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

The articles that include this uncommon annual quantity for the *Advances in Mechanics and Mathematics* sequence were written in honor of Gilbert Strang, an international popular mathematician and unheard of individual. Written by way of top specialists in complementarity, duality, international optimization, and quantum computations, this assortment unearths 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 cloth, together with a few of the methodologies, is written for nonexperts and is meant to stimulate graduate scholars and younger college to enterprise into this wealthy area of study; it's going to additionally profit researchers and practitioners in different parts of utilized arithmetic, mechanics, and engineering.

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The four arcs would fit together in a circle of radius r. With L = 1 − 2r, the optimal cut solves the Cheeger problem: tmax = h(Ω) = min perimeter of S 4(1−2r) + 2πr . 23) The derivative of that ratio is zero when (1 − 4r2 + πr2 )(8 − 2π) = (4 − 8r + 2πr)(8r − 2πr). √ Cancel 8 − 2π to reach 1 − 4r + (4 − π)r2 = 0. 265. The Cheeger constant h(Ω) is the ratio |∂S|/|S| = 1/r = 2 + π. A prize of √ 10,000 yen was oﬀered in [30] for the flow field that achieves div v = 2 + π with |v| ≤ 1. Lippert [20] and Overton [24] have the strongest 1 Maximum Flows and Minimum Cuts in the Plane r S L = 1−2r R S L = 1−2R r |∂S| = 4L + 2πr 9 S L=1 R |∂S|∞ = 4L + 4R Fig.

50 (1983), 80—145. [3] G. Auchmuty, Saddle Point Methods, and Algorithms, for Non-Symmetric Linear Equations, Numer. Funct. Anal. Optim. 16 (1995), 1127—1142. [4] G. Auchmuty, Min-Max Problems for Non-Potential Operator Equations, Optimization Methods in Partial Diﬀerential Equations (S. Cox and I. ), Contemporary Mathematics, vol. 209, American Mathematical Society, Providence, 1997, pp. 19—28. [5] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland, Amsterdam, 1976.

R. Nozawa, Examples of max-flow and min-cut problems with duality gaps in continuous networks, Math. Program. 63 (1994) 213—234. 23. R. Nozawa, Max-flow min-cut theorem in an anisotropic network, Osaka J. Math. 27 (1990) 805—842. 24. M. L. Overton, Numerical solution of a model problem from collapse load analysis, Computing Methods in Applied Science and Engineering VI, R. Glowinski and J. L. , Elsevier, 1984. 25. S. T. Rachev and L. R¨ uschendorf, Mass Transportation Problems I, II, Springer (1998).