We prove various extensions of the Local Flatness Criterion over a Noetherian local ring R with residue field k. For instance, if Ω is a complete R-module of finite projective dimension, then Ω is flat if and only if Torn (Ω, k) = 0 for all n = 1, . . . , depth(R). In low dimensions, we have the following criteria. If R is onedimensional and reduced, then Ω is flat if and only if Tor1 (Ω, k) = ...

For k ∈ {1, 2, 3, . . . }, we construct an even compactly supported piecewise polynomial ψk whose Fourier transform satisfies Ak(1 + ω ) ≤ b ψk(ω) ≤ Bk(1 + ω ), ω ∈ R, for some constants Bk ≥ Ak > 0. The degree of ψk is shown to be minimal, and is strictly less than that of Wendland’s function φ1,k−1 when k > 2. This shows that, for k > 2, Wendland’s piecewise polynomial φ1,k−1 is not of minima...

This paper is concerned with the union sp (j,k) Ω Tn(a) of all possible spectra that may emerge when perturbing a large n× n Toeplitz band matrix Tn(a) in the (j, k) site by a number randomly chosen from some set Ω. The main results give descriptive bounds and, in several interesting situations, even provide complete identifications of the limit of sp (j,k) Ω Tn(a) as n → ∞. Also discussed are ...

For an interconnection network G, the ω-wide diameter dω(G) is the least l such that any two vertices are joined by ω internally-disjoint paths of length at most l, and the (ω − 1)-fault diameter Dω(G) is the maximum diameter of a subgraph obtained by deleting fewer than ω vertices of G. The enhanced hypercube Qn,k is a variant of the well-known hypercube. Yang, Chang, Pai, and Chan gave an upp...

We consider the equation −ε2∆u+u = u in Ω ⊆ R , where Ω is open, smooth and bounded, and we prove concentration of solutions along k-dimensional minimal submanifolds of ∂Ω, for N ≥ 3 and for k ∈ {1, . . . , N − 2}. We impose Neumann boundary conditions, assuming 1 < p < N−k+2 N−k−2 and ε → 0+. This result settles in full generality a phenomenon previously considered only in the particular case ...

Let Ω = ω × R where ω ⊂ R be a bounded domain, and V : Ω −→ R a bounded potential which is 2π-periodic in the variable x3 ∈ R. We study the inverse problem consisting in the determination of V , through the boundary spectral data of the operator u 7→ Au := −∆u+ V u, acting on L(ω × (0, 2π)), with quasi-periodic and Dirichlet boundary conditions. More precisely we show that if for two potentials...

Theorem. Suppose that an א0-presentable K is almost Galois ω-stable. If K has only countably many models in א1, then K is Galois ω-stable.

Let Ω be a convex domain in ℂn satisfying certain restrictions, and let f function holomorphic continuous $$ \overline{\Omega},\kern0.5em f\kern0.5em \in \kern0.5em {H}^{r+\omega}\left(\overline{\Omega}\right) for an appropriate modulus of continuity ω. Then there exist polynomials PN, deg PN ≤ N, such that \left|f(z)-{P}_N(z)\right|\le {cN}^{-r}\omega \left(\frac{1}{N}\right),z\kern0.5em \over...