Let $f\in \mathbb Z[X]$ be a polynomial of positive degree.How to show that there are infinitely many prime numbers $p$ such that the polynomial $f$ has a zero in $\mathbb Z/p \mathbb Z$ ?  I have no idea of how to start even.  Need help
Let $g(x)\in \mathbb{Z}[x]$, a nonconstant polynomial. Show that the set of primes $p$ such that $p\mid g(n)$ for some $n\in \mathbb{Z}$ is infinite.  I don't know how to start. I have tried asuming that the set is finite but i haven't reached any contradiction.
How to Customize the Email template on chatter posts In some literature, I have read that a regression with multiple explanatory variables, if in different units, needed to be standardized.  (Standardizing consists in subtracting the mean and dividing by the standard deviation.)  In which other cases do I need to standardize my data?  Are there cases in which I should only center my data (i.e., without dividing by standard deviation)? I am looking for the definition of when I am allowed to do forward declaration of a class in another class's header file:  Am I allowed to do it for a base class, for a class held as a member, for a class passed to member function by reference, etc. ? This is result of who -uH:  NAME     LINE         TIME             IDLE          PID COMMENT me    :0           2013-08-29 01:29   ?          1686 (:0) me    pts/0        2013-08-29 01:42  old         2905 (:0) me    pts/1        2013-08-29 01:45  old         2905 (:0) me    pts/2        2013-08-29 01:52  old         2905 (:0) me    pts/3        2013-08-29 02:26  old         2905 (:0) me    pts/4        2013-08-30 18:37 00:07        2905 (:0) me    pts/5        2013-08-29 10:34  old         2905 (:0) me    pts/6        2013-08-29 11:05  old         2905 (:0) me    pts/7        2013-08-29 13:34  old         2905 (:0) me    pts/8        2013-08-30 15:56 02:45        2905 (:0) me    pts/9        2013-08-30 18:27 00:17        2905 (:0) me    pts/10       2013-08-30 17:34   .          2905 (:0) me    pts/11       2013-08-30 17:37 00:13        2905 (:0) me    pts/12       2013-08-30 18:28 00:17        2905 (:0) me    pts/13       2013-08-30 18:38 00:07        2905 (:0)   Why it lists my username in 15 lines? How do I know If someone has connected to my system remotely? Isomorphism between dual space and bilinear forms Studying from Roman's Advanced Linear Algebra, I want to prove that $$U^* \otimes V^* \cong (U \otimes V)^* \cong \hom(U,V;\mathbb{F})$$ The author proves that $U^* \otimes V^* \cong (U \otimes V)^*$ by showing that there is an unique linear transformation   $$\theta:U^* \otimes V^* \to (U \otimes V)^*$$  defined by $\theta(f \otimes g)=f \odot g$ where $(f \odot g)(u \otimes v)=f(u)g(v)$  I want to prove that $U^* \otimes V^* \cong \hom(U,V;\mathbb{F})$. I tried to use the universal property by fixing $f \in U^*, g\in V^*$ and defining $S: U^*\times V^* \to \hom(U,V;\mathbb{F})$ as $S(f,g)=F_{f,g}$, where $F_{f,g}(u,v)=f(u)g(v)$, but I got stuck. SharePoint 2013 Jan 2016 Updates Is anyone here experience problems in SharePoint after installing the latest JAN 2016 on 13th Jan 2016 (Western Australian Time) updates for SharePoint 2013. I am guessing it could be KB3114503 update giving me problems.  I am getting the same problem on other machines after running the same updates too.  The problem I am receiving is, on every list view I get this error message below:  TypeError: Unable to get property 'replace' of undefined or null reference   TypeError: Unable to get property 'replace' of undefined or null referenceSys.ArgumentNullException: Sys.ArgumentNullException: Value cannot be null. Parameter name: element Is there a difference between installing an application via Ubuntu Software Center or a terminal? How can I get the current working directory? I want to have a script that takes the current working directory to a variable. The section that needs the directory is like this dir = pwd. It just prints pwd how do I get the current working directory into a variable? What are the norms in Ito isometry? Itō isometry from :     Let $W : [0, T] \times \Omega \to \mathbb{R}$ denote the canonical   real-valued Wiener process defined up to time $T &gt; 0$, and let $X :  [0, T] \times \Omega \to \mathbb{R}$ be a stochastic process that is   adapted to the natural filtration $\mathcal{F}_{*}^{W}$ of the Wiener   process. Then $$      \mathbb{E} \left[ \left( \int_{0}^{T} X_{t} \, \mathrm{d} W_{t} \right)^{2} \right] = \mathbb{E} \left[ \int_{0}^{T} X_{t}^{2} \,  \mathrm{d} t \right], $$ where $\mathbb{E}$ denotes expectation with   respect to  $\gamma$. In other words, the Itō   stochastic integral, as a function, is an isometry of normed vector   spaces with respect to the norms induced by the inner products $$      ( X, Y )_{L^{2} (W)} := \mathbb{E} \left( \int_{0}^{T} X_{t} \, \mathrm{d} W_{t} \int_{0}^{T} Y_{t} \, \mathrm{d} W_{t} \right) =  \int_{\Omega} \left( \int_{0}^{T} X_{t} \, \mathrm{d} W_{t}  \int_{0}^{T} Y_{t} \, \mathrm{d} W_{t} \right) \, \mathrm{d} \gamma  (\omega) $$ and $$      ( A, B )_{L^{2} (\Omega)} := \mathbb{E} ( A B ) = \int_{\Omega} A(\omega) B(\omega) \, \mathrm{d} \gamma (\omega). $$   I was wondering what the two normed spaces are and what their norms are, so that the two normed spaces are isometric wrt their norms?  Thanks and regards! Intuition for Rayleigh PDF We have ground-truth data $\mathbf{x}^* = (0,0)^T \in \mathbb{R}^{2}$. Furthermore, we have $N$ measurements $\mathbf{x}_i \in \mathbb{R}^{2}, i\in \{1,\ldots,N\}$. We calculate the $N$ error vectors $\mathbf{e}_i = \mathbf{x}_i - \mathbf{x}^*$. We assume that the two components of $\mathbf{e}_i = (e_{x,i}, e_{y,i})^T $ are independent of each other and normally distributed with equal mean $\mathbf{\mu} = (0,0)^T$ and variance $\sigma^2$. The absolute errors $e_i = \sqrt {e_{x,i}^2 + {e}_{y,i}^2} $ follow a Rayleigh distribution, is that correct?  The probability density function (PDF) of the Rayleigh function is zero at a value of zero. Why is this the case (intuitively)? I would assume that our errors should most likely be 0 (i.e. PDF should be highest at 0)... Keyboard Shortcuts no longer working on GNOME 3.4.1 Quadratic Formula in Complex Variables Long boot delay on Ubuntu loading/splash screen following regular dist-upgrade on clean SSD install (18.04) I am trying to make a callout to AWS S3 using named credentials and Apex  The named credential is set up as follows  Label: AWS S3 Name: AWS_S3 URL: https://bucket-name.s3.amazonaws.com - Certificate: Identity Type: Named Principal Authentication Protocol: AWS Signature Version 4 AWS Access Key: AKIAxxxxxx AWS Secret Access Key: xxxxxx AWS Region: ap-southeast-2 AWS Service: s3 - Generate Authorization header: true Merge fields in header: false Merge fields in body: false   The Apex class is very simple  HttpRequest req = new HttpRequest(); req.setEndpoint('callout:AWS_S3'); req.setMethod('GET');                        Http http = new Http(); HTTPResponse res = http.send(req);                 System.debug(res.getBody()); return res.getBody();   The response body contains the following:     SignatureDoesNotMatchThe request signature we calculated does not match the signature you provided. Check your key and signing method.AKIAxxxxx   I have used the same bucket, endpoint and credentials using Postman and am able to successfully return the list of objects in the bucket so it is not an IAM permissions issue  There have been a couple of similar questions on stackexchange but unfortunately they do not provide any resolution or assistance in this case:    I've tried looking for code samples or further pointers in the Apex documentation but it doesn't really give any further guidance on how to proceed or correctly set this up so I'm a bit stuck Do messages in archive count to your inbox space? Pretty much as the title says, do messages in archive count to your total inbox space? Is a differentiable function always continuous? Continuous Functions are not Always Differentiable. But can we safely say that if a function f(x) is differentiable within range $(a,b)$ then it is continuous in the interval $[a,b]$ . If so , what is the logic behind it ? Why does the zoom sometimes stop at a point? I often find that I can zoom in to a point, but then cannot zoom any more, even when there is a lot of room to zoom to the meshes on the scene.   This is particularly annoying in the following scenario:  I often land up left clicking on the scene and the 3D cursor shifts there. I use Shift+C to shift the 3D cursor back to the origin. However, this also centers the view on the cursor and shifts the view to show all objects, and sometimes this zooms things way out &ndash; at this point, if I try to zoom in, I can zoom in to a point (using the pinch/zoom on trackpad), but then it stops zooming in.   How do I zoom in after that? Is there anything else I am doing wrong in this case? Integrating $\int\frac{x^3}{\sqrt{9-x^2}}dx$ via trig substitution What I have done so far: Substituting $$x=3\sin(t)\Rightarrow dx=3\cos(t)dt$$ converting our integral to  $$I=\int\frac{x^3}{\sqrt{9-x^2}}dx=\int \frac{27\sin^3(t) dt}{3\sqrt{\cos^2(t)}}3\cos(t)dt\\ \Rightarrow \frac{I}{27}=\int \sin^3(t)dt=-\frac{\sin^2(t)\cos(t)}{3}+\frac{2}{3}\int\sin(t)dt\\ \Rightarrow I=18\cos(t)-9\sin^2(t)\cos(t)+c$$ Which is where I'm stuck. I can substitute back in for the $\sin(t)$ terms but don't know hot to deal with the $\cos(t)$ terms and back out information about $x$.   edited: because I miswrote the problem, I am sorry. Variations in the pronunciation of "the" Although there are rather simple rules determining the pronunciation of "the", native speakers quite often deviate from these rules (including, e.g., TV shows). According to the Longman Pronunciation Dictionary,     The EFL learner is advised to use [ðə] before a consonant sound (the   boy, the house), [ði] before a vowel sound (the egg, the hour). Native   speakers, however, sometimes ignore this distribution, in particular   by using [ðə] before a vowel (which is in turn usually reinforced by a   preceding ʔ), or by using [ði:] in any   environment, though especially before a hesitation pause. Furthermore,   some speakers use stressed [ðə] as a strong form, rather than the   usual [ði:].   My question is: when native speakers use [ðə] instead of [ði] before a vowel sound, do they do it on purpose or accidentally? If it is on purpose, how do they (typically) decide which pronunciation to use? What is a valid reason to use [ðə] before a vowel sound? Drush from GitHub stopped working I have a  that has been working since it moved there.  I did a git pull this morning, and am now getting errors:     $ drush      Warning: require(/home/mpd/drush/vendor/autoload.php): failed to open   stream: No such file or directory in   /home/mpd/drush/includes/bootstrap.inc on line 1007      Fatal error: require(): Failed opening required   '/home/mpd/drush/vendor/autoload.php' (include_path='.:') in   /home/mpd/drush/includes/bootstrap.inc on line 1007   How can I fix this?