Related papers: Irrational Base Counting
Writing for a general mathematical audience, we provide elementary upper and lower bounds on the growth (as a function of N) of the sum \sum_{n=1}^N (-1)^{\floor{n x}} for various fixed x. For example, if x is a quadratic irrational, then…
We study normalising reduction strategies for infinitary Combinatory Reduction Systems (iCRSs). We prove that all fair, outermost-fair, and needed-fair strategies are normalising for orthogonal, fully-extended iCRSs. These facts properly…
We construct a Beurling generalized number system satisfying the Riemann hypothesis and whose integer counting function displays extremal oscillation in the following sense. The prime counting function of this number system satisfies…
We present a new topological proof of the infinitude of prime numbers with a new topology. Furthermore, in this topology, we characterize the infinitude of any non-empty subset of prime numbers.
This paper presents an innovative approach, the Adaptive Orthogonal Basis Method, tailored for computing multiple solutions to differential equations characterized by polynomial nonlinearities. Departing from conventional practices of…
This paper investigates $\exists\mathbb{R}(r^{\mathbb{Z}})$, that is the extension of the existential theory of the reals by an additional unary predicate $r^{\mathbb{Z}}$ for the integer powers of a fixed computable real number $r > 0$. If…
For each positive integer n greater than or equal to 2, a new approach to expressing real numbers as sequences of nonnegative integers is given. The n=2 case is equivalent to the standard continued fraction algorithm. For n=3, it reduces to…
Suppose we have n algorithms, quantum or classical, each computing some bit-value with bounded error probability. We describe a quantum algorithm that uses O(sqrt{n}) repetitions of the base algorithms and with high probability finds the…
We extend the classical Ostrowski numeration systems, closely related to Sturmian words, by allowing a wider range of coefficients, so that possible representations of a number $n$ better reflect the structure of the associated Sturmian…
In this paper we explicitly compute finite bases of disjunctive identities and finite bases of regular representations for a number of interesting finite groups.
Evaluating performance across optimization algorithms on many problems presents a complex challenge due to the diversity of numerical scales involved. Traditional data processing methods, such as hypothesis testing and Bayesian inference,…
Set out here are some fundamental theories that may be regarded as newly discovered metamathematics of the odd integers in relation to the Collatz conjecture (also called the 3x+1 problem). Originally motivated by the requirement to invent…
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 give three elementary proofs of a nice equality of definite integrals, which arises from the theory of bivariate hypergeometric functions, and has connections with irrationality proofs in number theory. We furthermore provide a…
We present the Lukasiewicz logic as a viable system for an implication algebra on a system of qubits. Our results show that the three valued Lukasiewicz logic can be embedded in the stabilized space of an arbitrary quantum error correcting…
We describe an approximate rational arithmetic with round-off errors (both absolute and relative) controlled by the user. The rounding procedure is based on the continued fraction expansion of real numbers. Results of computer experiments…
Let $n$ be a positive integer, and let $R$ be a (possibly infinite dimensional) finitely presented algebra over a computable field of characteristic zero. We describe an algorithm for deciding (in principle) whether $R$ has at most finitely…
Anomalous cancellation of fractions is a mathematically inaccurate method where cancelling the common digits of the numerator and denominator correctly reduces it. While it appears to be accidentally successful, the property of anomalous…
We use recurrences of integrals to give new and elementary proofs of the irrationality of pi, tan(r) for all nonzero rational r, and cos(r) for all nonzero rational r^2. Immediate consequences to other values of the elementary…
This paper investigates the problem of bounding possible output from a counterfactual query given a set of observational data. While various works of literature have described methodologies to generate efficient algorithms that provide an…