Narrow Results

Let Ω ⊂ ℝⁿ be a bounded open set. If a sequence f_k : Ω → ℝ^N converges to f in L^∞ in a certain "controlled" manner while bounded in W^1,p (1 < p < + ∞) or BV, we show that f ∈ W^1,p (respectively, f ∈ BV) and ∇f_k → ∇f almost everywhere, where ∇f_k and ∇f are the usual gradients if f_k ∈ W^1,p (respectively, the absolutely continuous part of the gradient measures if f_k ∈ BV). Our main theorem generalizes results for Lipschitz mappings. We show by an example that when p = 1, the limit of a sequence of increasing functions may fail to be in W^1,1 and can even be nowhere C¹.

Multiple regression's coefficients define change in the dependent variable due to a predictor's change while all other predictors are constant. Rearranging data to paired differences of observations and keeping only biggest changes yield a matrix of a single variable change, which is close to orthogonal design, so there is no impact of multicollinearity on the regression. A similar approach is used for meaningful coefficients of nonlinear regressions with coefficients of half-elasticity, elasticity, and odds' elasticity due the gradients in each predictor. In contrast to regular linear and nonlinear regressions, the suggested technique produces interpretable coefficients not prone to multicollinearity effects.

Part I of this article provides a status update on the ongoing projects for both high-beta and low-beta applications. Some of these projects are already under production, others are perfecting prototypes and future plans. We first cover the funded projects and continue with the planned projects. The update naturally captures the state-of-the-art for superconducting RF (SRF) performance for applications in progress. Part II goes on to present a vision for future prospects for performance progress in the field, along with some advice about the likely fruitful R&D paths to follow. In general, the R&D paths chosen for discussion will benefit most SRF-based accelerators.

Part I of this article provides a status update on the ongoing projects for both high-beta and low-beta applications. Some of these projects are already under production, others are perfecting prototypes and future plans. We first cover the funded projects and continue with the planned projects. The update naturally captures the state-of-the-art for superconducting RF (SRF) performance for applications in progress. Part II goes on to present a vision for future prospects for performance progress in the field, along with some advice about the likely fruitful R&D paths to follow. In general, the R&D paths chosen for discussion will benefit most SRF-based accelerators.