Let us consider a bump function $\phi: \mathbb{R} \longrightarrow \mathbb{R}$, smooth, with compact support. The most common examples are built from the function $$ \psi(x) = \begin{cases} \exp ( \frac {1}{x^2 - 1}) &amp; \lvert x \rvert &lt; 1 \\ 0 &amp; \text{otherwise} \end{cases}.$$ Although this function behaves very well, its derivatives become arbitrarily large inside the unit ball. I wonder --- do there exist bump functions with "small" derivatives?     Stated more precisely, does there exist a bump function $\phi \in C^\infty_c(\mathbb{R})$ and some $M &lt; \infty$ such that $\phi^{(k)} \leq M$ for all $k$th derivatives?   Intuitively, I feel the answer is no, as a sort of cost of vanishing entirely.
Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ having the following properties?   $f(x) = 0$ for all $x \le 0$. $f(x) = 1$ for all $x \ge 1$. For $0 &lt; x &lt; 1$, $f$ is strictly increasing. $f$ is everywhere $C^\infty$. The sequence of $L^\infty$ norms $\langle \left\lVert f \right\rVert_\infty, \left\lVert f' \right\rVert_\infty, \left\lVert f'' \right\rVert_\infty, \dots \rangle$ is bounded.   If we impose only the first four conditions, there is : for $0 &lt; x &lt; 1$, define $f(x)$ by $$ f(x) = \frac{e^{-1/x}}{e^{-1/x} + e^{-1/(1-x)}} = \frac{1}{1 + e^{1/x - 1/(1-x)}} $$ However, the derivatives of this function appear to grow quite rapidly. (I'm not sure how to verify this, but it seems at least exponential to me.)  If such a function does not exist, what is the smallest order of asymptotic growth that the sequence $\langle \left\lVert f \right\rVert_\infty, \left\lVert f' \right\rVert_\infty, \left\lVert f'' \right\rVert_\infty, \dots \rangle$ can have?
Tracking Object Position - Firing on a Trajectory Let $A$ be a real square matrix en let $I$ be the identity matrix with the same size. Let $n&gt;0$ be the smallest natural number for which there exist  $a_0,a_1,...a_{n-1} \in {R}$ such that $A^n+a_{n-1}A^{n-1}+....+a_1A+a_0I=O$.  Show that A is invertible if and only if $a_0\neq0 $.  So i need to proof    $A$ invertible $\rightarrow$$a_0 \neq 0$ $a_0 \neq0 \rightarrow$ $A$ invertible    I don't really know where to start with 1, and with 2 I tried to prove it with a contradiction but I don't think it's right either. I'm looking into the technique of 'hanging' meat, and whether there are 'home' applications that would be safe and have a beneficial effect.  Now, there are a lot of (possible) synonyms that muddle the waters (for me). There's hanging, dry-aging, faisandage (for game birds), and more. What I'm looking for is the 1-3 day (near) room-temperature storage of meat and the spoiling / enzyme process that will cause a flavor change. Feel free to help me along by clarifying the answer.  Basically:   Is there any (good) use for this technique / process for supermarket / butcher cuts of meat? What should be taken in consideration for safe execution? $$|a\sin x+b \cos x|\leq \sqrt{a^2+b^2}$$   I have tried: $$|a\sin x+b \cos x|\leq |a+b|\leq \sqrt{a^2+b^2}$$  enough to prove: $$|a+b|\leq \sqrt{a^2+b^2}$$  But I can find how to continue from here I have a few questions regarding CAN bus load  Is there any derived equation/formula to calculate the bus load?  I need transmit below frames(Assume Standard frame format) at baud rate 500 kbps on the bus 5ms  ----- 50 frames. 10ms ----- 10 frames. 100ms ---- 30 frmaes. Event messages ---- 10 frames.   How to calculate bus load, does anyone knows kindly illustrate the same?  What is the maximum bus load on CAN bus? I'm trying to get through my first SalesForce mod and it's definitely been a rocky path. I have one task I'm trying to do, but this section is holding me back.  I have two custom fields.  One a pick list and the other a text box. I have added them to the Approve/Reject Approval Request section in the Opportunity page. I added the fields through the Select Fields to Display on Approval Page Layout and they DO show up on the Approve/Reject Approval Request section. I added them in like ().  However, just the label for the fields show, the field itself is nowhere to be found and can't be used.    You can see and even use the fields when you open the Opportunity page itself, but when you go to use the Approve/Reject, you can't do anything with them or see any values.  I was never a SalesForce user and got this tossed in my lap and really don't know my way around yet. I appreciate any help on this!!  Thanks!!  Kurt Why not make it mandatory to leave a reason or at least a remark of some sort whenever down voting?  You see more and more often questions being down voted while many times they are simply perfectly legitimate, as questions/as people being unable to achieve some desirable result. I am not talking about "Hey, please do this and that for me, thanks :D " posts.  I would like to know what reasons there are for not making it mandatory to leave a reason for a down vote. Keras CNN with low/constant accuracies n-dimensional vector space isomorphic to the n-th product of a field as abelian groups Let $V$ be a finite dimensional vector space (of dimension n) over a field $F$. I need to show that $V$ is isomorphic to $F^n$ as abelian groups. However, I don't really understand what does "isomorphic as abelian groups" mean, my (poor) attempt to solve this was:  Let $f:V\to F^n$, $f((v_1,v_2,\ldots,v_n)):=v_1+\ldots+v_n$, clearly $f$ is a group homomorphism, but it is not biyective since, for example, in the case where $dim(V)=2$ and $F=\mathbb{R}$ we have $f((2,-2))=0=f((3,-3))$, I've also tried it by defining $f$ to be the product of a vector's entries but obviously it didn't work so I'm stuck!   I think this should be an easy problem but since I don't quite understand it, it's giving me problems. Thank you for your help. What physical properties of materials trigger the capacitive touch screen? I have watched some youtube videos about capacitive touch screen      The videos show that a conductor(finger), in which charges can be induced, can be used to trigger the capacitive touch screen. However, I tried to wear a plastic glove to hold a metal rod to touch the screen and no detection is triggered.  I really want to know the physics theory about the capacitive sensing.  Can someone tell me what physical properties or shape trigger the capacitive touch screen?  Is there any examples that something WITHOUT conducting to human body can trigger the screen?  (P.S. I have read other posts about capacitive touch screen here and do not find a good answer so this is not a duplicate) ant deploy error - Maximum size of request reached I used to deploy metadata using ant migration tool successfully but from last night I am getting the below error:  John@John:~/mm_workspace/TEST_Project/deploy$ ant deployUnpackaged  Buildfile: /home/john/mm_workspace/TEST_Project/deploy/build.xml  deployUnpackaged:  BUILD FAILED /home/john/mm_workspace/TEST_Project/deploy/build.xml:72: Failed to submit request to https://cs19.salesforce.com/services/Soap/m/35.0/00D293456000Qaf for user john@test.com.dev. Error: {http://schemas.xmlsoap.org/soap/envelope/}Client - Maximum size of request reached. Maximum size of request is 52428800 bytes.  Total time: 2 minutes 10 seconds   I think this is because of too much metadata that I am trying to deploy. For now, I am deploying classes and then VF pages and so on but I am looking for an option to deploy everything in one go without running into the above error.  I am sure that people are working on more complex projects than mine with lots of metadata and I am wondering how are they deploying with out hitting this error?  Thanks Testing random variate generation algorithms Which methods are used for testing random variate generation algorithms? How to connect to Oracle as "SYS" from SQL*Plus in Java Is there a technical reason Android update story is so bad? Is there a technical reason why Android phones general lag a lot behind the stable released version of Android?    This applies to both new phones, for example a recent trip to the phone shop everything was 4.0 to 4.2.1, KitKat could not be seen. And also old phones, my girlfriend's iPhone 4 had an OTA update to the latest OS, this is a July 2010 phone, whilst a July 2010 Android is basically a paperweight.    Presumably Android itself doesn't concern itself with the actual hardware, and thus talks to the OS via abstractions? I guess then the handset maker just provides the drivers for the specific hardware, thus I can't see the issue with pushing OTA updates if the ABI remains stable (my phone doesn't grow new hardware features). Landscape topography from grayscale image How do I pass variables and data from PHP to JavaScript? Sum of points and rank according to points Bash is not finding a program even though it's on my path I have a program on my path.  The program runs when executed with a full path specified.  But the program cannot be found when I run it with just its name.    Essentially, I want to understand how the below output is possible, and how to fix it so that my program can actually be found without a full path specified:  root:/usr/local/bin# ./siege **************************************************** siege: could not open /usr/local/bin/etc/siegerc run 'siege.config' to generate a new .siegerc file **************************************************** root:/usr/local/bin# echo $PATH /usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin:/usr/games root:/usr/local/bin# siege bash: /usr/bin/siege: No such file or directory root:/usr/local/bin# wtf!?!?   I'm on Ubuntu 12.04 using bash.  Also please note the warning output from siege is not relevant for the purposes of this question, as I am only interested in whether or not the program can be found and invoked. What's the keyboard shortcut for keyboard layout switch in Mojave? Shift + Control + Space doesn't seem to work anymore. Apt-get 403 Forbidden, but accessible in the browser