![]() ![]() We also explore the competence of complex hyperbolic geometry on the multitree structure and $1$-$N$ structure. Through experiments on synthetic and real-world data, we show that our approach improves over the hyperbolic embedding models significantly. ![]() The unit ball model based embeddings have a more powerful representation capacity to capture a variety of hierarchical structures. Specifically, we propose to learn the embeddings of hierarchically structured data in the unit ball model of the complex hyperbolic space. To address this limitation of hyperbolic embeddings, we explore the complex hyperbolic space, which has the variable negative curvature, for representation learning. However, many real-world hierarchically structured data such as taxonomies and multitree networks have varying local structures and they are not trees, thus they do not ubiquitously match the constant curvature property of the hyperbolic space. Due to the constant negative curvature, the hyperbolic space resembles tree metrics and captures the tree-like properties naturally, which enables the hyperbolic embeddings to improve over traditional Euclidean models. Of the Dirichlet-to-Neumann operator.Learning the representation of data with hierarchical structures in the hyperbolic space attracts increasing attention in recent years. In the same geometric setting, it is proved that the spectral isoperimetric inequality holds for the lowest eigenvalue ![]() The constraint of fixed perimeter, the geodesic disk of constant curvature K ○ maximizes the lowest Robin eigenvalue. We prove that in the sub-class of manifolds with the Gauss curvature bounded from above by a constant K ○≥0 and under In this paper we investigate analytic functions of unbounded type on a complex infinite dimensional Banach space X. ( 1.20 ) whereas the complex spin factor ( 1.15 ) yields the ' Lie ball ' N. Laplacian on a compact, simply-connected two-dimensional manifold with boundary subject to an attractive Robin boundary condition. For a general hermitian Jordan triple Z, the unbounded phase space II is a so. The spectrum of any bounded linear operator on a Banach space is a nonempty compact subset of the complex numbers. PDF We consider the model describing the vertical motion of a ball falling with constant acceleration on a wall and elastically reflected. The Poincar Ball model is (Bn,g1), where Bn is the Euclidean unit ball in Rn. We consider the problem of geometric optimization of the lowest eigenvalue for the The unit ball model is a projective geometry based model to identify the complex hyperbolic space. On the other hand, the closed unit ball of the dual of a normed space is compact for the weak- topology. In this paper we study the geometry and topology of unbounded domains in. optimize (Constraints) yalmiptime: 0.1859 solvertime: 0.2381 info: Infeasible problem (MOSEK) problem: 1. Hence, if the problem without objective is feasible, the problem is in the objective and not in the constraints. X Spectral isoperimetric inequalities for Robin Laplacians on 2-manifolds and unbounded cones If that is the case, you would have to debug your unbounded model. We generally consider complex vector spaces most of the theory holds word-to-word also for real spaces. have studied the damped wave equation in an unbounded manifold and with a. theory of Hilbert and Banach spaces, the linear operators that areconsideredacting on such spaces orfrom one such space to another are taken to be bounded, i.e. On the isoperimetric inequality for the magnetic Robin Laplacian with negative boundary parameter the semilinear models, distinguishing the free wave and Klein-Gordon equations. Thickness is varying and is determined by the function $\mathbb)) \geq 1$, is infinite and discrete. Which is a multiple of the characteristic function of an unbounded strip-shaped region, whose In this paper we consider the two-dimensional Schr\"odinger operator with an attractive potential X Bound states of weakly deformed soft waveguides If the solver says it is unbounded, the standard trick is to artificially bound the solution-space and solve the problem.
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