Violation of normality of residuals in glmm
Resolving heteroscedasticity in Poisson GLMM
Zero to the zero power – is $0^0=1$? Could someone provide me with a good explanation of why $0^0=1$?  My train of thought:  $x&gt;0$  $0^x=0^{x-0}=0^x/0^0$, so  $0^0=0^x/0^x=\,?$  Possible answers:   $0^0\cdot0^x=1\cdot0^0$, so $0^0=1$ $0^0=0^x/0^x=0/0$, which is undefined   PS. I've read the  on mathforum.org, but it isn't clear to me. What is the size of this microcontroller? Having trouble reading a data sheet I am trying to find a small enough microcontroller for a project, but I am not familiar with the notation used to denote the sizes in this image:  What does that mean when it says 12,20/11,80 SQ and 7,50 TYP.  The image is from .  OR is this not where I find the physical dimensions? If these do not denote the physical dimensions of the part, then where in the datasheet is that written? When trying to execute a symbolic link which points to an exe file (on Windows 7), the "Open with" windows appears. When renaming it to a ".exe" file, it says "The specified path does not exist. Check the path, and then try again.".  Symlinks to other destinations seem to work fine.  So is there any way to execute a symlink?  EDIT: In the command prompt, I execute the following and the MyProgram.exe runs both times.  However, if I double-click on the symbolic link in Windows Explorer it fails with the above error message.  C:\Temp&gt;mklink SomeProgram.exe MyProgram.exe symbolic link created for Backup.exe &lt;&lt;===&gt;&gt; Backup-1.0.1.exe  C:\Temp&gt;SomeProgram.exe  C:\Temp&gt;del SomeProgram.exe  C:\Temp&gt;mklink SomeProgram.exe C:\Temp\MyProgram.exe symbolic link created for Backup.exe &lt;&lt;===&gt;&gt; C:\Temp\Backup-1.0.1.exe  C:\Temp&gt;SomeProgram.exe I am reading  that finite-dimensional subspaces of normed vector spaces have closed direct sum complements. This is the proof:      Let $\{e_1, ..., e_n\}$ be a basis for $\mathcal M$. Every $x \in   \mathcal M$ has then a unique representation $$x = \alpha_1(x)e_1 +...+  \alpha_n(x)e_n.$$ Each $\alpha_i$ is a continuous linear functional on    $\mathcal M$ (a linear map from  finite dimensional space is always    continuous) which extends to a member of $\mathcal X^*$,  by the    Hahn-Banach theorem ($\mathcal X^*$ is the dual of $\mathcal X$). Let    $\mathcal N$ be the intersection of the null spaces of these exten­sions. Then $\mathcal X = \mathcal M\oplus \mathcal N$.   I follow it until the last sentence. Clearly $\mathcal N$ is closed, as it is the intersection of closed sets (the inverse of $0$ under any continuous function, and in particular any linear functional on a vector space, is closed). Also it is easy to see that $\mathcal M \cap \mathcal N= \{0\}$, since if some $m\in\mathcal M$ is in every null space, it must have the zero coefficient for every basis vector, and so it must be zero.  My question: Why do we have $\mathcal X =\mathcal M + \mathcal N$? This seems to only hold if the extension given by Hahn-Banach is zero off of $\mathcal M$, but I see no reason why this should be the case. What Is Energy? Where did it come from? Cosine of the sum of two solutions of trigonometric equation $a\cos \theta + b\sin \theta = c$ Question: If $\alpha$ and $\beta$ are the solutions of $a\cos \theta + b\sin \theta = c$, then show that: $$\cos (\alpha + \beta) = \frac{a^2 - b^2}{a^2 + b^2}$$  No idea how to even approach the problem. I tried taking two equations, by substituting $\alpha$ and $\beta$ in place of $\theta$ in the equation and manipulating them, but that didn't get me anywhere. Please help! May $y=e^x$ be satisfied with both $x$ and $y$ be positive integers?  I think it is not possible as $e$ ,a transcendental number, when multiplied by itself would never result in rational number.  Am I right? How to force nginx to resolve DNS (of a dynamic hostname) everytime when doing proxy_pass? I am using nginx/0.7.68, running on CentOS, with the following configuration:  server {     listen       80;     server_name ***;     index index.html index.htm index.php default.html default.htm default.php;      location / {             root   /***;             proxy_pass   http://***:8888;             index  index.html index.htm;     }     # where *** is my variables   The proxy_pass is to a DNS record whose IP changes frequently. Nginx caches the outdated IP address, resulting in a request to the wrong IP address.  How can I stop nginx from caching the IP address, when it is outdated? Prove that $\sum_{n=0}^\infty \frac{(-1)^n}{3n+1} = \frac{\pi}{3\sqrt{3}}+\frac{\log 2}{3}$ I have several identical elements with different attributes that I'm accessing with SimpleXML:  &lt;data&gt;     &lt;seg id="A1"/&gt;     &lt;seg id="A5"/&gt;     &lt;seg id="A12"/&gt;     &lt;seg id="A29"/&gt;     &lt;seg id="A30"/&gt; &lt;/data&gt;   I need to remove a specific seg element, with an id of "A12", how can I do this?  I've tried looping through the seg elements and unsetting the specific one, but this doesn't work, the elements remain.  foreach($doc-&gt;seg as $seg) {     if($seg['id'] == 'A12')     {         unset($seg);     } } An instructor has an exam of 5 questions and a question bank of 50 questions. Are there $50\choose 5$ or $(50)_5$ possible exams? Where $(50)_5 = 50! / 45! $ and $50\choose 5$ $= 50!/(5!45!)$   I've narrowed it down to these two possibilities (though even these may be wrong), but I think I'm confusing myself as to which one I use. Can anyone suggest a good way of thinking of this problem that will make it apparent whether I need use use the falling factorial or the binary coefficient?   Edit: thank you for all the answers! Indeed, I meant $50!/45$ as the falling factorial. Also, the question doesn't specify whether it was meant to be ordered or unordered, but I think I understand my confusion now! I know that who is used when asking about a subject (Who is at the door?) and whom is used when asking about an object (By whom was the door opened?). How do you determine which one to use when the object becomes the subject of another clause?   For example, take the sentence:      I gave the prize to ____ deserved it most.   Would it be whomever because it's the object of "gave," or would it be whoever because it's the subject of "deserved"? Is their a hard-and-fast rule to rely on in situations like this? A smooth vector field on $S^2$ vanishing at a single point SE Android app for tablet has bad formatting on feed On OpenBSD and with OpenBSD's  and , is it possible to store the  of a user in a  repository of the same user?  What would be the proper way to accomplish something like this?  (To steer the answer into the correct direction, I would not be opposed to changing some permissions in the system, although I'd rather not have to recompile the binaries, nor violate any good security paradigms.) Sugar daddy wants my online banking login info to make deposits — is it a scam? Let $ D = 2^\mathbb{N} $, i.e., D is the set of all sets of natural numbers.   What's the meaning of this definition? Intuitively, I would suggest that $ D = \{1,2,4,...\} $ but the explanation "set of all sets" leads me to the guess that this is wrong. Does higher education exist in Harry Potter? Does the closedness of a subgroup $H$ of $G$ imply that for $h\in H, g\in G \setminus H: hg \not \in H$?  I'm new to abstract algebra and this would appear to be a useful result. However, by definition, a subgroup is nearly closed, and I don't think this directly implies that for any element $g$ from $G$ not in $H$, the composition $gh \not \in H$.  Is this in fact a true statement? Is there any direct relation among first, third, fourth order moments?