How to show that $\prod_{d/n} d = n^{\frac{\tau(n)}{2}}$
How to prove $ \prod_{d|n} d= n^{\frac{\tau (n)}{2}}$
If the dual spaces are isometrically isomorphic are the spaces isomorphic? How to disable the "Your site has updated to WordPress x.y.z" admin email? Using the automatic update of my self-hosted WordPress blogs, I configured automatic updates of both the WordPress core as well as WordPress plugins and themes ().  Whenever a core update occurs, I get an email like:     Your site has updated to WordPress 3.9.2      Howdy! Your site at  has been updated automatically   to WordPress 3.9.2.      No further action is needed on your part. For more on version 3.9.2,   see the About WordPress screen:       If you experience any issues or need support, the volunteers in the   WordPress.org support forums may be able to help.         The WordPress Team   I understand that I can  but I would prefer to only disable the above notification.  Therefore my question is:  How to disable update notification emails in WordPress? Use $\delta-\epsilon$ to show that $\lim_{n\to\infty} a^{\frac{1}{n}} = 1$? Hope this is a meaningful question, but I'm curious if is possible to show that   $$\lim_{n\to\infty} a^{\frac{1}{n}}=1, \text{where  }a&gt;0$$  using $\delta-\epsilon$ directly or other methods. One method that I am aware of is to use the following:     If $\{s_n\}$ is a nonzero sequence, then $\liminf\bigl|\frac{s_{n+1}}{s_n}\bigr|\le \liminf |s_n|^{\frac{1}{n}}\le \limsup |s_n|^{\frac{1}{n}}\le\limsup\bigl|\frac{s_{n+1}}{s_n}\bigr|$ How to output a counter with leading zeros? I can output the value of a counter in a LaTeX document via:  \setcounter{mycounter}{1} \arabic{mycounter}   Which inserts a 1. But how do I format the output by, for example, adding a leading zero if mycounter is less than 10 in a way similar to what  lets you do by specifying %02d as your format string? How can I cast int to enum? How can an int be cast to an enum in C#? Change email login to another id Can all real polynomials be factored into quadratic and linear factors? Cardinality of the set of bijective functions on $\mathbb{N}$? How can I execute `date` inside of a cron tab job? I want to create a log file for a cron script that has the current hour in the log file name. This is the command I tried to use:  0 * * * * echo hello &gt;&gt; ~/cron-logs/hourly/test`date "+%d"`.log   Unfortunately I get this message when that runs:  /bin/sh: -c: line 0: unexpected EOF while looking for matching ``' /bin/sh: -c: line 1: syntax error: unexpected end of file   I have tried escaping the date part in various ways, but without much luck. Is it possible to make this happen in-line in a crontab file or do I need to create a shell script to do this? Real-life examples of common distributions 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. Yu Yuphoria bricked after installing custom recovery and boot file My YU YUPHORIA got bricked while I was working with it. I want to install custom ROM into it for that I am installing custom recovery and boot file but the file that is found on Internet was not working and that's why I am stuck with company logo. I can't go into recovery mod, neither it boot, and USB debugging mode is turned off. I tried to install fresh ROM from fastboot but after installing every driver properly command prompt is showing "waiting for the device". Please sugest some way to get rid of this problem. Consider an irreducible but possibly periodic Markov chain on a finite state space with transition matrix $P$. We know there exists a unique stationary distribution $\pi$. If the Markov chain were aperiodic, we would have $P^n_{ij} \to \pi(j)$ as $n \to \infty$. This fails if the chain is periodic, but we have convergence of the Cesaro averages: $$\frac{1}{n}\sum_{k=1}^n P^k_{ij} \to \pi(j) \text{ as } n \to \infty.$$ Can anyone point me to a reference that states this fact? Every reference I've seen only considers convergence for aperiodic chains, or "fixes" periodicity by considering a lazy version of the chain. Alternatively, is there a simple way to obtain this result using the result for aperiodic chains? Restore deleted photos Is it possible to increase Output Field Name length beyond 10 characters? When creating a new Column, the default value for the length of the name is 10. Using the Field Calculator via the Model Builder, I want to include the names of regional areas into the columns which has more than 10 characters.  Can the length of the Output Field Name be increased? When I turn 13 years old, will some of my Roblox account restrictions unlock automatically? Prove by induction that for all $n \geq 3$: $n^{n+1} > (n+1)^n$ I just browsed through this wonderful book: . The irony is, I am still not able to understand the motivation for using multivariate (regression) models instead of separate univariate (regression) models. I went through stats.statexchange posts  and  that explain (a) difference between multiple and multivariate regression and (b) interpretation of multivariate regression results, but I am not able to tweak out the use of multivariate statistical models from all the information I get online about them.   My questions are:   Why do we need multivariate regression? What is the advantage of considering outcomes simultaneously rather than individually, in order to draw inferences. When to use multivariate models and when to use multiple univariate models (for multiple outcomes).  Take an example given in the  with three outcomes: locus of control, self-concept, and motivation. With respect to 1. and 2., can we compare the analysis when we do three univariate multiple regression versus one multivariate multiple regression? How to justify one over another? I haven't come across many scholarly papers that utilize multivariate statistical models. Is this because of the multivariate normality assumption, the complexity of model fitting/interpretation or any other specific reason? Use "does" in front of another verb in a question? What time does sound good to you?     What time sounds good to you?    Is there a difference? Why are Harry's GoF dreams in the third-person?