Related papers: Surreal Arithmetic, Lazily
We obtain simple quadratic recurrence formulas counting bipartite maps on surfaces with prescribed degrees (in particular, $2k$-angulations), and constellations. These formulas are the fastest known way of computing these numbers. Our work…
This paper focuses on greedy expansions, one possible representation of numbers, and on arithmetical operations with them. Performing addition or multiplication some additional digits can appear. We study bounds on the number of such digits…
In this article, we consider a simple representation for real numbers and propose top-down procedures to approximate various algebraic and transcendental operations with arbitrary precision. Detailed algorithms and proofs are provided to…
Interestingly, LLMs yet struggle with some basic tasks that humans find trivial to handle, e.g., counting the number of character r's in the word "strawberry". There are several popular conjectures (e.g., tokenization, architecture and…
Speedable numbers are real numbers which are algorithmically approximable from below and whose approximations can be accelerated nonuniformly. We begin this article by answering a question of Barmpalias by separating a strict subclass that…
A real random variable admits median(s) and quantiles. These values minimize convex functions on $\mathbb R$. We show by "Convex Analysis" arguments that the function to be minimized is very natural. The relationship with some notions about…
Optimization problems with convex quadratic cost and polyhedral constraints are ubiquitous in signal processing, automatic control and decision-making. We consider here an enlarged problem class that allows to encode logical conditions and…
A critical challenge for any intelligent system is to infer structure from continuous data streams. Theories of event-predictive cognition suggest that the brain segments sensorimotor information into compact event encodings, which are used…
Inspired by a recent beautiful construction of Armin Straub and Wadim Zudilin, that 'tweaked' the sum of the $s^{th}$ powers of the $n$-th row of Pascal's triangle, getting instead of sequences of numbers, sequences of rational functions,…
In the literature, we have various ways of proving irrationality of a real number. In this survey article, we shall emphasize on a particular criterion to prove irrationality. This is called nice approximation of a number by a sequence of…
Quantum computers will be able solve important problems with significant polynomial and exponential speedups over their classical counterparts, for instance in option pricing in finance, and in real-space molecular chemistry simulations.…
The paper is an extensive and systematic study of cardinal invariants we call slalom numbers, describing the combinatorics of sequences of sets of natural numbers. Our general approach, based on relational systems, covers many such cardinal…
A description of the properties of \L with complex characters is given. By using these, together with the more familiar \L with real characters, it is shown how certain two dimensional lattice sums, which previously could not be put into…
A generalization of the classical Lipschitz summation formula is proposed. It involves new polylogarithmic rational functions constructed via the Fourier expansion of certain sequences of Bernoulli--type polynomials. Related families of…
Our results can be viewed as applications of algebraic combinatorics in random matrix theory. These applications are motivated by the predictive power of random matrix theory for the statistical behavior of the celebrated Riemann…
After a short review of the historical milestones on normal numbers, we introduce the Borel numbers as the reals admitting a probability function on their different bases representations. In this setting, we provide two probabilistic…
It's the age-old recurrence with a twist: sum the last two terms and if the result is composite, divide by its smallest prime divisor to get the next term (e.g., 0, 1, 1, 2, 3, 5, 4, 3, 7, ...). These sequences exhibit pseudo-random…
We deliver a call to arms for probabilistic numerical methods: algorithms for numerical tasks, including linear algebra, integration, optimization and solving differential equations, that return uncertainties in their calculations. Such…
N.Katz's middle convolution algorithm provides a description of rigid connections on the projective line with regular singularities. We extend the algorithm by adding the Fourier transform to it. The extended algorithm provides a…
We present a way to capture high-information posteriors from training sets that are sparsely sampled over the parameter space for robust simulation-based inference. In physical inference problems, we can often apply domain knowledge to…