Related papers: Kolmogorov Complexity and Solovay Functions

Uniform upper bounds and the asymptotic expansion with an explicit remainder term are established for the Macdonald function $K_{i\tau}(x)$. The results can be applied, for instance, to study the summability of the divergent…

Let $f \in S_k(\Gamma_1(N))$ be a primitive holomorphic form of arbitrary weight $k$ and level $N$. We show that the completed $L$-function of $f$ has $\Omega\left(T^\delta\right)$ simple zeros with imaginary part in $\left[-T, T\right]$,…

We prove a Kolmogorov complexity variant of the birthday paradox. Sufficiently sized random subsets of strings are guaranteed to have two members x and y with low K(x/y). To prove this, we first show that the minimum conditional Kolmogorov…

We introduce orderings between total functions f,g: N -> N which refine the pointwise "up to a constant" ordering <=cte and also insure that f(x) is often much less thang(x). With such orderings, we prove a strong hierarchy theorem for…

This manuscript includes some classical results we select apart from the new results we've found on the Analysis of Boolean Functions and Fourier-Entropy-Influence conjecture. We try to ensure the self-completeness of this work so that…

In an attempt to demonstrate that local hidden variables are mathematically possible, Pitowsky constructed "spin-$\frac12$ functions" and later "Kolmogorovian models", which employs a nonstandard notion of probability. We describe…

A real number is called left-computable if there exists a computable increasing sequence of rational numbers converging to it. In this article we are investigating a proper subset of the left-computable numbers. We say that a real number…

In this paper, we study questions of definability and decidability for infinite algebraic extensions ${\bf K}$ of $\mathbb{F}_p(t)$ and their subrings of $\mathcal{S}$-integral functions. We focus on fields ${\bf K}$ satisfying a local…

This paper introduces a new generalized superfactorial function (referable to as $n^{th}$- degree superfactorial: $sf^{(n)}(x)$) and a generalized hyperfactorial function (referable to as $n^{th}$- degree hyperfactorial: $H^{(n)}(x)$), and…

We introduce a notion of algorithmic randomness for algebraic fields. We prove the existence of a continuum of algebraic extensions of $\mathbb{Q}$ that are random according to our definition. We show that there are noncomputable algebraic…

In this article, we prove the existence of extremal functions in higher-order affine Sobolev inequalities. Proofs rely on concentration-compactness methods in spaces of integer or fractional regularity. The tools we use, available in spaces…

In this paper we present a new class of complexity measures, induced by a new data structure for representing $k$-valued functions (operations), called minor decision diagram. The results are presented in terms of Multi-Valued Logic…

By Kolmogorov Complexity,two number-theoretic problems are solved in different way than before,one problem is Maxim Kontsevich and Don Bernard Zagier's Problem 3 \emph{Exhibit at least one number which does not belong to} $ \mathcal{P}$…

Let No be Conway's class of surreal numbers. I will make explicit the notion of a function f on No recursively defined over some family of functions. Under some "tameness" and uniformity condition, f must satisfy some interesting…

Let G be a connected, real, semisimple Lie group contained in its complexification G_C, and let K be a maximal compact subgroup of G. We construct a K_C-G double coset domain in G_C, and we show that the action of G on the K-finite vectors…

Although information content is invariant up to an additive constant, the range of possible additive constants applicable to programming languages is so large that in practice it plays a major role in the actual evaluation of K(s), the…

We consider classical representations of integers: Church's function iterators, cardinal equivalence classes of sets, ordinal equivalence classes of totally ordered sets. Since programs do not work on abstract entities and require formal…

We investigate the computational properties of basic mathematical notions pertaining to $\mathbb{R}\rightarrow \mathbb{R}$-functions and subsets of $\mathbb{R}$, like finiteness, countability, (absolute) continuity, bounded variation,…

The objective of this study is a better understanding of the relationships between reduction and continuity. Solovay reduction is a variation of Turing reduction based on the distance of two real numbers. We characterize Solovay reduction…

A clone on a set X is a set of finitary functions on X which contains the projections and which is closed under composition. The set of all clones on X forms a complete algebraic lattice Cl(X). We obtain several results on the structure of…