## Suppose $f(0)=f'(0)=0$ and $f''(x)0$ everywhere, prove $f(x)0$ for all non-zero $x$. February 10    3

Any help would be appreciated. Not quite sure how to prove this: Suppose $f: \mathbb{R}\rightarrow \mathbb{R}$, $f(0)=f'(0) = 0$ and $f''$ is everywhere positive. Prove $f(x)>0$ for all nonzero $x$ without referring to concavity. $f'$ is strictly inc

## Problem 10 of Section 1.2 from Hatcher. February 10

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 ...

## Find the smallest positive value taken by $a^3+b^3+c^3-3abc$ February 10    1

Find the smallest positive value taken by $a^3+b^3+c^abc$ for positive integers $a,b,c$. Find all integers $a,b,c$ which give the smallest value. Since it is generally hard to find the minimum of a multivariate polynomial, I tried factoring it at

## How do I prove that if $2\notn$ then $2(n+1)$ February 10

I'd like to prove a very simple fact, but it's stumping me: namely, that if $2 \not| n$ then $2|(n+1)$.How would this usually be done?

## Writing a product of transpositions as a 3-cycle. February 10

For the case that we have two transpositions equal to each other, say (a b) (a b) then how can I write the product as a product of 3 cycles?

## 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

## $f(x) = 2x \mod 1$ not equal to zero for all $x$ February 10

If any number $\mod 1$ is zero, then how can $f(x) = 2x \mod 1$ be a Baker's map? For any $x\in \mathbb{R}$, shouldn't $f(x)=0$?

## Chain rule for discrete/finite calculus February 10    4

In the context of discrete calculus, or calculus of finite differences, is there a theorem like the chain rule that can express the finite forward difference of a composition $∆(f\circ g)$ in simplified or otherwise helpful terms? It's probably not p

## Why isn't finite calculus more popular February 10

I'm reading through Concrete Math, and learning about finite calculus. I've never heard of it anywhere else, and a Google search found very few relevant sources. It seems to me an incredibly powerful tool for evaluating sums, essentially a systematiz

## What would be a good cartesian equation to represent the shape of a wine glass February 10

I want to find the volume of a wine glass by using either the disk or shell method (solids of revolutions). The wine glass doesn't have to be of any particular dimensions, however it should roughly resemble an actual wine glass.The cartesian equation

## 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

## nilpotent endomorphism on finitely generated modules over a domain February 10

If $R$ is an domain and $f: R^n \to R^n$ is an $R$-module endomorphism. Suppose $f^m = 0$ for some $m> 0$. Show that $f^n = 0$.The cases $m \le n$ is trivial. When $m>n$, I don't have much idea how to start. I tried to apply the Cayley-Hamilton the

## 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 $\cos t-|- You Might Also Like • I have two functions that I'm working on. The first is:$\begin{align} \cos x &= (\cos 1)^3 \cos(3-x) \ ... • How AreY_{lm}$and$Y^m_l$related? How are$Y_{lm}^*$and$Y_{lm}$related? I am very confused seeing dif ... • Theorem to prove:Let$\{v_1,\dots,v_n\}$be a linearly independent set in a finite-dimensional vector space ... • Prove that:$n!>2^n$for$n \ge 4$.So in my class we are learning about induction, and the difference bet ... • Find the subgroup of$GL(2,\mathbb{C})$generated by the matrices$A$and$B$, where$A=\begin{pmatrix} 1 &a ...
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