Journal•ISSN: 0002-9920
Notices of the American Mathematical Society
American Mathematical Society
About: Notices of the American Mathematical Society is an academic journal published by American Mathematical Society. The journal publishes majorly in the area(s): Computer science & Medicine. It has an ISSN identifier of 0002-9920. Over the lifetime, 1100 publications have been published receiving 14071 citations. The journal is also known as: Notices of the AMS & AMS Notices.
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TL;DR: A survey of the concepts, methods, and applications of community detection can be found in this article, where the authors provide a guide to available methodology and open problems, and discuss why scientists from diverse backgrounds are interested in these problems.
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.
807 citations
TL;DR: In this paper, Popescu et al. discuss necessary and sufficient conditions for circulant matrices to be non-singular, and various distinct representations they have, goals that allow us to lay out the rich mathematical structure that surrounds them.
Abstract: Some mathematical topics, circulant matrices, in particular, are pure gems that cry out to be admired and studied with different techniques or perspectives in mind. Our work on this subject was originally motivated by the apparent need of one of the authors (IK) to derive a specific result, in the spirit of Proposition 24, to be applied in his investigation of theta constant identities [9]. Although progress on that front eliminated the need for such a theorem, the search for it continued and was stimulated by enlightening conversations with Yum-Tong Siu during a visit to Vietnam. Upon IK’s return to the US, a visit by Paul Fuhrmann brought to his attention a vast literature on the subject, including the monograph [4]. Conversations in the Stony Brook Mathematics’ common room attracted the attention of the other author, and that of Sorin Popescu and Daryl Geller∗ to the subject, and made it apparent that circulant matrices are worth studying in their own right, in part because of the rich literature on the subject connecting it to diverse parts of mathematics. These productive interchanges between the participants resulted in [5], the basis for this article. After that version of the paper lay dormant for a number of years, the authors’ interest was rekindled by the casual discovery by SRS that these matrices are connected with algebraic geometry over the mythical field of one element. Circulant matrices are prevalent in many parts of mathematics (see, for example, [8]). We point the reader to the elegant treatment given in [4, §5.2], and to the monograph [1] devoted to the subject. These matrices appear naturally in areas of mathematics where the roots of unity play a role, and some of the reasons for this to be so will unfurl in our presentation. However ubiquitous they are, many facts about these matrices can be proven using only basic linear algebra. This makes the area quite accessible to undergraduates looking for research problems, or mathematics teachers searching for topics of unique interest to present to their students. We concentrate on the discussion of necessary and sufficient conditions for circulant matrices to be non-singular, and on various distinct representations they have, goals that allow us to lay out the rich mathematical structure that surrounds them. Our treatement though is by no means exhaustive. We expand on their connection to the algebraic geometry over a field with one element, to normal curves, and to Toeplitz’s operators. The latter material illustrates the strong presence these matrices have in various parts of modern and classical mathematics. Additional connections to other mathematics may be found in [11]. The paper is organized as follows. In §2 we introduce the basic definitions, and present two models of the space of circulant matrices, including that as a
792 citations
Journal Article•
TL;DR: A simplified version of RSA encryption is described and a malicious attacker wishing to eavesdrop or tamper with the communication between Alice and Bob is used, to illustrate the dangers of improper use of RSA.
Abstract: Introduction The RSA cryptosystem, invented by Ron Rivest, Adi Shamir, and Len Adleman [18], was first publicized in the August 1977 issue of Scientific American. The cryptosystem is most commonly used for providing privacy and ensuring authenticity of digital data. These days RSA is deployed in many commercial systems. It is used by Web servers and browsers to secure Web traffic, it is used to ensure privacy and authenticity of e-mail, it is used to secure remote login sessions, and it is at the heart of electronic credit card payment systems. In short, RSA is frequently used in applications where security of digital data is a concern. Since its initial publication, the RSA system has been analyzed for vulnerability by many researchers. Although twenty years of research have led to a number of fascinating attacks, none of them is devastating. They mostly illustrate the dangers of improper use of RSA. Indeed, securely implementing RSA is a nontrivial task. Our goal is to survey some of these attacks and describe the underlying mathematical tools they use. Throughout the survey we follow standard naming conventions and use “Alice” and “Bob” to denote two generic parties wishing to communicate with each other. We use “Marvin” to denote a malicious attacker wishing to eavesdrop or tamper with the communication between Alice and Bob. We begin by describing a simplified version of RSA encryption. Let N = pq be the product of two large primes of the same size (n/2 bits each). A typical size for N is n = 1024 bits, i.e., 309 decimal digits. Each of the factors is 512 bits. Let e, d be two integers satisfying ed = 1 mod φ(N) where φ(N) = (p − 1)(q − 1) is the order of the multiplicative group ZN. We call N the RSA modulus, e the encryption exponent, and d the decryption exponent. The pair 〈N, e〉 is the public key. As its name suggests, it is public and is used to encrypt messages. The pair 〈N,d〉 is called the secret key or private key and is known only to the recipient of encrypted messages. The secret key enables decryption of ciphertexts. A message is an integer M ∈ ZN. To encrypt M, one computes C =Me mod N . To decrypt the ciphertext, the legitimate receiver computes Cd mod N. Indeed, Cd =Med =M mod N,
620 citations
TL;DR: The On-Line Encyclopedia of Integer Sequences (OEIS) as mentioned in this paper is a database of some 130000 number sequences, which is freely available on the Web (http://wwwresearchattcom/~njas/sequences/) and is widely used.
Abstract: The On-Line Encyclopedia of Integer Sequences (or OEIS) is a database of some 130000 number sequences It is freely available on the Web (http://wwwresearchattcom/~njas/sequences/) and is widely used
There are several ways in which it benefits research: 1 It serves as a dictionary, to tell the user what is known about a particular sequence There are hundreds of papers which thank the OEIS for assistance in this way
1 The associated Sequence Fans mailing list is a worldwide network which has evolved into a powerful machine for tackling new problems
1 As a direct source of new theorems, when a sequence arises in two different contexts
1 As a source of new research, when one sees a sequence in the OEIS that cries out to be analyzed
The 40-year history of the OEIS recapitulates the story of modern computing, from punched cards to the internet
The talk will be illustrated with numerous examples, emphasizing new sequences that have arrived in the past few months Many open problems will be mentioned
Because of the profusion of books and journals, volunteers play an important role in maintaining the database If you come across an interesting number sequence in a book, journal or web site, please send it and the reference to the OEIS (You do not need to be the author of the sequence to do this) There is a web site for sending in "Comments" or "New sequences"
Several new features have been added to the OEIS in the past year Thanks to the work of Russ Cox, searches are now performed at high speed, and thanks to the work of Debby Swayne, there is a button which displays plots of each sequence Finally, a "listen" button enables one to hear the sequence played on a musical instrument (try Recamaan's sequence A005132!)
480 citations
Journal Article•
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TL;DR: From Manifolds to Fractals Analysis on manifolds has been one of the central areas of mathematical research in the twentieth century as discussed by the authors, and it has attracted mathematicians with diverse expertise and points of view, including topology, differential equations, differential geometry, functional and harmonic analysis and probability theory.
Abstract: From Manifolds to Fractals Analysis on manifolds has been one of the central areas of mathematical research in the twentieth century. Rooted in the foundational work of the nineteenth century, with its rigorous theory of multidimensional calculus and the visionary ideas of Riemann, it has flowered into a richly layered mathematical tapestry. It has attracted mathematicians with diverse expertise and points of view, including topology, differential equations, differential geometry, functional and harmonic analysis, and probability theory. This heady mix of ideas has produced a vast body of work and a seemingly endless supply of challenging problems that should keep mathematicians busy well into the next century. At the same time it has become apparent that many phenomena in the real world are best modeled by geometric structures that are much more irregular. The theory of fractals, as B. Mandelbrot [Ma] has so forcefully argued, seeks to provide the mathematical framework for such development. A theory of analysis on fractals is now emerging and is perhaps poised for the kind of explosive and multilayered expansion that has characterized analysis on manifolds. This article will explain some of what has been accomplished and where it might lead. The central character in the theory of analysis on manifolds is the Laplacian. Thus the starting point for analysis on fractals will be the construction of an analogous operator on a class of fractals. This will not be a genuine differential operator, of course, but it will have quite a few of the features we have come to expect from anything labeled “Laplacian”. It will be a local operator, and in fact ∆f (x) will be a limit in a suitable renormalized sense of the difference between an average value of f in a neighborhood of x and f (x). We will be imitating the weak formulation of the Laplacian, so that ∆u = f will be interpreted to mean
419 citations