Q: Let $A_{n\times n}$ square matrix. Prove that there is an invertible matrix $B$, that $BA$ is a triangular matrix.  How can you prove it?   I thought that you could use that "every row reduction matrix, is an triangular matrix".  But wasn't sure on how to use it.
I'm given a matrix A, its dimensions are n x n.  I am required to prove that an invertible matrix B exists, such that the product of the matrices BA is triangular.  Any help?
Newton's Law of Graviation: Why $G$ and not e.g. $\dfrac{1}{4\pi G_0}$? Seo: Change the title of a page Lets say my website looks like this:  News -Hello world -Foo -Bar  So the parent page will list the last 3 news messages. For each news message there is a sub-page. For the parent page I was thinking of setting the page title to: News - Hello world, Foo, Bar.  Now if somebody adds a new message the title tag will also change. I googled this and read a few blog posts where people said that changing the title actually hurts the ranking of that page.  So is this true? Should I just set a static title tag for all my pages? Java error: Implicit super constructor is undefined for default constructor Show that $$\int_{0}^{\infty} \frac{x - \sin(x)}{x^3} \, dx = \frac{\pi}{4}$$   My attempt is as follows: Let $$f(z) = \frac{z - i e^{iz}}{z^3}$$ and consider the contour on $[\epsilon, R]$ followed by a semicircular arc in the counter clockwise direction, then on $[-R, -\epsilon]$, then the semicircular clockwise contour avoiding the origin. We have, then, that  $$0 = \int_{\Gamma} f(z) dz = \int_{[\epsilon, R]} f(t) dt + \int_{C_R}f(Re^{it})Rie^{it}dt + \int_{[-R, -\epsilon]}f(t)dt + \int_{C_{\epsilon}}f(\epsilon e^{-it})\epsilon i e^{-it} dt$$  Then the first and third integrals ( $I_1$ and $I_3$) combine so that  $$I_1 + I_3 = 2\int_{\epsilon}^R \frac{t - \sin{t}}{t^3}\,dt$$  Further,   $$|I_{C_R}| \leq \int_0^\pi \left|\frac{Re^{it} - ie^{-R\sin{t}}e^{iRcos{t}}}{R^2 e^{2it}} \right|dt \rightarrow 0 \text{ as } R\rightarrow \infty$$  (I've omitted the details, it isn't too bad to bound)  However, I'm having trouble computing the limit  $$\lim_{\epsilon \rightarrow 0}\int_{C_{\epsilon}} f(\epsilon e^{-it})\epsilon i e^{-it} dt$$  No matter which way I look at it, it seems like this limit does not exist. Perhaps I'm seeing something wrong or have I chosen a bad $f(z)$? I'm trying to create globally-unique identifiers in JavaScript.  I'm not sure what routines are available on all browsers, how &quot;random&quot; and seeded the built-in random number generator is, etc. The GUID / UUID should be at least 32 characters and should stay in the ASCII range to avoid trouble when passing them around. Find out BIOS Version from Windows I am trying to install chrome browser via command line.  I tried this:  $ apt-cache search chrome browser   The results show that the proper term is "chrome-browser," so I tried that:  $ sudo apt-get install chrome-browser   And then "Y" for the Y/n question.  But the installation threw errors.  Does someone see anything wrong with the commands I issued? Power series of a function I am wondering if there are any functions $f(x)$ such that it cannot be expressed as a power series of $x$? This might turn out to be a silly question, but I can't think of one at the moment.   Thanks! Does the new finding on "reversing a quantum jump mid-flight" rule out any interpretations of QM? new finding by Minev et al. seems to suggest that transitions between atomic states are not instantaneous, but continuous processes wherein a superposition smoothly adjusts from favoring one state to another (if I understand it correctly). The authors also claim to be able to catch a system "mid-jump" and reverse it. Popular articles are  and .  I am curious if this finding rules out any interpretations of QM. It seems to generally go against the Copenhagen attitude, which describes measurements as collapsing physical systems into a definite classical state. The popular articles indeed claim that the founders of QM would have been surprised by the new finding.  The link with the asterisk mentions that something called "quantum trajectories theory" predicts what was observed. Is this an interpretation, or a theory? And are they implying that other interpretations/theories don't work? System.LimitException: Too many query rows: 50001 What I am trying to do is to set a date in my date field and also update the object. I am using the class below to do so which works just fine in my sanbox but in production I am getting this error:      System.LimitException: Too many query rows: 50001   public with sharing class updateObject{     public  List&lt;NRProducts__c&gt;  values;     public PageReference updateObject() {         values=[SELECT Refreshed_Date__c FROM NRProducts__c ];         for (NRProducts__c s :values){            s.Refreshed_Date__c =Date.today();          }          update values;         return null;                       }     }   VF Page:  &lt;apex:page showheader="false" controller="updateObject" sidebar="false"&gt; &lt;apex:form&gt;  &lt;apex:commandButton action="{!updateObject}" value="Update"/&gt; &lt;/apex:form&gt; &lt;/apex:page&gt; Convert Latex to MathJax-HTML Is there any tool that can convert Latex documents to HTML documents, so that all math is presented with ?  (Obviously, it is fairly easy to hack together a quick Perl script that is able to do a crude conversion, but I am interested in a bit more robust solutions — at least it should get all cross-references right automatically.) Convert hex string to int in Python How do I convert a hex string to an int in Python?   I may have it as "0xffff" or just "ffff". Someone who does not like excess Integration by partial fractions; how and why does it work? How to answer questions in a conference talk that I should know the answer to, but don't? When is an Integer a Rational Number, and are All Ratios Rational, Even $\frac{\sqrt{7}}{2}$? $$\Bbb{Q} = \left\{\frac ab \mid \text{$a$ and $b$ are integers and $b \ne 0$} \right\}$$ In other words, a rational number is a number that can be written as one integer over another. For an integer, the denominator is $1$ in that case. For example, $5$ can be written as $\dfrac 51$.  Is $5$ a rational number? Or is $\dfrac 51$  a rational number? I'm not able to figure out what the definition is actually saying. What are the numbers that cannot be written as one integer over another?  Irrational numbers are the numbers that cannot be written as one integer over another. Roots of numbers that are not perfect squares are examples of irrational numbers.  However, what is this then: $\dfrac {\sqrt 7} {2}$? When I log in on a Windows 8 machine for the first time with my Microsoft Account (by providing an e-mail address), a new user folder is created:  C:\Users\dzinx_000   In previous versions of Windows, I could choose the name of this folder (it was equal to my username). As I use the commandline often, it is quite annoying to me.    Can I somehow rename it to dzinx?  Does simply renaming the folder break everything? Automatic left and right commands How do I get useful SQL Server database performance statistics? We currently have several shared SQL Server servers and instances in our organization. They are going with a different architecture where business units will get their own SQL Server instances. We have SQL Server 2008R2.  In the past, databases were put on servers in a more or less random fashion -- business use and disk space were the main drivers.  I've been tasked with coming up with a plan for allocating our 50+ databases across 3 servers (one instance per server). My criteria for divvying them up is as follows:   Business critical db's should be spread across the servers. i.e. avoid lumping a bunch of business critical db's all one server.   For the remainder of the non-mission critical db's, consider: a. Disk space usage b. I/O usage -- memory, CPU, etc.     1 and 2a are easy for me to figure out. I know how to get data file sizes.   My question is with that last bit - 2b. I do not know how relevant this is, what specific stats to look at, or how to get them.   Frankly, I'm not 100% sure how to phrase this question! I think I basically want to know is, "How 'busy' is this db -- however you define 'busy'?" Or maybe, "How does this db impact the performance of the server?"  I mean, disk space is important -- but it could (in theory) have a ton of rows or binary objects in there that are not written very often, so it has little impact on the performance of the server. I think I would want to know how much impact a db within a date range.  Is there a way to get these kind of general stats? I'm not looking for expensive queries, detailed user stats, or query optimization here. I'm aware of the SSMS reports, but they seem to be more real-time. The end goal is to divide up our current databases into evenly balanced buckets, and I would think performance is one way to do that.  Thoughts? Wisdom? Any other ideas on how to divvy up the db's?  Thanks! Tom How can we prove that for every pair $N \in \mathbb{N}$, and natural number $\beta\in [2, \infty)$ there exists a unique set of integers $x_i \in [0, \beta -1]$, $k\in [0,\infty)$ such that:  $$N = \sum_{i=0}^{k}\beta^ix_{i}, \quad $$  It's a question if every positive integer has its unique representation in every base $\beta$. Is the validity of above statement somehow related to Horner's rule?