Entropies of Sets of Functions of Bounded Variation

Clements, G. F.

doi:10.4153/cjm-1963-045-3

Cited by 16 publications

(7 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…There is evidence for the 6th through 13th caustic rings in that there are sudden rises in the inner rotation curve published in ref. [8] at radii very near those listed above. Similarly there is some evidence for the 2d and 3d caustic rings in the averaged outer rotation curve published in ref.…”

mentioning

confidence: 71%

Caustic rings of dark matter

1998

View full text Add to dashboard Cite

There are compelling reasons to believe that the dominant component of the dark matter of the universe is non-baryonic collisionless particles [l]. The leading candidates are axions, WIMPs and massive neutrinos. The word "collisionless" signifies that the particles are so weakly interacting that they have moved purely under the influence of gravity since their decoupling at a very early time (of order lo-* sec for axions, of order 1 sec for neutrinos and WIMPs). In the limit where the primordial velocity dispersion of the particles is neglected, they all lie on the same 3-dim. 'sheet' in 6-dim. phase-space. Their phase-space evolution must obey Liouville's theorem. This implies that the 3-dim. sheet cannot tear and hence that it satisfies certain topological constraints.Let us assume that collisionless dark matter (CDM) exists. Usually, CDM means 'cold dark matter' (e.g., axions or WIMPs) but because massive neutrinos (m, >> eV) behave, from the point of view of this paper, in a similar way we include them in our definition as well. Because their phase-space sheet cannot tear, CDM particles must be present everywhere in space, including specifically intergalactic space. The space density may be reduced by stretching of the phase-space sheet but it cannot vanish. Moreover, the average space density is recovered as soon as the average is taken over distances larger than the distance CDM may have locally moved away from perfect Hubble flow. In a region which is sparsely populated with galaxies, this distance is much smaller than the distance between galaxies. The implication is that isolated galaxies are surrounded by unseen CDM and hence, because of gravity, CDM keeps falling onto isolated galaxies continuously from all directions. If the galaxy merges with other galaxies to form a cluster, infall onto the galaxy gets shut off because of lack of material but infall onto the cluster continues assuming that the cluster is itself isolated. In an open universe (R < 11, the infall process eventually turns off because the universe becomes very dilute. However, even if our own universe is open, we are far from having reached the turn-off time.It has been shown [2] that, under a wide range of circumstances, CDM infall onto an isolated galaxy produces a halo whose density falls off like 5 for large T , where T is the distance to the galactic center. Such a halo implies a flat galactic rotation curve [3]. It has

show abstract

mentioning

confidence: 71%

Caustic rings of dark matter

1998

View full text Add to dashboard Cite

show abstract

“…Since the L λ metric is smaller than the uniform metric, the estimate H z (Fq) ^ (l/ε) w/(? is immediate from Kolmogorov's result (1). To get the reverse estimate we show the existence of a large number of ε/M-distinguishable functions in F?…”

Section: H S (A φ ) ^ Hm-\e)mentioning

confidence: 99%

“…If one now selects / (1) arbitrarily from U and with it all functions of tf/i (1) , •• ,/r ( ( 1 i ) ), r(l)^ k(e), which satisfy (14) with/ = / (1) , and then from the remaining functions of U selects / (2) arbitrarily and with it all functions of Ufl 2) , . ,/ r ( ( %, r(2) ^ fc(e), which satisfy (14) with / = / (2) , and so on until U is exhausted, one obtains at least t -[ (2 s )l(k(ε) + 1)] groups of functions.…”

Section: H S (A φ ) ^ Hm-\e)mentioning

confidence: 99%

“…,/ r ( ( %, r(2) ^ fc(e), which satisfy (14) with / = / (2) , and so on until U is exhausted, one obtains at least t -[ (2 s )l(k(ε) + 1)] groups of functions. The functions / (1) ,/ (2) , ,/ where…”

Section: H S (A φ ) ^ Hm-\e)mentioning

confidence: 99%

“…For an exposition of this and other topics related to entropy see [5]. For other entropy calculations by the present author see [1]. The Kolmogorov-Vituskin result makes use of the following entropy calculation:…”

mentioning

confidence: 99%

See 2 more Smart Citations

Entropies of several sets of real valued functions

Clements¹

1963

Pacific J. Math.

View full text Add to dashboard Cite

Leading Strategies in Competitive On-Line Prediction

Vovk

2006

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Competitive on-line prediction (also known as universal prediction of individual sequences) is a strand of learning theory avoiding making any stochastic assumptions about the way the observations are generated. The predictor's goal is to compete with a benchmark class of prediction rules, which is often a proper Banach function space. Metric entropy provides a unifying framework for competitive on-line prediction: the numerous known upper bounds on the metric entropy of various compact sets in function spaces readily imply bounds on the performance of on-line prediction strategies. This paper discusses strengths and limitations of the direct approach to competitive on-line prediction via metric entropy, including comparisons to other approaches.

show abstract

Entropies of Sets of Functions of Bounded Variation

Cited by 16 publications

References 4 publications

Caustic rings of dark matter

Caustic rings of dark matter

Entropies of several sets of real valued functions

Leading Strategies in Competitive On-Line Prediction

Contact Info

Product

Resources

About