I want to know which article to be used with the word attachment and which is correct is these two sentences &quot;Please find the attachment&quot; Or &quot;Please find an attachment&quot;
I can’t for the life of me figure out where to use a and where to use the — and where there is no article at all. Is there a simple rule of thumb to memorize?   The standard rule you always hear:     “If a person knows which item you are talking about then use "the"   . . . doesn’t clear things up for me, as I have no idea whether or not they know.
Prove: $\gcd(a,b) = \gcd(a, b + at)$. There is a $C > 0$ such that $\vert f(z)-f(w)\vert \leq C\vert z-w\vert$ for a holomorphic map into $B(0, R)$. Let $0&lt;\delta&lt;r$ and $R&gt;0$, and $a\in\mathbb C$. I want to show that there exists $C&gt;0$ such that for all holomorphic functions $f\colon B(a,r)\to B(0,R)$, we have $$ \vert f(z)-f(w)\vert\leq C\vert z-w\vert $$ for $z,w\in B(a,\delta)$.  I'm not sure how to approach this. I was wondering if I could use that $$ \left\vert\frac{f^{(n)}(a)}{n!}\right\vert\leq\frac{\Vert f\Vert_{C(a,\rho)}}{\rho^n} $$ for some $0&lt;\rho&lt;r$. I also have an expression from biholomorphic maps from $B(0,1)$ to itself, but I'm not sure if I could use that here. Any hints? What is a good reference for rigorous Electromagnetism and Electrodynamics? Is there any good book on Electromagnetism from a more mathematical point of view? By this I mean a book which makes use of differential forms and maybe De Rham cohomology. I was also searching for something that would include E&amp;M in the big picture of Gauge theories. I am a kid trying to teach myself Calculus in order to prepare for next year.   I have the expression  $$\lim_{x\to 2}\frac{\cos(\frac \pi x)}{x-2} $$  There is a hint that says to substitute t for $(\frac \pi2 - \frac \pi x)$ and WolframAlpha evaluates this expression as $\frac \pi4$. However, I got the answer of $1$. Can someone clarify the steps to solving this problem. I am very new to LaTeX and I am writing my thesis. In a chapter, I want to show my codes, which are in Mathematica.   I use the listings package as follows:  \documentclass{article}   \usepackage{listings}   \begin{document}   \lstset{language=Mathematica}    \begin{lstlisting}   Do[i^j,{i,1,10},{j,1,10}]   \end{lstlisting}    \end{document}   However, the output is much different to what the code looks like in the Mathematica IDE (even if I use other languages in \lstset{language=&lt;name&gt;}):    I read some questions on the site but, in some of them, the font or color of the codes were not the same. Could you please help me to display something like the picture below in LaTeX?  If it is important, I am using the BiDi package. Find PostgreSQL server hostname on which it runs I have start learning Selenium WebDriver, but i am confused between Java and Python. Which language should i choose? I do have basic ideas of Core Java but haven't wrote a single piece of code with python.  So please suggest me which is the better language for Selenium WebDriver? I want to create a math problem-solution sheet. The coefficients a, b, c must be assigned on the fly at random so the generated pdf will show a numeric quadratic equation rather than a symbolic quadratic equation. Note a must not be zero.  \documentclass{article} \begin{document} \section{Problem} Let \(x_1\) and \(x_2\) be the roots of \(a x^2 +bx +c=0\). Find  \(x_1x_2\).  \section{Solution} \[ x_1x_2=\frac{c}{a} \] \end{document}   Please kindly make the solution get automatically simplified. For example: \frac{c}{a}=3 when a=2 and c=6.    Actually the real case is more complex than this one. Not only the equations must be generated at random, but also the associated graphic plots that I write in PSTricks.     I chose Paulo's answer because using Maxima as the computation engine is very very great idea. Other answers are also good and creative, especially Altermundus'. how can i automatically record every terminal session? Proof that any self-complementary graph has to have $4k$ or $4k+1$ vertex, for some $k \in \mathbb N$ this code always returns 0 in PHP 5.2.5 for microseconds:  &lt;?php $dt = new DateTime(); echo $dt-&gt;format("Y-m-d\TH:i:s.u") . "\n"; ?&gt;   Output:  [root@www1 ~]$ php date_test.php 2008-10-03T20:31:26.000000 [root@www1 ~]$ php date_test.php 2008-10-03T20:31:27.000000 [root@www1 ~]$ php date_test.php 2008-10-03T20:31:27.000000 [root@www1 ~]$ php date_test.php 2008-10-03T20:31:28.000000   Any ideas? Uniform convergence of $\sum_{n=1}^{\infty}\frac{1}{n(1+(x-n)^2)}$ Is $$\sum_{n=1}^{\infty}\frac{1}{n(1+(x-n)^2)}$$ uniformly convergent on $(0,+\infty)$? Sadly, Weierstrass' criterion nor Cauchy's condition don't seem to work here and I just ran out of ideas. Peggy would like to prove to Victor that she knows the discrete logarithm of $y$ based $g$; that is, she knows $x$ such that $y = g^x \bmod p$. One round of the interactive proof protocol consists of the following steps.   Peggy picks random $k \in \mathbb Z/(p−1)\mathbb Z$, computes $t = g^k \bmod p$, and sends $t$ to Victor. Victor picks random $h \in \mathbb Z/(p−1)\mathbb Z$ and sends $h$ to Peggy. Peggy computes $r = (k − hx) \bmod (p − 1)$ and sends $r$ to Victor. Victor verifies that $t = g^r y^h \bmod p$.   The interactive protocol can be converted into a noninteractive zero-knowledge proof by choosing and making public a collision-resistant hash function $H$, and changing the second step of the interactive protocol to the following: Peggy computes $h = H(y, t)$. Then the noninteractive proof consists of $(t, h, r)$, which can be verified as follows: $$h = H(y, t), \qquad t \stackrel?= g^r y^h \bmod p.$$   What is the problem if in the non-interactive proof the hash $h$ depends only on $y$? That is, $h = H(y)$, and the proof consists of $(t, h, r)$, which can be verified as follows: $$h = H(y), \qquad t \stackrel?= g^r y^h \bmod p.$$ What is the problem if in the non-interactive proof the hash $h$ depends only on $t$? That is, $h = H(t)$, and the proof consists of $(t, h, r)$, which can be verified as follows: $$h = H(t), \qquad t \stackrel?= g^r y^h \bmod p.$$ Is it possible and if so how can you pass flags to the ShareLaTeX compiler which one would have passed as parameters to the LaTeX compiler?  I am using the minted package and need to pass the -shell-escape flag. Why or how did Agen Kolar and Saesee Tiin seem to die so easily without any resistance? Why does Quantum Field Theory use usually Lagrangians rather than Hamiltonains?  I heard many reasons, but I'm not sure which is true.   Some say it's just a matter of beauty, so Lagrangians are more beautiful because they don't break/separate the space-time variables (so space-time is a single variable, like in the Klein-Gordon Lagrangian and Hamiltonian).  Some say that Hamiltonians are not always Lorentz invariant.  Could someone explain in a little bit more details? Compare the statistical significance of the difference between two polynomial regressions in R So first of all I did some research on this forum, and I know   questions have been asked but they usually haven't been answered properly or sometimes the answer are simply not detailed enough for me to understand. So this time my question is : I have two sets of data, on each, I do a polynomial regression like so :  Ratio&lt;-(mydata2[,c(2)]) Time_in_days&lt;-(mydata2[,c(1)]) fit3IRC &lt;- lm( Ratio~(poly(Time_in_days,2)) )   The polynomial regressions plots are:    The coefficients are :  &gt; as.vector(coef(fit3CN)) [1] -0.9751726 -4.0876782  0.6860041 &gt; as.vector(coef(fit3IRC)) [1] -1.1446297 -5.4449486  0.5883757    And now I want to know, if there is a way to use an R function to do a test that would tell me whether or not there is a statistical significance in the difference between the two polynomials regression knowing that the relevant interval of days is [1,100].  From what I understood I can not apply directly the anova test because the values come from two different sets of data nor the AIC, which is used to compare model/true data.  I tried to follow the instructions given by @Roland in the related question but I probably misunderstood something when looking at my results :  Here is what I did :   I combined both my datasets into one.  f is the variable factor that @Roland talked about. I put 1s for the first set and 0s for the other one.  y&lt;-(mydata2[,c(2)]) x&lt;-(mydata2[,c(1)]) f&lt;-(mydata2[,c(3)])  plot(x,y, xlim=c(1,nrow(mydata2)),type='p')  fit3ANOVA &lt;- lm( y~(poly(x,2)) )  fit3ANOVACN &lt;- lm( y~f*(poly(x,2)) )   My data looks like this now :    The red one is fit3ANOVA which is still working but I have a problem with the blue one fit3ANOVACN the model has weird results. I don't know if the fit model is correct, I do not understand what @Roland meant exactly.  Considering @DeltaIV solution I suppose that in that case :  The models are significantly different even though they overlap. Am I right to assume so ? Example of a group $G$ such that $G = N_1\cdots N_n$ and $N_i \cap N_j = \{e\}$ for all $i \neq j$ but $G$ is not the internal direct product of them. I was trying to find an example of a group $G$ and normal subgroups $N_1 , \ldots , N_n$ such that $G = N_1\cdots N_n$ and $N_i \cap N_j = \{e\}$ for all $i \neq j$ and yet $G$ is not the internal direct product of  $N_1, \ldots , N_n$ .  It is easy to show that G is the internal direct product of normal subgroups $N_1 , \ldots , N_n$ iff  (i) $G = N_1\cdots N_n$  (ii) $N_i\cap (N_1 \cdots N_{i-1}\cdots N_{i+1} \cdots N_n) = \{e\}$ for $i = 1, 2, \ldots, n$.  So I guess the question boils down to produce an example of a group $G$ and its normal subgroups  $N_1 , \ldots , N_n$ such that   (1) $G = N_1\cdots N_n$  (2) $N_i \cap N_j = \{e\}$ for all $i \neq j$   but     (3) $N_i \cap (N_1 \cdots N_{i-1} N_{i+1} \cdots N_n) \neq \{e\}$ for at least one $1 \le i \le n.$ Thanks in advance for help.... How can I reliably and accurately identify the passive voice in writing or speech? Foobar2000-like music player?