Currently I'm learning about Brownian motion. In the lecture slides the following definition is given. Definition: A Wiener process $W_{t}, t \geq 0,...

Let $f:\mathbb{R}^N\to\mathbb{R}$ be strongly convex and its gradient is Lipschitz continuous, i.e., for some $l>0$ and $L>0$ we have $$f(y)\geq f(x...

For the past couple of days I have been encountering the word "Torus" quite often. I would like to know what special properties does the Torus possess tha...

For $f \in C[a,b], g\ge0$ (or $g\le0$) on $[a,b]$, then there exists $c\in(a,b)$ $\int _a^b f(x)g(x)dx = f(c)\int_a^bg(x)dx$. It is MVT for integral. Why ...

For instance, we know that odd numbers behave like: $$x = 2y + 1 \quad\text{where}\quad x,y\in\mathbb Z$$ For even numbers: $$a = 2b \quad\text{where}\qua...

How would I find the inverse of $$\ln(8x-64)?$$ I've tried put $8x-64$ as the power to the base of $e$, I don't know what to do from there on, t...

Show that the pair of equations $$x^2-y^2-u^3+v^2+4=0\;,\;\; 2xy+y^2-2u^2+3v^4+8=0$$ Determine local functions $x(u,v)$ and $y(u,v)$ defined for $(u,v)$ n...

Could somebody please verify or help me along with my answers? Thank you. My updated work(maybe someone can help me a little more now):

I've been trying to figure out the intuition behind the closed formula: $$\sum i^{2} = \frac{(n)(n+1)(2n+1)}{6}$$ This is not hard to prove via induc...

Determine whether or not the graph of the function has a vertical tangent or vertical cusp at the indicated point c. $f(x) = (x+2)^7/3$ $c=-2$ I took the ...