How to make a node tree the context? (MTree implementation)
How to find out why "context is incorrect"?
Is there any uncountably infinite set that does not generate the reals? Does there exist an uncountably infinite set $X \subseteq \mathbb R$ such that $\mathbb R \neq \left&lt;X\right&gt;$? I can't think of any, but I'm also having trouble trying to prove that no such subset exists.  For example: $\mathbb R$ is uncountable and obviously $\mathbb R = \left&lt;\mathbb R\right&gt;$. The Cantor set $C$ is uncountable, and we know that $C - C = [0, 1]$, so then since $\mathbb R = \left&lt;[0, 1]\right&gt;$ we know that $C$ also generates $\mathbb R$. Also the set of irrationals $\mathbb R \setminus \mathbb Q$ is uncountable, but we can generate all the rational numbers by fixing one irrational number $\alpha$ and then saying the any rational number $x$ shall be $(\alpha + x) - \alpha$, since both $\alpha + x$ and $\alpha$ are irrational.  So the examples that quickly come to mind all generate the reals. Is there a simple counterexample? Determine if two rectangles overlap each other? I am trying to write a C++ program that takes the following inputs from the user to construct rectangles (between 2 and 5): height, width, x-pos, y-pos. All of these rectangles will exist parallel to the x and the y axis, that is all of their edges will have slopes of 0 or infinity.  I've tried to implement what is mentioned in  question but I am not having very much luck.  My current implementation does the following:  // Gets all the vertices for Rectangle 1 and stores them in an array -&gt; arrRect1 // point 1 x: arrRect1[0], point 1 y: arrRect1[1] and so on... // Gets all the vertices for Rectangle 2 and stores them in an array -&gt; arrRect2  // rotated edge of point a, rect 1 int rot_x, rot_y; rot_x = -arrRect1[3]; rot_y = arrRect1[2]; // point on rotated edge int pnt_x, pnt_y; pnt_x = arrRect1[2];  pnt_y = arrRect1[3]; // test point, a from rect 2 int tst_x, tst_y; tst_x = arrRect2[0]; tst_y = arrRect2[1];  int value; value = (rot_x * (tst_x - pnt_x)) + (rot_y * (tst_y - pnt_y)); cout &lt;&lt; "Value: " &lt;&lt; value;     However I'm not quite sure if (a) I've implemented the algorithm I linked to correctly, or if I did exactly how to interpret this?  Any suggestions? Can principal component analysis be applied to datasets containing a mix of continuous and categorical variables? I have a dataset that has both continuous and categorical data. I am analyzing by using PCA and am wondering if it is fine to include the categorical variables as a part of the analysis. My understanding is that PCA can only be applied to continuous variables. Is that correct? If it cannot be used for categorical data, what alternatives exist for their analysis? In the sentence:     All the information you need is in these folders.   I would have expected to find:     All the information that/which you need is in these folders.   Why are those omitted? I've had a promising Skype interview with a great professor from a top-level university in US. Our research interests are closely related and he encouraged me to apply BUT he says that all decisions are finally made by admission committee there.  The question is "Is a professor in such a high-level university able to affect admission committee if he really wants a student?" or "Does admission committee decision-making process become easier if a professor really wants an applicant?"  The main point is the difference between my transcript (which is a real catastrophe!) and my research background which is pretty good so the professor's decision might be different from admission committee. Salesforce Toolkit for .Net with silverlight 5 I am creating a application with Silverlight 5 in which I have to communicate with Salesforce APIs.  Currently I used Salesforce toolkit for .net (version 1.2.1) by I am gettting error after using authentication method as -     The base class or interface 'System.Object' in assembly   'System.Runtime, Version=4.0.0.0, Culture=neutral,   PublicKeyToken=b03f5f7f11d50a3a' referenced by type   'Salesforce.Common.AuthenticationClient' could not be resolved.   System.runtime dll is referenced my application.   And Latest version of toolkit is even not installing in application. There is some compatibility issue with silverlight 5.  As per github and blogs Salesforce Toolkit is compatible with Silverlight 5. I installed it using nuget package manager.  Any idea to fix this ?  Thanks, Finding Radii of Convergence for $\sum a_n z^{2n}$ and $\sum a_n^2 z^n$ Setting: Let $\sum a_n z^n$ have radius of convergence $R$.  We have that  $$ R = \frac{1}{\underset{n \rightarrow \infty}{\limsup} \left|a_n \right|^{1/n}} $$  via Hadamard's formula for the radius of convergence.  Question: What are the radii of convergence for (i) $\sum a_n z^{2n}$ and (ii) $\sum a_n^2 z^n$?  Attempt for $\sum a_n^2 z^n$:   For ease of notation, let $R_1$ and $R_2$ denote the radii of convergence for power series $\sum a_n z^{2n}$ and $\sum a_n^2 z^n$ respectively. From Hadamard's formula, we then have  $$ R_2 = \frac{1}{\underset{n \rightarrow \infty}{\limsup} \left|a_n^2 \right|^{1/n}} = \left( \frac{1}{\underset{n \rightarrow \infty}{\limsup} \left|a_n \right|^{1/n}} \right)^2 = R^2 $$   But what about $R_1$? How can I make environment variables "exported" in a shell script stick around? After reading , the most popular answer use the identity $$\sum_{t=0}^n \binom{t}{k} = \binom{n+1}{k+1}.$$  What's the name of this identity? Is it the identity of the  modified.  How can we prove it? I tried by induction, but without success. Can we also prove it algebraically?  Thanks for your help.    EDIT 01 : This identity is known as the  because, on Pascal's triangle, when the addends represented in the summation and the sum itself are highlighted, a hockey-stick shape is revealed. Converse to Hilbert basis theorem Prove the converse to Hilbert basis theoren:     If the polynomial ring $R[x]$ is Noetherian, then $R$ is noetherian. What does “would not have been possible had I remained” mean? What does this mean?      I am sure this would not have been possible had I remained a typical Anglophone North American.   I have never heard that before and I really find it strange.  I deduce that it means      I am sure this would not have been possible if I kept remaining Anglophone North American.   But I might be wrong in this. Where do I find system sounds? After installing gm-notify I see the option of hearing a sound every time a new e-mail arrives in my Gmail inbox, but I don't know where Ubuntu (10.4) stores the system sounds to assign one. Any ideas? Thanks. The question is to show that for any $x,y \in\Re^n$  $||x+y||\cdot||x-y||\le||x||^2+||y||^2$  with equality if and only if $\langle x,y \rangle= 0.$  I have that $||x+y||^2=||x||^2+2 \langle x,y \rangle+||y||^2$ , which helps me prove equality but I'm lost on how to show this and show that the inequality part. Using the field of an object as a generic Dictionary key If I want to use objects as the keys for a Dictionary, what methods will I need to override to make them compare in a specific way?  Say I have a a class which has properties:  class Foo {     public string Name { get; set; }     public int FooID { get; set; }      // elided }    And I want to create a:  Dictionary&lt;Foo, List&lt;Stuff&gt;&gt;   I want Foo objects with the same FooID to be considered the same group.  Which methods will I need to override in the Foo class?  To summarize: I want to categorize Stuff objects into lists, grouped by Foo objects. Stuff objects will have a FooID to link them to their category. From 2012 onward the SQL Server docs show they support  which I'm trying to use instead of a LIMIT.  The following works fine in PostgreSQL to sample a result set,  SELECT * FROM ( VALUES (1),(2),(3) ) AS t(x) OFFSET 0 ROWS FETCH NEXT 1 ROWS ONLY;   However, with SQL Server, I get  Msg 153, Level 15, State 2, Line 4 Invalid usage of the option FIRST in the FETCH statement.   What's going on here? Does SQL Server support the standardized OFFSET .. FETCH? If M is a non-orientable closed connected 3 manifold prove H1(M) is an infinite group. Why is it that they decided to make String immutable in Java and .NET (and some other languages)? Why didn't they make it mutable? "provide" vs. "provide with" I am wondering if the following sentence is correct:     We add the information their study provides with to our article.   The context is: their study provides with some information. And we add the information to our article.  I want to keep the word "add", and someone told me that "provides with to" sounds wired... Prove the inequality $n! \geq 2^n$ by induction I'm having difficulty solving an exercise in my course.  The question is:      Prove that $n!\geq 2^n$.   We have to do this by induction. I started like this:   The lowest natural number where the assumption is correct is $4$ as: $4! \geq 2^4 \iff 24 \ge 16$. The assumption is: $n! \ge 2^n$.   Now proof for $(n+1)$ which brings me to: $(n+1)! \ge 2^{(n+1)}$  I think I can rewrite it somehow like this:  $$ {(n+1)} \times {n!}  \stackrel{\text{(definition of factorial)}}{\ge} 2^n \times 2 $$  $$ (n+1) \times 2^n \ge 2^n \times 2 $$  Then I think I can eliminate the $2^n$ and have something like this: $n+1 \ge 2$, or $n \ge 1$.  But I think I'm wrong here some where and was hoping somebody has some advice on this. How can I prove the above assumption?  Any help would be appreciated, kind regards. There have been several questions already posted with specific questions about , such as when to use it and what frameworks are there for it. However,  What is dependency injection and when/why should or shouldn't it be used?