On Commuting Exponentials in Low Dimensions

We show that the discrete complex, and numerous hypercomplex, Fourier transforms defined and used so far by a number of researchers can be unified into a single framework based on a matrix exponential version of Euler's formula $e^{j\theta}=\cos\theta+j\sin\theta$, and a matrix root of -1 isomorphic to the imaginary root $j$. The transforms thus defined can be computed using standard matrix multiplications and additions with no hypercomplex code, the complex or hypercomplex a...

This note presents an analytic technique for proving the linear independence of certain small subsets of real numbers over the rational numbers. The applications of this test produce simple linear independence proofs for the subsets of triples $\{1, e, \pi\}$, $\{1, e, \pi^{-1}\}$, and $\{1, \pi^r, \pi^s\}$, where $1\leq r<s $ are fixed integers.

In the first paper we proved that on the cyclic groups of odd order d, there exist non zero functions whose convolution square f*f(2t) is proportional to their square f(t)^2 when the proportionality constant is an odd algebraic integer of norm d whose both real and imaginary part are square roots of integers. We show here that the function f can be chosen to be equal to the conjugate of its Fourier transform.

In this paper, we determine the complex-valued solutions of the following functional equations \[g(x\sigma (y)) = g(x)g(y)+f(x)f(y),\quad x,y\in S,\]\[f(x\sigma (y)) = f(x)g(y)+f(y)g(x),\quad x,y\in S,\]\[f(x\sigma (y)) = f(x)g(y)+f(y)g(x)-g(x)g(y),\quad x,y\in S,\]\[f(x\sigma(y))=f(x)g(y)+f(y)g(x)+\alpha g(x\sigma(y)),\quad x,y\in S,\]\[f(x\sigma(y))=f(x)g(y)-f(y)g(x)+\alpha g(x\sigma(y)),\quad x,y\in S,\] where $S$ is a semigroup, $\alpha \in \mathbb{C}\backslash \lbrace 0\...

The Fourier Transform is one of the most important linear transformations used in science and engineering. Cooley and Tukey's Fast Fourier Transform (FFT) from 1964 is a method for computing this transformation in time $O(n\log n)$. From a lower bound perspective, relatively little is known. Ailon shows in 2013 an $\Omega(n\log n)$ bound for computing the normalized Fourier Transform assuming only unitary operations on pairs of coordinates is allowed. The goal of this documen...

The paper is devoted to vector fields on the spaces R^2 and R^3, their flow and invariants. Attention is plaid on the tensor representations of the group GL(2,R) and on fundamental vector fields. The rotation group on R^3 is generalized to rotation groups with arbitrary quadrics as orbits.

The leading idea of the paper is to treat the theorem of Wigner with methods inspired by geometry. The exercise mentionned in the title has two functions: On the one hand it can serve as a pedagogical text in order to make the reader acquainted with the essential features of the theorem and its proof. On the other hand it will turn out to be the core of the general proof.

It is well-known that the canonical commutation relation $[x,p]=i$ can be realized only on an infinite-dimensional Hilbert space. While any finite set of experimental data can also be explained in terms of a finite-dimensional Hilbert space by approximating the commutation relation, Occam's razor prefers the infinite-dimensional model in which $[x,p]=i$ holds on the nose. This reasoning one will necessarily have to make in any approach which tries to detect the infinite-dimen...

The goal of this expository article is to present a proof that is as direct and elementary as possible of the fundamental theorem of complex multiplication (Shimura, Taniyama, Langlands, Tate, Deligne et al.). The article is a revision of part of my manuscript, Complex Multiplication, April 7, 2006.

We formulate an exponential Diophantine equation, which is is some sense one order higher that Fermat's Last Theorem. We also give three examples of solutions to this exponential Diophantine equation and formulate a conjecture.