MathOverflow

I derived the following area formula for a quadrilateral using two opposite sides and four angles:(When a=AB and c=CD,)
S=a^2/(2(cotA+cotB))+c^2/(2(cotC+cotD))
It applies under conditions A+B≠π, C+D≠π, and includes convex, concave, and the paticular self-intersecting cases.I divided the proof into cases and want to check logical validity. I divided the proof into several cases....

Given a partially ordered set (poset) $(P,\leq)$, is there a lattice $L_P$ and an order-preserving map $h_P:P\to L_P$ with the following property?
Whenever $L$ is a lattice and $f: P\to L$ is an order homorphism, there is a lattice homomorphism $h:L_P\to L$ such that $f = h \circ h_P$....

Using only the functional equation $\xi(s) = \xi(1-s)$ and the Hadamard product for $\xi(s)$, I derive that for $\sigma > 1/2$,
$$\sum_{\gamma} \frac{\sigma - 1/2}{(\sigma - 1/2)^2 + (t - \gamma)^2} > 0$$
where the sum is over all zeros $\gamma = \beta + i\gamma$ of $\xi(s)$....

Let $N$ be an odd integer, $n\in\mathbb{N}$, and $-\frac{2T}{NR^2}\leq a\leq0$ for given $R,T\in\mathbb{R}$ with $\frac{T}{NR^2}\leq\frac{\pi}{2}$.
Now consider the quadratic form $\Omega(a)=\sum_{l\in\mathbb{Z}/N\mathbb{Z}}(\frac{1}{\tan{a}}q_l^2-\frac{1}{\sin{a}}q_lq_{l+1})$ on $(\mathbb{R}^n)^N$ where $q_l\in\mathbb{R}^n$ for $l\in\mathbb{Z}/N\mathbb{Z}$ when $a\neq 0$, and $\Omega(0)=0$ defined...