By Gilbert Strang (auth.), David Y. Gao, Hanif D. Sherali (eds.)

The articles that contain this individual annual quantity for the *Advances in Mechanics and Mathematics* sequence were written in honor of Gilbert Strang, a global popular mathematician and unparalleled individual. Written by way of top specialists in complementarity, duality, worldwide optimization, and quantum computations, this assortment finds the wonderful thing about those mathematical disciplines and investigates fresh advancements in worldwide optimization, nonconvex and nonsmooth research, nonlinear programming, theoretical and engineering mechanics, huge scale computation, quantum algorithms and computation, and knowledge theory.

Much of the fabric, together with a few of the methodologies, is written for nonexperts and is meant to stimulate graduate scholars and younger school to enterprise into this wealthy area of study; it's going to additionally profit researchers and practitioners in different components of utilized arithmetic, mechanics, and engineering.

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A summary account of these can be found in 3 Finite Element Solution of Variational Inequalities with Applications 29 [39]. In [61, 60], the technique of the duality theory was used to derive a posteriori error estimates of the finite element method in solving boundary value problems of some nonlinear equations. In these papers, the error bounds are shown to converge to zero in the limit; however, no eﬃciency analysis of the estimates is given. For convenience, we recall here a representative result on the duality theory (see [30]).

Ford Jr. and D. R. Fulkerson, Flows in Networks, Princeton University Press, 1962. 14. L. R. Ford Jr. and D. R. Fulkerson, Maximal flow through a network, Canad. J. Math. 8 (1956) 399—404. 15. W. Gangbo and R. R. Acad. Sci. Paris. Ser. I. Math. 325 (1995) 1653—1658. 16. D. Grieser, The first eigenvalue of the Laplacian, isoperimetric constants, and the max flow min cut theorem, Archiv der Mathematik 87 (2006) 75—85. 17. T. C. Hu, Integer Programming and Network Flows, Addison-Wesley, 1969. 18. M.

1a as the set where |v| = 1. The eigenfunctions of the Laplacian on the unit square are (sin πx)(sin πy) and the lowest eigenvalue is λ1 = 2π2 . Cheeger’s inequality λ1 ≥ h2 /4, which other authors have tested earlier, is far from tight. 56 . 24) The second challenge problem has |v1 | + |v2 | ≤ 1 leading to the measure |∂S|∞ of the perimeter in the dual. Now the unconstrained isoperimetric √ problem is solved by a diamond with |n1 | = |n2 | = 1/ 2 on all edges. 1b is a union of boundary √ edges.