Showing papers in "Notices of the American Mathematical Society in 2009"

Communities in networks

Abstract: We survey some of the concepts, methods, and applications of community detection, which has become an increasingly important area of network science. To help ease newcomers into the field, we provide a guide to available methodology and open problems, and discuss why scientists from diverse backgrounds are interested in these problems. As a running theme, we emphasize the connections of community detection to problems in statistical physics and computational optimization.

A clicker approach to teaching calculus

Abstract: Clickers (electronic voting systems) are all the buzz in higher education these days as many universities and colleges invest significant sums of money to integrate these systems into their classrooms. But what are clickers? Are they just another high-tech gimmick, or can they really be used to improve learning? Can clickers be used efficiently in calculus classes? In an ideal world, students would take calculus for the sheer love of it. The reality is, however, that calculus is a service course, and most students take it to fulfill university requirements. Engaging students in calculus classes thus can be a challenge. Furthermore, in traditional lectures, students passively take notes, at times barely processing the material as they struggle to keep their attention focused for an hour. Students often fear that they are alone in not understanding the material and are at times afraid of asking questions in class. At the end of class, both students and instructor may leave without knowing whether the material has been understood. Clickers can be used to address these challenges. The term “clickers” refers to the student input devices (see Figure 1) of electronic voting systems. These systems allow

Mathematical Models in Science and Engineering

Abstract: M athematical modeling aims to describe the differentaspectsof the real world, their interaction, and their dynamics through mathematics. It constitutes the third pillar of science and engineering, achieving the fulfillment of the two more traditional disciplines, which are theoretical analysis and experimentation. Nowadays, mathematical modeling has a key role also in fields such as the environment and industry, while its potential contribution in many other areas is becoming more and more evident. One of the reasons for this growing success is definitely due to the impetuous progress of scientific computation; this discipline allows the translation of a mathematical model—which can be explicitly solved only occasionally—into algorithms that can be treated and solved by ever more powerful computers. See Figure 1 for a synthetic view of the whole process leading from a problem to its solution by scientific computation. Since 1960 numerical analysis—the discipline that allows mathematical equations (algebraic, functional, differential, and integrals) to be solved through algorithms—had a leading role in solving problems linked to mathematical modeling derived from engineering and applied sciences. Following this success, new disciplines started to use mathematical modeling, namely information and communication technology, bioengineering, financial engineering, and so on. As a matter of fact, mathematical models offer new possibilities to manage the increasing complexity of technology, which is at the basis of modern industrial

What is Stanley depth

Abstract: History and Background Richard P. Stanley is well known for his fundamental and important contributions to combinatorics and its relationship to algebra and geometry, in particular in the theory of simplicial complexes. Two kinds of simplicial complexes play central roles in combinatorics: partitionable complexes and Cohen-Macaulay complexes. Stanley posed a central conjecture relating these two notions: Are all Cohen-Macaulay simplicial complexes partitionable? In a 1982 Inventiones Mathematicae paper [4], Stanley defined what is now called the Stanley depth of a graded module over a graded commutative ring. Stanley depth is a geometric invariant of a module that, by a conjecture of Stanley, relates to an algebraic invariant of the module, called simply the depth. It is shown in [2] that this conjecture implies his conjecture about partitionable Cohen-Macaulay simplicial complexes. Our aim here is to introduce the notion of the Stanley depth. Let K be a field and S = K[x1, . . . , xn] the K-algebra of polynomials over K in n indeterminates x1, . . . , xn. We may write x = {x1, . . . , xn} and denote S by K[x] for convenience. A monomial in S is a product xa = x1 1 . . . xn n for a vector a = (a1, . . . , an) ∈ωn of nonnegative integers. The

Climate change and the mathematics of transport in sea ice

Abstract: A s the boundary layer between the ocean and atmosphere in the polar regions, sea ice is a critical component of the global climate system. As temperatures on Earth have warmed, the Arctic sea ice pack in particular has exhibited a dramatic decline in its summer extent. Indeed, the polar sea ice packs are harbingers of climate change. Predicting what may happen over the next ten, fifty, or one hundred years requires extensive modeling of critical sea ice processes and the role that sea ice plays in global climate. Currently, large-scale climate models in general do not realistically treat a number of sea ice processes that can significantly affect predictions. Moreover, monitoring the state of Earth’s sea ice packs, in particular their thickness distributions, provides key data on the impact of global warming. Mathematics is currently playing an important role in addressing these fundamental issues and will likely play an even greater role in the future. Here we give a brief bullet-by-bullet summary of the contents of this article.

On the concept of genus in topology and complex analysis

Abstract: T he English word “genus” hails from biology, where it is used to connote a grouping of organisms having common characteristics. In mathematics the word is also used to group objects with common characteristics. The concept of genus arises in various mathematical contexts, such as number theory, as well as in the areas we consider in this article, topology and complex analysis. Even within the latter two areas there are various notions of genus that historically originated with the genus of an oriented surface. We begin with these origins and afterward treat generalizations and modifications. We provide no detailed definitions and proofs; rather, our goal is to give the reader an intuitive feeling for the concept of genera.