The exponential map is chaotic: An invitation to transcendental dynamics

We present an elementary and conceptual proof that the complex exponential map is "chaotic" when considered as a dynamical system on the complex plane. (This result was conjectured by Fatou in 1926 and first proved by Misiurewicz 55 years later.) The only background required is a first undergraduate course in complex analysis.Comment: 22 pages, 4 figures. (Provisionally) accepted for publication the American Mathematical Monthly. V2: Final pre-publication version. The article has been revised, corrected and shortened by 14 pages; see Version 1 for a more detailed discussion of further properties of the exponential map and wider transcendental dynamic

Group Testing with Pools of Fixed Size

In the classical combinatorial (adaptive) group testing problem, one is given two integers $d$ and $n$, where $0\le d\le n$, and a population of $n$ items, exactly $d$ of which are known to be defective. The question is to devise an optimal sequential algorithm that, at each step, tests a subset of the population and determines whether such subset is contaminated (i.e. contains defective items) or otherwise. The problem is solved only when the $d$ defective items are identified. The minimum number of steps that an optimal sequential algorithm takes in general (i.e. in the worst case) to solve the problem is denoted by $M(d, n)$. The computation of $M(d, n)$ appears to be very difficult and a general formula is known only for $d = 1$. We consider here a variant of the original problem, where the size of the subsets to be tested is restricted to be a fixed positive integer $k$. The corresponding minimum number of tests by a sequential optimal algorithm is denoted by $M^{\lbrack k\rbrack}(d, n)$. In this paper we start the investigation of the function $M^{\lbrack k\rbrack}(d, n)$

Semi-supervised Local Cluster Extraction by Compressive Sensing

Local clustering problem aims at extracting a small local structure inside a graph without the necessity of knowing the entire graph structure. As the local structure is usually small in size compared to the entire graph, one can think of it as a compressive sensing problem where the indices of target cluster can be thought as a sparse solution to a linear system. In this paper, we propose a new semi-supervised local cluster extraction approach by applying the idea of compressive sensing based on two pioneering works under the same framework. Our approves improves the existing works by making the initial cut to be the entire graph and hence overcomes a major limitation of existing works, which is the low quality of initial cut. Extensive experimental results on multiple benchmark datasets demonstrate the effectiveness of our approach

The Kolmogorov Superposition Theorem can Break the Curse of Dimensionality When Approximating High Dimensional Functions

We explain how to use Kolmogorov Superposition Theorem (KST) to overcome the curse of dimensionality in approximating multi-dimensional functions and learning multi-dimensional data sets by using neural networks of two hidden layers. That is, there is a class of functions called $K$-Lipschitz continuous in the sense that the K-outer function $g$ of $f$ is Lipschitz continuous and can be approximated by a ReLU network of two layers with $(2d+1)dn, dn$ widths to have an approximation order $O(d^2/n)$. In addition, we show that polynomials of high degree can be reproduced by using neural networks with activation function $\sigma_\ell(t)=(t_+)^\ell$ for $\ell\ge 2$ with multiple layers and appropriate widths. More layers of neural networks, the higher degree polynomials can be reproduced. Furthermore, we explain how many layers, weights, and neurons of neural networks are needed in order to reproduce high degree polynomials based on $\sigma_\ell$. Finally, we present a mathematical justification for image classification by using the convolutional neural network algorithm