A general class of discrete, nonlinear renewal equations containing a real parameter is studied. Bifurcation theory methods are used to prove the existence of nontrivial periodic solutions and asymptotically periodic ...
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A general class of discrete, nonlinear renewal equations containing a real parameter is studied. Bifurcation theory methods are used to prove the existence of nontrivial periodic solutions and asymptotically periodic solutions. Fundamental to the approach is the “limit equation” whose periodic solutions are shown to be asymptotic limits of solutions of the renewal equation. An application is made to a model of age-structured population dynamics in which the bifurcation of nontrivial equilibria and 2-cycles is shown to occur with increasing inherent net reproductive value.
A nonparametric formulation is set up for selecting the best one of k populations. “Best” is defined as the one with the smallest inter(a,e)-range, a measure of dispersion defined by the difference of the 6th quanti...
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The classic Beverton-Holt (discrete logistic) difference equation, which arises in population dynamics, has a globally asymptotically stable equilibrium (for positive initial conditions) if its coefficients are consta...
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The dynamic behavior of RMSprop and Adam algorithms is studied through a combination of careful numerical experiments and theoretical explanations. Three types of qualitative features are observed in the training loss...
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Neural population activity exhibits complex, nonlinear dynamics, varying in time, over trials, and across experimental conditions. Here, we develop Conditionally Linear Dynamical System (CLDS) models as a general-purp...
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Let N = [N(t), t ≥ 0] be a point process, and let T1 and T2 be two positive random variables that are independent of N. Define Mk = N(Tk), k = 1,2. In this paper we study conditions on the process N under which some ...
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We prove that the gradient descent training of a two-layer neural network on empirical or population risk may not decrease population risk at an order faster than t−4/(d−2) under mean field scaling. The loss functiona...
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This paper contributes to the exploration of a recently introduced computational paradigm known as second-order flows, which are characterized by novel dissipative hyperbolic partial differential equations extending a...
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This paper contributes to the exploration of a recently introduced computational paradigm known as second-order flows, which are characterized by novel dissipative hyperbolic partial differential equations extending accelerated gradient flows to energy functionals defined on Sobolev spaces, and exhibiting significant performance particularly for the minimization of nonconvex energies. Our approach hinges upon convex-splitting schemes, a tool which is not only pivotal for clarifying the well-posedness of second-order flows, but also yields a versatile array of robust numerical schemes through temporal (and spatial) discretization. We prove the convergence to stationary points of such schemes in the semidiscrete setting. Further, we establish their convergence to time-continuous solutions as the timestep tends to zero. Finally, these algorithms undergo thorough testing and validation in approaching stationary points of representative nonconvex variational models in scientific computing.
We study the evolution of the dark energy parameter within a Bianchi type-I cosmological model filled with barotropic fluid and dark energy. The solutions have been obtained for power law and exponential forms of the ...
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We study the evolution of the dark energy parameter within a Bianchi type-I cosmological model filled with barotropic fluid and dark energy. The solutions have been obtained for power law and exponential forms of the expansion parameter （they correspond to a constant deceleration parameter in general relativity）. After a long time, the models tend to be isotropic under certain conditions.
A rigorous gain scheduling approach is developed to design non-linear state-feedback controllers for sampled continuous-time non-linear systems. All the information needed in the design can be obtained from measuremen...
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A rigorous gain scheduling approach is developed to design non-linear state-feedback controllers for sampled continuous-time non-linear systems. All the information needed in the design can be obtained from measurements of the continuous-time plant about selected equilibrium points. The performance of the resulting controllers is explored using two examples.
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