Related papers: Adding a random real number and its effect on Mart…
Considered will be properties of the set of real numbers $\Re$ generated by an operator that has form of an exponential function of Gelfond-Schneider type with rational arguments. It will be shown that such created set has cardinal number…
The arithmetic partial derivative (with respect to a prime $p$) is a function from the set of integers that sends $p$ to 1 and satisfies the Leibniz rule. In this paper, we prove that the $p$-adic valuation of the sequence of higher order…
A proof that the set of real numbers is denumerable is given.
We prove a Ramsey theorem for finite sets equipped with a partial order and a fixed number of linear orders extending the partial order. This is a common generalization of two recent Ramsey theorems due to Soki\'c. As a bonus, our proof…
We characterize some major algorithmic randomness notions via differentiability of effective functions. (1) As the main result we show that a real number z in [0,1] is computably random if and only if each nondecreasing computable function…
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…
We establish a sharp upper estimate for the order of a canonical system in terms of the Hamiltonian. This upper estimate becomes an equality in the case of Krein strings. As an application we prove a conjecture of Valent about the order of…
Providing explanations about how machine learning algorithms work and/or make particular predictions is one of the main tools that can be used to improve their trusworthiness, fairness and robustness. Among the most intuitive type of…
It is well known to generalize the meagre ideal replacing aleph_0 by a (regular) cardinal lambda > aleph_0 and requiring the ideal to be lambda^+-complete. But can we generalize the null ideal? In terms of forcing, this means finding a…
This chapter introduces the \emph{random-order model} in online algorithms. In this model, the input is chosen by an adversary, then randomly permuted before being presented to the algorithm. This reshuffling often weakens the power of the…
We investigate which infinite binary sequences (reals) are effectively random with respect to some continuous (i.e., non-atomic) probability measure. We prove that for every n, all but countably many reals are n-random for such a measure,…
Based on continued fractions with subtractions, we identify the set of real numbers with the set of infinite integer sequences with all terms but the first one greater or equal to two. Each such sequence produces in a canonical way a unique…
This article explores the model-dependent nature of set cardinality, emphasizing that cardinality is not absolute but varies across different axiomatic frameworks. Although Cantor's diagonal argument shows the real numbers are…
We discuss the relationship between perfect sets of random reals, dominating reals, and the product of two copies of the random algebra B. Recall that B is the algebra of Borel sets of 2^omega modulo the null sets. Also given two models M…
We perturb a real matrix $A$ of full column rank, and derive lower bounds for the smallest singular values of the perturbed matrix, in terms of normwise absolute perturbations. Our bounds, which extend existing lower-order expressions,…
We study pairs $(V, V_1)$ of models of $ZFC$ such that adding $\kappa$-many random reals over $V_1$ adds $\lambda$-many random reals over $V$, for some $\lambda > \kappa.$
We study the arithmetic property which allows to sharpen number-theoretic estimates. Previous results on this property are, as a rule, quantitive. The application of our general qualitive theorems to generalized hypergeometric functions…
Our main theorem is about iterated forcing for making the continuum larger than aleph_2. We present a generalization of math.LO/0303294 which is dealing with oracles for random, etc., replacing aleph_1, aleph_2 by lambda,lambda^+ (starting…
Many questions in number theory concern the nonvanishing of determinants of square matrices of logarithms (complex or p-adic) of algebraic numbers. We present a new conjecture that states that if such a matrix has vanishing determinant,…
We study randomness beyond $\Pi^1_1$-randomness and its Martin-L\"of type variant, introduced in \cite{MR2340241} and further studied in \cite{Continuous-higher-randomness}. The class given by the infinite time Turing machines (\ITTM s),…