How to prove $n$ is prime February 10    2

Let $n \gt 1$ and $$\left\lfloor\frac n 1\right\rfloor + \left\lfloor\frac n2\right\rfloor + \ldots + \left\lfloor\frac n n\right\rfloor = \left\lfloor\frac{n-1}{1}\right\rfloor + \left\lfloor\frac{n-1}{2}\right\rfloor + \ldots + \left\lfloor\frac{n-

Problem 10 of Section 1.2 from Hatcher. February 10

Problem 10 of Section 1.2 from Hatcher.
Problem. Consider two arcs $\alpha$ and $\beta$ embedded in $D^2\times I$ as shown in the figure. The loop $\gamma$ is obviously nullhomotopic in $D^2\times I$, but show that there is no nul ...

Probability of an even number of sixes February 10

We throw a fair die $n$ times, show that the probability that there are an even number of sixes is $\frac{1}{2}[1+(\frac{2}{3})^n]$. For the purpose of this question, 0 is even. I tried doing this problem with induction, but I have problem with induc

Instantaneous rate of change help please February 10

USING ALTERNATIVE DEFINITION FORM OF LIMF(x) = x/x-1, given x=2Im stuck mid way through of simplfying! Someone help!I plugged in -(x/x-1) - 2/ which is all over x-2 -common dinominator -expanding and now im stuck on how to continue

Runge-Kutta 4 failure February 10

Say we want to solve numerically y'(x) = f(x) * y, with y0 = y(x=0) = 0 and applying RK4 method with step dx = h:k1 = f(0) * y(0) * h = 0 k2 = f(0+h/2) * (y0 + k1/2) * h = f(h/2) * 0 = 0 k3 = f(h/2) * (y0 + k2/2) *h = 0 k4 = f(c+h) * (y0 + k3) * h =

Exponential of a matrix with elements $\cos t \& \sin t$ February 10    3

I want to calculate $e^{A}$ of the matrix $A$: $$\left ( \begin{array}{cc} \cos t & \sin t \\ -\sin t & \cos t \end{array} \right )$$I tried to use $e^{At}=P\ \mbox{diag}(e^{\lambda t}) P^{-1}$, but from there I obtain the eigenvalue as $\cos t-|-

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