RANDOMIZED URYSOHN–TYPE INEQUALITIES

We show that the third eigenvalue of the Neumann Laplacian in hyperbolic space is maximal for the disjoint union of two geodesic balls, among domains of given volume. This extends a recent result by Bucur and Henrot in Euclidean space, while providing a new proof of a key step in their argument

Komori and Umemoto detected combinatorial shapes of Dirichlet polyhedra for simplex groups acting on hyperbolic space. We generalize their result to simplex groups acting on Euclidean or spherical spaces.

Let Ω be some domain in the hyperbolic space Hn (with n≥2), and let S1 be a geodesic ball that has the same first Dirichlet eigenvalue as Ω. We prove the Payne-Polya-Weinberger (PPW) conjecture for Hn, namely, that the second Dirichlet eigenvalue on Ω is smaller than or equal to the second Dirichlet eigenvalue on S1. We also prove that the ratio of the first two eigenvalues on geodesic balls is a decreasing function of the radius

The metrics induced on free boundary minimal surfaces in geodesic balls in the upper unit hemisphere and hyperbolic space can be characterized as critical metrics for the functionals $\Theta _{r,i}$ and $\Omega _{r,i}$ , introduced recently by Lima, Menezes, and the second author. In this article, we generalize this characterization to free boundary minimal submanifolds of higher dimension in the same spaces. We also introduce some functionals of the form different from $\Theta _{r,i}$ and show that the critical metrics for them are the metrics induced by free boundary minimal immersions into a geodesic ball in the upper unit hemisphere. In the case of surfaces, these functionals are bounded from above and not bounded from below. Moreover, the canonical metric on a geodesic disk in a 3-ball in the upper unit hemisphere is maximal for this functional on the set of all Riemannian metric of the topological disk.

The PPW conjecture in curved spaces

Hyperbolic hierarchical knowledge graph embeddings for biological entities

Detecting events and their evolution through time is a crucial task in natural language understanding. Recent neural approaches to event temporal relation extraction typically map events to embeddings in the Euclidean space and train a classifier to detect temporal relations between event pairs. However, embeddings in the Euclidean space cannot capture richer asymmetric relations such as event temporal relations. We thus propose to embed events into hyperbolic spaces, which are intrinsically oriented at modeling hierarchical structures. We introduce two approaches to encode events and their temporal relations in hyperbolic spaces. One approach leverages hyperbolic embeddings to directly infer event relations through simple geometrical operations. In the second one, we devise an end-to-end architecture composed of hyperbolic neural units tailored for the temporal relation extraction task. Thorough experimental assessments on widely used datasets have shown the benefits of revisiting the tasks on a different geometrical space, resulting in state-of-the-art performance on several standard metrics. Finally, the ablation study and several qualitative analyses highlighted the rich event semantics implicitly encoded into hyperbolic spaces.

On an analytic Riemannian manifold (M,g), several authors have studied the Taylor expansion for the volume of geodesic balls under the exponential mapping. In the foregoing paper [1] we studied a more general structure (M,D,g), where D is a torsion-free and Ricci-symmetric connection. We calculated the Taylor expansion up to order (n+4) for the volume of what we called a generalized geodesic ball under the exponential mapping in case that all metric notions are Riemannian, while the exponential mapping is induced from the connection D. For the structure % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A! $(M,D,{\cal G})$ the coefficients of the Taylor expansion are much more complicated than in the Riemannian case. It is one of the main objectives of the present paper to study centroaffine hypersurfaces in Euclidean space, their geometric invariants which appear in the very complicated coefficient of order (n+4), and their behaviour under polarization (inversion at the unit sphere). Our results complement applications in the foregoing paper [1], where mainly the coefficients up to order (n+2) and geometric consequences have been studied.

In this paper, we consider the Universe deep inside of the cell of uniformity. At these scales, the Universe is filled with inhomogeneously distributed discrete structures (galaxies, groups and clusters of galaxies), which disturb the background Friedmann model. We propose mathematical models with conformally flat, hyperbolic and spherical spaces. For these models, we obtain the gravitational potential for an arbitrary number of randomly distributed inhomogeneities. In the cases of flat and hyperbolic spaces, the potential is finite at any point, including spatial infinity, and valid for an arbitrary number of gravitating sources. For both of these models, we investigate the motion of test masses (e.g., dwarf galaxies) in the vicinity of one of the inhomogeneities. We show that there is a distance from the inhomogeneity, at which the cosmological expansion prevails over the gravitational attraction and where test masses form the Hubble flow. For our group of galaxies, it happens at a few Mpc and the radius of the zero-acceleration sphere is of the order of 1 Mpc, which is very close to observations. Outside of this sphere, the dragging effect of the gravitational attraction goes very fast to zero.

Higher order Brezis-Nirenberg problem on hyperbolic spaces: Existence, nonexistence and symmetry of solutions

Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. In Euclidean space it is possible for every circle in such a packing to have integer radius of curvature, and we call such a packing an integral Apollonian circle packing. There are infinitely many different integral packings; these were studied in Part I (J. Number Theory 100, 1–45, 2003). Integral circle packings also exist in spherical and hyperbolic space, provided a suitable definition of curvature is used and again there are an infinite number of different integral packings. This paper studies number-theoretic properties of such packings. This amounts to studying the orbits of a particular subgroup ${\mathcal{A}}$ of the group of integral automorphs of the indefinite quaternary quadratic form $Q_{{\mathcal{D}}}(w,x,y,z)=2(w^{2}+x^{2}+y^{2}+z^{2})-(w+x+y+z)^{2}$ . This subgroup, called the Apollonian group, acts on integer solutions $Q_{{\mathcal{D}}}(w,x,y,z)=k$ . This paper gives a reduction theory for orbits of ${\mathcal{A}}$ acting on integer solutions to $Q_{{\mathcal{D}}}(w,x,y,z)=k$ valid for all integer k. It also classifies orbits for all k≡0 (mod 4) in terms of an extra parameter n and an auxiliary class group (depending on n and k), and studies congruence conditions on integers in a given orbit.

This is an overview of some of the similarities and differences between structures such as frameworks and cabled tensegrities in the hyperbolic plane and hyperbolic space on the one hand and the Euclidean plane, the sphere and Euclidean space on the other hand.KeywordsHyperbolic planeconfigurationsframeworktensegrity

Improved Moser–Trudinger type inequalities in the hyperbolic space [formula omitted

A new proof is given of an inequality of J. Bokowski and E. Sperner [1] referring to the product of the volume of the two parts into which a convex body is divided by a plane. The proof, which is given for dimensionsn=2, 3 uses known formulas of Integral Goemetry and is generalized to convex bodies of the elliptic and hyperbolic spaces.

The systematic study of various mean-values of nonnegative valued functions is due in large part to the work of J. L. W. V. Jensen [12], A. Kolmogoroff [11], and B. Jessen [13]. In this paper we present a similar theory of certain means of functions whose values are taken from a collection of convex bodies in a finite dimensional Euclidean space. The range of nonnegative valued functions is totally ordered while the collection of convex bodies to be considered is partially ordered by set inclusion; corresponding to inequalities, we shall have inclusions, here to be called inequalities, between mean-values. Thus the discussion will furnish an example of a partially ordered system whose algebraic and topological structure is sufficient to admit a fairly elaborate theory of inequalities analogous to that of the real numbers. The first section is a resume of those parts of the work of Jensen, Kolmogoroff, and Jessen for which analogues will be developed. ?2 treats of pertinent material about star and convex bodies. Certain families of star bodies and of convex bodies are defined in the third section; these latter play the role of nonnegative valued functions. Two systems of power means of such families are described. Each system is, in its way, anialogous to the power means of nonnegative valued functions. In ?4 we discuss some special cases of power means including elementary means and the Riemaiin-Minkowski integrals of A. Dinghas [2], [3]. Also some rotation invariants of a convex body are described as power means of special families determined by the convex body. The fifth section begins with crucial properties of the means and contains an extremal characterization for the two systems of means defined in the third section. Analogues of Jensen's and Jessen's inequalities make up ?6. We also mentioni the limiting cases of power means as the power index becomes infinite, positively and negatively. As an applicationl of Jessen's inequality, a Brunn-Minkowski type theorem is proved. In the final section we discuss some further systems of meanls of convex bodies. 1. The definition of the elementary power means extends in a natural way to include power means of certain functions f which have finite, nonnegative values and whose domain T is a compact topological group equipped with its Haar measure. Iff is bounded and measurable over its domain, its pth- power mean is defined by