Python 2D array - appending row into a list of rows without Numpy
List of lists changes reflected across sublists unexpectedly
I just read on wikipedia that the the complex projective line is homeomorphic to the riemann sphere. How do I prove this? But, before that I have an extremely silly doubt that has been eating me. In the complex projective line, antipodal points are identified, but in the Riemann sphere they are clearly distinct. Then how can the two be homeomorphic.   I found out the following map from the complex projective line to $S^2$. But how do I prove this is a homeomorphism? Finding the inverse of this functions seems complicated, and then how do I prove that the inverse map of a open subset is a open subset, and vice versa?  $f:\mathbb{C}\mathbb{P}^1 \to S^2 = [z:w] \to \frac{(2Re(w\bar{z}), 2Im(w\bar{z}),|w|^2-|z|^2)}{(|w|^2+|z|^2)} $ Convergence of $\sum _{n=1}^{\infty }\left\{ 1-n\log \frac {2n+1} {2n-1}\right\} $ Solving "MySQL server has gone away", mysql error 2006 Transit Visa in Hong Kong for 15 hour layover between India and USA? We are (2) Indian nation and travelling from India to USA via Hong Kong airport and about 15 hours layover in Hong Kong airport. Do we need a transit visa even if only changing planes? And can we apply in Hong Kong Airport? How to convert from one grading scheme to another? Is it possible to show our custom message in QGIS Status bar using python? Why do the planets' orbital distances fall on an exponential curve? Background: I was recently reading a book on the planets to my son and I noticed a pattern in the distributions of the planets.  The planets' distances roughly follow an exponential distribution.  Below is a plot of scaled, log orbital distances $$ \tilde{d_n} = \frac{\log(d_n/d_1)}{\log(d_2/d_1)} $$ with the line $an+b$:   Where $d_1$ corresponds to Mercury and so on. Ceres is included, Pluto is excluded.  By linear regression, $a = 0.90, b = -1.06$.  For the statistically minded, the data has a Pearson's correlation of 0.996.  Note that this is a well known phenomenon, see  and references.  The code used to generate the plot may be provided on request.  Question: What is the mechanism that leads to this distribution?  Aside: Is there a good introductory text on solar system formation for the mathematically inclined?  Update: This is also known as Titius–Bode law. Is it safe to connect a battery in series to wall via adapter?  I have an AC adapter which gives me 10 vDC from the wall (220 volts). I've measured the adapter and it outputs 10 vDC. I have a 3 volt battery. I would like to connect in series to get 13 vDC.   Is this possible? Is this safe? If not, what can go wrong? What communication can be intercepted by network I'm connected over VPN to? I have to connect to client's network over VPN, and I was wondering if the client can intercept (and read) my Google Talk messages (or any other IM communication) while I'm VPN-ing.  My main concern is them having access to our internal conversations.  Suggest workarounds if you have any. After doing a "sudo su -" on an Ubuntu 12.04 notebook I did a "crontab -e", added this:   * * * * * env DISPLAY=:0.0 /usr/bin/gnome-calculator   and waited for minutes. Nothing Happened. I don't have any external monitors and if I run this command "env DISPLAY=:0.0 /usr/bin/gnome-calculator" in the terminal, it just works. But not from cron. Why?   The syslog only contains this:   May 24 14:37:01 localhost cron[1227]: (root) RELOAD (crontabs/root) May 24 14:37:01 localhost CRON[16432]: (root) CMD (env DISPLAY=:0.0 /usr/bin/gnome-calculator )   And I already tried an "xhost +localhost".   [root@NOTEBOOK /var/log] xhost access control enabled, only authorized clients can connect INET:localhost.localdomain SI:localuser:USERNAME [root@NOTEBOOK /var/log]    So the solution for another question like this on askubunut didn't worked. Best way to start investing, for a young person just starting their career? I am young, just started my career, and looking to begin investing so I can make my money work for me. I'm still living like I did in college (occasional purchase on myself, but most of it goes into savings) and I would like to take the extra money and invest it.   What would you suggest are the best options for me to begin? I was considering getting a personal investor [a financial advisor] for perhaps a year or so, but I've heard that they really are not worth the money. I wanted to get the input of the great minds here. Uniform convergence of alternating series What's the downtime limit of a domain controller (DC)? On my VMware  environment there are 10 VMs running. One of the VMs is the domain controller (). The other VMs are configured with a static DHCP and DNS IP address in their network settings.  How long is the downtime limit of my domain controller? I'm having issues unlocking my Nexus 7 2013 WiFi's bootloader, and it was suggested that I try fastboot oem unlock-go instead of fastboot oem unlock. Both seem to act the same on my device, however both also fail on my device so it is possible that the difference appears after the failure on my end comes. Regardless, I would like to know: What is the difference between these two commands? I'm not managing to find documentation. Basic circuit to keep LED either on or off depending on night/day How can I have a LED either on or off (not in between) depending on night or day with basic electronics? I have created the circuit below, but it only shines and in darkness it dims.    The photo-resistor is about 3kΩ when light shines and about 1MΩ when dark.  I want my LED to shine at night and be off at day.  What is the simplest way I can achieve this? What is the relativistic calculation of travel time to Proxima Centauri? It has already been  how fast a probe would have to travel to reach Alpha Centauri within 60 years.  NASA has done  into a probe that would take 100 years to make the trip.  But I am interested in this question from a slightly different perspective.  I would like to know how long the trip from our solar system to Proxima Centauri would take for both a ship and for an observer on Earth while accounting for the effects of general relativity on a .  The following assumptions may be made:   The ship starts and ends at 0 km/s relative to the Earth observer. The ship experiences a constant force of acceleration of 20 g until the half-way point and then a constant force of deceleration of 20 g until the destination. The path traveled is such that accleration is in only one spatial direction. The distance traveled is 4.25 light years.   Calculate (accounting for general relativistic effects):   Time of trip for Earth-based observer. Time of trip for ship-based observer. Maximum speed.   You don't need to consider mass, or life-support systems, or method of propulsion; I am really only interested in the effects of relativity on the perceived time of the trip. Vieta Jumping: Related to IMO problem 6, 1988: If $ab + 1$ divides $a^2 + b^2$ then $ab + 1$ cannot be a perfect square. The famous IMO problem 6 states that if $a,b$ are positive integers, such that $ab + 1$ divides $a^2 + b^2$, then $\frac{ a^2 + b^2}{ab + 1 }$ is a perfect square, namely, $gcd(a,b)^2$.  How about a modification of this problem:    If $a,b$ are (strictly) positive integers, such that $ab + 1$ divides $a^2 + b^2$, then $ab + 1$ cannot be a perfect square.    I am looking for a proof of the claim above, or a counter-example.  One possible approach towards a proof is the following:   Suppose there is such a pair $(a,b)$ as above, then by the famous IMO problem 6 from 1988, $\frac{ a^2 + b^2}{ab + 1 } = g^2$ where $g = gcd(a,b)$. Since $ab + 1$ is a perfect square, $a^2 + b^2 = c^2$ for some integer $c$. So that $(a,b,c)$ is a Pythagorean tripple, therefore there exists positive integers $n,m,l$ with $n$ coprime to $m$, such that $a = l(n^2 - m^2)$ and $b = 2lnm$ - by Euclids formula.  Then by plugging in $a$ and $b$ in terms of $n,m,l$ into the original equation, and solving for $l$, it is possible to obtain the following:  There exists positive coprime integers $n,m$ such that   $\frac{(n^2 + m^2 + 1)(n^2 + m^2 - 1)}{2mn(n+m)(n-m)}$ is a perfect square.  If we put the quotient above into a program, as in this python code snipet:  N = 1000  for n in range(1,N):     for m in range(n+1, N):         A = (n*n + m*m + 1)*(n*n + m*m - 1)         B = 2*m*n*(n+m)*(m-n)         if A % B == 0:             print(A/B)   The quotient always is 2, regardless of whether or not $n$ and $m$ are co prime! So if the very strong implication   $2mn(n+m)(n-m) \mid (n^2 + m^2 + 1)(n^2 + m^2 - 1) \implies \frac{(n^2 + m^2 + 1)(n^2 + m^2 - 1)}{2mn(n+m)(n-m)} = 2$    holds, then the original problem will be solved. Sometimes I need to look up certain words through all the manual pages. I am aware of apropos, but if I understand its manual right, it restricts search to the descriptions only.     Each manual page has a short description available within it. apropos searches the descriptions for instances of keyword.   For example, if I look up a word like 'viminfo', I get no results at all...  $ apropos viminfo viminfo: nothing appropriate.   ... although this word exists in a later section of the manual of Vim (which is installed on my system).      -i {viminfo}                When  using  the  viminfo file is enabled, this option sets the filename to use, instead of the default "~/.vim‐                info".  This can also be used to skip the use of the .viminfo file, by giving the name "NONE".    So how can I look up a word through every section of every manual? Newton's cradle Can you reject a postdoc offer after the PI has paid a large sum for flights/accommodation for your visit? I am being considered for a very nice industry scientist position but this involves many interviews and assessments and can take some time.  However I have also accepted an invitation to the US (I'm in Australia) to give a presentation in relation to high likelihood of employment as a postdoc in a research institute.  The postdoc supervisor has arranged for sightseeing and lab visits and hotels. I feel bad if somehow I wish to pursue the industry position.  Has anyone had a similar experience and have you turned down something like that before?