Given a space $X$, the topological cone $CX$ is defined as the quotient $X\times I/(X\times \{1\})$.  If $X$ is a subset of $\mathbb R^n$, choose a point in $\mathbb R^{n+1}$ that does not lie in $\mathbb R^n$. The geometrical cone $aX$ is defined as $\{ta+(1-t)x|x\in X, t\in I\}$ equipped with subspace topology.     Proposition: If $X\subset \mathbb R^n$ is compact, then $CX$ is homeomorphic to $aX$.   Question: I wonder if it is possible to drop the assumption that $X$ is compact.  My teacher guessed that the conclusion remains to be true if $X$ is merely bounded, but didn’t provide any proof or counterexample.
Let $K\subseteq \mathbb{R}^{n} \times \{ 0 \}\subseteq \mathbb{R}^{n+1} $ and $v=(0,0,\cdots,1) \in \mathbb{R}^{n+1}$.      For every $x \in K$,  let $$Lx=\left \{ tv+(1-t)x \; | \ t \in [0,1] \right \}$$      Define    $$ \text{Cone}(K)=\bigcup_{x \in K}Lx. $$   namely the union of all segments from $v$ to every point of $K$.      According to all definitions that I see on books, Cone is defined:       However the exercise I'm working on  asks to find under what circumstances  $$\text{Cone}(K)=\bigcup_{x \in K}Lx$$ is isomorphic to $(K \times [0,1]) /(K \times \{ 1 \})$ (quotient space).   For example if $$K=\{(1/n,0):n\in\mathbb{N}\} \cup (0,0)$$ I can see both of sets form the same graph but I'm still unable to derive a homeomorphism between these two objets.
If a prime divides a product then it must divide a factor of the product [Euclid's Lemma] We have $q, p$ prime and $tq = (p-1)(p+1)$ for some positive integer $t$. The book I'm reading says that because $q$ is prime, it divides $p-1$ or $p+1$.   Question: Why exactly does it matter that $q$ is prime here? Error-Handling in Swift-Language We have an instance of SQL Server 2008 installed on a customer server. The customer's IT department is in charge of the various backups of the machine.  The SQL Server log shows there is a full backup every day at 7 PM but we can't find evidence of any plan scheduled in SQL Server.  The technicians we contacted couldn't tell us if there was some sort of automatic backup, all they told is that the entire machine is being backup up.  Using a script I found in  I found out that the physical device name is a GUID and that this means that this is an external backup process:  USE [msdb] GO  SELECT      [bs].[database_name],      [bs].[backup_start_date],      [bs].[backup_finish_date],      [bs].Server_name,     [bs].user_name AS [BackupCreator] ,     [bmf].physical_device_name FROM msdb..backupset bs   INNER JOIN msdb..backupmediafamily bmf ON [bs].[media_set_id] = [bmf].[media_set_id]  ORDER BY [bs].[backup_start_date] DESC   This is a row from the above query:     db_name 2015-09-16 19:01:23.000   2015-09-16 19:01:28.000SERVER_NAME  NT AUTHORITY\SYSTEM{424F084A-F35D-4A66-8FC7-072268A89A77} 5   Moreover the backup start and finish date spans only for 5 seconds, so I guess it's clear it's not a job of sql server.  This is a line from the log:     2015-09-03 19:02:30.71 Backup   Database backed up.   Database: db_name, creation date(time): 2012/10/12(20:52:11), pages dumped: 3290021,   first LSN: 276028:152755:172, last LSN: 276028:152827:1, number of dump devices: 1,   device information:   (FILE=1, TYPE=VIRTUAL_DEVICE: {'{95380B0A-D50B-408F-B95F-1AB8975BA7F8}5'}).   This is an informational message only. No user action is required.   So, since they can't help us what can I do to track down the process responsible of the backups? We need this because we want to coordinate the backup in an ordered manner, and do the transaction log backups too in order to keep the log at a reasonable size (now we do shrink the log every week, and this is not the good way to go). How does matchmaking work in Play mode? How does Hearthstone match you with opponents in Play mode, for both Ranked and Unranked? Does your record in Ranked affect matchmaking in Unranked (and vice-versa)? Wrong posts number on Review section Lately it's happening to me this:  After completing one or more "First Posts" reviews, I go back to the Review Section main page:   And the post counter shows that there are still posts to review. I click the link to review them, but there is nothing.  Refreshing the page doesn't solve the issue: it shows the wrong number again. Do I make money in the stock market from other people losing money? $f$ strictly increasing does not imply $f'>0$ We know that a function $f: [a,b] \to \mathbb{R}$ continuous on $[a,b]$ and differentiable on $(a,b)$, and if $f'&gt;0 \mbox{ on} (a,b)$ , f is strictly increasing on $[a,b]$. Is there any counterexample that shows the converse fails?   I have been trying to come up with simple examples but they all involve functions that are discontinuous or has derivative $f'=0$ which does not agree with the hypothesis hmmm dpkg error: "trying to overwrite file, which is also in..." For example:  $ sudo apt-get install curl Reading package lists... Done Building dependency tree Reading state information... Done You might want to run 'apt-get -f install' to correct these: The following packages have unmet dependencies: bsh : Depends: libjline-java but it is not going to be installed groovy : Depends: libjline-java but it is not going to be installed rhino : Depends: libjline-java but it is not going to be installed E: Unmet dependencies.   Try 'apt-get -f install' with no packages (or specify a solution).   I get the same or similar errors when I attempt to install clojure1.3, leiningen, and several other packages.   When I try the suggestion made in the error message, this is what happens:  $ sudo apt-get -f install  Reading package lists... Done Building dependency tree        Reading state information... Done Correcting dependencies... Done The following packages were automatically installed and are no longer required:   diffstat linux-headers-3.2.0-26-generic linux-headers-3.2.0-26 dh-apparmor dkms html2text libmail-sendmail-perl libsys-hostname-long-perl Use 'apt-get autoremove' to remove them. The following extra packages will be installed:   libjline-java Suggested packages:   libjline-java-doc The following NEW packages will be installed:   libjline-java 0 upgraded, 1 newly installed, 0 to remove and 0 not upgraded. 23 not fully installed or removed. Need to get 0 B/72.0 kB of archives. After this operation, 129 kB of additional disk space will be used. Do you want to continue [Y/n]? Y (Reading database ... 226243 files and directories currently installed.) Unpacking libjline-java (from .../libjline-java_1.0-1_all.deb) ... dpkg: error processing /var/cache/apt/archives/libjline-java_1.0-1_all.deb (--unpack):  trying to overwrite '/usr/share/java/jline.jar', which is also in package scala 2.9.2-400 Errors were encountered while processing:  /var/cache/apt/archives/libjline-java_1.0-1_all.deb E: Sub-process /usr/bin/dpkg returned an error code (1)  $ sudo apt-get upgrade Reading package lists... Done Building dependency tree        Reading state information... Done You might want to run 'apt-get -f install' to correct these. The following packages have unmet dependencies:  bsh : Depends: libjline-java but it is not installed  groovy : Depends: libjline-java but it is not installed  rhino : Depends: libjline-java but it is not installed E: Unmet dependencies. Try using -f. I'm trying to typeset SuperCollider code using the listings package.  Any "freestanding" identifiers that begin with a capital letter are class names, and should be highlighted; by "freestanding", I mean outside of strings or comments. I would be tickled pink if listings could let you define identifiers in terms of regexp, e.g. [A-Z][A-Za-z0-9_]* but the manual doesn't suggests that as a possibility.  SuperCollider has some 2300 classes... I really don't relish listing all of them literally in my preamble. I guess I could extract them by hand for each individual listing, but I'd rather not have to.  Is there a way to do this without a really massive morekeywords expression? (Perhaps using another package?)  Sample SuperCollider code:  p.clear;  ~grains.addSpec(\tfreq, [1, 40, \exp]); ~grains.addSpec(\overlap, [0.1, 10, \exp]); ~grains.addSpec(\pos, [0, b.duration]);  // 3.43 is nice! ~grains.addSpec(\rate, [0.5, 2, \exp]); ~grains = { |tfreq = 25, overlap = 6, pan = 0, amp = 0.2, pos = 3.43,    rate = 1|    var trig = Impulse.ar(tfreq);    TGrains.ar(2, trig, b, rate, pos, overlap / tfreq, pan, amp) }; ~grains.play;   Impulse and TGrains should be highlighted in blue. I have dozens of other listings using different capitalized keywords. How can I stop Ubuntu Dock from extending to the edges? By default Ubuntu Dock in Ubuntu 17.10 and later releases always extends to the edges (top-bar-to-bottom if placed on the left or right and left-to-right when placed at the bottom) even when it's not filled with icons. Is there a way to prevent this behaviour (i.e. I want to make the dock-length flexible so that as the number of application icons in the dock increases, it should expand accordingly)?  ➡️➡️➡️  In other words, is there a way make Ubuntu Dock look like a &quot;proper dock&quot; instead of a side/bottom panel with full-height/width? For each of the following, give an example of a sequence of functions $f_n$ that converges to f  A. uniformly but not in the mean square sense.  B. in the mean square sense but pointwise nowhere.  I know that for part A the domain of the function cannot be bounded but other then that I'm not sure how to go about the problem I'd like to show that the class of all limit ordinals, LIM, is cofinal in the class of all ordinals ON.  I thought that I could do this by contradiction and assume that it wasn't cofinal in ON. Then there is an $\alpha \in$ ON such that for all $\beta \in $ LIM: $\beta &lt; \alpha$, i.e. $\beta \in \alpha$ and hence LIM $\subset \alpha$.  This means that LIM is a set so we can consider $\bigcup$ LIM. Then we have $\beta &lt; \bigcup$ LIM for all $\beta \in$ LIM by definition of $\bigcup$ LIM. So LIM $\subseteq \bigcup$ LIM.   $\bigcup$ LIM is either a limit or a successor ordinal.  If it is a limit ordinal then $\bigcup$ LIM $\in$ LIM and by transitivity of ON, $\bigcup$ LIM $\subseteq$ LIM, so $LIM = \bigcup$ LIM in this case which would mean that $\bigcup$ LIM $\in$ LIM $= \bigcup$ LIM which would be a contradiction because sets cannot contain themselves. Hence $\bigcup$ LIM cannot be a limit ordinal.  If $\bigcup$ LIM is a successor ordinal then $\bigcup$ LIM = $\beta \cup \{&#160;\beta \}$ for some $\beta \in $ ON.  Now I'm not sure how to proceed from here. I'm also not sure about whether I'm on the right track with this proof because I think there should be a shorter way  to prove this.  Thanks for your help. Monotone functions and non-vanishing of derivative How to print a date in a regular format? This is my code:  import datetime today = datetime.date.today() print(today)   This prints: 2008-11-22 which is exactly what I want.  But, I have a list I'm appending this to and then suddenly everything goes "wonky". Here is the code:  import datetime mylist = [] today = datetime.date.today() mylist.append(today) print(mylist)   This prints the following:   [datetime.date(2008, 11, 22)]   How can I get just a simple date like 2008-11-22? $a,b,m,n$ be positive integers such that g.c.d.$(a,b)=1$ and $a^m+b^m|a^n+b^n$ , then $m|n$? Let $a,b,m,n$ be positive integers such that g.c.d.$(a,b)=1$ and $a^m+b^m|a^n+b^n$ , then is it true that $m$ divides $n$ ? Help evaluating a gamma function Can the Artificer Artillerist's tiny-sized Eldritch Cannon that is not held in hand still ride or be carried elsewhere on the body? NSF IIS status date changed, status still pending I have association bonus on superuser.com. After that I've started a bounty worst of 50 rep and earned more than 10 rep. Now I have 89 rep (38 of which is earned at superuser which is more than 10) but still not able to answer protected questions.  So, currently new user with 10 rep can answer protected question and I with 89 (38) rep can't.   How does it work now: after registration on new Stack Exchange site you get 101 rep Association Bonus and are able to answer protected questions only when you have 111+ rep on that site. How should it work: after registration on new stackexchange site you get 101 rep Association Bonus and are able to answer protected questions only when you have earned 10+ rep on that site. Add days to JavaScript Date