历史与综述

Most work on computational complexity is concerned with time. However this course will try to show that program-size complexity, which measures algorithmic information, is of much greater philosophical significance. I'll discuss how one can…

This is a report on a talk to high school mathematics students. It gives some combinatorial connections for the entries of the boustrophedon triangle whose sides are the Taylor coefficients for the tangent and secant functions.

In this paper two conjectures are proposed based on which we can prove the first case of Fermat's Last Theorem(FLT) for all primes $p \equiv -1 (\bmod~6)$. With Pollaczek's result {\bf [1]} and the conjectures the first case of FLT can be…

Two philosophical applications of the concept of program-size complexity are discussed. First, we consider the light program-size complexity sheds on whether mathematics is invented or discovered, i.e., is empirical or is a priori. Second,…

Noncommutativity lays hidden in the proofs of classical dynamics. Modern frameworks can be used to bring it to light: *-products, groupoids, q-deformed calculus, etc.

For a sample set of 1024 values, the FFT is 102.4 times faster than the discrete Fourier transform (DFT). The basis for this remarkable speed advantage is the `bit-reversal' scheme of the Cooley-Tukey algorithm. Eliminating the burden of…

The paper opens with an overview of the discussion of international comparisons (including goals) in mathematics education. Afterwards, the two most important recent international studies, the PISA Study and TIMSS-Repeat, are described.…

In this paper the author presents some non-conventional thoughts on the complexity of the Universe and the algorithmic reproducibility of the human brain, essentially sparked off by the notion of algorithmic complexity. We must warn that…

This essay was invited for publication in Nieuw Archief voor Wiskunde; it will also appear in translation in the SMF Gazette and in the DMV Mitteilungen. I discuss the recent trends in scholarly communication in mathematics, the current…

We discuss views about whether the universe can be rationally comprehended, starting with Plato, then Leibniz, and then the views of some distinguished scientists of the previous century. Based on this, we defend the thesis that…

This paper presents mathematics as a general science of computation in a way different from the tradition. It is based on the radical philosophical standpoint according to which the content, meaning and justification of experience lies in…

We provide a brief review of some of the major research results arising from the method of the Inverse Scattering Transform.

This essay contains three parts. The first part of essay focuses on the hypothesis of the functional semantic constructions (FSC-Hypothesis). This hypothesis explains that a language, a number, a money are the functional semantic…

We give a short biographical sketch of Emmy Noether.

The theory of aperiodic order is concerned with the development of ideas stimulated by the discovery of quasicrystals. We give a gentle introduction to some mathematical aspects of aperiodic order, aimed at a more general audience.

This article discusses what can be proved about the foundations of mathematics using the notions of algorithm and information. The first part is retrospective, and presents a beautiful antique, Godel's proof, the first modern incompleteness…

We give a short biographical sketch of Karl Weierstrass.

a 10-page biography of Raoul Bott followed by a 25-page discussion of his major papers

This paper presents a graded hierarchy or chain of binary operations on the reals and the complex numbers. The operations are related distributively in the sense that any one of them distributes over the next lower operation in the chain.…

During the past 25 years there has been a controversy regarding the adequacy of Newton's proof of Prop. 1 in Book 1 of the {\it Principia}. This proposition is of central importance because its proof of Kepler's area law allowed Newton to…