Can someone provide me with a sketch of proof / hint for this exercise:  Let K $\subset$ $L^1([0,1]\,,\, \mu)$ be a closed linear subspace. If $\forall$ $f \in K$, $\exists \, p &gt; 1$ such that $f \in L^p$, then $\exists \, p &gt; 0 \Rightarrow\, K \subset L^p$.  I understand that the uniform boundedness principle should be applied here, but I have no clue how. Thanks for any help in advance.
Let $X \subset L^1(\mathbb{R})$ a closed linear subspace satisfying \begin{align} X\subset \bigcup_{p&gt;1} L^p(\mathbb{R})\end{align}    Show that $X\subset L^{p_0}(\mathbb{R})$ for some $p_0&gt;1.$   I guess the problem is that in infinite measure spaces the inclusion $L^p\subset L^q$ only holds for $p=q$. Is it maybe possbile to apply Baire's Theorem in some way?
I want Ctrl+Alt+B bring up a browser.  Ctrl+Alt+Delete is handy.  Can I set up Ctrl+Alt+B to bring up a browser (either default or specific)? Show that the Sorgenfrey line does not have a countable basis. When I analyze my variables in two separate (univariate) logistic regression models, I get the following:  Predictor 1:    B= 1.049,    SE=.352,    Exp(B)=2.85,    95% CI=(1.43, 5.69),    p=.003    Constant:    B=-0.434,    SE=.217,    Exp(B)=0.65,                            p=.046  Predictor 2:    B= 1.379,    SE=.386,    Exp(B)=3.97,    95% CI=(1.86, 8.47),    p&lt;.001    Constant:    B=-0.447,    SE=.205,    Exp(B)=0.64,                            p=.029   but when I enter them into a single multiple logistic regression model, I get:  Predictor 1:    B= 0.556,    SE=.406,    Exp(B)=1.74,    95% CI=(0.79, 3.86),    p=.171 Predictor 2:    B= 1.094,    SE=.436,    Exp(B)=2.99,    95% CI=(1.27, 7.02),    p=.012    Constant:    B=-0.574,    SE=.227,    Exp(B)=0.56,                            p=.012   Both predictors are dichotomous (categorical). I have checked for multicollinearity.  I am not sure if I have given enough info, but I cannot understand why predictor 1 has gone from being significant to non-significant and why the odds ratios are so different in the multiple regression model. Can anyone provide a basic explanation of what is going on? Expectation of the Maximum of iid Gumbel Variables I keep reading in economics journals about a particular result used in random utility models. One version of the result is: if $\epsilon_i \sim_{iid}, $ Gumbel($\mu, 1), \forall i$, then:   $$E[\max_i(\delta_i + \epsilon_i)] = \mu + \gamma + \ln\left(\sum_i \exp\left\{\delta_i \right\} \right),  $$  where $\gamma \approx 0.52277$ is the Euler-Mascheroni constant.  I've checked that this makes sense using R, and it does.  The CDF for the Gumbel$(\mu, 1)$ distribution is:  $$G(\epsilon_i) = \exp(-\exp(-(\epsilon_i - \mu)))$$  I'm trying to find a proof of this and I've had no success.  I've tried to prove it myself but I can't get past a particular step.   Can anyone point me to a proof of this?  If not, maybe I can post my attempted proof up to where I get stuck. Change \textwidth after first page in scrlttr2 I am looking for a way to add some string to the beginning of every line (same string for every line). Not something customizable but rather something that will be easy to remember and available on every POSIX-compliant platform (and every shell as well). Is it possible to install an earlier version of Skype on Windows 8? SQL Server replication subscriptions marked as inactive Is there any way to force SQL Server NOT to mark subscriptions as inactive, ever?  It happens sporadically when there are connection issues and I don't want to have to reinitialize the subscription every time.  Note, I'm not talking about the subscriptions being marked as expired...just as inactive.  We have retention set to 0 (infinite). I'd rather have a big distribution database than have to reinitialize a 15GB database to a subscriber over the WAN. Anyway, the real issue here is that they are getting marked as inactive within just a few hours sometimes...or sometimes a few days...it seems sporadic. Is there any way to tell WHY SQL decided in its wisdom to mark them inactive? I have Ubuntu on my MacBook 2007 with a Spanish keyboard, on MacOS the @-sign was typed using Alt (Option) + 2, but on Ubuntu I have no idea. I have to copy and paste it every time. How do I type an at sign? I was trying to prove this inequality using induction, but couldn't do.  Question: Suppose $a$ and $b$ are real numbers with $0 &lt; b &lt; a$. Prove that if $n$ is a positive integer, then:  $$a^n-b^n \leq na^{n-1}(a-b)$$ How do I have a camera follow my object in Unity? I have an object that automatically moves by itself and I want the main camera to automatically follow it.(Like in games such as geometry dash and jetpack joyride)  This is the code for the automatic moving object in case it is needed:  using UnityEngine; using System.Collections; public class automove : MonoBehaviour { public static int movespeed = 5; public Vector3 userDirection = Vector3.right;    public void Update() {     transform.Translate(userDirection * movespeed * Time.deltaTime);  } }   So does anyone know any good scripts I could add to the main camera to follow this object that automatically moves? Thanks in advance! Why are downloads destructive in the Star Trek universe? There are numerous places in various Star Trek episodes where references are made to the fact that "downloads" are a destructive operation: you download something from one place, and unless you explicitly made a copy beforehand, what you downloaded is gone from its original storage location and instead appears in some other location. Hence was born the term "copy and download".  There are of course out-of-universe, plot reasons for things happening that way which are exploited from time to time. There is also the out-of-universe explanation of how large media companies feel about people downloading copyrighted works, and a possible out-of-universe explanation that at the time, people in general may not have been intimitely familiar with the implications of the term "download", so they added "copy" to help describe what happens to at least portions of their viewership.  But is there a plausible in-universe explanation for why downloads would be destructive by default, and a specific copying operation would have to be made beforehand so as to not delete the data from its original location?  As for specific examples, these are two that I can think of right now (I'm quite sure there are more):   VOY "Twisted" (S2E06), right near the end, stated by Torres: "Our entire database has been copied and downloaded into somebody else's system." VOY "Message In A Bottle" (S4E14), a little more than five minutes into the episode, the EMH is "downloaded into the transceiver array" for transmission across an alien sensor network to the Prometheus when the latter is in the alpha quadrant, and some 20 minutes into the episode it's established that the EMH is not available in the sickbay and will return later. Real valued analytic function defined on a connected set is constant If Cooper wanted to stay, why did he send the NASA Coordinates? In Interstellar, after Cooper arrives in the Tesseract, Cooper tries to send the message STAY to his daughter in the past so she won't let him go on the mission. If that was his intention then why did he send the coordinates of NASA afterwards? If he didn't send the coordinates, he wouldn't have found NASA or go into space, he would have stayed. I have a problem with derivation of some function. This is related to computing the Poincare Map of Logistic Population model with Periodic harvesting. This function is:  $\phi(t,x_0) = x_0 + \int_{0}^{t} f(s,\phi(s,x_0)) ds $,  where $\phi(t,x_0)$ is a function: $\mathbb{R}  \times \mathbb{R} \rightarrow \mathbb{R}$.  I'm trying to differentiate  equation of this function both sides  with respect to $x_{0}$. In book from this equation comes from the result of this differentiation is:  $\frac{\partial \phi}{\partial x_0} (t,x_0) = 1 + \int_0^{t} \frac{\partial f}{\partial x_0} (s, \phi(s,x_0)) \cdot \frac{\partial \phi}{\partial x_0} (s, x_0)ds$.  I can't understand expression under the integral. In this book it is explained that it is clear from chain rule.Could someone explain to me this transformation in easy way? I will be grateful for your help Best regards Install suspension forks on non-suspension bike I have a 1990s Saracen PowerTrax. Is there a way I can install a suspension fork, or will that muck up the bike somehow? I'm actually interested in ways to raise the bars with respect to the seat, so that wouldn't be a problem. My toes rubbing in the wheel, in the other hand, would definitely be a problem. What are the best ways to improve a site's position in Google? What are the best ways to improve a site's position in the Google search results? Is there any way to erase personal information from my phone that has already been stolen? My phone has been stolen. I do not want the thief to have access to any of my personal information such as calls, texts, and information saved in the phones contents. Is there any possible way I can get this information erased from my phone? What Nikon lens to buy to make photos of birds? Non-Root Package Managers From my research, I seem to notice that all package managers insist on being used as a privileged user and must be installed into /.  Typically, what I like to do is create a throwaway account, compile some software, and install to $HOME for that account. I can try a variety of setups and then when I'm done, just destroy the account.  However, compiling software becomes tedious.  My experience is really just limited to yum, but I don't understand why I wouldn't be able to drop a repo file into ~/etc/yum.repos.d and have yum install everything into a home account.  Is there any reason why package managers must be used as a privleged user to install software?