Jesse P Francis

Let $l_{n}$ denote the number of those permutations $f$ on the set $A=\{1,2,....,n\}$ such that $f$ is the product of exactly two disjoint cycles. Show that $l_{5}=50.$ I tried a lot but reached nowhere around the answer. Any help will be appreciated. Thank you.

Let $u=u(x,t)$ be the solution of the Cauchy problem $$\frac{\partial u}{\partial t}+\left(\frac{\partial u}{\partial x}\right)^2=1, x\in\Bbb R, t>0$$ $$u(x,0)=-x^2$$ Then which of the following is/are true? $u(x,t)$ exists for all $x\in\Bbb R$ and $t>0$. $|u(x,t)|\to\infty$...

If the value of a particular stock increased by 10% every day of the first four days of a week.However, it value decreased by 30% at the end of fifth day compared to its value at the end of the fourth day .the value of stock at the end of fifth day was $56 what was the value of stock at the end o...

(12) The length L of an iron rod at the temperature T being given by L = lt[1+0:000012(T -􀀀t)],where lt is the length at the temperature t, finnd the rate of variation of the diameter D of an iron tyre suitable for being shrunk on a wheel, when the temperature T varies. Rate of variation of dia...

So I've been thinking through some test cases. If $a_n = n$ then $\sum_n \frac{a_{n+1} - a_n}{a_n}$ is the harmonic series which diverges. And if $a_n = \sum_{k=1}^n 1/k$ then $\sum_n \frac{a_{n+1} - a_n}{a_n}$ diverges like $\sum_n 1/(n \log n)$. So that got me thinking, if $a_n$ is a strictly i...

The question looks incomplete and unanswered for a long time: but guessing from OP's attempt, I think (s)he is trying to find the Wronskian. Here's couple of hints that will help you solve the question: $y_1(0) = 0, y_1'(0)=1,y_2(0) = 1, y_2'(0)=0 $. What does that tell you about $W_{(y_1,y_2)...

As Herebrij and David pointed out; if one of the is zero, say $b=0$, then we have a unique solution. Using Method of Characteristics, letting $x=x(t),y=y(t)$ and hence $u=u(t)$ $$\frac{du}{dt}=\frac{\partial u}{\partial x}\frac{dx}{dt}+\frac{\partial u}{\partial y}\frac{dy}{dt}$$ Comparing to ...

Since the fields considered are $GF(2)$ and $GF(3)$, easiest way will be to just see if all values ($\bar0,\bar1\in GF(2),\bar0,\bar1,\bar2\in GF(3)$) satisfy the polynomial in hand. Cohn's Irreducibility Criterion, since coefficients are $<2$.

To push it from unanswered queue: Notice that given polynomial is monic, hence by Rational Root Theorem, all rational solutions are integer, namely $\pm1,\pm11$. Also, Descartes Rule of Signs assures that there are no positive roots, making our list $-1,-11$, and its now easy to verify both won...