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. We obtained the Sobolev boundedness of Mb. In this paper, we introduce a class of Fourier multiplier operators Mb on ncomplex unit sphere, where the symbol b Hs(S). Specifically, this is true if the measure. CLASS OF UNBOUNDED FOURIER MULTIPLIERS ON THE UNIT COMPLEX BALL PENGTAO LI, JIANHAO LV, AND TAO QIAN Abstract. We generally consider complex vector spaces most of the theory holds word-to-word also for real spaces. every open convex set containing the function is unbounded for the -quasi-norm therefore, the vector does not possess a fundamental system of convex neighborhoods. 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. 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. If we use complex-valued functions, the space. There is no direct discussion of topological vectorspaces. Z.Abstract: Learning the representation of data with hierarchical structures in the hyperbolic space attracts increasing attention in recent years.ĭue 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. Paul Garrett: Hahn-Banach theorems (July 17, 2008) The result involves elementary algebra and inequalities (apart from an invocation of transnite induction) and is the heart of the matter. 1 Introduction We consider the model of a ball bouncing on an in nitely heavy racket that is moving in the vertical direction according to a given regular periodic function f(t). Marò, S.: Diffusion and chaos in a bouncing ball model. unbounded motions is known, it is possible to nd a class of functions f that allow both bounded and unbounded motions. Marò, S.: Chaotic dynamics in an impact problem. Marò, S.: A mechanical counterexample to KAM theory with low regularity. Marò, S.: Coexistence of bounded and unbounded motions in a bouncing ball model. The first result in this direction is due to Pustyl’nikov assuming that \(2\) mechanical counterexample to Moser’s twist theorem. More generally, the space C(K) of continuous functions. real-valued (or complex-valued) functions on a b with the sup-norm is a Banach space. Unbounded linear operators are also important in applications: for example. We understand that a motion is unbounded if the velocity of the ball tends to infinity. and study some properties of bounded linear operators. In this paper we are concerned with the existence of unbounded motions, supposing f real analytic. Moreover, for some f presenting some singularities it is possible to study statistical and ergodic properties. This model has inspired many authors as it represents a simple mechanical model exhibiting complex dynamics see for example where results on periodic or quasiperiodic motions are proved together with, in some case, topological chaos. Moreover, the mass of the racket is assumed to be large with respect to the mass of the ball so that the impacts do not affect the motion of the racket. The only force acting on the ball is the gravity, with acceleration g. The racket is supposed to move in the vertical direction according to a periodic function f( t) and the ball is reflected according to the law of elastic bouncing when hitting the racket. Complex hyperbolic space is now represented as an unbounded domain, a Siegel domain in C '. ball model of hyperbolic space was used to embed taxonomies and graphs with. The vertical dynamics of a free falling ball on a moving racket is considered. This model generalizes the upper half - plane model of H. v t e The shape of the universe, in physical cosmology, is the local and global geometry of the universe. model have unbounded numerical error as points get far from the origin.
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