This note discusses proofs for convergence of first-order methods based on simple potentialfunction arguments. We cover methods like gradient descent (for both smooth and non-smooth settings), mirror descent, and some accelerated variants.

This paper discusses optical beam control using adaptive optics for imaging spacecraft with large aperture telescopes. The segmented mirror alignment and active mirror surface control are achieved by novel wavefront sensing and control techniques. The experimental testbed and test results are also presented to demonstrate the adaptive optics techniques for large-aperture segmented mirrors.

This paper considers the p-Laplacian PDE for p = 1,∞ and some interesting new game theoretic interpretations, due to Kohn–Serfaty [K-S] and to Peres–Schramm–Sheffield–Wilson [P-S-S-W].

We define the thin fundamental categorical group P2(M, ∗) of a based smooth manifold (M, ∗) as the categorical group whose objects are rank-1 homotopy classes of based loops on M , and whose morphisms are rank2 homotopy classes of homotopies between based loops on M . Here two maps are rank-n homotopic, when the rank of the differential of the homotopy between them equals n. Let C(G) be a Lie c...

Mirror descent with an entropic regularizer is known to achieve shifting regretbounds that are logarithmic in the dimension. This is done using either a carefullydesigned projection or by a weight sharing technique. Via a novel unified analysis,we show that these two approaches deliver essentially equivalent bounds on a no-tion of regret generalizing shifting, adaptive, discount...

Infinitely many Casoratian identities are derived for the Wilson and Askey-Wilson polynomials in parallel to the Wronskian identities for the Hermite, Laguerre and Jacobi polynomials, which were reported recently by the present authors. These identities form the basis of the equivalence between eigenstate adding and deleting Darboux transformations for solvable (discrete) quantum mechanical sys...

We present a calculation of Wilson loops, and the static inter-quark potential to O(α) in lattice perturbation theory. This is carried out with the Wilson, Symanzik-Weisz, and Iwasaki gauge actions and the Wilson, Sheikholeslami-Wohlert, and Kogut-Susskind dynamical fermion action for small Wilson loops, and with the Wilson gauge action and each of the dynamical quark actions in the case of the...

Recently orthonormal Wilson bases with good time–frequency localization have been constructed by Daubechies, Jaffard, and Journé. We extend this construction to Wilson sets and frames with arbitrary oversampling (or redundancy). We state conditions under which dual Weyl–Heisenberg (WH) sets induce dual Wilson sets, and we formulate duality conditions in the time domain and frequency domain. We ...