Edward A. Bender

Bio: Edward A. Bender is an academic researcher from University of California, San Diego. The author has contributed to research in topics: Generating function & Asymptotic formula. The author has an hindex of 31, co-authored 121 publications receiving 7807 citations. Previous affiliations of Edward A. Bender include Harvard University & University of California, Los Angeles.

Concrete Mathematics: A Foundation for Computer Science.

Abstract: From the Publisher: This book introduces the mathematics that supports advanced computer programming and the analysis of algorithms. The primary aim of its well-known authors is to provide a solid and relevant base of mathematical skills - the skills needed to solve complex problems, to evaluate horrendous sums, and to discover subtle patterns in data. It is an indispensable text and reference not only for computer scientists - the authors themselves rely heavily on it! - but for serious users of mathematics in virtually every discipline. Concrete Mathematics is a blending of CONtinuous and disCRETE mathematics. \"More concretely,\" the authors explain, \"it is the controlled manipulation of mathematical formulas, using a collection of techniques for solving problems.\" The subject matter is primarily an expansion of the Mathematical Preliminaries section in Knuth's classic Art of Computer Programming, but the style of presentation is more leisurely, and individual topics are covered more deeply. Several new topics have been added, and the most significant ideas have been traced to their historical roots. The book includes more than 500 exercises, divided into six categories. Complete answers are provided for all exercises, except research problems, making the book particularly valuable for self-study. Major topics include: Sums Recurrences Integer functions Elementary number theory Binomial coefficients Generating functions Discrete probability Asymptotic methods This second edition includes important new material about mechanical summation. In response to the widespread use ofthe first edition as a reference book, the bibliography and index have also been expanded, and additional nontrivial improvements can be found on almost every page. Readers will appreciate the informal style of Concrete Mathematics. Particularly enjoyable are the marginal graffiti contributed by students who have taken courses based on this material. The authors want to convey not only the importance of the techniques presented, but some of the fun in learning and using them.

Perturbation Experiments in Community Ecology: Theory and Practice

Abstract: We analyze perturbation experiments performed on real and idealized ecological com- munities. A community may be considered as a black box in the sense that the individual species grow and interact in complicated ways that are difficult to discern. Yet, by observing the response (output) of the system to natural or human-induced disturbances (inputs), information can be gained regarding the character and strengths of species interactions. We define a perturbation as selective alteration of the density of one or more members of the community, and we distinguish two quite different kinds of perturbations. A PULSE perturbation is a relatively instantaneous alteration of species numbers, after which the system is studied as it "relaxes" back to its previous equilibrium state. A PRESS perturbation is a sustained alteration of species densities (often a complete elimination of particular species); it is maintained until the unperturbed species reach a new equilibrium. The measure of interest in PRESS perturbation is the net change in densities of the unperturbed species. There is a very important difference between these two approaches: PULSE experiments yield in- formation only on direct interactions (e.g., terms in the interaction matrix), while PRESS experiments yield information on direct interactions mixed together with the indirect effects mediated through other species in the community. We develop mathematical techniques that yield measures of ecolog- ical interaction between species from both types of experimental designs. Particular caution must be exercised in interpreting results from PRESS experiments, particularly when some species are lumped into functional categories and others are neglected altogether in the experimental design. We also suggest mathematical methods to deal with temporal and random variation during experiments. Fi- nally, we critically review techniques that rely on natural variation in numbers to estimate species interaction coefficients. The problems with such studies are formidable.

Asymptotic Methods in Enumeration

Abstract: This is an expository paper dealing with those tools in asymptotic analysis which are especially useful in obtaining asymptotic results in enumeration problems. Emphasis is on tools which are general, are easily applied, and give estimates of the form $a_n \sim f(n)$. Many examples are given to illustrate the usage of the various tools. It is assumed that a summation or a generating function for $a_n $ is explicitly or implicitly given.

The Structure and Function of Complex Networks

Abstract: Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.

Assortative mixing in networks.

Abstract: A network is said to show assortative mixing if the nodes in the network that have many connections tend to be connected to other nodes with many connections. Here we measure mixing patterns in a variety of networks and find that social networks are mostly assortatively mixed, but that technological and biological networks tend to be disassortative. We propose a model of an assortatively mixed network, which we study both analytically and numerically. Within this model we find that networks percolate more easily if they are assortative and that they are also more robust to vertex removal.

Network Medicine: A Network-Based Approach to Human Disease

Abstract: Given the functional interdependencies between the molecular components in a human cell, a disease is rarely a consequence of an abnormality in a single gene, but reflects the perturbations of the complex intracellular and intercellular network that links tissue and organ systems. The emerging tools of network medicine offer a platform to explore systematically not only the molecular complexity of a particular disease, leading to the identification of disease modules and pathways, but also the molecular relationships among apparently distinct (patho)phenotypes. Advances in this direction are essential for identifying new disease genes, for uncovering the biological significance of disease-associated mutations identified by genome-wide association studies and full-genome sequencing, and for identifying drug targets and biomarkers for complex diseases.

Analytic Combinatorics

Abstract: Analytic Combinatorics is a self-contained treatment of the mathematics underlying the analysis of discrete structures, which has emerged over the past several decades as an essential tool in the understanding of properties of computer programs and scientific models with applications in physics, biology and chemistry. Thorough treatment of a large number of classical applications is an essential aspect of the presentation. Written by the leaders in the field of analytic combinatorics, this text is certain to become the definitive reference on the topic. The text is complemented with exercises, examples, appendices and notes to aid understanding therefore, it can be used as the basis for an advanced undergraduate or a graduate course on the subject, or for self-study.