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Full-Text Articles in Applied Mathematics
Dynamic Function Learning Through Control Of Ensemble Systems, Wei Zhang, Vignesh Narayanan, Jr-Shin Li
Dynamic Function Learning Through Control Of Ensemble Systems, Wei Zhang, Vignesh Narayanan, Jr-Shin Li
Publications
Learning tasks involving function approximation are preva- lent in numerous domains of science and engineering. The underlying idea is to design a learning algorithm that gener- ates a sequence of functions converging to the desired target function with arbitrary accuracy by using the available data samples. In this paper, we present a novel interpretation of iterative function learning through the lens of ensemble dy- namical systems, with an emphasis on establishing the equiv- alence between convergence of function learning algorithms and asymptotic behavior of ensemble systems. In particular, given a set of observation data in a function learning task, we …
The Fundamental Limit Theorem Of Countable Markov Chains, Nathanael Gentry
The Fundamental Limit Theorem Of Countable Markov Chains, Nathanael Gentry
Senior Honors Theses
In 1906, the Russian probabilist A.A. Markov proved that the independence of a sequence of random variables is not a necessary condition for a law of large numbers to exist on that sequence. Markov's sequences -- today known as Markov chains -- touch several deep results in dynamical systems theory and have found wide application in bibliometrics, linguistics, artificial intelligence, and statistical mechanics. After developing the appropriate background, we prove a modern formulation of the law of large numbers (fundamental theorem) for simple countable Markov chains and develop an elementary notion of ergodicity. Then, we apply these chain convergence results …
Masked Instability: Within-Sector Financial Risk In The Presence Of Wealth Inequality, Youngna Choi
Masked Instability: Within-Sector Financial Risk In The Presence Of Wealth Inequality, Youngna Choi
Department of Applied Mathematics and Statistics Faculty Scholarship and Creative Works
We investigate masked financial instability caused by wealth inequality. When an economic sector is decomposed into two subsectors that possess a severe wealth inequality, the sector in entirety can look financially stable while the two subsectors possess extreme financially instabilities of opposite nature, one from excessive equity, the other from lack thereof. The unstable subsector can result in further financial distress and even trigger a financial crisis. The market instability indicator, an early warning system derived from dynamical systems applied to agent-based models, is used to analyze the subsectoral financial instabilities. Detailed mathematical analysis is provided to explain what financial …
A Primer On Noise-Induced Transitions In Applied Dynamical Systems, Eric Forgoston, Richard O. Moore
A Primer On Noise-Induced Transitions In Applied Dynamical Systems, Eric Forgoston, Richard O. Moore
Department of Applied Mathematics and Statistics Faculty Scholarship and Creative Works
Noise plays a fundamental role in a wide variety of physical and biological dynamical systems. It can arise from an external forcing or due to random dynamics internal to the system. It is well established that even weak noise can result in large behavioral changes such as transitions between or escapes from quasi-stable states. These transitions can correspond to critical events such as failures or extinctions that make them essential phenomena to understand and quantify, despite the fact that their occurrence is rare. This article will provide an overview of the theory underlying the dynamics of rare events for stochastic …
Generalized Thomas-Fermi Equations As The Lampariello Class Of Emden-Fowler Equations, Haret C. Rosu, S.C. Mancas
Generalized Thomas-Fermi Equations As The Lampariello Class Of Emden-Fowler Equations, Haret C. Rosu, S.C. Mancas
Publications
A one-parameter family of Emden-Fowler equations defined by Lampariello’s parameter p which, upon using Thomas-Fermi boundary conditions, turns into a set of generalized Thomas-Fermi equations comprising the standard Thomas-Fermi equation for p = 1 is studied in this paper. The entire family is shown to be non integrable by reduction to the corresponding Abel equations whose invariants do not satisfy a known integrability condition. We also discuss the equivalent dynamical system of equations for the standard Thomas-Fermi equation and perform its phase-plane analysis. The results of the latter analysis are similar for the whole class.
Modeling Contagion In The Eurozone Crisis Via Dynamical Systems, Giuseppe Castellacci, Youngna Choi
Modeling Contagion In The Eurozone Crisis Via Dynamical Systems, Giuseppe Castellacci, Youngna Choi
Department of Applied Mathematics and Statistics Faculty Scholarship and Creative Works
We recently (Castellacci and Choi, 2013) formulated a theoretical framework for the modeling of financial instability contagion using the theories of dynamical systems. Here, our main goal is to model the Eurozone financial crisis within that framework. The underlying system comprises many economic agents that belong to several subsystems. In each instantiation of this framework, the hierarchy and nesting of the subsystems is dictated by the nature of the problem at hand. We describe in great detail how a suitable model can be set up for the Eurozone crisis. The dynamical system is defined by the evolution of the wealths …