Related papers: Lowness for the class of Schnorr random reals
The low for random reals are characterized topologically, as well as in terms of domination of Turing functionals on a set of positive measure.
The notion of Schnorr randomness refers to computable reals or computable functions. We propose a version of Schnorr randomness for subcomputable classes and characterize it in different ways: by Martin L\"of tests, martingales or measure…
A real is called integer-valued random if no integer-valued martingale can win arbitrarily much capital betting against it. A real is low for integer-valued randomness if no integer-valued martingale recursive in A can succeed on an…
One of the main lines of research in algorithmic randomness is that of lowness notions. Given a randomness notion R, we ask for which sequences A does relativization to A leave R unchanged (i.e., R^A = R)? Such sequences are call low for R.…
In this paper, we will generalize the definition of partially random or complex reals, and then show the duality of random and complex, i.e., a generalized version of Levin-Schnorr's theorem. We also study randomness from the view point of…
When researchers carry out a null hypothesis significance test, it is tempting to assume that a statistically significant result lowers Prob(H0), the probability of the null hypothesis being true. Technically, such a statement is…
In this paper we study a particular class of polynomials. We study the distribution of their zeros, including the zeros of their derivatives as well as the interaction between this two. We prove a weak variant of the sendov conjecture in…
We show that positive measure domination implies uniform almost everywhere domination and that this proof translates into a proof in the subsystem WWKL$_0$ (but not in RCA$_0$) of the equivalence of various Lebesgue measure regularity…
We prove that a set is K-trivial if and only if it is not weakly ML-cuppable. Further, we show that a set below zero jump is K-trivial if and only if it is not ML-cuppable. These results settle a question of Ku\v{c}era, who introduced both…
We define here the notion of a {\it weakly reversible ring} $R$ saying that a non-zero element $a\in R$ is weakly reversible if there exists an integer $m>0$ depending on $a$ such that $a^m\neq 0$ is reversible, that is,…
We correct Miyabe's proof of van Lambalgen's Theorem for truth-table Schnorr randomness (which we will call uniformly relative Schnorr randomness). An immediate corollary is one direction of van Lambalgen's theorem for Schnorr randomness.…
In this expository note we present simple proofs of the lower bound of Ramsey numbers (Erd\"os theorem), and of the estimation of discrepancy. Neither statements nor proofs require any knowledge beyond high-school curriculum (except a minor…
We show how the presence of resonances close to the real axis implies exponential lower bounds on the norm of the cut-off resolvent on the real axis.
We show that there is a low T-upper bound for the class of K-trivial sets, namely those which are weak from the point of view of algorithmic randomness. This result is a special case of a more general characterization of ideals in the…
We compare two methods of proving separable reduction theorems in functional analysis -- the method of rich families and the method of elementary submodels. We show that any result proved using rich families holds also when formulated with…
Many situations in quantum theory and other areas of physics lead to quasi-probabilities which seem to be physically useful but can be negative. The interpretation of such objects is not at all clear. In this paper, we show that…
We investigate the strength of a randomness notion $\mathcal R$ as a set-existence principle in second-order arithmetic: for each $Z$ there is an $X$ that is $\mathcal R$-random relative to $Z$. We show that the equivalence between…
In a previous paper (see arXiv:1003.3411 [math.CA]), we investigated the existence of an element x of a quasi-Banach space X whose errors of best approximation by a given approximation scheme (A_n) (defined by E(x,A_n) = \inf_{a \in A_n}…
A strongly special subvariety of a Shimura variety $S$ is (essentially) a subvariety associated to a semi-simple sub-Shimura datum. We prove that the set of probability measures canonically associated to to strongly special subvarieties is…
We provide a lower bound on the probability that a binomial random variable is exceeding its mean. Our proof employs estimates on the mean absolute deviation and the tail conditional expectation of binomial random variables.