Related papers: Irrational Base Counting

In this study, an algorithm for computing the inverse of periodic k banded matrices, which are needed for solving the differential equations by using the finite differences, the solution of partial differential equations and the solution of…

We address a well-known problem in combinatorics involving the identification of counterfeit coins with a systematic approach. The methodology can be applied to cases where the total number of coins is exceedingly large such that brute…

A new characterization of provably recursive functions of first-order arithmetic is described. Its main feature is using only terms consisting of 0, the successor S and variables in the quantifier rules, namely, universal elimination and…

We analyze algorithms that output absolutely normal numbers digit-by-digit with respect to quality of convergence to normality of the output, measured by the discrepancy. We consider explicit variants of algorithms by Sierpinski, by Turing…

We consider the problem of how to verify the security of probabilistic oblivious algorithms formally and systematically. Unfortunately, prior program logics fail to support a number of complexities that feature in the semantics and…

Work in progress concerning alternative formalizations of arithmetic.

We prove the Oppenheim conjecture for indefinite ternary diagonal forms of the type $x^{2}+y^{2} -\alpha z^{2}$ where $ \alpha $ is an irrational number. Our method is explicit in the sense that we are able to construct a solution to the…

We construct a countable family of multi-dimensional continued fraction algorithms, built out of five specific multidimensional continued fractions, and find a wide class of cubic irrational real numbers a so that either (a, a^2) or (a,…

In this note we give a wellfoundedness proof of a computable notation system for first-order reflection.

This article concerns the $p$-basic set existence problem in the representation theory of finite groups. We show that, for any odd prime $p$, the alternating group $\A_n$ has a $p$-basic set. More precisely, we prove that the symmetric…

Short nonstandard proofs are given for some results about infinite systems of equations in infinitely many variables.

We extend the recently proposed order-N algorithms (cond-mat/9703224) for calculating linear- and nonlinear-response functions in time domain to the systems described by nonorthonormal basis sets.

We study a well-known technique of using absoluteness for giving choice-free proofs to some statements which are known to be provable with the axiom of choice. The idea is to reduce the problem to an inner model where the axiom of choice…

We describe and classify countable Boolean rings (which may or may not have a multiplicative identity) with finitely many distinguished ideals whose elementary theory is countably categorical. This extends the description by Macintyre and…

We describe the countable ordinals in terms of iterations of Mostowski collapsings. This gives a proof-theoretic bound of definable countable ordinals in the Zermelo-Fraenkel's set theory ZF.

In this paper we review a general proof for the irrationality property of numbers which take a certain form of infinite sums.

We discuss the use of matrices for providing sequences of rationals that approximate algebraic irrationalities. In particular, we study the regular representation of algebraic extensions, proving that ratios between two entries of the…

The increasing advancement of emerging device technologies that provide alternative basis logic sets necessitates the exploration of innovative logic design automation methodologies. Specifically, emerging computing architectures based on…

Sophisticated machine models are increasingly used for high-stakes decisions in everyday life. There is an urgent need to develop effective explanation techniques for such automated decisions. Rule-Based Explanations have been proposed for…

For alternate Cantor real base numeration systems we generalize the result of Frougny and~Solomyak on~arithmetics on the set of numbers with finite expansion. We provide a class of alternate bases which satisfy the so-called finiteness…