Moo Young Choi

In general, the rates of infection and removal (whether through recovery or death) are nonlinear functions of the number of infected and susceptible individuals. One of the simplest models for the spread of infectious diseases is the SIR model, which categorizes individuals as susceptible, infectious, recovered or deceased. In this model, the infection rate, governing the transition from susceptible to infected individuals, is given by a linear function of both susceptible and infected populations. Similarly, the removal rate, representing the transition from infected to removed individuals, is a linear function of the number of infected individuals. While nonlinear infection and removal rates have been extensively studied in deterministic epidemiological models, analytic results for stochastic dynamics with general nonlinear rates remain limited. This work presents an analytic expression for the number of infected individuals considering nonlinear infection and removal rates. In particular, we examine how the number of infected individuals varies as cases emerge and obtain the expression accounting for the number of infected individuals at each moment. This work paves the way for new quantitative approaches to understanding infection dynamics.

This research presents a novel algorithmic implementation to improve the analysis of visually controlled interception and accompanying motor action through the computational application of harmonic ratios and stimulation invariants. Unlike traditional models that focus mainly on psychological aspects, our approach integrates the relevant constructs into a practical mathematical framework. This allows for dynamic prediction of interception points with improved accuracy and real-time perception–action capabilities, essential for applications in neurorehabilitation and virtual reality. Our methodology uses stimulation invariants as key parameters within a mathematical model to quantitatively predict and improve interception outcomes. The results demonstrate the superior performance of our algorithms over conventional methods, confirming their potential for advancing robotic vision systems and adaptive virtual environments. By translating complex theories of visual perception into algorithmic solutions, this study provides innovative ways to improve motion perception and interactive systems. This study aims to articulate the complex interplay of geometry, perception, and technology in understanding and utilizing cross ratios at infinity, emphasizing their practical applications in virtual and augmented reality settings.

This research presents a novel algorithmic implementation to improve the analysis of visually controlled interception and accompanying motor action through the computational application of harmonic ratios and stimulation invariants. Unlike traditional models that focus mainly on psychological aspects, our approach integrates the relevant constructs into a practical mathematical framework. This allows for dynamic prediction of interception points with improved accuracy and real-time perception-action capabilities, essential for applications in neurorehabilitation and virtual reality. Our methodology uses stimulation invariants as key parameters within a mathematical model to quantitatively predict and improve interception outcomes. The results demonstrate the superior performance of our algorithms over conventional methods, confirming their potential for ad-vancing robotic vision systems and adaptive virtual environments. By translating complex theo-ries of visual perception into algorithmic solutions, this study provides innovative ways to im-prove motion perception and interactive systems. This study aims to articulate the complex inter-play of geometry, perception, and technology in understanding and utilizing cross-ratios at infin-ity, emphasizing their practical applications in virtual and augmented reality settings.

We demonstrate that the surface entropy given by the volume of an energy shell in the phase space can be the thermodynamically consistent entropy in a classical microcanonical ensemble if the thickness of the energy shell is not an arbitrary constant but a non-extensive function satisfying a specific differential equation. A particular form of the energy shell thickness as a possible solution to the differential equation converts the surface entropy into the volume entropy given by the phase-space volume bounded by a constant energy surface. However, such a form bears a problem: The temperature derived accordingly becomes extensive when the density of states is a non-monotonic function of energy. Based on the adiabatic invariance of the degeneracy of a quantum system and the Weyl correspondence, we propose an alternative solution: the energy shell thickness given by the energy level spacing in the quantum counterpart of the classical ensemble considered, which is illustrated by a few simple examples.

The Feynman–Kac formula establishes a link between parabolic partial differential equations and stochastic processes in the context of the Schrödinger equation in quantum mechanics. Specifically, the formula provides a solution to the partial differential equation, expressed as an expectation value for Brownian motion. This paper demonstrates that the Feynman–Kac formula does not produce a unique solution but instead carries infinitely many solutions to the corresponding partial differential equation.

We consider a system of clusters of various sizes or masses, subject to aggregation and fragmentation by collision with monomers or by self-disintegration. The aggregation rate for the cluster of size or mass k is given by a kernel proportional to k ^a , whereas the collision and disintegration kernels are given by λ k b and μ k a , respectively, with 0 ⩽ a , b ⩽ 1 and positive factors λ and µ. We study the emergence of oscillations in the phase diagram ( μ , λ ) for two models: ( a , b ) = ( 1 , 0 ) and ( 1 , 1 ) . It is shown that the monomer population satisfies a class of integral equations possessing oscillatory solutions in a finite domain in the plane ( μ , λ ) . We evaluate analytically this domain and give an estimate of the oscillation frequency. In particular, these oscillations are found to occur generally for small but nonzero values of the parameter µ, far smaller than λ.

Neuronal avalanches are cascades of bursts of activity observed primarily in the superficial cortical layers, the distribution of which fits a power law well. Motivated by the observation, we study how a power-law activity distribution emerges in a spiking neural network model. Specifically, we clarify the fundamentals of the phenomenon by applying a general theory of scale-free behavior, introduced to explain the power-law degree distribution in a brain network, and disclose that two kinds of fluctuations in spiking dynamics serve as the essential mechanism for the phenomenon. It is shown that the scale-free behavior arises from a Markov process or a Fokker-Planck diffusion in one dimension and how the power-law exponent of the activity distribution is determined depending on several factors, including the time bin. Finally, we also explain the scale-free behavior observed in the statistics of activity lifetimes.