Let $A$ be an $n \times n$ matrix with integer coefficients and nonzero determinant. Can we say something about $ \mathbb{Z}^n /\ \text{im}( \phi )$ (here $\phi : v \mapsto Av$ )?    This problem has arised as I was solving some problem in homology theory.
The following problem came up from a though I had while reading:  Let's say we have $M=\mathbb{Z}^n$ and we have another free $\mathbb{Z}$-module, $N$, inside of $M$ also with rank $n$.  We know we can make a matrix $A$ that changes $N$ to its representation in $M$ (ie has as columns the basis $N$ expressed in coordinates coming from the basis of $M$). If the index of $N$ in $M$ is $m$, I am pretty sure just from tinkering that the determinant of $A$ should also be $m$, however, I have not been able to show this.  It is unclear to me how to proceed. I though maybe I could do something related to the points in $\mathbb{Z}^n$ that are within the unit box of $N$, but I cannot really make sense of this.  Thank you for any help or direction.
What is the difference between object keys with quotes and without quotes? Closure of a Ball is Contained in Closed Ball (usual metric) Let $\overline{B_d (x, r)}$ be the closure of an open ball, and let $\hat B_d (x, r)$ be a closed ball. Show that in any metric space $(X, d)$, $\overline{B_d (x, r)} \subseteq \hat B_d (x, r)$ for all $x \in X$ and $r &gt; 0$.    I used the definition of closure: $\overline{B}$ = $B \bigcup B'$. If $x \in \overline{B}$, then $x \in B$ or $x \in \overline{B}$, or both. If $x \in B$, then clearly $x \in \hat B$ since $B \subseteq \hat B$. But, if $x \in B'$, then $x$ is an accumulation point. Since $\hat B$ is closed, it contains all accumulation points of $B$. Thus, $x \in \hat B$.  Is this approach correct, or at least in the right direction? Should I be using the closure definition, or try another one? Thank you. I would like to use Flash Player to show web contents but since today (12 Jan 2021) Adobe is blocking it. Is there a way to use it? I've tried Firefox, Chrome, Safari and Brave browsers. bash if -a vs -e option Both about -a and -e options in  is said:  -a file     True if file exists. -e file     True if file exists.    Trying to get what the difference is I ran the following script:  resin_dir=/Test/Resin_wheleph/Results  if [ -e ${resin_dir} ] ; then     echo "-e "; fi  if [ ! -e ${resin_dir} ] ; then     echo "! -e"; fi  if [ -a ${resin_dir} ] ; then     echo "-a"; fi  if [ ! -a ${resin_dir} ] ; then     echo "! -a"; fi   /Test/Resin_wheleph/Results exists and is a directory. And this is what I get:  -e -a ! -a   which seems to be a little strange (notice -a and  ! -a). But when I use double brackets (e. g. if [[ -e ${resin_dir} ]]) in the similar script it gives reasonable output:  -e -a   So:    What is a difference between -a and -e options? Why -a produces a strange result when used inside single brackets? How can I make my ad hoc iPhone application's icon show up in iTunes? I've got an iPhone app with icon file Icon.png.  This icon shows up properly when the app is on the phone itself, but it doesn't show up in the applications pane in iTunes.  What do I need to do to get it to show up properly? It is a general concept that horizontal forces can only influence physical quantities along horizontal direction; similarly vertical force can produce acceleration etc. only along vertical direction. This is the same concept that helps us study projectile motion as a combination of two independent motions. If this relation is universally true, how come, then friction depend on a force which normal to it? What does "Incompatible types: void cannot be converted to ..." mean? Can one construct a non-measurable set without Axiom of choice? Is axiom of choice required to show the existence of non-measurable sets? Is there a Lebesgue non-measurable set that can be constructed without axiom of choice?  Related question on MO says it is consistent: Why are there no Pokémon near me? I see pictures online of everyone having all these Pokémon to catch being everywhere around them, but it seems like I can only ever see one or two. I live in a rural area, but even walking miles through the farmland, it's rare to find any Pokémon.  Why is this? How can I find more Pokémon? As of today, a  has been found affecting versions 1.0.1 through 1.0.1f (inclusive) and 1.0.2-beta.  Since Ubuntu 12.04, we are all vulnerable to this bug. In order to patch this vulnerability, affected users should update to OpenSSL 1.0.1g.  How can every affected user apply this update now? I'm trying to install Windows 7 with a 4GB flash drive, but my error that comes up is that my hard drive I'm trying to install on is in ext4. I need to format it to read NTFS. I can't seem to find any topics on how to format an active hard drive. I found a topic that explains how to move Ubuntu to a new drive, but it's a bit confusing to me. Do the laws of physics evolve? Hubble's constant $a(t)$ appears to be changing over time. The fine stucture constant $\alpha$, like many others in QFT, is a running constant that varies, proportional to energy being used to measure it. Therefore, it could be agued that all running constants have 'evolved' over time as the Universe has expanded and cooled. Both the local and global curvature of the Universe changes over time implying that so too does the numerical value of $\pi$. All these things are however constants (well, let's say parameters since they are not really 'constant'.)   In  with astronomer Sir Fred Hoyle, Feynman said "what today, do we not consider to be part of physics, that may ultimately become part of physics?" He then goes on to say "..it's interesting that in many other sciences there's a historical question, like in geology - the question how did the Earth evolve into the present condition? In biology - how did the various species evolve to get to be the way they are? But the one field that hasn't admitted any evolutionary question - is physics."  So have the laws of physics remained form-invariant over the liftetime of the Universe? Does the recent understanding of the aforementioned not-so-constant constants somehow filter into the actual form of the equations being used? Has advances in astronomical observations, enabling us to peer back in time as far back as the CMB, given us any evidence to suggest that the laws of nature have evolved? If Feynman thinks that "It might turn out that they're not the same all the time and that there is a historical, evolutionary question." then this is surely a question worth asking.  NB/ To be clear: this is a question concerning purely physics, whether the equations therein change as the Universe ages, and whether there is any observational evidence for this. It is not intended as an oportunity for a philosophical discussion. SQL comma-separated row with Group By clause Android and setting width and height programmatically in dp units I have a number of measurements of the same quantity (in this case, the speed of sound in a material). Each of these measurements has their own uncertainty.  $$ v_{1} \pm \Delta v_{1} $$ $$ v_{2} \pm \Delta v_{2} $$ $$ v_{3} \pm \Delta v_{3} $$ $$ \vdots $$ $$ v_{N} \pm \Delta v_{N} $$  Since they're measurements of the same quantity, all the values of $v$ are roughly equal. I can, of course, calculate the mean:  $$ v = \frac{\sum_{i=1}^N v_{i}}{N}$$  What would the uncertainty in $v$ be? In the limit that all the $\Delta v_i$ are small, then $\Delta v$ should be the standard deviation of the $v_i$. If the $\Delta v_i$ are large, then $\Delta v$ should be something like $\sqrt{\frac{\sum_i \Delta v_i^2}{N}}$, right?  So what is the formula for combining these uncertainties? I don't think it's the one given in  (though I may be wrong) because it doesn't look like it behaves like I'd expect in the above limits (specifically, if the $\Delta v_i$ are zero then that formula gives $\Delta v = 0$, not the standard deviation of the $v_i$). In an older question from June 2011, , apparently revived by the system, a user is asking about the origin of the rotation of celestial bodies.  The  (with 17 votes) states the following:     When things collapse under their own gravity in space (i.e. clouds of gas and dust), any small amount of asymmetry in the collapse will be enough start it spinning.   And then it goes on saying:     Even if it spins by a tiny amount, as it collapses, angular momentum     conservation will mean it spins more and more quickly.   Is all this really correct?  I could try to ask in a comment to the answer. I actually did. But the author is no longer registered.  Hence I am asking it as a question.  But that is not the end of the story. As I was about to ask my question, I looked for other relevant questions, and there are some. I did not make an exaustive survey, but I did not find one that had satisfactory answers:    (december 2011) has just one accepted answer (0 votes) stating essentially that the solar system has angular momentum because it started from a cloud that had angular momentum. No comment.  (july 2011) is questionning about everything from galaxies to electrons. The answers are of a much higher quality, but while they seem to justify the possiblility of spin and angular momentum (and its conservation), there are not clear on how it appears, at least for celestial bodies.  (june 2013): That was my own very naive first question on this site. It is related but does not directly address spin.   One answer I found is that momentum appear from bodies in relative motion past each other. Nice, but not really an answer: where did they get their relative speed to begin with. Unfortunately the answer is: from the angular momentum of larger structures they belong to (well, it can be more comples). So this is just chicken and egg explanation. But how did it get started?  Note that my question is not a duplicate, since it is primarily whether the above statements are correct, which is not considered in these other questions.  Furthermore, these questions are fairly old, and given that none had received answers that seem to answer the question of the origine of existence of celestial angular momentum (not in a way I can understand anyway), it may be appropriate to attempt restarting the process, with the above discussion to motivate that an answer should explain the physical mechanism that started the existence of spin of celestial bodies. An awful identity We take place on $\mathbb C(x_1,...,x_r,x'_1,...,x'_p,u_0,...,u_r,u'_0,...,u'_p)$ with $r,p\in \mathbb N$     Show that :   $$\displaystyle{\sum_{i=1}^r \left(   \frac{\prod_{j=0}^r (u_j-x_i) \prod_{j=1}^p (x'_j-x_i) }{  \prod_{1\leq j\leq r, j\neq i     }(x_j-x_i) \prod_{j=0}^p (u'_j-x_i)}      \right)}-\displaystyle{\sum_{i=0}^p \left(   \frac{\prod_{j=0}^r (u'_i-u_j) \prod_{j=1}^p (u'_i-x'_j) }{  \prod_{0\leq j\leq p, j\neq i     }(u'_i-u'_j) \prod_{j=1}^r (u'_i-x_j)}      \right)}=\displaystyle{\sum_{i=1}^p x'_i +\sum_{i=0}^r u_i -\sum_{i=1}^r x_i-\sum_{i=0}^p u'_i}$$   I have seen this exercise in an old book at the library. Unfortunately there was no indication how can I solve the problem.   Anyway if someone can give me some ideas it will be greatful.   NB: Sorry for the title, I have not had other ideas Is it better to have an ErrorDocument 404 redirect back to the homepage or a standard 404 error page? Why does DMM show AC and DC current when source is a battery? (How to calculate power in this case?) The circuit is an altered Joule Thief of my design that is driven by an AA or AAA battery. I’m observing the current at the battery anode with my DMM in series. (Extech EX330) This circuit is extremely efficient (In relation to other Joule Thief I've build.) and does some weird stuff - like, it needs a pushbutton sometimes.    Measurements: It is drawing 9.21 mA DC and 6.48 mA AC in series with the battery. It shows 1.364 VDC and 0.010 VAC in parallel with the battery. Output on the LED is 2.572 VDC and 0.011 VAC with no measurable frequency. (with my DMM at least) All measurements are taken with the same configuration and voltage.    But how is it drawing AC and DC and how do I calculate power consumption? --> Could it be because the emitter pin of the NPN is directly connected to ground? --> Or is the circuit just "switching on and off"?     Download schematic:  (I tried to simulate the circuit in LTspice. Thanks for making me aware of this amazing program, Sredni Vashtar!)     What the pushbutton is used for: The pushbutton is used to "kickstart" the oscillation when it wont start by itself. For example: I have to manually "oscillate" using this button to start the circuit if the capacitor is too big and battery voltage is low. Pushing the button releases the energy in the transformer into the capacitor. If it reaches LED forward voltage the circuit keeps oscillating on it's own. It does work without manual intervention when then capacitor in parallel with the LED is not too big, like &lt;5000 µF, and it's a fresh battery.    . Thank you for your answers! I'm learning so much! :D Do electrons move around a circuit? We could imagine a simple electronic circuit composed by a power source and a resistor.   It is usual find descriptions as "The moving charged particles in an electric current are called charge carriers. In metals, one or more electrons from each atom are loosely bound to the atom, and can move freely about within the metal. These conduction electrons are the charge carriers in metal conductors." (wikipedia ). That made me imagine electrons going from power -, running by the wires, crossing the resistor and returning to power +, where they suffer a new kick to power -. All that at near the speed of light.  In this model, it is easy to define "electrical current intensity" as proportional to the number of electrons per second that cross one specific point of the circuit.  However, looking for the physics of resistivity, I can read the phrase "The actual drift velocity of electrons is very small, in the order of magnitude of a meter per hour." (wikipedia ). This is a complete different point of view, where electrons are more or less fixed around a fixed position (or moving very slowly), an electricity is a wave of "collisions" or "jumps" of electrons, like in a Newton's cradle (or "do the wave" as public in a football match). Obviously, we do not need severals hours from the switch on to the light on.  So, the question is, do electrons move around a circuit ? References to band theory are welcome.  A related question is: if electrons doesn't moves along the circuit, what is "electrical current intensity"?  Note: I known there are similar questions already posted, but answers of the ones I've read are not very detailed (could be I've not found the best ones). Please, be a few patient before "close vote".