In this article, semi-ring and semi-algebra of sets are formalized so as to construct a measure of a given set in the next step. Although a semi-ring of sets has already been formalized in [12], that is, strictly speaking, a definition of a quasi semi-ring of sets suggested in the last few decades [14]. We adopt a classical definition of a semi-ring of sets here to avoid such a confusion. Ring ...

We prove several dichotomy theorems which extend some known results on σ -bounded and σ -compactpointsets. In particular we show that, given a finite number of Δ1 equivalence relations F1 , . . . , Fn , any Σ1 set A of the Baire space either is covered by compact Δ1 sets and lightface Δ1 equivalence classes of the relations Fi , or A contains a superperfect subset which is pairwise Fi -inequiva...

The direct numerical solution of a non-convex variational problem (P ) typically faces the difficulty of the finite element approximation of rapid oscillations. Although the oscillatory discrete minimisers are properly related to corresponding Young measures and describe real physical phenomena, they are costly and difficult to compute. In this work, we treat the scalar double-well problem by n...

The generalized inverse A T ,S of a matrix A is a {2}-inverse of A with the prescribed range T and null space S. A representation for the generalized inverse A T ,S has been recently developed with the condition σ(GA|T ) ⊂ (0,∞), where G is a matrix with R(G) = T and N(G) = S. In this note, we remove the above condition. Three types of iterative methods for A T ,S are presented if σ(GA|T ) is a...

Amyloid load, as measured by florbetapir positron emission tomography (PET) standardized uptake value ratio (SUVr), has high specificity in the diagnosis of Alzheimer disease (AD). As the posterior cingulate cortex (PCC) represents densely amyloid-affected regions early in AD, we hypothesized that amyloid load within the key hubs of the default mode networks (DMN) may result in local or distant...

We obtain necessary and (different) sufficient conditions for a series summable | ¯ N, p n | k , 1 < k ≤ s < ∞, to imply that the series is summable |T | s , where (¯ N, p n) is a weighted mean matrix and T is a lower triangular matrix. As corollaries of this result, we obtain several inclusion theorems. Let a n be a given series with partial sums s n , (C, α) the Césaro matrix of order α. If σ...

Chemical imaging, thickness mapping, layer speciation and polarization dependence have been performed on single and multilayered (up to three layers and trilayered nanosheets overlapping to form 6 and 9 layers) hexagonal boron nitride (hBN) nanosheets by scanning transmission X-ray microscopy. Spatially-resolved XANES directly from freestanding regions of different layers has been extracted and...

We propose a pivotal method for estimating high-dimensional sparse linear regression models, where the overall number of regressors p is large, possibly much larger than n, but only s regressors are significant. The method is a modification of the lasso, called the square-root lasso. The method is pivotal in that it neither relies on the knowledge of the standard deviation σ or nor does it need...

BACKGROUND Allowable total error (TE(a)) goals for hemoglobin (Hb) A(1c) require minimal assay imprecision and bias and implementation of a robust QC monitoring program. Here, we compare the combined influence on the risk of reporting unreliable results of TE(a) goals, a routine QC practice, and assay performance characteristics of 6 Hb A(1c) instruments across 4 academic medical centers. MET...