Universality of deep convolutional neural networks

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.     

A Zinc Oxide Carbon Nanotube Based Sensor for In Situ Monitoring of Hydrogen Peroxide in Swimming Pools

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.     

Prospective elementary teachers’ claiming in responses to false generalizations

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.     

Maximum matchings in regular graphs

It was conjectured by Mkrtchyan, Petrosyan and Vardanyan that every graph $G$ with $\Delta \left(G\right)?\delta \left(G\right)\le 1$ 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 $4\le k\le 6$. All counterexamples for $k$-regular graphs $(k\ge 7)$ 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 $k\le 4$.     

Optimal rates for spectral algorithms with least-squares regression over Hilbert spaces

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.     

An acid-free Pictet–Spengler reaction using deep eutectic solvents (DES)

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.     

Finding Δ(Σ) for a surface σ of characteristic χ(Σ) = −5

For each surface Σ, we define Δ(Σ) = max{Δ(G)|Gis a class two graph of maximum degree Δ(G) that can be embedded in Σ}. Hence, Vizing's Planar Graph Conjecture can be restated as Δ(Σ) = 5 if Σ is a plane. In this paper, we show that Δ(Σ) = 9 if Σ is a surface of characteristic χ(Σ) = ?5. © 2010 Wiley Periodicals, Inc. J Graph Theory 68:148‐168, 2011     

A mild synthesis of bis(indolyl)methanes using a deep eutectic solvent

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.     

On certain spanning subgraphs of embeddings with applications to domination

We prove the existence of certain spanning subgraphs of graphs embedded in the torus and the Klein bottle. Matheson and Tarjan proved that a triangulated disc with n vertices can be dominated by a set of no more than n/3 of its vertices and thus, so can any finite graph which triangulates the plane. We use our existence theorems to prove results closely allied to those of Matheson and Tarjan, but for the torus and the Klein bottle.     

Smoothing schemes for reaction-diffusion systems with nonsmooth data

Cox and Matthews [S.M. Cox, P.C. Matthews, Exponential time differencing for stiff systems, J. Comput. Phys. 176 (2002) 430–455] developed a class of Exponential Time Differencing Runge–Kutta schemes (ETDRK) for nonlinear parabolic equations; Kassam and Trefethen [A.K. Kassam, Ll. N. Trefethen, Fourth-order time stepping for stiff pdes, SIAM J. Sci. Comput. 26 (2005) 1214–1233] have shown that these schemes can suffer from numerical instability and they proposed a modified form of the fourth-order (ETDRK4) scheme. They use complex contour integration to implement these schemes in a way that avoids inaccuracies when inverting matrix polynomials, but this approach creates new difficulties in choosing and evaluating the contour for larger problems. Neither treatment addresses problems with nonsmooth data, where spurious oscillations can swamp the numerical approximations if one does not treat the problem carefully. Such problems with irregular initial data or mismatched initial and boundary conditions are important in various applications, including computational chemistry and financial engineering. We introduce a new version of the fourth-order Cox–Matthews, Kassam–Trefethen ETDRK4 scheme designed to eliminate the remaining computational difficulties. This new scheme utilizes an exponential time differencing Runge–Kutta ETDRK scheme using a diagonal Padé approximation of matrix exponential functions, while to deal with the problem of nonsmooth data we use several steps of an ETDRK scheme using a sub-diagonal Padé formula. The new algorithm improves computational efficiency with respect to evaluation of the high degree polynomial functions of matrices, having an advantage of splitting the matrix polynomial inversion problem into a sum of linear problems that can be solved in parallel. In this approach it is only required that several backward Euler linear problems be solved, in serial or parallel. Numerical experiments are described to support the new scheme.