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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]

We present an explicit formula for calculating the volume of an arbitrary hyperbolic 4-simplex in terms of the coordinates of its vertices; by this formula, the volume can be expressed in terms of one-dimensional integrals of real-valued integrands over closed intervals of the real line. In addition, it is proved in the paper that the volume of a hyperbolic 5-simplex cannot be expressed as the double integral of an elementary function of the coordinates of its vertices (of edge lengths). [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]

In one of our recent papers, we considered a complete metric space of nonexpansive mappings taking a bounded and closed subset of a complete hyperbolic space into the space so that the boundary of this subset is mapped back into the subset itself. Using the Baire category approach, we proved that most of these mappings possess a unique fixed point which attracts all their iterates. In the present paper, we improve upon this result by showing that the complement of the set of mappings which have a fixed point is not only of the first Baire category, but also is s-porous. [ABSTRACT FROM AUTHOR]

Most existing representation learning models for heterogeneous graphs depend on meta-paths, which requires domain-specific prior knowledge and reduces model practicality. In addition, real-world graphs usually conform to power-law distributions, and conventional graph models defined in Euclidean space lead to high distortion for such data. In this paper, we propose a Multi-curvature Hyperbolic Heterogeneous Graph Convolutional Network (McH-HGCN) based on the graph's inherent type triplets. By selecting triplets as data units for message passing and defining the model in hyperbolic space, our model caters to the power-law properties of heterogeneous graphs while avoiding meta-paths dependence. To model the heterogeneity of the graph, we set distinct hyperbolic curvatures as learnable parameters for different types of nodes to obtain the optimal parameterized space mapping after training. Additionally, we introduce a dynamic heterogeneous attention mechanism to compute the attention weights for heterogeneous neighbor aggregation. Node classification and recommendation experiments with several heterogeneous graph datasets show that our model outperforms state-of-the-art methods on multiple datasets, achieving excellent performance without relying on meta-paths. [ABSTRACT FROM AUTHOR]

In this paper, we present a method of embedding physics data manifolds with metric structure into lower dimensional spaces with simpler metrics, such as Euclidean and Hyperbolic spaces. We then demonstrate that it can be a powerful step in the data analysis pipeline for many applications. Using progressively more realistic simulated collisions at the Large Hadron Collider, we show that this embedding approach learns the underlying latent structure. With the notion of volume in Euclidean spaces, we provide for the first time a viable solution to quantifying the true search capability of model agnostic search algorithms in collider physics (i.e. anomaly detection). Finally, we discuss how the ideas presented in this paper can be employed to solve many practical challenges that require the extraction of physically meaningful representations from information in complex high dimensional datasets. [ABSTRACT FROM AUTHOR]

Deep learning has been widely used in medical image segmentation, such as breast tumor segmentation, prostate MR image segmentation, and so on. However, the labeling of the data set takes a lot of time. Although the emergence of unsupervised domain adaptation fills the technical gap, the existing domain adaptation methods for breast segmentation do not consider the alignment of the source domain and target domain breast mass structure. This paper proposes a hyperbolic graph convolutional network architecture. First, a hyperbolic graph convolutional network is used to make the source and target domains structurally aligned. Secondly, we adopt a hyperbolic space mapping model that has better expressive ability than Euclidean space in a graph structure. In particular, when constructing the graph structure, we added the completion adjacency matrix, so that the graph structure can be changed after each feature mapping, which can better improve the segmentation accuracy. Extensive comparative and ablation experiments were performed on two common breast datasets(CBIS-DDSM and INbreast). Experiments show that the method in this paper is better than the most advanced model. When CBIS-DDSM and INbreast are used as the source domain, the segmentation accuracy reaches 89.1% and 80.7%. [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]

In this paper, we solve the asymptotic Plateau problem in hyperbolic space for constant \sigma _{n-1} curvature, i.e. the existence of a complete hypersurface in \mathbb {H}^{n+1} satisfying \sigma _{n-1}(\kappa)=\sigma \in (0,n) with a prescribed asymptotic boundary \Gamma. The key ingredient is the curvature estimates. Previously, this was only known for \sigma _0<\sigma

Predicting network abnormal events and behavior can enhance security situation awareness and the ability to infer attack intentions. Most of the existing abnormal event prediction methods usually rely on the temporal relationship features between events and the spatial relationship features between hosts. However, the existing spatio-temporal anomaly event prediction methods do not fully consider the spatial relationship between events and the cross-domain environment of the behavior, resulting in poor performance in practical applications. In addition, the existing methods are mostly based on Euclidean space and hyperbolic space in terms of feature space relationship representation and do not fully consider the complexity of the relationship structure of anomalous events. In this paper, we propose a cross-domain spatio-temporal abnormal events prediction method, referred to as CDSTAEP. This method divides the local event sequence based on the temporal behavior sequence of entities and realizes the graphical representation of the multi-domain event correlation relationship. In the mixed-curvature space, we realize the representation learning of the correlation relationship of complex events and combine the event mixed-curvature vector representation and attention-based long short-term memory (LSTM-ATT) to capture the spatial and temporal correlation characteristics of cross-domain events, and finally realize the prediction. In this paper the proposed CDSTAEP is verified with the live network data set collected by a national key research and development plan. The results demonstrate that CDSTAEP can retain more spatial relationship features between events, the area under roc curve (AUC) score is better than the result of single-space representation and is 4.53% and 6.699% higher than the baseline models such as LSTM and LSTM-ATT. [ABSTRACT FROM AUTHOR]

In this paper, we first study a mapping problem between indefinite hyperbolic spaces by employing the work established earlier by the authors. In particular, we generalize certain theorems proved by Baouendi-Ebenfelt-Huang [Amer. J. Math. 133 (2011), pp. 1633–1661] and Ng [Michigan Math. J. 62 (2013), pp. 769–777; Int. Math. Res. Not. IMRN 2 (2015), pp. 291–324]. Then we use these results to prove a rigidity result for proper holomorphic mappings between type I classical domains, which confirms a conjecture formulated by Chan [Int. Math. Res. Not., doi.org/10.1093/imrn/rnaa373] after the work of Zaitsev-Kim [Math. Ann. 362 (2015), pp. 639-677], Kim [ Proper holomorphic maps between bounded symmetric domains , Springer, Tokyo, 2015, pp. 207–219] and himself. [ABSTRACT FROM AUTHOR]

The paper shows that it is possible to construct quantum chromodynamics as a rigorous theory on the basis of employment of hyperbolic unitary group S U h (3) , which is a symmetry group for the three-dimensional complex space of the hyperbolic type. Such an approach allows researchers to discover a profound connection between conserved color charges of the quarks and the symmetries of the hyperbolic three-dimensional complex space. Further, it allows a correct introduction of the Hermitian operators to describe the eight gluons, which are carriers of strong interactions. [ABSTRACT FROM AUTHOR]

This paper is dedicated to the advancement of fixed-point results for multi-valued asymptotically non-expansive maps regarding convergence criteria in complete uniformly convex hyperbolic metric spaces that are endowed with a graph. The famous fixed-point theorems of Goebel and Kirk, Khamsi and Khan, along with other recent results in the literature can be obtained as corollaries of these main results. A nice graph and an interesting example are also provided in support of the hypothesis of the main results. [ABSTRACT FROM AUTHOR]

Embedding data in hyperbolic spaces has proven beneficial for many advanced machine learning applications. However, working in hyperbolic spaces is not without difficulties as a result of its curved geometry (e.g., computing the Fréchet mean of a set of points requires an iterative algorithm). In Euclidean spaces, one can resort to kernel machines that not only enjoy rich theoretical properties but that can also lead to superior representational power (e.g., infinite-width neural networks). In this paper, we introduce valid kernel functions for hyperbolic representations. This brings in two major advantages, 1. kernelization will pave the way to seamlessly benefit the representational power from kernel machines in conjunction with hyperbolic embeddings, and 2. the rich structure of the Hilbert spaces associated with kernel machines enables us to simplify various operations involving hyperbolic data. That said, identifying valid kernel functions on curved spaces is not straightforward and is indeed considered an open problem in the learning community. Our work addresses this gap and develops several positive definite kernels in hyperbolic spaces (modeled by a Poincaré ball), the proposed kernels include the rich universal ones (e.g., Poincaré RBF kernel), or realize the multiple kernel learning scheme (e.g., Poincaré radial kernel). We comprehensively study the proposed kernels on a variety of challenging tasks including few-shot learning, zero-shot learning, person re-identification, deep metric learning, knowledge distillation and self-supervised learning. The consistent performance gain over different tasks shows the benefits of the kernelization for hyperbolic representations. [ABSTRACT FROM AUTHOR]

Graph Convolutional Networks (GCNs) have attracted broad attention from industry and academia, for their excellent expressive power in terms of modeling the irregular data, e.g., skeletal data and graph-structured data. The most effective existing model may be the fully hyperbolic graph neural network. However, it involves a large number of parameters, thus consuming considerable computing resources. In this paper, we propose a model based on adaptive frequency filter and corresponding optimizer in hyperbolic space. The adaptive frequency can learn the different frequency components of the embeddings of the nodes in graph, which adaptively adjust the beneficial signals of high-frequency and low-frequency. And the optimizer is based on a subset of the orthogonal constraint, which is dedicated for the adaptive frequency with less parameters. Moreover, our model bridges the gap of hyperbolic space and the spectral space for exploring the underlying semantics of the node and relation embeddings of graph. Consequently, our model needs only to optimize the less parameters in hyperbolic space and meanwhile prevent the distortion caused by conventional manifold GCN. Experimental results show that our method achieves substantial improvements and outperforms the state-of-the-art performance in terms of node classification. [ABSTRACT FROM AUTHOR]

In this paper, our main aim is to develop some basic properties of hyperbolic valued metric spaces. We also establish the hyperbolic version of the Banach contraction principle. Further, we construct a hyperbolic valued metric on the space of all hyperbolic valued continuous functions and prove some results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the geometric properties of quasi-trees and prove some equivalent criteria. We give a general construction of a tree that approximates the ends of a geodesic space, and use this to prove that every quasi-tree is (1, C)-quasi-isometric to a simplicial tree. As a consequence, we show that Gromov's tree approximation lemma for hyperbolic spaces (1987) can be improved in the case of quasi-trees to give a uniform approximation for any set of points, independent of cardinality. From this we show that having uniform tree approximation for finite subsets is equivalent to being able to uniformly approximate the entire space by a tree. As another consequence, we note that the boundary of a quasi-tree is isometric to the boundary of its approximating tree under a certain choice of visual metric, and that this gives a natural extension of the standard metric on the boundary of a tree. [ABSTRACT FROM AUTHOR]

This paper studies the triangle similarity classes obtained by iterative application of the longest-edge trisection of triangles. The longest-edge trisection (3T-LE) of a triangle is obtained by joining the two points which divide the longest edge in three equal parts with the opposite vertex. This partition, as well as the longest-edge bisection (2T-LE), does not degenerate, which means that there is a positive lower bound to the minimum angle generated. However, unlike what happens with the 2T-LE, the number of similarity classes appearing by the iterative application of the 3T-LE to a single initial triangle is not finite in general. There are only three exceptions to this fact: the right triangle with its sides in the ratio 1: 2 : 3 and the other two triangles in its orbit. This result, although of a combinatorial nature, is proved here with the machinery of discrete dynamics in a triangle shape space with hyperbolic metric. It is also shown that for a point with an infinite orbit, infinite points of the orbit are in three circles with centers at the points with finite orbits. [ABSTRACT FROM AUTHOR]

Glaucoma is one of the most irreversible eye diseases that causes visual damage worldwide. For glaucoma screening, it is essential to accurately segment the optic disc (OD) from the optic cup (OC) in fundus images. However, most of the known segmentation methods are two-stage processes and cannot obtain satisfactory segmentations without performing OD localization in advance. To overcome this limitation, in this paper, we propose a novel one-stage framework named the multiple graph reasoning network (MGRNet) to segment ODs and OCs from different fundus image datasets. The MGRNet performs graph inference in the Euclidean space, channel space, and hyperbolic space. Euclidean space-based graph reasoning captures the spatial dependencies between pixels from the coordinate dimension. Channel space-based graph reasoning constructs the channel relationship between any pair of channel maps. Hyperbolic space-based graph reasoning provides pixel-level hierarchical embeddings in the hyperbolic space. The MGRNet exhibits obvious advantages over a strong baseline without preprocessing operations, including OD localization, region of interest (ROI) cropping, and data augmentation advantages, and achieves satisfactory results on both the REFUGE (0.9442 mean intersection over union (IoU)) and DRISHTI-GS (0.8792 mean IoU) datasets. Additionally, the MGRNet can also provide reliable glaucoma screening results by measuring the cup-to-disc ratio (CDR) based on segmentation results. [ABSTRACT FROM AUTHOR]

In this paper, we continue our investigations giving the characterization of weights for two-weight Hardy inequalities to hold on general metric measure spaces possessing polar decompositions. Since there may be no differentiable structure on such spaces, the inequalities are given in the integral form in the spirit of Hardy's original inequality. This is a continuation of our paper (Ruzhansky & Verma 2018. Proc. R. Soc. A475, 20180310 (doi:10.1098/rspa.2018.0310)) where we treated the case p ≤ q. Here the remaining range p > q is considered, namely, 0 < q < p, 1 < p < ∞. We give several examples of the obtained results, finding conditions on the weights for integral Hardy inequalities on homogeneous groups, as well as on hyperbolic spaces and on more general Cartan–Hadamard manifolds. As in the first part of this paper, we do not need to impose doubling conditions on the metric. [ABSTRACT FROM AUTHOR]

Complex system theory is increasingly applied to develop control protocols for distributed computational and networking resources. The paper deals with the important subproblem of finding complex connected structures having excellent navigability properties using limited computational resources. Recently, the two-dimensional hyperbolic space turned out to be an efficient geometry for generative models of complex networks. The networks generated using the hyperbolic metric space share their basic structural properties (like small diameter or scale-free degree distribution) with several real networks. In the paper, a new model is proposed for generating navigation trees for complex networks embedded in the two-dimensional hyperbolic plane. The generative model is not based on known hyperbolic network models: the trees are not inferred from the existing links of any network; they are generated from scratch instead and based purely on the hyperbolic coordinates of nodes. We show that these hyperbolic trees have scale-free degree distributions and are present to a large extent both in synthetic hyperbolic complex networks and real ones (Internet autonomous system topology, US flight network) embedded in the hyperbolic plane. As the main result, we show that routing on the generated hyperbolic trees is optimal in terms of total memory usage of forwarding tables. [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 the two preceding parts of this series of papers, we introduced and studied a recursion scheme for constructing joint eigenfunctions |$J_N(a_+, a_-,b;x,y)$| of the Hamiltonians arising in the integrable |$N$| -particle systems of hyperbolic relativistic Calogero–Moser type. We focused on the 1st steps of the scheme in Part I and on the cases |$N=2$| and |$N=3$| in Part II. In this paper, we determine the dominant asymptotics of a similarity-transformed function |$\textrm{E}_N(b;x,y)$| for |$y_j-y_{j+1}\to \infty $|⁠ , |$j=1,\ldots , N-1$| and thereby confirm the long-standing conjecture that the particles in the hyperbolic relativistic Calogero–Moser system exhibit soliton scattering. This result generalizes a main result in Part II to all particle numbers |$N>3$|⁠. [ABSTRACT FROM AUTHOR]

In this paper, we prove some fixed points properties and demiclosedness principle for mean nonexpansive mapping in uniformly convex hyperbolic spaces. We further propose an iterative scheme for approximating a common fixed point of two mean nonexpansive mappings and establish some strong and △-convergence theorems for these mappings in uniformly convex hyperbolic spaces. The results obtained in this paper extend and generalize corresponding results in uniformly convex Banach spaces, CAT(0) spaces and other related results in literature. [ABSTRACT FROM AUTHOR]

The formability limits of bulk metal forming in principal strain space and in the effective strain vs. stress-triaxiality space are characterized by an uncertainty region in which cracks may be triggered by tension (mode I of fracture mechanics) or by out-of-plane shear (mode III). The problem in obtaining experimental data in this region has been known for a long time and the main objective of this paper is to present a new upset formability test geometry that can effectively contribute to the characterization of the formability limits of bulk metal forming parts subjected to biaxial tension. Alongside with this objective, this paper also presents an analytical expression for converting the fracture forming limit line corresponding to crack opening by mode III in principal strain space into a hyperbolic fracture limit curve in the effective strain vs. stress-triaxiality space. The overall methodology employed by the authors combines experimentation along with analytical and numerical modelling, and the contents of the paper is a step towards diminishing the actual lack of knowledge regarding failure by fracture in bulk metal forming parts subject to stress-triaxiality values beyond uniaxial tension. Results show that a new uncoupled ductile fracture criterion built upon combination of the integrands of the Cockcroft-Latham and McClintock criteria can be successfully used to model the physics of the bulk metal forming limits for the entire range of stress-triaxiality values corresponding to cracking on free surfaces. [ABSTRACT FROM AUTHOR]

In this paper, we prove gap results for constant mean curvature (CMC) surfaces. First, we find a natural inequality for CMC surfaces that imply convexity for distance function. We then show that if Σ$\Sigma$ is a complete, properly embedded CMC surface in the Euclidean space satisfying this inequality, then Σ$\Sigma$ is either a sphere or a right circular cylinder. Next, we show that if Σ$\Sigma$ is a free boundary CMC surface in the Euclidean 3‐ball satisfying the same inequality, then either Σ$\Sigma$ is a totally umbilical disk or an annulus of revolution. These results complete the picture about gap theorems for CMC surfaces in the Euclidean 3‐space. We also prove similar results in the hyperbolic space and in the upper hemisphere, and in higher dimensions. [ABSTRACT FROM AUTHOR]

In this paper, we establish upper bounds on the length of the shortest conjugator between pairs of infinite order elements in a wide class of groups. We obtain a general result which applies to all hierarchically hyperbolic groups, a class which includes mapping class groups, right-angled Artin groups, Burger-Mozes-type groups, most 3-manifold groups, and many others. In this setting, we establish a linear bound on the length of the shortest conjugator for any pair of conjugate Morse elements. For a subclass of these groups, including, in particular, all virtually compact special groups, we prove a sharper result by obtaining a linear bound on the length of the shortest conjugator between a suitable power of any pair of conjugate infinite order elements. [ABSTRACT FROM AUTHOR]

In this paper, we modify the K F -iteration process into hyperbolic metric spaces where the symmetry condition is satisfied and establish the weak w 2 -stability and data dependence results for contraction mappings. We also prove some Δ -convergence and strong convergence theorems for generalized (α , β) -nonexpansive type 1 mappings. Finally, we offer a numerical example of generalized (α , β) -nonexpansive type 1 mappings and show that the K F -iteration process is more effective than some other iterations. Our results generalize and improve several relevant results in the literature. [ABSTRACT FROM AUTHOR]