718 research outputs found
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
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