Prove $\sum_{n= -\infty}^{\infty} \frac{1}{(t+n)^2} = \frac{\pi^2 }{\sin^2(\pi t)}$ using f(x)=1-|x| and Poisson summation formula
$\sum_{n=-\infty}^{\infty}\frac{1}{(n+\alpha)^2}=\frac{\pi^2}{(\sin \pi \alpha)^2}$?
Associated Prime Ideals in a Noetherian Ring; Exercise 6.4 in Matsumura Let $I$ and $J$ be ideals of a Noetherian ring $A$. If $JA_P\subseteq IA_P$ for every $P\in \operatorname{Ass}_A(A/I)$, then $J\subseteq I$.   I'm reading Matsumura's Commutative Ring Theory book on my own. This is exercise $6.4$ in that book, and I fail to prove it. Could you help me with this? Thanks in advance. DES Initialization Vector in C# "cron" processes spawning and piling up, but no crontab EDIT: I solved this, so see below for my own answer. While the bounty asks for a fix to the problem, I'd like to award it to someone who can explain why a cron process that ran for 6 years over multiple MacOS versions suddenly became a problem in Catalina.  First note, this is not a duplicate of  because I don't have active crontabs on my computer. Cron processes spawn, but they never finish, so they just keep piling up. I'm running MacOS Catalina 10.15.1  Cron jobs are spawning every couple of minutes and never dying. The oldest cron process always has a parent process "launchd", and I've searched for populated crontabs and can't find any. If I let this go on long enough, eventually I run into a scenario where I have hundreds of running cron processes and everything on my computer grinds to a halt, and I can't even open new apps (I get cryptic messages like ".app cannot be opened" when I try to open an app). My only solution at this point has been to keep activity monitor open and periodically killing off all active cron processes.  I've been able to create a new user account that seems to be immune from this issue. However, as soon as I logged into my apple ID on that account to set up iCloud drive, the problem started with that account, too. I logged out of my Apple ID and attempted uninstall iCloud drive, but the problem persisted. Clean installs of Catalina don't clear up the problem, I've reset SMC and PRAM multiple times, and repairing permissions (via this ). I've also ensured that cron has full disk access (via this ). I'm at a dead end with trying to troubleshoot this problem.  I'm not sure it's related, but my "system.log" file seems to get spammed the following few lines:  &lt;redacted&gt; AMPDeviceDiscoveryAgent[2132]: Entered:_AMMuxedDeviceDisconnected, mux-device:63 &lt;redacted&gt; AMPDeviceDiscoveryAgent[2132]: Entered:__thr_AMMuxedDeviceDisconnected, mux-device:63 &lt;redacted&gt; AMPDeviceDiscoveryAgent[2132]: tid:9a27 - Mux ID not found in mapping dictionary   ANY help is appreciated. What is the Ubuntu equivalent of this interface?    I am looking for a user-friendly method that the owner of a computer can use to log off other users without having to restart.  The case I'm concerned about is of the beginner-level user who needs administrative control over the computer because they are the owner, but probably isn't concerned with most of the tasks we would usually associate with system administration. Where can I find good literature for beginners? I am trying to find some database tutorials,books and similar, since I have started this course in college, and received some material, but it is really hard to manage through all the info on the net. I searched through the questions on the site but didn't find any. Thankfull for all your input. Where can I write a temp file from ASP.NET? Does borrowing from my 401(k) make sense in my specific circumstance? I know the standard answer and understand that I will be losing out on that compound interest, but I am young and only need to borrow a few thousand. here are some more details:   Need to borrow $6500 to pay off a car and two student loans ($300/month in payments combined) I am 22 and work full time, making about $70K annually I contribute 8% of my income with a 4% company match I increase that by 2% every year The loan would be over 48 months ($140/month) I live very frugally and have almost no discretionary spending that can be cut (don't eat out all the time, no gym membership/ Netflix/ Apple music etc.) I'm cash-poor: 90% of my salary is going to mortgage, groceries, students loans, car payment, newborn baby medical expenses, and commute. These are not my only debts, just the ones I am seeking to pay off in the short term. I work two jobs, one full and one part time. I started a business, and my wife has two jobs. Working more is not a good answer to the original question.   It seems to me that I would be reducing middle to high interest debt while also saving myself $150 per month.   Loans:     Student loan 1: $1200 @ 5%   Student loan 2: $1100 @ 4.25%   Car Loan: $4000 @ 6% What early-2000s show about dimension hopping has a character who can swap limbs? An earlier question asked about a show that made me think of another one I can't remember.  It was a live action show about exploring different dimensions, and I think the main character was searching for his dad.  I remember seeing it in the US, but I don't know if it was limited to that.  I'm a little fuzzy on the details, but I think the main character could swap his limbs or something like that. Should I mention my Canadian visa refusal in my UK visa application? $T(av_1 + bv_2) = aT(v_1) + bT(v_2)$  Why is this called linear? $f(x) =ax + b$, the simplest linear equation does not satisfy $f(x_1 + x_2) = f(x_1) + f(x_2)$.  Thank you. In a family with two children, what are the chances, if one of the children is a girl, that both children are girls? "Compared with" vs "Compared to"—which is used when? Is only one of them correct? Are they used in different situations? Or are they interchangeable? How to read ext4 partitions on Windows? Is there a way to read ext4 partitions from Windows? If so, what can I do? I have been reading Hamilton Chapter 13 and he has the following state space representation for an ARMA(p,q). Let $r = \max(p,q+1)$.Then the ARMA (p,q) process is as follows: $$ \begin{aligned}  y_t -\mu &amp;= \phi_1(y_{t-1} -\mu) + \phi_2(y_{t-2} -\mu) + ... + \phi_3(y_{t-3} -\mu) \\ &amp;+ \epsilon_t + \theta_1\epsilon_{t-1} + ... + \theta_{r-1}\epsilon_{t-r+1}.  \end{aligned} $$ Then, he defines the State Equation as follows:   $$  \xi_{t+1} = \begin{bmatrix}      \phi_1 &amp; \phi_2 &amp; \dots &amp; \phi_{r-1} &amp; \phi_r \\      1 &amp; 0  &amp; \dots &amp; 0 &amp; 0  \\      \vdots &amp; \vdots &amp; \dots &amp; 0 &amp; 0 \\      0 &amp; 0 &amp; \dots &amp;1 &amp;0 \end{bmatrix} \xi_t  +  \begin{bmatrix}      \epsilon_{t+1} \\     0\\\vdots  \\     0       \end{bmatrix}  $$  and the observation equation as:  $$y_t = \mu + \begin{bmatrix}      1 &amp; \theta_1 &amp; \theta_2 &amp; \dots &amp; \theta_{r-1} \\     \end{bmatrix} \xi_t. $$  I do not understand what the $\xi_t$ is in this case. Because in his AR(p) representation it is $\begin{bmatrix}      y_{t} - \mu \\     y_{t-1} - \mu\\\vdots  \\     y_{t-p+1} - \mu       \end{bmatrix}  $ and in his MA (1) representation it is $\begin{bmatrix}      \epsilon_{t} \\     \epsilon_{t-1}      \end{bmatrix}  $.  Could someone explain this to me a little bit better? How far can one get in analysis without leaving $\mathbb{Q}$? Suppose you're trying to teach analysis to a stubborn algebraist who refuses to acknowledge the existence of any characteristic $0$ field other than $\mathbb{Q}$.  How ugly are things going to get for him?  The algebraist argues that the real numbers are a silly construction because any real number can be approximated to arbitrarily high precision by the rational numbers - i.e., given any real number $r$ and any $\epsilon&gt;0$, the set $\left\{x\in\mathbb{Q}:\left|x-r\right|&lt;\epsilon\right\}$ is nonempty, thus sating the mad gods of algebra.  As @J.M. and @75064 pointed out to me in chat, we do start having some topology problems, for example that $f(x)=x^2$ and $g(x)=2$ are nonintersecting functions in $\mathbb{Q}$.  They do, however, come arbitrarily close to intersecting, i.e. given any $\epsilon&gt;0$ there exist rational solutions to $\left|2-x^2\right|&lt;\epsilon$.  The algebraist doesn't find this totally unsatisfying.     Where is this guy really going to start running into trouble?  Are there definitions in analysis which simply can't be reasonably formulated without leaving the rational numbers?  Which concepts would be particularly difficult to understand without the rest of the reals? I know the thread  However, there is no Linux anymore in the installation site .  Unsuccessful attempts   I do DriraWassim's proposal which does not add anything to apt-get and does not show the installed .deb package in the path/search  sudo apt-get install libxml2:i386  sudo apt-get update sudo apt-get upgrade  apt-cache search adobereader-enu   I download the newest .deb package  and run it.  I cannot find the application anywhere in the system. It is not in the path and not in Ubuntu's search. How can you start it?   System: Ubuntu 16.04, 64 bit Hardware: Macbook Air 2013-mid Linux kernel: 4.6 Cookie blocked/not saved in IFRAME in Internet Explorer I have two websites, let's say they're example.com and anotherexample.net. On anotherexample.net/page.html, I have an IFRAME SRC="http://example.com/someform.asp". That IFRAME displays a form for the user to fill out and submit to http://example.com/process.asp. When I open the form ("someform.asp") in its own browser window, all works well. However, when I load someform.asp as an IFRAME in IE 6 or IE 7, the cookies for example.com are not saved. In Firefox this problem doesn't appear.  For testing purposes, I've created a similar setup on  .  example.com uses cookie-based sessions (and there's nothing I can do about that), so without cookies, process.asp won't execute. How do I force IE to save those cookies?  Results of sniffing the HTTP traffic: on GET /someform.asp response, there's a valid per-session Set-Cookie header (e.g. Set-Cookie: ASPKSJIUIUGF=JKHJUHVGFYTTYFY), but on POST /process.asp request, there is no Cookie header at all.  Edit3: some AJAX+serverside scripting is apparently capable to sidestep the problem, but that looks very much like a bug, plus it opens a whole new set of . I don't want my applications to use a combination of bug+security hole just because it's easy.  Edit: the P3P policy was the root cause, full explanation below. How to prove or disprove $\forall x\in\Bbb{R}, \forall n\in\Bbb{N},n\gt 0\implies \lfloor\frac{\lfloor x\rfloor}{n}\rfloor=\lfloor\frac{x}{n}\rfloor$. How to prove or disprove $\forall x\in\Bbb{R}, \forall n\in\Bbb{N},n\gt 0\implies \left\lfloor\frac{\lfloor x\rfloor}{n}\right\rfloor=\left\lfloor\frac{x}{n}\right\rfloor$.  So we want to prove $\left\lfloor\frac{\lfloor x\rfloor}{n}\right\rfloor\ge\left\lfloor\frac{x}{n}\right\rfloor$ and $\left\lfloor\frac{\lfloor x\rfloor}{n}\right\rfloor\le\left\lfloor\frac{x}{n}\right\rfloor$  Since $\lfloor x\rfloor\le x$, we can just start from here and prove  $\left\lfloor\frac{\lfloor x\rfloor}{n}\right\rfloor\le\left\lfloor\frac{x}{n}\right\rfloor$  But for $\left\lfloor\frac{\lfloor x\rfloor}{n}\right\rfloor\ge\left\lfloor\frac{x}{n}\right\rfloor$, I have no idea how to start. I would like to upload a file asynchronously with jQuery.     $(document).ready(function () {     $("#uploadbutton").click(function () {         var filename = $("#file").val();          $.ajax({             type: "POST",             url: "addFile.do",             enctype: 'multipart/form-data',             data: {                 file: filename             },             success: function () {                 alert("Data Uploaded: ");             }         });     }); }); &lt;script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.2.0/jquery.min.js"&gt;&lt;/script&gt; &lt;span&gt;File&lt;/span&gt; &lt;input type="file" id="file" name="file" size="10"/&gt; &lt;input id="uploadbutton" type="button" value="Upload"/&gt;     Instead of the file being uploaded, I am only getting the filename. What can I do to fix this problem? I tried to find an answer to this a few years ago, but it seemed it wasn't possible then (I suspect it isn't now either! - but what the hell!)   I have a three monitor system, that I mostly use for remote desktop. However occasionally it would be useful to have remote desktop on two monitors and use the other monitor for the local machine.   I know it is possible to have a single monitor full screen, or to stretch an RDP session across monitors, but this loses things like snapping etc.   P.S. this is a duplicate of  thread, but that was a couple of years ago, I'm hoping things have changed.