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In this short paper, we derive a new result on Umehara algebra. As a consequence, we prove that an indefinite complex hyperbolic space and an indefinite complex projective space do not share a common complex submanifold with induced metrics, answering a question raised in Cheng et al. [ABSTRACT FROM AUTHOR]

Wang and Ye [Adv. Math. 230 (2012), pp. 294–320] prove a Hardy–Moser–Trudinger inequality in dimension two which improves both the classical Moser–Trudinger inequality and the classical Hardy inequality in the unit disc. In this paper, we generalize their result both to the higher dimensional unit ball and to the singular weighted cases as well. More precisely, we prove that \[ \sup _{\substack {u\in W^{1,n}_0(\mathbb {B}^n),\\ \int _{\mathbb {B}^n} |\nabla u|^n dx -\left (\frac {2(n-1)}n\right)^n \int _{\mathbb {B}^n} \frac {|u|^n}{(1-|x|^2)^n} dx \leq 1}}\int _{\mathbb {B}^n} e^{(1-\frac \beta n)\alpha _n |u|^{\frac n{n-1}}} |x|^{-\beta } dx \leq C \] for any \beta \in [0,n), n\geq 3 where \alpha _n = n \omega _{n-1}^{\frac 1{n-1}} and \omega _{n-1} is the surface area of the n-1 dimensional unit sphere. The proof of Wang and Ye is based on the blow-up analysis method which seems not work in the higher dimensions. In this paper, we propose a new approach based on the method of transplantation of Green's functions to prove our inequality. As a consequence, we obtain a singular Moser–Trudinger inequality in the hyperbolic spaces which confirms affirmatively a conjecture by Mancini, Sandeep and Tintarev [Adv. Nonlinear Anal., 2 (2013), pp. 309–324]. [ABSTRACT FROM AUTHOR]

We intend to continue our previous papers on dense ball packing hyperbolic space H³ by equal balls, but here with centres belonging to different orbits of the fundamental group Cw(2z, 3 ≤ z ∈ N, odd number), of our new series of tube or cobweb manifolds Cw = H³/Cw with z-rotational symmetry. As we know, Cw is a fixed-point-free isometry group, acting on H³ discontinuously with appropriate tricky fundamental domain Cw, so that every point has a ball-like neighbourhood in the usual factor-topology. Our every Cw(2z) is minimal, i.e. does not cover regularly a smaller manifold. It can be derived by its general symmetry group W(u; v;w = u) that is a complete Coxeter orthoscheme reflection group, extended by the half-turn h (0 ↔ 3, 1 ↔ 2) of the complete orthoscheme A0A1A2A3 ∼ b0b1b2b3 (see Figure 1). The vertices A0 and A3 are outer points of the (Beltrami-Cayley-Klein) B-C-K model of H³, as 1/u+1/v ≤ 1/2 is required, 3 ≤ u = w, v for the above orthoscheme parameters. For the above simple manifold-construction we specify u = v = w = 2z. Then the polar planes a0 and a3 of A0 and A3, respectively, make complete with reflections a0 and a3 the Coxeter reflection group, where the other reflections are denoted by b0, b¹, b², b³ in the sides of the orthoscheme b0b1b2b3. The situation is described first in Figure 1 of the half trunc-orthoscheme W and its usual extended Coxeter diagram, moreover, by the scalar product matrix (bij) = (⟨bi, bj⟩) in formula (1) and its inverse (Ajk) = (⟨Aj, Ak⟩) in (3). These will describe the hyperbolic angle and distance metric of the half trunc-orthoscheme W, then its ball packings, densities, then those of the manifolds Cw(2z). As first results we concentrate only on particular constructions by computer for probable material model realizations, atoms or molecules by equal balls, for general W(u; v;w = u) as well, summarized at the end of our paper. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to establish the higher order Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities on quaternionic hyperbolic spaces and on the Cayley hyperbolic plane using the Helgason-Fourier analysis on symmetric spaces. A crucial part of our work is to establish appropriate factorization theorems on these spaces, which can be of independent interest. To this end, we need to identify and introduce the "quaternionic Geller operators" and the "octonionic Geller operators", which have been absent on these spaces. Combining the factorization theorems and the Geller type operators with the Helgason-Fourier analysis on symmetric spaces, some precise estimates for the heat and the Bessel-Green-Riesz kernels, and the Kunze-Stein phenomenon for connected real simple groups of real rank one with finite center, we succeed to establish the higher order Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities on quaternionic hyperbolic spaces and on the Cayley hyperbolic plane. The kernel estimates required to prove these inequalities are also sufficient to establish the Adams and Hardy-Adams inequalities on these spaces. This paper, together with our earlier works on real and complex hyperbolic spaces, completes our study of the factorization theorems, higher order Poincaré-Sobolev, Hardy-Sobolev-Maz'ya, Adams and Hardy-Adams inequalities on all rank one symmetric spaces of noncompact type. [ABSTRACT FROM AUTHOR]

In the first part of this paper, we consider a partially overdetermined mixed boundary value problem in space forms and generalize the main result in [11] to the case of general domains with partial umbilical boundary in space forms. Precisely, we prove that a partially overdetermined problem in a domain with partial umbilical boundary admits a solution if and only if the rest part of the boundary is also part of an umbilical hypersurface. In the second part of this paper, we prove a Heintze–Karcher–Ros-type inequality for embedded hypersurfaces with free boundary lying on a horosphere or an equidistant hypersurface in the hyperbolic space. As an application, we show an Alexandrov-type theorem for constant mean curvature hypersurfaces with free boundary in these settings. [ABSTRACT FROM AUTHOR]

In this paper, we study the blow-up of solutions for hyperbolic equations involving the fractional Laplacian operator with damping and source terms. We obtain the global existence results. Then, we observe the blow-up of solutions using the concavity method. Finally, we present some numerical simulation results. [ABSTRACT FROM AUTHOR]

In this paper, some concepts of convex analysis on hyperbolic spaces are studied. We first study properties of the intrinsic distance, for instance, we present the spectral decomposition of its Hessian. Next, we study the concept of convex sets and the intrinsic projection onto these sets. We also study the concept of convex functions and present first- and second-order characterizations of these functions, as well as some optimization concepts related to them. An extensive study of the hyperbolically convex quadratic functions is also presented. [ABSTRACT FROM AUTHOR]

In the present paper, we first establish and verify a new sharp hyperbolic version of the Michael-Simon inequality for mean curvatures in hyperbolic space H n + 1 based on the locally constrained inverse curvature flow introduced by Brendle, Guan and Li ("An inverse curvature type hypersurface flow in H n + 1 ," (Preprint)) as follows (0.1) ∫ M λ ′ f 2 E 1 2 + | ∇ M f | 2 − ∫ M ∇ ̄ f λ ′ , ν + ∫ ∂ M f ≥ ω n 1 n ∫ M f n n − 1 n − 1 n provided that M is h-convex and f is a positive smooth function, where λ′(r) = coshr. In particular, when f is of constant, (0.1) coincides with the Minkowski type inequality stated by Brendle, Hung, and Wang in ("A Minkowski inequality for hypersurfaces in the anti-de Sitter-Schwarzschild manifold," Commun. Pure Appl. Math., vol. 69, no. 1, pp. 124–144, 2016). Further, we also establish and confirm a new sharp Michael-Simon inequality for the kth mean curvatures in H n + 1 by virtue of the Brendle-Guan-Li's flow ("An inverse curvature type hypersurface flow in H n + 1 ," (Preprint)) as below (0.2) ∫ M λ ′ f 2 E k 2 + | ∇ M f | 2 E k − 1 2 − ∫ M ∇ ̄ f λ ′ , ν ⋅ E k − 1 + ∫ ∂ M f ⋅ E k − 1 ≥ p k ◦ q 1 − 1 ( W 1 (Ω)) 1 n − k + 1 ∫ M f n − k + 1 n − k ⋅ E k − 1 n − k n − k + 1 provided that M is h-convex and Ω is the domain enclosed by M, pk(r) = ωn(λ′)k−1, W 1 (Ω) = 1 n | M | , λ′(r) = coshr, q 1 (r) = W 1 S r n + 1 , the area for a geodesic sphere of radius r, and q 1 − 1 is the inverse function of q1. In particular, when f is of constant and k is odd, (0.2) is exactly the weighted Alexandrov–Fenchel inequalities proven by Hu, Li, and Wei in ("Locally constrained curvature flows and geometric inequalities in hyperbolic space," Math. Ann., vol. 382, nos. 3–4, pp. 1425–1474, 2022). [ABSTRACT FROM AUTHOR]

In this paper, we study the following prescribed Gaussian curvature problem: K = f ~ ⁢ (θ) ϕ ⁢ (ρ) α − 2 ⁢ ϕ ⁢ (ρ) 2 + | ∇ ¯ ⁢ ρ | 2 , a generalization of the Alexandrov problem ( α = n + 1 ) in hyperbolic space, where f ~ is a smooth positive function on S n , 휌 is the radial function of the hypersurface, ϕ ⁢ (ρ) = sinh ⁡ ρ and 퐾 is the Gauss curvature. By a flow approach, we obtain the existence and uniqueness of solutions to the above equations when α ≥ n + 1 . Our argument provides a parabolic proof in smooth category for the Alexandrov problem in H n + 1 . We also consider the cases 2 < α ≤ n + 1 under the evenness assumption of f ~ and prove the existence of solutions to the above equations. [ABSTRACT FROM AUTHOR]

Visualizing graphs and networks in non‐Euclidean space can have benefits such as natural focus+context in hyperbolic space and the familiarity of interactions in spherical space. Despite work on these topics going back to the mid 1990s, there is no survey, or a part of a survey for this area of research. In this paper we review and categorize over 60 relevant papers and analyze them by geometry, (e.g., spherical, hyperbolic, torus), by contribution (e.g., technique, evaluation, proof, application), and by graph class (e.g., tree, planar, complex). [ABSTRACT FROM AUTHOR]

Gaussian random fields are completely characterised by their mean value and covariance function. Random fields on hyperbolic spaces have been studied to a limited extent only, namely for the case of scalar-valued fields that are not evolving over time. This paper challenges the problem of the second-order characteristics of multivariate (vector-valued) random fields that evolve temporally over hyperbolic spaces. Specifically, we characterise the continuous space–time covariance functions that are isotropic (radially symmetric) over space (the hyperbolic space) and stationary over time (the real line). Our finding is the analogue of recent findings that have been shown for the case where the space is either the n-dimensional sphere or more generally a two-point homogeneous space. Our main result can be read as a spectral representation theorem, and we also detail the main result for the subcase of covariance functions having a spectrum that is absolutely continuous with respect to the Lebesgue measure (technical details are reported below). [ABSTRACT FROM AUTHOR]

In general, it is difficult to measure distances in the Weil–Petersson metric on Teichmüller space. Here we consider the distance between strata in the Weil–Petersson completion of Teichmüller space of a surface of finite type. Wolpert showed that for strata whose closures do not intersect, there is a definite separation independent of the topology of the surface. We prove that the optimal value for this minimal separation is a constant δ 1 , 1 and show that it is realized exactly by strata whose nodes intersect once. We also give a nearly sharp estimate for δ 1 , 1 and give a lower bound on the size of the gap between δ 1 , 1 and the other distances. A major component of the paper is an effective version of Wolpert's upper bound on 〈 ∇ ℓ α , ∇ ℓ β 〉 , the inner product of the Weil–Petersson gradient of length functions. We further bound the distance to the boundary of Teichmüller space of a hyperbolic surface in terms of the length of the systole of the surface. We also obtain new lower bounds on the systole for the Weil–Petersson metric on the moduli space of a punctured torus. [ABSTRACT FROM AUTHOR]

In this paper, we construct signal constellations from lattices in complex hyperbolic spaces. To construct a hyperbolic lattice, we identify an arithmetic Fuchsian group with the group of units O1 of a natural quaternion order O ⊂ V, in which V is some quaternion algebra over an algebraic number field. The arithmetic Fuchsian group, G, is isomorphic to the fundamental group of a regular hyperbolic polygon P, and an oriented compact surface arises from the pairwise identification of its opposite edges. The polygon P is the fundamental region associated with a regular tessellation {p, q}. The main contribution of this paper is to employ the Reidemeister-Schreier rewriting process to use proper decompositions of the full symmetry group of a tessellation {p, q}, which allows the matching of uniform signal constellations to quotient groups of G. In this direction, we consider the labelling of phaseshift keying (PSK) and amplitude-phase keying (APK) signal constellations diagrams via three approaches: cyclic quotient groups, a direct product of cyclic quotient groups and semi-direct product of cyclic groups (the dihedral group case). [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of principal angles between subspaces of the same signature in a (real, complex or quaternionic) pseudo-euclidean space of index 1. We show that these determine the relative position of an important class of pairs of hyperbolic and elliptic subspaces (Theorems 3.9 and 4.10). Also, as an application, we will see that these angles can be used to study the relative position of pairs of an important class of totally geodesic submanifolds of the (real, complex or quaternionic) hyperbolic space (Theorem 6.3). As a consequence we obtain a nice interpretation of these principal angles as geometric invariants of the hyperbolic space. [ABSTRACT FROM AUTHOR]

Intelligent vehicular networks can not only connect various smart terminals to manned or unmanned vehicles but also to roads and people's hands. In order to support diverse vehicle-to-everything (V2X) applications in dynamic, intelligent vehicular networks, efficient and flexible routing is fundamental but challenging. Aimed to eliminate routing voids in traditional Euclidean geographic greedy routing strategies, we propose a hyperbolic-embedding-aided geographic routing strategy (HGR) in this paper. By embedding the network topology into a two-dimensional Poincaré hyperbolic disk, greedy forwarding is performed according to nodes' hyperbolic coordinates. Simulation results demonstrated that the proposed HGR strategy can greatly enhance the routing success rate through a smaller stretch of the routing paths, with little sacrifice of routing computation time. [ABSTRACT FROM AUTHOR]

For bounded domains of Euclidean spaces with piecewise smooth boundary, the integral of outward unit normal vectors of the boundary is zero. In this paper we consider a similar theorem on Minkowski spaces (Minkowski spaces does not mean finite dimensional Banach spaces but finite dimensional vector spaces with pseudo-inner products). We also consider Gauss' divergence theorem on Minkowski spaces, which implies above. Remark that this theorem implies easily the equation to calculate a kind of centroids of hyperbolic simplices. [ABSTRACT FROM AUTHOR]

The object of the present paper is to characterize two classes of almost Kenmotsu manifolds admitting almost η-Ricci solitons. In this context we have shown that in a (k, μ) and (k, μ)′-almost Kenmotsu manifold admitting an almost η-Ricci soliton the curvature conditions (i) the manifold is Einstein, (ii) the manifold is Ricci symmetric (∇S = 0), (iii) the manifold is Ricci semisymmetric (R · S = 0) and (iv) the manifold is projective Ricci semisymmetric (P · S = 0) are equivalent. Also we have shown that the curvature condition Q · P = 0 in a (k, μ)-almost Kenmotsu manifold admitting an almost η-Ricci soliton holds if and only if the manifold is locally isometric to the hyperbolic space H2n+1(-1) and if a (k, μ)′-almost Kenmotsu manifold admitting an almost η-Ricci soliton satisfies the curvature condition Q · R = 0, then it is locally isometric to the Riemannian product Hn+1(-4) × Rn. [ABSTRACT FROM AUTHOR]

Graph neural networks (GNNs) are powerful tools for data mining on graph-structured data in various domains, such as social science, finance, and biology. However, most existing GNNs operate in Euclidean space and may fail to preserve the intrinsic network properties, such as self-similarity and hierarchy, that characterize many real-world graphs. Hyperbolic graph neural networks (HGNNs) address this limitation by embedding graphs into hyperbolic space, which can better capture the hierarchical structures of networks. However, HGNNs often involve complex computations in hyperbolic space or its tangent space during training, which may hinder their efficiency. In this paper, we propose hyperbolic attention graph neural networks (HA-GNN), which can leverage both network structure and node features for graph representation learning in an efficient way. Specifically, we design a structural properties attention mechanism that measures the structural connection between nodes based on their hyperbolic embeddings. We also design a node features attention mechanism that quantifies the feature similarity between nodes. We then combine these two attentions to obtain a hyperbolic attention that weights the relevance between all connected nodes. We conduct extensive experiments on five real-world networks and demonstrate that our model consistently and significantly outperforms other state-of-the-art methods. For example, on the Cora network, our model achieves an accuracy of 83.1 (± 0.4) on node classification tasks, which is 1.6% higher than the best baseline method in Euclidean space. [ABSTRACT FROM AUTHOR]

The set of equivalence classes of cobounded actions of a group G$G$ on different hyperbolic metric spaces carries a natural partial order. Following Abbott–Balasubramanya–Osin, the group G$G$ is H$\mathcal {H}$‐accessible if the resulting poset has a largest element. In this paper, we prove that every nongeometric 3‐manifold has a finite cover with H$\mathcal {H}$‐inaccessible fundamental group and give conditions under which the fundamental group of the original manifold is H$\mathcal {H}$‐inaccessible. We also prove that every Croke–Kleiner admissible group (a class of graphs of groups that generalizes fundamental groups of three‐dimensional graph manifolds) has a finite index subgroup that is H$\mathcal {H}$‐inaccessible. [ABSTRACT FROM AUTHOR]

In this paper, we study the following prescribed Gaussian curvature problem: K = f ~ ⁢ (θ) ϕ ⁢ (ρ) α − 2 ⁢ ϕ ⁢ (ρ) 2 + | ∇ ¯ ⁢ ρ | 2 , a generalization of the Alexandrov problem ( α = n + 1 ) in hyperbolic space, where f ~ is a smooth positive function on S n , 휌 is the radial function of the hypersurface, ϕ ⁢ (ρ) = sinh ⁡ ρ and 퐾 is the Gauss curvature. By a flow approach, we obtain the existence and uniqueness of solutions to the above equations when α ≥ n + 1 . Our argument provides a parabolic proof in smooth category for the Alexandrov problem in H n + 1 . We also consider the cases 2 < α ≤ n + 1 under the evenness assumption of f ~ and prove the existence of solutions to the above equations. [ABSTRACT FROM AUTHOR]

In this paper, we study the stability of the volume preserving mean curvature flow of closed hypersurfaces in the hyperbolic space. We prove that an L 2 -almost umbilical hypersurface will be deformed to a totally umbilical hypersurface along the flow. Our result removes the assumption on the mean curvature in the theorems of Huang-Lin-Zhang [Peking J. Math. (2023)] and Leng-Xu-Zhao [Int. J. Math. (2014)]. [ABSTRACT FROM AUTHOR]

In this paper, we give sufficient conditions for Cauchy-completeness of Kobayashi hyperbolic domains in complex manifolds. The first result gives a sufficient condition for completeness for relatively compact domains in several large classes of manifolds. This follows from our second result, which may be of independent interest, in a much more general setting. This extends a result of Gaussier to the setting of manifolds. [ABSTRACT FROM AUTHOR]

We consider the question of whether the high-energy eigenfunctions of certain Schrödinger operators on the d-dimensional hyperbolic space of constant curvature −κ2 are flexible enough to approximate an arbitrary solution of the Helmholtz equation Δh + h = 0 on R d , over the natural length scale O ( λ − 1 / 2 ) determined by the eigenvalue λ ≫ 1. This problem is motivated by the fact that, by the asymptotics of the local Weyl law, approximate Laplace eigenfunctions do have this approximation property on any compact Riemannian manifold. In this paper we are specifically interested in the Coulomb and harmonic oscillator operators on the hyperbolic spaces H d (κ). As the dimension of the space of bound states of these operators tends to infinity as κ ↘ 0, one can hope to approximate solutions to the Helmholtz equation by eigenfunctions for some κ > 0 that is not fixed a priori. Our main result shows that this is indeed the case, under suitable hypotheses. We also prove a global approximation theorem with decay for the Helmholtz equation on manifolds that are isometric to the hyperbolic space outside a compact set, and consider an application to the study of the heat equation on H d (κ). Although global approximation and inverse approximation results are heuristically related in that both theorems explore flexibility properties of solutions to elliptic equations on hyperbolic spaces, we will see that the underlying ideas behind these theorems are very different. [ABSTRACT FROM AUTHOR]

This paper introduces two novel scores for detecting local perturbations in networks. For this, we consider a non-Euclidean representation of networks, namely, their embedding onto the Poincaré disk model of hyperbolic geometry. We numerically evaluate the performances of these scores for the detection and localization of perturbations on homogeneous and heterogeneous network models. To illustrate our approach, we study latent geometric representations of real brain networks to identify and quantify the impact of epilepsy surgery on brain regions. Results suggest that our approach can provide a powerful tool for representing and analyzing changes in brain networks following surgical intervention, marking the first application of geometric network embedding in epilepsy research. [ABSTRACT FROM AUTHOR]

Actions on hyperbolic metric spaces are an important tool for studying groups, and so it is natural, but difficult, to attempt to classify all such actions of a fixed group. In this paper, we build strong connections between hyperbolic geometry and commutative algebra in order to classify the cobounded hyperbolic actions of numerous metabelian groups up to a coarse equivalence. In particular, we turn this classification problem into the problems of classifying ideals in the completions of certain rings and calculating invariant subspaces of matrices. We use this framework to classify the cobounded hyperbolic actions of many abelian‐by‐cyclic groups associated to expanding integer matrices. Each such action is equivalent to an action on a tree or on a Heintze group (a classically studied class of negatively curved Lie groups). Our investigations incorporate number systems, factorization in formal power series rings, completions, and valuations. [ABSTRACT FROM AUTHOR]

In this paper, we prove sharp isoperimetric inequalities for lower order eigenvalues of Neumann Laplacian on bounded domains in both compact and noncompact rank-1 symmetric spaces. Our results generalize the work of Wang and Xia for bounded domains in the hyperbolic space (Xia and Wang in Math Ann 385(1–2):863–879, 2023), and Szegö–Weinberger inequalities in rank-1 symmetric spaces obtained by Aithal and Santhanam (Trans Am Math Soc 348(10):3955–3965, 1996). [ABSTRACT FROM AUTHOR]

In this paper, we prove some fixed point results for relatively nonexpansive mappings using Thakur iteration in uniformly convex W − Hyperbolic spaces. Our results generalize some of the existing results in the literature. We illustrate our results with an example and shown the convergence of Thakur iteration to the fixed points of relatively nonexpansive mappings using Matlab code. [ABSTRACT FROM AUTHOR]

The problem of the prescribed curvature measure is one of the important problems in differential geometry and nonlinear partial differential equations. In this paper, we consider the prescribed curvature measure problem in the hyperbolic space. We obtain the existence of star‐shaped k‐convex bodies with prescribed (n‐k)‐th curvature measures (k

This paper focuses on the geometric properties of quasihyperbolic geodesics in centered John metric spaces. Our contribution is to provide a necessary and sufficient condition in centered John spaces such that quasihyperbolic geodesics terminating at the center to be cone arcs. As an application, we prove that for a centered John space which carries a Gromov hyperbolic quasihyperbolization, every quasihyperbolic geodesic terminating at the center is a cone arc. [ABSTRACT FROM AUTHOR]

A classical result of Cheng states that the bottom spectrum of complete manifolds of fixed dimension and Ricci curvature lower bound achieves its maximal value on the corresponding hyperbolic space. The paper establishes an analogous result for three-dimensional complete manifolds with scalar curvature lower bound subject to some necessary topological assumptions. The rigidity issue is also addressed and a splitting theorem is obtained for such manifolds with the maximal bottom spectrum. [ABSTRACT FROM AUTHOR]

In this manuscript, we introduce a new class of uniformly L-Lipschitzian and total asymptotically quasi-nonexpansive mappings, which is significantly more general than many known nonexpansive mappings that appeared before. We establish some convergence and Δ-convergence theorems for two infinite families of such mappings in the setting of CAT(κ) spaces. Our results refine, generalize, and improve several corresponding results in the existing literature. [ABSTRACT FROM AUTHOR]

We show that if 퐺 is an admissible group acting geometrically on a CAT ⁡ (0) space 푋, then 퐺 is a hierarchically hyperbolic space and its 휅-Morse boundary (∂ κ G , ν) is a model for the Poisson boundary of (G , μ) , where 휈 is the hitting measure associated to the random walk driven by 휇. [ABSTRACT FROM AUTHOR]

Hyperbolic spaces have recently achieved acceleration in the context of machine learning of their high capacity and tree-likeliness structures, taxonomies, text, and graphs. With the same dimension, a hyperbolic vector can represent richer information than a Euclidean vector. In this paper, we propose Metaheuristic optimization algorithms in hyperbolic space. Considering that the most popular optimization tools have not been generalized in hyperbolic space, we design optimization algorithms according to the specific property of the hyperbolic manifold. However, a major bottleneck here is the obscurity of hyperbolic space and a better comprehension of its gyrovector operations. We aim to introduce researchers and practitioners in the metaheuristic community to the hyperbolic equivariant of the Euclidean operations necessary to tackle their application to Metaheuristic optimization. We conduct experiments on various metaheuristic algorithms in hyperbolic space. [ABSTRACT FROM AUTHOR]

In this paper, we consider the α -Gauss curvature flow for complete convex graphs over horosphere in the hyperbolic space. We show that for all positive power α > 0 , if the initial hypersurface is smooth, complete non-compact uniformly convex graph over R n and bounded by two horospheres, then the solution of the flow exists for all time. Moreover, the evolution of horospheres act as barriers along the flow. [ABSTRACT FROM AUTHOR]

This paper proposes a novel 3D point cloud segmentation network called EDGCNet. Structurally, the network combines the encoder–decoder structure and graph convolution to improve the processing efficiency and the exploration of point-to-point correlation. In addition, it contains two sub-modules: the dynamic hyperbaric graph convolution module and the dual squeeze-and-attention module. To mitigate the issue of incorrect inference in existing graph convolution networks for multi-category hybrid regions, the dynamic hyperbolic graph convolution module maps the features captured in the Euclidean space to the hyperbolic space to aggregate the neighborhoods and dynamically update the edge weights. Compared to Euclidean geometry, hyperbolic embedding with learnable curvature can deeply fit the geometric topology of point cloud by transforming the manifold shape to enrich feature representation. The dual squeeze-and-attention module recalibrates the original features by modeling the correlation between channels from both global and local perspectives to improve the feature shift caused by the deepening of the convolution. Empirical experiments on three public datasets verify that EDGCNet has an excellent segmentation effect on both artificial shapes and real scenes. Furthermore, ablation studies and confirmatory experiments show that the modules in EDGCNet are contributing, low-complexity and robust. • Our network covers the advantages of encoder–decoder structure and graph convolution network. • We are the first to use dynamic hyperbolic manifolds to represent the correlation of 3D point clouds. • The proposed network EDGCNet can monitor and correct the biased features in time. • We improve Feature Propagation model which always be ignored by others. • Empirical experiments show that EDGCNet has excellent segmentation performance. [ABSTRACT FROM AUTHOR]

Given two circle patterns of the same combinatorics in the plane, the Möbius transformations mapping circumdisks of one to the other induce a PSL(2,\mathbb {C})-valued function on the dual graph. Such a function plays the role of an osculating Möbius transformation and induces a realization of the dual graph in hyperbolic space. We characterize the realizations and obtain a one-to-one correspondence in the cases that the two circle patterns share the same discrete conformal structure. These correspondences are analogous to the Weierstrass representation for surfaces with constant mean curvature H\equiv 1 in hyperbolic space. We further establish convergence on triangular lattices. [ABSTRACT FROM AUTHOR]

Tree-like structures, characterized by hierarchical relationships and power-law distributions, are prevalent in a multitude of real-world networks, ranging from social networks to citation networks and protein–protein interaction networks. Recently, there has been significant interest in utilizing hyperbolic space to model these structures, owing to its capability to represent them with diminished distortions compared to flat Euclidean space. However, real-world networks often display a blend of flat, tree-like, and circular substructures, resulting in heterophily. To address this diversity of substructures, this study aims to investigate the reconstruction of graph neural networks on the symmetric manifold, which offers a comprehensive geometric space for more effective modeling of tree-like heterophily. To achieve this objective, we propose a graph convolutional neural network operating on the symmetric positive-definite matrix manifold, leveraging Riemannian metrics to facilitate the scheme of information propagation. Extensive experiments conducted on semi-supervised node classification tasks validate the superiority of the proposed approach, demonstrating that it outperforms comparative models based on Euclidean and hyperbolic geometries. [ABSTRACT FROM AUTHOR]

We consider the volume preserving flow of smooth, closed and convex hypersurfaces in the hyperbolic space \mathbb {H}^{n+1} (n\geq 2) with the speed given by arbitrary positive power \alpha of the Gauss curvature. We prove that if the initial hypersurface is convex, then the smooth solution of the flow remains convex and exists for all positive time t\in [0,\infty). Moreover, we apply a result of Kohlmann which characterises the geodesic ball using the hyperbolic curvature measures and an argument of Alexandrov reflection to prove that the flow converges to a geodesic sphere exponentially in the smooth topology. This can be viewed as the first result for non-local type volume preserving curvature flows for hypersurfaces in the hyperbolic space with only convexity required on the initial data. [ABSTRACT FROM AUTHOR]

Given the prolonged timelines and high costs associated with traditional approaches, accelerating drug development is crucial. Computational methods, particularly drug-target interaction prediction, have emerged as efficient tools, yet the explainability of machine learning models remains a challenge. Our work aims to provide more interpretable interaction prediction models using similarity-based prediction in a latent space aligned to biological hierarchies. We investigated integrating drug and protein hierarchies into a joint-embedding drug-target latent space via embedding regularization by conducting a comparative analysis between models employing traditional flat Euclidean vector spaces and those utilizing hyperbolic embeddings. Besides, we provided a latent space analysis as an example to show how we can gain visual insights into the trained model with the help of dimensionality reduction. Our results demonstrate that hierarchy regularization improves interpretability without compromising predictive performance. Furthermore, integrating hyperbolic embeddings, coupled with regularization, enhances the quality of the embedded hierarchy trees. Our approach enables a more informed and insightful application of interaction prediction models in drug discovery by constructing an interpretable hyperbolic latent space, simultaneously incorporating drug and target hierarchies and pairing them with available interaction information. Moreover, compatible with pairwise methods, the approach allows for additional transparency through existing explainable AI solutions. [ABSTRACT FROM AUTHOR]

For 0 < k < 1, a finite-type k-surface in 3-dimensional hyperbolic space is a complete, immersed surface of finite area and of constant extrinsic curvature equal to k. In Smith (2021), we showed that such surfaces have finite genus and finitely many cusp-like ends. Each of these cusps is asymptotic to an immersed cylinder of exponentially decaying radius about a complete geodesic and terminates at an ideal point which we call the extremity of the cusp. We show that every cusp of any finite-type k-surface has a well-defined axis, which we will call the Steiner geodesic of the cusp. One of the end-points of this axis is the extremity, and we will call the other, which constitutes new geometric data, the Steiner point of the cusp. We prove a new identity involving extremities and Steiner points in terms of Möbius invariant vector fields over the Riemann sphere. We define two new functionals over the space of finite-type k-surfaces. The first, which will be called the generalized volume, is defined by the integral of a certain well-chosen form, and extends to the non-embedded case the concept of volume of the set bounded by the surface. The second, which will be called the renormalized energy, is related to the integral of the mean curvature of the surface, and is well-defined up to a choice of Busemann function. Upon describing natural parametrizations of the strata of the space of finite-type k-surfaces by open complex manifolds, we prove a new Schläfli-type formula relating the extremities and Steiner points to the first order variations of the generalized volume and the renormalized energy. In particular, Möbius invariance of this formula yields the aforementioned identity. We conclude by studying some applications of this identity and Schläfli-type formula. [ABSTRACT FROM AUTHOR]

We study relations between maps between relatively hyperbolic groups/spaces and quasisymmetric embeddings between their boundaries. More specifically, we establish a correspondence between (not necessarily coarsely surjective) quasi-isometric embeddings between relatively hyperbolic groups/spaces that coarsely respect peripherals, and quasisymmetric embeddings between their boundaries satisfying suitable conditions. Further, we establish a similar correspondence regarding maps with at most polynomial distortion. We use this to characterise groups which are hyperbolic relative to some collection of virtually nilpotent subgroups as exactly those groups which admit an embedding into a truncated real hyperbolic space with at most polynomial distortion, generalising a result of Bonk and Schramm for hyperbolic groups. [ABSTRACT FROM AUTHOR]

We prove a comparison theorem for certain types of polyhedra in a 3-manifold with its scalar curvature bounded below by -6. The result confirms in some cases the Gromov dihedral rigidity conjecture in hyperbolic 3-space. [ABSTRACT FROM AUTHOR]

A novel video summarisation method called the Hyperbolic Graph Convolutional Network (HVSN) is proposed, which addresses the challenges of summarising edited videos and capturing the semantic consistency of video shots at different time points. Unlike existing methods that use linear video sequences as input, HVSN leverages Hyperbolic Graph Convolutional Networks (HGCNs) and an adaptive graph convolutional adjacency matrix network to learn and aggregate features from video shots. Moreover, a feature fusion mechanism based on the attention mechanism is employed to facilitate cross‐module feature fusion. To evaluate the performance of the proposed method, experiments are conducted on two benchmark datasets, TVSum and SumMe. The results demonstrate that HVSN achieves state‐of‐the‐art performance, with F1‐scores of 62.04% and 50.26% on TVSum and SumMe, respectively. The use of HGCNs enables the model to better capture the complex spatial structures of video shots, and thus contributes to the improved performance of video summarisation. [ABSTRACT FROM AUTHOR]

We first briefly summarize several well-known properties of regular tessellations of the three two-dimensional maximally symmetric manifolds, E 2 , S 2 , and H 2 , by bounded regular tiles. For instance, there exist infinitely many regular tessellations of the hyperbolic plane H 2 by curved hyperbolic equilateral triangles whose vertex angles are 2 π / d for d = 7 , 8 , 9 , ... On the other hand, we prove that there is no curved hyperbolic regular tetrahedron which tessellates the three-dimensional hyperbolic space H 3 . We also show that a regular tessellation of H 3 can only consist of the hyperbolic cubes, hyperbolic regular icosahedra, or two types of hyperbolic regular dodecahedra. There exist only two regular hyperbolic space-fillers of H 4 . If n > 4 , then there exists no regular tessellation of H n . [ABSTRACT FROM AUTHOR]

In this work we start from near vector spaces, which we endow with some additional properties that allow convex analysis. The seminormed structure used here will also be improved by adding properties such as the null condition and null equality, thus resulting in a new type of space, which is still weaker than the conventional Banach structures: pre-convex regular near-Banach space. On the newly defined structure, we introduce the concept of uniform convexity and analyze several resulting properties. The major outcomes prove a remarkable resemblance to the classical properties resulting from uniform convexity on hyperbolic metric spaces or modular function spaces, including the famous Browder-Gohde fixed point theorem. [ABSTRACT FROM AUTHOR]

We investigate existence and nonexistence of global in time nonnegative solutions to the semilinear heat equation, with a reaction term of the type e μ t u p (μ ∈ R , p > 1) , posed on cones of the hyperbolic space. Under a certain assumption on µ and p, related to the bottom of the spectrum of − Δ in H n , we prove that any solution blows up in finite time, for any nontrivial nonnegative initial datum. Instead, if the parameters µ and p satisfy the opposite condition we have: (a) blow-up when the initial datum is large enough, (b) existence of global solutions when the initial datum is small enough. Hence our conditions on the parameters µ and p are optimal. We see that blow-up and global existence do not depend on the amplitude of the cone. This is very different from what happens in the Euclidean setting (Bandle and Levine 1989 Trans. Am. Math. Soc. 316 595–622), and it is essentially due to a specific geometric feature of H n . [ABSTRACT FROM AUTHOR]

We introduce the notion of locally visible and locally Gromov hyperbolic domains in \mathbb {C}^d. We prove that a bounded domain in \mathbb {C}^d is locally visible and locally Gromov hyperbolic if and only if it is (globally) visible and Gromov hyperbolic with respect to the Kobayashi distance. This allows to detect, from local information near the boundary, those domains which are Gromov hyperbolic and for which biholomorphisms extend continuously up to the boundary. [ABSTRACT FROM AUTHOR]

Event sequences with spatiotemporal characteristics have been rapidly produced in various domains, such as earthquakes in seismology, electronic medical records in health care, and transactions in the financial market. These data are discrete events that are continuous and often continue for weeks, months, or years, and past events may trigger subsequent events. In this context, modeling spatiotemporal event sequences and forecasting the occurrence time and marker of the next event has become a hot topic. However, existing models either fail to capture the long-term temporal dependencies or ignore the essential spatial information between sequences. Moreover, existing models learn the influence of past events and predicted future events in Euclidean space, which has been shown to cause a significant distortion in hierarchical structure data. To correctly predict future events from historical events and design interventions and controls to guide the event dynamics to the desired results and inspired by the high capacity of modeling data in hyperbolic space, we proposed a novel hyperbolic graph transformer Hawkes process (HGTHP) model to capture the long-term temporal dependencies and spatial information from historical events with a hierarchical structure. The core concept of the HGTHP is to integrate the learned spatial information into the event embedding as auxiliary information and capture long-short term temporal dependencies from event sequences in non-Euclidean space by a hyperbolic self-attention mechanism. Numerous experiments on synthetic and real-world datasets proved that the proposed model obtains spatiotemporal information from hyperbolic space, and its predictions outperform those of state-of-the-art baselines in both time and marker, proving the proposed model's effectiveness. [ABSTRACT FROM AUTHOR]