Does the Universe spin?
What if the universe is rotating as a whole?
$$ 3x+6y+5z=7 $$ The general solution to this linear Diophantine equation is as described    $$ x = 5k+2l+14 $$ $$ y = -l $$ $$ z = -7-k $$ $$ k,l \in \mathbb{Z} $$  If I plug the original equation into  the solution is: $$ y = 5n+2x+2  $$ $$ z =-6n-3x-1 $$ $$ n \in \mathbb{Z} $$  I can rewrite this as:  $$ x = l $$ $$ y = 5k+2l+2 $$ $$ z = -6k-3l-1 $$ $$ k,l \in \mathbb{Z} $$  However now two equations depend on two variables ($k,l$) and one on one variable $l$. In the first solution one equation depends on two variables and two on one variable.  Questions:  How can I come from a representation like the one from wolfram alpha for the general solution to one where all equations depend on one distinct variable except one equation.  Is there always such a representation? Using single quote by curly braces in String.format in Dynamic SOQL I'm struggling to use String.format to replace a token with a value when the token is within single quotes:  Example:  String query = String.format('SELECT Id FROM Availability WHERE Person__c = \'{0}\'', personId);   String query is equal to:  SELECT Id FROM Availability WHERE Person__c = {0}   Rather than:  SELECT Id FROM Availability WHERE Person__c = '234567890' Separate long math text under sum symbol into different lines? Reflection to Identify Extension Methods Does a creature with Lifelink provide lifegain equal to its power, or the defender's toughness? I have recently began playing a deck that has a number of creatures with lifelink in it.  I want to be certain I understand the mechanic and the following question occurred to me.  Which damage total do you use to determine the life gain from lifelink?  I had assumed that if  a 3/3 creature with lifelink was blocked by a 2/2 creature that the life gain total would be 2, the maximum that the blocker can absorb before dying.  This seems to be incorrect, as when I read the example attached to CR 119.4c      [...] attacks with a 3/3 creature with wither and lifelink. It’s blocked by a 2/2 creature, [...]. The damage event starts out as [3 damage is dealt to the 2/2 creature, 2 damage is dealt to the 3/3 creature]   Have I been short changing myself? Why don't C++ compilers define operator== and operator!=? I am a big fan of letting the compiler do as much work for you as possible.  When writing a simple class the compiler can give you the following for 'free':   A default (empty) constructor A copy constructor A destructor An assignment operator (operator=)   But it cannot seem to give you any comparison operators - such as operator== or operator!=.  For example:  class foo { public:     std::string str_;     int n_; };  foo f1;        // Works foo f2(f1);    // Works foo f3; f3 = f2;       // Works  if (f3 == f2)  // Fails { }  if (f3 != f2)  // Fails { }   Is there a good reason for this?  Why would performing a member-by-member comparison be a problem?  Obviously if the class allocates memory then you'd want to be careful, but for a simple class surely the compiler could do this for you? Render an InputText based on Selected Option In my VF page I have an apex:selectList with a List of Vendors (e.g. Alcatel,Cisco ec..), the last option is "Other".   When the user select "Other" an apex:inputText should be rendered  where he can writes the Vendor. I'm not able to render the inputText Field   My VF page:  &lt;apex:outputPanel id="vendor2"&gt; &lt;p&gt;&lt;b&gt; Vendor: &lt;/b&gt;&lt;/p&gt; &lt;apex:selectList value="{!vendor}" multiselect="False" size="1"&gt;     &lt;apex:actionSupport event="onchange" action="{!specifyVendor}" rerender="otherPanel"/&gt;     &lt;apex:selectOptions value="{!vendors}"/&gt; &lt;/apex:selectList&gt;&lt;p/&gt; &lt;/apex:outputPanel&gt; &lt;br/&gt; &lt;apex:outputPanel id="otherPanel"&gt;     &lt;apex:inputText value="{!otherVendor}" rendered="{!vendor=='Other'}"&gt; Other: &lt;/apex:inputText&gt; &lt;/apex:outputPanel&gt;     &lt;br/&gt;&lt;br/&gt;    The custom controller:      public with sharing class ExcelPageController {          public String vendor {get; set;}//Vendor Name         public String otherVendor {get; set;}//Vendor Name          public List&lt;SelectOption&gt; getVendors(){             System.debug('getVendors');                 List&lt;SelectOption&gt; options = new List&lt;SelectOption&gt;();             options.add(new SelectOption('Cisco','Cisco'));             options.add(new SelectOption('Alcatel','Alcatel'));             options.add(new SelectOption('Tibco','Tibco'));             options.add(new SelectOption('Other','Other'));             return options;         }     public PageReference specifyVendor(){         System.debug(vendor);         return null;     } } Why do objects in front and back view look reversed? Whenever I model objects that has different look at front and back. Viewing it from front perspective or orthographic the model displays back of the object. Generate a random variable with a defined correlation to an existing variable(s) For a simulation study I have to generate random variables that show a predefined (population) correlation to an existing variable $Y$.  I looked into the R packages copula and CDVine which can produce random multivariate distributions with a given dependency structure. It is, however, not possible to fix one of the resulting variables to an existing variable.  Any ideas and links to existing functions are appreciated!  &nbsp;  Conclusion: Two valid answers came up, with different solutions:   An R  by caracal, which calculates a random variable with an exact (sample) correlation to a predefined variable An R  I found myself, which calculates a random variable with a defined population correlation to a predefined variable     [@ttnphns' addition: I took the liberty to expand the question title from single fixed variable case to arbitrary number of fixed variables; i.e. how to generate a variable having predefined corretation(s) with some fixed, existing variable(s)] Writing in bash while it is working When you are performing time consuming operations in bash like installing new software, is it possible to write text that will appear at the prompt when the operation has finished.  E.g   Imagine running:  apt-get install eclipse-platform   Then it will use quite a while to finish while you see the installation log, meanwhile I want to create new folders (workspaces) which Eclipse later will use.  Is this possible without opening a new terminal (or tab)? Stopping resetting of linked locators in manipulate How do I remove the space between inline/inline-block elements? I've never really understood why math induction is supposed to work.  You have these 3 steps:   Prove true for base case (n=0 or 1 or whatever) Assume true for n=k.  Call this the induction hypothesis. Prove true for n=k+1, somewhere using the induction hypothesis in your proof.   In my experience the proof is usually algebraic, and you just manipulate the problem until you get the induction hypothesis to appear.  If you can do that and it works out, then you say the proof holds.  Here's one I just worked out,  Show $\displaystyle\lim_{x\to\infty} \frac{(\ln x)^n}{x} = 0$  So you go:   Use L'Hospital's rule. $\displaystyle\lim_{x\to\infty} \frac{\ln x}{x} = 0$. Since that's $\displaystyle\lim_{x\to\infty} \frac{1}{x} = 0$. Assume true for $n=k$. $\displaystyle\lim_{x\to\infty} \frac{(\ln x)^k}{x} = 0$. Prove true for $n=k+1$.  You get $\displaystyle\lim_{x\to\infty} \frac{(\ln x)^{k+1}}{x} = 0.$   Use L'Hospital again: $\displaystyle\lim_{x\to\infty} \frac{(k+1)(\ln x)^{k}}{x} = 0$.  Then you see the induction hypothesis appear, and you can say this is equal to $0$.  What I'm not comfortable with is this idea that you can just assume something to be true ($n=k$), then based on that assumption, form a proof for $n=k+1$ case.  I don't see how you can use something you've assumed to be true to prove something else to be true. Define the Lie algebra $\mathfrak{g}$ of a Lie group G to be the set of all left-invariant vector fields of $G$. I want to prove that   $$          f: \mathfrak{g} \rightarrow T_eG \\              X \mapsto X(e) $$ is a linear isomorphism.  The only thing that I have left to prove is surjectivity.Letting $v\in T_eG$, we set $X\in \mathcal{X}(G)$ such that $\forall x\in G,\,X(x)=L_{x_{*,e}}(v)$ (I still have to show that such an $X$ is indeed a vector field on $G$). We get that  $$ f(X)=X(e)=L_{e_{*,e}}(v)=v $$  where the last equality holds because  $$       L_{e_{*,e}}(v)f = v(f \circ L_e) = v(f \circ id_G) = v(f) $$ for any $f\in C^\infty(G)$. (I am thinking of the tangent vectors as derivations).  Now all we have left to do is prove that such an $X$ is left-invariant, and it is therefore an element of $\mathfrak{g}$, but I do not know how. I tried to prove that  $$            \forall f\in C^\infty(G),\, L_{{x}_*,y}(X(y))f=X(L_x(y))f $$ but with no success. Questions about insects eating flour and living in the cracks of a wooden cutting boards I have a large wooden cutting board which is used in my kitchen mainly for preparing egg pasta. The wood type is Populus (poplar or aspen or cottonwood).  The main ingredients used with the cutting board are: flour (mainly soft wheat flour), hens' eggs, white sugar, butter, and baking powder.  After the food preparation, the cutting board is cleaned by scraping away the ingredients remains.  The board has some cracks and some small unknown insects are living inside the cracks.   The board has also a lot of holes due to woodboring beetles but I think the woodworms operated long time ago and they no longer live in the cutting board.  The unknown insect has an approximate length of 1 mm (along the direction of their movements) and a width of 0.3 mm. The unknown insects do not have wings, they just walk. I am not able to see how many pairs of legs they have.  The color of the insect is pale brown.  The cutting board was free from these unknown insects some months ago but then they appeared.  My questions:   What insects are they? Are these insects poisonous to humans? How can I get rid of them?   Update  The unknown insects could be Acarus siro, I will try to get a picture of the insects.  Update 2  Here is a picture. The bottom part of the image is a one Euro coin, the top part of the image is a caliper (each bright area is one mm tick).   Some more pictures: Incubated but not hatched egg I got the message "oh" and it showed my egg. Before I could do anything the server went down. When I got back on my egg was gone. Is there anything I can do to retrieve my incubated but not hatched egg? How to prove and interpret $\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B))$? I read somewhere that companies are not required to pay dividends to shareholders (this is correct, yes?).   If so, then if company A never pays dividends to its shareholders, then what is the point of owning company A's stock? Surely the right to one vote for company A's Board can't be that valuable.   What is it that I'm missing? Why is Facebook PHP SDK getUser always returning 0? C# compiler error: "cannot have instance field initializers in structs"