Show that $I = \{a +\sqrt2b \in \Bbb Z[\sqrt2] : a$ and $b$ are both multiple of $5\}$ is a maximal ideal.
Let $I = \{a +ib \in \Bbb Z[i] : a$ and $b$ are both multiple of $5\}$.
I'm always on the hunt for free stock images to use while I design, what are some good resources for images that are under a a royalty free or creative commons licence? Maximum Likelihood Estimator for $\theta$ when $X_1,\dots, X_n \sim U(-\theta,\theta)$ Exercise :     Calculate a Maximum Likelihood Estimator for the model $X_1,\dots, X_n \; \sim U(-\theta,\theta)$.   Solution :  The distribution function $f(x)$ for the given Uniform model is :  $$f(x) = \begin{cases} 1/2\theta, \; \; -\theta \leq x \leq \theta \\ 0 \quad \; \; , \quad\text{elsewhere} \end{cases}$$  Thus, we can calculate the likelihood function as :  $$L(\theta)=\bigg(\frac{1}{2\theta}\bigg)^n\prod_{i=1}^n\mathbb I_{[-\theta,\theta]}(x_i)= \bigg(\frac{1}{2\theta}\bigg)^n\prod_{i=1}^n \mathbb I_{[0,\theta]}(|x_i|) $$  $$=$$  $$\bigg(\frac{1}{2\theta}\bigg)^n\prod_{i=1}^n \mathbb I_{[-\infty,\theta]}(|x_i|)\prod_{i=1}^n \mathbb I_{[0, +\infty]}(|x_i|)$$  $$=$$  $$\boxed{\bigg(\frac{1}{2\theta}\bigg)^n\prod_{i=1}^n \mathbb I_{[-\infty,\theta]}(\max|x_i|)}$$  Question : How does one derive the final expression in the box from the previous one ? I can't seem to comprehend how this is equal to the step before.  Other than that, to find the maximum likelihood estimator you need a $\theta$ sufficiently small but also $\max |x_i| \leq \theta$ which means that the MLE is : $\hat{\theta} = \max |x_i|$. What happens if a person can't die of a heart attack? This is a  about Monitoring Software. Also Related:   I need to monitor my servers; what do I need to consider when deciding on a monitoring solution? Proving that $ 2 $ is the only real solution of $ 3^x+4^x=5^x $ I would like to prove that the equation $ 3^x+4^x=5^x $ has only one real solution ($x=2$)  I tried to study the function $ f(x)=5^x-4^x-3^x $ (in order to use the intermediate value theorem) but I am not able to find the sign of $ f&#39;(x)= \ln(5)\times5^x-\ln(4)\times4^x-\ln(3)\times3^x $ and I can't see any other method to solve this exercise... Why are "baked" and "naked" not pronounced the same? Multiple outputs for beamer handouts I'm currently generating some presentations with beamer for students of my course. Some like to print the material, so I created a handout. Some students prefer to have 2,3,4,6 etc. pages per sheet of A4 paper. How can I go about to automatize the following code, so that it outputs 4 different files in 1x2,1x3,2x2,2x3 format?  I'd like to avoid having to compile it again everytime for the different formats.  \documentclass[a4paper]{article} \usepackage{pdfpages}  \begin{document} \includepdf[pages=1-last,nup=1x2,landscape=false,frame=true,             noautoscale=false,scale=1,delta=0mm 5mm]{Pres7Chap13-Handout.pdf} \end{document} I've lost a command block, how can I find it? I've placed a command block making it day all the time. I want to turn it off, but I've lost it. What's the quickest way to find it? Proving set of interiors is open Every finite set contains its supremum: proof improvement. Every finite subset of $\mathbb R$ contains its supremum (and its infimum)   Proof Let $A=\{a_1,...,a_n\}$ be a finite subset of $\mathbb{R}$. Since it is non-empty and it is bounded ($\max A$ is an upper bound), it has supremum, that is $\exists \sup A$ and by definition $\forall a \in A \;\, a \leq \sup A$. Let's suppose that $\sup A  \not\in A$ then, since $\max A \in A$ we have that $\max A &lt; \sup A$. But considering that $\mathbb Q$ is dense in $\mathbb R$ we can conclude that $\exists r \in \mathbb Q$ s.t. $\max A &lt; r &lt; \sup A$, but this is absurd since $r$ is an upper bound of $A$ and it is lower than the supremum. Necessarily, $\sup A\in A$.  Is there anything wrong? Is there any way to prove this without using density of $\mathbb Q$ or another property? Thanks in advance. Why is the Principle of Superposition true in EM? Does it hold more generally? In the theory of electromagnetism (EM), why is the principle of superposition true? Can we read it off from Maxwell's equations directly?  Does it have any limit of applicability or is it a fundamental property of nature? I was wondering why the software center doesn't download all the requested apps and install one by one as they finish to download. Or instead why the software center doesn't download an app while the other is installing I know I need to have (although I don't know why) a GROUP BY clause on the end of a SQL query that uses any aggregate functions like count, sum, avg, etc:  SELECT count(userID), userName FROM users GROUP BY userName   When else would GROUP BY be useful, and what are the performance ramifications? 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? River flowing down a slope Maximum and minimum value of Determinant of $3 \times 3$ Matrix with entries $\pm 1$ Find Maximum value of Determinant of $3 \times 3$ Matrix  with  entries $\pm 1$  My try:  I considered a matrix as :  $$A=\begin{bmatrix} 1 &amp;-1  &amp;-1 \\  -1 &amp;1  &amp;-1 \\   -1&amp;-1  &amp;1  \end{bmatrix}$$  we have $$Det(A)=-4$$ and maximum is $4$, but how can we show that these are max and min values?   I also tried as follows:  By definition Determinant of a matrix is dot product of elements of any row with corresponding Cofactors  hence  $$Det(A)=a_{11}C_{11}+a_{12}C_{12}+a_{13}C_{13}$$  By cauchy Scwartz Inequality we have  $$a_{11}C_{11}+a_{12}C_{12}+a_{13}C_{13} \le \left(\sqrt{a_{11}^2+a_{12}^2+a_{13}^2}\right)\left(\sqrt{C_{11}^2+C_{12}^2+C_{13}^2}\right)$$  any way to proceed here? Does infinity exist in physical systems? Sometimes people fail at asking a question by being too broad, unclear like . So I'll take a stab at what I believe to be the same question, but more concise and clearly stated:     Does infinity exist in either terms of structure, parameters, or   measurement in any physical systems?   Can infinity have any real connection with reality?  Or is infinity purely a mathematical concept just used by physicists as a convenient way to describe the very large, an approximation?  I have heard that if you model a physical system (), and you run into infinity as a solution, then you have either made an error in your calculations or your model is wrong. I have a content type with multiple fields. Some of the fields are accessible to everyone and there are other fields for which the access is controlled.   Example: There is a content type called Reports with fields like title, summary, body etc. Title and summary should be visible to everyone but the access to body field is controlled.  For implementing this I need to have 2 view modes. one of them will have the fields title and summary displayed with the rest of the fields hidden. The other view mode will have only the body field visible. I am planning to implement the hook hook_node_access to derive the access permissions for the node. So how can I get the view mode in this hook so that I can have my access check based on the view mode. Ignore .DS_Store and Icon files in a folder with Cocoa NSFIleManager I am trying to solve an integral in $k$-space where I want to calculate the ratio of the average energy of an arbitrary electronic band ($E(k)=k^n)$ to the square of average group velocity of electrons($v_g   = k^{n-1})$ approximately in the form of this expression given by $$L=\frac{\frac{\sum_{\mathbf{k}}k^{n}}{\sum_{\mathbf{k}}1}}{\left(\frac{\sum_{\mathbf{k}}k^{n-1}}{\sum_{\mathbf{k}}1}\right)^2}\;.$$  Now, I want to do this for $d$-dimensions, so I convert this sum to integral.  $$L=\frac{\dfrac{\int_{0}^{k_f}k^{n} d^d k}{\int_{0}^{k_f}1d^d k}}{\left(\dfrac{\int_{0}^{k_f}k^{n-1}d^d k}{\int_{0}^{k_f}1d^d k}\right)^2}$$ where $d^dk  = \dfrac{d \pi^{d/2}}{\Gamma(\frac{d}{2}+1)}k^{d-1}dk$ is the volume element in $d$-dimensional space.  I want to get a simple expression for $L$ and how it depends with $k_f,n$ and $d$. any physical insight into this integral could also be helpful.