We develop a recursive method for computing the $r$-removed $P$-orderings and $P$-orderings of order $h,$ the characteristic sequences associated to these and limits associated to these sequences for subsets $S$ of a Dedekind domain $D.$ This method is applied to compute these objects for $S=\mathbb{Z}$ and $S=\mathbb{Z}\setminus p\mathbb{Z}$.

Every relatively convex-compact convex subset of a locally convex space is contained in a Banach disc. Moreover, an upper bound for the class of sets which are contained in a Banach disc is presented. If the topological dual $E^{\text{'}}$ of a locally convex space $E$ is the $\sigma (E^{\text{'}},E)$-closure of the union of countably many $\sigma (E^{\text{'}},E)$-relatively countably compacts sets, then every weakly (relatively) convex-compact set is weakly (relatively) compact.

We develop a new combinatorial method to deal with a degree estimate for subalgebras generated by two elements in different environments. We obtain a lower bound for the degree of the elements in two-generated subalgebras of a free associative algebra over a field of zero characteristic. We also reproduce a somewhat refined degree estimate of Shestakov and Umirbaev for the polynomial algebra, which plays an essential role in the recent celebrated solution of the Nagata conjecture and the strong...

Let ${\Delta }_{n,d}$ (resp. ${\Delta }_{n,d}^{\text{'}}$) be the simplicial complex and the facet ideal $I_{n,d}=(x_1\cdots x_d,x_{d-k+1}\cdots x_{2d-k},...,x_{n-d+1}\cdots x_n)$ (resp. $J_{n,d}=(x_1\cdots x_d,x_{d-k+1}\cdots x_{2d-k},...,x_{n-2d+2k+1}\cdots x_{n-d+2k},x_{n-d+k+1}\cdots x_n{x}_1\cdots x_k)$). When $d\ge 2k+1$, we give the exact formulas to compute the depth and Stanley depth of quotient rings $S/J_{n,d}$ and $S/I_{n,d}^t$ for all $t\ge 1$. When $d=2k$, we compute the depth and Stanley depth of quotient rings $S/J_{n,d}$ and $S/I_{n,d}$, and give lower bounds for the depth and Stanley depth of quotient rings $S/I_{n,d}^t$ for all $t\ge 1$.

This note summarizes a presentation made at the Third International Meeting on Integer Valued Polynomials and Problems in Commutative Algebra. All the work behind it is joint with Scott T. Chapman, and will appear in [2]. Let $\mathrm{Int}\left(\mathbb{Z}\right)$ represent the ring of polynomials with rational coefficients which are integer-valued at integers. We determine criteria for two such polynomials to have the same image set on $\mathbb{Z}$.