33,279 research outputs found
Research Achievements Review Series no. 20 - Mathematics and computation research
Computational mathematics, perturbed orbit three-body problem, and periodic trajectories solutions through computer method
Rigid body dynamics
Encyclopedia of Applied and Computational Mathematics, SpringerFull entry in Encyclopedia of Applied and Computational Mathematics, Springe
Stochastic ordinary differential equations in applied and computational mathematics
Using concrete examples, we discuss the current and potential use of stochastic ordinary differential equations (SDEs) from the perspective of applied and computational mathematics. Assuming only a minimal background knowledge in probability and stochastic processes, we focus on aspects that distinguish SDEs from their deterministic counterparts. To illustrate a multiscale modelling framework, we explain how SDEs arise naturally as diffusion limits in the type of discrete-valued stochastic models used in chemical kinetics, population dynamics, and, most topically, systems biology. We outline some key issues in existence, uniqueness and stability that arise when SDEs are used as physical models, and point out possible pitfalls. We also discuss the use of numerical methods to simulate trajectories of an SDE and explain how both weak and strong convergence properties are relevant for highly-efficient multilevel Monte Carlo simulations. We flag up what we believe to be key topics for future research, focussing especially on nonlinear models, parameter estimation, model comparison and multiscale simulation
The Expected Norm of a Sum of Independent Random Matrices: An Elementary Approach
In contemporary applied and computational mathematics, a frequent challenge
is to bound the expectation of the spectral norm of a sum of independent random
matrices. This quantity is controlled by the norm of the expected square of the
random matrix and the expectation of the maximum squared norm achieved by one
of the summands; there is also a weak dependence on the dimension of the random
matrix. The purpose of this paper is to give a complete, elementary proof of
this important, but underappreciated, inequality.Comment: 20 page
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