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1.
Deep learning has been widely applied and brought breakthroughs in speech recognition, computer vision, and many other domains. Deep neural network architectures and computational issues have been well studied in machine learning. But there lacks a theoretical foundation for understanding the approximation or generalization ability of deep learning methods generated by the network architectures such as deep convolutional neural networks. Here we show that a deep convolutional neural network (CNN) is universal, meaning that it can be used to approximate any continuous function to an arbitrary accuracy when the depth of the neural network is large enough. This answers an open question in learning theory. Our quantitative estimate, given tightly in terms of the number of free parameters to be computed, verifies the efficiency of deep CNNs in dealing with large dimensional data. Our study also demonstrates the role of convolutions in deep CNNs.  相似文献   
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A simple method for the functionalization of closo‐borates [closo‐B10H10]2? ( 1 ), [closo‐1‐CB9H10]? ( 2 ), [closo‐B12H12]2? ( 3 ), [closo‐1‐CB11H12]? ( 4 ), and [3,3′‐Co(1,2‐C2B9H11)2]? ( 5 ) is described. Treatment of the anions and their derivatives with ArI(OAc)2 gave aryliodonium zwitterions, which were sufficiently stable for chromatographic purification. The reactions of these zwitterions with nucleophiles provided facile access to pyridinium, sulfonium, thiol, carbonitrile, acetoxy, and amino derivatives. The synthetic results are augmented by mechanistic considerations.  相似文献   
4.
Dong Ye 《Discrete Mathematics》2018,341(5):1195-1198
It was conjectured by Mkrtchyan, Petrosyan and Vardanyan that every graph G with Δ(G)?δ(G)1 has a maximum matching M such that any two M-unsaturated vertices do not share a neighbor. The results obtained in Mkrtchyan et al. (2010), Petrosyan (2014) and Picouleau (2010) leave the conjecture unknown only for k-regular graphs with 4k6. All counterexamples for k-regular graphs (k7) given in Petrosyan (2014) have multiple edges. In this paper, we confirm the conjecture for all k-regular simple graphs and also k-regular multigraphs with k4.  相似文献   
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In this paper, we study regression problems over a separable Hilbert space with the square loss, covering non-parametric regression over a reproducing kernel Hilbert space. We investigate a class of spectral/regularized algorithms, including ridge regression, principal component regression, and gradient methods. We prove optimal, high-probability convergence results in terms of variants of norms for the studied algorithms, considering a capacity assumption on the hypothesis space and a general source condition on the target function. Consequently, we obtain almost sure convergence results with optimal rates. Our results improve and generalize previous results, filling a theoretical gap for the non-attainable cases.  相似文献   
6.
Zinc oxide nanoparticles were tethered to carbon nanotubes to form an electrochemical sensing composite, which was characterized using transmission electron microscopy, and X‐ray photoelectron and attenuated total reflectance infrared spectroscopies. The sensor′s electrocatalytic response for measuring hydrogen peroxide was investigated using cyclic voltammetry and chronoamperometry in a dynamic, well defined swimming pool environment. A wide, linear response in the concentration range of 0.025–7.0 mM at a potential of ?0.360 V was shown, with rapid response time (<5 s). The sensor had excellent reproducibility, exhibited stability and selectivity, and was able to measure concentrations in a dynamic environment as they varied.  相似文献   
7.
When faced with a false generalization and a counterexample, what types of claims do prospective K-8 teachers make, and what factors influence the type and prudence of their claims relative to the data, observations, and arguments reported? This article addresses that question. Responses to refutation tasks and cognitive interviews were used to explore claiming. It was found that prospective K-8 teachers’ claiming can be influenced by knowledge of argumentation; knowledge and use of the mathematical practice of exception barring; perceptions of the task; use of natural language; knowledge of, use of, and skill with the mathematics register; and abilities to technically handle data or conceptual insights. A distinction between technical handlings for developing claims and technical handlings for supporting claims was made. It was found that prudent claims can arise from arguer-developed representations that afford conceptual insights, even when searching for support for a different claim.  相似文献   
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This paper sets the scene for a special issue of ZDMThe International Journal on Mathematics Education—by tracing key elements of the fields of teacher and didactician/teacher-educator learning related to the development of opportunities for learners of mathematics in classrooms. It starts from the perspective that joint activity of these two groups (teachers and didacticians), in creation of classroom mathematics, leads to learning for both. We trace development through key areas of research, looking at forms of knowledge of teachers and didacticians in mathematics; ways in which teachers or didacticians in mathematics develop their professional knowledge and skill; and the use of theoretical perspectives relating to studying these areas of development. Reflective practice emerges as a principal goal for effective development and is linked to teachers’ and didacticians’ engagement with inquiry and research. While neither reflection nor inquiry are developmental panaceas, we see collaborative critical inquiry between teachers and didacticians emerging as a significant force for teaching development. We include a summary of the papers of the special issue which offer a state of the art perspective on developmental practice.  相似文献   
9.
Deep eutectic solvents (such as the combination of urea and choline chloride) are effective solvents/organocatalysts for the condensation of indole and aryl or alkenyl aldehydes to form bis(indolyl)methanes. The reaction conditions are quite mild and do not require additional Bronsted or Lewis acid catalyst, though they fail with ketones or aliphatic aldehydes. Given the inexpensive, non-toxic, and recyclable nature of the DES, these reaction conditions are simple and highly environmentally friendly.  相似文献   
10.
Deep eutectic solvents (such as the combination of either urea or glycerol with choline chloride) are effective solvents/organocatalysts for Pictet–Spengler condensations to form carbolines. The reaction conditions are quite mild and do not require additional Bronsted or Lewis acid catalyst. Given the inexpensive, non-toxic, and recyclable nature of the DES, these reaction conditions are simple and highly environmentally friendly.  相似文献   
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