### Citations

2957 |
The analysis of linear partial differential operators. III Pseudodifferential operators
- Hörmander
- 1985
(Show Context)
Citation Context ...2)α/2ϕ̂ ∈ L2(Rn)} equipped with the norm ‖ϕ‖Hα := √∫ Rn ( 1 + ‖x‖2)α ∣∣∣ϕ̂(x)∣∣∣2 dx. (See e.g. [4, Section 7.9]; our normalization for the Fourier transform—identified in (5.1)— differs from that of =-=[4]-=-, but the Hα norms agree.) When α ≥ 0, Hα ⊆ L2, and so Hα is MAGNITUDE, &C. 11 actually a space of functions. For each α > 0, the space S of Schwartz functions is dense in Hα, and Hα and H−α act as du... |

2566 |
and Complex Analysis
- Real
- 1987
(Show Context)
Citation Context ...larity theory; see e.g. [18, Corollary 4.5]. For a more elementary treatment in the case that n is odd (so that (I−∆)(n+1)/2 is actually a differential operator), see the corollary to Theorem 8.12 in =-=[15]-=-. Proposition 5.9. Let A ⊆ Rn be compact with potential function h. Then the weighting w of A is the distribution w = 1n!ωn (I −∆)(n+1)/2h. Proof. This follows from Corollary 5.2, (5.2), and the fac... |

1287 |
Theory of reproducing kernels
- Aronszajn
- 1950
(Show Context)
Citation Context ...h(x) = 〈h, δx〉 = 〈 h, e−d(x,·) 〉 H . MAGNITUDE, &C. 7 That is,H is the reproducing kernel Hilbert space (RKHS) onX with the reproducing kernel e−d(x,y). (Readers unfamiliar with RKHSs are referred to =-=[2]-=-.) We have chosen to define first the weighting space W, and then define H in terms of it, since this more closely parallels Leinster’s original definition of the magnitude of a finite metric space. A... |

1221 | Introduction to Fourier analysis in Euclidean spaces - Stein, Weiss - 1971 |

250 | Introduction to pseudo-differential and Fourier integral operators, Vol. 2: Fourier integral operators - Trèves - 1980 |

215 | Asymptotic Theory of Finite-Dimensional Normed - Milman, Schechtman - 1986 |

57 |
On the Measurement of Biological Diversity”,
- Solow, St, et al.
- 1993
(Show Context)
Citation Context ...de. A second surprise, in a completely different direction, is that the magnitude of a finite metric space has been introduced in the literature before, in connection with quantifying biodiversity in =-=[16]-=-. Although the theory of magnitude was not developed at all in [16], the relationship between magnitude and diversity has been investigated more fully in [7]. The present work grew out of the author’s... |

42 | Analysis, volume 14 of Graduate Studies in Mathematics - Lieb, Loss - 2001 |

35 | 2008: The Euler Characteristic of a Category
- Leinster
(Show Context)
Citation Context ...troduction The magnitude of a metric space is a numerical isometric invariant introduced by Leinster in [9]. From the perspective of geometry, its definition was motivated in a rather unusual way. In =-=[6]-=-, Leinster had defined the Euler characteristic of a finite category, which generalizes the Euler characteristic of a topological space or of a poset. This notion of Euler characteristic can be natura... |

25 |
Function spaces and potential theory, volume 314 of Grundlehren der
- Adams, Hedberg
- 1996
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Citation Context ...sis of the growth of magnitude functions in Section 7 below. Many of the definitions and results of potential theory have complicated histories of successive generalizations. We will rely on the book =-=[1]-=- as a source, referring the reader there for original references. For α > 0, the Bessel kernel Gα : R n → R is defined as the function such that (6.1) Ĝα(x) = (2π) −n/2 ( 1 + ‖x‖2)−α/2; see [1, Secti... |

21 |
Metric Spaces, Generalized Logic, and Closed
- Lawvere
- 1973
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Citation Context ...ical space or of a poset. This notion of Euler characteristic can be naturally generalized from categories to to enriched categories, a family of algebraic structures which, as observed by Lawvere in =-=[5]-=-, includes metric spaces; in this context the generalization of Euler characteristic is named “magnitude”. Specialized then to metric spaces, one obtains Leinster’s definition of the magnitude of a fi... |

12 | Measuring diversity: the importance of species similarity, submitted
- Leinster, Cobbold
- 2010
(Show Context)
Citation Context ...ty of the given collection of species when one considers all possible relative abundances. We remark that the reciprocal of (4.5) is just one of an infinite family of diversity measures introduced in =-=[7, 10]-=-. It is shown in [7] that, under certain conditions, they all have the same maximum value. See [10] for a thorough discussion of these diversity measures in the context of theoretical ecology, and [7]... |

10 | The magnitude of metric spaces.
- Leinster
- 2010
(Show Context)
Citation Context ...nitude dimension considered by Leinster and Willerton is equal to the Minkowski dimension. 1. Introduction The magnitude of a metric space is a numerical isometric invariant introduced by Leinster in =-=[9]-=-. From the perspective of geometry, its definition was motivated in a rather unusual way. In [6], Leinster had defined the Euler characteristic of a finite category, which generalizes the Euler charac... |

9 | On the asymptotic magnitude of subsets of Euclidean space, arXiv:0908.1582 (2009), available from http://arxiv.org
- Leinster, Willerton
(Show Context)
Citation Context ...then to metric spaces, one obtains Leinster’s definition of the magnitude of a finite metric space, stated in Definition 2.1 below. Magnitude was extended to compact metric spaces in multiple ways in =-=[9, 11, 19, 20]-=-, which were shown by the author in [13] to agree with each other for many spaces (specifically, for so-called positive definite spaces, which include all compact subsets of Euclidean space). Given th... |

7 | A maximum entropy theorem with applications to the measurement of biodiversity, arXiv preprint
- Leinster
(Show Context)
Citation Context ...nnection with quantifying biodiversity in [16]. Although the theory of magnitude was not developed at all in [16], the relationship between magnitude and diversity has been investigated more fully in =-=[7]-=-. The present work grew out of the author’s search for a more satisfactory definition of magnitude for compact metric spaces. The approach of [13] was to introduce yet another definition, a measure-th... |

6 | Positive definite metric spaces
- Meckes
(Show Context)
Citation Context ...ition of the magnitude of a finite metric space, stated in Definition 2.1 below. Magnitude was extended to compact metric spaces in multiple ways in [9, 11, 19, 20], which were shown by the author in =-=[13]-=- to agree with each other for many spaces (specifically, for so-called positive definite spaces, which include all compact subsets of Euclidean space). Given this exotic provenance, it may come as a s... |

5 | Heuristic and computer calculations for the magnitude of metric spaces, arXiv preprint
- Willerton
(Show Context)
Citation Context ...then to metric spaces, one obtains Leinster’s definition of the magnitude of a finite metric space, stated in Definition 2.1 below. Magnitude was extended to compact metric spaces in multiple ways in =-=[9, 11, 19, 20]-=-, which were shown by the author in [13] to agree with each other for many spaces (specifically, for so-called positive definite spaces, which include all compact subsets of Euclidean space). Given th... |

4 | On the magnitude of spheres, surfaces and other homogeneous spaces, arXiv preprint
- Willerton
(Show Context)
Citation Context ... the deeper consequence is that magnitude dimensions and Minkowski dimensions always agree in Euclidean space, fully explaining the various such agreements observed both rigorously and numerically in =-=[9, 11, 19, 20]-=-. Corollary 7.4. If A ⊆ Rn is compact, then dimMagA = dimMinkA and dimMagA = dimMinkA. Consequently, dimMagA is defined if and only if dimMinkA is defined, and in that case dimMag A = dimMinkA. Proof.... |

1 |
Pri la funkcia ekvacio f(x+ y) = f(x) + f(y). L’Enseignement Mathématique
- Fréchet
- 1913
(Show Context)
Citation Context ...tion + in x+ y is the same operation appearing in the triangle inequality (the tensor product in the enriching category). If Φ is to be Lebesgue measurable, we must have Φ(x) = αx for some α ≥ 0 (see =-=[3]-=-); the choice of α = e−1 is the arbitrary choice of scale in Definition 2.1 which is addressed by considering magnitude functions. To adapt Definition 2.1 to ultrametric spaces, we instead need a func... |

1 |
The magnitude of an enriched category. Post at The n-Category Café, http://golem.ph.utexas.edu/category/2011/06/the magnitude of an enriched c.html
- Leinster
- 2011
(Show Context)
Citation Context ...is known as Euler characteristic, and is related to more classical invariants of that name) and of metric spaces, the magnitude of enriched categories has mostly not yet been very fully explored (see =-=[8]-=- for a discussion). In this section, we work out another special case, that of ultrametric spaces. We will see that this leads to an extremely simple notion of the size of an ultrametric space, whose ... |