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Sending email in .NET through Gmail
If $q$ is coprime to $a$ then $a\mid nq-1,\,$ so $q$ is invertible mod $a$ I think the following is true:  If $q$ is coprime to $a$ then $a\mid (nq-1)$ for some $n\in\mathbb Z$.  I have started to first reducing $n\in\mathbb Z$ to $n\in\{1,\ldots,a-1\}$ and then considering all the cases $q = ka + j$ for $j=1,2,3,\ldots,a-1$. I stopped at $j=3$ and found that this is kinda messy. Does anyone have an elegant proof for this (maybe using some results from number theory)?  [Note: question has been generalized from prime $q\nmid a$ to $\,q$ coprime to $a$] ¿Can you help me with axiom of regularity? Changing the resistor to be in front or behind an LED doesn't affect brightness? Note: I am very new to the more advanced side of electricity. I do not have a firm grasp on the exact mechanics.  Lets say I have an example circuit:  5V Power Supply -&gt; LED -&gt; Resistor -&gt; Ground  Now the LED is receiving 20ma and 5V. If I were to switch the LED and the resistor however, the LED should receive less voltage but the same current, so wouldn't it be dimmer? Or would it receive the same voltage?  I've looked at these two threads and their answer doesn't help. Why are there $|G/G'|$ 1-dimensional representations of $G$? Let $G'$ be the derived subgroup of a finite group $G$.   We have a correspondence $\{\mathrm{reps \ of \ G/G'}\} \longleftrightarrow \{\mathrm{reps \ of \ G \ with \ kernel \ containing \ G' }\} $  If we restrict to 1-dimensional reps, we get:  $\{\mathrm{1\ dimensional \ reps \ of \ G/G'}\} \longleftrightarrow \{\mathrm{1 \ dimensional \ reps \ of \ G \ with \ kernel \ containing \ G' }\} $  Now my notes say that there are $|G/G'|$ 1-dimensional reps of $G$. Since there are $|G/G'|$ 1-dimensional reps of $G/G'$, this must mean that all 1-dimensional reps of $G$ have kernel containing $G'$. Why is this so?  Thanks Setting git default flags on commands How do I secure my house against intruders? What are some suggestions for securing your house from an intruder? Right now we have a GE Simon XT alarm system, but it doesn't cover breaking a window. We put some PVC pipes in the tracks of the lower floor windows as well. What other suggestions do you have?  EDIT: I also would like to know what structural changes I should be looking at. A simple one is to ensure that the deadbolt goes into a stud (they don't always!). Any other suggestions like this would be great. Plotting a function with an argument in degrees I want to plot a function. The function is, F is proportional to x/cos(x). Here x is rotation in degree and F is the force. I have to plot the graph where the x axis will be from 0 to 70 degree and the y axis will be linear value 0 to 5. I am not able to find any option to draw graphs where x axis is with rotational degree. When to use a GAM vs GLM I realize this may be a potentially broad question, but I was wondering whether there are assumptions that indicate the use of a GAM (Generalized additive model) over a GLM (Generalized linear model)? Someone recently told me that GAMs should only be used when I assume the data structure to be &quot;additive&quot;, i.e. I expect additions of x to predict y. Another person pointed out that a GAM does a different type of regression analysis than a GLM, and that a GLM is preferred when linearity can be assumed. In the past I have been using a GAM for ecological data, e.g.:  continuous time-series when the data did not have a linear shape I had multiple x to predict my y that I thought to have some nonlinear interaction that I could visualize using &quot;surface plots&quot; together with a statistical test  I obviously don't have a great understanding of what a GAM does different than a GLM. I believe it's a valid statistical test, (and I see an increase in the use GAMs, at least in ecological journals), but I need to know better when its use is indicated over other regression analyses. Is tipping mandatory at Restaurants and Bars in Germany? In Bavaria, I paid for my drink, I asked her for "Ruckgeld", but only got the integer part of the money back, the rest of it, she kept it for tip! I looked her in the eye and she understood that I wasn't happy with that "initiative", but wouldn't blink an eye!  My company advised me to let it go, mentioning that if I was going to ask for the change, she would swear.  Notice how she didn't return me the correct amount back and then stare at me, demanding for tip, but she would decide for me!  Since, I will be around for a week or so, I was thinking of paying the exact amount of money next time. Will I get in an uncomfortable situation of having the waiter stare at me, demanding a tip? In other words, is tipping mandatory in Germany?    PS: Having experienced the tipping-cruelty in USA, and having tipped generously waiters across the Mediterranean for their supreme behavior, this Saturday night came as a shock, since I had the impression that this awful habit was only a USA-thing (you know, just like the fake audience laughing they have in USA TV series :) ). I have been attending a number of academic / professional conferences recently and one thing that really awed me was the flawless and natural way in which the speakers presented on their topics.  Sure, they had the occasional mispronunciations or awkward pauses but they spoke with authority and confidence.   Some of them used powerpoint slides but were not reading from the slides; they were just talking on the points and they had so much to say on each point that I felt they could not do it without notes, but they were not even looking at their notes most of the time.  (The advantages of not looking at their notes too much was that they could maintain eye contact and could use very interesting slides that were not crammed with the dot points of their speeches.)  I hate to think that they had memorised their speech notes but some of the presentations went for more than an hour.  I know rehearsal is important but I wonder how could anyone remember so much in a nervous situation. (Its one thing to know everything on your topic; its another thing to present that 'everything'!)  Question:  What background / foreground things do presenters do that make their presentations flawless and natural? Do I have to re-enter the Schengen area to activate my second visa when traveling on back-to-back Schengen visas? I currently have a valid multiple entry Schengen visa that expires on Feb 10th (issued by the Spanish consulate). I need to visit the Czech Republic from Feb 9th to Feb 15th. For this, I am getting a visa starting Feb 11th from the Czech consulate as I was advised that I cannot have two simultaneous visas.   Is it possible for me to enter Czech republic on the current visa on Feb 9th, and stay past its expiry up to Feb 15th using the new visa without leaving the Czech Republic? Will there be any likely problem either while entering or exiting Schengen zone (I am entering Schengen in Frankfurt)? If $|f(z)|\le |g(z)|$, can we say $f$ is constant? Let $f$ and $g$ be entire functions and $g(z)\neq 0$ for all $z\in \mathbb{C}$. If $|f(z)|\le |g(z)|$, can we say $f$ is constant?  Liouville theorem says that an entire bounded function is constant, but g is not given bounded here. So I think it should be false. What else can we say about $f$? BasicTeX - how to identify packages that need to be added after an upgrade Is it possible to get a list of most recently installed packages? How do I color intersection of two ellipses in different color. For instances, the ellipses should themselves be colored as yellow and their intersection as red. My code for drawing ellipses is given below  \draw (-4,0) ellipse (8 and 3); \draw (8,0) ellipse (8 and 3);   UPDATE: In the comments to individual answers, I have requested to show it for three ellipses. Any open subset of $\Bbb R$ is a countable union of disjoint open intervals GTA V Online Character Transfer PS3 to PS4 I need help identyfying this movie I once saw late night on TNT, late 90s or early 2000s.  I don't have very clear recollection about it. The movie was very strange. It all took place in a single room of a spaceship or an escape pod that had lost power. A woman and a man, or several men, were stuck in a ship or a spacepod that was somehow damaged or didn't have power.   The movie was entirely made of the dialogs between the woman and the man (or men). They would talk about what was happening outside, how to contact help, about their lives. During their conversation details about the world around them emerge. In this movie people have colonized the solar system but not moved outside it, and there is some sort of a war.  Other things I remember are that it was dark in that room. I think they used only a lightbulb or even direct starlight and sunlight (if they were far away from the sun), because there was no power. Also it was getting colder, becaus the ship's heating system was not functioning, and so it was losing its heat. The people would constantly complain about how cold it's getting. I think that the woman was the captain of the ship. She would sometimes go to the control cabin and try to do something to help their situation, either restore power or contact someone so they can be rescued.  What impressed me about it was how they managed to make it realistic. Because the actors would talk and act like it was all real, it made it somehow feel very realistic to me as well.  That's about it as far as I remember. Hope someone has watched this movie and can tell me what it is.  Update  I want to add a few more details about the movie. While I watched it in the late 90s/early 2000s the movie looked much older. It looked like a vintage B-Movie made in the style of the 70s or 80s. In particular the decor looked more like Alien 1 or Cosmos 1999.   I just watched the firefly episode and it can't be it. It has too much action, and the atmosphere is not dark enough. How do we control web page caching, across all browsers? I assume it doesn't connect to anything (other than the satelite I guess), is this right? Or it does and has some kind of charge?