User j.l. nelson - MathOverflow

Answer by J.L. Nelson for Hilbert Nullstellsatz and Non-Complete Fields

2011-12-18T06:55:07Z

First, I would like to offer a more rigorous statement of Hilbert's Nullstellensatz from Dummit and Foote's Abstract Algebra:

Let $E$ be an algebraically closed field. Then $\mathcal{I}(\mathcal{Z}(I)) = \mathop{\mathrm{rad}} I$ for every ideal $I$ of $E[x_1, \ldots, x_n].$ Moreover the maps $\mathcal{Z}$ and $\mathcal{I}$ in the correspondence $$\{ \mbox{affine algebraic sets} \} \xleftarrow[\mathcal{Z}]{\xrightarrow{\mathcal{I}}} \{\mbox{radical ideals} \}$$ are bijections of each other.

Now, it should be absolutely obvious why the bijection breaks down when the field considered is $\mathbb{R}$. $\mathbb{R}$ is not algebraically closed (consider $x^2+1 \in \mathbb{R}[x]$, which has a variety that includes $i\not\in \mathbb{R}$) so the Nullstellensatz does not apply and the bijection does not happen.

As for whether or not this is a failure to be injective or surjective, that depends on whether you are talking about $\mathcal{Z}$ or $\mathcal{I}.$

Maiden Names vs. Married Names

2011-08-30T19:36:00Z

Is there a set convention for which name (maiden name or married name) a female married mathematician should use?

While this question addresses women's maiden name it applies equally to men's maiden name when it differs from their married name. The question seeks for an advice for the dilemma: whether to use the maiden name or the new married name.

For example, Fan Chung is married to Ron Graham, but she publishes under "Fan Chung." Vera T. Sós is another married woman who continued to use her maiden name, but the T. stands for Turán. Yet, I'm pretty sure that Emma Lehmer (née Trotskaia) published under her married name.

Does it have something to do with the name under which the woman first publishes or the name under which name she receives her Ph.D.?

Answer by J.L. Nelson for Relationship between quasicrystals and PV numbers

2011-09-01T19:46:39Z

In her book Quasicrystals and geometry Majorie Senechal discusses the relationship between quasicrystals and PV number on pp. 126-128. She cites Pisot (1946) and Cassels (1965).

First she presents an alternative characterization of PV numbers:

Theorem 4.1 Let $\mu_1>1$ be a real algebraic integer. Then $\mu_1$ is a PV number if and only if there exist nonzero $q \in \mathbb{R}$ such that $$\lim_{m\rightarrow \infty} \mu_1^m q = 0 \mod \mathbb{Z}.$$

Thus it follows that

Theorem 4.2 The diffraction condition is satisfied for an $\mathcal{A}$ sequence if and only if the leading eigenvalue $\lambda_1$ of $\mathcal{A}$ is a PV number.

I note that she defined an $\mathcal{A}$ sequence to be "any sequence of points $\Lambda = \{x_n\}$ such that, for all $n,$ $x_n - x_{n-1} \in \{\alpha_1, \ldots, \alpha_n\}$ and $\Lambda$ has suitably defined predecessors of all orders with respect to the [linear map which may be represented by a primitive matrix] $\mathcal{A}.$"

It's a nice book, easy to read. Highly recommended.

Oh, the references are:

C. Pisot (1946), Repartition (mod 1) des puissances successives des nombres reels, Commentarii Mathematici Helvetici, Vol. 19, 153-60.

J.W.S. Cassells (1965), Diophantine Approximation, Cambridge Tracts in Mathematics and Mathematical Physics No. 45, Combridge University Press.

Comment by J.L. Nelson

2011-12-18T07:16:35Z

(1) I am pretty sure that the OP is hoping to improve something like the result in en.wikipedia.org/wiki/… (2) OP may wish to look up the very beautiful and extensive study of "quadrature formulas." This area deals with selecting appropriate points in order to best approximate given families of functions. It would be interesting to see if Chebyshev points give a better rate of approximation than uniformly distributed points. However, the result given on Wikipedia is so classical, I would strongly suggest a literature search first.

Comment by J.L. Nelson

2011-09-29T00:38:27Z

When I took Model Theory the required textbook was [Wilfred Hodge's *A Shorter Model Theory.*][1] I found it to be a very clear and concise introduction to the key theorems in the subject for someone who has had some basic Abstract Algebra at the graduate level. [1]: amazon.com/Shorter-Model-Theory-Wilfrid-Hodges/dp/…