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 given signature. 7. Existentially closed extensions. 8. The elementary theory of convex subgroups. 9. The elementary theory of cuts. 10. Counter examples. 1. Introduction. Let X be a totally ordered set. A (De...

In this paper, we consider a non-self-adjoint, singular, nonlinear fourth order boundary value problem which arises in the theory of epitaxial growth. It is possible to reduce the fourth order equation to a singular boundary value problem of second order given by w''-1/r w'=w^2/(2r^2 )+1/2 λ r^2. The problem depends on the parameter λ and admits multiple solutions. Therefore, it is difficult to...

For a connected graph G, the r-th extraconnectivity κr(G) is defined as the minimum cardinality of a cutset X such that all remaining components after the deletion of the vertices of X have at least r+1 vertices. The standard connectivity and superconnectivity correspond to κ0(G) and κ1(G), respectively. The minimum r-tree degree of G, denoted by ξr(G), is the minimum cardinality of N(T ) taken...

A Dedekind cut of an ordered abelian group G is a pair (X, Y) of nonempty subsets of G where Y=G−X and every member of X precedes every member of Y. A Dedekind cut (X, Y) is said to be continuous if X has a greatest member or Y has a least member, but not both; if every Dedekind cut of G is a continuous cut, G is said to be (Dedekind) continuous. The ordered abelian group R of real numbers is, ...

Given a graph G = (V;E), a weight function w : E ! R+ and a parameter k we examine a family of maximization problems arising naturally when considering a subset U V of size exactly k. Speci cally we consider the problem of nding a subset U V of size k that maximizes : Max-k-Vertex Cover : the weight of edges incident with vertices in U . Max-k-Dense Subgraph : the weight of edges in the subgrap...

Opening and closing of a CFTR Cl(-) channel is controlled by PKA-mediated phosphorylation of its cytoplasmic regulatory (R) domain and by ATP binding, and likely hydrolysis, at its two nucleotide binding domains. Functional interactions between the R domain and the two nucleotide binding domains were probed by characterizing the gating of severed CFTR channels expressed in Xenopus oocytes. Expr...

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