Hyperbolic Lines Research Articles

The solar type II radio bursts of 7 March 2012: Detailed simulation analyses

AbstractType II solar radio bursts are often indicators for impending space weather events at Earth. They are consequences of shock waves driven by coronal mass ejections (CMEs) that move outward from the Sun. We simulate such type II radio bursts by combining elaborate three‐dimensional (3‐D) magnetohydrodynamic (MHD) predictions of realistic CMEs near the Sun with an analytic kinetic radiation theory developed recently. The simulation approach includes the reconstruction of initial solar magnetic fields, the dimensioning of the initial flux rope of the CME with STEREO spacecraft data, and the launch of the CME into an empirical data‐driven corona and solar wind. In this paper, we simulate a complicated double CME event (a very fast CME followed by a slower CME without interaction) and the related coronal and interplanetary type II radio bursts that occurred on 7 March 2012. We extend our previous work to show harmonic and interplanetary emission as well as the simulation's surprising ability (for these events at least) for predicting emission for two closely spaced CMEs leaving the same active region. We demonstrate that the theory predicts well the observed fundamental and harmonic emission from ∼20 MHz to 50 kHz or from the high corona to near 1 AU. Specifically, the theory predicts flux, frequency, and time variations that are consistent with the presence or absence of observed type II emissions when interfering emissions are absent and are not inconsistent with observations when interfering type III bursts are present. The predicted and observed type II emission is predominantly fundamental for these two events. Harmonic emission occurs for the second CME only for a short time interval, when an extended shock has developed that can drive flank emission. The coronal and interplanetary emission follow closely hyperbolic lines in frequency‐time space, consisting of a succession of islands of emission with varying intensity. The islands develop due to competition between the shock moving through varying coronal and solar wind magnetic field structures (e.g., loops and streamers), growth of the driven radio source due to the spherical expansion of the shock, and movement of the active radio sources from the shock's nose to its flanks.

Fan triangulations of a hyperbolic plane of positive curvature

We study the families (ℱλ) of normal partitions of a 3-(1)-contour F of a hyperbolic plane \(\hat H\) of positive curvature into simple 4-contours whose hyperbolic diagonal lines are parallel to the base of F. A 3-(1)-contour with a given partition from a family (ℱλ) (or some its normal subpartition) is called a fan. We construct fan partitions Pe, Ph, and Pp of \(\hat H\) whose symmetry groups are generated by a shift along an elliptic (respectively, hyperbolic and parabolic) straight line. It is proved that the partitions Ph and Pp are normal. The partitions Ph and Pp whose cells are trihedrals present examples of the first triangulations of \(\hat H\).

Periodic perturbation of quadratic systems with two infinite heteroclinic cycles

We study periodic perturbations of planar quadratic vector fieldshaving infinite heteroclinic cycles, consisting of an invariantstraight line joining two saddle points at infinity and an arc oforbit also at infinity. The global study concerning the infinity ofthe perturbed system is performed by means of the Poincarécompactification in polar coordinates, from which we obtain a systemdefined on a set equivalent to a solid torus in $\mathbb{R}^3$, whoseboundary plays the role of the infinity. It is shown that forcertain type of periodic perturbation, there exist twodifferentiable curves in the parameter space for which the perturbedsystem presents heteroclinic tangencies and transversalintersections between the stable and unstable manifolds of twonormally hyperbolic lines of singularities at infinity. Thetransversality of the manifolds is proved using the Melnikov methodand implies, via the Birkhoff-Smale Theorem, in a complex dynamicalbehavior of the perturbed system solutions in a finite part of thephase space. Numerical simulations are performed for a particularexample in order to illustrate this behavior, which could be called'the chaos arising from infinity', because it depends on theglobal structure of the quadratic system, including the points atinfinity.

Fissioned triangular schemes via sharply 3-transitive groups

In [D. de Caen, E.R. van Dam, Fissioned triangular schemes via the cross-ratio, European J. Combin. 22 (2001) 297–301], de Caen and van Dam constructed a fission scheme FT(q+1) of the triangular scheme on PG(1,q). This fission scheme comes from the naturally induced action of PGL(2,q) on the 2-element subsets of PG(1,q). The group PGL(2,q) is one of two infinite families of finite sharply 3-transitive groups. The other such family M(q) is a “twisted” version of PGL(2,q), where q is an even power of an odd prime. The group PSL(2,q) is the intersection of PGL(2,q) and Mq(q). In this paper, we investigate the association schemes coming from the actions of PSL(2,q), Mq(q) and PML(2,q), respectively. Through the conic model introduced in [H.D.L. Hollmann, Q. Xiang, Association schemes from the actions of PGL(2,q) fixing a nonsingular conic, J. Algebraic Combin. 24 (2006) 157–193], we introduce an embedding of PML(2,q) into PML(3,q). For each of the three groups mentioned above, this embedding produces two more isomorphic association schemes: one on hyperbolic lines and the other on hyperbolic points (via an orthogonal parity) in a 3-dimensional orthogonal geometry. This embedding enables us to treat these three isomorphic association schemes simultaneously.

Barriers in the transition to global chaos in collisionless magnetic reconnection. II. Field line spectroscopy

The transitional phase from local to global chaos in the magnetic field of a reconnecting current layer is investigated. The identification of the ridges in the field of the finite time Lyapunov exponent as barriers to the field line motion is carried out adopting the technique of field line spectroscopy to analyze the radial position of a field line while it winds its way through partial stochastic layers and to compare the frequencies of the field line motion with the corresponding frequencies of the distinguished hyperbolic field lines that are the nonlinear generalizations of linear X-lines.

On incidence structures of nonsingular points and hyperbolic lines of ovoids in finite orthogonal spaces

We study the point-line incidence structures of nonsingular points and hyperbolic secant lines associated with ovoids in finite orthogonal spaces. We show that these incidence structures frequently produce partial linear spaces and the parameters of the bipartite graphs (called ovoidal graphs) associated with these structures produce simple and effective isomorphism invariants to distinguish non-isomorphic ovoids. We prove explicit formulas for these isomorphism invariants for a number of infinite families of 2-transitive ovoids.

On the hyperbolic unitary geometry

Hans Cuypers (Preprint) describes a characterisation of the geometry on singular points and hyperbolic lines of a finite unitary space--the hyperbolic unitary geometry--using information about the planes. In the present article we describe an alternative local characterisation based on Cuypers' work and on a local recognition of the graph of hyperbolic lines with perpendicularity as adjacency. This paper can be viewed as the unitary analogue of the second author's article (J. Comb. Theory Ser. A 105:97---110, 2004) on the hyperbolic symplectic geometry.

Hyperplanes and Motions of Dimension-Free Hyperbolic Geometry

In Section 1 we characterize hyperbolic hyperplanes of dimension–free hyperbolic geometry over the real inner product space X of arbitrary (finite or infinite) dimension greater than 1 by euclidean hyperplanes ∋ 0 of \(X {\otimes} \mathbb{R}\) intersecting the surface \(\{(x,\sqrt{1 + x^2}) \mid x \in X\}\). This is for \(X = {\mathbb{R}}^2\) a well-known result of the classical theory, since the Weierstras model of plane hyperbolic geometry defines hyperbolic lines via the euclidean planes of \({\mathbb{R}}^3\) through 0 intersecting the surface in question. – In Section 2 formulas will be derived representing dimension–free hyperbolic motions of X as well as products consisting of two such factors. – Finally, in Section 3, coordinates will be considered describing naturally the action of hyperbolic translations on points.

Spatial structure of internal tidal waves generated on weak topographies

The generation of internal gravity waves by barotropic tidal flow passing over a two-dimensional topography is investigated. Rather than calculating the conversion of tidal energy, this study focuses on delineating the geometric characteristics of the spatial structure of the resulting internal wave fields (i.e., the configurations of the internal beams and their horizontal projections) which have usually been ignored. It is found that the various possible wave types can be demarcated by three characteristic frequencies: the tidal frequency, ω 0; the buoyancy frequency, N; and the vertical component of the Coriolis vector or earth's rotation, f. When different possibilities arising from the sequence of these frequencies are considered, there occur 12 kinds of wave structures in the full 3D space in contrast to the 5 kinds identified by the 2D theory. The constant wave phase lines may form as ellipses or hyperbolic lines on the horizontal plane, provided the buoyancy frequency is greater or less than the tidal frequency. The effect that stems from the consideration of the basic flow is also found, which not only serves as the reason for the occurrence of higher harmonics but also increases the wave strength in the direction of basic flow.

Commuting polarities and maximal partial ovoids of H(4,q2)

AbstractIn PG(4,q2), q odd, let Q(4,q2) be a non‐singular quadric commuting with a non‐singular Hermitian variety H(4,q2). Then these varieties intersect in the set of points covered by the extended generators of a non‐singular quadric Q0 in a Baer subgeometry Σ0 of PG(4,q2). It is proved that any maximal partial ovoid of H(4,q2) intersecting Q0 in an ovoid has size at least 2(q2+1). Further, given an ovoid O of Q0, we construct maximal partial ovoids of H(4,q2) of size q3+1 whose set of points lies on the hyperbolic lines 〈P,X〉 where P is a fixed point of O and X varies in O\{P}. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 307–313, 2009

Two new classes of hyperplanes of the dual polar space [formula omitted] not arising from the Grassmann embedding

Let Γn(q) denote the geometry of the hyperbolic lines of the symplectic polar space W(2n-1,q),n⩾2. We show that every hyperplane of Γn(q) gives rise to a hyperplane of the Hermitian dual polar space DH(2n-1,q2). In this way we obtain two new classes of hyperplanes of DH(2n-1,4) which do not arise from the Grassmann embedding of DH(2n-1,4).

Karhunen–Loève spectral analysis in multiresolution decomposition

In this work we develop a method for pinpointing the scales in a multiresolution decomposition where coherent structures appear. A sequence of images yielded from the wavelet multiresolution analysis of seismic signals are analyzed in the Karhunen–Loeve (KL) space. Using KL decomposition we distinguish two coherent structures in the scale–images: the ground roll and the hyperbolic lines characteristic of geologic layers. Moreover, the KL spectrum also points out the high frequency noise. In this way the method allow us to split three energy mode regimes in the data and, as a consequence, to select the relevant geologic information. The method can be extended to other problems where coherent structures need to be recognized.

Temperature field and heat flow around an elliptical lava tube

We study the temperature distribution around a cylindrical lava tube with an elliptical cross section. The steady-state heat equation is solved by assuming an unbounded medium and a uniform temperature of the tube wall. An analytical solution for the temperature field is obtained by the conformal mapping technique. It is found that the isothermal lines on the planes perpendicular to the tube are confocal ellipses. The heat flow in the medium surrounding the tube has hyperbolic field lines, while the curves on which the magnitude of heat flow is constant are Cassini ovals. At the tube wall, the heat flow density is proportional to the cubic root of the curvature of the wall. A solution is also obtained for a tube embedded in a half-space, when the size of the tube is small with respect to its depth. Formulae are given relating the heat flow at the surface of the half-space to the eccentricity and to the orientation of the major axis of the tube cross section, as well as to the temperature and the depth of the tube. For eccentricity equal to zero, all the formulae are reduced to those for a circular tube.

Observation of singularities in multiply scattered microwave fields

Speckle patterns of arbitrary resolution are obtained by applying the sampling theorem to measurements of two orthogonal components of the microwave field transmitted through multiply scattering samples. Core structures of phase singularities, phase critical points, and polarization singularities are explored. We find that equiphase lines connect phase singularities with opposite topological signs except for the bifurcation lines, which run through a phase saddle point, in agreement with predictions by Freund [Phys. Rev. E25, 2348 (1995)]. We observe hyperbolic equiphase lines near phase saddle points and elliptical equiphase lines around phase extrema. Polarization singularities of the vector field with the three morphologies predicted are observed.

Hyperbolic Bending of Vortex Lines with Finite Number andLength in Rotating Trapped Bose–Einstein Condensates

The minimal energy configurations of hyperbolic bending vortex lines inthe rotating trapped Bose–Einstein condensates are investigated byusing a variational ansatz and numerical simulation. The theoreticalcalculation of the energy of the vortex lines as a function of therotation frequency gives self-consistently vortex number, curvature andconfiguration. The numerical results show that bending is more stablethan straight vortex line along the z-axis, and the vortexconfiguration in the xy-plane has a little expansion by increasing z.

Association schemes from the action of PGL(2, q) fixing a nonsingular conic in PG(2, q)

The group PGL(2,q) has an embedding into PGL(3,q) such that it acts as the group fixing a nonsingular conic in PG(2,q). This action affords a coherent configuration $${\cal R}$$ (q) on the set $${\cal L}$$ (q) of non-tangent lines of the conic. We show that the relations can be described by using the cross-ratio. Our results imply that the restrictions $${\cal R}$$ +(q) and $${\cal R}$$ ?(q) of $${\cal R}$$ (q) to the set $${\cal L}$$ +(q) of secant (hyperbolic) lines and to the set $${\cal L}$$ ?(q) of exterior (elliptic) lines, respectively, are both association schemes; moreover, we show that the elliptic scheme $${\cal R}$$ ?(q) is pseudocyclic. We further show that the coherent configurations $${\cal R}$$ (q 2) with q even allow certain fusions. These provide a 4-class fusion of the hyperbolic scheme $${\cal R}$$ +(q 2), and 3-class fusions and 2-class fusions (strongly regular graphs) of both schemes $${\cal R}$$ +(q 2) and $${\cal R}$$ ?(q 2). The fusion results for the hyperbolic case are known, but our approach here as well as our results in the elliptic case are new.

Commutative [formula omitted]-algebras of Toeplitz operators and quantization on the unit disk

A family of recently discovered commutative C ∗ -algebras of Toeplitz operators on the unit disk can be classified as follows. Each pencil of hyperbolic straight lines determines a set of symbols consisting of functions which are constant on the corresponding cycles, the orthogonal trajectories to lines forming a pencil. The C ∗ -algebra generated by Toeplitz operators with such symbols turns out to be commutative. We show that these cases are the only possible ones which generate the commutative C ∗ -algebras of Toeplitz operators on each weighted Bergman space.

Relationship between albumin excretion rate and aortic stiffness in untreated essential hypertensive patients.

To evaluate, in a group of nondiabetic essential hypertensive patients with normal renal function, the relationship between albumin excretion rate (AER) and carotid-femoral pulse wave velocity (PWV), as an index of aortic stiffness. Cross-sectional study. Outpatient hypertension clinic. Seventy patients with mild-to-moderate essential hypertension, aged 42 +/- 8 years, never pharmacologically treated. All subjects underwent routine laboratory tests, 24-h ambulatory blood pressure (BP) monitoring, measurement of carotid-femoral PWV, by means of a computerized method, and AER. Microalbuminuric patients (AER > or = 20 microg min(-1); n = 19), when compared with normoalbuminuric subjects, showed more elevated 24-h BP (136/88 +/- 10/10 vs. 128/83 +/- 7/6 mmHg; P < 0.001 and P = 0.013, for systolic and diastolic BP respectively) and higher values of carotid-femoral PWV (10.4 +/- 2 m s(-1) vs. 9.2 +/- 1.3; P = 0.006). This latter difference remained statistically significant, even after correction by ancova for 24-h systolic and diastolic BP, and body mass index (BMI, P = 0.016). Univariate regression analysis disclosed a tight correlation between AER and carotid-femoral PWV (r = 0.42; P = 0.0003). This association was confirmed in a multiple regression model (beta = 0.35; P = 0.009) in which, as independent variables, besides PWV, 24-h BP, age, serum glucose values, smoking status, gender and BMI, were added. Our results seem to confirm that microalbuminuria may represent the early renal manifestation of a widespread vascular dysfunction, and therefore it is an integrated marker of cardiovascular risk.

On the hyperbolic symplectic geometry

The present article provides a new characterization of the geometry on the points and hyperbolic lines of a non-degenerate symplectic polar space. This characterization is accomplished by studying the family of subspaces obtained when considering the polars of all hyperbolic lines.

Hyperbolic reflections as fundamental building blocks for multilayer optics.

We re-elaborate on the basic properties of lossless multilayers by using bilinear transformations. We study some interesting properties of the multilayer transfer function in the unit disk, showing that hyperbolic geometry turns out to be an essential tool for understanding multilayer action. We use a simple trace criterion to separate multilayers into three classes that represent rotations, translations, or parallel displacements. Moreover, we show that these three actions can be decomposed as a product of two reflections in hyperbolic lines. Therefore, we conclude that hyperbolic reflections can be considered as the basic pieces for a deeper understanding of multilayer optics.