Hyperbolic Cone Research Articles

Jordan automorphisms and derivatives of symmetric cones

Hyperbolicity cones, and in particular symmetric cones, are of great interest in optimization. Renegar showed that every hyperbolicity cone has a family of derivative cones that approximate it. Ito and Lourenço found the automorphisms of those derivatives when the original cone is generated by rank-one elements, as symmetric cones happen to be. We show that the derivative automorphisms of a symmetric cone are closely related to the automorphisms of its associated Euclidean Jordan algebra. In the process, we find the automorphism group of the quaternion positive-semidefinite cone and list explicitly the Jordan-automorphisms of the quaternion Hermitian matrices. We also address the path-connectedness of the simple Euclidean Jordan-automorphism groups.

The dusty heart of Circinus

In this paper we present the first-ever L- and M-band interferometric observations of Circinus, building upon a recent N-band analysis. We used these observations to reconstruct images and fit Gaussian models to the L and M bands. Our findings reveal a thin edge-on disk whose width is marginally resolved and is the spectral continuation of the disk imaged in the N band to shorter wavelengths. Additionally, we find a point-like source in the L and M bands that, based on the LMN-band spectral energy distribution fit, corresponds to the N-band point source. We also demonstrate that there is no trace of direct sightlines to hot dust surfaces in the circumnuclear dust structure of Circinus. By assuming the dust is present, we find that obscuration of AV ≳ 250 mag is necessary to reproduce the measured fluxes. Hence, the imaged disk could play the role of the obscuring “torus” in the unified scheme of active galactic nuclei. Furthermore, we explored the parameter space of the disk + hyperbolic cone radiative transfer models and identify a simple modification at the base of the cone. Adding a cluster of clumps just above the disk and inside the base of the hyperbolic cone provides a much better match to the observed temperature distribution in the central aperture. This aligns well with the radiation-driven fountain models that have recently emerged. Only the unique combination of sensitivity and spatial resolution of the VLTI allows such models to be scrutinized and constrained in detail. We plan to test the applicability of this detailed dust structure to other MATISSE-observed active galactic nuclei in the future.

An In-depth Comparative Analysis of Conical Curve Variations: Exploring the Distinctive Properties and Applications of Elliptic and Hyperbolic Cones

An In-depth Comparative Analysis of Conical Curve Variations: Exploring the Distinctive Properties and Applications of Elliptic and Hyperbolic Cones

Hyperbolic cones for casual diffeomorphisms

This paper begins with introducing a new class of diffeomorphisms on pseudo-Riemannian manifolds which are called casual diffeomorphisms. We deduce a new pseudo-Riemannian metric via a casual diffeomorphism. We introduce a class of cones for a diffeomorphism, and we prove the existence of such cones for each p in a hyperbolic set of a casual diffeomorphism. As an example we find a hyperbolic casual diffeomorphism on a Schwarzschild pseudo-Riemannian manifold, and show that the whole manifold is a hyperbolic set for it.

Automorphisms of Rank-One Generated Hyperbolicity Cones and Their Derivative Relaxations

A hyperbolicity cone is said to be rank-one generated (ROG) if all its extreme rays have rank 1, where the rank is computed with respect to the underlying hyperbolic polynomial. This is a natural class of hyperbolicity cones which are strictly more general than the ROG spectrahedral cones. In this work, we present a study of the automorphisms of ROG hyperbolicity cones and their derivative relaxations. One of our main results states that the automorphisms of the derivative relaxations are exactly the automorphisms of the original cone fixing a certain direction. As an application, we completely determine the automorphisms of the derivative relaxations of the nonnegative orthant and of the cone of positive semidefinite matrices. More generally, we also prove relations between the automorphisms of a spectral cone and the underlying permutation-invariant set, which might be of independent interest.

Restriction estimates for hyperbolic cone in high dimensions

In this paper, we obtain the L p L^p restriction estimates for the truncated conic surface Σ = { ( ξ ′ , ξ n , − ξ n − 1 ⟨ ξ ′ , N ξ ′ ⟩ ) : ( ξ ′ , ξ n ) ∈ B n − 1 ( 0 , 1 ) × [ 1 , 2 ] } \begin{equation*} \Sigma =\big \{(\xi ’,\xi _n,-\xi _n^{-1}\langle \xi ’,N\xi ’\rangle ): (\xi ’,\xi _n)\in B^{n-1}(0,1)\times [1,2]\big \} \end{equation*} with N = I n − 1 − m ⊕ ( − I m ) N=I_{n-1-m}\oplus (-I_m) for m ≤ ⌊ n − 3 2 ⌋ m\leq \lfloor \tfrac {n-3}2\rfloor provided p > 2 ( n + 3 ) n + 1 p>\tfrac {2(n+3)}{n+1} . The main ingredients of the proof are the bilinear estimates of strongly separated property and a geometric distribution about caps.

Relative Reshetikhin–Turaev Invariants, Hyperbolic Cone Metrics and Discrete Fourier Transforms I

We propose the Volume Conjecture for the relative Reshetikhin–Turaev invariants of a closed oriented 3-manifold with a colored framed link inside it whose asymptotic behavior is related to the volume and the Chern–Simons invariant of the hyperbolic cone metric on the manifold with singular locus the link and cone angles determined by the coloring. We prove the conjecture in the case that the ambient 3-manifold is obtained by doing an integral surgery along some components of a fundamental shadow link and the complement of the link in the ambient manifold is homeomorphic to the fundamental shadow link complement, for sufficiently small cone angles. Together with Costantino and Thurston’s result that all compact oriented 3-manifolds with toroidal or empty boundary can be obtained by doing an integral surgery along some components of a suitable fundamental shadow link, this provides a possible approach of solving Chen–Yang’s Volume Conjecture for the Reshetikhin–Turaev invariants of closed oriented hyperbolic 3-manifolds. We also introduce a family of topological operations (the change-of-pair operations) that connect all pairs of a closed oriented 3-manifold and a framed link inside it that have homeomorphic complements, which correspond to doing the partial discrete Fourier transforms to the corresponding relative Reshetikhin–Turaev invariants. As an application, we find a Poisson Summation Formula for the discrete Fourier transforms.

Absolutely k -incoherent quantum states and spectral inequalities for the factor width of a matrix

We investigate the set of quantum states that can be shown to be $k$-incoherent based only on their eigenvalues (equivalently, we explore which Hermitian matrices can be shown to have small factor width based only on their eigenvalues). In analogy with the absolute separability problem in quantum resource theory, we call these states "absolutely $k$-incoherent", and we derive several necessary and sufficient conditions for membership in this set. We obtain many of our results by making use of recent results concerning hyperbolicity cones associated with elementary symmetric polynomials.

Linear Principal Minor Polynomials: Hyperbolic Determinantal Inequalities and Spectral Containment

Abstract A linear principal minor polynomial or lpm polynomial is a linear combination of principal minors of a symmetric matrix. By restricting to the diagonal, lpm polynomials are in bijection with multiaffine polynomials. We show that this establishes a one-to-one correspondence between homogeneous multiaffine stable polynomials and PSD-stable lpm polynomials. This yields new construction techniques for hyperbolic polynomials and allows us to find an explicit degree 3 hyperbolic polynomial in six variables some of whose Rayleigh differences are not sums of squares. We further generalize the well-known Fisher–Hadamard and Koteljanskii inequalities from determinants to PSD-stable lpm polynomials. We investigate the relationship between the associated hyperbolicity cones and conjecture a relationship between the eigenvalues of a symmetric matrix and the values of certain lpm polynomials evaluated at that matrix. We refer to this relationship as spectral containment.

Degeneration of 3-dimensional hyperbolic cone structures with decreasing cone angles

For 3-dimensional hyperbolic cone structures with cone angles θ \theta , local rigidity is known for 0 ≤ θ ≤ 2 π 0 \leq \theta \leq 2\pi , but global rigidity is known only for 0 ≤ θ ≤ π 0 \leq \theta \leq \pi . The proof of the global rigidity by Kojima is based on the fact that hyperbolic cone structures with cone angles at most π \pi do not degenerate in deformations decreasing cone angles to zero. In this paper, we give an example of a degeneration of hyperbolic cone structures with decreasing cone angles less than 2 π 2\pi . These cone structures are constructed on a certain alternating link in the thickened torus by gluing four copies of a certain polyhedron. For this construction, we explicitly describe the isometry types on such a hyperbolic polyhedron.

Hyperbolic cone metrics and billiards

A negatively curved hyperbolic cone metric is called rigid if it is determined (up to isotopy) by the support of its Liouville current, and flexible otherwise. We provide a complete characterization of rigidity and flexibility, prove that rigidity is a generic property, and parameterize the associated deformation space for any flexible metric. As an application, we parameterize the space of hyperbolic polygons with the same symbolic coding for their billiard dynamics, and prove that generically this parameter space is a point.

The radiation field on product cones

We consider the wave equation on a product cone and find a joint asymptotic expansion for solutions near null and future infinities. The rates of decay seen in the expansion at future infinity are the resonances of a hyperbolic cone and were computed by the authors in [2]. The expansion treats an asymptotic regime not considered in the influential work of Cheeger and Taylor [5,6]. The main result follows the blueprint laid out in the [3,4] with key new elements including propagation estimates near the conic singularities. The proof of the propagation estimates extends prior work of Melrose–Vasy–Wunsch [19] and Gannot–Wunsch [10].

Hyperbolic Relaxation of $k$-Locally Positive Semidefinite Matrices

A successful computational approach for solving large-scale positive semidefinite (PSD) programs is to enforce PSD-ness on only a collection of submatrices. For our study, we let $\mathcal{S}^{n,k}$ be the convex cone of $n\times n$ symmetric matrices where all $k\times k$ principal submatrices are PSD. We call a matrix in this $k$-locally PSD. In order to compare $\mathcal{S}^{n,k}$ to the cone of PSD matrices, we study eigenvalues of $k$-locally PSD matrices. The key insight in this paper is that there is a convex cone $H(e_k^n)$ so that if $X \in \mathcal{S}^{n,k}$, then the vector of eigenvalues of $X$ is contained in $H(e_k^n)$. The cone $H(e_k^n)$ is the hyperbolicity cone of the elementary symmetric polynomial $e_k^n$ (where $e_k^n(x) = \sum_{S \subseteq [n] : |S| = k} \prod_{i \in S} x_i$) with respect to the all ones vector. Using this insight, we are able to improve previously known upper bounds on the Frobenius distance between matrices in $\mathcal{S}^{n,k}$ and PSD matrices. We also study the quality of the convex relaxation $H(e^n_k)$. We first show that this relaxation is tight for the case of $k = n -1$, that is, for every vector in $H(e^n_{n -1})$ there exists a matrix in $\mathcal{S}^{n, n -1}$ whose eigenvalues are equal to the components of the vector. We then prove a structure theorem on nonsingular matrices in $\mathcal{S}^{n,k}$ all of whose $k\times k$ principal minors are zero, which we believe is of independent interest. This result shows shows that for $1< k < n -1$ “large parts” of the boundary of $H(e_k^n)$ do not intersect with the eigenvalues of matrices in $\mathcal{S}^{n,k}$.

Terracini convexity

We present a generalization of the notion of neighborliness to non-polyhedral convex cones. Although a definition of neighborliness is available in the non-polyhedral case in the literature, it is fairly restrictive as it requires all the low-dimensional faces to be polyhedral. Our approach is more flexible and includes, for example, the cone of positive-semidefinite matrices as a special case (this cone is not neighborly in general). We term our generalization Terracini convexity due to its conceptual similarity with the conclusion of Terracini’s lemma from algebraic geometry. Polyhedral cones are Terracini convex if and only if they are neighborly. More broadly, we derive many families of non-polyhedral Terracini convex cones based on neighborly cones, linear images of cones of positive-semidefinite matrices, and derivative relaxations of Terracini convex hyperbolicity cones. As a demonstration of the utility of our framework in the non-polyhedral case, we give a characterization based on Terracini convexity of the tightness of semidefinite relaxations for certain inverse problems.

Determinant of Friedrichs Dirichlet Laplacians on 2-dimensional hyperbolic cones

We explicitly express the spectral determinant of Friedrichs Dirichlet Laplacians on the 2-dimensional hyperbolic (Gaussian curvature -1) cones in terms of the cone angle and the geodesic radius of the boundary.

Binaural Synthetic Aperture Imaging of the Field of Audition as the Head Rotates and Localisation Perception of Monophonic Sound Listened to through Headphones

A human listening to monophonic sound through headphones perceives the sound to emanate from a point inside the head at the auditory centre at effectively zero range. The extent to which this is predicted by synthetic-aperture calculation performed in response to head rotation is explored. The instantaneous angle between the auditory axis and the acoustic source, lambda, for the zero inter-aural time delay imposed by headphones is 90°. The lambda hyperbolic cone simplifies to the auditory median plane, which intersects a spherical surface centred on the auditory centre, along a prime meridian lambda circle. In a two-dimensional (2-D) synthetic-aperture computation, points of intersection of all lambda circles as the head rotates constitute solutions to the directions to acoustic sources. Geometrically, lambda circles cannot intersect at a point representing the auditory centre; nevertheless, 2-D synthetic aperture images for a pure turn of the head and for a pure lateral tilt yield solutions as pairs of points on opposite sides of the head. These can reasonably be interpreted to be perceived at the sums of the position vectors of the pairs of points on the acoustic image, i.e., at the auditory centre. But, a turn of the head on which a fixed lateral tilt of the auditory axis is concomitant (as in species of owl) yields a 2-D synthetic-aperture image without solution. However, extending a 2-D synthetic aperture calculation to a three-dimensional (3-D) calculation will generate a 3-D acoustic image of the field of audition that robustly yields the expected solution.

Alfonso: Matlab Package for Nonsymmetric Conic Optimization

We present alfonso, an open-source Matlab package for solving conic optimization problems over nonsymmetric convex cones. The implementation is based on the authors’ corrected analysis of a method of Skajaa and Ye. It enables optimization over any convex cone as long as a logarithmically homogeneous self-concordant barrier is available for the cone or its dual. This includes many nonsymmetric cones, for example, hyperbolicity cones and their duals (such as sum-of-squares cones), semidefinite and second-order cone representable cones, power cones, and the exponential cone. Besides enabling the solution of problems that cannot be cast as optimization problems over a symmetric cone, algorithms for nonsymmetric conic optimization also offer performance advantages for problems whose symmetric cone programming representation requires a large number of auxiliary variables or has a special structure that can be exploited in the barrier computation. The worst-case iteration complexity of alfonso is the best known for nonsymmetric cone optimization: [Formula: see text] iterations to reach an ε-optimal solution, where ν is the barrier parameter of the barrier function used in the optimization. Alfonso can be interfaced with a Matlab function (supplied by the user) that computes the Hessian of a barrier function for the cone. A simplified interface is also available to optimize over the direct product of cones for which a barrier function has already been built into the software. This interface can be easily extended to include new cones. Both interfaces are illustrated by solving linear programs. The oracle interface and the efficiency of alfonso are also demonstrated using an optimal design of experiments problem in which the tailored barrier computation greatly decreases the solution time compared with using state-of-the-art, off-the-shelf conic optimization software. Summary of Contribution: The paper describes an open-source Matlab package for optimization over nonsymmetric cones. A particularly important feature of this software is that, unlike other conic optimization software, it enables optimization over any convex cone as long as a suitable barrier function is available for the cone or its dual, not limiting the user to a small number of specific cones. Nonsymmetric cones for which such barriers are already known include, for example, hyperbolicity cones and their duals (such as sum-of-squares cones), semidefinite and second-order cone representable cones, power cones, and the exponential cone. Thus, the scope of this software is far larger than most current conic optimization software. This does not come at the price of efficiency, as the worst-case iteration complexity of our algorithm matches the iteration complexity of the most successful interior-point methods for symmetric cones. Besides enabling the solution of problems that cannot be cast as optimization problems over a symmetric cone, our software can also offer performance advantages for problems whose symmetric cone programming representation requires a large number of auxiliary variables or has a special structure that can be exploited in the barrier computation. This is also demonstrated in this paper via an example in which our code significantly outperforms Mosek 9 and SCS 2.

Ideal simplices and double-simplices, their non-orientable hyperbolic manifolds, cone manifolds and orbifolds with Dehn type surgeries and graphic analysis

In connection with our works in Molnár (On isometries of space forms. Colloquia Math Soc János Bolyai 56 (1989). Differential geometry and its applications, Eger (Hungary), North-Holland Co., Amsterdam, 1992), Molnár (Acta Math Hung 59(1–2):175–216, 1992), Molnár (Beiträge zur Algebra und Geometrie 38/2:261–288, 1997) and Molnár et al. (in: Prékopa, Molnár (eds) Non-Euclidean geometries, János Bolyai memorial volume mathematics and its applications, Springer, Berlin, 2006), Molnár et al. (Symmetry Cult Sci 22(3–4):435–459, 2011) our computer program (Prok in Period Polytech Ser Mech Eng 36(3–4):299–316, 1992) found 5079 equivariance classes for combinatorial face pairings of the double-simplex. From this list we have chosen those 7 classes which can form charts for hyperbolic manifolds by double-simplices with ideal vertices. In such a way we have obtained the orientable manifold of Thurston (The geometry and topology of 3-manifolds (Lecture notes), Princeton University, Princeton, 1978), that of Fomenko–Matveev–Weeks (Fomenko and Matveev in Uspehi Mat Nauk 43:5–22, 1988; Weeks in Hyperbolic structures on three-manifolds. Ph.D. dissertation, Princeton, 1985) and a nonorientable manifold M_{c^2} with double simplex {widetilde{{mathcal {D}}}}_1, seemingly known by Adams (J Lond Math Soc (2) 38:555–565, 1988), Adams and Sherman (Discret Comput Geom 6:135–153, 1991), Francis (Three-manifolds obtainable from two and three tetrahedra. Master Thesis, William College, 1987) as a 2-cusped one. This last one is represented for us in 5 non-equivariant double-simplex pairings. In this paper we are going to determine the possible Dehn type surgeries of M_{c^2}={widetilde{{mathcal {D}}}}_1, leading to compact hyperbolic cone manifolds and multiple tilings, especially orbifolds (simple tilings) with new fundamental domain to {widetilde{{mathcal {D}}}}_1. Except the starting regular ideal double simplex, we do not get further surgery manifold. We compute volumes for starting examples and limit cases by Lobachevsky method. Our procedure will be illustrated by surgeries of the simpler analogue, the Gieseking manifold (1912) on the base of our previous work (Molnár et al. in Publ Math Debr, 2020), leading to new compact cone manifolds and orbifolds as well. Our new graphic analysis and tables inform you about more details. This paper is partly a survey discussing as new results on Gieseking manifold and on M_{c^2} as well, their cone manifolds and orbifolds which were partly published in Molnár et al. (Novi Sad J Math 29(3):187–197, 1999) and Molnár et al. (in: Karáné, Sachs, Schipp (eds) Proceedings of “Internationale Tagung über geometrie, algebra und analysis”, Strommer Gyula Nemzeti Emlékkonferencia, Balatonfüred-Budapest, Hungary, 1999), updated now to Memory of Professor Gyula Strommer. Our intention is to illustrate interactions of Algebra, Analysis and Geometry via algorithmic and computational methods in a classical field of Geometry and of Mathematics, in general.

Kippenhahn's Theorem for Joint Numerical Ranges and Quantum States

Kippenhahn's theorem asserts that the numerical range of a matrix is the convex hull of a certain algebraic curve. Here, we show that the joint numerical range of finitely many Hermitian matrices is similarly the convex hull of a semialgebraic set. We discuss an analogous statement regarding the dual convex cone to a hyperbolicity cone and prove that the class of bases of these dual cones is closed under linear operations. The result offers a new geometric method to analyze quantum states.

Cauchy and uniform temporal functions of globally hyperbolic cone fields

We study a class of time functions called uniform temporal functions in the general context of globally hyperbolic closed cone fields. We prove some existence results for uniform temporal functions, and establish the density of uniform temporal functions in Cauchy causal functions.