We study the coordinate ring of an $L$-convex polyomino, determine its regularity in terms maximal number rooks that can be placed polyomino. also characterize Gorenstein polyominoes and those which are on punctured spectrum, compute Cohen–Macaulay type any polyomino rectangles covering it. Though main results algebraic nature, all proofs combinatorial.

Contents 1. Introduction. 2. Heirs. 3. The invariance group of a cut. 4. Review of T-convex valuation rings. 5. The invariance valuation ring of a cut. 6. A method for producing cuts with prescribed signature. 7. Existentially closed extensions. 1. Introduction. Let M be a totally ordered set. A (Dedekind) cut p of M is a couple (p L , p R) of subsets p L , p R of M such that p L ∪ p R = M and ...

The main purpose of this work is to study and apply generalized contact distributions of (inhomogeneous) Boolean models Z with values in the extended convex ring. Given a convex body L ⊂ Rd and a gauge body B ⊂ Rd such a generalized contact distribution is the conditional distribution of the random vector (dB(L,Z), uB(L,Z), pB(L,Z), lB(L,Z)) given that Z ∩L = ∅, where Z is a Boolean model, dB(L...

In order to get estimates on the solutions of the equation ∂̄u = ω on Stein manifold, we introduce a new method the ”raising steps method”, to get global results from local ones. In particular it allows us to transfer results form open sets in C to open sets in a Stein manifold. Using it we get L − L results for solutions of equation ∂̄u = ω with a gain, s > r, in strictly pseudo convex domains i...