The multidimensional truncated moment problem: Carathéodory numbers from Hilbert functions

In this paper we improve the bounds for the Carathéodory number, especially on algebraic varieties and with small gaps (not all monomials are present). We provide explicit lower and upper bounds on algebraic varieties, $$\mathbb {R}^n$$ R n , and $$[0,1]^n$$ [ 0 , 1 ] n . We also treat moment problems with small gaps. We find that for every $$\varepsilon >0$$ ε > 0 and $$d\in \mathbb {N}$$ d ∈ N there is a $$n\in \mathbb {N}$$ n ∈ N such that we can construct a moment functional $$L:\mathbb {R}[x_1,\cdots ,x_n]_{\le d}\rightarrow \mathbb {R}$$ L : R [ x 1 , ⋯ , x n ] ≤ d → R which needs at least $$(1-\varepsilon )\cdot \left( {\begin{matrix} n+d\\ n\end{matrix}}\right) $$ ( 1 - ε ) · n + d n atoms $$l_{x_i}$$ l x i . Consequences and results for the Hankel matrix and flat extension are gained. We find that there are moment functionals $$L:\mathbb {R}[x_1,\cdots ,x_n]_{\le 2d}\rightarrow \mathbb {R}$$ L : R [ x 1 , ⋯ , x n ] ≤ 2 d → R which need to be extended to the worst case degree 4d, $$\tilde{L}:\mathbb {R}[x_1,\cdots ,x_n]_{\le 4d}\rightarrow \mathbb {R}$$ L ~ : R [ x 1 , ⋯ , x n ] ≤ 4 d → R , in order to have a flat extension.

On Nonnil-S-Noetherian Rings

Let R be a commutative ring with identity, and let S be a (not necessarily saturated) multiplicative subset of R. We define R to be a nonnil-S-Noetherian ring if each nonnil ideal of R is S-finite. In this paper, we study some properties of nonnil-S-Noetherian rings. More precisely, we investigate nonnil-S-Noetherian rings via the Cohen-type theorem, the flat extension, the faithfully flat extension, the polynomial ring extension, and the power series ring extension.

4-Dimensional manifolds with nonnegative scalar curvature and CMC boundary

In this paper, we will consider 4-dimensional manifolds with nonnegative scalar curvature and constant mean curvature (CMC) boundary. For compact manifolds with boundary, the influence of the nonnegativity of the region scalar curvature to the geometry of the boundary is considered. Some inequalities are established for manifolds with inner boundary and outer boundary. Even for compact manifolds without inner boundary, we can obtain some inequalities involving the geometric quantities of the boundary and give some obstruction. We also discuss the 4-dimensional asymptotically flat extension of the 3-dimensional Bartnik data with CMC boundary and provide the upper bound of the Bartnik mass.

Non-flat extension of flat vector bundles

We construct a pair [Formula: see text], where [Formula: see text] is a holomorphic vector bundle over a compact Riemann surface and [Formula: see text] a holomorphic subbundle, such that both [Formula: see text] and [Formula: see text] admit holomorphic connections, but [Formula: see text] does not.

Deformations of rings

Let A and A0 be rings with a surjective homomorphism A → A0. Given a flat extension B0 of A0, a deformation of B0/A0 over A is a flat extension B of A such that B ⊗AA0 is isomorphic to B0. We show that such a deformation will exist if A0 is an Artin local ring, A is noetherian, and the homological dimension of B0 over A0 is ≤ 2. We also show that a deformation will exist if the kernel of A is nilpotent and if A0 is a finitely generted A0-algebra whose defining ideal is a local complete intersection.

m-full ideals

An ideal a of a local ring (R, m) is called m-full if am: y = a for some y in a certain faithfully flat extension of R. The definition is due to Rees (unpublished) and he had obtained some elementary results (also unpublished). The present paper concerns some basic properties of m-full ideals. One result is the characterization of m-fullness in terms of the minimal number of generators of ideal, generalizing his result in a low dimensional case (Theorem 2, § 2).