We introduce and study a class of determinantal probability measures generalising the discrete point processes. These live on Grassmannian real, complex, or quaternionic inner product space that is split into pairwise orthogonal finite-dimensional subspaces. They are determined by positive self-adjoint contraction space, in way equivariant under action group isometries preserve splitting.

We use Hadamard's determinantal inequality and its generalization to prove some upper bounds on the energy of a graph in terms degrees, average 2-degrees number common neighbors vertices. Also, we an relating one arbitrary subgraph it.

Let $X$ be a generic determinantal affine variety over perfect field of characteristic $p \geq 0$ and $P \subset X$ standard prime divisor generator $\mathrm{Cl}(X) \cong \mathbb{Z}$. We prove that the pair $(X,P)$ is purely $F$-regular if $p>0$ so log terminal (PLT) $p=0$ $\mathbb{Q}$-Gorenstein. In general, using recent results Z. Zhuang S. Lyu, we show PLT-type, i.e. there $\mathbb{Q}$-divis...