What exactly is the cosmological constant? I know that the  was developed as an addition to the Einstein Field Equation as an anti-gravity force: $$R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} + \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu}$$  so that he could get a static universe. But later it was discarded. And now it is related to dark energy.  But what exactly does the cosmological constant mean. Is it a measure of the amount of dark energy? If so, how can we measure it? Please note that I want to know the physical meaning of the constant in our universe (but some mathematical exposure can be helpful anyway)  NOTE: The answer given here:  is more inclined towards the fact that the cosmological constant was developed as a tool to keep the universe static, or that it represents curvature where $T_{\mu \nu} = 0$, but what I am looking for is what do we think of the cosmological constant today? I am not asking how or why it crops up in the EFE, or why it was developed to make sense of a solution to the EFE, but simply what it really is, physically?
What does the 'cosmological constant' represent? Newtons theory of gravity involves a gravitational constant $G$, however one does not refer to it directly, we speak instead of gravity or the force of gravity.   Now Einsteins introduced a  in his equation for GR, and ever since it's been referred to as such; however, this usage seems strange to me, given the above parallel; it seems to me we ought to be referring to something physical, which can then be quantified and whose quantification involves the cosmological constant.   Does the cosmological constant then refer to a cosmological force? A force that is, contra gravity, repulsive? And which is even weaker than gravity, kicking in on the scale of galaxies rather than solar systems?
how to Query parent to child I have process in salesforce in such a way opportunity we can relate this opportunity to this lookup field parent_opportunity__c . Which relates opportunity to other parent opportunity.  I want to write a query parent to child :  select id,(select id from opportunity where parent_opportunity__c!=null) from opportunity where parent_opportunity__c=null   I have worked with different scenarios i cant able to crack it  select id,name,(select id,name from parent_opportunity__r) from opportunity this is showing error and   select id,name,(select id,name from opportunity) from parent_opportunity__c this is also error Make terminal use pageup/pagedown instead of shift-pageup/shift-pagedown? I've grown quite weary of shift-pgup to page up in gnome terminal, the command line, terminator, and guake, etc.  Any way to easily make these apps use the straight page-up and page-down keys for paging around? How to keep a constant baselineskip when using minipages (or \parboxes)? When a minipage is used to box a text paragraph, the first line of the minipaged text and the last line of the previous text paragraph are closer than two consecutive lines in a standard text paragraph; analogously, the last line of text inside the minipage and the first line of text after the minipage are also closer.  The following MWE illustrates this:  \documentclass{article} \usepackage{lipsum}  \newcommand\Text{Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Ut purus elit, vestibulum ut, placerat ac, adipiscing vitae, felis. Curabitur dictum gravida mauris. Nam arcu libero, nonummy eget, consectetuer id, vulputate a, magna.\textbullet}  \begin{document}  \Text  \noindent\begin{minipage}{\linewidth} \Text \end{minipage}  \Text  \end{document}   How can I guarantee a constant baseline-skip even if minipages (or \parboxes) are used? Does an Echo Knight fighter's echo provoke an opportunity attack when it moves? If GCD $(a_1,\ldots, a_n)=1$ then there's a matrix in $SL_n(\mathbb{Z})$ with first row $(a_1,\ldots, a_n)$ Let $n \geq 2$. Let $a_1, a_2, \ldots, a_n$ be $n$ integers such that $\gcd\left(a_1, a_2, \ldots, a_n\right) = 1$. Prove that there exists a matrix in $\operatorname{SL}_n\left(\mathbb{Z}\right)$ whose first row is $\left(a_1, a_2, \ldots, a_n\right)$.   Since the gcd of the integers $a_1,\ldots, a_n$ is $1$, there exists weights $x_i \in \mathbb{Z}$ such that $a_1x_1+\cdots+ a_nx_n=1$. My two ideas are (a) to brute force construct an $n\times n$ matrix with first row $a_1,\ldots ,a_n$ and to construct the remaining rows such that the determinant is $\sum a_ix_i=1$ or (b) to use induction.  (a) (Constructive) This is tedious since once I find a way to construct the remaining $n-1$ rows to ensure that $a_1x_1$ appears in the determinant, I am not sure how to modify these $n-1$ rows to ensure  that only the terms $a_ix_i$ appear in the cofactor expansion. If such a matrix exists, I'd like to see it.  (ii) (Non-constructive) If I proceed by induction then the base case $n=2$ is settled since I can choose the 2nd row to be $-x_2, x_1$ so that the determinant is $a_1x_1-a_2(-x_2)=1$. However, I'm not sure how to use the inductive hypothesis to show that if I can construct such an $n\times n$-matrix then I can construct an $\left(n+1\right) \times \left(n+1\right)$-matrix with the desired property. In particular, if the gcd $(a_1,\ldots ,a_{n+1})$ is $1$, it is not necesarry that the gcd of any $n$ of these terms is $1$, so induction may not even apply here.  How can I construct such a matrix or prove that one exists (without necessarily constructing it)? How can I prove $\sqrt{2} ^{\sqrt{2}}$ is irrational? Reputation widget failure on Android The title says it all. When I try to place the reputation widget on my home screen, I get this:  When I try again, I get this error, presumably because the Stack Exchange service has stopped.  Is this a bug? If so, where should I report it?  I am running Android 5.1 armeabi on a DigiLand DL718M tablet. How to break out of multiple loops? Given the following code (that doesn't work):  while True:     #snip: print out current state     while True:         ok = get_input("Is this ok? (y/n)")         if ok.lower() == "y": break 2 #this doesn't work :(         if ok.lower() == "n": break     #do more processing with menus and stuff   Is there a way to make this work? Or do I have do one check to break out of the input loop, then another, more limited, check in the outside loop to break out all together if the user is satisfied? Let $a_n$ be a sequence of positive reals, such that the partial sums $S_n = \sum_{i = 1} ^ n a_i$ diverge to $\infty$. For given $\epsilon &gt; 0$ do we have $$\sum_{n = 1} ^ \infty \frac{a_n}{S_n^{1 + \epsilon}} &lt; \infty?$$  For $\epsilon \ge 1$ we can resolve this quickly by noting $$\frac{a_n}{S_n ^ 2} \le \frac 1 {S_{n - 1}} - \frac 1 {S_n}$$ so for sufficiently large $n$ we can bound $\frac{a_n}{S_n^{1 + \epsilon}}$ by $\frac 1 {S_{n - 1}} - \frac 1 {S_n}$ as well. I'm wondering if this is true for arbitrary $\epsilon &gt; 0$. I know that the series in question diverges for $ \epsilon = 0$, so all that is missing is what happens in $(0, 1)$. How to access the correct `this` inside a callback? How to insert page number in Beamer navigation symbols? I think it would be a very useful and quality-improving feature.  As for me, when I speak on something, I often have a visual representation of how it works, or something. It's obvious there are many things (in algorithms, or, say, UI) which are much easier understood with pictures and diagrams, than from scarce text.  So I'd really like here to be a special answer form, where you can attach some visual things to better explain your ideas.  I imagine this form as just a canvas with few painting tools and some basic blocks and patterns. Triggering actions when a variable is set Firefox application associations not working in the 'Downloads' window I'm getting the following error message:     System.QueryException: Didn't understand relationship 'AccountContactRelation' in FROM part of query call. If you are attempting to use a custom relationship, be sure to append the '__r' after the custom relationship name. Please reference your WSDL or the describe call for the appropriate names.   Select Email, Id, (Select ContactId, AccountId From AccountContactRelation) From Contact  In AccountContactRelation object there are two lookup fields for Account and for Contact.  Please advise how to avoid such error message? Disable flagging on old chat messages In response to the effect  is referring, I am proposing one of two possible scenarios:   Disallow flags on chat messages older than X hours, or Remove old flagged posts from the transcript, but no suspension occurs (as in the aforementioned question). Power of a generator generates the group iff this power is coprime to the group order. Let $\langle g\rangle$ be a cyclic group of order $n$. Suppose $1\leq q \leq n-1$, I want to show that $g^q$ generates $\langle g\rangle$ if and only if $\gcd(n,q)=1$.  Suppose $g^q$ generates $\langle g\rangle$, then   $$ 1,g^q,g^{2q},...,g^{(n-1)q} $$  are all distinct and $g^{nq}=1$, since $n$ is the order of the group. Note that this is the lowest multiple of $q$ for which this is the case. Therefore $\operatorname{lcm}(q,n)=qn$ and it follows that $q$ and $n$ are coprime.  On the other hand, if $\gcd(q,n)=1$, then $\operatorname{lcm}(q,n)=qn$, we have $g^{nq}=1$ and hence all of  $$ 1,g^q,g^{2q},...,g^{(n-1)q} $$  are distinct again and $g^q$ is a generator.  Is my proof correct at all and is there maybe a more elegant argument? Is it possible to find all php files within a certain directory that have been modified on a certain date  I'm using   find /var/www/html/dir/ -mtime -28 | grep '\.php'    to get files modified within the last 28 days, but I only need files that have been modified on the following date: 2011-02-08 I can't understand where the final keyword is really handy when it is used on method parameters.    If we exclude the usage of anonymous classes, readability and intent declaration then it seems almost worthless to me.   Enforcing that some data remains constant is not as strong as it seems.    If the parameter is a primitive then it will have no effect since the parameter is passed to the method as a value and changing it will have no effect outside the scope.  If we are passing a parameter by reference, then the reference itself is a local variable and if the reference is changed from within the method, that would not have any effect from outside of the method scope.    Consider the simple test example below. This test passes although the method changed the value of the reference given to it, it has no effect.  public void testNullify() {     Collection&lt;Integer&gt; c  = new ArrayList&lt;Integer&gt;();           nullify(c);     assertNotNull(c);            final Collection&lt;Integer&gt; c1 = c;     assertTrue(c1.equals(c));     change(c);     assertTrue(c1.equals(c)); }  private void change(Collection&lt;Integer&gt; c) {     c = new ArrayList&lt;Integer&gt;(); }  public void nullify(Collection&lt;?&gt; t) {     t = null; } Make Firefox save new bookmarks in the Bookmarks Menu (not Other Bookmarks) Recently, Firefox has changed the default folder for newly created bookmarks (using Ctrl+D) from the Bookmarks Menu to "Other Bookmarks".  Is there some way to change that back? Or some alternate method instead of Ctrl+D?  EDIT  A native way to do this (without an add-on) would be best.