Finding isosceles triangles
Number of isosceles triangle.
Find all the numbers $n$ such that $\frac{4n-5}{60-12n}$ can't be reduced. Immutability, Association and typing What do I need to know about temperature and food safety? What general rules do I need to follow to keep my food safe? How do I know what temperature to cook something to, or whether my food is safe at room temperature? How to initialize private static members in C++? What is the best way to initialize a private, static data member in C++?  I tried this in my header file, but it gives me weird linker errors:  class foo {     private:         static int i; };  int foo::i = 0;   I'm guessing this is because I can't initialize a private member from outside the class.  So what's the best way to do this? I have a psd that has rounded rectangle. I need to create a new rounded rectangle with the same radius. How do I determine the radius used on that said rounded rectangle?   I feel like this is such an awful question but thanks in advance for answering. Why $\dim U+\dim U^\perp=\dim V$? Say you take a dice and you roll it twice so that you have a pair (X,Y) where X represents the first roll, Y represents the second.   When you have a distribution like max(X,Y), what type of distribution is that? It's not uniform, obviously, because the outcomes of max(X,Y) are clearly more favorable towards the higher numbers. I'm being asked to "describe" the distribution. I know that we can look at each distinct outcome, like:  max(1,1) = 1  max(1,2) = 2  max(1,3) = 3  max(1,4) = 4  max(1,5) = 5  max(1,6) = 6  max(2,1) = 2  max(2,2) = 2  max(2,3) = 3  max(2,4) = 4  max(2,5) = 5  max(2,6) = 6  max(3,1) = 3  max(3,2) = 3  max(3,3) = 3  max(3,4) = 4  max(3,5) = 5  max(3,6) = 6  max(4,1) = 4  max(4,2) = 4  max(4,3) = 4  max(4,4) = 4  max(4,5) = 5  max(4,6) = 6  max(5,1) = 5  max(5,2) = 5  max(5,3) = 5  max(5,4) = 5  max(5,5) = 5  max(5,6) = 6  max(6,1) = 6  max(6,2) = 6  max(6,3) = 6  max(6,4) = 6  max(6,5) = 6  max(6,6) = 6  But is that the most effective way to look at a distribution like this? Is there a faster and more concise way to describe this distribution other than saying the probability of each outcome?  i.e.,  P(max = 1) = 1/36  P(max = 2) = 3/36 = 1/12  P(max = 3) = 5/36  P(max = 4) = 7/36  P(max = 5) = 9/36 = 1/4  P(max = 6) = 11/36  So, obviously, I understand how to LOOK at a discrete distribution and see what the probabilities are, but not until I look at each possible outcome. But what does it mean to "describe" a distribution? This isn't uniform or any sort of discrete distribution we're taught. Can someone provide clarity? How to allow Page Up in GNU Screen? I use Vim to edit text files on Linux. Commonly I connect to remote Linux machines using PuTTY from Windows.  I use GNU Screen commonly as well.  When using Vim inside Screen I am unable to use the Page Up key because Screen interprets it as a command to start copy mode.    I would like to use Page Up for its regular up-paging functionality.  How can I configure Screen to do this? Please check my simplification for x? False proof that $ρe^{iθ} = ρ$ and so complex numbers do not exist? My professor showed the following false proof, which showed that complex numbers do not exist. We were told to find the point where an incorrect step was taken, but I could not find it. Here is the proof: (Complex numbers are of the form $\rho e^{i\theta}$, so the proof begins there) $$\large\rho e^{i\theta} = \rho e^{\frac{2 \pi i \theta}{2\pi}} = \rho (e^{2\pi i})^{\frac{\theta}{2\pi}} = \rho (1)^{\frac{\theta}{2\pi}} = \rho$$ $$Note: e^{i\pi} = -1, e^{2\pi i} = (-1)^2 = 1$$ Since we started with the general form of a complex number and simplified it to a real number (namely, $\rho$), the proof can claim that only real numbers exist and complex numbers do not. My suspicion is that the error occurs in step $4$ to $5$ , but I am not sure if that really is the case. What is a contrast matrix? What exactly is contrast matrix (a term, pertaining to an analysis with categorical predictors) and how exactly is contrast matrix specified? I.e. what are columns, what are rows, what are the constraints on that matrix and what does number in column j and row i mean? I tried to look into the docs and web but it seems that everyone uses it yet there's no defition anywhere. I could backward-engineer the available pre-defined contrasts, but I think the definition should be available without that.  &gt; contr.treatment(4)   2 3 4 1 0 0 0 2 1 0 0 3 0 1 0 4 0 0 1 &gt; contr.sum(4)   [,1] [,2] [,3] 1    1    0    0 2    0    1    0 3    0    0    1 4   -1   -1   -1 &gt; contr.helmert(4)   [,1] [,2] [,3] 1   -1   -1   -1 2    1   -1   -1 3    0    2   -1 4    0    0    3 &gt; contr.SAS(4)   1 2 3 1 1 0 0 2 0 1 0 3 0 0 1 4 0 0 0 Entity Query field condition for Webform submissions I'm trying to query webform submissions based on a field value. The field is called "ticket_for". Somethings I tried and do not work:      -&gt;condition('ticket_for',$adnummer)     -&gt;condition('webform_submission_data:ticket_for',$adnummer,'CONTAINS') ...   What kind of condition should I be using in a query like this?  $query = \Drupal::entityQuery('webform_submission') -&gt;condition('webform_id', 'ticket') -&gt;condition('entity_id', $eventID) -&gt;condition('webform_submission_data:ticket_for',$adnummer,'CONTAINS'); How can we recover the Newtonian gravitational potential from the metric of general relativity? The Newtonian description of gravity can be formulated in terms of a potential function $\phi$ whose partial derivatives give the acceleration:  $$\frac{d^2\vec{x}}{dt^2}=\vec{g}=-\vec{\nabla}\phi(x)=\left(\frac{\partial\phi}{\partial x}\hat{x}+\frac{\partial\phi}{\partial y}\hat{y}+\frac{\partial\phi}{\partial z}\hat{z}\right)$$  However, in general relativity, we describe gravity by means of the metric. This description is radically different from the Newtonian one, and I don't see how we can recover the latter from the former. Could someone explain how we can obtain the Newtonian potential from general relativity, starting from the metric $g_{\mu\nu}$? Computing the cardinality of a combinatorial set Define two sets: $S_1$ and $S_2$ where we have that $a$ is an odd natural number and $b\in \mathbb{N}$, $S_1 = \Big\{ \frac{2}{a},\frac{2}{a-2},\frac{2}{a-4},...,\frac{2}{a-(a-1)},$ $\frac{2+4}{a+2},\frac{2+4}{a},\frac{2+4}{a-2},...,\frac{2+4}{a-(a-1)},$ $\frac{2+8}{a+4},...,\frac{2+8}{a-2},...,\frac{2+8}{a-(a-1)},$ $\vdots$ $\frac{2+4\lfloor\frac{b}{2}\rfloor}{a+\lfloor\frac{b}{2}\rfloor},\frac{2+4\lfloor\frac{b}{2}\rfloor}{a+\lfloor\frac{b}{2}\rfloor-2},...,\frac{2+4\lfloor\frac{b}{2}\rfloor}{a-(a-1)},$ $\frac{2+4(\lfloor\frac{b}{2}\rfloor+1)}{a+\lfloor\frac{b}{2}\rfloor},\frac{2+4(\lfloor\frac{b}{2}\rfloor+1)}{a+\lfloor\frac{b}{2}\rfloor-2},...,\frac{2+4(\lfloor\frac{b}{2}\rfloor+1)}{a-(a-1),}$ $\frac{2+4(\lfloor\frac{b}{2}\rfloor+2)}{a+\lfloor\frac{b}{2}\rfloor-2},\frac{2+4(\lfloor\frac{b}{2}\rfloor+2)}{a+\lfloor\frac{b}{2}\rfloor-4},...,\frac{2+4(\lfloor\frac{b}{2}\rfloor+2)}{a-(a-1)}$ $\vdots$ $\frac{2+4b}{a},\frac{2+4b}{a-2},...,\frac{2+4b}{a-(a-1)} \Big\}$ and $S_2 = \Big\{ \frac{2}{\lceil\frac{a}{2}\rceil}, \frac{2}{\lceil\frac{a}{2}\rceil-1},\frac{2}{\lceil\frac{a}{2}\rceil-2},...,\frac{2}{1}$ $\frac{2+2}{\lceil\frac{a}{2}\rceil+1},\frac{2+2}{\lceil\frac{a}{2}\rceil},\frac{2+2}{\lceil\frac{a}{2}\rceil-1},...,\frac{2+2}{1},$ $\vdots$ $\frac{2+2(\lfloor\frac{b}{2}\rfloor-1)}{\lceil\frac{a}{2}\rceil+(\lfloor\frac{b}{2}\rfloor-1)},\frac{2+2(\lfloor\frac{b}{2}\rfloor-1)}{\lceil\frac{a}{2}\rceil+(\lfloor\frac{b}{2}\rfloor-2)},...,\frac{2+2(\lfloor\frac{b}{2}\rfloor-1)}{1}$ $\frac{2+2\lfloor\frac{b}{2}\rfloor}{\lceil\frac{a}{2}\rceil+\lfloor\frac{b}{2}\rfloor},...,\frac{2+2\lfloor\frac{b}{2}\rfloor}{1},$ $\frac{2+2(\lfloor\frac{b}{2}\rfloor+1)}{\lceil\frac{a}{2}\rceil+\lfloor\frac{b}{2}\rfloor-1},...,\frac{2+2(\lfloor\frac{b}{2}\rfloor+1)}{1},$ $\frac{2+(\lfloor\frac{b}{2}\rfloor+2)}{\lceil\frac{a}{2}\rceil+\lfloor\frac{b}{2}\rfloor-2},...,\frac{2+(\lfloor\frac{b}{2}\rfloor+2)}{1}$ $\vdots$ $\frac{2+2b}{\lceil\frac{a}{2}\rceil-1},...,\frac{2+2b}{1} \Big\}$ The question is, how do I determine the cardinality of the intersection of these two sets i.e. how would I compute the value $|S_1 \cap S_2|$? Note that duplicates may appear within both sets such as $\frac{1}{2}=\frac{2}{4}$ etc. Therefore we must make sure that for each fraction $\frac{n}{m}$ we have $\text{gcd}(n,m)=1$. I am aware of the existence of , but not entirely sure if that identity would be useful here. throwing exceptions out of a destructor Most people say never throw an exception out of a destructor - doing so results in undefined behavior. Stroustrup makes the point that "the vector destructor explicitly invokes the destructor for every element. This implies that if an element destructor throws, the vector destruction fails... There is really no good way to protect against exceptions thrown from destructors, so the library makes no guarantees if an element destructor throws" (from Appendix E3.2).   seems to say otherwise - that throwing destructors are more or less okay.  So my question is this - if throwing from a destructor results in undefined behavior, how do you handle errors that occur during a destructor?  If an error occurs during a cleanup operation, do you just ignore it? If it is an error that can potentially be handled up the stack but not right in the destructor, doesn't it make sense to throw an exception out of the destructor?  Obviously these kinds of errors are rare, but possible. Labeling Multiple Equations Per Row of a Table I am trying to create a table with two columns of independently labelled equations (1-6). Unfortunately, everything I have found so far either has not worked or would number them as sub-equations (1a, 1b, etc).  My current attempt is:  \documentclass{article} \usepackage{graphicx, verbatim, tabularx} \usepackage{amsfonts} \usepackage{amsmath, amssymb}  \newcommand{\grad}{\vec{\nabla}}       \newcommand{\prtl}[2]{\frac{\partial #1}{\partial #2}}  \newcommand{\curl}{\grad\times} \begin{document}  \begin{tabular}{p{5cm}p{5cm}}  $\curl\vec{E} = -i\omega\mu\vec{H}$ &amp;   $\curl\vec{H} = i\omega\mu\vec{E}$ \\  \begin{equation}  \label{eq:curlHx}   \prtl{E_z}{y} +\gamma E_y= -i\omega\mu H_x  \end{equation}  &amp;  \begin{equation}  \label{eq:curlEx}    \prtl{H_z}{y} +\gamma H_y= i\omega\mu E_x  \end{equation}\\  \begin{equation}  \label{eq:curlHy}    -\prtl{E_z}{x} -\gamma E_x= -i\omega\mu H_y  \end{equation}  &amp;  \begin{equation}  \label{eq:curlEy}    -\prtl{H_z}{x} -\gamma H_x= -i\omega\mu E_y  \end{equation}  \\  \begin{equation}  \label{eq:curlHz}   \prtl{E_y}{x}-\prtl{E_x}{y} = -i\omega\mu H_z  \end{equation}  &amp;  \begin{equation}  \label{eq:curlEz}    \prtl{H_y}{x}-\prtl{H_x}{y} = -i\omega\mu E_z  \end{equation}  \end{tabular}  \end{document}   Is there an environment other than tabular that would make this work? Tabularx came close (got the numbering correct), but the formatting/spacing was horrific. Why are the power transmission/distribution systems AC and not DC? if $x+2y+z=4$ and $x,y,z$ are real number. then find maximum value of $xy+yz+zx$  putting $x+z=4-2z$ in $y(x+z)+zx = y(4-2z)+zx = 4y-2yz+zx$  i wan,t be able to  go further,could some help me with this How do I install drivers for a Canon LBP-2900 printer? I seen many times how people describe how to's about LBP-2900, and 10-12 steps from start to end of how to download, unpack, build driver, fix driver, make it work... Could be there some .deb package with ready-to-go driver for Ubuntu? This printer is very popular... Implementation of "show" for function