I know how to prove $0 \times a \cong 0$ in categories admitting exponentiation (and initial objects, of course). But does this hold for categories that don't have exponentiation?  I don't have enough examples in my mental categories library to give a counter-example, nor have I managed to prove that $0 \times a \cong 0$ based on just the notion of products.
In $Sets$, the initial object is $0 = \emptyset$. We have $C \times \emptyset = \emptyset$ for any set $C$. As there are no maps to the empty set, we don't get much of the universal property of the product.  I am now wondering what is special about arbitrary categories $\mathscr{C}$ with $C \times 0 \cong 0$ for all objects $C$ of $\mathscr{C}$, where $0$ is the initial object of $\mathscr{C}$.   From Sheaves in Geometry and Logic by MacLane (p.194), in any topos $\mathscr{C}$, any arrow $k:C \to 0$ is an isomorphism. The universal property of the product would then also not yield very much.   Are there any categories of higher interest?
Prove $\lim\limits_{n\to\infty}\frac{1}{\sqrt[n]{n!}}=0$ CDN - Content Delivery Networks. How do they work and why would I want to use one? What exactly is a Content Delivery Network (CDN), how does it work, and why would I want to use one for my web site?  Which are the well-known CDNs out there? What's the right way to hard boil eggs? Google AdSense Policy Non Compliant "Site Behaviour: Navigation" "Washroom", "restroom", "bathroom", "lavatory", "toilet" or "toilet room" I've always been confused by the terms washroom, restroom, bathroom, lavatory, toilet and toilet room. My impression is that Canadians would rather say washroom while Americans would probably say bathroom or Saint John's in the same situation.  I guess the difference here is not only in different kinds of English, but also in whether one is referring to a room in their house or in some public place.  Which do you usually use? Please specify the difference if you use more than two from those six with different meanings, and also where you are from (i.e. what type of English you speak). Why are only certain shutter speeds or apertures available? Will ArcGIS for Desktop run on Windows 10? Will ArcGIS 10 run on the new windows 10 technical preview?  I am setting up an interface on my Mac and there is the free option of downloading the Windows 10 technical preview, some sort of beta version, I'm assuming.   Just wanted to see if 10.1 will run ok on Windows 10 (ignoring the Mac aspect), as this is all I am using my windows interface for? Using notify-send with cron I think that all the zeros of the Riemann-Zeta function ${\zeta}( z ) = \frac{1}{1-2^{1-z}} \sum_{n = 0}^{\infty} \frac{1}{2^{n+1}} \sum_{k = 0}^{n} (-1)^k \binom{n}{k} (k+1)^{-z}$  have real part smaller than 1, but I don't know how to prove it. If Re(z) > 1 than  ${\zeta}( z ) = \sum_{n = 1}^{\infty} \left( \frac{1}{n} \right)^z$ Let $a,b,c$ are non-negative numbers, such that $a+b+c = 3$.  Prove that $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$    Here's my idea:  $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$  $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) \ge 2(ab + bc + ca)$  $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) - 2(ab + bc + ca) \ge 0$  $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) - ((a+b+c)^2 - (a^2 + b^2 + c^2) \ge 0$  $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) + (a^2 + b^2 + c^2) - (a+b+c)^2 \ge 0$  $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) + (a^2 + b^2 + c^2) \ge (a+b+c)^2$  $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) + (a^2 + b^2 + c^2) \ge 3^2 = 9$  And I'm stuck here.  I need to prove that:   $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) + (a^2 + b^2 + c^2) \ge (a+b+c)^2$ or  $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) + (a^2 + b^2 + c^2) \ge 3(a+b+c)$, because $a+b+c = 3$  In the first case using Cauchy-Schwarz Inequality I prove that:  $(a^2 + b^2 + c^2)(1+1+1) \ge (a+b+c)^2$  $3(a^2 + b^2 + c^2) \ge (a+b+c)^2$  Now I need to prove that:   $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) + (a^2 + b^2 + c^2) \ge 3(a^2 + b^2 + c^2)$   $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) \ge 2(a^2 + b^2 + c^2)$   $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge a^2 + b^2 + c^2$   I need I don't know how to continue.  In the second case I tried proving:  $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) \ge 2(a+b+c)$ and  $a^2 + b^2 + c^2 \ge a+b+c$  Using Cauchy-Schwarz Inequality I proved:  $(a^2 + b^2 + c^2)(1+1+1) \ge (a+b+c)^2$  $(a^2 + b^2 + c^2)(a+b+c) \ge (a+b+c)^2$  $a^2 + b^2 + c^2 \ge a+b+c$  But I can't find a way to prove that $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) \ge 2(a+b+c)$  So please help me with this problem.  P.S   My initial idea, which is proving:  $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) + (a^2 + b^2 + c^2) \ge 3^2 = 9$  maybe isn't the right way to prove this inequality. Is there a reason, beyond historical reasons, that three phases has become the dominant number of phases?  I am aware of the advantages against one phase and two phase, namely the reduced amount of conductor needed, and that motors can provide torque when stalled (and less pulsation).  Is this solely due to diminishing returns, with only a small increase in smoothness of torque application, at the cost of increased complexity (increased number of wires (albeit of smaller CSA)).  To be clear, the phases are all evenly distributed, that is, five phases separated by 72 degrees. Sum over multiple indices ASP.NET Stress Testing Is there a way to test an application where you simulate a hundred different clients connecting to a IIS server and asking the same data? At the customer where our project is running they have 400 computers and they often do stress tests with all computers. I on the other hand have only got my laptop... (and a development server).  (In my case the data is asked through a WebORB Gateway). Can a single molecule have a temperature? Is there an open source alternative to "Adobe Digital Editions"? For the moment I run Adobe Digital Editions under Wine to download ebooks burdened with ditigal restriction management. I only use it to open .acsm files supplied by ebook-shops, which contain authentication information required to download the actual DRM'ed ebook files, which I then transfer to my reader-device with calibre. My question is, if there exist open source alternatives to Adobe Digital Editions. They don't need much functionality. I just want to put the .acsm-file in and get the .epub out (no, I don't want to remove the drm, that'd be illegal afaik). How do I block unwanted callers without jailbreaking? I'm trying to access an XML file within a jar file, from a separate jar that's running as a desktop application.  I can get the URL to the file I need, but when I pass that to a FileReader (as a String) I get a FileNotFoundException saying "The file name, directory name, or volume label syntax is incorrect."  As a point of reference, I have no trouble reading image resources from the same jar, passing the URL to an ImageIcon constructor.  This seems to indicate that the method I'm using to get the URL is correct.  URL url = getClass().getResource("/xxx/xxx/xxx/services.xml"); ServicesLoader jsl = new ServicesLoader( url.toString() );   Inside the ServicesLoader class I have   XMLReader xr = XMLReaderFactory.createXMLReader(); xr.setContentHandler( this ); xr.setErrorHandler( this ); xr.parse( new InputSource( new FileReader( filename )));   What's wrong with using this technique to read the XML file? Carry over long sequence of missing values with Postgres I have a table like this:  create table foo (foo_label text, foo_price int, foo_date date);  insert into foo (           values           ('aaa', 100,  '2017-01-01'),           ('aaa', NULL, '2017-02-01'),           ('aaa', NULL, '2017-03-01'),           ('aaa', NULL, '2017-04-01'),           ('aaa', 140,  '2017-05-01'),           ('aaa', NULL, '2017-06-01'),           ('aaa', 180,  '2017-07-01')         );   As you can see a few values on the foo_price column are missing.  What I need is that missing values get filled up with the "previous" available value in this way:   foo_label | fixed_foo_price | foo_date -----------+-----------------+------------  aaa       | 100             | 2017-01-01  aaa       | 100             | 2017-02-01  aaa       | 100             | 2017-03-01  aaa       | 100             | 2017-04-01  aaa       | 140             | 2017-05-01  aaa       | 140             | 2017-06-01  aaa       | 180             | 2017-07-01   My attempt:  select      foo_label,      (case when foo_price is null then previous_foo_price else foo_price end) as fixed_foo_price,     foo_date from (   select        foo_label,        lag(foo_price) OVER (PARTITION BY foo_label order by foo_date::date) as previous_foo_price,        foo_price,       foo_date       from foo ) T;   As you can see from here:    It doesn't fill completely the '100' series.  Any idea how can I get the wanted result? How do I enable automatic updates? Is there an easier way to install Chromium than what is described ?  Furthermore, is there an easier way to stay up to date with current Chromium releases rather than having to re-build Chromium each release? (a script which automatically grabs and installs the latest build possibly?).  Related: