Survey and Review
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Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. In this issue’s Survey and Review section, we feature two articles. The first paper is about controlling quantum systems. This is a hot area of research which has applica- tions in chemistry, physics, and photonics. The article, written by Bouten, van Handel, and James, is a tremendous resource for someone who wants to work in this field. It can, in fact, serve as a self-contained course material at the graduate level. This being the case, it can still be accessed for a quick read. In a feedback control problem, one needs to have measurements of the system in order to exercise appropriate control. The heart of the problem in quantum systems is that sensing is very challenging when there is so much uncertainty. The authors develop a finite state model of observation which leads to a filtering equation for estimating the state. Dynamic feedback control is considered and its connections to the classical theory of stochastic control are made. The article provides a valuable outlook on quantum control. It does indeed serve as an invitation, a “discreet” one, for mathematicians to get into this quickly developing field. The second paper, by Brandts, Korotov, Kˇ r´ ıˇ zek, and ˇ Solc, presents several prob- lems in discrete geometry. For instance, they pose a fascinating puzzle which can be easily visualized in two dimensions. Given a polygon, we know that we can triangulate it. In nonobtuse simplicial partition, we make a further requirement that the angles in any of the triangles cannot exceed 90 degrees. Such a partition is called a “nonobtuse simplicial partition.” If we require that all angles are acute, we obtain an “acute simpli- cial partition.” A natural question one may ask is how many triangles would one need. The authors review questions such as this, and more. The paper offers a comprehensive survey of the subject of nonobtuse simplicial partition. While a lot is known about this discrete geometry problem, there is still much to be discovered. The article carefully goes through known results and states open conjectures. In addition it provides a colorful history of the subject, including Aristotle’s “contribution,” which in the Middle Ages was proven to be incorrect. A nice description of how nonobtuse simplicial partitions play an important role in finite element approximations of solutions of partial differential equations is highlighted. This short, and thought-provoking, review is a pleasure to read. Fadil Santosa Section Editor [email protected] 237 Downloaded 11/17/14 to 99.251.251.153. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
Transcript of Survey and Review
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
In this issue’s Survey and Review section, we feature two articles. The first paperis about controlling quantum systems. This is a hot area of research which has applica-tions in chemistry, physics, and photonics. The article, written by Bouten, van Handel,and James, is a tremendous resource for someone who wants to work in this field. Itcan, in fact, serve as a self-contained course material at the graduate level. This beingthe case, it can still be accessed for a quick read.
In a feedback control problem, one needs to have measurements of the system inorder to exercise appropriate control. The heart of the problem in quantum systems isthat sensing is very challenging when there is so much uncertainty. The authors developa finite state model of observation which leads to a filtering equation for estimatingthe state. Dynamic feedback control is considered and its connections to the classicaltheory of stochastic control are made.
The article provides a valuable outlook on quantum control. It does indeed serveas an invitation, a “discreet” one, for mathematicians to get into this quickly developingfield.
The second paper, by Brandts, Korotov, Krızek, and Solc, presents several prob-lems in discrete geometry. For instance, they pose a fascinating puzzle which can beeasily visualized in two dimensions. Given a polygon, we know that we can triangulateit. In nonobtuse simplicial partition, we make a further requirement that the angles inany of the triangles cannot exceed 90 degrees. Such a partition is called a “nonobtusesimplicial partition.” If we require that all angles are acute, we obtain an “acute simpli-cial partition.” A natural question one may ask is how many triangles would one need.The authors review questions such as this, and more.
The paper offers a comprehensive survey of the subject of nonobtuse simplicialpartition. While a lot is known about this discrete geometry problem, there is stillmuch to be discovered. The article carefully goes through known results and statesopen conjectures. In addition it provides a colorful history of the subject, includingAristotle’s “contribution,” which in the Middle Ages was proven to be incorrect. Anice description of how nonobtuse simplicial partitions play an important role in finiteelement approximations of solutions of partial differential equations is highlighted. Thisshort, and thought-provoking, review is a pleasure to read.
Fadil SantosaSection Editor
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