Related papers: Normality, Relativization, and Randomness
The theory of quasirandomness has greatly expanded from its inaugural graph theoretical setting to several different combinatorial objects such as hypergraphs, tournaments, permutations, etc. However, these quasirandomness variants have…
Randomness is a crucial resource for a broad range of important applications, such as Monte Carlo simulation and computation, generative artificial intelligence and cryptography. But what is randomness? A widely accepted definition has…
In the present article, real number representations, that are generalizations of classical positive and alternating representations of numbers, are introduced and investigated. The main metric relation, properties of cylinder sets are…
We investigate the role of continuous reductions and continuous relativisation in the context of higher randomness. We define a higher analogue of Turing reducibility and show that it interacts well with higher randomness, for example with…
We use the notion of Borel generators to give alternative methods for computing standard invariants, such as associated primes, Hilbert series, and Betti numbers, of Borel ideals. Because there are generally few Borel generators relative to…
Despite the fact that almost all real numbers are absolutely normal---that is, the digits in their expansions to any base occur in all possible configurations with the expected frequency---not one specific example of an absolutely normal…
In Monoidal Computer I, we introduced a categorical model of computation where the formal reasoning about computability was supported by the simple and popular diagrammatic language of string diagrams. In the present paper, we refine and…
In this project we further investigate the idea of reducing the dimensionality of datasets using a Borel isomorphism with the purpose of subsequently applying supervised learning algorithms, as originally suggested by my supervisor V.…
This paper defines a new notion of bounded computable randomness for certain classes of sub-computable functions which lack a universal machine. In particular, we define such versions of randomness for primitive recursive functions and for…
The threshold network model is a type of finite random graphs. In this paper, we introduce a generalized threshold network model. A pair of vertices with random weights is connected by an edge when real-valued functions of the pair of…
We study a bijective map from integer partitions to the prime factorizations of integers that we call the "supernorm" of a partition, in which the multiplicities of the parts of partitions are mapped to the multiplicities of prime factors…
We introduce a series of numbers which serve as a generalization of Bernoulli, Euler numbers and binomial coefficients. Their properties are applied to solve a probability problem and suggest a statistical test for independence and…
The subject of this paper is regularity-preserving aggregation of regular norms on finite-dimensional linear spaces. Regular norms were introduced in [5] and are closely related to ``type 2'' spaces [9, Chapter 9] playing important role in…
Unlike Martin-L\"of randomness and Schnorr randomness, computable randomness has not been defined, except for a few ad hoc cases, outside of Cantor space. This paper offers such a definition (actually, several equivalent definitions), and…
Quasirandomness is a general mathematical concept meant to encapsulate several characteristics usually satisfied by random combinatorial objects, and which we regard as describing when a given object 'looks random'. In this survey we…
We propose a new family of fairness definitions for classification problems that combine some of the best properties of both statistical and individual notions of fairness. We posit not only a distribution over individuals, but also a…
In the paper I study properties of random polynomials with respect to a general system of functions. Some lower bounds for the mathematical expectation of the uniform and recently introduced integral-uniform norms of random polynomials are…
A natural connection between rational functions of several real or complex variables, and subspace collections is explored. A new class of function, superfunctions, are introduced which are the counterpart to functions at the level of…
The results of the renormalization group are commonly advertised as the existence of power law singularities near critical points. The classic predictions are often violated and logarithmic and exponential corrections are treated on a…
Hartle and Srednicki have suggested that standard quantum theory does not favor our typicality. Here an alternative version is proposed in which typicality is likely, Eventual Quantum Mechanics. This version allows one to calculate…