Suppose   $E=\{f\in C^1[0,1]$:  $f(0)=0$,$f(1)=1\}$.   Find the maximal $u\in \mathbb{R}$,such that forall $f\in E$,   \begin{align*} \int_0^1|f'(x)-f(x)|dx \geqslant  u. \end{align*}   I can prove that $u\geqslant \frac1e$,since \begin{align*} \frac1e=e^{-x}f(x)\Big|_0^1=\int_0^1e^{-x}(f'(x)-f(x))dx\leqslant \int_0^1|f'(x)-f(x)|dx . \end{align*} But I don't know how to find the  maximal $u$.
Let $C$ be the class of all real-valued continuously differentiable functions $f$ on the interval $[0,1]$ with $f(0)=0$ and $f(1)=1$. How to show that $$\inf_{f\in C}\int_0^1|f'(x)-f(x)|dx=\frac{1}{e}?$$  I have been able to show that $1/e$ is a lower bound. Indeed, $$\begin{align*} \int_0^1|f'(x)-f(x)|dx &amp;= \int_0^1|f'(x)e^{-x}-f(x)e^{-x}|e^xdx \\ &amp;\geq \int_0^1\left(f'(x)e^{-x}-f(x)e^{-x}\right) dx \\ &amp;= \int_0^1 \frac{d\left(f(x)e^{-x}\right)}{dx}dx  \\ &amp;= f(1)e^{-1}-f(0)e^{0}\\ &amp;=\frac{1}{e}. \end{align*}$$ But how to show this is the infimum? Is there a function $f\in C$ such that we get $\int_0^1|f'(x)-f(x)|dx=1/e$?
Is it ethical to use a recommendation letter from a deceased professor? She was a really nice professor and she wrote me a good recommendation letter that resulted in a lot of good results with applications. Recently, this professor died from brain cancer, which was a fast and shocking death for me.  Tomorrow is the deadline for a grant. What should I do? Is it professional to use letters from the deceased? Can we access class properties dynamically with Object class? Is it possible to access Apex class properties dynamically? While we can create Object's, and assign any other class to an Object, I'm not sure how I can access the properties of these Object's. Can anyone provide some documentation or references to code which would allow me to do this?   Example:  public class XYZ {    public string s1 { get; set; }    public string s2 { get; set; } }  XYZ testXyzObj = new XYZ(); Object obj = testXyzObj;  // Method does not exist or incorrect signature: [Object].get(String) String s1 = obj.get('s1'); Why does find -mtime +1 only return files older than 2 days? I'm struggling to wrap my mind around why the find interprets file modification times the way it does.  Specifically, I don't understand why the -mtime +1 doesn't show files less than 48 hours old.   As an example test I created three test files with different modified dates:   [root@foobox findtest]# ls -l total 0 -rw-r--r-- 1 root root 0 Sep 25 08:44 foo1 -rw-r--r-- 1 root root 0 Sep 24 08:14 foo2 -rw-r--r-- 1 root root 0 Sep 23 08:14 foo3   I then ran find with the -mtime +1 switch and got the following output:  [root@foobox findtest]# find -mtime +1 ./foo3   I then ran find with the -mmin +1440 and got the following output:   [root@foobox findtest]# find -mmin +1440 ./foo3 ./foo2   As per the man page for find, I understand that this is expected behavior:   -mtime n         File’s  data was last modified n*24 hours ago.  See the comments         for -atime to understand how rounding affects the interpretation         of file modification times.   -atime n        File  was  last  accessed n*24 hours ago.  When find figures out        how many 24-hour periods ago the file  was  last  accessed,  any        fractional part is ignored, so to match -atime +1, a file has to        have been accessed at least two days ago.   This still doesn't make sense to me though.  So if a file is 1 day, 23 hours, 59 minutes, and 59 seconds old, find -mtime +1 ignores all that and just treats it like it's 1 day, 0 hours, 0 minutes, and 0 seconds old?  In which case, it's not technically older that 1 day and ignored?   Does... not... compute. Is there a symmetry associated to the conservation of information? Conservation of information seems to be a deep physical principle. For instance, Unitarity is a key concept in Quantum Mechanics and Quantum Field Theory.  We may wonder if there is an underlying symmetry, in some space, which may explain this conservation of information. Moving linux install to a new computer I know that it can, in some circumstances, be difficult to move a Windows installation from one computer to another (physically move the hard drive), but how does that work on linux? Aren't most of the driver modules loaded at bootup? So theoretically would it be that much of a hassle?  Obviously xorg configs would change and proprietary ATI drivers and such would have to be recompiled (maybe?). Is there more to it than I'm thinking of?  Assume the 2 computers are from the same era, ie both i7s but slightly different hardware.  Update: Thanks for the answers. This is mostly for my own curiosity. I have my linux system up and running at work, but eventually I'd like to move to a computer that I can get dual video cards into so I can run more than 2 monitors. But not any time soon What is the reason behind the multiple Spider-Man reboots? Extracting phone number from string Does the cross-entropy cost make sense in the context of regression? Does the cross-entropy cost make sense in the context of regression (as opposed to classification)? If so, could you give a toy example through TensorFlow? If not, why not?   I was reading about cross-entropy in  by Michael Nielsen and it seems like something that could naturally be used for regression as well as classification, but I don't understand how you'd apply it efficiently in TensorFlow since the loss functions take logits (which I don't really understand either) and they're listed under classification For privacy and performance reasons, I'd like to prevent my computer from ever communicating with certain Internet hosts. I'd like to do this at the OS level, not through web browser plug-ins (not broad enough), or even through my home router (my laptop travels). I'd prefer to use built-in tools.  I'm running Windows 7 Professional 64-bit. I'm all too familiar with the Windows hosts file - and its limitations. There seems to be some confusion over whether hosts supports wildcards, but my own tests confirm that it does not.  I have experimented with Windows Firewall and the IP Security Policy MMC snap-in. So far as I can tell, both require numeric addresses. I don't want to block specific IP addresses, or ranges of addresses, in part because IP-address-to-hostname mappings can and do change.  Is there any tool in Windows 7 Professional with which I can block (or route to 0.0.0.0) communication with Internet hosts by hostname mask? If not, are there any good, free, third-party tools? What are XML namespaces for? This is something that I always find a bit hard to explain to others:  Why do XML namespaces exist?  When should we use them and when should we not? What are the common pitfalls when working with namespaces in XML?   Also, how do they relate to XML schemas? Should XSD schemas always be associated with a namespace? Let $n$ be an integer such that if $d | n$ then $d + 1 | n + 1.$ Show that $n$ is a prime number Let $n$ be an integer such that for any integer $d$, if $d | n$ then $d + 1 | n + 1$.   Show that $n$ is a prime number or equal to $1$  My workout...  Suppose $n = xy$ for some postive integers $x$ and $y. x|n$, and hence we assume without loss of generality $x + 1|n + 1.$   Then $x|xy + 1$. However, $xy + x$ is divisible by $x$.  This implies $x − 1$ is divisible by $x$, which is an impossibility if $x$ is not equal to $1$.  Therefore whenever $n$ is decomposed into two factors, one must turn out to be $1$. Hence $n$ must be a prime.  I am looking for any other method which solves this quickly and more quickly... How can I upgrade to Java 1.8 on an Amazon Linux Server? I tried sudo yum update but it just keeps java "1.7.0_75". I need 1.8 for it to work with another application but can't figure out how to upgrade it. Do I need to manually install it somehow? There's not much information on this on the internet as far as I can see. Specs:   java version "1.7.0_75" OpenJDK Runtime Environment (amzn-2.5.4.0.53.amzn1-x86_64 u75-b13) OpenJDK 64-Bit Server VM (build 24.75-b04, mixed mode)   When I try update now:     [ec2-________]$ sudo yum update Loaded plugins: priorities, update-motd, upgrade-helper amzn-main/latest                                         | 2.1 kB     00:00 amzn-updates/latest                                      | 2.3 kB     00:00 No packages marked for update   Is there anything else I need to do? Thanks. tar compression without directory structure Proving a direct product of groups is a group I am trying to prove the following and am looking toward the math.stackexchange community to comment on whether I am on the right track or not. Thank you in advance.    Let $n \geq 1$ be an integer. Suppose $\{G_1,...,G_n\}$ is a finite set of groups. Define the set $G = G_1 \times G_2 \times ... \times G_n$ of ordered n-tuples $(g_1,g_2,...,g_n)$ where $g_i \in G_i$. Show that the operation $\bullet$: $G \times G \rightarrow G$ given by $(g_1,g_2,...,g_n) \bullet (g'_1,g'_2,...,g'_n) = (g_1g'_1,g_2g'_2,...,g_ng'_n)$ makes $G$ into a group.    Proof of associativity: $((g_1,g_2,...,g_n) \bullet (g'_1,g'_2,...g'_n)) \bullet (g''_1,g''_2,...,g''_n)$ = $(g_1g'_1,g_2g'_2,...,g_ng'_n) \bullet (g''_1,g''_2,...,g''_n) $ = $(g_1g'_1g''_1,g_2g'_2g''_2,...,g_ng'_ng''_n)$ = $(g_1,g_2,...,g_n) \bullet (g'_1g''_1,g'_2g''_2,...,g'_ng''_n) = (g_1,g_2,...,g_n) \bullet ((g'_1,g'_2,...,g'_n) \bullet (g''_1,g''_2,...,g''_n))$  \  Proof of the existence of an identity For each $G_i \in G$, take its identity element $e_i$ and multiply it as follows: $(g_1,g_2,...,g_n) \bullet (e_1,e_2,...,e_3)$ = $(g_1,g_2,...,g_n)$.  \  Proof of the existence of an inverse: Since each $G_i \in G$ is a group, simply takes its multiplicative inverse $g_i^{-1}$ and multiply it: $(g_1,g_2,...,g_n) \bullet (g_1^{-1},g_2^{-1},...,g_n^{-1}) = (e_1,e_2,...,e_n)$. $$\overline{z}=z^n$$ where $n\in \mathbb{N}$   So I have started with moving the polar representation as the expression is in the n-th power  $$rcis(-\theta)=r^ncis(n\theta)$$   I can not multiply both sides in $r^{(-n)}$ as $n\in \mathbb{N}$ how should I continue? Text won't appear in interface? There is one problem, it seems that text won't appear on my program. The three images show the errors and also my laptop specs (I believe I just have very small amount of laptop power that makes Blender glitch).  Everything else works fine (zooming, panning, rotating, etc.) and the buttons are still there and do what they're supposed to do, but the text just won't appear.  Might anybody know how to fix this?        The drive update didn't help either. I know a way around it though, does anybody know a page which shows screenshots of more or less the buttons on blender Unable to install any updates through update manager/apt-get upgrade How do I install iTunes on Ubuntu? Different font color for question status => closed <= When I stared at  () and I even thought [question refined] as offical status of the post.    &lt;somebody  it now, So, I've edited this question to get same effect>  How about official status [closed] as   different font color or  different font size or put the text [closed] outside the question link   Or   put a special tag as suggested in the    to clarify from those? Don't include SE site in SE search filter? I couldn't find this question here, but I would like to be able to search all sites in the SE network except some sites. In my case, I want to remove the thousands of results from SO on a popular project for devs so I can see the handful of results on other sites from admins and users.  The current form only allows all sites or listed sites.  The answer linked does not tell me how to use the information in the answer. It is a poor answer for an average user.  As @Laurel describes in the comment below, the answer at  no longer works.