Is there a term for a smooth $\mathbb R\to\mathbb R^+$ function that is unimodal (has one local maximum), and whose $n^{\text{th}}$ derivative has $(n+1)$ local extrema? An example of such a function is $\exp(-x^2)$. An example of a unimodal function that does not fit these criteria is $\frac1{1+x^4}$.

A) List all proper, non-trivial subgroups of $U_6$ where $U_6$ is the group of $6$th roots of unity: So 6th roots of unity are $\pm1, \pm \frac{1}{2} \pm \frac{\sqrt{3}}2i$ How do these relate to the subgroups? B) List the generators: If I am not mistaken the generators of the 6th roots of unity are $\frac{1}{2} […]

Bruce Springsteen, aka The Boss, has just finished a large arena show. Since he has won $20$ Grammy Awards, he decides he will toss $20$ guitar picks into the crowd as souvenirs. He has a pile of $7$ identical red picks, $7$ identical yellow picks, $7$ identical gray picks, $7$ identical green picks, and $7$ […]

Lets suppose a ≥ 1. Prove that if a | s + 3 & a | s + 7 then a = 1, 2 or 4.

I wrote a really cool code sample which I will include at the end of this question. It completes in O(N) time in the javascript engine, but solves the boolean satisfiability problem, which is in the family of problems NP-Complete. I kind of freaked out when I realized the efficiency of this algorithm which I […]

## Free vs. torsion-free modules

I’m trying to show that if $M$ is a nonzero cyclic module, then $M$ is free if and only if $M$ is torsion-free. How would a proof for this look like?

## inital value problem, Linear algebra

I’m sitting on the following problem for the past days and have no clue how to solve it: Consider the initial value problem: $y”-y’-2y = 0$ with the inital conditions $y_0 = 3$ and $y’_0 = 3$. where $y_0 = y(0)$. a) Express the differential equation in the form $\frac{d\bf{u}}{dt} = A\bf{u}$ where $\bf{u} = […]

Does there exist an infinite series with sequence {$a_k$}$ \implies f(n,k)$ s.t for some positive integer $n$: $$\sum \limits_{k=0}^{\infty}f(n,k) = \frac{(n)^2(n+1)^2}{4}$$the right hand side being the sum of sequence of cubes for some positive integer $n$. For instance: $$\sum \limits_{k=0}^{\infty}f(2,k) = 9$$ $$\sum \limits_{k=0}^{\infty}f(3,k) = 36$$

I have no idea how to prove that $$\lim_{x\rightarrow 4} (\sqrt x-2 )= 0$$ using $\delta$ and $\varepsilon$ My attempt after the hint ( Please, I barely know what I’m doing this could all be gibberish and I wouldn’t be any wiser): $0 < (x-4) < \delta$ If $|x-4| < 1$ then $x < 5$ […]

## Disproving Multivariable Limit

I just started learning about multivariable functions, and I was wondering if there is something like Heine for disproving the limit of multivariable functions. For example: I know that we can approach multivariable functions from different directions to see if the function keep the same value, but is this enough? Is there a more rigorous […]