Show that $m^3\le2^m$ for $m\ge10$  My try:  Base case is true for $m=10$  Inductive Hypothesis: Assume $P(k)$ is true $\implies k^3\le2^k$  Now showing that $P(k+1)$ is true  $(k+1)^3\le2^{k+1}$  $\implies (k+1)^3\le k^3+1+3k^2+3k$  $\le 2^{k+1}+3k^2+3k+1($ from inductive hypothesis$)$  From here I could not proceed.  Can anyone explain how to proceed from here.
Problem  For what natural numbers is $n^3 &lt; 2^n$?  Attempt @ Solution   For $n=1$, $1 &lt; 2$ Suppose $n^3 &lt; 2^n$ for some $n = k \ge 1$ It looks like the inequality is true for $n = 0$, $n = 1$ and $n\ge10$ But, how can I prove this through induction?
Is it possible to train a neural network without backpropagation? Many neural network books and tutorials spend a lot of time on the backpropagation algorithm, which is essentially a tool to compute the gradient.  Let's assume we are building a model with ~10K parameters / weights. Is it possible to run the optimization using some gradient free optimization algorithms?   I think computing the numerical gradient would be too slow, but how about other methods such as Nelder-Mead, Simulated Annealing or a Genetic Algorithm?  All the algorithms would suffer from local minima, why obsessed with gradient? Please provide a mobile-optimized captcha page Probability that median falls within a range? If I draw 5 samples from a distribution that is cont., what is the probability that the median of the population is between the xmin and xmax of the draw samples?  What is the formula to apply here? I have two systems installed previously Windows 7 and Ubuntu. I was trying to reinstall Ubuntu and format the Ubuntu OS partition using a tool in Windows 7.   When I have Ubuntu ready on a USB stick and restart my laptop, it prompts me with the GRUB rescue screen. If I boot from USB to install Ubuntu, it just hangs.   Anyway, I messed up everything. So my question is how can I boot Windows 7 now? Consider the sentence      E-books are on the rise, but they haven't suppressed paper books though.    This usage seems to be quite common, but when I learned English  I was taught to use "however" where I now read "though". My questions on this use of "though" are:     Is this a new trend? Is it restricted to American English? Is "though"  considered more colloquial than "however" ?    And on a syntactic level: Do you put a comma before "though"? How can I specify the default save directory for gnome-screenshot? Are there any sort of configuration options for specifying the default save location for gnome-screenshot, or is this hard-coded into the source code? It used to be ~/Desktop, which seems to have changed to ~/Pictures (in 12.04). The only possible solution I've seen is about  (as it includes time stamp information now instead of simply 'Screenshot#'), but that solution doesn't really seem ideal to me. Also,  post suggested that the last save location is remembered the next time you take a screenshot, but in my experience, this doesn't seem to be the case. And in any case, following on from that, that entry in gconf-editor doesn't even seem to accurately reflect the last location, so more than likely an entry related to an older version of gnome-screenshot. Table of contents: I'm a newbie in WordPress. I'm trying to understand how it generates the XHTML pages. I guess it's the function of do_action( 'hook-name' ) , but I'm stuck with the understanding of the instruction isset( $wp_filter['all] ) when reading the function do_action because I don't understand what is the meaning of the word all.  Is it a PHP feature or a WordPress one? How to break a Quagmire 3 cipher? Why is homology invariant under deformation retraction/homotopy equivalence? What is the order when doing $x^{y^z}$ and why? Does $x^{y^z}$ equal $x^{(y^z)}$? If so, why?   Why not simply apply the order of the operation from left to right? Meaning $x^{y^z}$ equals $(x^y)^z$?  I always get confused with this and I don't understand the underlying rule. Any help would be appreciated! Cant install via Ubuntu Software- "apt transaction returned result exit-failed" I'm using 17.04 ( Clean Install ), and as far as I remember, every try to install any sofware- it fails with that error code.  I tried sudo apt-get update &amp;&amp; apt-get upgrade and problem persists.  any idea why?   edit: Using software Updater gave this result :   edit : output of update  guy@LG-LAPTOP:~$ sudo apt-get update &amp;&amp; apt-get upgrade [sudo] password for guy:  Ign:1 http://il.archive.ubuntu.com/ubuntu zesty InRelease Ign:2 http://il.archive.ubuntu.com/ubuntu zesty-updates InRelease               Ign:3 http://il.archive.ubuntu.com/ubuntu zesty-backports InRelease             Err:4 http://il.archive.ubuntu.com/ubuntu zesty Release                           503  Service Unavailable Get:5 http://security.ubuntu.com/ubuntu zesty-security InRelease [89.2 kB]      Err:6 http://il.archive.ubuntu.com/ubuntu zesty-updates Release                   503  Service Unavailable Get:7 http://archive.canonical.com/ubuntu zesty InRelease [10.2 kB]             Err:8 http://il.archive.ubuntu.com/ubuntu zesty-backports Release                 503  Service Unavailable Get:9 http://archive.canonical.com/ubuntu zesty/partner Sources [1,416 B]       Get:10 http://archive.canonical.com/ubuntu zesty/partner amd64 Packages [1,616 B] Get:11 http://archive.canonical.com/ubuntu zesty/partner i386 Packages [1,612 B] Get:12 http://archive.canonical.com/ubuntu zesty/partner Translation-en [708 B] Get:13 http://security.ubuntu.com/ubuntu zesty-security/main amd64 DEP-11 Metadata [11.7 kB] Get:14 http://security.ubuntu.com/ubuntu zesty-security/main DEP-11 64x64 Icons [9,219 B] Get:15 http://security.ubuntu.com/ubuntu zesty-security/universe amd64 DEP-11 Metadata [14.4 kB] Get:16 http://security.ubuntu.com/ubuntu zesty-security/universe DEP-11 64x64 Icons [31.0 kB] Reading package lists... Done                                          E: The repository 'http://il.archive.ubuntu.com/ubuntu zesty Release' does no longer have a Release file. N: Updating from such a repository can't be done securely, and is therefore disabled by default. N: See apt-secure(8) manpage for repository creation and user configuration details. E: The repository 'http://il.archive.ubuntu.com/ubuntu zesty-updates Release' does no longer have a Release file. N: Updating from such a repository can't be done securely, and is therefore disabled by default. N: See apt-secure(8) manpage for repository creation and user configuration details. E: The repository 'http://il.archive.ubuntu.com/ubuntu zesty-backports Release' does no longer have a Release file. N: Updating from such a repository can't be done securely, and is therefore disabled by default. N: See apt-secure(8) manpage for repository creation and user configuration details. guy@LG-LAPTOP:~$    output of apt get  guy@LG-LAPTOP:~$ sudo apt-get install putty Reading package lists... Done Building dependency tree        Reading state information... Done The following additional packages will be installed:   putty-tools Suggested packages:   putty-doc The following NEW packages will be installed:   putty putty-tools 0 upgraded, 2 newly installed, 0 to remove and 0 not upgraded. Need to get 661 kB of archives. After this operation, 2,713 kB of additional disk space will be used. Do you want to continue? [Y/n]  Err:1 http://il.archive.ubuntu.com/ubuntu zesty/universe amd64 putty-tools amd64 0.67-3   503  Service Unavailable Err:2 http://il.archive.ubuntu.com/ubuntu zesty/universe amd64 putty amd64 0.67-3   503  Service Unavailable E: Failed to fetch http://il.archive.ubuntu.com/ubuntu/pool/universe/p/putty/putty-tools_0.67-3_amd64.deb  503  Service Unavailable E: Failed to fetch http://il.archive.ubuntu.com/ubuntu/pool/universe/p/putty/putty_0.67-3_amd64.deb  503  Service Unavailable E: Unable to fetch some archives, maybe run apt-get update or try with --fix-missing? guy@LG-LAPTOP:~$ Fortunately, I can still login with this email address that I have lost access to.  I have been searching around here for a way to update it to the current one I use.  I have found many answers.  But they all give instructions to click on something not displayed or in one case, a page that prompts me to login via another email I no longer have the password for. I need current up to date instructions on how to change my email address or add another email-based login for this same user name that I am posting this with. Why my sql query run faster than my stored procedure? I am trying to write a script in MYSQL Workbench to update my database every day using stored procedure but I find it too slow compared normal query or temporary tables. If i use temporary table the time of execution : 6 second but my stored procedure 370 second.  I really need help.        CREATE DEFINER=`datasll`@`...` PROCEDURE `product`()       BEGIN       EXPLAIN SELECT                cs.id,              cs.fk_c,              cs.so,               cs.pr,              cs.date,              cs.o_p,              cs.sh_c,              cs.x,              LEFT(cs.x,2),              cs.doc,              cs.fk_op,              cc.id_c_c,              cc.name,              cc.descrip,              left(cc.n_o,3) as bon,              cc.fk_c_br,              cc.de,              cc.m_mat,              cc.color,               cc.fk_c_s       FROM             data_pro.catas cs        LEFT OUTER JOIN             data_pro.c_co cc ON  data_pro.cs.fk_c_c = data_pro.cc._id_cc       WHERE             data_pro.cc.pap = 1       AND   data_pro.cc.stati = 'b';       END   When i use EXPLAIN this's the result :   Many Thanks for any help. Roman numerals with over- and underline This question led to a new package:      (using a variation of egreg's code, and stating this in the manual; the package isn't really new now, sorry for the delayed announcement)  I have read about "How to get Roman numerals" but asked myself how to get Roman numerals formatted as in     i.e. with fitted over- and underline (without manually using boxes and such)? I'm curious for your answer(s)! A characterisation of quadratic extensions contained in cyclic extensions of degree 4 Let $D$ be a squarefree integer. I am trying to prove that $\mathbb Q[\sqrt D]$ is contained in a Galois extension of $\mathbb Q$ with Galois group $\mathbb Z/4$ if and only if $D$ is the sum of two squares, $D = a^2 + b^2$ with $a,b\in \mathbb Q$.   The hint in the exercise in Dummit and Foote suggests considering the extension $\mathbb Q[\sqrt{s + s\sqrt D}]$ for the forward direction. However, I am unsure how to show that this extension is Galois, much less that its Galois group is cyclic of order 4.   For the reverse direction, I expressed the given Galois extension as $\mathbb Q[\sqrt{a+b\sqrt D}]$ for some $a,b\in \mathbb Q$. I showed that this extension has $\mathbb Q[\sqrt{c^2 - d^2 D}]$ as a subfield for some $c,d \in \mathbb Q$. From this, I deduced that the $\sqrt D$ and $\sqrt{c^2 - d^2 D}$ generate the same extension, so $\sqrt D = x \sqrt{c^2 - d^2 D}$ for some $x$, and hence that $D = \frac{x^2 c^2}{1 + x^2 d^2} $. It seems that there should be a way to decompose this as a sum of two squares, but I did not see how.   I did not tag this as homework as I am studying it independently. There is an alignment issue in the stackexchange.com search page .  The bottom bar is missing in the search input box.  Sample URL:   Screen shot for reference:    Firefox version: 54.0b8 (32-bit) OS Version: Windows 10 Pro  Also happens with Chrome latest version. Best way to select a subset of items in a long list One of our clients wants us to build a web application to share information about the magazines he publishes. Each edition of a magazine contains a variable amount of advertisers. These advertisers come from a central pool of advertisers. This pool contains (at the moment) about 150 advertisers. Each edition of a magazine contains a subset (about 70-80) of these advertisers.  Our client wants us to come up with a way for him to add 70-80 advertisers from this list of 150 to an edition of a magazine. It needs to be as fast as possible, as easy as possible and as intuitive as possible.  What would be an ideal solution here? I was thinking about something in the lines of the image below.   List of available advertisers. List of selected advertisers. Advertisers have a delete button. Textbox with auto-suggest for advertisers in the list on the left. &lt;Enter> removes the advertiser from the left list and adds it to the right one. Textbox gets cleared and remains focus.   My colleagues suggested drag-and-drop or just one list with a checkbox for each advertiser, but those approaches force the user to do a lot of searching and clicking. With my approach the user can just keep on typing.  Any other suggestions for this situation? How to know where shell variables and functions are set? When I type the set command in my system I've got this extract out :  __colormgr_commandlist='     create-device     create-profile     delete-device     delete-profile     device-add-profile     device-get-default-profile     device-get-profile-for-qualifier     device-inhibit     device-make-profile-default     device-set-kind     device-set-model     device-set-serial     device-set-vendor     find-device     find-device-by-property     find-profile     find-profile-by-filename     get-devices     get-devices-by-kind     get-profiles     get-sensor-reading     get-sensors     get-standard-space     profile-set-filename     profile-set-qualifier     sensor-lock     sensor-set-options     ' __grub_script_check_program=grub-script-check _backup_glob='@(#*#|*@(~|.@(bak|orig|rej|swp|dpkg*|rpm@(orig|new|save))))' _xspecs=([freeamp]="!*.@(mp3|ogg|pls|m3u)" [cdiff]="!*.@(dif?(f)|?(d)patch)?(.@([gx]z|bz2|lzma))" [bibtex]="!*.aux" [rgview]="*.@(o|so|so.!(conf|*/*)|a|[rs]pm|gif|jp?(e)g|mp3|mp?(e)g|avi|asf|ogg|class)" [oowriter]="!*.@(sxw|stw|sxg|sgl|doc?([mx])|dot?([mx])|rtf|txt|htm|html|?(f)odt|ott|odm)" [chromium-browser]="!*.@(?([xX]|[sS])[hH][tT][mM]?([lL]))" [tex]="!*.@(?(la)tex|texi|dtx|ins|ltx|dbj)" [netscape]="!*.@(?([xX]|[sS])[hH][tT][mM]?([lL]))" .../.. _xinetd_services ()  {      local xinetddir=/etc/xinetd.d;     if [[ -d $xinetddir ]]; then         local restore_nullglob=$(shopt -p nullglob);         shopt -s nullglob;         local -a svcs=($( printf '%s\n' $xinetddir/!($_backup_glob) ));         $restore_nullglob;         COMPREPLY+=($( compgen -W '${svcs[@]#$xinetddir/}' -- "$cur" ));     fi } dequote ()  {      eval printf %s "$1" 2&gt; /dev/null } quote ()  {      local quoted=${1//\'/\'\\\'\'};     printf "'%s'" "$quoted" } quote_readline ()  {      local quoted;     _quote_readline_by_ref "$1" ret;     printf %s "$ret" }   I checked all files in my knowledge as /etc/profile, /etc/environment, and/or ~/.bashrc. I didn't find any generation  script or code calling.   Do you have any advice where is come from ? Is there a way to recognize destructible environment in XCOM:Enemy Unknown? Format a number with optional decimal places in Excel I have numbers in cells in Excel. I want the numbers formatted so that if they have decimal places they show to a maximum of two, and if they have no decimal places it doesn't show any.  For example.    15 should be formatted as 15, NOT 15.00 14.3453453 should be formatted as 14.35 12.1 should be formatted as 12.1  0 should be formatted as 0   The closest custom format code I've come up with is 0.##. Unfortunately this formats 15.00 as 15. (note the extra decimal point).  To further complicate the issue, the spreadsheet is a result of an export from SQL Server Reporting Services. So no macros are possible. Oh well, it looks like 0.## is my best bet, and they can just live with the extra period.