Evaluate the integral $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} (\cos(t)+\sqrt{1+t^2} \cos^3(t)\sin^3(t) )dt$
Evaluate $\int_{-\pi/4}^{\pi/4}(\cos(t)+\sqrt{1+t^2}\cos(t)^3\sin (t)^3)dt$
When does a Möbius transformation map $\Im(z)>0$ to itself? Show Möbius transformation which maps $\Im(z)&gt;0$ to itself iff  $$ f(z)= \frac{az+b}{cz+d}\,,\,\,ad-bc&gt;0$$ and $a,b,c,d$ are real. Recovering Outlook .pst password? I am trying to recover an Outlook .pst file password.  Is there some way to do this?  I am using Outlook 2007 and Windows 7. Given two increasing continuous functions $f,g$ prove that $(b-a) \int^b_a f(x)g(x) dx > \int^b_a f(x) dx \int^b_a g(x) dx$ Given two monotonically increasing continuous functions $f,g$ prove that  $$(b-a) \int^b_a f(x)g(x) dx &gt; \int^b_a f(x) dx \int^b_a g(x) dx,\; b&gt;a$$  what I have tried:  Let $$h(x) = (x-a)\int^x_a f(t)g(t) dt - \int^x_a f(t)dt\int^x_a g(t)dt $$ and trying to figure out that $h'&gt;0$ using MVT to reform $h'(x)$ as   $$h'(x) = (x-a)(-f(x)g(\xi_1) - g(x)f(\xi_2) + f(x)g(x) + f(\xi_3)g(\xi_3))$$ where $\xi_{1,2,3} \in (a,x)$ (That method does not seem to work) Ball motion with air resistance coupled differential equation for fourth-order Runge-Kutta I've created a MATLAB function for solving coupled differential equation with the fourth-order Runge-Kutta method based on what is provided  (Simultaneous Equations of First Order). Here the function:  function [y] = rk4_c(f, g, h, x, y, z, n) % Runge-Kutta % Implementation of the fourth-order method for coupled equations % h = dt % x is the time here  for ii=1:(n-1)     k1 = h * f(x(ii), y(ii), z(ii));     l1 = h * g(x(ii), y(ii), z(ii));     k2 = h * f(x(ii) + 0.5*h, y(ii) + 0.5*k1, z(ii) + 0.5*l1);     l2 = h * g(x(ii) + 0.5*h, y(ii) + 0.5*k1, z(ii) + 0.5*l1);     k3 = h * f(x(ii) + 0.5*h, y(ii) + 0.5*k2, z(ii) + 0.5*l2);     l3 = h * g(x(ii) + 0.5*h, y(ii) + 0.5*k2, z(ii) + 0.5*l2);     k4 = h * f(x(ii) + h, y(ii) + k3, z(ii) + l3);     l4 = h * g(x(ii) + h, y(ii) + k3, z(ii) + l3);     y(ii+1) = y(ii) + (1/6)*(k1 + 2*k2 + 2*k3 + k4);     z(ii+1) = z(ii) + (1/6)*(l1 + 2*l2 + 2*l3 + l4); end   I would like to use this function for solving motion equations for a ball with air resistance. So starting with the basic equations (I hope they are correct): $$md^2x/dt^2 = F_{D_x}$$ and $$md^2y/dt^2 = F_{D_y}-mg$$ with $$F_{D_x} = -Dvvcos(\theta)$$ and $$F_{D_y} = -Dvvsin(\theta)$$ where $D$ is a drag coefficient. From my understanding, and from what I read, both second-order equation can be converted to two coupled equations that could be used with my coupled Runge-Kutta method function. Can someone help me understand how to convert my equation to a set of coupled equations that could be used with my MATLAB function?  EDIT  I was able to use my rk4_c function, however it only works for solving the velocities. I'm now trying to see how I could solve for the positions x and y. I'm kind of learning about Runge-Kutta methods at the same time, so sorry for the questions that may sound trivials.  % Runge Kutta code to solve projectile motion with quadratic drag % dVx/dt = -(D/m)*vx*sqrt(vx^2+vy^2) % dVy/dt = -(D/m)*vy*sqrt(vx^2+vy^2) - g  clc clear all  % Constant D = 0.24; % m = 2; % kg g = 9.80665; % m/s^2  % Define function handles fVx = @(t,vx,vy) -(D/m)*vx*sqrt(vx^2+vy^2); fVy = @(t,vx,vy) -(D/m)*vy*sqrt(vx^2+vy^2) - g;  % Initial conditions v0 = 200; % m/s theta = 30*pi/180; % rad t(1) = 0; vx(1) = v0*cos(theta); vy(1) = v0*sin(theta);  % Step size h = 0.01; % s tFinal = 2; N = ceil(tFinal/h);  % RK4 simultaneous coupled loop for ii = 1:N     % Update time     t(ii+1) = t(ii) + h;      % Update vx and vy     k1 = h * fVx(t(ii), vx(ii), vy(ii));     l1 = h * fVy(t(ii), vx(ii), vy(ii));     k2 = h * fVx(t(ii) + 0.5*h, vx(ii) + 0.5*k1, vy(ii) + 0.5*l1);     l2 = h * fVy(t(ii) + 0.5*h, vx(ii) + 0.5*k1, vy(ii) + 0.5*l1);     k3 = h * fVx(t(ii) + 0.5*h, vx(ii) + 0.5*k2, vy(ii) + 0.5*l2);     l3 = h * fVy(t(ii) + 0.5*h, vx(ii) + 0.5*k2, vy(ii) + 0.5*l2);     k4 = h * fVx(t(ii) + h, vx(ii) + k3, vy(ii) + l3);     l4 = h * fVy(t(ii) + h, vx(ii) + k3, vy(ii) + l3);     vx(ii+1) = vx(ii) + (1/6)*(k1 + 2*k2 + 2*k3 + k4);     vy(ii+1) = vy(ii) + (1/6)*(l1 + 2*l2 + 2*l3 + l4); end  % Plot the solution figure(1) plot(t, vx, t,vy) xlabel('Time (s)') ylabel('Velocities (m/s)') legend('Vx', 'Vy') How do I address unchecked cast warnings? How do I install applications in Ubuntu? How can I install applications like Google Chrome on Ubuntu?  Are there any commands to install an application?    Index of answers: Could there be a bit of reputation leakage between the sites, please? Riemann and Darboux Integral of a product of two functions Is there a way to exclude other particle systems from the effects of a Wind effect? I have a scene with a number of particle systems. I'd like to have a separate wind effect for my main scene and another for an extra: an emitter + wind-effect to blow leaves into my scene. I can't find any way to exclude, so here I am. At the very worst, I can just render the main scene and then render the wind effect and composite it, but I figured I'd check first. Making sense of principal component analysis, eigenvectors & eigenvalues In today's pattern recognition class my professor talked about PCA, eigenvectors and eigenvalues.   I understood the mathematics of it. If I'm asked to find eigenvalues etc. I'll do it correctly like a machine. But I didn't understand it. I didn't get the purpose of it. I didn't get the feel of it.        I strongly believe in the following quote:     You do not really understand something unless you can explain it to your grandmother. -- Albert Einstein   Well, I can't explain these concepts to a layman or grandma.   Why PCA, eigenvectors &amp; eigenvalues? What was the need for these concepts? How would you explain these to a layman? How to mix three different shaders on the same material? Recovering data off of a dead hard drive My hard drive died in my macbook and it no longer will start up. It will not mount in an external enclosure but disk utility is still reading that the hard drive is there, and I tried to repair disk and I am getting this error: (Disk Utility can’t repair this disk. Back up as many of your files as possible, reformat the disk, and restore your backed-up files.) So my question is how do I backup my files if the hard drive won't mount? The only way that I have been able to see the hard drive is through disk utility. I just want to be clear that I am using mac os x snow leapard. Let $A [a_{i,j}]$  and $B =[b_{ij}]$   be two diagonal matrices in   $M_n(F)$. Let $i \ne j \in \left\{1, \ldots, n \right\}$. Using the   product rule, we obtain      $$(AB)_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}$$      We claim that $a_{ik}b_{kj} =0$ for all $k \in \left\{1, \ldots, n \right\}$ thus $(AB)_{ij}=0$ for all $i \ne j$ and $AB$ is diagonal. To    prove the claim, note that since $i \ne j$ , we have $i \ne k$ or $    k\ne j$. Thus either $a_{ik} = 0$ (since $A$ is diagonal) or $b_{kj} =  0$ (since $B$ is diagonal), thus in all cases $a_{ik}b_{kj}=0$ and the   claim is proved.   Where does the bit "we have $i \ne k$ or $k\ne j$" come from? I understand that since $i \ne j$, and $k$ and $j$ run over the same set of natural numbers, we also have $i \ne k$. But the "or $k \ne j$" baffles me. I understand that $k \ne j$ implies that we can have $i=k$. But that's from the other direction ($\Longleftarrow).$  Also, when it says $a_{ik}=0$ or $b_{kj}=0$ it's talking about when $i \ne k$ or $k \ne j$ right? Many thanks. Pluralizing Numbered Items Biblatex-apa style is missing a comma in the references. why? I have been trying to solve this problems for hours, but there is no solution in sight.  Until a few weeks ago, my biblatex-apa style was working like a charm. Although, after installing the updates and Biber, the reference section became a problem. It should display a journal article by two authors with a comma after the first author, but it doesn't:     Weber, M. &amp; Ruch, W. (2012). The role of character strengths in adolescent romantic relationships: An initial study on partner selection and mates’ life satisfaction. Journal of adolescence, 35(6), 1537–1546. doi:10.1016/j.adolescence.2012.06.002   it should look like this:     Weber, M., &amp; Ruch, W. (2012). The role of character strengths in adolescent romantic relationships: An initial study on partner selection and mates’ life satisfaction. Journal of adolescence, 35(6), 1537–1546. doi:10.1016/j.adolescence.2012.06.002   This is how my document looks like:   \documentclass[12pt, onecolumn]{apa6}   \usepackage[utf8]{inputenc} \usepackage[american]{babel} \usepackage{csquotes} \usepackage[style= apa, backend=biber]{biblatex} \bibliography{testbib} \DeclareLanguageMapping{american}{american-apa}   \begin{document}  Some text \parencite{Weber2012}.  \printbibliography \end{document}   And the BibTeX file:  @article{Weber2012, abstract = {}, author = {Weber, M. and Ruch, W.}, doi = {10.1016/j.adolescence.2012.06.002}, file = {}, issn = {1095-9254}, journal = {Journal of adolescence}, number = {6}, pages = {1537--1546}, pmid = {22749517}, publisher = {Elsevier Ltd}, title = {{The role of character strengths in adolescent romantic relationships: An initial        study on partner selection and mates' life satisfaction.}}, url = {http://www.ncbi.nlm.nih.gov/pubmed/22749517}, volume = {35}, year = {2012} }    I have tried various things so far, but none of it helped: e.g,   \renewcommand*{\finalnamedelim}{ \finalandcomma}   or  \renewcommand\finalandcomma{\addcomma}   Even in the biblatex manual, the reference examples of two authors is without the comma (it works for three or more authors). There should be a comma, according to . Why is that comma not there? And how could I insert that into my references? How can I remove snaps without losing GUI? I want to know how to check how many snaps programs/apps I'm running. Also, how to replace them (if possible) without losing GUI? What to do when my 3-year-old holds poop in and won't go in diaper or on potty? What is the status icon that looks like a house? Show f(x) is a Lipschitz function Let $(X,d)$ be a metric space and $ a \in X $ Show that $f(x) = d(x,a)$ is a Lipschitz function from X to $\Re$ Use this fact to show $S$ is an open subset of $X$: Let $(X,d)$ be a metric space and $a \in X, r &gt; 0.$ Let $S$={$x \in X : d(a,x) &gt; r$}  &gt;Lipschitz Defn:  Let $f$ be a function from $(X,d)$ to $(Y,\rho)$ $f$ is said to be Lipschitz if: $\exists$ K $\geq$ 0 s.t. $\rho( f(x_1) , f(x_2) ) \leq$ Kd($x_1$,$x_2$) $\forall$ $x_1 x_2 \in X$  &gt;Continuity Defn:  Let $(X,d)$ and $(Y,\rho)$ be metric spaces, and $f: X \to Y$ Then $f$ is continuous $iff$ f$^{-1}$($B$) = { $x \in X : f(x) \in B $} is an open subset of $X$, whenever $B$ is an open subset of $Y$  &gt;Triangle Inequality for a Metric:  $d(x,z) \leq d(x,y) + d(y,z)$   My first question is, am I thinking about this correctly? If $f(x)$ is a Lipschitz function, it is uniformly continuous If $f(x)$ is uniformly continuous, it is continuous at every point If $f(x)$ is continuous at every point $\Rightarrow$ the inverse images of open sets are open Hence $S$ is an open subset of $X$.  Secondly, how would you formally construct this? We know $f(x) = d(x,a)$ s.t. $f:X \to \Re$ By definition of Lipschitz we have $\exists  K \geq 0$  s.t. $\rho(f(x_1),f(x_2)) \leq Kd(x_1,x_2) \forall x_1,x_2 \in X$ &amp; since our codomain is $\Re$ $\rho(f(x_1),f(x_2)) = | d(x_1,a) - d(x_2,a) | \leq Kd(x_1,x_2)$ Taking $K=1$ $\ldots$ Help in finishing this off would be appreciated. Thanks. What does the (unary) * operator do in this Ruby code?