Related papers: The Ouroboros Goodstein Principle
The query model offers a concrete setting where quantum algorithms are provably superior to randomized algorithms. Beautiful results by Bernstein-Vazirani, Simon, Aaronson, and others presented partial Boolean functions that can be computed…
We consider an elementary discrete process which starts from purely random configuration and leads to well-ordered and stable state. Complete analytical solution to this problem is presented.
We prove various theorems on approximation using polynomials with integer coefficients in the Bernstein basis of any given order. In the extreme, we draw the coefficients from $\{ \pm 1\}$ only. A basic case of our results states that for…
A 1971 conjecture of Graham (later repeated by Erd\H{o}s and Graham) asserts that every set $A \subseteq \mathbb{F}_p \setminus \{0\}$ has an ordering whose partial sums are all distinct. We prove this conjecture for sets of size $|A|…
We propose the novel augmented Gaussian random field (AGRF), which is a universal framework incorporating the data of observable and derivatives of any order. Rigorous theory is established. We prove that under certain conditions, the…
Let $B_H=\{B_H(t):t\in\mathbb R\}$ be a fractional Brownian motion with Hurst parameter $H\in(0,1)$. For the stationary storage process $Q_{B_H}(t)=\sup_{-\infty<s\le t}(B_H(t)-B_H(s)-(t-s))$, $t\ge0$, we provide a tractable criterion for…
In the study of random access machines (RAMs) it has been shown that the availability of an extra input integer, having no special properties other than being sufficiently large, is enough to reduce the computational complexity of some…
We propose a new model of computation based on nonstandard analysis. Intuitively, the role of "algorithm" is played by a new notion of finite procedure, called Omega-invariance and inspired by physics, from nonstandard analysis. Moreover,…
A methodology on making the variational principle well-posed in degenerate systems is constructed. In the systems including higher-order time derivative terms being compatible with Newtonian dynamics, we show that a set of position…
Let $U_{n}\subset C^{n}[ a,b] $ be an extended Chebyshev space of dimension $n+1$. Suppose that $f_{0}\in U_{n}$ is strictly positive and $% f_{1}\in U_{n}$ has the property that $f_{1}/f_{0}$ is strictly increasing. We search for…
In the article 'Ordinal Logics and the Characterizations of the Informal Concept of Proof', Georg Kreisel poses the problem of assigning unique notations to recursive ordinals, and additionally suggests that the methods which are developed…
We study the recursion-theoretic complexity of Positive Almost-Sure Termination ($\mathsf{PAST}$) in an imperative programming language with rational variables, bounded nondeterministic choice, and discrete probabilistic choice. A program…
Many Gibbs measures with mean field interactions are known to be chaotic, in the sense that any collection of $k$ particles in the $n$-particle system are asymptotically independent, as $n\to\infty$ with $k$ fixed or perhaps $k=o(n)$. This…
In this paper we give a new proof of the Ne\v{s}et\v{r}il-R\"odl Theorem, a deep result of discrete mathematics which is one of the cornerstones of the structural Ramsey theory. In contrast to the well-known proofs which employ intricate…
We motivate and study an infinite sequence of binary operations on the ordinal numbers, extending the standard arithmetic on the ordinals to higher degrees of iteration. Connections to the hyperoperations on the natural numbers are…
This paper introduces several new classes of mathematical structures that have close connections with physics and with the theory of dynamical systems. The most general of these structures, called indivisible stochastic processes,…
Bayesian inference for rank-order problems is frustrated by the absence of an explicit likelihood function. This hurdle can be overcome by assuming a latent normal representation that is consistent with the ordinal information in the data:…
We prove that omega^2 strictly bounds the iterations required for modal definable functions to reach a fixed point across all countable structures. The result corrects and extends the previously claimed result by the first and third authors…
Two models were recently proposed to explore the robust hardness of Gr\"obner basis computation. Given a polynomial system, both models allow an algorithm to selectively ignore some of the polynomials: the algorithm is only responsible for…
In mathematical logic there are two seemingly distinct kinds of principles called "reflection principles." Semantic reflection principles assert that if a formula holds in the whole universe, then it holds in a set-sized model. Syntactic…