Let Mn be an n-dimensional complete connected and oriented hypersurface in a hyperbolic space H（n＋1）（c） with non-zero constant mean curvature H and two distinct principal curvatures. In this paper, we show that ...Let Mn be an n-dimensional complete connected and oriented hypersurface in a hyperbolic space H（n＋1）（c） with non-zero constant mean curvature H and two distinct principal curvatures. In this paper, we show that （1） if the multiplicities of the two distinct principal curvatures are greater than 1,then Mn is isometric to the Riemannian product Sk（r）×H（n-k）（-1/（r2 ＋ ρ2））, where r 〉 0 and 1 〈 k 〈 n - 1;（2）if H2 〉 -c and one of the two distinct principal curvatures is simple, then Mn is isometric to the Riemannian product S（n-1）（r） × H1（-1/（r2 ＋ρ2）） or S1（r） × H（n-1）（-1/（r2 ＋ρ2））,r 〉 0, if one of the following conditions is satisfied （i） S≤（n-1）t22＋c2t（-2）2 on Mn or （ii）S≥ （n-1）t21＋c2t（-2）1 on Mn or（iii）（n-1）t22＋c2t（-2）2≤ S≤（n-1）t21＋c2t（-2）1 on Mn, where t_1 and t_2 are the positive real roots of （1.5）.展开更多
In this article, by solving a nonlinear differential equation, we prove the existence of a one parameter family of constant mean curvature hypersurfaces in the hyperbolic space with two ends. Then, we study the stabil...In this article, by solving a nonlinear differential equation, we prove the existence of a one parameter family of constant mean curvature hypersurfaces in the hyperbolic space with two ends. Then, we study the stability of these hypersurfaces.展开更多
In this paper,we consider quasi Einstein hypersurfaces in a hyperbolic space. The following theorem is obtained. Theorem Quasi Einstein hypersurfaces of a hyperbolic space are of constant curvature,where the dimension...In this paper,we consider quasi Einstein hypersurfaces in a hyperbolic space. The following theorem is obtained. Theorem Quasi Einstein hypersurfaces of a hyperbolic space are of constant curvature,where the dimension is large enough.展开更多
We introduce a one-step implicit iterative method for two finite families of asymptotically nonexpansive mappings in a hyperbolic space and use it to approximate common fixed points of these families. The results pres...We introduce a one-step implicit iterative method for two finite families of asymptotically nonexpansive mappings in a hyperbolic space and use it to approximate common fixed points of these families. The results presented in this paper are new in the setting of hyperbolic spaces. On top, these are generalizations of several results in literature from Banach spaces to hyperbolic spaces. At the end of the paper, we give an example to validate our results.展开更多
In this paper, we study upper bounds of the first eigenvalue of a complete noncompact submanifold in an （n ＋p）-dimensional hyperbolic space H^n＋p. In particular, we prove that the first eigenvalue of a complete su...In this paper, we study upper bounds of the first eigenvalue of a complete noncompact submanifold in an （n ＋p）-dimensional hyperbolic space H^n＋p. In particular, we prove that the first eigenvalue of a complete submanifold in H^n＋p with parallel mean curvature vector H and finite L^q（q ≥ n） norm of traceless second fundamental form is not more than （n-1）2（1-（H）^2）/4 . We also prove that the first eigenvalue of a complete hypersurfaces which has finite index in H^n＋1（n ≤ 5） with constant mean curvature vector H and finite.展开更多
In this paper, we introduce the notion of a （2＋1）-dimenslonal differential equation describing three-dimensional hyperbolic spaces （3-h.s.）. The （2＋1）-dimensional coupled nonlinear Schrodinger equation and its...In this paper, we introduce the notion of a （2＋1）-dimenslonal differential equation describing three-dimensional hyperbolic spaces （3-h.s.）. The （2＋1）-dimensional coupled nonlinear Schrodinger equation and its sister equation, the （2＋1）-dimensional coupled derivative nonlinear Schrodinger equation, are shown to describe 3-h.s, The （2 ＋ 1 ）-dimensional generalized HF model：St=（1/2i[S,Sy]＋2iσS）x,σx=-1/4i tr（SSxSy）, in which S ∈ GLc（2）/GLc（1）×GLc（1）,provides another example of （2＋1）-dimensional differential equations describing 3-h.s. As a direct con-sequence, the geometric construction of an infinire number of conservation lairs of such equations is illustrated. Furthermore we display a new infinite number of conservation lairs of the （2＋1）-dimensional nonlinear Schrodinger equation and the （2＋1）-dimensional derivative nonlinear Schrodinger equation by a geometric way.展开更多
In this article, new visual and intuitive interpretations of Lorentz transformation and Einstein velocity addition are given. We first obtain geometric interpretations of isometries of vertical projection model of hyp...In this article, new visual and intuitive interpretations of Lorentz transformation and Einstein velocity addition are given. We first obtain geometric interpretations of isometries of vertical projection model of hyperbolic space, which are the analogues of the geometric construction of inversions with respect to a circle on the complex plane. These results are then applied to Lorentz transformation and Einstein velocity addition to obtain geometric illustrations. We gain new insights into the relationship between special relativity and hyperbolic geometry.展开更多
In recent years,the rapid advancement of mega-constellations in Low Earth Orbit(LEO)has led to the emergence of satellite communication networks characterized by a complex interplay between high-and low-altitude orbit...In recent years,the rapid advancement of mega-constellations in Low Earth Orbit(LEO)has led to the emergence of satellite communication networks characterized by a complex interplay between high-and low-altitude orbits and by unprecedented scale.Traditional network-representation methodologies in Euclidean space are insufficient to capture the dynamics and evolution of high-dimensional complex networks.By contrast,hyperbolic space offers greater scalability and stronger representational capacity than Euclidean-space methods,thereby providing a more suitable framework for representing large-scale satellite communication networks.This paper aims to address the burgeoning demands of large-scale space-air-ground integrated satellite communication networks by providing a comprehensive review of representation-learning methods for large-scale complex networks and their application within hyperbolic space.First,we briefly introduce several equivalent models of hyperbolic space.Then,we summarize existing representation methods and applications for large-scale complex networks.Building on these advances,we propose representation methods for complex satellite communication networks in hyperbolic space and discuss potential application prospects.Finally,we highlight several pressing directions for future research.展开更多
We consider the dynamic property of the volume preserving mean curvature flow.This flow was introduced by Huisken(J Reine Angew Math 382:35–48,1987)who also proved it converges to a round sphere of the same enclosed ...We consider the dynamic property of the volume preserving mean curvature flow.This flow was introduced by Huisken(J Reine Angew Math 382:35–48,1987)who also proved it converges to a round sphere of the same enclosed volume if the initial hypersurface is strictly convex in Euclidean space.We study the stability of this flow in hyperbolic space.In particular,we prove that if the initial hypersurface is hyperbolically mean convex and close to an umbilical sphere in the L^(2)-sense,then the flow exists for all time and converges exponentially to an umbilical sphere.展开更多
We obtain new complete minimal surfaces in the hyperbolic space H3, by using Ribaucour transformations. Starting with the family of spherical catenoids in H^3 found by Mori(1981), we obtain 2-and 3-parameter families ...We obtain new complete minimal surfaces in the hyperbolic space H3, by using Ribaucour transformations. Starting with the family of spherical catenoids in H^3 found by Mori(1981), we obtain 2-and 3-parameter families of new minimal surfaces in the hyperbolic space, by solving a non trivial integro-differential system. Special choices of the parameters provide minimal surfaces whose parametrizations are defined on connected regions of R^2 minus a disjoint union of Jordan curves. Any connected region bounded by such a Jordan curve, generates a complete minimal surface, whose boundary at infinity of H^3 is a closed curve. The geometric properties of the surfaces regarding the ends, completeness and symmetries are discussed.展开更多
The author constructs a sequence of cubes in the infinitely dimensional hyperbolic space H∞ which is equi-coarsely equivalent to Z2n. As a corollary, it is proved that the infinitely dimensional hyperbolic space H∞ ...The author constructs a sequence of cubes in the infinitely dimensional hyperbolic space H∞ which is equi-coarsely equivalent to Z2n. As a corollary, it is proved that the infinitely dimensional hyperbolic space H∞ does not have property A.展开更多
In this paper, we reprove a theorem of M. Anderson [Invent. Math., 69 （1982）, pp. 477-494] which established the existence of a minimal hypersurface in the hyperbolic space with prescribed asymptotic boundary with n...In this paper, we reprove a theorem of M. Anderson [Invent. Math., 69 （1982）, pp. 477-494] which established the existence of a minimal hypersurface in the hyperbolic space with prescribed asymptotic boundary with non-negative mean curvature in the non-parametric case. We use the mean curvature flow method.展开更多
In this paper, we establish a singular Trudinger-Moser inequality for the whole hyperbolic space H^n：u∈W^1,n（H^n）,^sup,fH^n｜ H^nu｜^ndμ≤1∫H^n ea｜u｜n/n-1-∑^n-2a^k｜u｜nk/n-1 k=0 k!/ρβ dμ〈∞ a/an＋β/n≤1...In this paper, we establish a singular Trudinger-Moser inequality for the whole hyperbolic space H^n：u∈W^1,n（H^n）,^sup,fH^n｜ H^nu｜^ndμ≤1∫H^n ea｜u｜n/n-1-∑^n-2a^k｜u｜nk/n-1 k=0 k!/ρβ dμ〈∞ a/an＋β/n≤1,where α 〉 0,β E [0,n）, ρ and dμ are the distance function and volume element of H^n respectively.展开更多
It is important to study the propagation and interaction of progressing waves of nonlinear equations in the class of piecewise smooth function. However, there has not been many works on that in multidimensional case. ...It is important to study the propagation and interaction of progressing waves of nonlinear equations in the class of piecewise smooth function. However, there has not been many works on that in multidimensional case. In 1985, J, Rauch & M. Reed have provad the existence and uniqueness of piecewise smooth solution for展开更多
For an analytic function f on the hyperbolic domain Ω in C,the following conclusions are obtained: (i)f∈B(Ω)=BMOA(Ω,m)if and only if Ref∈B(?)(Ω)=BMOH(Ω,m).(ii)QB_h(Ω)=B_h(Ω) (BMOH,(Ω,m)=BMOH(Ω,m)if and only...For an analytic function f on the hyperbolic domain Ω in C,the following conclusions are obtained: (i)f∈B(Ω)=BMOA(Ω,m)if and only if Ref∈B(?)(Ω)=BMOH(Ω,m).(ii)QB_h(Ω)=B_h(Ω) (BMOH,(Ω,m)=BMOH(Ω,m)if and only if C(Ω)=inf{Z_o(z)·δ_o(z)·z≡Ω}>0,Also some applica- lions to automorphic function are considered.展开更多
In this paper we study the interaction of strong and weak singularities for hyperbolic system of conservation laws in multidimensional space. Under the assumption of transversal intersect of the shock front with the b...In this paper we study the interaction of strong and weak singularities for hyperbolic system of conservation laws in multidimensional space. Under the assumption of transversal intersect of the shock front with the bicharacteristics bearing weak singularities we proved a theorem on regularity propagation across the shock front.展开更多
CMC-H surfaces satisfying H≥1 are studied.It is proved that these surfaces possess a natural representation which is a generalization of the Bryant’s result.
In the metric-based meta-learning detection model,the distribution of training samples in the metric space has great influence on the detection performance,and this influence is usually ignored by traditional meta-det...In the metric-based meta-learning detection model,the distribution of training samples in the metric space has great influence on the detection performance,and this influence is usually ignored by traditional meta-detectors.In addition,the design of metric space might be interfered with by the background noise of training samples.To tackle these issues,we propose a metric space optimisation method based on hyperbolic geometry attention and class-agnostic activation maps.First,the geometric properties of hyperbolic spaces to establish a structured metric space are used.A variety of feature samples of different classes are embedded into the hyperbolic space with extremely low distortion.This metric space is more suitable for representing tree-like structures between categories for image scene analysis.Meanwhile,a novel similarity measure function based on Poincarédistance is proposed to evaluate the distance of various types of objects in the feature space.In addition,the class-agnostic activation maps(CCAMs)are employed to re-calibrate the weight of foreground feature information and suppress background information.Finally,the decoder processes the high-level feature information as the decoding of the query object and detects objects by predicting their locations and corresponding task encodings.Experimental evaluation is conducted on Pascal VOC and MS COCO datasets.The experiment results show that the effectiveness of the authors’method surpasses the performance baseline of the excellent few-shot detection models.展开更多
Utilizing graph neural networks for knowledge embedding to accomplish the task of knowledge graph completion(KGC)has become an important research area in knowledge graph completion.However,the number of nodes in the k...Utilizing graph neural networks for knowledge embedding to accomplish the task of knowledge graph completion(KGC)has become an important research area in knowledge graph completion.However,the number of nodes in the knowledge graph increases exponentially with the depth of the tree,whereas the distances of nodes in Euclidean space are second-order polynomial distances,whereby knowledge embedding using graph neural networks in Euclidean space will not represent the distances between nodes well.This paper introduces a novel approach called hyperbolic hierarchical graph attention network(H2GAT)to rectify this limitation.Firstly,the paper conducts knowledge representation in the hyperbolic space,effectively mitigating the issue of exponential growth of nodes with tree depth and consequent information loss.Secondly,it introduces a hierarchical graph atten-tion mechanism specifically designed for the hyperbolic space,allowing for enhanced capture of the network structure inherent in the knowledge graph.Finally,the efficacy of the proposed H2GAT model is evaluated on benchmark datasets,namely WN18RR and FB15K-237,thereby validating its effectiveness.The H2GAT model achieved 0.445,0.515,and 0.586 in the Hits@1,Hits@3 and Hits@10 metrics respectively on the WN18RR dataset and 0.243,0.367 and 0.518 on the FB15K-237 dataset.By incorporating hyperbolic space embedding and hierarchical graph attention,the H2GAT model successfully addresses the limitations of existing hyperbolic knowledge embedding models,exhibiting its competence in knowledge graph completion tasks.展开更多
基金supported by NSF of Shaanxi Province (SJ08A31)NSF of Shaanxi Educational Committee (2008JK484+1 种基金2010JK642)Talent Fund of Xi'an University of Architecture and Technology
文摘Let Mn be an n-dimensional complete connected and oriented hypersurface in a hyperbolic space H（n＋1）（c） with non-zero constant mean curvature H and two distinct principal curvatures. In this paper, we show that （1） if the multiplicities of the two distinct principal curvatures are greater than 1,then Mn is isometric to the Riemannian product Sk（r）×H（n-k）（-1/（r2 ＋ ρ2））, where r 〉 0 and 1 〈 k 〈 n - 1;（2）if H2 〉 -c and one of the two distinct principal curvatures is simple, then Mn is isometric to the Riemannian product S（n-1）（r） × H1（-1/（r2 ＋ρ2）） or S1（r） × H（n-1）（-1/（r2 ＋ρ2））,r 〉 0, if one of the following conditions is satisfied （i） S≤（n-1）t22＋c2t（-2）2 on Mn or （ii）S≥ （n-1）t21＋c2t（-2）1 on Mn or（iii）（n-1）t22＋c2t（-2）2≤ S≤（n-1）t21＋c2t（-2）1 on Mn, where t_1 and t_2 are the positive real roots of （1.5）.
基金supported by the King Saud University D.S.F.P program
文摘In this article, by solving a nonlinear differential equation, we prove the existence of a one parameter family of constant mean curvature hypersurfaces in the hyperbolic space with two ends. Then, we study the stability of these hypersurfaces.
文摘In this paper,we consider quasi Einstein hypersurfaces in a hyperbolic space. The following theorem is obtained. Theorem Quasi Einstein hypersurfaces of a hyperbolic space are of constant curvature,where the dimension is large enough.
基金King Fahd University of Petroleum and Minerals for supporting the research project IN121055Higher Education Commission (HEC) of Pakistan for financial support
文摘We introduce a one-step implicit iterative method for two finite families of asymptotically nonexpansive mappings in a hyperbolic space and use it to approximate common fixed points of these families. The results presented in this paper are new in the setting of hyperbolic spaces. On top, these are generalizations of several results in literature from Banach spaces to hyperbolic spaces. At the end of the paper, we give an example to validate our results.
基金Supported by the National Natural Science Foundation of China(Grant No.11261038)the Natural Science Foundation of Jiangxi Province(Grant Nos.2010GZS0149+1 种基金20132BAB201005)Youth Science Foundation of Eduction Department of Jiangxi Province(Grant No.GJJ11044)
文摘In this paper, we study upper bounds of the first eigenvalue of a complete noncompact submanifold in an （n ＋p）-dimensional hyperbolic space H^n＋p. In particular, we prove that the first eigenvalue of a complete submanifold in H^n＋p with parallel mean curvature vector H and finite L^q（q ≥ n） norm of traceless second fundamental form is not more than （n-1）2（1-（H）^2）/4 . We also prove that the first eigenvalue of a complete hypersurfaces which has finite index in H^n＋1（n ≤ 5） with constant mean curvature vector H and finite.
基金The project partially supported by National Natural Science Foundation of China
文摘In this paper, we introduce the notion of a （2＋1）-dimenslonal differential equation describing three-dimensional hyperbolic spaces （3-h.s.）. The （2＋1）-dimensional coupled nonlinear Schrodinger equation and its sister equation, the （2＋1）-dimensional coupled derivative nonlinear Schrodinger equation, are shown to describe 3-h.s, The （2 ＋ 1 ）-dimensional generalized HF model：St=（1/2i[S,Sy]＋2iσS）x,σx=-1/4i tr（SSxSy）, in which S ∈ GLc（2）/GLc（1）×GLc（1）,provides another example of （2＋1）-dimensional differential equations describing 3-h.s. As a direct con-sequence, the geometric construction of an infinire number of conservation lairs of such equations is illustrated. Furthermore we display a new infinite number of conservation lairs of the （2＋1）-dimensional nonlinear Schrodinger equation and the （2＋1）-dimensional derivative nonlinear Schrodinger equation by a geometric way.
文摘In this article, new visual and intuitive interpretations of Lorentz transformation and Einstein velocity addition are given. We first obtain geometric interpretations of isometries of vertical projection model of hyperbolic space, which are the analogues of the geometric construction of inversions with respect to a circle on the complex plane. These results are then applied to Lorentz transformation and Einstein velocity addition to obtain geometric illustrations. We gain new insights into the relationship between special relativity and hyperbolic geometry.
文摘In recent years,the rapid advancement of mega-constellations in Low Earth Orbit(LEO)has led to the emergence of satellite communication networks characterized by a complex interplay between high-and low-altitude orbits and by unprecedented scale.Traditional network-representation methodologies in Euclidean space are insufficient to capture the dynamics and evolution of high-dimensional complex networks.By contrast,hyperbolic space offers greater scalability and stronger representational capacity than Euclidean-space methods,thereby providing a more suitable framework for representing large-scale satellite communication networks.This paper aims to address the burgeoning demands of large-scale space-air-ground integrated satellite communication networks by providing a comprehensive review of representation-learning methods for large-scale complex networks and their application within hyperbolic space.First,we briefly introduce several equivalent models of hyperbolic space.Then,we summarize existing representation methods and applications for large-scale complex networks.Building on these advances,we propose representation methods for complex satellite communication networks in hyperbolic space and discuss potential application prospects.Finally,we highlight several pressing directions for future research.
基金The research of Z.Huang is partially supported by a PSC-CUNY GrantThe research of Z.Zhang is partially supported by ARC Future Fellowship FT150100341.
文摘We consider the dynamic property of the volume preserving mean curvature flow.This flow was introduced by Huisken(J Reine Angew Math 382:35–48,1987)who also proved it converges to a round sphere of the same enclosed volume if the initial hypersurface is strictly convex in Euclidean space.We study the stability of this flow in hyperbolic space.In particular,we prove that if the initial hypersurface is hyperbolically mean convex and close to an umbilical sphere in the L^(2)-sense,then the flow exists for all time and converges exponentially to an umbilical sphere.
基金supported by a Post-Doctoral Fellowship offered by CNPqpartially supported by CNPq, Ministry of Science and Technology, Brazil (Grant No. 312462/2014-0)
文摘We obtain new complete minimal surfaces in the hyperbolic space H3, by using Ribaucour transformations. Starting with the family of spherical catenoids in H^3 found by Mori(1981), we obtain 2-and 3-parameter families of new minimal surfaces in the hyperbolic space, by solving a non trivial integro-differential system. Special choices of the parameters provide minimal surfaces whose parametrizations are defined on connected regions of R^2 minus a disjoint union of Jordan curves. Any connected region bounded by such a Jordan curve, generates a complete minimal surface, whose boundary at infinity of H^3 is a closed curve. The geometric properties of the surfaces regarding the ends, completeness and symmetries are discussed.
基金supported by the National Natural Science Foundation of China(No.10731020)the Shanghai Pujiang Program(No.08PJ14006)
文摘The author constructs a sequence of cubes in the infinitely dimensional hyperbolic space H∞ which is equi-coarsely equivalent to Z2n. As a corollary, it is proved that the infinitely dimensional hyperbolic space H∞ does not have property A.
文摘In this paper, we reprove a theorem of M. Anderson [Invent. Math., 69 （1982）, pp. 477-494] which established the existence of a minimal hypersurface in the hyperbolic space with prescribed asymptotic boundary with non-negative mean curvature in the non-parametric case. We use the mean curvature flow method.
文摘In this paper, we establish a singular Trudinger-Moser inequality for the whole hyperbolic space H^n：u∈W^1,n（H^n）,^sup,fH^n｜ H^nu｜^ndμ≤1∫H^n ea｜u｜n/n-1-∑^n-2a^k｜u｜nk/n-1 k=0 k!/ρβ dμ〈∞ a/an＋β/n≤1,where α 〉 0,β E [0,n）, ρ and dμ are the distance function and volume element of H^n respectively.
基金This paper is supported by the National Foundations.
文摘It is important to study the propagation and interaction of progressing waves of nonlinear equations in the class of piecewise smooth function. However, there has not been many works on that in multidimensional case. In 1985, J, Rauch & M. Reed have provad the existence and uniqueness of piecewise smooth solution for
基金This research was supported by the Doctoral Program Foundation of Institute of Higher Education.
文摘For an analytic function f on the hyperbolic domain Ω in C,the following conclusions are obtained: (i)f∈B(Ω)=BMOA(Ω,m)if and only if Ref∈B(?)(Ω)=BMOH(Ω,m).(ii)QB_h(Ω)=B_h(Ω) (BMOH,(Ω,m)=BMOH(Ω,m)if and only if C(Ω)=inf{Z_o(z)·δ_o(z)·z≡Ω}>0,Also some applica- lions to automorphic function are considered.
文摘In this paper we study the interaction of strong and weak singularities for hyperbolic system of conservation laws in multidimensional space. Under the assumption of transversal intersect of the shock front with the bicharacteristics bearing weak singularities we proved a theorem on regularity propagation across the shock front.
基金supported by the National Natural Science Foundation of China and SFECC
文摘CMC-H surfaces satisfying H≥1 are studied.It is proved that these surfaces possess a natural representation which is a generalization of the Bryant’s result.
基金National Natural Science Foundation of China,Grant/Award Number:61602157Henan scientific and technological project,Grant/Award Number:242102210020Basal Research Fund,Grant/Award Number:NSFRF240618。
文摘In the metric-based meta-learning detection model,the distribution of training samples in the metric space has great influence on the detection performance,and this influence is usually ignored by traditional meta-detectors.In addition,the design of metric space might be interfered with by the background noise of training samples.To tackle these issues,we propose a metric space optimisation method based on hyperbolic geometry attention and class-agnostic activation maps.First,the geometric properties of hyperbolic spaces to establish a structured metric space are used.A variety of feature samples of different classes are embedded into the hyperbolic space with extremely low distortion.This metric space is more suitable for representing tree-like structures between categories for image scene analysis.Meanwhile,a novel similarity measure function based on Poincarédistance is proposed to evaluate the distance of various types of objects in the feature space.In addition,the class-agnostic activation maps(CCAMs)are employed to re-calibrate the weight of foreground feature information and suppress background information.Finally,the decoder processes the high-level feature information as the decoding of the query object and detects objects by predicting their locations and corresponding task encodings.Experimental evaluation is conducted on Pascal VOC and MS COCO datasets.The experiment results show that the effectiveness of the authors’method surpasses the performance baseline of the excellent few-shot detection models.
基金the Beijing Municipal Science and Technology Program(No.Z231100001323004).
文摘Utilizing graph neural networks for knowledge embedding to accomplish the task of knowledge graph completion(KGC)has become an important research area in knowledge graph completion.However,the number of nodes in the knowledge graph increases exponentially with the depth of the tree,whereas the distances of nodes in Euclidean space are second-order polynomial distances,whereby knowledge embedding using graph neural networks in Euclidean space will not represent the distances between nodes well.This paper introduces a novel approach called hyperbolic hierarchical graph attention network(H2GAT)to rectify this limitation.Firstly,the paper conducts knowledge representation in the hyperbolic space,effectively mitigating the issue of exponential growth of nodes with tree depth and consequent information loss.Secondly,it introduces a hierarchical graph atten-tion mechanism specifically designed for the hyperbolic space,allowing for enhanced capture of the network structure inherent in the knowledge graph.Finally,the efficacy of the proposed H2GAT model is evaluated on benchmark datasets,namely WN18RR and FB15K-237,thereby validating its effectiveness.The H2GAT model achieved 0.445,0.515,and 0.586 in the Hits@1,Hits@3 and Hits@10 metrics respectively on the WN18RR dataset and 0.243,0.367 and 0.518 on the FB15K-237 dataset.By incorporating hyperbolic space embedding and hierarchical graph attention,the H2GAT model successfully addresses the limitations of existing hyperbolic knowledge embedding models,exhibiting its competence in knowledge graph completion tasks.
