How to prepare oneself to show a polyline based on latitude and longitude information in a csv/json file with heatmap information too?
How to Start Web Mapping?
For every set $A$, the empty set is a subset of $A$. The empty set is a set. Therefore, the empty set has a cardinality $\geq 1\ldots$ I have only recently been exposed to sets. According to , as seen on the first bullet mark of the link, $ \forall A: \emptyset \subseteq A$  Does this mean that $\emptyset$ is an element of all sets? (This is False, thank you those that answered)  Is the empty set also a set itself?  Assuming these statements are true, then the empty set therefore an element of the empty set. This does not sound right, please clarify for me. Thank you. Making all row sums and column sums non-negative by a sequence of moves The number of books in economics that we need to buy (is/are) three 1) The number of books in economics that we need to buy (is/are) three.   or       2) A number of books in economics that we can buy for the course (is/are) available in the college book store.   Possible answers:  a)is  b)are  My approach:  I am confused here why is and are are used in the 1 and 2 sentence.  What they exactly mean according to subject verb agreement rules. Functions satisfying $f:\mathbb{N}\rightarrow\ \mathbb{N}$ and $f(f(n))+f(n+1)=n+2$ Find all functions $f$ such that $f:\mathbb{N}\rightarrow\  \mathbb{N}$ and $f(f(n))+f(n+1)=n+2$   Let us plug in $n=1$  $f(f(1))+f(2)=3$  Since the function is from $\mathbb{N}$ to $\mathbb{N}$, $f(2)$ can only take the values $1,2$. Now we divide the problem into cases.  Case-1: $f(f(1))=2,f(2)=1$  We can assume that $f(1)=c$ for the time being. Then plugging in $n=3$ and using $f(2)=1$ gives $$f(3)=4-c$$ and again since the range of the function is positive integers,then $4-c$ has to be positive and hence $c$ belongs to  {$1,2,3$}. Now, $$f(1)=c$$$$\implies f(f(1))=f(c)$$$$\implies 2=f(c)$$ by the assumption that $f(f(1))=2$ . Now,since $c$ can only take the values $1,2,3$,we start treating cases. If $c=1$,$$f(c)=2$$$$\implies f(1)=2$$ but we know from the deinition of $c$ that $f(1)=c=1$,a contradiction.If $c=2$,then $2=f(c)=f(2)$ but $f(2)=1$ by assumption. Finally,if $c=3$ $$2=f(c)=f(3)$$ but $$f(3)=4-c=4-3=1$$ which is once again a contradiction. Therefore there are no such functions in this case.  Case-2: $f(f(1))=1,f(2)=2$  Again assuming $f(1)=c$ and using $f(n)\le n$ along with plugging $n=c-1$ will give us that $f(1)=1$ and then it is easy to prove that such a function exists by recursion. I can only give a "sort of recursive" way to define the function. Here it goes $$f:\mathbb{N}\rightarrow \mathbb{N}$$$$f(1)=1$$$$f(n)=n+1-f(f(n-1))$$  But this case is harder to deal with.Some help will be appreciated. Lookup Tables Best Practices: DB Tables... or Enumerations If we have to store the available positions at a company (i.e. Manager, Team Lead, ... etc). What are the best practices for storing it? I have two opinions with comments... "sure, welcoming yours"        Storing it as DB table with columns ID and Name, and deal with it using queries and joins.     Storing it as Enum and forget about the DB table.    In my opinion, I will choose the first solution if I have changing items. So that I won't hard code these options as Enum. I may choose the Enum solution, if I have no doubt that data won't change (for example, Gender: Male, Female).  NOTE: I code in English, and the UI Culture may be Arabic. If I will work with the Enum Solution, I will hard code the culture-based strings in the presentation layer, is it okay from the best practices perspective!!!!  I would like to know your opinions and if my thoughts correspond to what is most recommended "Best Practices"?? Limits, find a limit that exist in absolute value but not outside the absolute value. How to calculate: $\sum_{n=1}^{\infty} n a^n$ I've tried to calculate this sum:  $$\sum_{n=1}^{\infty} n a^n$$  The point of this is to try to work out the "mean" term in an exponentially decaying average.  I've done the following:  $$\text{let }x = \sum_{n=1}^{\infty} n a^n$$ $$x = a + a \sum_{n=1}^{\infty} (n+1) a^n$$ $$x = a + a (\sum_{n=1}^{\infty} n a^n + \sum_{n=1}^{\infty} a^n)$$ $$x = a + a (x + \sum_{n=1}^{\infty} a^n)$$ $$x = a + ax + a\sum_{n=1}^{\infty} a^n$$ $$(1-a)x = a + a\sum_{n=1}^{\infty} a^n$$  Lets try to work out the $\sum_{n=1}^{\infty} a^n$ part:  $$let y = \sum_{n=1}^{\infty} a^n$$ $$y = a + a \sum_{n=1}^{\infty} a^n$$ $$y = a + ay$$ $$y - ay = a$$ $$y(1-a) = a$$ $$y = a/(1-a)$$  Substitute y back in:  $$(1-a)x = a + a*(a/(1-a))$$ $$(1-a)^2 x = a(1-a) + a^2$$ $$(1-a)^2 x = a - a^2 + a^2$$ $$(1-a)^2 x = a$$ $$x = a/(1-a)^2$$  Is this right, and if so is there a shorter way?  Edit:  To actually calculate the "mean" term of a exponential moving average we need to keep in mind that terms are weighted at the level of $(1-a)$. i.e. for $a=1$ there is no decay, for $a=0$ only the most recent term counts.  So the above result we need to multiply by $(1-a)$ to get the result:  Exponential moving average "mean term" = $a/(1-a)$  This gives the results, for $a=0$, the mean term is the "0th term" (none other are used) whereas for $a=0.5$ the mean term is the "1st term" (i.e. after the current term). How to make listings code indentation remain unchanged when copied from PDF? So I am using  package for code examples. But there is a problem. When I copy the code examples from the final PDF document, they are usually not the same - the indentation is gone, there are random spaces after parenthesis, etc. Does anyone know how to make sure that the code that I copy from the final document is the same as I wrote it in the LaTeX document?  Here is what I mean. Compile this document:  \documentclass[12pt,oneside]{memoir}  \usepackage{listings} \usepackage[T1]{fontenc} \usepackage{xcolor} \usepackage{textcomp}  \definecolor{codebg}{HTML}{EEEEEE} \definecolor{codeframe}{HTML}{CCCCCC}  \lstset{language=Awk} \lstset{backgroundcolor=\color{codebg}} \lstset{frame=single} \lstset{framesep=10pt} \lstset{rulecolor=\color{codeframe}} \lstset{upquote=true} \lstset{basicstyle=\ttfamily} \lstset{showstringspaces=false}  \begin{document}  This code example prints out all users on your system:  \begin{lstlisting}[language=c] #include &lt;stdio.h&gt; #include &lt;stdlib.h&gt; #include &lt;string.h&gt;  #define MAX_LINE_LEN 1024  int main() {     char line[MAX_LINE_LEN];     FILE *in = fopen("/etc/passwd", "r");     if (!in) exit(EXIT_FAILURE);      while (fgets(line, MAX_LINE_LEN, in) != NULL) {         char *sep = strchr(line, ':');         if (!sep) exit(EXIT_FAILURE);         *sep = '\0';         printf("%s\n", line);     }     fclose(in);     return EXIT_SUCCESS; } \end{lstlisting}  \end{document}   Now compile it, open it, select and copy the code example and paste it in a text editor. Here is the result I get:  # include &lt;stdio .h&gt; # include &lt;stdlib .h&gt; # include &lt;string .h&gt; # define MAX_LINE_LEN 1024 int main () { char line [ MAX_LINE_LEN ]; FILE *in = fopen ("/ etc / passwd ", "r"); if (! in) exit ( EXIT_FAILURE ); while ( fgets (line , MAX_LINE_LEN , in) != NULL ) { char * sep = strchr (line , ':'); if (! sep ) exit ( EXIT_FAILURE ); * sep = '\0 '; printf ("%s\n", line ); } fclose (in ); return EXIT_SUCCESS ; }   Jeez! What's this... The indentation is gone. There are spaces in included headers &lt;stdio .h&gt;, there are spaces after parenthesis [ MAX_LINE_LEN ].  Anyone knows how to fix this? I'd like the code examples in my book to be copy/paste-able into a text editor so people can try them out easily. dpkg: error: dpkg status database is locked by another process I just wondered if somebody could tell me what this means:  E: dpkg was interrupted, you must manually run 'sudo dpkg --configure -a' to correct the problem. reidsr@ubuntu:~$ sudo dpkg --configure -a dpkg: error: dpkg status database is locked by another process reidsr@ubuntu:~$    How do I run sudo dpkg --configure -a  manually? Evaluating string "3*(4+2)" yield int 18 Is there a function the .NET framework that can evaluate a numeric expression contained in a string and return the result? F.e.:  string mystring = "3*(2+4)"; int result = EvaluateExpression(mystring); Console.Writeln(result); // Outputs 18   Is there a standard framework function that you can replace my EvaluateExpression method with? Is there a reason why we gain ghost reputation on the meta of an SE website? Characterize the groups $G$ for which the map $\iota: G \to G$, sending $x \mapsto x^{-1}$ for all $x \in G$, is an automorphism of $G$ I have trouble on this question: "Characterize the groups G for which the map $$\iota: G \to G$$ sending $$x \mapsto x^{-1}$$ for all $x \in G$, is an automorphism of $G$." From  post I have seen that $G$ must be abelian. Can anyone propose another characteristic of $G$ that I can derive from the given conditions besides abelian? Thank you so much!! The Medibuntu Project has come to an end - what do I do now? $$|a\sin x+b \cos x|\leq \sqrt{a^2+b^2}$$   I have tried: $$|a\sin x+b \cos x|\leq |a+b|\leq \sqrt{a^2+b^2}$$  enough to prove: $$|a+b|\leq \sqrt{a^2+b^2}$$  But I can find how to continue from here Question  I calculated the classical heat capacity of a diatomic gas as $C_V = (9/2)Nk_B$, however the accepted value is $C_V = (7/2)Nk_B$.  I assumed the classical Hamiltonian of two identical atoms bound together as $$ H = \dfrac{1}{2m}( |\bar{p}_2|^2 + |\bar{p}_2|^2)+ \dfrac{\alpha}{2} |\bar{q}_1-\bar{q}_2|^2. $$ I calculated the partition function of $N$ particles as $$ Z = \left( \iiint_{-\infty}^{\infty} \iiint_{-\infty}^{\infty} \iiint_{-\infty}^{\infty} \iiint_{-\infty}^{\infty} e^{-\beta H} ~d^3q_1~d^3p_1~d^3q_2~d^3p_2 \right)^N \propto V^N T^{(9/2)N}. $$ I calcuated the heat capacity as $$ C_V = \dfrac{\partial }{\partial T} \left( k_B T^2 \dfrac{\partial \ln(Z)}{\partial T} \right) = \dfrac{9}{2}k_BN. $$  Why does the classical argument fail?  Classical Derivation  The partition function is \begin{align} Z &amp;=&amp; \left( \frac{1}{h^6} \int \mathrm{e}^{- \beta H(\bar{q}_1,\bar{q}_2,\bar{p}_1,\bar{p}_2)} ~d^{3}q_1 ~d^{3}q_2 ~d^{3}p_1   ~d^{3}p_2 \right)^N \\&amp;=&amp; \left( \frac{1}{h^6} \int \mathrm{e}^{- \beta ((|\bar{p}_1|^2+|\bar{p}_2|^2)/(2m)+\alpha |\bar{q}_1-\bar{q}_2|^2/2)} ~d^{3}q_1 ~d^{3}q_2 ~d^{3}p_1   ~d^{3}p_2 \right)^N \end{align} A useful gaussian integral \begin{align} \int_{-\infty}^{\infty} e^{-\gamma (x-x_0)^2}dx = \sqrt{\dfrac{\pi}{\gamma}} \end{align} The partition function can be evaluated using separated integrals \begin{align} \iiint_{-\infty}^{\infty} \mathrm{e}^{- \beta |\bar{p}_1|^2} ~d^{3}p_1 = \iiint_{-\infty}^{\infty} \mathrm{e}^{- \beta |\bar{p}_2|^2} ~d^{3}p_2 = \left(\sqrt{\dfrac{\pi}{\beta}}\right)^3 \end{align} and \begin{align} \iiint_{-\infty}^{\infty} \iiint_{-\infty}^{\infty} \mathrm{e}^{- \beta \alpha |\bar{q}_1-\bar{q}_2|^2/2 } ~d^{3}q_1 ~d^{3}q_2 = \left( \sqrt{\dfrac{\pi}{\beta \alpha/2}} \right)^3 \iiint_{-\infty}^{\infty}  ~d^{3}q_1 = \left( \sqrt{\dfrac{\pi}{\beta \alpha/2}} \right)^3 V \end{align} The last set of integrals are improper integrals. One has to take the limit as the space approaches infinite containment. In that limit, integrating one set of variables $d^3q_2$ approaches the limit of a finite Gaussian term, while the other $d^3q_1$ approaches the diverging value of the total volume of the gas.  The partition function is \begin{align} Z &amp;=&amp; \left( h^{-6} \left(\sqrt{\dfrac{\pi}{\beta}}\right)^3 \left(\sqrt{\dfrac{\pi}{\beta}}\right)^3 \left( \sqrt{\dfrac{\pi}{\beta \alpha/2}} \right)^3 V \right)^N \\&amp;=&amp; \left( h^{-6} \left(k_B T \pi\right)^{9/2} \left( \dfrac{2}{\alpha} \right)^{3/2} V \right)^N \\&amp;=&amp; \left( h^{-6} \left(k_B \pi\right)^{9/2} \left( \dfrac{2}{\alpha} \right)^{3/2} \right)^N V^N T^{9N/2} \end{align} What are the differences between the original Kaito Kid specials vs. the new Kaito Kid 1412 series? I am currently watching . Should I watch the other  anime films?  What's the difference between this version and the 1412 version?  And another question, what's the meaning of 1412? I am trying to get a better feel for the topology of ordinals and just received a great answer to this  where the Cantor-Bendixson rank and degree turn out to be a complete homeomorphism invariant of countable compact spaces. Now I am curious about what is going on within homeomorphism classes of ordinals. In particular, I want to know how to characterize countable ordinals $\alpha$ which are minimal among all ordinals homeomorphic to $\alpha$.   Certainly finite ordinals and $\omega$ are easy. Next, if $\alpha\cong \omega+1$, then it seems like we would have to have $\omega+1\leq \alpha&lt;\omega +\omega$ since there can only be one limit point in $\alpha$. Things become a little fuzzier for me with ordinals that have infinitely many limit points.  For instance, suppose $\alpha$ is an ordinal homeomorphic to $\omega^{\omega}$. Is it necessarily true that $\alpha\geq \omega^{\omega}$? Is there a general rule for knowing when an ordinal is the smallest among all ordinals homeomorphic to it? This is kind of a weird request. For some reason whoever owns safeandbuy.com has pointed their domain at my IP address. The reason it's a problem is that I'm having all kinds of crawlers that are trying to crawl my site with that domain name.   Is there anything I can do about this? I have an itemized list:  \begin{itemize}     \item Lorem     \item Ipsum     \item Sit     \item Dolor     \item Amet \end{itemize}   I want to create an accolade on the right side of 'Lorem' and 'Ipsum', and add a bit of text at the point of the accolade. The level of this text should be between the first and the second line. How can I get this done? Infinite algebraic extensions of $\mathbb{Q}$ I need help to solve the following exercise:   If $K$ is an algebraic extension of $\mathbb{Q}$ (finite or infinite), then we let $O_{K}$ denote the ring of algebraic integers in $K.$ If $\frak{P}$ is a prime ideal of $O_{K}$ then $\frak{P}∩\mathbb{Z}$ is a prime ideal of $\mathbb{Z}.$ Moreover, if $\frak{P}$ is non-zero, then so is $\frak{P}∩\mathbb{Z}$ . Thus $\frak{P}∩\mathbb{Z}=$$ p\mathbb{Z}$ for some prime $p.$ Show that $O_{K}/\frak{P}$  is a field, and is an algebraic extension of $\mathbb{F}_{p}$; then show that it is Galois over $\mathbb{F}_{p}$ with abelian Galois group.  Now suppose that $K/F$ is a (finite or infinite) Galois extension. Let $\frak{P},$  $O_{K}$ be as before and put $\frak{p}=\frak{P} \cap $$O_{F}$, a prime ideal of $O_{F}.$  Show that $\mathrm{Gal}(K/F)$ acts transitively on the set of prime ideals of $O_{K}$ above $\frak{p}.$   Thanks in advance