Universal autohomeomorphisms of ℕ Hart, Klaas Pieter; van Mill, Jan
Proceedings of the American Mathematical Society. Series B,
3/2022, Volume:
9, Issue:
8
Journal Article
Peer reviewed
Open access
We study the existence of universal autohomeomorphisms of
N
∗
\mathbb {N}^*
. We prove that the Continuum Hypothesis (
C
H
\mathsf {CH}
) implies there is such an autohomeomorphism and show that ...there are none in any model where all autohomeomorphisms of
N
∗
\mathbb {N}^*
are trivial.
For a simply connected domain G, we study the problem of disjoint universality for the sequences of operators Tα,n:H(G)→H(G), defined by Tα,n(f)(z)=∑j=0nf(j)(z)j!(αz)j, where α∈C∖{0}. Note that ...Tα,n(f)(z) is the nth partial sum of the Taylor expansion of f around z on (α+1)z. The motivation to study such sequences comes from universal Taylor series, by changing the role of the center of expansion.
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.
We consider Dyson Brownian motion for classical values of β with deterministic initial data V. We prove that the local eigenvalue statistics coincide with the GOE/GUE in the fixed energy sense after ...time t≳1/N if the density of states of V is bounded above and below down to scales η≪t in a window of size L≫t. Our results imply that fixed energy universality holds for essentially any random matrix ensemble for which averaged energy universality was previously known. Our methodology builds on the homogenization theory developed in 16 which reduces the microscopic problem to a mesoscopic problem. As an auxiliary result we prove a mesoscopic central limit theorem for linear statistics of various classes of test functions for classical Dyson Brownian motion.
Skyrmion Motion
Understanding skyrmion motion is essential for the efficient operation of skyrmionic devices. In article number 2203275, Chanyong Hwang, Kyoung‐Whan Kim, Albert Min Gyu Park, Kab‐Jin ...Kim, and co‐workers uncover a universal scaling behavior in the current‐driven motion of skyrmions. The topology of the shape of the interface drags the skyrmions and causes them to show hopping properties distinct from the creep motion expected for magnetic domains. The universal scaling can provide insights for implementing skyrmion devices and understanding the dynamics of moving bodies with a closed interface.
Recently, harmonic functions and frequently universal harmonic functions on a tree T have been studied, taking values on a separable Fréchet space E over the field C or R. In the present paper, we ...allow the functions to take values in a vector space E over a rather general field F. The metric of the separable topological vector space E is translation invariant and instead of harmonic functions we can also study more general functions defined by linear combinations with coefficients in F. We don't assume that E is complete and therefore we present an argument avoiding Baire's theorem.
Psychology is currently facing a multilayered crisis stemming from the fact that the results of many psychological studies cannot be replicated (replication crisis), that psychological research has ...neglected cross-cultural and cross-temporal variation (universality crisis), and that many psychological theories are ill-developed and underspecified (theory crisis). In the present article, we use ideas derived from debates in theoretical and philosophical psychology as a basis for responding to all three crises. In short, we claim that psychological concepts are inherently vague in the sense that their meanings and the rules for their application are indeterminate. This does not imply that psychological concepts are ineffable or lack meaning. It implies, however, that hoping to arrive at a finite set of necessary and sufficient criteria that define psychological concepts once and for all is an illusion. From this, we deduce four recommendations for responding to psychology's crises. First, we argue that the replication crisis could be approached by paying more attention to the context conditions under which psychological realities and knowledge about these realities are being created. Second, we claim that the universality crisis can be alleviated by putting more effort into exploring variability across times and cultures. Third, we contend that acknowledging the language dependence of psychological research could be a fruitful way of addressing the theory crisis. Last, we show that embracing theoretical and methodological pluralism would be an antidote against psychology's crises in general. (PsycInfo Database Record (c) 2024 APA, all rights reserved).
Coupled Neural P Systems Peng, Hong; Wang, Jun
IEEE transaction on neural networks and learning systems,
06/2019, Volume:
30, Issue:
6
Journal Article
Inspired by Eckhorn's neuron model that emulates a mammal's visual cortex, this paper proposes a new kind of neural-like P system, called a coupled neural P (CNP) system. The CNP system consists of ...some coupled neurons, each with three components: receptive field, modulation, and output module. CNP systems are a kind of distributed parallel-computing model with a directed graph structure like spiking neural P systems. Moreover, CNP systems have a nonlinear coupled-modulation characteristic and a dynamic threshold mechanism. The computational power of CNP systems is discussed. Specifically, it is proved that CNP systems as number-generating devices are Turing universal. Moreover, we provide a small universal CNP system for function-computing devices.
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