Survey and Review
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Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. Mathematical objects often arise in investigation of practical, applied problems. Both articles in the Survey and Review section in this issue are studies of such ob- jects. In the first article, J.-B. Hiriart-Urruty and A. Seeger provide an exposition on copositive matrices which show up in control theory and in mathematical economics. These mathematical objects are very easy to describe. In their simplest form, they are square matrices A with real entries such that x T Ax ≥ 0 for all x in the nonnegative orthant. If a matrix associated with a control problem is copositive, then one can make some qualitative statements about the behavior of the system. The difficulty is that copositivity is hard to check. The article begins with a review of methods for checking copositivity. These methods include recursive algorithms and methods involving the eigenvalues of the matrix. Then the authors dig a little deeper and study the geometric properties of sets associated with copositive matrices. They point out interesting connections and generalizations. A particularly unexpected application of this theory is the solution of several NP-hard optimization problems by formulating them as linear programs over sets defined by copositivity. The second article, by C. M. Linton, is about a mathematical object called lattice sums. Lattice sums show up in the study of periodic structures such as crystals and photonic bandgap devices. The focus of the paper is the Green’s function that arises in the study of scalar, time-harmonic wave equations. Of interest is the strength of the field at a coordinate point caused an infinite array of point sources which are laid out in a periodic manner. When each source is phased according to a prescribed pattern, the resulting sum is a quasi-periodic Green’s function. Quasi-periodic Green’s functions are especially important for solving wave propa- gation problems in composites and for modeling optical gratings. Lattice sums are the basis for evaluating these Green’s functions efficiently and accurately. The author pro- vides a comprehensive review of various techniques of dealing with the infinite lattice sums, rewriting them in such a way that convergence rates of these sums are improved. Both papers in this section show that mathematics can be beautiful and useful. Furthermore, these articles illustrate two important phenomena. First, there are prob- lems arising in applications which cannot be solved without new mathematics. Second, the field of mathematics is enriched when it interacts with other disciplines. Fadil Santosa Section Editor [email protected] 591 Downloaded 11/20/14 to 99.229.175.189. 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.
Mathematical objects often arise in investigation of practical, applied problems.Both articles in the Survey and Review section in this issue are studies of such ob-jects. In the first article, J.-B. Hiriart-Urruty and A. Seeger provide an exposition oncopositive matrices which show up in control theory and in mathematical economics.These mathematical objects are very easy to describe. In their simplest form, they aresquare matrices A with real entries such that xT Ax ≥ 0 for all x in the nonnegativeorthant. If a matrix associated with a control problem is copositive, then one canmake some qualitative statements about the behavior of the system. The difficulty isthat copositivity is hard to check.
The article begins with a review of methods for checking copositivity. Thesemethods include recursive algorithms and methods involving the eigenvalues of thematrix. Then the authors dig a little deeper and study the geometric properties ofsets associated with copositive matrices. They point out interesting connections andgeneralizations. A particularly unexpected application of this theory is the solution ofseveral NP-hard optimization problems by formulating them as linear programs oversets defined by copositivity.
The second article, by C. M. Linton, is about a mathematical object called latticesums. Lattice sums show up in the study of periodic structures such as crystals andphotonic bandgap devices. The focus of the paper is the Green’s function that arisesin the study of scalar, time-harmonic wave equations. Of interest is the strength of thefield at a coordinate point caused an infinite array of point sources which are laid outin a periodic manner. When each source is phased according to a prescribed pattern,the resulting sum is a quasi-periodic Green’s function.
Quasi-periodic Green’s functions are especially important for solving wave propa-gation problems in composites and for modeling optical gratings. Lattice sums are thebasis for evaluating these Green’s functions efficiently and accurately. The author pro-vides a comprehensive review of various techniques of dealing with the infinite latticesums, rewriting them in such a way that convergence rates of these sums are improved.
Both papers in this section show that mathematics can be beautiful and useful.Furthermore, these articles illustrate two important phenomena. First, there are prob-lems arising in applications which cannot be solved without new mathematics. Second,the field of mathematics is enriched when it interacts with other disciplines.
Fadil SantosaSection Editor
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