Topic
Zero set
About: Zero set is a research topic. Over the lifetime, 1036 publications have been published within this topic receiving 16574 citations.
Papers published on a yearly basis
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Papers
TL;DR: The two-parameter Poisson-Dirichlet distribution with a single parameter is known as the size-biased random permutation (SBNP) as discussed by the authors, which was introduced by Engen in the context of species diversity and rediscovered by Perman and the authors in the study of excursions of Bessel processes.
Abstract: The two-parameter Poisson-Dirichlet distribution, denoted $\mathsf{PD}(\alpha, \theta)$ is a probability distribution on the set of decreasing positive sequences with sum 1. The usual Poisson-Dirichlet distribution with a single parameter $\theta$, introduced by Kingman, is $\mathsf{PD}(0, \theta)$. Known properties of $\mathsf{PD}(0, \theta)$, including the Markov chain description due to Vershik, Shmidt and Ignatov, are generalized to the two-parameter case. The size-biased random permutation of $\mathsf{PD}(\alpha, \theta)$ is a simple residual allocation model proposed by Engen in the context of species diversity, and rediscovered by Perman and the authors in the study of excursions of Brownian motion and Bessel processes. For $0 < \alpha < 1, \mathsf{PD}(\alpha, 0)$ is the asymptotic distribution of ranked lengths of excursions of a Markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index $\alpha$. Formulae in this case trace back to work of Darling, Lamperti and Wendel in the 1950s and 1960s. The distribution of ranked lengths of excursions of a one-dimensional Brownian motion is $\mathsf{PD}(1/2, 0)$, and the corresponding distribution for a Brownian bredge is $\mathsf{PD}(1/2, 1/2)$. The $\mathsf{PD}(\alpha, 0)$ and $\mathsf{PD}(\alpha, \alpha)$ distributions admit a similar interpretation in terms of the ranked lengths of excursions of a semistable Markov process whose zero set is the range of a stable subordinator of index $\alpha$.
1,449 citations
Book•
1 Jan 1982TL;DR: In this article, the authors introduce the subject of harmonic analysis and its applications, including the solution of the Levi problem, the zero set of a holomorphic function, and the invariant metrics.
Abstract: An introduction to the subject Some integral formulas Subharmonicity and its applications Convexity Hormander's solution of the $\bar\partial$ equation Solution of the Levi problem and other applications of $\bar\partial$ techniques Cousin problems, cohomology, and sheaves The zero set of a holomorphic function Some harmonic analysis Constructive methods Integral formulas for solutions to the $\bar\partial$ problem and norm estimates Holomorphic mappings and invariant metrics Manifolds Area measures Exterior algebra Vectors, covectors, and differential forms List of notation Bibliography Index.
1,358 citations
19 Jun 2002
TL;DR: This work addresses the problem of building watertight 3D models from surfaces that contain holes by constructing a signed distance function, the zero set of which defines the surface, and applies a diffusion process to extend this function through the volume until its zero set bridges whatever holes may be present.
Abstract: We address the problem of building watertight 3D models from surfaces that contain holes - for example, sets of range scans that observe most but not all of a surface. We specifically address situations in which the holes are too geometrically and topologically complex to fill using triangulation algorithms. Our solution begins by constructing a signed distance function, the zero set of which defines the surface. Initially, this function is defined only in the vicinity of observed surfaces. We then apply a diffusion process to extend this function through the volume until its zero set bridges whatever holes may be present. If additional information is available, such as known-empty regions of space inferred from the lines of sight to a 3D scanner, it can be incorporated into the diffusion process. Our algorithm is simple to implement, is guaranteed to produce manifold non-interpenetrating surfaces, and is efficient to run on large datasets because computation is limited to areas near holes.
628 citations
TL;DR: In this paper, the authors prove sharp regularity theorems for minimisers of a class of variational integrals whose integrand switches between two different types of degenerate elliptic phases, according to the zero set of a modulating coefficient.
Abstract: We prove sharp regularity theorems for minimisers of a class of variational integrals whose integrand switches between two different types of degenerate elliptic phases, according to the zero set of a modulating coefficient \({a(\cdot)}\) The model case is given by the functional
$$ w \mapsto \int (|Dw|^p + a(x)|Dw|^q) \, {\rm d}x,$$
where q > p and \({a(\cdot) \geqq 0}\)
563 citations
TL;DR: The principal result of this paper demonstrates that a variant of control Lyapunov functions that enforce rapid exponential convergence to the zero dynamics surface, Z, can be used to achieve exponential stability of the periodic orbit in the full-order dynamics, thereby significantly extending the class of stabilizing controllers.
Abstract: This paper addresses the problem of exponentially stabilizing periodic orbits in a special class of hybrid models-systems with impulse effects-through control Lyapunov functions. The periodic orbit is assumed to lie in a C1 submanifold Z that is contained in the zero set of an output function and is invariant under both the continuous and discrete dynamics; the associated restriction dynamics are termed the hybrid zero dynamics. The orbit is furthermore assumed to be exponentially stable within the hybrid zero dynamics. Prior results on the stabilization of such periodic orbits with respect to the full-order dynamics of the system with impulse effects have relied on input-output linearization of the dynamics transverse to the zero dynamics manifold. The principal result of this paper demonstrates that a variant of control Lyapunov functions that enforce rapid exponential convergence to the zero dynamics surface, Z, can be used to achieve exponential stability of the periodic orbit in the full-order dynamics, thereby significantly extending the class of stabilizing controllers. The main result is illustrated on a hybrid model of a bipedal walking robot through simulations and is utilized to experimentally achieve bipedal locomotion via control Lyapunov functions.
519 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2024 | 4 |
| 2023 | 5 |
| 2022 | 14 |
| 2021 | 48 |
| 2020 | 61 |
| 2019 | 69 |