I'm trying to construct a quotient map $f$ from the $n$-sphere $S^n$ to the $n$-disk $D^n$, in which $f(x_1,x_2,\ldots,x_n,x_{n+1})=f(x_1,x_2,\ldots,x_n,-x_{n+1})$. The goal is to show that $S^n/\sim_f$ is homeomorphic to $D^n$.  (Is this the right way to go about it?)  I've been looking at the stereographic projection for inspiration, but haven't come up with anything. I only just started learning about topology. Any hints?
Let $\sim$ denote the equivalence relation on the $n$-sphere $S^n$ defined via   $$ (x_1,\dots,x_n,x_{n+1})\sim(x_1,\dots,x_n,−x_{n+1})\:\text{ for all }\: (x_1,\dots,x_{n+1})\in S^n. $$      Show that the quotient space $S^n/\sim$ is homeomorphic to the $n$-disc $D^n$.   I know that Ii have to show that there exists a bijective function $f: (S^n/\sim) \to D^n$ that is continuous and that the pre-image $f^{-1}$ is continuous as well, but I can't advance from here, any tips?
Derivative of $(y-XB)' h(y-XB)$ with respect to $B$ In a table of contents that has been generated with LaTeX's \tableofcontents command, I would like to add a vertical space before a Chapter item in the table of contents. One way that I have in my notes is to use  \addtocontents{toc}{\protect\vspace{10pt}}   at a desired location in the document so that this gets processed by \tableofcontents to include an appropriate vertical space in the table of contents. I have two questions:   Is the above the correct way to get a vertical space in a table of contents? Why is it necessary to use the \protect command here?   Thank you... A good approximation of $(1+x)^n$ is $1+xn$ when $|x|n &lt;&lt; 1$.  Does this approximation have a name?  Any leads on estimating the error of the approximation? How to change programmatically exposed filters in views I have a page view with exposed filters. One of the filters is term vocabulary, and that's dropdown list, user can select only one category (term).   I want to set programmatically one term from dropdown list when I came from specific URL. I tried with views_query_alter where $category is tid.    $set_exposed_input  = array();  $set_exposed_input['field_category_tid'] = $category;   if ($set_exposed_input) {     $view-&gt;set_exposed_input($set_exposed_input);  } A zero-dimensional ring is Noetherian? Proposition: Let $A$ be a non-zero ring that is not a field. Suppose $A$ is zero dimensional. Then it is Noetherian.   Proof: Let $p$ be a prime ideal of $A$. If $p$ is not maximal, then $p \subsetneq m$ for a maximal ideal $m$. Hence $dim A \ge 1$, contradiction. Hence every prime ideal is maximal. Consequently, every increasing (or decreasing) chain of prime ideals will be trivial in the sense that it will have length $0$. Hence the ring is Noetherian (and Artinian).  Is there a problem with the above proof? I have a vf page that is a custom button. That's it.  this VF page is on a STD Page Layout. When the user clicks on the VF page, the parent window needs to be redirected to the new URL.  The way this is working, is that the new page is opening in the vf section of the page layout.   &lt;apex:page standardController="Opportunity"  standardStylesheets="true" extensions="OpportunityController" &gt; &lt;base target="_parent" /&gt; &lt;body style="background:#f3f3ec" &gt; &lt;apex:form &gt; &lt;center&gt;  &lt;apex:commandButton action="{!redirect}"  value="Change Record Type" rendered="{!render}"/&gt; &lt;/center&gt;   &lt;/apex:form&gt;    &lt;/body&gt; &lt;/apex:page&gt; Run a batch file in a completely hidden way How do I test a private function or a class that has private methods, fields or inner classes? How do I unit test (using xUnit) a class that has internal private methods, fields or nested classes? Or a function that is made private by having  (static in C/C++) or is in a private () namespace?  It seems bad to change the access modifier for a method or function just to be able to run a test. Was the chip real or fake? I'm talking about the movie  Spoilers ahead.  Walter (Daniel Radcliffe) gave the task or blackmailed the Four Horsemen to steal a chip from a lab which can access any digital device. So they did steal the chip. And when they rescue Dylan Rhodes (Mark Ruffalo) from drowning in the water, Jack Wilder said that the chip is fake.     Jack: Wait something is not right.      Atlas: What's wrong?      Jack: It's fake.   Then they assumed that Walter did not know that the chip is fake. So they came with a plan to bluff. They pretended that they had the real chip. After they got caught by Walter, they gave the fake chip to Walter. Walter checked the chip and declared it as real. The Four Horsemen were surprised.     Walter: I'm sorry to interrupt. But I have to say congratulations and   thank you. It's real.      Lula: What? It's real!!!!!      Atlas: What? Is it real?      Jake: What is he talking about?   What happened? I didn't get it. What did I miss? Was the chip real? Was Walter pretending it to be real? If so, then why? If not, who switched the real chip with the fake one? Who had the real chip before? Are there cheap GPS or other navigator devices with downloadable maps for the whole world? I don't need a recommendation or a list but my knowledge of GPS and "navi"s and the differences between them is an utter mystery to me.  But often I'm lost in backstreets in random countries where access to something like a live Google Maps would be awesome. I absolutely cannot afford a smartphone with worldwide roaming though.  Can GPS's do what I want or am I barking up the wrong tree? If they can is it only high end ones than can? Flag abusive users I think the community does a great job policing questions/answers, etc. and the community has some good tools available to do this.  Between flagging questions with some reasons behind it and editing posts, closing them, and deleting them, the only thing that seems to be missing to me is the ability to flag a user.  Reasons to flag a user as abusive:   Duplicate User Abusive Behavior Offensive user name or profile User profile consists of spam Others? What is a good algorithm for estimating the median of a huge read-once data set? Function to determine if two numbers are nearly equal when rounded to n significant decimal digits I have been asked to test a library provided by a 3rd party. The library is known to be accurate to n significant figures. Any less-significant errors can safely be ignored. I want to write a function to help me compare the results:  def nearlyequal( a, b, sigfig=5 ):   The purpose of this function is to determine if two floating-point numbers (a and b) are approximately equal. The function will return True if a==b (exact match) or if a and b have the same value when rounded to sigfig significant-figures when written in decimal.   Can anybody suggest a good implementation? I've written a mini unit-test. Unless you can see a bug in my tests then a good implementation should pass the following:  assert nearlyequal(1, 1, 5)  assert nearlyequal(1.0, 1.0, 5)  assert nearlyequal(1.0, 1.0, 5)  assert nearlyequal(-1e-9, 1e-9, 5)  assert nearlyequal(1e9, 1e9 + 1 , 5)  assert not nearlyequal( 1e4, 1e4 + 1, 5)  assert nearlyequal( 0.0, 1e-15, 5 )  assert not nearlyequal( 0.0, 1e-4, 6 )    Additional notes:   Values a and b might be of type int, float or numpy.float64. Values a and b will always be of the same type. It's vital that conversion does not introduce additional error into the function. Lets keep this numerical, so functions that convert to strings or use non-mathematical tricks are not ideal. This program will be audited by somebody who is a mathematician who will want to be able to prove that the function does what it is supposed to do. Speed... I've got to compare a lot of numbers so the faster the better. I've got numpy, scipy and the standard-library. Anything else will be hard for me to get, especially for such a small part of the project. I have a front end module that generates an (ECG) signal that varies from +/-2.5 V. I want to shift this signal to 0 - 5V. What is the best way to do this? Would a summing amplifier like the below circuit good enough? With R1 = R2 and V1 = 2.5V, V2 = my signal, V3 = V4 = GND Calculating the limit $\lim((n!)^{1/n})$ Find $\lim_{n\to\infty} ((n!)^{1/n})$. The question seemed rather simple at first, and then I realized I was not sure how to properly deal with this at all. My attempt: take the logarithm,  $$\lim_{n\to\infty} \ln((n!)^{1/n}) =  \lim_{n\to\infty} (1/n)\ln(n!) =  \lim_{n\to\infty} (\ln(n!)/n)$$ Applying L'hopital's rule:  $$\lim_{n\to\infty} [n! (-\gamma + \sum(1/k))]/n! = \lim_{n\to\infty} (-\gamma + \sum(1/k))=  \lim_{n\to\infty} (-(\lim(\sum(1/k) - \ln(n)) + \sum(1/k))  = \lim_{n\to\infty} (\ln(n) + \sum(1/k)-\sum(1/k)  = \lim_{n\to\infty} (\ln(n))$$ I proceeded to expand the $\ln(n)$ out into Maclaurin form $$\lim_{n\to\infty} (n + (n^2/2)+...) = \infty$$ Since I $\ln$'ed in the beginning, I proceeded to e the infinity  $$= e^\infty  = \infty$$  So am I write in how I approached this or am I just not on the right track? I know it diverges, I was just wanted to try my best to explicitly show it. "Neither Michael nor Albert is correct" or "Neither Michael nor Albert are correct"? What is the correct sentence?  Neither Michael nor Albert is correct.   Neither Michael nor Albert are correct. Can you make a custom mouse cursor for the game engine? I would like the mouse to have a custom look like most games and not just the default white.  Is there a way I can make an object as the mouse? Address is invalid on local machine [on windows 8 only] For some reasons my brush stroke suddenly looks like this:     As the picture showed, the edges are jagged and pixelated. It doesn't happen before and it's just one of the built-in inking brush.  I know there are a lot of similar questions here but please don't mark this as duplicate. I double checked all the thing the other answers suggested: It's brush tool, not pencil, and my mode of brush and layer is Normal, not Dissolve. It's pixelated even if I draw with mouse so it's not tablet driver issue. I disabled GPU render but nothing changed. How do I fix this? What conservation law corresponds to Lorentz boosts? Noether's Theorem is used to relate the invariance of the action under certain continuous transformations to conserved currents. A common example is that translations in spacetime correspond to the conservation of four-momentum.  In the case of angular momentum, the tensor (in special relativity) has 3 independent components for the classical angular momentum, but 3 more independent components that, as far as I know, represent Lorentz boosts. So, what conservation law corresponds to invariance under Lorentz boosts?