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This paper presents a new abstract method for proving lower bounds in computational complexity. Based on the notion of topological and measurable entropy for dynamical systems, it is shown to generalise three previous lower bounds results from the literature in algebraic complexity. We use it to prove that $\mathtt{maxflow}$, a $\mathrm{Ptime}$ complete problem, is not computable in polylogarithmic time on parallel random access machines (prams) working with real numbers. Thi...

The arithmetic of natural numbers has a natural and simple encoding within sets, and the simplest set whose structure is not that of any natural number extends this set-theoretic representation to positive and negative integers. The operation that implements addition when applied to sets that represent natural numbers yields both addition and subtraction when used with the sets that encode integers. The encoding of the integers naturally extends beyond them and identifies set...

We introduce in this section an Algebraic and Combinatorial approach to the theory of Numbers. The approach rests on the observation that numbers can be identified with familiar combinatorial objects namely rooted trees, which we shall here refer to as towers. The bijection between numbers and towers provides some insights into unexpected connexions between Number theory, combinatorics and discrete probability theory.

To achieve systematic generalisation, it first makes sense to master simple tasks such as arithmetic. Of the four fundamental arithmetic operations (+,-,$\times$,$\div$), division is considered the most difficult for both humans and computers. In this paper we show that robustly learning division in a systematic manner remains a challenge even at the simplest level of dividing two numbers. We propose two novel approaches for division which we call the Neural Reciprocal Unit (...

Science and mathematics help people better to understand world, eliminating different fallacies and misconceptions. One of such misconception is related to arithmetic, which is so important both for science and everyday life. People think that their counting is governed by the rules of the conventional arithmetic and that other kinds of arithmetic do not exist and cannot exist. It is demonstrated in this paper that this popular image of the situation with integer numbers is i...

Our number system is a magnificent tool. But it is far from perfect. Can it be improved? In this paper some possibilities are discussed, including the use of a different base or directed (negative as well as positive) numerals. We also put forward some suggestions for further research.

This article is an expanded version of my talk at the Gathering for Gardner, 2012.

I propose a class of numeral systems where numbers are represented by Dyck words, with the systems arising from a generalization of prime factorization. After describing two proper subsets of the Dyck language capable of uniquely representing all natural and rational numbers respectively, I consider "Dyck-complete" languages, in which every member of the Dyck language represents a number. I conclude by suggesting possible research directions.