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## ALGORITHMIC ASPECTS OF LIPSCHITZ FUNCTIONS

Citations: | 9 - 4 self |

### Citations

2136 | An introduction to Kolmogorov complexity and its applications. 2nd edition, Addison-Wesley
- Li, Vitányi
- 1997
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Citation Context ...≤ y for each n. Then by continuity f(y)−f(x) is the sup of the values f(qn−2−n)−f(pn+ 2−n) where pn+ 2−n ≤ qn−2−n. This is left-r.e. in the Cauchy names by hypothesis. See [Nie09, Ch. 2] or [DH10] or =-=[LV08]-=- for background on prefix-free machines and prefixfree complexity K. For a set B ⊆ 2<ω let [B]≺ denote the open set {X ∈ 2ω : ∃nX n∈ B}. Let S be a prefix-free machine. We identify a binary string γ ... |

1330 | Probability: theory and examples - Durrett - 1996 |

324 | Weakly Differentiable Functions, - Ziemer - 1989 |

248 |
Algorithmic Randomness and Complexity.
- Downey, Hirshfeldt
- 2010
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Citation Context ...nd qn−2−n ≤ y for each n. Then by continuity f(y)−f(x) is the sup of the values f(qn−2−n)−f(pn+ 2−n) where pn+ 2−n ≤ qn−2−n. This is left-r.e. in the Cauchy names by hypothesis. See [Nie09, Ch. 2] or =-=[DH10]-=- or [LV08] for background on prefix-free machines and prefixfree complexity K. For a set B ⊆ 2<ω let [B]≺ denote the open set {X ∈ 2ω : ∃nX n∈ B}. Let S be a prefix-free machine. We identify a binary... |

174 |
Zufalligheit und Wahrscheinlichkeit
- Schnorr
- 1970
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Citation Context ...ith M . Then |ν[0.τm0 , a)| ≤ µ[0.τm0 , a)→ 0 as m→∞, and hence |ν[0.σ, a)| ≤∑∞i=1 2−(s+i) = 2−s. 4. Computable randomness and Lipschitz functions 4.1. Characterizing computable randomness. Schnorr =-=[Sch71]-=- introduced the following notion. Definition 4.1. A sequence of bits Z is called computably random if no computable martingale succeeds on Z. A real z ∈ [0, 1] is called computably random if a binary ... |

38 | Computability of probability measures and Martin-Löf randomness over metric spaces - Hoyrup, Rojas |

30 | Computability and randomness, Oxford Logic Guides, - Nies - 2009 |

20 |
and Cristóbal Rojas, Computability of probability measures and Martin-Löf randomness over metric spaces
- Hoyrup
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Citation Context ...this as a theorem joint with Rute at the end of Section 3. One can also ask whether a similar result holds in the context of effective measure theory developed by Gács, Hoyrup, Rojas and others (see =-=[HR09]-=-). For instance, is every lower semicomputable measure on [0, 1]n without point masses the variation, in the sense of [Rud87, Section 6.1], of a computable signed measure? Rute has pointed out that ou... |

16 |
Untersuchungen uber systeme integrierbarer funktionen,”Mathematische Annalen,
- Riesz
- 1910
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Citation Context ...∣p |ti+1 − ti|p−1 : x ≤ t1 < t2 < . . . < tn ≤ y } . Let ||f ||Vp = |f(0)|+ (Vp(f, [0, 1]))1/p, and let Ap[0, 1] denote the class of functions f defined on [0, 1] with f(0) = 0 and ||f ||Vp <∞. Riesz =-=[Rie10]-=- showed that each function in Ap[0, 1] is absolutely continuous. In analogy to the isometry of Banach spaces above, he also showed that the map g → λx. ∫ x0 g yields an isometry (Lp[0, 1], ||·||p)→ (A... |

15 |
The differentiability of constructive functions of weakly bounded variation on pseudo
- Demuth
- 1975
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Citation Context ...mplexity of exception null sets, and to characterize algorithmic randomness notions via computable analysis. Much earlier, the constructivist Demuth already observed this connection in papers such as =-=[Dem75]-=-. Our purpose is to carry out some of this program in the setting of Lipschitz analysis. We briefly discuss some basic concepts in computable analysis. A sequence (qn)n∈N of rationals is called a Cauc... |

13 | Schnorr randomness and the Lebesgue differentiation theorem
- Pathak, Rojas, et al.
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Citation Context ...k Lebesgue points. In dimension 1 they will later on be translated to results on differentiability of Lipschitz functions that are effective in a strong sense. Theorem 1.3 (Pathak, Rojas, and Simpson =-=[PRS14]-=- and Rute [Rut13]). Let z ∈ [0, 1]n. Then z is Schnorr random ⇔ z is a weak Lebesgue point of every L1-computable function. In Theorem 5.1 we give a proof of the implication “⇐” in Theorem 1.3, which ... |

10 |
Randomness, martingales and differentiability.
- Rute
- 2011
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Citation Context .... In dimension 1 they will later on be translated to results on differentiability of Lipschitz functions that are effective in a strong sense. Theorem 1.3 (Pathak, Rojas, and Simpson [PRS14] and Rute =-=[Rut13]-=-). Let z ∈ [0, 1]n. Then z is Schnorr random ⇔ z is a weak Lebesgue point of every L1-computable function. In Theorem 5.1 we give a proof of the implication “⇐” in Theorem 1.3, which we obtained indep... |

9 | A computational aspect of the Lebesgue differentiation theorem - Pathak |

3 |
Differentiability of polynomial time computable functions, to appear
- Nies
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Citation Context ...ity 1 at z. Andrews, Cai, Diamondstone, Lempp and Miller in unpublished work (2012) have shown that this randomness notion is equivalent to nonoscillation of left-r.e. martingales (see [Nie13]). Nies =-=[Nie14]-=- has shown that z is density random if and only if all interval-r.e. functions are differentiable at z. Oberwolfach randomness implies density randomness as shown in [BGK+12]. It is unknown whether th... |

2 |
Lectures on Lipschitz analysis, Rep
- Heinonen
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Citation Context ...om the well-known theorem of Lebesgue that every real function of bounded variation is differentiable almost everywhere. In higher dimensions one uses arguments particular to Lipschitz functions. See =-=[Hei05]-=- and the references given there for more background on Lipschitz functions. Computable analysis seeks algorithmic analogues of theorems from analysis when effectiveness conditions are imposed on the f... |

1 | and André Nies, Randomness and differentiability, submitted 2011 - Brattka, Miller |