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The aim of this expository article is to present recent developments in the centuries-old discussion on the interrelations between continuous and differentiable real valued functions of one real variable. The truly new results include, among others, the Dn-Cn interpolation theorem: For every n-times differentiable f : ℝ → ℝ and perfect P ⊂ ℝ, there is a Cn function g : ℝ → ℝ such that f ↾ P and g ↾ P agree on an uncountable set and an example of a differentiable function F : ℝ → ℝ (which can be nowhere monotone) and of compact perfect X ⊂ ℝ such that F'(x) = 0 for all x ∈ X while F[X] = X. Thus, the map f = F ↾ X is shrinking at every point though, paradoxically, not globally. However, the novelty is even more prominent in the newly discovered simplified presentations of several older results, including a new short and elementary construction of everywhere differentiable nowhere monotone h: ℝ → ℝ and the proofs (not involving Lebesgue measure/integration theory) of the theorems of Jarník--Every differentiable map f : P → ℝ, with P ⊂ ℝ perfect, admits differentiable extension F : ℝ → ℝ--and of Laczkovich--For every continuous g : ℝ → ℝ there exists a perfect P ⊂ ℝ such that g ↾ P is differentiable. The main part of this exposition, concerning continuity and first-order differentiation, is presented in a narrative that answers two classical questions: To what extent must a continuous function be differentiable? and How strong is the assumption of differentiability of a continuous function? In addition, we give an overview of the results concerning higher-order differentiation. This includes the Whitney extension theorem and the higher-order interpolation theorems related to the Ulam-Zahorski problem. Finally, we discuss the results concerning smooth functions that are independent of the standard axioms of ZFC set theory. We close with a list of currently open problems related to this subject. [ABSTRACT FROM AUTHOR]

The covering property axiom CPA is consistent with ZFC: it is satisfied in the iterated perfect set model. We show that CPA implies that for every ν∈ω∪{∞} there exists a family Fν⊂Cν(R) of cardinality ω1

Purpose:Recently, clinical radiological research and practice are becoming increasingly quantitative. Further, images continue to increase in size and volume. For quantitative radiology to become practical, it is crucial that image segmentation algorithms and their implementations are rapid and yield practical run time on very large data sets. The purpose of this paper is to present a parallel version of an algorithm that belongs to the family of fuzzy connectedness (FC) algorithms, to achieve an interactive speed for segmenting large medical image data sets. Methods: The most common FC segmentations, optimizing an ℓ∞-based energy, are known as relative fuzzy connectedness (RFC) and iterative relative fuzzy connectedness (IRFC). Both RFC and IRFC objects (of which IRFC contains RFC) can be found via linear time algorithms, linear with respect to the image size. The new algorithm, P-ORFC (for parallel optimal RFC), which is implemented by using NVIDIA's Compute Unified Device Architecture (CUDA) platform, considerably improves the computational speed of the above mentioned CPU based IRFC algorithm. Results: Experiments based on four data sets of small, medium, large, and super data size, achieved speedup factors of 32.8×, 22.9×, 20.9×, and 17.5×, correspondingly, on the NVIDIA Tesla C1060 platform. Although the output of P-ORFC need not precisely match that of IRFC output, it is very close to it and, as the authors prove, always lies between the RFC and IRFC objects. Conclusions: A parallel version of a top-of-the-line algorithm in the family of FC has been developed on the NVIDIA GPUs. An interactive speed of segmentation has been achieved, even for the largest medical image data set. Such GPU implementations may play a crucial role in automatic anatomy recognition in clinical radiology. [ABSTRACT FROM AUTHOR]