## A Long-Lost NASA Spacecraft Comes Back Alive

A NASA spacecraft IMAGE (Imager for Magnetopause-to-Aurora Global Exploration) was launched in March 2000 to study Earth’s magnetosphere where charged particles from solar winds trapped by Earth’s magnetic field. In December 2005, it suddenly stopped talking and since then it was thought to be lost in space. Recently a signal from IMAGE was detected by a Canadian amateur astronomer after 13 years of long silence. I cannot imagine what goes on in the minds of former IMAGE project members hearing this news. More details can be read here.

## 107 cancer papers retracted due to peer review fraud

There was a report at ars Technica that the journal Tumor Biology is retracting staggering 107 research papers due to fraudulent peer review process. It appears that such practice has been an on-going business. According to the report, last year 58 papers were retracted from 7 different journals (the report does not specify what they are but I suspect mostly biology journals) and 25 of them came from Tumor Biology. What happened was that when authors submitted their manuscripts for review the editorial office asked the authors to recommend reviewers (the editors perhaps assumed that the authors would suggest best possible reviewers with no conflict of interest who would give a fair and professional review). Apparently the process was abused by some authors and they recommended people they know personally who would give their papers a favorable review regardless of their academic value. I would say those journals are also responsible for the mayhem as they basically let it happen. If people think that scientists would always conduct their research and academic activities with honesty, pride and integrity, they would be flatly naive and wrong. Research publication is directly tied to tenure, promotion and also grants (especially in biological sciences grants come with a big money). Never think that academia would never spoil. Scientists are also men with many flaws. Unfortunately for some often these things weigh more than their pride and academic integrity.

Theoretical physics journals have the same kind of review process but there is little room for such fraudulent practice as it is relatively easier for any third party experts to verify the results and academic merits of a paper in theoretical physics. While theoretical physics research often comes with hypes of all levels, it is relatively much more honest area compared with certain experimental sciences. Without any bias, mathematics is certainly the most honest area. If you are not honest about what you do, you really cannot be a good mathematician. Also there is no room for cooking up your results as everything is readily verifiable by experts. In mathematics, most common malpractices are abusing citation and plagiarism. But such malpractices have never been a big issue in mathematics community as no serious mathematicians would even think of committing them. Whoever commit such things could/would never be regarded as a mathematician.

## Are We Alone in the Universe After All?

In the movie Contact (which is based on the novel with the same title by Carl Sagan), Ellie Arroway (played by Jodie Foster) was speaking to a group of children: “I’ll tell you one thing about the universe, though. The universe is a pretty big place. It’s bigger than anything anyone has ever dreamed of before. So if it’s just us… seems like an awful waste of space. Right?” For us scientists, perhaps I should say most of us, it is almost like faith that there should be an intelligent life other than us somewhere out there in the universe. In 1961, an astronomer Frank Drake proposed the Drake equation (which is not really a mathematical equation but a probabilistic argument) $$N=R_\ast\cdot f_p\cdot n_e\cdot f_\ell\cdot f_i\cdot f_c\cdot L,$$ where

1.  $N$ is the number of active, communicative extraterrestrial civilizations.
2. $R_\ast$ is the average rate of star formation in our galaxy.
3. $f_p$ is the fraction of formed stars that have planets.
4. $n_e$ is the average number of planets per star that has planets.
5. $f_\ell$ is the fraction of those planets that actually develop life.
6. $f_i$ is the fraction of planets bearing life on which intelligent, civilized life has developed.
7. $f_c$ is the fraction of these civilizations that have developed communications, i.e., technologies that release detectable signs into space.
8. $L$ is the length of time over which such civilizations release detectable signals.

The original estimate for $N$ given by Frank Drake in 1961 is somewhere between 1000 and 100,000,000 civilizations in the Milky Way galaxy alone. (See here for more details on Drake’s educated guesses on the above quantities 1-8.) I remember back in the 80’s the Drake equation was a great talking point for astronomers to argue why we should support SETI (Search for Extraterrestrial Intelligence).

These new findings tell us that after all the Earth is a very special place in the universe contrary to Copernican principle. To me personally it is still inconceivable that we human beings on the Earth are the only intelligent life in the universe. If it were true, I would probably feel really lonely and sad though there are some bright sides. We would never have to worry about alien invasions and for us the whole universe is up for grabs. Perhaps then it is our sole and sacred duty to go out, explore, colonize, cultivate, and populate the universe for the sake of our own survival and of the preservation of human civilization.

Update: A related article here which is about a paper on the arXiv titled “Dissolving the Fermi Paradox.” In the paper the authors attempt to resolve the Fermi Paradox by examining highly uncertain parameters of the Drake equation.

Update: Ethan Siegel has his taken on the arXiv paper “Dissolving the Fermi Paradox” here.

## A Notable Quote on Richard Feynman

From Freeman Dyson, “Disturbing the Universe”, Harper & Row, New York, 1979:

“Dick was also a profoundly original scientist. He refused to take anybody’s word for anything. This meant that he was forced to rediscover or reinvent for himself almost the whole of physics. It took him five years of concentrated work to reinvent quantum mechanics. He said that he couldn’t understand the official version of quantum mechanics that was taught in textbooks, and so he had to begin afresh from the beginning. This was a heroic enterprise. He worked harder during those years than anybody else I ever knew. At the end he had a version of quantum mechanics that he could understand.”

This quote is, of course, about Richard Feynman‘s path integral formulation of quantum mechanics.

## $\sum_{n=0}^\infty e^{nix}$ is Cesàro Summable

Back when I was a Ph.D. student, a friend of mine (he was a Ph.D. student in physics studying laser optics) asked me if the series $\sum_{n=0}^\infty e^{nix}$ converges. I vaguely remember that his advisor needed to have a finite value for the infinite sum for whatever reason I don’t remember. At that time, I didn’t know other summability methods and I only knew the conventional definition of infinite sums. The series is a geometric series with $r=e^{ix}$ and since $|e^{ix}|=1$, I bluntly told him that you can’t have a finite value for the series. It diverges!

I don’t know what motivated me but I just thought about the series $\sum_{n=0}^\infty e^{nix}$ and I was wondering if it is Cesàro summable. I was able to show that indeed it is. To my pleasant surprise, the Cesàro sum of $\sum_{n=0}^\infty e^{nix}$ is $\frac{1}{1-e^{ix}}$. I am sure that this is well-known and I am just being ignorant about it. I would appreciate if someone can tell me any reference where the Cesàro sum of $\sum_{n=0}^\infty e^{nix}$ appears.

The $(n+1)$-th partial sum $S_{n+1}$ of $\sum_{n=0}^\infty e^{nix}$ is
\begin{align*}
S_{n+1}&=\sum_{k=0}^n e^{nix}\\
&=1+e^{ix}+e^{2ix}+\cdots+e^{nix}\\
&=\frac{1-e^{(n+1)ix}}{1-e^{ix}}
\end{align*}
provided $x\ne 2m\pi$, $m\in\mathbb{Z}$. Now the sum of the first $n$ partial sums is calculated to be
$$\sum_{k=0}^{n-1}S_k=\frac{1}{1-e^{ix}}\left[(n+1)-\frac{1-e^{(n+1)ix}}{1-e^{ix}}\right].$$
The Cesàro sum of $\sum_{n=0}^\infty e^{nix}$ is
\begin{align*}
\sum_{n=0}^\infty e^{nix}&=\lim_{n\to\infty}\frac{\sum_{k=0}^{n-1}S_k}{n}\\
&=\frac{1}{1-e^{ix}}
\end{align*}
as $\lim_{n\to\infty}\frac{1}{n}\frac{1-e^{(n+1)ix}}{1-e^{ix}}=0$. For $x=2m\pi$ with $m\in\mathbb{Z}$, $\sum_{n=0}^\infty e^{nix}=1+1+1+\cdots$ is not Cesàro summable as its Cesàro sum is \begin{align*}
1+1+1+\cdots&=\lim_{n\to\infty}\frac{\sum_{k=1}^n S_k}{n}\\
&=\lim_{n\to\infty}\frac{\frac{n(n+1)}{2}}{n}\\
&=\infty.
\end{align*}
However, it can be shown that $1+1+1+\cdots=-\frac{1}{2}$ using zeta function regularization. (I also discussed it here.)

The real part of Cesaro sum with n=300

This figure shows the real part of the Cesàro sum $\sum_{n=0}^\infty e^{nix}$ with $x=\frac{\pi}{6}$ converging to the real part of $\frac{1}{1-e^{ix}}=\frac{1}{2}$.

The imaginary part of Cesaro sum with n=300

This figure shows the imaginary part of the Cesàro sum $\sum_{n=0}^\infty e^{nix}$ with $x=\frac{\pi}{6}$ converging to the imaginary part of $\frac{1}{1-e^{ix}}=\frac{\sin x}{2(1-\cos x)}$.

Cesaro sum with n=100

Finally this figure shows the Cesàro sum $\sum_{n=0}^\infty e^{nix}$ with $x=\frac{\pi}{6}$ converging to $\frac{1}{1-e^{ix}}$.

Posted in Summability Methods | 5 Comments