In the mathematics of flows and shocks, Cathleen Morawetz, Olga Ladyzhenskaya, and Olga Oleinik together form a singular legacy, an interconnected legacy (Early Memories of Olga Ladyzhenskaya and Olga Oleinik). One senses a process and procedure of inheritance indifferent to the act of birth (Women in Mathematics: The Legacy of Ladyzhenskaya and Oleinik). “Of course, many discoveries have something of an accident about them,” said Morawetz. And, perhaps, Hölderlin would have responded: “By that mysterious yearning toward the chasm; / Chaotic deeps attract, and whole peoples too”
Maryam Mirzakhani died today. Terrance Tao writes about her. Elsewhere, Derrida – writing about Deleuze (I’ll Have to Wander All Alone): “Too much to say, and I don’t have the heart for it today.” It is because:
death takes from us not only some particular life within the world, some moment that belongs to us, but, each time, without limit, someone through whom the world, and first of all our own world, will have opened up in a both finite and infinite—mortally infinite—way.
In a beautiful article, The Mathematics of Doodling, Ravi Vakil poses two problems:
- Given a random curve, like a doodle, in what sense do closed curves successively drawn around the doodle become more and more circular?
- How do geometric invariants, like area and volume, of the closed curves relate with the original shape and with each other?
Given a weighted connected component, what is the weight of of the maximum independent set, and how many different sets have this weight? A Petersen graph GP(5, 2) with zero vertex weights has three maximal subsets by size, 0 maximum set weight but 76 distinct independent sets with this weight. Continue reading
Gus wants to open franchises of his restaurant, Los Pollos Hermanos, along Central Avenue. There are n possible locations for franchises, where location i is at mile i on Central. Each location i > 1, is thus a distance of 1 mile from the previous one. There are two rules.
- At each location, there can be at most one restaurant, and the profit of a restaurant at location i is p_i.
- Any two restaurants must be at least 2 miles apart.
Min-max heaps were introduced in [ASSS86] as an efficient way to support heap operations for both minimum and maximum values. Structurally, the min-max heap levels alternate between min-heap condition and max-heap, and hence evaluates grandchildren/grandparents during insertion or search. Min-max heaps can also be generalized to find the k-th smallest element in O(1) time.
In addition to inventory policy, Scarf had at least four more major breakthroughs. Perhaps his most famous discovery is the Scarf algorithm. Arrow and Debreu had proved that the equations describing economic equilibrium always have a solution when goods are divisible, but they were baffled by the problem of how to find one, except in special cases. The Scarf algorithm always finds an equilibrium, no matter how complicated the economy. This gave applied economists the ability to work with much more realistic models of the economy and thus to predict the consequences of major policy reforms including NAFTA and the U.S. tax system.