Analysis for Applied Mathematics

By E. Cheney. 2001

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Control functionals for Monte Carlo integration (2014, background): “reira and Menegatto, 2009). Moreover hTg;g 0i H+ = hg;gi L 2(ˇ) for all g2H +, g 02L 2(ˇ) (Prop. 2.6 of Ferreira and Menegatto, 2013). The spectral theorem for self-adjoint compact operators (p.93 of Cheney, 2001) then shows that there exists a countable orthonormal system fv jg1 j=1 L 2(ˇ)\C0(X;R) ”
On Robustness Properties of Convex Risk Minimization Methods for Pattern Recognition (2004, unknown): “For this the following theorem which is known as the Fredholm Alternative, (Cheney, 2001) turns out to be very helpful: is equal to the closure of the convex hull of the set of all extreme points of K. Appendix B: Proofs of the theorems.”
A coupling of mixed and discontinuous Galerkin finite-element methods for poroelasticity (2008, background): “It is well known that the set of invertible matrices forms an open and dense set, O ⊂ R n p×n p (see, e.g., Cheney [ 9 ]).”
Pseudo-Hermitian Representation of Quantum Mechanics (2008, methodology): “Given a basis {χn} of a separable Hilbert space H, one can apply the Gram-Schmidt process [198, 66] to construct an orthonormal basis, i.”
Linear vs. nonlinear selection for the propagation speed of the solutions of scalar reaction‐diffusion equations invading an unstable equilibrium (2004, methodology): “Thus, applying the Lagrange multiplier theorem (see, e.g., [ 9, section 3.5 ]), we obtain”