Isolated 2-computably enumerable Q-degrees

In this work we prove that for every pair of computably enumerable degrees a<Q b there exists a properly 2-computably enumerable degree d such that a <Q d <Q b, a isolates d from below, and b isolates d from above. Two corollaries follow from this result. First, there exists a 2-computably enumerable degree which is Q-incomparable with any nontrivial (different from 0 and 0′) computably enumerable degree. Second, every nontrivial computably enumerable degree isolates some 2-computably enumerable degree from below and some 2-computably enumerable degree from above. © 2010 Allerton Press, Inc

Quasi-completeness and functions without fixed-points

We prove a completeness criterion for quasi-reducibility and generalize it to higher levels of the arithmetical hierarchy. As an application of the criterion we obtain Q-completeness of the set of all pairs (x, n) such that the prefix-free Kolmogorov complexity of x is less than n. © 2006 WILEY-VCH Verlag GmbH & Co. KGaA

Non-isolated quasi-degrees

We show that non-isolated from below 2-c.e. Q-degrees are dense in the structure of c.e. Q-degrees. We construct a 2-c.e. Q-degree, which can't be isolated from below not only by c.e. Q-degrees, but by any Q-degree. We also prove that below any c.e. Q-degree there is a 2-c.e. Q-degree, which is non-isolated from below and from above. © 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Lexicographical estimates of likelihood with universal boundaries. II

Properties of negation operations, not being involutions in the general case, are studied. Arbitrary linear-ordered set of likelihood values is considered as a set of likelihood values. Properties of negation operations over various sets of lexicographical likelihood estimates, being the extensions of the assumed negation operation, are studied. Over sets of the Λ-estimates and V-estimates of likelihood, it is possible to introduce only minimum extensions of negation operation from the assumed likelihood scale. Over a set of (V, Λ)-estimates of likelihood the introduction of several different negation operations is possible

Lexicographical estimates of likelihood with universal boundaries. II

Properties of negation operations, not being involutions in the general case, are studied. Arbitrary linear-ordered set of likelihood values is considered as a set of likelihood values. Properties of negation operations over various sets of lexicographical likelihood estimates, being the extensions of the assumed negation operation, are studied. Over sets of the Λ-estimates and V-estimates of likelihood, it is possible to introduce only minimum extensions of negation operation from the assumed likelihood scale. Over a set of (V, Λ)-estimates of likelihood the introduction of several different negation operations is possible

Q-reducibility and m-reducibility on computably enumerable sets

© 2014, Pleiades Publishing, Ltd. We study the distinctions between Q-reducibility and m-reducibility on computably enumerable sets. We construct a noncomputable m-incomplete computably enumerable set B such that all computably enumerable sets A ≤QB satisfy A ≤mB. We prove that for every noncomputable computably enumerable set A there exists a computably enumerable set B such that A ≤QB but A ≰mB. We prove that for every simple set B there exists a computably enumerable set A such that A ≤QB but A ≰mB. The last result implies in particular that the Q-degree of every simple set contains infinitely many computably enumerable m-degrees

Irreducible, Singular, and Contiguous Degrees

© 2017, Springer Science+Business Media, LLC. We study structures of degrees of stronger algorithmic reducibilities inside the degrees of weaker algorithmic ones. Results in this area are reviewed for algorithmic reducibilities m-, 1-, tt-, wtt-, T-, e-, s-, Q-, and we formulate questions that are still not settled for these. A computably enumerable Q-degree which consists of one computably enumerable m-degree is constructed

Positive and negative local trend association patterns in analysis of associations between time series

The paper introduces new time series shape association measures based on Euclidean distance. The method of analysis of associations between time series based on separate analysis of positively and negatively associated local trends is discussed. The examples of application of the proposed measures and methods to analysis of associations between historical prices of securities obtained from Google Finance are considered. An example of time series with inverse associations between them is discussed. © 2014 Springer International Publishing

Time series shape association measures and local trend association patterns

© 2015 Elsevier B.V. All rights reserved. The paper gives the new definition of non-statistical time series shape association measures that can measure positive and negative shape associations between time series. The local trend association measures based on linear regressions in sliding window are considered. The methods of extraction and presentation of positive and negative local trend association patterns from the pairs of time series are described. Examples of application of these methods to analysis of associations between securities data from Google Finance and between exchange rates are discussed. It was shown on the benchmark example and in the analysis of real time series that the correlation coefficient in spite of its fundamental role in statistics does not useful here and can cause confusion in analysis of time series shape similarity and shape associations