What does the term "arbitrary number" mean in math?
What does "arbitrary number" means in math? I've often seen this vocabulary but I can't find the meaning of it. For example, I'll see phras...
Category: General
Precalc factoring $x^2+10x+25-9y^2$
Factor $$x^2+10x+25-9y^2$$ The solution is $$(x+5-3y)(x+5+3y)$$ I understand how to factor when there is only one variable $x$ but I am not sure how to co...
Complex Analysis - Liouville's Theorem application and polynomial degree
Exercise : Let $f:\mathbb C \to \mathbb C$ be an entire function. (a) If $|f(z)| \geq 1$ $\forall z \in \mathbb C$, show that the function $f$ is constant...
Minimum velocity
There is a particle moving along the $x$-axis at any time $t \geq 0$. The velocity of this particle is given by $$v(t) = \cos(πt) - t(6-2\pi).$$ I am supp...
Proof of Clarkson's Inequality
Trying to find a proof for Clarkson's inequality, which states that if $2 \leq p < \infty$, then for any $f, g \in L^p$, we have that $$\left|\lef...
Dice Game - Probability Theory
There are many dice problems on here so there could be duplicate. If there is I apologize. Here is the question. The game is given as follows: you are giv...
Proving the inverse (if any) of a lower triangular matrix is lower triangular
The inverse of a non-singular lower triangular matrix is lower triangular. Construct a proof of this fact as follows. Suppose that $L$ is a non-singular l...
On the proof of Fleury's algorithm. (Question 2)
On pages 42-43 in [1], it says: We conclude our introduction to Eulerian graphs with an algorithm for constructing an Eulerian trail in a give Eulerian gr...
Proving $\int \csc^2 x\, dx$
According to the standard integral results, it is known that $\int \csc^2 x \, dx = -\cot x$ since $\frac{d}{dx} \cot x = - \csc^2 x$. However, supposing ...
Proof that every polygon with an inscribed circle is convex?
In many elementary (and not-so-elementary) Euclidean geometry texts, a (simple) polygon is said to be tangential if it is convex and has an inscribe...
