planes of position, one for each of the triads of elements of /spl Delta//spl circ/. Both algorithmic steps are least squares (LS) and finite. Numerical comparisons in simulation show a substantial improvement in location error variances.

Symplectic geometries, transvection groups, and modules

We show that any connected partial linear space in which there is a line with at least four points and that has the property that any pair of intersecting lines is contained in a subspace isomorphic to a symplectic plane is isomorphic to the geometry of hyperbolic lines in some symplectic geometry. As a corollary to this result we obtain a characterization of the subgroups of the symplectic groups that are generated by transvection subgroups. Also a characterization of the natural modules for these groups is obtained.

�ber die Leitfl�chen konstanten Mittenabstandes von Geradenkongruenzen des einfach isotropen Raumes

In the simply isotropic space J3(1) let S be a congruence of straight lines with the unit vector giving the direction of its lines\(\vec e\)(u,v) and with middle surface\(\vec p\)(u,v). A surface of reference\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {p} _t (u,v): = \vec p(u,v) + t(u,v) \vec e(u,v)\) has constant distance from\(\vec p\)(u,v) if t= const. In this paper we first investigate the mapping, which is established by the lines of S between two arbitrary surfaces of the above type. Next we give a new characterization of the focal points of some hyperbolic lines of the congruence S. Finally, we study some associated ruled surfaces of S, which are connected to the surfaces\(\vec p_t\)(u,v), t ∈ ℝ.

Observation of loss boundaries and ion flux spectra of end-loss ions in the tandem mirror GAMMA 10

Loss boundaries and ion flux spectra of end-loss ions have been observed by means of a new diagnostic device in the tandem mirror GAMMA 10 [Nucl. Fusion Suppl. 1, 429 (1983)]. In consequence of measuring simultaneously both the pitch angle profile and energy of the end-loss ions, the loss boundaries were directly ascertained to be composed of two hyperbolic lines characterized by the electrostatic plug potential in the velocity space, and also high outer mirror throat potential was found. The flux spectra were compared with the two types of theoretical results, and discussed from a viewpoint of Coulomb collision.

Automatic Generation of Hyperbolic Tilings

than Euclid's other assumptions The author discusses the (such as the one that states fundamentals of plane hyperbolic that two points determine exgeometry and describes a method for generating hyperbolic tilings by actly one line), and for a long computer, based on the recently detime people tried to prove its veloped theory of automatic groups. redundancy within the Euclidean framework. However, by the 1820s at least three people had independently discovered that a selfconsistent geometry that does not satisfy the parallel axiom could be built: Janos Bolyai in Hungary, Carl Friedrich Gauss in Germany and Nikolai Ivanovich Lobachevskii in Russia [4]. Their creation became known as Lobachevskiian geometry until Felix Klein, at the turn of this century, introduced the now more common term hyperbolic geometry. (The term non-Euclidean geometry is also commonly used, but it encompasses geometries other than the hyperbolic.) The denial of one of Euclid's axioms remained a profoundly disturbing idea, and it was not until the second half of the century that the general mathematical public started to get a clear grasp of the nature of hyperbolic geometry. This understanding was much aided by the construction by Eugenio Beltrami [5], in 1866, of an explicit model of hyperbolic geometry-something like a map of hyperbolic space that can be drawn on a piece of Euclidean paper. This model, now known as the projective, or Klein model, after its popularizer, represents n-dimensional hyperbolic space by an open ball {(Xl.,'Xn)ERn X+ 2 <* I+<} in Euclidean space, and hyperbolic lines by Euclidean straight-line segments in the ball. Figure 1 shows that, given a line L and a point P outside of L, there are infinitely many Silvio Levy (mathematician, editor), Geometry Center, University of Minnesota, 1300 S. Second St., Minneapolis, MN 55454, U.S.A. Received 13 March 1991. LEONARDO, Vol. 25, No. 3, pp. 349-354, 1992 349 This content downloaded from 157.55.39.255 on Mon, 01 Aug 2016 05:43:04 UTC All use subject to http://about.jstor.org/terms Fig. 2. A tiling of the hyperbolic plane, shown in the Klein model. All the pentagons shown here are congruent and rightangled. The sum of the angles of a hyperbolic polygon is always less than what it would be for a Euclidean polygon of the same number of sides, and the difference is equal to the polygon's hyperbolic area. Fig. 3. The same tiling of Fig. 2, shown in the Poincare disk model. parallels to L through P, that is, lines that are coplanar with L but do not intersect it. (Sometimes only lines that meet at infinity, that is, on the circle that bounds the Klein model, are called parallel, while lines that meet outside infinity are called ultraparallel. I will not enforce this distinction.) Although the Klein model is projectively correct-that is, it represents straight lines by straight lines-it distorts shapes drastically. As one moves away from the center of the disk, an object of constant hyperbolic size quickly starts looking very small in the model and also gets flattened against the edge. Figure 2 shows a number of identical regular pentagons, which tile the hyperbolic plane much as squares tile the Euclidean plane. The tile at the center appears much bigger than its closest neighbors, and just a few layers out the triangles are already too small to make out. The figure also makes clear how the Klein model Fig. 4. Circles centered at Pare mapped to Fig. 5. The locus of themselves by rotations around P. Notice from a fixed hyperb that a hyperbolic circle in the Poincare appears as an arc of model appears round, although other model, but it does ni shapes are preserved only approximately. orthogonally, therefo poi1 olic

Damage location system for a tanker ship

Hydrophones are installed in each tank of a liquid cargo tanker ship. In the unfortunate event that the tanker collides with an obstacle in the water, the hydrophones detect the shock wave due to the collision. The relative first arrival times of the shock wave at the respective hydrophones are measured and the arrival-time differences between two hydrophones of any two different selected pairs of hydrophones is calculated. Hyperbolic lines of constant time difference are projected on a map of the locations of the hydrophones on the ship. The intersection of the lines of position representing the calculated first arrival time differences marks the location of the impact of the ship with the obstacle.

The dual of Pasch's axiom

We consider partial linear spaces that satisfy the dual of Pasch's axiom. We give a uniform proof of some old and new characterizations of partial linear spaces and graphs related to projective spaces and the hyperbolic lines of symplectic spaces. Furthermore, we use these results to classify a class of groups that are generated by a conjugacy class of subgroups of order 3.

The Cauchy Integral Formula

Our purpose in this note is to illustrate the Cauchy integral formula and the concept of winding number. The software we use is fiz), ver. 4.0, Martin Lapidus, Lascaux Graphics, Bronx, NY, 1988. Many mathematicians and most students are at a loss to picture the graph of even the most elementary complex function. This tutorial and graphics program enhances geometric intuition about functions of a complex variable by graphing lines, circles, or other figures in a z plane and simultaneously graphing their images in a w =/(z) plane. Other domain options include hyperbolic lines on the Poincare plane and circles on the Riemann sphere. You can construct and save a page with two or more windows, sized, scaled, and positioned as you wish, and with domain options, functions, and operations specified. Once you have con? structed a page, you can request that the operations be carried out at the press of a draw key. The Cauchy integral formula states that

Hyperbolic reflections on pell's equation

Let p be a prime э 1 mod(4). This paper shows that arithmetic properties of the fundamental solution ( x 0, y 0) of the Pell equation x 2 − py 2 = −1 are given by geometric properties of the surface H Γ(p) ⌢ Γ 0(p 2) and hyperbolic reflection lines thereon. In particular the length of one such line determines the magnitude of y 0 and the center of the surface either has a cylinder running around it or is vacant, being surrounded by such cylinders, according as p ∤ y 0 or p | y 0.

Impedance transformation using a hyperbolic cosine-squared tapered transmission line

Abstract A closed-form equation for the impedance transformation of an arbitrarily complex load, using a hyperbolic cosine-squared tapered transmission line, is presented. Design charts are also given. A numerical example is considered and results indicate a tremendous reduction in size, in comparison to other forms of lines, when hyperbolic cosine-squared lines are employed to match a large complex impedance to a real 50 Ω impedance.

The hyperbolic lines of finite symplectic spaces

1. INTRODUCTION The classical theorem of Veblen and Young [14] characterizes the Desarguesian projective spaces of dimension at least three as those connec- ted partial linear spaces in which each pair of intersecting lines lies in a proper subspace which is a projective plane. In this article we present a similar result.

Hyperbolic Grids and TM Coordinates

Procedures for converting hyperbolic to TM coordinates and for preparing charts with hyperbolic lattices are discussed. The presented transformation is simple, it does not require any significant computational effort and can easily be executed on a microcomputer in real time. A computed example illustrates the discussion. A method of charting hyperbolic lines of position is also shown.

The conformal geometry of Minkowski space: Hyperbolic world lines

There is a discussion of (i) the transformation properties of hyperbolic hypersurfaces in Minkowski 4-space, (ii) the association of hyperbolic world lines in Minkowski 2-space with uniformly accelerated motion.