Picard Iteration: Convergence of system I want to prove that, for any $t$, a solution exists in the interval $[0,T]$, when $T&gt;0$.  $x'(t)=A(t)x(t)$  My question is quite similar to this one , but with one small difference. My $A(t)$ is a function of $t$, whereas in the other question $A$ seems to be constant.  Picard iteration: $x^{[k+1]}=x_0+\int^t_0A(τ)x^{[k]}(τ) dτ$. Where $x_0$ is arbitrary.  I want tor prove that $x^{[k+1]}$ converges on the given interval and that it satisfies the equation below.  $|x^{[k+1]}(t)-x^{[k]}(t)|\le |x_0| $$\frac{π^{k+1}(t)}{(k+1)!}$ where $π(t)=\int^t_0||A(τ)||dτ$.  I'm not sure how to account for the variant function $A(t)$ in my proof, so I was hoping someone could show me what it look like
First Order Time-Variant System : Picard Method I looking at a time variant first order system. I am trying to prove that a sequence of functions $x^{[k]}(t)$ generated using Picard iterations converges uniformly on some interval $[0,T]$.  Given:  $\dot{x}(t)=A(t)x(t)$ where $x(0)=x_0$  $x(t)=x∈R^n$ where $A(t)$ is a $n×n$ matrix  Assume there exists $M$ such that $||A(t)||\le M$ for all $t\ge 0$ and that $A(t)$ is a piecewise continuous function. Show that, for any $T &gt; 0$, a solution exists in the interval $[0,T]$.  I know that $x(t)=x_0+\int^t_0A(τ)x(τ) dτ$ must be satisfied if the solution does indeed exist.  I am supposed to use Picard iterations to prove this, which I am not familiar with yet. But I know that they are of the form: $x^{[k+1]}=x_0+\int^t_0A(τ)x^{[k]}(τ) dτ$. Where $x^{[0]}(t)=x_0$  Can I use the standard Picard proof to prove this or do I need to start somewhere else?  I was also asked to show that this method satisfies  $|x^{[k+1]}(t)-x^{[k]}(t)|\le |x_0| $$\frac{π^{k+1}(t)}{(k+1)!}$ where $π(t)=\int^t_0||A(τ)||dτ$.  I have no idea where to start with this, but hopefully it will become clearer once I finish the first part.
Why does Linux require that a user be root/using sudo/specifically authorized per mount in order to mount something? It seems like the decision as to whether to allow a user to mount something should be based on their access rights to the source volume/network share and to the mount point. A couple of uses for non-root mounting are mounting file-system images to a user owned direction and mounting a network share to a user owned directory. It seems like if the user has control over both sides of the mount equation everything should be cool.  Clarification of access restriction:  I feel I should be able to mount anything that the user otherwise would have access to to a mount-point that the user is the owner of.  For instance, on my computer /dev/sda1 is owned by user root and group disk with permissions brw-rw----. Therefore non-root users can't mess with /dev/sda1 and clearly mount shouldn't allow them to mount it. However if the user owns /home/my_user/my_imagefile.img and mount point /home/my_user/my_image/ why shouldn't they be able to mount that image file on that mount point with:  mount /home/my_user/my_imagefile.img /home/my_user/my_image/ -o loop   As kormac pointed out there is a suid problem. So some restrictions would have to be added to prevent suid from being a problem as well as potentially some other issues. Perhaps one way to do this would be to make the OS treat all files as belonging to the user that did the mounting. However for simple read/write/execute, I don't see why this would be a problem.  Use case:  I have an account in a lab where my home space is restricted to 8GB. This is tiny and very very annoying. I would like to mount an nfs volume from my personal server to essentially increase the amount of room that I have. However, because Linux does not allow such things I'm stuck with scp'ing files back and forth to stay under the 8GB limit. Why do the Localization of a Ring This question may be a bit vague, but neverthless, i would like to see an answer. Wikipedia tells me that:   In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring.   It is clear that for integral domains, we have the Field of Fractions, and we work on it. It's obvious that a Field has multiplicative inverses. Now if we consider any arbitrary ring $R$, by the definition of localization, it means that we are adding multiplicative inverses to $R$ thereby wanting $R$ to be a division ring or a field. My question, why is it so important to look at this concept of Localization. Or how would the theory look like if we never had the concept of Localization. My public website name and AD domain name are the same. How can I get to my external website from inside my network? I need help with this exercise. What I've done so far is prove the exercise when $n=1$. So:  $$n=1$$ $$1!\ge2^{(1-1)}$$ $$1\ge2^0$$ $$1\ge1$$ Which is true  Therefore, now that I assume that the assumption is correct, I want to prove that with $n+1$, it will also be true. So, what I've done now is:  $$P(n) \implies P(n+1) $$ $$n!+n+1\ge 2^{(n-1)}...$$ And my  problem is that I do not know how to add the $n+1$ in the right side of the equation, therefore I'm not been able to finish the exercise.  Thank you so much for your help. If something's not very clear, please let me know. I'll try to be clearer next time :) When do you use at the office? And when do you use in the office? What's the difference between the phrases? How can I write multiple lines in a subscript? Is there a way to force a line break inside a math mode text?  My example:  \[   \underbrace{....}_\text{Some long text that should be multiline} \]   Trying  \text{Some long text that\\ should be multiline}   Didn't do the trick. Omitting "and" in a sentence He called her, emailed her, texted her, tweeted her—all to no use.   Strictly speaking, I would need to write texted her and tweeted her, but I'm omitting and to convey a rhythm and sense of urgency. What do you call this kind of construction, and is it accepted grammar (at least for creative writing, if not in an academic context)? Proving that without the axiom of choice there is a set with an accumulation point that isn'tthe limit of a convergeant sequence I've been reading the simple parts of Thomas Jech's book on the Axiom of Choice and came across the following proof on page 141. The proof assumes a mathematical model without the axiom of choice.     Lemma. $X$ is D-finite if and only if $X$ does not have a countable subset.       Theorem. There is a set $S$ of real numbers and a real number $a \not \in S$ such that a is in the closure of $S$ but there is no sequence $(x_n)^{\infty}_{n=0} \subset S$ such that $(x_n)^{\infty}_{n=0} \longrightarrow a$.       Proof. Let $D$ be a D-finite infinite set of reals. We claim that the set $D$ must have an accumulation point. To prove this claim suppose that all $d \in D$ are not accumulation points, then let $\{I_n:n=0,1,...\}$ be a fixed enumeration of open intervals with rational endpoints. Assign the least $n$ such that $I_n \cap D={d}$ to each $d \in D$. This makes $D$ countable and by the above lemma gives us our contradiction. So $D$ has an accumulation point $a$. Let $S=D \setminus\{a\}$ and since $S$ is D-finite, every convergent sequence in $S$ is eventually constant.   Could somebody explain this proof? Why is the last sentence true/ Why does this mean every convergent sequence is eventually constant?  I think the following theorem is important     Theorem. There is a model of ZF which has an infinite set of real numbers without a countable subset Duplicate:                      The number of residents has grown.    The number of residents have grown.      Which of the above sentences is grammatical? Since has should stick with singular nouns and have with plural ones, I guess (2) should be correct, but I'm not sure. How do I disable the volume control overlay? When changing the volume on a Mac, you get a big ugly gray bezel showing the change in volume. This is super annoying when (say) watching a movie. Is there any way to disable this? ViewExpiredException in Firefox and IE after migrating to Jboss 7.0 AS Where are the outliner toggles in 2.8? How many arrangement of the letters in MISSISSIPPI have at least 2 adjacent S's?  I was thinking that I can glue two of the S's together, so there will be 9 letters plus the special letter SS, and the number of arrangements will be $10!/(2!2!4!)$, this is not the correct answer, but what is wrong with my reasoning.  (the correct answer is $11!/(4!4!2!)-7!/(4!2!)\binom{8}{4}$, I understand the solution) Pokemon Go Supported Devices Can't connect to Android device My Android device won't connect. I've never got MTP to work, I've only ever got it to connect by USB mass storage, which was how it was working most recently. But now, since 13.04, it seems that option has been taken away entirely.  I need to be able to connect by USB to transfer files.  When I plug in the device I get an error that says Unable to open MTP device '[usb:002,003]':    After some web searching, I found that I might need a program called go-mtpfs, so I tried that, and it failed:  $ go-mtpfs /media/MyAndroid 2013/07/16 16:41:43 compiled against libmtp 1.1.5 Device 0 (VID=04e8 and PID=685e) is a Samsung Galaxy Y. 2013/07/16 16:41:43 found device Samsung: Galaxy Y (04e8:685e) @ bus 2, dev 3 :  LIBMTP PANIC: Unable to find interface &amp; endpoints of device 2013/07/16 16:41:43 rdev.open failed: open: open returned nil   How do I get my USB device to just connect, automatically, like it used to? How to create a bootable Ubuntu USB flash drive from terminal? Is there any way to create a bootable Ubuntu USB flash drive from the terminal without using any third-party applications like , , ,  etc.  I tried to create a bootable Ubuntu flash drive with dd method,  sudo umount /dev/sdb sudo dd if=/path/to/ubuntu.iso of=/dev/sdb bs=1M   It creates files on the USB disk, but when I try to boot the USB disk it shows an Operating System Not Found error. It's time for drag and drop! I often find myself wishing SO supported drag and drop. For example I drag a tag to my favs and it would add it, or when searching by tag I would like to drag the tag to the search box and get [asp.net] added in the search box.   Does it make sense to add this feature or is it just too much work for so little use? I've seen a couple questions around here like , which mentions:     Unfortunately that same browser won't allow me to test HTTP PUT, DELETE, and to a certain degree even HTTP POST.   I've also heard that browsers support only GET and POST, from some other sources like:        However, a few quick tests in Firefox show that sending PUT and DELETE requests works as expected -- the XMLHttpRequest completes successfully, and the request shows up in the server logs with the right method. Is there some aspect to this I'm missing, such as cross-browser compatibility or non-obvious limitations? Apparently, .  How the heck did that happen? Surprisingly many network interfaces on macOS: are these legitimate? ifconfig reveals a surprisingly long list of network interfaces on my MacBook, which is running macOS High Sierra (10.13.6): lo0, gif0, stf0, XHC20, en0, p2p0, awdl0, en1, en2, bridge0, utun0.  There is an F5 VPN client installed, but I've rebooted the MacBook since the last VPN connection.  Are all of these network interfaces legitimate or is this something I should worry about?