Education

• Autores: Louis F. Rossi
• Localización: SIAM Review, ISSN 0036-1445, Vol. 55, Nº. 4, 2013, págs. 749-750
• Idioma: inglés
• Texto completo no disponible (Saber más ...)
• Resumen

  • The car I drove in college was assembled in the year before I was born, so keeping a few essential tools in the trunk and understanding the systems in an automobile was as important as knowing how to drive. Cars are more reliable now, but it's still very useful to understand what is happening under the hood. Indeed, in recent times, every generation has adapted to the miracles of a new age, and this is especially true in this time of reliable, easy-to-use, networked, digital devices. Instead of being covered by a hood, the latest technological magic is covered by glass and carbon. Curious instructors and students will enjoy the two contributions to this SIAM Review Education section, which delve into some of the mathematics underlying our digital age. The first article is an introduction to robust codes and coding theory in general, the linchpin of all digital communication and storage. The second article takes a deep dive into matrix exponentials to explore the properties of one aspect of fiber optic communication.

    In our first article, �An Introduction to Robust Codes over Finite Fields,� authors Shlomo Engelberg and Osnat Keren introduce readers to robust codes, an application of coding theory. Error correction is often what comes to mind when one thinks of applications of coding theory. However, in this paper, the authors look into a variation on this theme where a malicious third party seeks to distort a communication consisting of a stream of words between a sender and a receiver. The central issue is whether there is a coding scheme whereby the receiver can know when the third party is tampering with some of the words. To quantify the problem, a word is a string of bits, and in coding theory each of these words is considered a polynomial of the field $Z/2$. Thus, adding two words together amounts to performing an exclusive-or operation. It is important to note that not every string of bits is a valid word in the code, and if the receiver receives an invalid word, it will know that the third party has tampered with the message. The simplest type of attack is where an element is added to each word of the sent message. A robust code is one which is optimally resistant to these simple attacks. This article is accessible to students and instructors with little or no prior knowledge of coding theory and carefully takes the reader on a tour of linear and nonlinear codes, deriving important results on the suitability of these codes for resisting attacks. It would work well with an advanced undergraduate linear algebra or abstract algebra course, or for a graduate-level course.

    The second article, �An Application of Matrix Theory to the Evolution of Coupled Modes,� comes from the 27th Annual Mathematical Problems in Industry workshop. While many who live in the mid-Atlantic associate early summer with the northern migration of ruby-throated hummingbirds, the third week of June is also a migratory period for a cadre of loyal mathematicians who gather each year to spend a week together tackling leading-edge problems contributed by industry and government scientists. It is the oldest such �Oxford style� workshop in North America. In 2011, Alcatel-Lucent contributed a problem to model and understand coupled mode equations that describe amplifiers in fiber optic systems. The resulting linear system is a wonderful application of linear algebra and ordinary differential equations that would fit well in an advanced undergraduate or graduate course in linear algebra. The authors relate the properties of the physical system to the structure of the transfer matrix of the coupled mode equations. What is unusual about this contribution is that it is not a computational example of matrix algebra. Instead, the authors use concepts from linear algebra to explain the properties of the amplifier.

    Both articles in this Education section take us on a detailed and exciting tour of processes that are both miraculous and ubiquitous in our digital age. Recent advances make our devices so reliable that we do not need to think about how a code works or how data travels into our home along a glass filament. However, underneath the glass and carbon, mathematics is still the essential discipline, and maybe the day has come when a working knowledge of vector spaces is just as practical as knowing how to check the oil in your car