Hyperbolic Case Research Articles

No evidence for interstellar fireballs in the CNEOS database

Context. The detection of interstellar meteors, especially meteorite-dropping meteoroids, would be transformative, as this would enable direct sampling of material from other stellar systems on Earth. One candidate is the fireball observed by U.S. government sensors on January 8, 2014. It has been claimed that fragments of this meteoroid have been recovered from the ocean floor near Papua New Guinea and that they support an extrasolar origin. Based on its parameters reported in the Center for Near Earth Object Studies (CNEOS) catalog, the fireball exhibits a hyperbolic excess velocity that indicates an interstellar origin; however, the catalog does not report parameter uncertainties. Aims. To achieve a clear confirmation of the fireball’s interstellar origin, we assessed the underlying error distributions of the catalog data. Our aim was also to confirm whether the fragments of this meteoroid survived passage through the atmosphere and assess all conditions needed to unambiguously determine the fragments’ origin. Methods. We approached the investigation of the entire catalog using statistical analyses and modeling, and we provide a comprehensive analysis of the individual hyperbolic CNEOS cases. Results. We have developed several independent arguments indicating substantial uncertainties in the velocity and radiant position of the CNEOS events. We determined that all the hyperbolic fireballs exhibit significant deviations from the majority of the events in one of their velocity components, and we show that such mismeasurements can produce spurious parameters. According to our estimation of the speed measurement uncertainty for the catalog, we found that it is highly probable that such a catalog containing only Sun-bound meteors would show at least one event that appears highly unlikely to be Sun-bound. We also establish that it is unlikely that any fragments from a fireball traveling at the high inferred velocities could survive passage through the atmosphere. When assuming a much lower velocity, some fragments of this meteoroid could survive; however, they would be of a common Solar System origin and thus highly probable to be indistinguishable from the quantity of other local micrometeorites that have gradually accumulated on the sea floor. Conclusions. We conclude that there is no evidence in the CNEOS data to confirm or reject the interstellar origin of any of the nominally hyperbolic fireballs in the CNEOS catalog. Therefore, the claim of an interstellar origin for the fireball recorded over Papua New Guinea in 2014 remains unsubstantiated. We have also gathered arguments that refute the claim that the collected spherules from the sea floor originated in the body of this fireball.

On the Hardy number of Koenigs domains

This work studies the Hardy number of hyperbolic planar domains satisfying Abel’s inclusion property, which are usually known as Koenigs domains. More explicitly, we prove that the Hardy number of a Koenings domains whose complement is non-polar is greater than or equal to 1/2, and this lower bound is sharp. In contrast to this result, we provide examples of general domains whose Hardy numbers are arbitrarily small. Additionally, we outline the connection of the aforementioned class of domains with the discrete dynamics of the unit disc and obtain results on the range of Hardy number of Koenigs maps, in the hyperbolic and parabolic case.

The bicomplex tensor product and a bicomplex Choi theorem

In this paper we extend the concept of tensor product to the bicomplex case and use it to prove the bicomplex counterpart of the classical Choi theorem in the theory of complex matrices and operators. The concept of hyperbolic tensor product is also discussed, and we link these results to the theory of quantum channels in the bicomplex and hyperbolic case.

Entanglement structures from modified IR geometry

We investigate a new proposal connecting the geometry at various radial scales in asymptotic AdS spacetime with entanglement structure at corresponding real-space length scales of the boundary theory. With this proposal, the bulk IR geometry encodes the long-scale entanglement structure of the dual quantum system. We consider two distinct types of IR geometries, namely the spherical case and the hyperbolic case, which are intimately related to the physics of differential entropy and brane-world holography separately. We explore the corresponding change in the dual long-scale entanglement structures, utilizing the tools of the Ryu-Takayanagi formula, conditional mutual information, and partial entanglement entropy. The results indicate that modifying the IR geometry leads to a redistribution of entanglement at scales longer than a critical length determined by the location of the IR region, with the two modified IR geometries corresponding to two opposite ways of redistribution. Furthermore, we establish the maximum amount of entanglement that can be modified, which is proportional to the area of the IR region.

Outer billiards in the spaces of oriented geodesics of the three‐dimensional space forms

AbstractLet be the three‐dimensional space form of constant curvature , that is, Euclidean space , the sphere , or hyperbolic space . Let be a smooth, closed, strictly convex surface in . We define an outer billiard map on the four‐dimensional space of oriented complete geodesics of , for which the billiard table is the subset of consisting of all oriented geodesics not intersecting . We show that is a diffeomorphism when is quadratically convex. For , has a Kähler structure associated with the Killing form of . We prove that is a symplectomorphism with respect to its fundamental form and that can be obtained as an analogue to the construction of Tabachnikov of the outer billiard in defined in terms of the standard symplectic structure. We show that does not preserve the fundamental symplectic form on associated with the cross product on , for . We initiate the dynamical study of this outer billiard in the hyperbolic case by introducing and discussing a notion of holonomy for periodic points.

Hyperbolic radial spanning tree

We define and analyze an extension to the d-dimensional hyperbolic space of the Radial Spanning Tree (RST) introduced by Baccelli and Bordenave in the two-dimensional Euclidean space (2007). In particular, we will focus on the description of the infinite branches of the tree. The properties of the two-dimensional Euclidean RST are extended to the hyperbolic case in every dimension: almost surely, every infinite branch admits an asymptotic direction and each asymptotic direction is reached by at least one infinite branch. Moreover, the branch converging to any deterministic asymptotic direction is unique almost surely. To obtain results for any dimension, a completely new approach is considered here. Our strategy mainly involves the two following ingredients, that rely on the hyperbolic Directed Spanning Forest (DSF) introduced and studied in Flammant (2019). First, the hyperbolic metric allows us to obtain fine control of the branches’ fluctuations in the hyperbolic DSF without using planarity arguments. Then, we couple the hyperbolic RST with the hyperbolic DSF and conclude thanks to the precise estimates mentioned before.

On Euler number of symplectic hyperbolic manifold

In this article, we introduce a class of closed 2n-dimensional almost Kähler manifold X which called the special symplectic hyperbolic manifold. Those manifolds include Kähler hyperbolic manifolds. We study the spaces of L2-harmonic forms on the universal covering space of X. We then prove the Singer conjecture on special symplectic hyperbolic case. As an application, we can show that the Euler number of a special symplectic manifold satisfies the inequality (−1)nχ(X)>0.

A fundamental sequences method with time-reduction for one-dimensional lateral Cauchy problems

A fundamental sequences method is derived for the numerical solution of an ill-posed one-dimensional lateral Cauchy problem for a hyperbolic damped wave equation, including as a special case the parabolic heat equation. Either the Laguerre transform or the Houbolt finite difference scheme is applied to reduce the time-dependent lateral Cauchy problem to a sequence of second-order ordinary differential equations (ODEs) with function values and derivatives specified at the right endpoint of a finite space interval. A set of fundamental solutions is constructed, termed a fundamental sequence, to the differential equations. The solution of the obtained ODEs is approximated by a linear combination of elements in the fundamental sequence. Source points are placed outside of the solution interval in space, and by collocating at the endpoints of this interval a sequence of linear equations is obtained for finding the unknown coeficients. Tikhonov regularization is used to render a stable solution to the obtained systems of linear equations. Numerical results both for the parabolic and hyperbolic case confirm the efficiency of the proposed method including noisy data. The presented results complement the higher-dimensional case initiated in our previous researches.

Lyapunov Functions for Linear Hyperbolic Systems

We develop new methods to construct a Lyapunov function for 1-D linear hyperbolic equations with variable coefficients. The main focus is on the nonstrictly hyperbolic case for which we give an example demonstrating that existing approaches cannot provide sufficient conditions for the asymptotic stability, but our approach does. Sufficient conditions for exponential <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L^{2}$</tex-math></inline-formula> -stability for a connected <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$2 \times 2$</tex-math></inline-formula> system of linear 1-D hyperbolic systems are obtained. By means of examples we compare the capabilities of our approach with the existing ones.

Comparative analysis to study the Darcy–Forchheimer Tangent hyperbolic flow towards cylindrical surface using artificial neural network: An application to Parabolic Trough Solar Collector

Solar thermal collectors convert sunlight into useful thermal energy by absorbing its incoming radiation. Concentrated solar power technologies use the parabolic trough solar collector to collect solar energy with temperatures ranging from 325–700 K. The tangent hyperbolic fluid model is one of the most important non-Newtonian fluid models. Laboratory studies demonstrate that this model accurately predicts the shear thinning phenomenon. In addition, tangent hyperbolic fluid has a better heat transfer performance due to its rheological bearing at various shear rates. The current study investigates the heat transmission performance of Darcy–Forchheimer tangent hyperbolic radiative inclined cylindrical film movement in parabolic trough solar collector with an irregular heat sink/source utilizing the Levenberg–Marquardt technique and backpropagated neural networks. Through the implementation of required transformations, this system is turned into an equivalent nonlinear ordinary differential system. The findings are investigated for Newtonian and tangent hyperbolic fluid cases to understand the rheological characteristics. The outcomes are considered using graphical and mathematical evaluations. Fluids featuring tangent hyperbolic rheological conductivity are obligatory for active rate of heat diffusion. As a consequence, these fluids may be employed in Parabolic Trough Solar Collector for increased heat transmission rate and operational usage of solar energy. Furthermore, We create a dataset using the Runge–Kutta fourth-order shooting technique to create the proposed multilayer perceptron artificial neural network. The data points representing the MoD values are observed to be closely clustered around the zero deviation line. Additionally, it is important to highlight that these data points have relatively small numerical values. Moreover, when calculating the average MoD values for each output, it becomes evident that they are consistently very low.

Low regularity ill-posedness for elastic waves driven by shock formation

abstract: In this paper, we construct counterexamples to the local existence of low-regularity solutions to elastic wave equations in three spatial dimensions (3D). Inspired by the recent works of Christodoulou, we generalize Lindblad's classic results on the scalar wave equation by showing that the Cauchy problem for 3D elastic waves, a physical system with multiple wave-speeds, are ill-posed in $H^3(\Bbb{R}^3)$. We further prove that the ill-posedness is caused by instantaneous shock formation, which is characterized by the vanishing of the inverse foliation density. The main difficulties of the 3D case come from the multiple wave-speeds and its associated non-strict hyperbolicity. We obtain the desired results by designing and combining a geometric approach and an algebraic approach, equipped with detailed studies and calculations of the structures and coefficients of the corresponding non-strictly hyperbolic system. Moreover, the ill-posedness we depict also applies to 2D elastic waves, which corresponds to a strictly hyperbolic case.

A split special Lagrangian calibration associated with frame vorticity

Abstract Let M be an oriented three-dimensional Riemannian manifold. We define a notion of vorticity of local sections of the bundle SO ⁢ ( M ) → M {\mathrm{SO}(M)\rightarrow M} of all its positively oriented orthonormal tangent frames. When M is a space form, we relate the concept to a suitable invariant split pseudo-Riemannian metric on Iso o ⁢ ( M ) ≅ SO ⁢ ( M ) {\mathrm{Iso}_{o}(M)\cong\mathrm{SO}(M)} : A local section has positive vorticity if and only if it determines a space-like submanifold. In the Euclidean case we find explicit homologically volume maximizing sections using a split special Lagrangian calibration. We introduce the concept of optimal frame vorticity and give an optimal screwed global section for the three-sphere. We prove that it is also homologically volume maximizing (now using a common one-point split calibration). Besides, we show that no optimal section can exist in the Euclidean and hyperbolic cases.

Small volume 3-manifolds constructible from string Coxeter groups of rank 4

It is well known that various compact 3-manifolds of small volume can be constructed via the determination of torsion-free subgroups of minimum possible index in certain rank 4 string Coxeter groups. Examples include the Poincaré homology sphere and the Weber-Seifert and Gieseking manifolds, obtainable from the [3,3,5], [5,3,5] and [3,3,6] Coxeter groups respectively. Constructions were developed in some notes by Milnor on computing volumes in the late 1970s, and in papers by Lorimer (1992) and Everitt (2004) using the identification of faces of a Platonic solid. The notes by Milnor dealt only with hyperbolic cases, and did not resolve all of them, and also a subsequent paper by Lorimer (2002) unsuccessfully attempted to deal with the case of the [4,3,5] Coxeter group. We complete and extend these pieces of work by determining the smallest volume Euclidean or hyperbolic 3-manifolds constructible from torsion-free subgroups of minimum possible index in all of the infinite [p,q,r] Coxeter groups for which p,q,r≥3 and 1/p+1/q≥1/2 and 1/q+1/r≥1/2.

The Asymptotic Structure of the Centred Hyperbolic 2-Monopole Moduli Space

We construct an asymptotic metric on the moduli space of two centred hyperbolic monopoles by working in the point particle approximation, that is treating well-separated monopoles as point particles with an electric, magnetic and scalar charge and re-interpreting the dynamics of the 2-particle system as geodesic motion with respect to some metric. The corresponding analysis in the Euclidean case famously yields the negative mass Taub-NUT metric, which asymptotically approximates the L&lt;sup&gt;2&lt;/sup&gt; metric on the moduli space of two Euclidean monopoles, the Atiyah-Hitchin metric. An important difference with the Euclidean case is that, due to the absence of Galilean symmetry, in the hyperbolic case it is not possible to factor out the centre of mass motion. Nevertheless we show that we can consistently restrict to a 3-dimensional configuration space by considering antipodal configurations. In complete parallel with the Euclidean case, the metric that we obtain is then the hyperbolic analogue of negative mass Taub-NUT. We also show how the metric obtained is related to the asymptotic form of a hyperbolic analogue of the Atiyah-Hitchin metric constructed by Hitchin.

Wada boundaries on a hyperbolic pair of pants

In this paper, the geodesics of an open multiply connected hyperbolic manifold are presented from the dynamical system point of view. The approach is completely numerical. Similar to the closed hyperbolic case there is a zero-measure set of periodic orbits. The difference is that now, most of the geodesics escape to infinity through one of the topologically distinct channels, and initial conditions are identified to which channel the orbit escapes. The initial condition mesh reveals basins of attraction to each channel. We verified that the basin boundary points are of the Wada type by two independent methods. We have also calculated the basin entropy of the system and verified that it depends on the opening angle of the escape channels. The manifold chosen is the pair of pants together with the leaves, so there are 3 distinct exit channels.

Abolitionist Techniques of Weathering

Abolitionist Techniques of Weathering

Weighted monotonicity theorems and applications to minimal surfaces of ℍⁿ and 𝕊ⁿ

We prove that in a Riemannian manifold M M , each function whose Hessian is proportional the metric tensor yields a weighted monotonicity theorem. Such function appears in the Euclidean space, the round sphere S n {S}^n and the hyperbolic space H n \mathbb {H}^n as the distance function, the Euclidean coordinates of R n + 1 \mathbb {R}^{n+1} and the Minkowskian coordinates of R n , 1 \mathbb {R}^{n,1} . Then we show that weighted monotonicity theorems can be compared and that in the hyperbolic case, this comparison implies three S O ( n , 1 ) SO(n,1) -distinct unweighted monotonicity theorems. From these, we obtain upper bounds of the Graham–Witten renormalised area of a minimal surface in term of its ideal perimeter measured under different metrics of the conformal infinity. Other applications include a vanishing result for knot invariants coming from counting minimal surfaces of H n \mathbb {H}^n and a quantification of how antipodal a minimal submanifold of S n S^n has to be in term of its volume.

Consistency of the Bäcklund Transformation for the Spin Calogero–Moser System

We prove the consistency of the Bäcklund transformation (BT) for the spin Calogero–Moser (sCM) system in the rational, trigonometric, and hyperbolic cases. The BT for the sCM system consists of an overdetermined system of ordinary differential equations; to establish our result, we construct and analyze certain functions that measure the departure of this overdetermined system from consistency. We show that these functions are identically zero and that this allows for a unique solution to the initial value problem for the overdetermined system.

On the Nature of Bondi–Metzner–Sachs Transformations

This paper investigates, as a first step, the four branches of BMS transformations, motivated by the classification into elliptic, parabolic, hyperbolic and loxodromic proposed a few years ago in the literature. We first prove that to each normal elliptic transformation of the complex variable ζ used in the metric for cuts of null infinity, there is a corresponding BMS supertranslation. We then study the conformal factor in the BMS transformation of the u variable as a function of the squared modulus of ζ. In the loxodromic and hyperbolic cases, this conformal factor is either monotonically increasing or monotonically decreasing as a function of the real variable given by the modulus of ζ. The Killing vector field of the Bondi metric is also studied in correspondence with the four admissible families of BMS transformations. Eventually, all BMS transformations are re-expressed in the homogeneous coordinates suggested by projective geometry. It is then found that BMS transformations are the restriction to a pair of unit circles of a more general set of transformations. Within this broader framework, the geometry of such transformations is studied by means of its Segre manifold.

Time‐continuous and time‐discontinuous space‐time finite elements for advection‐diffusion problems

AbstractWe construct four variants of space‐time finite element discretizations based on linear tensor‐product and simplex‐type finite elements. The resulting discretizations are continuous in space, and continuous or discontinuous in time. In a first test run, all four methods are applied to a linear scalar advection‐diffusion model problem. Then, the convergence properties of the time‐discontinuous space‐time finite element discretizations are studied in numerical experiments. Advection velocity and diffusion coefficient are varied, such that the parabolic case of pure diffusion (heat equation), as well as, the hyperbolic case of pure advection (transport equation) are included in the study. For each model parameter set, the error at the final time is computed for spatial and temporal element lengths ranging over several orders of magnitude to allow for an individual evaluation of the methods' spatial, temporal, and space‐time accuracy. In the parabolic case, particular attention is paid to the influence of time‐dependent boundary conditions. Key findings include a spatial accuracy of second order and a temporal accuracy between second and third order. The temporal accuracy tends toward third order depending on how advection‐dominated the test case is, on the choice of the specific discretization method, and on the time‐(in)dependence and treatment of the boundary conditions. Additionally, the potential of time‐continuous simplex space‐time finite elements for heat flux computations is demonstrated with a piston ring pack test case and a subtractive manufacturing test case.