User talk:LutzL

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Welcome!

Hello, LutzL, and welcome to Wikipedia! Thank you for your contributions. I hope you like the place and decide to stay. Here are a few good links for newcomers:

I hope you enjoy editing here and being a Wikipedian! Please sign your name on talk pages using four tildes (~~~~); this will automatically produce your name and the date. If you need help, check out Wikipedia:Where to ask a question, ask me on my talk page, or place {{helpme}} on your talk page and someone will show up shortly to answer your questions. Again, welcome! 

A good mathematical resource is also Wikipedia:WikiProject Mathematics and its talk page. Enjoy! Oleg Alexandrov 17:53, 27 May 2005 (UTC)

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Oleg Alexandrov 17:00, 30 May 2005 (UTC)

[edit] More welcome

You may also be interested in the discussions at Wikipedia:WikiProject Mathematics -- linas 03:00, 8 December 2005 (UTC)

[edit] Gröbner basis and Hironaka

Hi. Concerning your recent edit of Gröbner basis, I am not sure that Hironaka's theory of "standard bases" is exactly the same. See for example Joachim Apel, Division of entire functions by polynomial ideals, in Proc. AAECC 11, LNCS 948 (1995), pp. 82-95. So if you agree but think the addition is nevertheless important, I propose that you change the text into something like: "In 1964, at almost the same time and independently, Heisuke Hironaka had developed a closely related theory, which he called standard bases." I'm not sure how close this is to your areas of expertise, but I you have interest, time and patience, perhaps you could also add some of this to the Hironaka article, putting it in context, and if possible cite the references as provided by Apel's paper. Cheerio. LambiamTalk 13:46, 3 May 2006 (UTC)

As I understand it now, the standard bases are defined for ideals in the ring of Puisseux series. Thus they contain Gröbner bases as a special case. The real contribution of Buchberger to the fundamentals is the proof that his algorithm stops in finite time. Perhaps it should be mentioned somewhere that the idea comes from the non-constructive proof by Hilbert of the Basissatz (Kronecker had a constructive proof, but only formulated for bivariate polynomials) -- No, I'm not an expert in the whole of the topic of Gröbner bases. They are interesting to me in the aspect of their inefficiency. I got most of my knowledge from Gethgen/vonzur Gathen: Modern Algebra, where they mention H. Hironaka in the second sentence of the introduction. And from M. Guisty: Bases standard, élimination et complexité, notes of a lecture given at X, where some propositions are attributed to Hironaka.--LutzL 07:29, 5 May 2006 (UTC)

I'm not an expert either and I don't have access to a library, so I wouldn't be comfortable making any changes of substance. I'll copy this exchange to the talk page of the article in the hope that a next reader can do something with it. --LambiamTalk 14:43, 5 May 2006 (UTC)

[edit] Images of Daubechy spectrum

Would you be so kind to provide me with source code or short description of what you've done used to obtain these: Image:Daubechies4-spectrum.png

I need to get fourier transforms of some wavelet functions and I'm kind of stuck with it. I'm totally new to numerical computation of FT and I don't know what I do wrong. I'm not interested in generation of wavelet functions, just the part which does FT.

If I have sampled mother wavelet into vector v of length N, what is right way to compute power spectrum? I'm doing FFT on v and then take positive frequency terms of the resulting vector. Then I i take abs^2 of these positive frequency terms, but the resulting image is quite different:(

I've found that FFT estimates spectrum with errors and the result drastically varies when second FFT parameter (N) changes. I can imagine that this FFT estimation does not converge to spectrum when sampling rate goes to infty.

But your images are just perfect. Have you used some special methods, windows(Hamming e t.c.) ?

Just as you assumed: Compute a sufficiently long vector of values and apply FFT. One should first stabilize the values at integer points via the cascade algorithm (or directly via the pointwise refinement equations) before computing values for smaller step sizes. The values drawn are the absolute values of the complex numbers, so the curves intersect at height 0.71..=sqrt(2). The curves were drawn with gnuplot using thick lines and converted with ImageMagick using antialiasing.--LutzL 09:54, 2 October 2006 (UTC)

[edit] Subsampling

LutzL, long time, no see. At Talk:Sampling (signal_processing)#Sampling_rate_for_bandpass_signals we're talking about changing some math at Sampling (signal_processing)#IF/RF (bandpass) sampling that you worked on back in 2005, originally at Nyquist–Shannon_sampling_theorem#Sampling of non-baseband signals. I'd like to make it simpler, and closer to the sources. You indicated that you thought it was "hopefully simplified" this way, so I'd like to hear why. Comments? Dicklyon (talk) 01:45, 7 January 2008 (UTC)

[edit] Fundamental theorem of algebra

Yesterday, I deleted this sentence from the article about the fundamental theorem of algebra:

But every complex polynomial of degree n is the characteristic polynomial of some complex n × n matrix, for instance, its companion matrix.

You decided to undo my revision, saying “what is wrong about this statement? It is central to this proof.” Concerning this, I have two observations:

  1. I claim that the sentence that I deleted is false. Indeed, I wrote just that when I edited the article (“Elimination of a false sentence”). But if you claim that it is true, then please provide a square matrix whose characteristic polynomial is 2z.
  2. My deletion did not undermine the proof, because it still contained the sentence “So to establish that every complex polynomial of degree n > 0 has a zero, it suffices to show that every complex square matrix of size n > 0 has a (complex) eigenvalue” and the footnote that comes after it, which links to a proof of the fact that, given a field F, if, for every natural number n, every endomorphism of Fn has some eigenvalue, then F is algebraically closed. JCSantos (talk) 09:07, 13 June 2008 (UTC)
Ok, so it should say "normed polynomial". Since this theorem is concerned with fields, the zeros do not depend if a polynomial or some multiple of it is concerned.--LutzL (talk) 10:31, 13 June 2008 (UTC)
On the other hand, the introduction to the proof section already restricts the polynomials to have "dominant coefficient 1". Added the division by the leading coefficient.--LutzL (talk) 10:44, 13 June 2008 (UTC)
The result of all this is that, at present, the text contains a complete proof of the fact that, in order to prove that every monic polynomial has a root it is enough to prove that every endomorphism of Cn has some eigenvalue and a footnote telling the reader where to find such a proof. In my opinion, one of them should be eliminated. The reason why I chose the first one is because (1) the proof exists already on Wikipedia and (2) the proof doesn't seem to me to be important for someone who is trying to learn proofs of the fundamental theorem of algebra. What is your opinion? JCSantos (talk) 22:13, 14 June 2008 (UTC)