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 ...

We call a module M , quasi-catenary if for each pair of quasi-prime submodules K and L of M with K L all saturated chains of quasi-prime submodules of M from K to L have a common finite length. We show that any homomorphic image of a quasi-catenary module is quasi-catenary. We prove that if M   is a module with following properties: (i) Every quasi-prime submodule of M has finite quasi-height;...

In this paper, for any two bipartite ordered sets P and Q; we de ne the convolution P Q of P and Q: For dim(P ) = s and dim(Q) = t; we prove that s+ t (U + V ) 2 dim(P Q) s+t (U+V )+2; where U+V is the max-min integer of the certain realizers. In particular, we also prove that dim(Pn;k) = n+k b n+k 3 c for 2 k n < 2k and dim(Pn;k) = n for n 2k; where Pn;k = Sn Sk is the convolution of two stand...

For a fmite ordered set P, let c(P) denote the cardinality of the largest subset Q such that the induced suborder on Q satisfies the Jordan-Dedekind chain condition (JDCC), i.e., every maximal chain in Q has the same cardinality. For positive integers n, let be the minimum of c(P) over all ordered sets P of cardinahty n. We prove: &r 1 <f(n) < 4e AhfS (MOS) subject ckwiflcation (1980). 06AO5.

The order polytope O(P ) and the chain polytope C(P ) associated to a partially ordered set P are studied. In this paper, we introduce the convex polytope Γ(O(P ),−C(Q)) which is the convex hull of O(P ) ∪ (−C(Q)), where both P and Q are partially ordered sets with |P | = |Q| = d. It will be shown that Γ(O(P ),−C(Q)) is a normal and Gorenstein Fano polytope by using the theory of reverse lexico...

We firstly construct generalized Baskakov operators V n,α,q (f; x) and their truncated sum B n,α,q (f; γ n , x). Secondly, we study the pointwise convergence and the uniform convergence of the operators V n,α,q (f; x), respectively, and estimate that the rate of convergence by the operators V n,α,q (f; x) is 1/n. Finally, we study the convergence by the truncated operators B n,α,q (f; γ n , x) ...

We explore connections between covariance representations, Bismut-type formulas and Stein's method. First, using the theory of closed symmetric forms, we derive representations for several well-known probability measures on $\mathbb{R}^d$, $d \geq 1$. When strong gradient bounds are available, these immediately lead to $L^p$-$L^q$ estimates, all $p \in (1, +\infty)$ $q = p/(p-1)$. Then, revisit...

We study the convergence of method reflections for Stokes equations in domains perforated by countably many spherical particles with boundary conditions typical suspension rigid particles. prove that a relaxed version is always convergent H ˙ 1 under mild separation condition on Moreover, we optimal rates W , q < ∞ and L terms particle volume fraction stronger