Corrigendum to “A Schanuel Condition for Weierstrass Equations”

I prove a version of Schanuel's conjecture for Weierstrass equations in differential fields, answering a question of Zilber, and show that the linear independence condition in the statement cannot be relaxed

A note on the axioms for Zilber's pseudo-exponential fields

We show that Zilber's conjecture that complex exponentiation is isomorphic to his pseudo-exponentiation follows from the a priori simpler conjecture that they are elementarily equivalent. An analysis of the first-order types in pseudo-exponentiation leads to a description of the elementary embeddings, and the result that pseudo-exponential fields are precisely the models of their common first-order theory which are atomic over exponential transcendence bases. We also show that the class of all pseudo-exponential fields is an example of a non-finitary abstract elementary class, answering a question of Kes\"al\"a and Baldwin.Comment: 10 pages, v2: substantial alteration

The rational field is not universally definable in pseudo-exponentiation

We show that the field of rational numbers is not definable by a universal formula in Zilber's pseudo-exponential field

On Quasiminimal Excellent Classes

A careful exposition of Zilber's quasiminimal excellent classes and their categoricity is given, leading to two new results: the L_w1,w(Q)-definability assumption may be dropped, and each class is determined by its model of dimension aleph_0.Comment: 16 pages. v3: correction to the statement of corollary 5.

Exponential algebraicity in exponential fields

I give an algebraic proof that the exponential algebraic closure operator in an exponential field is always a pregeometry, and show that its dimension function satisfies a weak Schanuel property. A corollary is that there are at most countably many essential counterexamples to Schanuel's conjecture.Comment: 12 pages; v2 minor change to proof of lemma 6.

Which Wavelet Best Reproduces the Fourier Power Spectrum?

The article compares the radially averaged Fourier power spectrum against the global wavelet power spectrum ('global scalogram') for seven continuous, two-dimensional wavelets: Derivative of Gaussian, Halo, Morlet, Paul, Perrier and Poisson wavelets, and a new wavelet based on a superposition of rotated Morlet wavelets, named the 'Fan' wavelet. This wavelet is complex, yet is able to give quasi-isotropic wavelet phase spectra. All seven wavelets were applied to synthetic and real data to test their ability to reproduce the Fourier spectrum: the Fan, Halo and Morlet wavelets reproduced this spectrum exactly; the Poisson wavelet performed very poorly. However, in tests of the space domain resolution of these wavelets with real and synthetic data, the Poisson wavelet gave by far the best results

Around Zilber's quasiminimality conjecture

This is an extended abstract for a survey talk given in Oberwolfach on 1st December 2022, slightly updated in June 2023. I survey some work around the notion of quasiminimality and some of the progress towards Zilber's conjecture from the last 25 years.Comment: 6 page