scholarly journals Effective aspects of Bernoulli randomness

2019 ◽  
Vol 29 (6) ◽  
pp. 933-946
Author(s):  
Christopher P Porter
Keyword(s):  
Random Sequence ◽  
Turing Degrees ◽  
Random Sequences ◽  
Bernoulli Measure ◽  
Computable Measure

Abstract In this paper, we study Bernoulli random sequences, i.e. sequences that are Martin-Löf random with respect to a Bernoulli measure $\mu _p$ for some $p\in [0,1]$, where we allow for the possibility that $p$ is noncomputable. We focus in particular on the case in which the underlying Bernoulli parameter $p$ is proper (i.e. Martin-Löf random with respect to some computable measure). We show for every Bernoulli parameter $p$, if there is a sequence that is both proper and Martin-Löf random with respect to $\mu _p$, then $p$ itself must be proper, and explore further consequences of this result. We also study the Turing degrees of Bernoulli random sequences, showing, for instance, that the Turing degrees containing a Bernoulli random sequence do not coincide with the Turing degrees containing a Martin-Löf random sequence. Lastly, we consider several possible approaches to characterizing blind Bernoulli randomness, where the corresponding Martin-Löf tests do not have access to the Bernoulli parameter $p$, and show that these fail to characterize blind Bernoulli randomness.

scholarly journals Computing from projections of random points

2019 ◽  
Vol 20 (01) ◽  
pp. 1950014
Author(s):  
Noam Greenberg ◽  
Joseph S. Miller ◽  
André Nies
Keyword(s):  
Cost Function ◽  
Random Sequence ◽  
Turing Degrees ◽  
Orthogonal Projections ◽  
Open Set ◽  
Proper Subclass ◽  
The Family ◽  
Function Characterization ◽  
The Cost ◽  
The Ideal

We study the sets that are computable from both halves of some (Martin–Löf) random sequence, which we call [Formula: see text]-bases. We show that the collection of such sets forms an ideal in the Turing degrees that is generated by its c.e. elements. It is a proper subideal of the [Formula: see text]-trivial sets. We characterize [Formula: see text]-bases as the sets computable from both halves of Chaitin’s [Formula: see text], and as the sets that obey the cost function [Formula: see text]. Generalizing these results yields a dense hierarchy of subideals in the [Formula: see text]-trivial degrees: For [Formula: see text], let [Formula: see text] be the collection of sets that are below any [Formula: see text] out of [Formula: see text] columns of some random sequence. As before, this is an ideal generated by its c.e. elements and the random sequence in the definition can always be taken to be [Formula: see text]. Furthermore, the corresponding cost function characterization reveals that [Formula: see text] is independent of the particular representation of the rational [Formula: see text], and that [Formula: see text] is properly contained in [Formula: see text] for rational numbers [Formula: see text]. These results are proved using a generalization of the Loomis–Whitney inequality, which bounds the measure of an open set in terms of the measures of its projections. The generality allows us to analyze arbitrary families of orthogonal projections. As it turns out, these do not give us new subideals of the [Formula: see text]-trivial sets; we can calculate from the family which [Formula: see text] it characterizes. We finish by studying the union of [Formula: see text] for [Formula: see text]; we prove that this ideal consists of the sets that are robustly computable from some random sequence. This class was previously studied by Hirschfeldt [D. R. Hirschfeldt, C. G. Jockusch, R. Kuyper and P. E. Schupp, Coarse reducibility and algorithmic randomness, J. Symbolic Logic 81(3) (2016) 1028–1046], who showed that it is a proper subclass of the [Formula: see text]-trivial sets. We prove that all such sets are robustly computable from [Formula: see text], and that they form a proper subideal of the sets computable from every (weakly) LR-hard random sequence. We also show that the ideal cannot be characterized by a cost function, giving the first such example of a [Formula: see text] subideal of the [Formula: see text]-trivial sets.

scholarly journals Von Mises' definition of random sequences reconsidered

1987 ◽  
Vol 52 (3) ◽  
pp. 725-755 ◽  
Author(s):  
Michiel van Lambalgen
Keyword(s):  
Statistical Tests ◽  
Random Sequence ◽  
The Other ◽  
Selection Procedures ◽  
Random Sequences ◽  
Definition Of ◽  
Von Mises ◽  
The Relationship

AbstractWe review briefly the attempts to define random sequences (§0). These attempts suggest two theorems: one concerning the number of subsequence selection procedures that transform a random sequence into a random sequence (§§1–3 and 5); the other concerning the relationship between definitions of randomness based on subsequence selection and those based on statistical tests (§4).

scholarly journals RECYCLE OF RANDOM SEQUENCES

1993 ◽  
Vol 04 (03) ◽  
pp. 569-590 ◽  
Author(s):  
NOBUYASU ITO ◽  
MACOTO KIKUCHI ◽  
YUTAKA OKABE
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Monte Carlo Simulations ◽  
Random Sequence ◽  
Random Sequences

The correlation between a random sequence and its transformed sequences is studied. In the case of a permutation operation or, in other words, the shuffling operation, it is shown that the correlation can be so small that the sequences can be regarded as independent random sequences. The applications to the Monte Carlo simulations are also given. This method is especially useful in the Ising Monte Carlo simulation.

scholarly journals Lowness for effective Hausdorff dimension

2014 ◽  
Vol 14 (02) ◽  
pp. 1450011 ◽  
Author(s):  
Steffen Lempp ◽  
Joseph S. Miller ◽  
Keng Meng Ng ◽  
Daniel D. Turetsky ◽  
Rebecca Weber
Keyword(s):  
Hausdorff Dimension ◽  
Random Sequence ◽  
Effective Dimension ◽  
Random Sequences ◽  
Central Notion

We examine the sequences A that are low for dimension, i.e. those for which the effective (Hausdorff) dimension relative to A is the same as the unrelativized effective dimension. Lowness for dimension is a weakening of lowness for randomness, a central notion in effective randomness. By considering analogues of characterizations of lowness for randomness, we show that lowness for dimension can be characterized in several ways. It is equivalent to lowishness for randomness, namely, that every Martin-Löf random sequence has effective dimension 1 relative to A, and lowishness for K, namely, that the limit of KA(n)/K(n) is 1. We show that there is a perfect [Formula: see text]-class of low for dimension sequences. Since there are only countably many low for random sequences, many more sequences are low for dimension. Finally, we prove that every low for dimension is jump-traceable in order nε, for any ε > 0.

scholarly journals Intrinsic density, asymptotic computability, and stochasticity

2021 ◽  
Vol 27 (2) ◽  
pp. 220-220
Author(s):  
Justin Miller
Keyword(s):  
Turing Degrees ◽  
Closure Properties ◽  
Computable Set ◽  
Open Questions ◽  
Bernoulli Measure ◽  
Almost All ◽  
E Mail

AbstractThere are many computational problems which are generally “easy” to solve but have certain rare examples which are much more difficult to solve. One approach to studying these problems is to ignore the difficult edge cases. Asymptotic computability is one of the formal tools that uses this approach to study these problems. Asymptotically computable sets can be thought of as almost computable sets, however every set is computationally equivalent to an almost computable set. Intrinsic density was introduced as a way to get around this unsettling fact, and which will be our main focus.Of particular interest for the first half of this dissertation are the intrinsically small sets, the sets of intrinsic density $0$ . While the bulk of the existing work concerning intrinsic density was focused on these sets, there were still many questions left unanswered. The first half of this dissertation answers some of these questions. We proved some useful closure properties for the intrinsically small sets and applied them to prove separations for the intrinsic variants of asymptotic computability. We also completely separated hyperimmunity and intrinsic smallness in the Turing degrees and resolved some open questions regarding the relativization of intrinsic density.For the second half of this dissertation, we turned our attention to the study of intermediate intrinsic density. We developed a calculus using noncomputable coding operations to construct examples of sets with intermediate intrinsic density. For almost all $r\in (0,1)$ , this construction yielded the first known example of a set with intrinsic density r which cannot compute a set random with respect to the r-Bernoulli measure. Motivated by the fact that intrinsic density coincides with the notion of injection stochasticity, we applied these techniques to study the structure of the more well-known notion of MWC-stochasticity.Abstract prepared by Justin Miller.E-mail: [email protected]: https://curate.nd.edu/show/6t053f4938w

scholarly journals Random graphs, weak coarse embeddings, and higher index theory

2015 ◽  
Vol 07 (03) ◽  
pp. 361-388 ◽  
Author(s):  
Rufus Willett
Keyword(s):  
Geometric Property ◽  
Random Sequence ◽  
Index Theory ◽  
Weak Form ◽  
Bounded Degree ◽  
Random Sequences ◽  
Finite Graphs ◽  
Coarse Embedding ◽  
Property T ◽  
Assembly Map

This paper studies higher index theory for a random sequence of bounded degree, finite graphs with diameter tending to infinity. We show that in a natural model for such random sequences the following hold almost surely: the coarse Baum–Connes assembly map is injective; the coarse Baum–Connes assembly map is not surjective; the maximal coarse Baum–Connes assembly map is an isomorphism. These results are closely tied to issues of expansion in graphs: in particular, we also show that such random sequences almost surely do not have geometric property (T), a strong form of expansion.The key geometric ingredients in the proof are due to Mendel and Naor: in our context, their results imply that a random sequence of graphs almost surely admits a weak form of coarse embedding into Hilbert space.

Goodness-of-fit tests for random sequences incorporating several components

2017 ◽  
Vol 25 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Tetiana O. Ianevych ◽  
Yuriy V. Kozachenko ◽  
Viktor B. Troshki
Keyword(s):  
Covariance Function ◽  
Goodness Of Fit ◽  
Random Sequence ◽  
Random Variables ◽  
Gaussian Random Variables ◽  
Random Sequences ◽  
Goodness Of Fit Tests

AbstractIn this paper we have constructed the goodness-of-fit tests incorporating several components, like expectation and covariance function for identification of a non-centered univariate random sequence or auto-covariances and cross-covariances for identification of a centered multivariate random sequence. For the construction of the corresponding estimators and investigation of their properties we utilized the theory of square Gaussian random variables.

scholarly journals Random Sequences Rapidly Evolve into de novo Promoters

2017 ◽  
Author(s):  
Avihu H. Yona ◽  
Eric J. Alm ◽  
Jeff Gore
Keyword(s):  
Protein Interactions ◽  
De Novo ◽  
Random Sequence ◽  
Lac Operon ◽  
Protein Protein Interactions ◽  
Random Sequences ◽  
E Coli ◽  
Promoter Recognition ◽  
Motif Recognition ◽  
Active Promoter

AbstractHow do new promoters evolve? To follow evolution of de novo promoters, we put various random sequences upstream to the lac operon in Escherichia coli and evolved the cells in the presence of lactose. We found that a typical random sequence of ~100 bases requires only one mutation in order to enable growth on lactose by increasing resemblance to the canonical promoter motifs. We further found that ~10% of random sequences could serve as active promoters even without any period of evolutionary adaptation. Such a short mutational distance from a random sequence to an active promoter may improve evolvability yet may also lead to undesirable accidental expression. We found that across the E. coli genome accidental expression is minimized by avoiding codon combinations that resemble promoter motifs. Our results suggest that the promoter recognition machinery has been tuned to allow high accessibility to new promoters, and similar findings might also be observed in higher organisms or in other motif recognition machineries, like transcription factor binding sites or protein-protein interactions.

scholarly journals Upper bound of probability of receiving error of aperiodic pseudo-random sequence at jamming

2020 ◽  
Vol 2020 (1) ◽  
pp. 64-72
Author(s):  
Iscandar Maratovich Azhmukhamedov ◽  
Evgeny Melnikov
Keyword(s):  
Communication Systems ◽  
Communication Channel ◽  
Random Sequence ◽  
Upper Bounds ◽  
Error Distribution ◽  
Reliable Operation ◽  
Broadband Communication ◽  
Worst Case ◽  
Random Sequences ◽  
Synchronizing Sequence

The article discusses the obtained estimate of the upper bounds for the probable receiving error of the synchronizing sequence during sensor phasing of aperiodic pseudo-random sequences (CRR) in broadband communication systems in the channels of low quality with strong disturbances of natural and organized structure. The obtained results allow to design synchronization systems of pseudo-random sequences for the worst case, which guarantees their reliable operation in low-quality channels, and, unlike the well-known methods, the estimation of synchronization of aperiodic pseudo-random sequence sensors doesn’t depend on the error distribution in the communication channel and the period of sequence. The appointed differences simplify the evaluation of synchronization in the operation of broadband communication systems in low-quality channels.

scholarly journals Yield of Indeterminate, Small-vine Cowpea Cultivars Unaffected by Uniformity of Within-row Spacing

HortScience ◽  
1995 ◽  
Vol 30 (5) ◽  
pp. 994-996 ◽  
Author(s):  
Brian A. Kahn ◽  
Peter J. Stoffella ◽  
Daniel I. Leskovar ◽  
James R. Cooksey
Keyword(s):  
Seed Yield ◽  
Vigna Unguiculata ◽  
Harvest Index ◽  
Random Sequence ◽  
Dry Weight ◽  
Row Spacing ◽  
Random Sequences ◽  
Cowpea Cultivars ◽  
Seed Spacing ◽  
Precision Planting

Cowpea [Vigna unguiculata (L.) Walp.] planters can produce variable within-row seed spacing. We determined whether precision planting of cowpea would produce a yield advantage over more random planting at the same rate. Studies were conducted from May 1992 to Feb. 1993 at three locations: Uvalde, Texas; Bixby, Okla.; and Fort Pierce, Fla. Seeds of the indeterminate, small-vine cowpea cultivars Mississippi Silver and Pinkeye Purplehull BVR were hand-planted at 42 per 3.15 m of row. Seeds within rows were either spaced uniformly at 7.5 cm [control, with sd = 0] or in one of two random sequences (sd = 4.8). At harvest, in Oklahoma and Florida, mean within-row spacings were similar, but sd values of random-sequence plots remained greater than those of control plots. Control plots averaged four more plants at harvest than random-sequence plots in Texas. However, seed yield (seed dry weight per hectare) and harvest index were unaffected by uniformity of within-row spacing at all three locations. Thus, precision seeding of indeterminate, small-vine cowpea cultivars seems unlikely to produce a yield advantage over more random planting at the same rate.

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