I have just started studying symplectic geometry, and one fact has me completely stumped.  Apparently a symplectic manifold $V$ such that $\mathrm{dim}V = n$ has volume form: $\frac{\omega^n}{n!}$.  Now I recall a volume form is a non-vanishing, top-dimensional form on $V$.  However I have no clue how to show $\frac{\omega^n}{n!}$ is either non-vanishing or a top-dimensional form. This last condition in particular seems like something that will have an 'obvious' explanation, but one that I can't figure out.  Most of these definitions have been floating around in my head barely a day. I'm not far into the topic, and my differential geometry/experience with differential forms is still fairly basic, so the simpler an explanation someone can offer, the better.
I have the following problem: Let $(M, \omega)$ be a symplectic manifold. How can I show $$\omega^n=\underbrace{\omega\wedge \ldots\wedge \omega}_{n-times},$$ satisfies $\omega^n(p)\neq 0$ for all $p\in M$. I believe that is not too dificult but I'm not used with exterior product.. Any help will be valuable...  This is important because the nondegeneracy condition on $ω$ is equivalent to the condition that M has an even dimension $2n$ and the top wedge product $\omega^n$ is nowhere vanishing on M, i.e., $ω^n$ is a volume form. In particular, $M$ must be orientable and is canonically oriented by $\omega^n$.
Subdivision surface modifier problem Hello!   I've recently started learning Blender and faced one problem.  After applying subdivision surface modifier I have plenty of extra faces. I've watched some teaching videos and I suppose additional faces shouldn't be added. Is it OK ? How to avoid this problem if it isn't. How to test if these two samples are from the same distribution? (These are tweets located near some tourist attractions.     If yes, is there a simple implementation in Python? Proving $\frac{1}{n+1} + \frac{1}{n+2}+\cdots+\frac{1}{2n} &gt; \frac{13}{24}$ for $n&gt;1,n\in\Bbb N$  To solve it I used induction but it is leading me nowhere my attempt was as follows:  Lets assume the inequality is true for $n = k$ then we need to prove that it is true for $k+1$  so we need to prove $\frac1{k+2} + \frac1{k+3}+\cdots+\frac1{2(k+1)} &gt; 13/24$  I don't know where to go from here please help. A childish question (literally) -    My 8 year old asked me this morning: "Dad, what does 'physical' mean?" - and I found myself at loss for an ordinary language answer.   Every answer I could come up with ended up sounding like a definition of empiricism: "physical things are those things that can be observed", "physical means something that can be measured", "whatever can be verified and tested", etc....   Claiming that things that are physical are those things which are material or things that are described by physics seems circular.   Google:     phys·i·cal   ˈfizik(ə)l/   adjective   1.   relating to the body as opposed to the mind.   "a whole range of physical and mental challenges"   synonyms:   bodily, corporeal, corporal, somatic; More   2.   relating to things perceived through the senses as opposed to the mind; tangible or concrete.   "pleasant physical environments"   synonyms:   material, concrete, tangible, palpable, solid, substantial, real, actual, visible   "the physical universe"   And various dictionaries (here Merriam-Webster):      Definition of physical   1   a :  of or relating to natural science   b (1) :  of or relating to physics (2) :  characterized or produced by the forces and operations of physics   2   a :  having material existence :  perceptible especially through the senses and subject to the laws of nature   everything physical is measurable by weight, motion, and resistance — Thomas De Quincey   b :  of or relating to material things     lead to similar definitions.   How can one define "physical" in ordinary language terms? Is there a way of defining "physical" that doesn't presuppose empiricism? What is the rationalist definition of 'physical'?   Is there a definition that doesn't commit to either dualism or empiricism or to any epistemology? What does echelon mean? When you solve a system of linear equations, you write down the augmented matrix and reduce this to reduced row echelon form. What is the meaning of the word echelon? I think the title says it. Did expansion of the universe steal the energy somehow? scope of clause within complex sentence If I said, &quot;I will run, not if it is snowing, if I my shoes are comfortable.&quot; Does the &quot;if my shoes are comfortable&quot; modify the &quot;I will run&quot; clause? I agree the sentence would be clearer with the addition of &quot;and&quot; in place of the comma. However, without the &quot;and&quot; is it still clear that the &quot;if my shoes are comfortable&quot; clause modifies the &quot;I will run&quot; clause? Merging two files, one column at a time We need a profanity filter I searched SE.Meta and was unable to find such question.  Someone just swore at us in a  as an  (warning: profanity, may be NSFW), sure, Arqade (profanity in phrases in games) or some SE (ELU, ELL or Anime.SE maybe?) can sometimes contain profanity as a small parts of question.  I think that a part should be added to the review system that will automatically hide questions/answers with profanity, just like the close vote system, the question/answer will be on hold and will be voted to be shown or be deleted.  Tip, if this gets implemented, u can be typed as \/ or v, and most other can be written as leet (l = 1 etc). Maybe it can be called "suspicious activity", and can be triggered when there is too many number or Unicode characters. (I can write hi as |-| |, |-|I, |-|l, l-li, hî, 1-1i, H1 and many more) How to make terminal transparent while still having its own background image? Is there a way to give terminal specific background image and make it 50% transparent at the same time? Broken screen while debug mode was disabled. How can I re-enable adb? The screen of my phone is physically broken, so it doesn't display anything, but the touch aspect does seem to be working, as I can unlock my phone, and get haptic feedback.  Before I go to get it replaced, I'd like to back up some personal data from the built-in memory to the SD card. I'm fairly certain that if I can get adb to recognize the device, then I can use  to see where I am touching the screen and do what I need to do. Problem is, when I plug it in, it's not recognized, while my emulator is.  I suspect that I might not have USB debugging enabled, which I think is necessary for adb to work. If this is the case, can someone give me screenshots of their Droid X (running 2.3.X) going from the home screen, through all the menus, to activating debug mode? Hopefully I can do it by dead reckoning and adb from there.  Edit: I haven't changed anything about the ROM or launcher; it's all stock. Kill the [bugs] tag? I'm getting an error - didnt understand relationship when running this query in dev console. I have set the child relationship the lookup field to IncEvents. The lookup field is on the MyRule object -   select Id, Name, (select Id from IncEvents_r) from Namespace_MyRule__c  I noticed that the editor is displaying a single underscore between the namespace and the object name but I did enter a double underscore. Looking for a short story about an artist and a forger Back in the late 1970s to early 1980s, I read a short story about an artist and a forger.  Between the two of them, they managed to start a revolution that brought down a corrupt dictatorship.  It was science fiction:   The dictatorship was interplanetary, maybe interstellar.   The artist used some kind of advanced technology to make his art works.   Plot:  The artist is one of the best (if not the best) at what he does.  The dictator wants the artist to make what amounts to a family portrait in whatever medium it is he works in.  Think of it like painting - you create the image manually (not a photograph) but it is much more than just a flat image.    The artist doesn't like the dictator - maybe on general principles, maybe because he knows the dictator personally - and refuses to make the "portrait."  The dictator has the artist arrested, and will only let the artist free again if he makes the portrait.  The artists makes the portrait.  He does the work while in the prison or jail.  That's where the forger comes into things. The forger is also in jail for something or other.  The forger makes copies of art works.  He of course recognizes the artist, and knows what the artist works with.  The forger can somehow see what the artist is working on.  The artist finishes the portrait.  At first the dictator is thrilled.  It looks exactly the way he wanted it.  As time goes by, though, the people looking at the picture see that it represents the vileness and corruption of the dictator.  The more you look at it, the more of the evilness of the dictator you see in it.  That is the genius of the artist, and whatever medium he is using.  Once you notice the hidden levels of the portrait, you can't unsee them and you see more and more of it as you look.  I think the effect carries over into reality as well.  Once you've seen the dictator's corruption in the portrait you also begin to see it in the dictator and his works as well.  This of course seriously pisses off the dictator, who has the artist killed and the portrait destroyed.  While in prison, the forger watched as the artist created his masterpiece, and recognized it for the subversive thing it is.  He also recognized it as the ultimate rebellion from the artist - the artist knows what will be seen, and that he will be killed for it.  All of that inspires the forger to do something more meaningful with his life than make forgeries for the money.  The forger is released from prison.  He saw the completed portrait, and knows how it was made.  He begins making forgeries of the portrait - like, hundreds if not thousands of them.  These are distrubuted throughout the nation/empire.  The people realize what evil is controlling them once they see it in the copies of the portrait, and eventually a revolution rises and brings down the dictatorship.  It was probably older than the 1970s.  I expect earlier than 1960.  Any ideas? how to launch Bluez on Lubuntu Trying to launch a BT adapter/manager(default installed on Lubu)a little window informs me that it cannot because there's no Bluez launched.  In terminal: sudo apt install bluez, it answers: Bluez is already the new version (5.46-0ubuntu3) Then invites me to remove a couple of obsolete packages.  So where's Bluez Daemon to launch?  Writing:systemctl status bluetooth.service ; hcitool dev  Answer:   bluetooth.service - Bluetooth service    Loaded: loaded (/lib/systemd/system/bluetooth.service; enabled; vendor preset    Active: inactive (dead)      Docs: man:bluetoothd(8)   Command:lspci -k | grep -A3 -i "wireless\|network\|bluetooth" Answer: 04:00.0 Ethernet controller: Qualcomm Atheros AR242x / AR542x Wireless Network Adapter (PCI-Express) (rev 01)     Subsystem: Quanta Microsystems, Inc AR242x / AR542x Wireless Network Adapter (PCI-Express)     Kernel driver in use: ath5k     Kernel modules: ath5k 0f:06.0 CardBus bridge: Texas Instruments PCIxx12 Cardbus Controller oem@oem-Extensa-5220:~$ Multiple Sub Query using child parent relationship Consider in my basket, I have a package (Main Product-just a holder) which will have 2 standalone products (child) and again the standalone products will have child products (Grand Parent). All these products are from same object called Product Configuration.   We have a child relationship to itself(Parent Configuration) using a custom lookup field. The problem here is, I could only fetch details only up to 1 level (i.e.,package and child details) and I was not able to drill deeper to grand child products.  Query I used to get 1 level of relation ships is below:  SELECT id, name, Access_Type__c, Circuit_Bandwidth_Value__c, QuoteType__c,     Upstream_Speed_Value__c, IP_Address_Type__c, Care_Level__c, Base_Plan_Value__c, Site_Address__c,      cscfga__Quantity__c, Quantity_of_Ip_Address__c, cscfga__Contract_Term__c, Connection_Price__c,      Monthly_Rental_Conga__c, Total_Annual_Rental__c,      (         SELECT id, name,cscfga__Product_Family__c,Upstream_Speed_Value__c, cscfga__Quantity__c, Quantity_of_Ip_Address__c,              Connection_Price__c, Monthly_Rental_Conga__c,cscfga__Parent_Configuration__r.Circuit_Bandwidth_Value__c,Circuit_Bandwidth_Value__c          FROM cscfga__Related_Configurations__r         WHERE cscfga__Parent_Configuration__c &lt;&gt; ''          ORDER BY name     )  FROM cscfga__Product_Configuration__c  WHERE cscfga__Product_Basket__r.cscfga__Opportunity__c='{pv0}'      AND cscfga__Product_Basket__r.csordtelcoa__Synchronised_with_Opportunity__c=True     AND cscfga__Parent_Configuration__c = ''  ORDER BY name   Not sure how to write another sub query to get grand child details. When I tried the below query:  SELECT id, name, Access_Type__c, Circuit_Bandwidth_Value__c, QuoteType__c, Upstream_Speed_Value__c,      IP_Address_Type__c, Care_Level__c, Base_Plan_Value__c, Site_Address__c, cscfga__Quantity__c,      Quantity_of_Ip_Address__c, cscfga__Contract_Term__c, Connection_Price__c,      Monthly_Rental_Conga__c, Total_Annual_Rental__c,     (         SELECT id, name, cscfga__Product_Family__c, Upstream_Speed_Value__c, cscfga__Quantity__c,              Quantity_of_Ip_Address__c, Connection_Price__c, Monthly_Rental_Conga__c, cscfga__Parent_Configuration__r.Circuit_Bandwidth_Value__c, Circuit_Bandwidth_Value__c,         FROM cscfga__Related_Configurations__r         WHERE cscfga__Parent_Configuration__c &lt;&gt; ''          ORDER BY name     ),     (         SELECT id, name, cscfga__Product_Family__c          FROM cscfga__Related_Configurations__r         WHERE cscfga__Parent_Configuration__c &lt;&gt; ''          ORDER BY name     ) FROM cscfga__Product_Configuration__c  WHERE cscfga__Product_Basket__r.cscfga__Opportunity__c='0066E000002Xrr0'      AND cscfga__Product_Basket__r.csordtelcoa__Synchronised_with_Opportunity__c=True      AND cscfga__Parent_Configuration__c = ''  ORDER BY name   I got below error:     MALFORMED_QUERY:  from cscfga__Related_Configurations__r ^ ERROR at   Row:6:Column:6 Cannot follow the same aggregate relationship twice:   cscfga__Related_Configurations__r   Can someone please help here. Note: I am executing these queries in Workbench. Integral $\int\!\sqrt{\cot x}\,dx $ How do I run a sudo command needing password input in the background? I have recently disabled  so that it now prompts me for a password every time.  And though this is good for security, it has caused one slight problem which I haven't been able to figure out a solution for, I am unable to run commands which go along the lines of:   sudo &lt;command&gt; &amp;   In the past I would have run a sudo command before that, it would have cached my password and allowed me to run sudo commands without prompt for the next few minutes, and thus it would allow me to run the command.  But when I run it now as there is no caching before hand and as it starts a new thread immediately and sudo doesn't even prompt me for a password, I am unable to run it in this way.  So unless I run sudo -i before it I am unable to run a command in this format which is becoming rather annoying. So I was wondering if there is any way to get around this and still run programs and commands in this way?  I am running Ubuntu GNOME 15.10 with GNOME 3.18, and specifically the program I want to run in this fashion is etherape if that makes any difference, but I would really like the solution to work for all programs and commands. Display SharePoint Online in an iframe on different domain We would like to display pages from SharePoint Online in an iframe of a Page hosted somewhere else, e.g. in an Azure Web App:  &lt;iframe src="https://site.sharepoint.com/"&gt;&lt;/iframe&gt;   When we do this (in Chrome for example), we get the following error:     Refused to display " ⏎                      Home.aspx" in a frame because it set 'X-Frame-Options' to 'SAMEORIGIN'.   A Page from SharePoint 2013 (on premise) can be displayed in an iframe from a Page hosted on another domain, if the following control is added to the head of the master page:  &lt;WebPartPages:AllowFraming runat="server" /&gt;   However if this control is not added to the master-page, then we also get the above mentioned error.  As far as I know there is no possibility to adjust the master-page in a SharePoint Online Team Site. Is that correct?   In a Publishing Site you are allowed to adjust the master-page, or even add a new one. But I can't find anything similar in Team Sites.  Does anybody know if it is somehow possible to configure domains which do not fall under the "X-FRAME-OPTIONS: SAMEORIGIN"-policy in a SharePoint Online environment? Or if there is another possibility to display SharePoint Online content in an iframe on a different domain. I checked google, checked the Site settings and checked the Office 365 admin center, but didn't find anything. existence of sequence of polynomial Is there a sequence of polynomials $P_n$ such that   $\displaystyle\lim_{n \to \infty} P_n(z) = \begin{cases} 1, &amp; \text{Im } z &gt; 0\\ 0, &amp; z \text{ is real,} \\ -1, &amp; \text{Im } z &lt; 0 \end{cases}$  I have no clue where to start from. Please provide some hints. Thanks in advance.