Degenerate measure on a metric space
If a measure only assumes values 0 or 1, is it a Dirac's delta?
Conditional typesetting / build Let's say I have .tex file which is a set of problems and solutions, of which I want to make two pdf files, one which includes the solutions and one which doesn't. so I need to build the .tex file under two different modes, and specify that parts of the document be ignored under one of the modes.  This is something common in programming context. e.g. in C++ i can code  #ifdef DEBUG   // do this additional stuff ... #endif   How can I do something equivalent in latex? References for multivariable calculus Due to my ignorance, I find that most of the references for mathematical analysis (real analysis or advanced calculus) I have read do not talk much about the "multivariate calculus". After dealing with the single variable calculus theoretically, it usually directly goes to the topic of measure theory.    After reading the wiki article "", I'd like to find the rigorous proof for this test. The first book comes to my mind is Courant's Introduction to Calculus and Analysis which includes the multivariate case in the second volume.   Motivated by this, I'd like to put the question here:     What are the usual references for the   theoretical treatment for   ? Graduate school admission with a degree in a different field What are Multi-Zone/Matrix metering, Center-Weighted metering, and Spot metering? What about Partial?  Is there a good rule of thumb, or a few pointers, on when best to use each exposure metering mode? I am having an access problem to some files and directories that I have tracked down to the group assigned to access them.    When I manually change the name of the group in the properties/permissions menu for a directory from the default setting (MyGroup) to www-data used by my Drupal Website, the directory error messages I get are gone.    There are a large number of files and directories that require this change.  If believe using chmod will change the owner which is set correctly and being new to Ubuntu, I am reluctant to experiment without a better understanding of how owner and groups work in the permissions settings.   What command changes the Group setting for a directory? $\int_{\frac{1}{3}\pi}^{\frac{2}{3}\pi} {\sin(x)\;dx}$ using Riemann sums? What function satisfies $ f(x)+f\left(\frac{1-x}x\right)=x$ ? I just went to install Java 8 on my ancient laptop that runs Windows XP SP3.  Java pops up a message:     Java 8 requires a newer version of Windows. You may continue with the installation, but for Java to work as specified we recommend upgrading your computer's operating system.   Will Java fail to work properly on Windows XP? Or is this just Oracle trying to be secure? To be clear on this, I know what is the definition of an inner product space and some properties and theorems about them. What I am asking for is an intuition for this definition in the complex case. In the real case, the intuition (or at least one of them) is  geometric: The inner product of two vectors is the length of the projection of the first to the second scaled by the norms of both vectors so that it is symmetric (modulo some details). In particular I (and everybody else) think of "inner product zero" as geometric orthogonality and of orthonormal bases as, well, orthonormal bases and so on. The question is, what should I think about when working with complex (or should I say hermitian?) inner product spaces? what is the "meaning" of the complex number associated to two vectors called their inner product?  I will be happy to hear all kinds of answers. For example, what physical phenomena does it model or in what mathematical situations does in "naturally" appear. Answers that stress the "nice structure" resulting are also welcome, yet I feel that by itself  it is a bit unsatisfying. It is well known that compactness implies pseudocompactness; this follows from . I know that the converse does not hold, but what is a counterexample?  (A  is a topological space $S = \langle X,{\mathfrak I}\rangle$ such that every continuous function $f:S\to\Bbb R$ has bounded range.) Returning multiple values from a C++ function Triangle Inequality question with fractions Given $a, b \in \mathbb{R}$, prove that   $$\frac{|a + b|}{1 + |a + b|} \le \frac{|a|}{1 + |a|} + \frac{|b|}{1+|b|}$$  When does equality hold?  The only useful thing I could get (using the triangle inequality) is:  $$\begin{align}1 + \left|\left(a + \frac{ab}{2}\right) + \left(b - \frac{ab}{2}\right)\right| &amp;\le 1 + \left|a + \frac{ab}{2}\right| + \left|b - \frac{ab}{2}\right|\\ &amp;\le 1 + |a| + |b| + |ab|\\ &amp;=(1 + |a|)(1 + |b|)\end{align}$$  But it seems to be inapplicable! Any ideas? Suppose someone from "Harry's world" married a Muggle, and they want to escort their child. Can the Muggle parent get through to the Hogwarts platform? Repeating "to" and "will" in enumerations of verbs Should I use the second "will" in constructions like this one: "it will definitely help you and will make the text more readable"  And should I write "to" before every infinitive in enumeration, or only before the first one, e.g. "it helps to develop and test" or "it helps to develop and to test"? Could this example be influenced by the fact that "help" can go with to-invinitive and bare infinitive, and with other verbs the rule would be different? Can I run PC steam games on a Mac? Related questions with over 99 score display incorrectly The score for the 4 questions are 111, 1218, 261 and 5 respectively, but they display 2 digits per line. This is an obvious bug in design. I've just read a proof of the statement:  Let $R$ be a commutative ring. If $R[X]$ is a principal ideal domain, then $R$ is a field.  In one part of the proof there is a step which I don't understand. I'll copy the proof: Let $u\in R$ be non-zero. Then using the principal ideal property, for some $f\in R[X]$ we have $\langle u,X\rangle = \langle f \rangle \subseteq R[X]$. Therefore, for some $p,q∈R[X], u=fp$ and $X=fq$. By properties of degree we conclude that $f=a$ and $q=b+cX$ for some $a,b,c\in R$. Substituting into the equation $X=fq$ we obtain $X=ab+acX$ which implies that $ac=1$, i.e. $a\in R^\times$, the group of units of $R$. Therefore, $\langle f\rangle =\langle 1\rangle=R[X]$. Therefore, there exist $r,s\in R[X]$ such that $ru+sX=1$ and if $d$ is the constant term of $r$, then we have $ud=1$. Therefore $u\in R^\times$. Our choice of $u$ was arbitrary, so this shows that $R^{\times}⊇R \setminus \{0\}$, which says precisely that $R$ is a field. I don't understand this: &quot;$X=ab+acX$ which implies that $ac=1$, i.e. $a\in R^\times$, the group of units of $R$. Therefore, $\langle f\rangle = \langle 1\rangle =R[X]$.&quot; Why does the fact that $a=f$ is a unit implies that $\langle f\rangle = \langle 1\rangle$? I would appreciate if someone could explain this to me. Thanks in advance. Uniform Scattering with vector I'm attempting to create a random spread of dots to create a gradient look. I'm getting really close to the effect I want by blending two scatter brushes. The issue I'm having is that the blended brushes creates unwanted lines, and the dots run into each other. I keep adjusting the settings with no luck.  Here's the effect I'm wanting to achieved.    Smooth transition  On Circular path Dots don't converge Scattered at random-ish     Here's how close I'm able to get:   Blended scatters create unwanted lines Dots converge Spacing isn't gradual from edge to center Why did the snake wink at Harry at the zoo? This is mostly a memory question - I'm having a hard time remembering this and I can't read the book right now.  On the first book, a snake at the zoo winks at Harry. For the sake of the question, we'll assume that snakes can in fact wink.  I have a fuzzy memory here - I think that it winked before Harry had any kind of interaction with the snake (was this before or after Harry broke the glass?)  So, why did the snake wink at Harry? Given that they had not met ever before.  I suppose that the only reason it would wink at Harry would be because it knows Harry can communicate with it. But this would imply that snakes are capable of sensing such ability on wizards - is there (other) evidence that this is true? 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)?