# Tag Archives: Mathematics

## Question: Do you think I’m able to pass the drug test?

I use to smoke maybe 4 days out the week, no more than 6-8 puffs (it don’t take many puffs to get me high). I stopped for 2 months, and ended up smoking again for my bday with only 4 puffs, stopped for maybe 2 weeks and ended up smoking again with only 4 puffs.… Read More »

## Can I calculate the product over the primes efficiently and with high precision?

I want to calculate the following product : \$\$prod_{p prime , ple 10^{10}} 1-frac{1}{p}\$\$ I know the approximation formula \$\$frac{e^{-gamma}}{ln(10^{10})}\$\$ where \$gamma\$ is the Euler-Mascheroni-constant. The result should be good to \$12\$ decimal digits. Is there an efficient way (not brute force by determining all primes, which takes long with PARI/GP) to calculate this product… Read More »

## How to prove that an angle is 90 degrees without using a protractor or having reciprocal slopes as a proof?

So the question I have to answer is this: An enlarged view of the daycare patio is shown below that contains two congruent triangles, one of which has been rotated. Using this image, prove that perpendicular lines have opposite and reciprocal slopes. I know that the slopes are reciprocals, but how do I prove that… Read More »

## Question: Find components of point Q?

B = (-1, 2), C = (5, 5), D = (2, -1). Q is on CD where the length of CQ is double the length of QD.

## How fast does a matrix expression go to zero?

With \$0 This is coming from a Markov chain problem which I solved up to here. We can see that \$X_nleq X_{n-1}\$ and \$Y_nleq Y_{n-1}\$. I am thinking of using the results of this, but it does not seem easy. Any idea how to continue?

## Does \$K\$ form a vector space? [on hold]

Does \$K\$ form a vector space? \$\$K =left{f:[0,1] rightarrow mathbb R text{ and } f text{ has a local maximum at } x = frac12 text{ over } mathbb Rright}\$\$ I think yes if we take \$fleft(frac 12right) = 0 \$ as zero function.

## How can I plot this nonlinear system using MATLAB

I am trying to plot this servomechanism system in MATLAB. The system is shown in figure. \$V\$(voltage) is input of the system and \$theta_L\$ is output of the system. How can I plot \$V(t)\$ and \$theta_L(t)\$ ? Equations of the system is shown below. \$\$dot{omega}_L=-512θ_L-25θ_M-10omega_L\$\$ \$\$dot{omega}_M=V-10omega_M-128θ_L-6θ_M \$\$ \$\$\$\$ So, \$\$ theta_L”=-512theta_L+25theta_M-10theta_L’ \$\$ \$\$theta”_M=V-10theta’_M+128theta_L-6theta_M \$\$ So… Read More »

## Sheaves on the product of algebraic varieties

Let \$X, Y\$ be smooth algebraic varieties. I am trying to figure out which are the relations between sheaves on \$X\$,\$Y\$ and on \$Z = X times Y\$. I am particularly interested in the structure sheaves, the sheaves of differential operators and the sheaves of differential forms. I think I have proved the relations below:… Read More »

## Does \$max_{ile n} big|Z_ibig|=frac 1 n sum_{ile n} big|Z_ibig|\$ as \$n to infty\$? (Convergence of R.V.s sequence \$Z_1,Z_2,Z_3,…\$)

\$Z_1,Z_2,Z_3,…\$ are integrable R.V.s that are independent and identically distributed. Show that the following expression converges to zero in probability: \$frac 1 n max_limits{ile n} left|Z_iright|\$. What does \$max_limits{ile n} big|Z_ibig|\$ mean? Is it the biggest \$Z_i\$ for \$ile n\$? If so, the law of large numbers state that \$frac 1 n sum_limits{ile n} big|Z_ibig|… Read More »

## show that the solution is a local martingale iff it has zero drift

Most financial maths textbook state the following: Given an \$n\$-dimensional Ito-process defined by begin{equation} X_t = X_0 + int_0^{t} alpha_s ,d W_s + int_0^{t} beta_s ,d s, end{equation} where \$(alpha_t)_{t geq0}\$ is a predictable process that is valued in the space of \$n times d\$ matrices and \$(W_t)_{t geq 0}\$ is a \$d\$-dimensional Brownian motion,… Read More »