Archive for September, 2007

CPU cost comparison

I’m sorry for a long time no update, today I wanna give a figure and point out the surprise of CPU cost comparison between addition and multiplication under modern CPU chip. Even though it’s a fact, I still cannot trust it, just because the integer’s addition computation has became as fast as multiplication. In other words, with the development on hardware integration and CPU chip manufacture, addition computation hasn’t any advantage comparing with multiplication so far as now. According to our researchful requirement, I figured out this comparison in the following figure.

画像をダウンロードするには、ここを右クリックします。プライバシー保護を促進するため、この画像はインターネットから自動的にダウンロードされません。 Addition VS Multiplication
What do you think about this surprised result? Please leave your comments, thanks. Maybe your good advices will help our research toward well progressing.

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Crazy hyperball

I have been figuring out some experimental results for the target paper these days. To give a simple introduction, please permit me to tell something about my research field on data mining.

In this field, we usually research data clustering, information retrieval, keyword analysis, document searching and so on. Although they are extremely abroad, I wanna strongly recommend their a certain point that a excellent distance function will take the most important role in similarity computation. Unfortunately, human brain is not enough imaginative to apply the distance computation over a non-Euclidean space in our research field. If this problem was solved by someone, it must become a crazy message immediately. Of course people have figured out a general formula for distance computation in a certain non-Euclidean space, which is often called as Lp norm or Lp metric. It’s well known as the formula in the following.

|X|_p = {(\sum_{i}^{D}{\left|X_i\right|^p}})^{1/p}

Considering this Lp norm, I found a crazy phenomenon when was computing the volume of unit radius ball in high-dimensional Euclidean space. Here, it’s usually called hyperball or nball. What can you image the sphere of this unit nball ? And what can you guess the appearance of its circumscribed hypercube? To tell it exactly, I can hardly trust that it’s crazy because I cannot carry out my imagination. Please go though as follows.

hypercube

Fig.1 hypercube in high-dimensional space

Besides, I want to refer to the formula of volume computation for unit hyperball in the following.

V_{ball}(n) = \frac{\pi^{n/2}}{\Gamma(\frac{1}{2}n+1)} (n is the number of dimension)

And then, what’s the curve of this formula about the change of hyperball’s volume with growing dimension? To see here.

curve of hyperball

Well, what do you think about this unimaginable thing?

Referrence:

1. http://en.wikipedia.org/wiki/Lp_spac
2. http://mathworld.wolfram.com/Ball.html
3. http://mathworld.wolfram.com/Hypercube.html

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Bitter work with LaTeX

Have you heard about \LaTeX? If never, it doesn’t matter indeed.

Why? Mostly because it’s honesty that \LaTeX has been widely considered as an outstanding paper editor. Furthermore, these groups’ authors strongly gave the recommendation that \LaTeX can do what MS office cannot do, such as the most beautiful performance of mathematical formula, the absolutely equal line length and width, the easily adjustable size and position of figures, and so on.

In spite of these advantages referred above, I would also tell the ture that it’s a bitter work with \LaTeX. Ok, it’s fully due to its against the idea WYSIWYG (What You See Is What You Get), and it must be patient to remember many tag commands and special expressional format.

For example, you see here, a very beautiful formula:

\lim_{d\to\infty}{E\left[ \frac {|PA_d - PB_d|} {d^{1/k-1/2}} \right]} = C \cdot \frac{\sigma_{{\cal F}, k}}{k \cdot \sigma_{{\cal F}, k}^{(k-1)/k}}

However what can you image about its expression in \LaTeX ? Woo! It’s crazy as follows.

\begin{equation}
\lim_{d\to\infty}{E\left[ \frac {|PA_d - PB_d|} {d^{1/k-1/2}} \right]} =
C \cdot \frac{\sigma_{{\cal F}, k}}{k \cdot \sigma_{{\cal F}, k}^{(k-1)/k}}
\end{equation}

Oh no, it proofs that “No pain no gain”! In order to write a smooth paper giving a comfortable sense in future, no matter what a student or a researcher must we master \LaTeX as soon as possible. I just wanna strongly recommend this point.

Now, let me share some useful links and materials in the following please.

1.wikipedia LaTeX: http://ja.wikipedia.org/wiki/LaTeX
2.LaTeX in WordPress: http://www.sixthform.info/steve/wordpress/
3.Online LaTeX editor: http://test.izyba.com/equationeditor/equationeditor.php, http://www.sitmo.com/latex/
4.Full package of LaTeX editor for windows: WinShell 3.1 (Japanese supported)
5.Usage manual for LaTeX: A Guide to LaTeX.pdf (EN), JSME LaTex Manual.pdf (JP)
6.if you are using Linux operating system, extra editor about LaTeX is unnecessary, which has been contained by the system.

Thanks and enjoy these above.

Comments (1)

HP updated

Hey,

Home page has been updated so far as now. There maybe something changed, but please don’t mind give your good advice. I am looking forward to your attention and appreciate what you will do. Thank you!

Now, you can register your favourite account, reply your comments freely, and send mail to me. Please enjoy yourself!

I think I can and I will frequently update my homepage, especially main articles concerning recent researches will be posted here as usual blog. By the way, I would share my codes, packages, figures, and tools referred to research, study, or software development. Not only but also, afford some useful linux technical materials as soon as possible according to my experience.

OK, keep in touch please.

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