How can I find the center of ellipse given an Arc and ellipse radii?
How to find the center of an ellipse?
Find out which process is locking a file or folder in Windows How can I find out which process is locking a file or folder in Windows?  For instance, when trying to delete a folder, Windows reports this:     The action can't be completed because the folder is open in another program   Happens the same with a file, but how do I find out what program or application is currently using it and preventing me from deleting the file or folder? Given a recurrence relation:   $$ x_{n+2} = px_{n+1} + qx_n + r $$   Find the general term $x_n$ given initial conditions $x_1 = a$ and $x_2 = b$, where $a,b,p,q,r$ are some given numbers, $n \in \mathbb N$.   Since this is a non-homogeneous linear recurrence relation i've started with solving for $x_n^h$ which is a solution for homogeneous relation:  $$ x_{n+2} - px_{n+1} - qx_n = 0 \\ \lambda^2-p\lambda - q = 0 $$  I'm assuming here that two distinct roots $\lambda_{1,2}$ exist for this equation (single root case is handled similarly). So the general term is in the form:  $$ x_n = C_1\lambda_1^{n-1}+C_2\lambda_2^{n-1} $$  With the initial condition it's possible to find the values for $C_1$ and $C_2$:  $$ C_1 + C_2 = a \\ C_1\lambda_1 + C_2\lambda_2 = b $$  After solving the system of equation with respect to $C_1$ and $C_2$:  $$ C_1 = \frac{b-a\lambda_2}{\lambda_1 - \lambda_2} \\ C_2 = \frac{b-a\lambda_1}{\lambda_2 - \lambda_1} \\ $$  Which eventually results in:  $$ x_n^h = \frac{(a\lambda_2 - b)\lambda_1^{n-1} - (a\lambda_1-b)\lambda_2^{n-1}}{\lambda_2 - \lambda_1} $$  Final solution consists of a sum of particular and homogeneous solutions, but that is where I got stuck. What is the way to find the particular solution for the given problem? I'm pretty new to such kind of problems so any details are very much appreciated.  upd:  Based on the technique in @rtybase answer here are my findings:  $$ \begin{align} x_1 &amp;= a \\ x_2 &amp;= b \\ x_3 &amp;= pb - qa + r \end{align} $$  Writing the system for $c_1, c_2, c_3$:  $$ \begin{align} a &amp;= c_1 + c_2 + c_3 \\ b &amp;= c_1\lambda_1 + c_2\lambda_2 + c_3 \\ pb+qa+r &amp;= c_1\lambda_1^2 + c_2\lambda_2^2 + c_3 \end{align} $$  Now solving for the coefficients gives a huge formula. The answer to this problem suggests that:  $$ x_n = \frac{(\lambda_2(a+\gamma) - b -\gamma)\lambda_1^{n-1} - (\lambda_1(a + \gamma) - (b-\gamma))\lambda_2^{n-1}}{\lambda_2 - \lambda_1} - \gamma $$  Where:  $$ \gamma = \frac{r}{p+q-1} $$  Unfortunately I was not able to arrive at exactly the same result, but testing my solution (not posting since it's monstrously big) using SameQ in Mathematica with the given answer shows that they are equivalent. Those transformations are not very obvious to me, so i skipped them since the answer matches the given one (yet the symbolic expressions are different) Types of polynomial functions. Quadratic, cubic, quartic, quintic, ...,? I would very much like to have a complete list of the types of polynomial functions. I know that theres:  Quadratic :                      (AX^2 + BX + C) Cubic     :               (AX^3 + BX^2 + CX + D) Quartic   :        (AX^4 + BX^3 + CX^2 + DX + E) Quintic   : (AX^5 + BX^4 + CX^3 + DX^2 + EX + F)   What are the names of polynomial functions to the further powers? Sexstic? Septic? Octic? I'm not sure, so if someone could enlighten me, then that would be great. Please provide a list that goes at least to the seventh power, but it would be nice if you could go further. Thanks. I'm wondering if there's a simple way to rewrite this in terms of $k$ and $x$, especially as a polynomial. It seems to me to crop up every so often, especially for $k=2$, when I integrate with trig substitution. But $k=2$ is not so bad, because I can use the double angle formula; it's the prospect of higher values of $k$ that motivates this question.   I think the law of sines may help? Or maybe even De Moivre's theorem, to find the length of the hypotenuse as the length of the angle changes, if we think of the right triangle drawn from $\arcsin(x)$ with side1 = $x$, hypotenuse = $1$, and side2 = $\sqrt{1 - x^2}$ as a complex number, though I'm not sure how that might work. Complex numbers equation: $z^4 = -16$ Save Illustrator as PDF with transparent background What is the JVM thread scheduling algorithm? How to arrive at $\int_{0}^{1} f^3(x) \, dx < (\int_{0}^{1} f(x) \, dx)^2$? Define $f : [0,1] \to \mathbb{R} $ differentiable satisfying $f(0)=0$ and $0&lt;f'(x)&lt;1$. Prove:  $\displaystyle\int_{0}^{1} f^3(x) \, dx &lt; \left(\displaystyle\int_{0}^{1} f(x) \, dx\right)^2$ I found the following answer,  Define $F(x)=\displaystyle\int_{0}^{x} f^3(t) \, dt - \left(\displaystyle\int_{0}^{x} f(t) \, dt\right)^2$ We have $$F'(x)=f^3(x)-2 \left(\displaystyle\int_{0}^{x} f(t) \, dt\right)f(x)=f(x)\left[f^2(x)-2 \left(\displaystyle\int_{0}^{x} f(t) \, dt\right)\right]$$ and $$\left[f^2(x)-2 \left(\displaystyle\int_{0}^{x} f(t) \, dt\right)\right]'=2f(x)\left[f'(x)-1\right]$$ from which we can arrive at the conclusion.  But I felt a little confused. In the past I usually proved the inequality by classical inequalities and something like Taylor expansion to use the condition about derivative. But this answer is little out of my expectation. Do we have other proofs or ideas to arrive at $\displaystyle\int_{0}^{1} f^3(x) \, dx &lt; \left(\displaystyle\int_{0}^{1} f(x) \, dx\right)^2$? And other thoughts about solving these types of issues? I think when we talk about $\left(\displaystyle\int_{0}^{1} f(x) \, dx\right)^2 &gt;\ldots \,$ it's a little difficult to deal with. I don't quite know what tools to use. Uniform convergence on two separate intervals Test the sequence of functions for uniform convergence       $$f_n(x)=\frac{nx}{1+n^2x^2}$$      $a) \text{on} \ [0,1]$      $b) \text{on} \ [1,{\infty})$   a)  If we differentiate this with respect to x we get the following :  $$(\frac{nx}{1+n^2x^2})'=\frac{n(1-n^2x^2)}{(1+n^2x^2)^2}$$  This is equal to $0$ for $x=^{+}_{-}\frac{1}{n}$  From here we can either check these two values in the original function and take the larger one as the supremum or we can find the second derivative and check whether it's negative. In either case we find that for $x=\frac{1}{n}$ we get the value $f_n(x)=\frac{1}{2}$ and for $x=-\frac{1}{n}$ we get $f_n(x)=-\frac{1}{2}$. Note that $x=-\frac{1}{n}$ doesn't even belong in our interval. Now we can finish the problem.  $$\lim_{n\to \infty}{\frac{nx}{1+n^2x^2}}=0$$  So it is piecewise converging to $f(x)=0$  $$\lim_{n\to \infty}\sup{\left \lvert{f_n(x)-f(x)}\right \rvert}=\lim_{n\to \infty}{\frac{n\frac{1}{n}}{1+n^2 \frac{1}{n^2}}}=\frac{1}{2}$$  So $f_n(x)$ does not convergence uniformly on $[0,1]$ to $f(x)=0$  b)  Now we need to test the on the interval $[1,\infty)$.From the first derivative we notice that if $x\in [1,\infty)$ then the numerator seems to be negative for all $x&gt;1$. So $f_n(x)$ is monotonically decreasing. That means that the supremum will be at the first point of the interval or $x=1$. Taking the limit :  $$\lim_{n\to \infty}{\frac{n}{1+n^2}}=0$$  We can see that the function convergences uniformly on this interval.  Is there something wrong with either of these solutions?  I'm having doubts mostly about the second interval since $x=\frac{1}{n}$ is a supremum but only for $n=1$. However I believe that we need to have the supremum that's satisfied for all $n \in N$. Pointing even the tiniest mistake in my solutions would be very helpful. Thank you. If I create a custom environment based on the equation environment, I get a spacing issue: the next line of text after my environment closes doesn't start on the left margin.  Example code:  \documentclass{article} \begin{document} \newenvironment{myenv}{\begin{equation}}{\end{equation}} \noindent $| \leftarrow$ this is the left margin. \begin{myenv} e ^{i \pi} + 1 = 0 \end{myenv} $| \leftarrow$ this isn't the left margin. \end{document}   Output:    Removing the newline after \end{myenv} solves the spacing problem, but the resulting code is not readable.  Of course, in this example I could just use the equation environment itself. My real issue involves a more complex custom environment that suffers from the same problem, and which also includes an equation environment within it. Cannot set LC_CTYPE to default locale: No such file or directory How does the US federal income tax system work? General question here about US personal income federal tax rates.  Is it a cumulative effect where your income is first taxed at the lower tranches and move up, or is it one specific percent for your entire taxable income? Should I set the title attribute for content DIV's to explain what they contain? I have a number of DIV's that break up a page into logical elements for styles and positioning, etc.  As part of optimizing my site for search engine spiders, should I include a title attribute within these DIV's?  (Yes I have a title as part of the whole page, I'm curious on individual structual elements)  Ex  ... &lt;div id="SearchOptions" title="Search filters and options - Type in a name or pick a business and Click 'Search' to get started" &gt;   .... &lt;/div&gt; ... How to get the native version of Spotify running? What is a virtual terminal for? Algorithm to match one input file with given numbers of file If $m,n\in N$ Prove that there is such a positive integer k, such that $(\sqrt{m}+\sqrt{m+1})^n=\sqrt{k}+\sqrt{k+1}$ If $m,n\in N$ Prove that there is such a positive integer k, such that $(\sqrt{m}+\sqrt{m+1})^n=\sqrt{k}+\sqrt{k+1}$ I attempted to solve this question using binomial coefficients, saying: $(\sqrt{m}+\sqrt{m+1})^n=\sqrt{k}+\sqrt{k+1}=\sum\limits_{a=0}^n {n\choose a}*\sqrt{m}^{n-a}*\sqrt{m+1}^a$ and from here I was thinking that I had to do something with $\sqrt{m}^{n-a}$ with $\sqrt{k}$ and do something else with $\sqrt{k+1}$ and $\sqrt{m+1}^a$. Unfortunately I couldn't think of what to do with this circumstance. I then thought of maybe taking $(\sqrt{k}+\sqrt{k+1})^2=(\sum\limits_{a=0}^n {n\choose a}*\sqrt{m}^{n-a}*\sqrt{m+1}^a)^2$, but this immediately over-complicated the question. Could you please explain to me how to solve this question and how to solve similar questions in the future? The  makes use of the term "the author."  When Bob makes a post, it's obvious Bob "owns" the content, but what happens when he checks the Community Wiki button? Does it become the property of the Stack Overflow community? Even if Bob is the only one to have edited it?  Say I were to use a Community Wiki answer for something or other &#8212; say, in a book &#8212; who would I attribute the content to? Many To many for Bulk records I have created two custom objects (A and B)and a junction object(AB). i.e. Both A and B have Many to Many relationship. Now I need to insert 5000 records. How can I automatically populate the relation in Junction object. DO I have to write any trigger or Do I have to do Manually mapping the records? Why is $(2, 1+\sqrt{-5})$ not principal? Why is $(2, 1+\sqrt{-5})$ not principal in $\mathbb{Z}[\sqrt{-5}]$?   Say $(2,1+\sqrt{-5})=(\alpha)$, then since $2\in(2,1+\sqrt{-5})$ we have $2\in (\alpha)$, so $\alpha\mid2$ in $\mathbb Z[\sqrt{-5}]$. Writing $2=\alpha\beta$ in $\mathbb Z[\sqrt{-5}]$ and taking norms, $4=N(\alpha)N(\beta)$ in $\mathbb Z$, so $N(\alpha)\mid4$ in $\mathbb Z$. Similarly, since $\sqrt{-5}\in(\alpha)$ we get $N(\alpha)\mid5$, thus $N(\alpha)$ is a common divisor of $4$ and $5$, therefore $N(\alpha)=(1)$, so $\alpha$ is a unit. But that means $1\in (2,1+\sqrt{-5})=(\alpha)$, so it must be the whole ring, but it cannot to reach a contradiction, so how can I find an element, which is not in $(2, 1+\sqrt{-5})$.  Or is there a simpler method ? (Maybe not UFD would imply not PID)