Let $S^1$ be the unit circle in $\mathbb{R}^2$, then for a continuous map from $S^1$ into $\mathbb{R}$, there exists $x \in S^1$ such that $f(x) = f(-x)$.    I know that $S^1$ is path connected and so connected.  And the intermediate value theorem.  For any choice of $-x, x$, and any $r$ between $f(-x)$ and $f(x)$ there is $c \in S^1$ such that $f(c) = r$.  Let $\psi : [0,2\pi] \rightarrow S^1, \ x \mapsto e^{i(a + x)}$ for any $a\in \mathbb{R}$ be a circular path in $S^1$ identified with the circle group as a subset of $\mathbb{C}$.  Let $g(x) = f(x) - f(-x)$.  Let $a\in S^1$ be any point such that there exist points in the image of $g$ to the right of $g(a)$ for at least a run of $\epsilon$.  Now translate everything to near the origin with $h(x) = g(x) - g(a) - \epsilon/2$  Can't figure this one out.
Question is to prove :  $f : S^1 \to \mathbb R$ is continuous then $f(x)=f(-x)$ for some $x\in S^1$  I guess it would be helpful to use intermediate value theorem  Assuming $f(x)\neq f(-x)$ then given any $p\in (f(-x),f(x))$ (assuming $f(-x)&lt;f(x)$) there exists $y\in (-x, x)$ such that $f(y)=p$  I am not very sure of how to use this..   It would be helpful if someone can give some hint which would help me to solve this..  Thank you.
Apache on MacOS Sierra: AH00557: httpd: apr_sockaddr_info_get() failed for Macbook-Pro.local AH00558: httpd: Could not reliably determine… I updated MacOS Sierra's Apache and PHP, and now whenever I $ sudo apachectl restart I get this:  AH00557: httpd: apr_sockaddr_info_get() failed for My-MacBook-Pro.local AH00558: httpd: Could not reliably determine the server's fully qualified domain name, using 127.0.0.1. Set the 'ServerName' directive globally to suppress this message   Things I've tried so far that didn't work:   Setting ServerName localhost in /etc/httpd.conf Setting ServerName localhost:80 in /etc/httpd.conf Setting ServerName 127.0.0.1 in /etc/httpd.conf Setting ServerName 127.0.0.1:80 in /etc/httpd.conf Ensuring 127.0.0.1       localhost is set in /etc/hosts Repeatedly restarting Apache How to simulate artificial data for logistic regression? I know I'm missing something in my understanding of logistic regression, and would really appreciate any help.  As far as I understand it, the logistic regression assumes that the probability of a '1' outcome given the inputs, is a linear combination of the inputs, passed through an inverse-logistic function. This is exemplified in the following R code:  #create data: x1 = rnorm(1000)           # some continuous variables  x2 = rnorm(1000) z = 1 + 2*x1 + 3*x2        # linear combination with a bias pr = 1/(1+exp(-z))         # pass through an inv-logit function y = pr &gt; 0.5               # take as '1' if probability &gt; 0.5  #now feed it to glm: df = data.frame(y=y,x1=x1,x2=x2) glm =glm( y~x1+x2,data=df,family="binomial")   and I get the following error message:      Warning messages:   1: glm.fit: algorithm did not converge    2: glm.fit: fitted probabilities numerically 0 or 1 occurred    I've worked with R for some time now; enough to know that probably I'm the one to blame.. what is happening here? Why too many capacitors in parallel for Vdd supply net? Can't we just add all to replace with one big capacitor? Combining two sorted lists in Python I have two lists of objects. Each list is already sorted by a property of the object that is of the datetime type. I would like to combine the two lists into one sorted list. Is the best way just to do a sort or is there a smarter way to do this in Python? How to find sum of power series $\sum_{n=0}^\infty\frac{1}{n!(n+2)}$ by differentiation and integration? Let $$ S = \sum_{n=0}^\infty \frac{1}{n!(n+2)} $$  Integrate the Taylor Series of $xe^x$ to show that S = 1  Also, Differentiate the Taylor series of $\frac{e^x - 1}{x}$ to show that S = 1  For the integration one, I got $\int xe^x dx$ = $x^2\sum_{n=0}^\infty \frac{1}{n!(n+2)} x^n$ and pretty much stuck. I can see the S term but i have no idea how to move on from there Suppose $k$ is a field that is finitely generated as a ${\mathbb Z}$-algebra.  (That is, $k$ is a quotient of ${\mathbb Z}[X_1,\dots,X_n]$ for some $n$). Does it follow that $k$ is finite? Why do non-hydrogen atomic orbitals have the same degeneracy structure as hydrogen orbitals? Is every sufficiently large positive integer of the form $ab + ac + bc + 1$? Is every sufficiently large positive integer $A$ of the form $ab + ac + bc + 1$   where $a,b,c$ are some positive integers larger than some given positive integer $d$ ?  How large is sufficiently large , in other words how do we know our number $A$ is sufficiently large ( as a function of $d$ ) ?    Update :  I read the following about positive integers that are not of the form $ ab + ac + bc $:    And I quote about that sequence above : " Note that for n in this sequence, n+1 is either a prime, twice a prime, the square of a prime, 8 or 16 "  This is why I considered the weird +1 in $ab + ac + bc + 1$  So composites larger than 16 not a prime , twice a prime or the square of a prime are of the form $ab + ac + bc + 1$.  And I wonder why !  I guess that relates strongly to the Original posted problem. Maybe I should have mentioned that before , sorry.  Will Jagy's answer gave the same OEIS link which I already knew. This also explains some of the comments.  Im no expert in genus theory or Galois theory but I assume this has a simple answer. Obtain the joint PDF of $\theta_{1} - \theta_{2}$ and $\theta_{1}$ where $\theta_{1}$ and $\theta_{2}$ are i.i.d $\text{Uniform}[0, 2\pi)$ I have 2 i.i.d Uniform random variables $\theta_{1}$ and $\theta_{2}$ on $[0, 2\pi)$. Here the problem pushes us to consider that all operations are made modulo $2\pi$. My question is how to obtain the joint PDF of $\theta_{1} - \theta_{2}$ and $\theta_{1}$ and then obtain the conditional PDF of $\theta_{1}$ given $\theta_{1} - \theta_{2}$??. For the second part It can be deduced from the first by writing it as the joint PDF divided by the PDF of $\theta_{1}$. But for the first part, A common idea that I saw is to derive the $\text{CDF}(x, y) = P(\theta_{1} - \theta_{2} \le x$, $\theta_{1} \le y$) and then perform derivations respect to $x$ and $y$. Is this a right approach? If yes, how to proceed with CDF especially with our uniform distribution restrictions? Can you flank with a reach weapon? The Variant: Flanking rules in the DMG (page 251) require adjacency, but is the intent that the Enemy has combatants on opposite sides and thus someone with Reach (and using the reach) could still flank even if not directly adjacent?  Looking for both RAW as well as Intent of the flanking rule. How do you write a case insensitive query for both MySQL and Postgres? I'm running a MySQL database locally for development, but deploying to Heroku which uses Postgres. Heroku handles almost everything, but my case-insensitive Like statements become case sensitive. I could use iLike statements, but my local MySQL database can't handle that.  What is the best way to write a case insensitive query that is compatible with both MySQL and Postgres? Or do I need to write separate Like and iLike statements depending on the DB my app is talking to? how do you make clothes move along with your model? Do you rig the model first and then parent it to the body of the character then click cloth simulation? I am a little bit confused with the process. Thanks for the help. Dodecagon Area Question The distance between two opposite vertices of the dodecagon is 2. Find the area of the dodecagon.  Is there any way to do this without trigonometry?  And could you include a proof also? :O HTML encoding strikes again (but with a twist) - Chat messages inbox notifications aren&#39;t looking good What to look for in a starter-level off-camera flash? I have zero experience with flash and want to learn how to use it from zero, for example by following the Strobist blog.  What are the absolutely necessary specifications I should look for when shopping for a flash, and can I find something in a price range less than $200.00?  My camera is a Nikon DSLR. I washed and dried my passport by mistake, do I need a new one? Around two years ago I washed and dried my passport by mistake at the laundry room. My ID, my old US student visa and my stamps from european countries are still readable but the pages are not "flat" anymore and the front cover is kind of erasing (some parts of the Brazil symbol is gone).   In April I'll travel to US for a conference, I'll need to get a new US visa (B1/tourism visa) but I'm worried if my passport is not "good to go" anymore because it was "washed and dried". Apparently there is an electronic chip in it and I don't know if it is still working.  Should I get a new passport or can I save some money and time and use this one? It expires only in 2019. Is it possible to do something like this  #ifdef SOMETHING #define foo // #else #define foo MyFunction #endif   The idea is that if SOMETHING is defined, then calls to foo(...) become comments (or something that doesn't get evaluated or compiled), otherwise it becomes a call to MyFunction.  I've seen __noop used, but I don't believe I can use that.  EDIT(s):  I don't think I can really use a macro here, because MyFunction takes a variable number of arguments.  Also, I'd like to make it so the arguments are NOT evaluated!  (So doing something like commenting out the body of MyFunction doesn't really give me what I need, as the arguments will still be evaluated) How can I withdraw a publication from a predatory (fake) journal and resubmit to a legitimate journal? I have published a paper in a fake journal. I want to withdraw the paper from that journal.   After successful withdrawal from the fake journal, can I can submit the same paper to a legitimate journal? In $\mathbb{Z}\left [ x \right ]$, let $I = \left \{ f\left ( x \right ) \in \mathbb{Z}\left [ x \right ]: f\left ( 0 \right ) \text{ is an even integer} \right \}$   In fact, $I=\left \langle x,2 \right \rangle$  How do I show that the order of $\mathbb{Z}\left [ x \right ]/I$ is 2? Cartesian Product of Two Countable Sets is Countable