Fix any $\lambda\in\mathbb{C}$. We say that a set $S\subseteq\mathbb{C}$ is $\lambda$-$convex$ if, whenever $a$ and $b$ are in $S$, the point $(1-\lambda)a+\lambda b$ is also in $S$. If $S$ is also (topologically) closed, then we say that $S$ is $\lambda$-$clonvex$. We investigate the properties of $\lambda$-convex and $\lambda$-clonvex sets and prove a number of facts about them. Letting $R_\lambda\subseteq\mathbb{C}$ be the least $\lambda$-clonvex superset of $\{0,1\}$, we show that if $R_\lambda$ is convex in the usual sense, then $R_\lambda$ must be either $[0,1]$ or $\mathbb{R}$ or $\mathbb{C}$, depending on $\lambda$. We investigate which $\lambda$ make $R_\lambda$ convex, derive a number of conditions equivalent to $R_\lambda$ being convex, and give several conditions sufficient for $R_\lambda$ to be convex or not convex; in particular, we show that $R_\lambda$ is either convex or uniformly discrete. Letting $\mathcal{C} := \{\lambda\in\mathbb{C}\mid \mbox{$R_\lambda$ is convex}\}$, we show that $\mathbb{C}\setminus\mathcal{C}$ is closed, discrete and contains only algebraic integers. We also give a sufficient condition on $\lambda$ for $R_\lambda$ and some other related $\lambda$-convex sets to be discrete by introducing the notion of a strong PV number. These conditions give rise to a number of periodic and aperiodic Meyer sets (the latter sometimes known as quasicrystals). The paper is in four parts. Part I describes basic properties of $\lambda$-convex and $\lambda$-clonvex sets, including convexity versus uniform discreteness. Part II explores the connections between $\lambda$-convex sets and quasicrystals and displays a number of such sets, including several with dihedral symmetry. Part III generalizes a result from Part I about the $\lambda$-convex closure of a path, and Part IV contains our conclusions and open problems.

You have 5 muffins and 3 (as usual) hungry students. They ask that you divide the muffins equitably between them. Do you cut every muffin into 3 parts, and give 5 separate pieces of size 1/3 to each student? Well no, because that's too easy, and more importantly no student wants any piece that's less than a third of a muffin. Can this be done? Indeed it can! It turns out you can guarantee that everyone gets a piece of size at least 5/12. Can everyone be guaranteed a bigger piece than that? Alas, the answer is no. That's the bad news. The good news is that one could write a book about it, and these guys wrote a great one.

This book is a definitive account of a field of research that has blossomed remarkably over the past two decades. Many of its chief developments are presented here, indeed in many cases having been obtained by its authors.

This book is a definitive account of a field of research that has blossomed remarkably over the past two decades. Many of its chief developments are presented here, indeed in many cases having been obtained by its authors.

"It may only be a slight exaggeration to claim that in the 1930s we started to understand what is and is not effectively computable and that in the 1970s we started to understand what is and is not practically or feasibly computable. There is no doubt that the results about what can and cannot be effectively computed or formalized in mathematics have had a profound influence on mathematics, and, even more broadly, they have influenced our view of our scientific methods. We believe that the results about what can and cannot be practically computed will also have a major influence on computer science, mathematics, and even though more slowly, will affect other research areas and influence how we think about scientific theories." - Juris Hartmanis (from the Introduction)

The future prospects for anyone falling into a black hole are bleak. For one thing, there is no chance (according to our present state of knowledge) of ever getting out again. Worse, one is facing certain destruction when one meets the "singularity" (or its inconceivably dense physical manifestation, whatever that may be) inside. However, there is an "event horizon," the point of no return, separating the overly curious infalling astronaut from the doom he or she faces at the singularity. Suppose Alice the Astronaut wants to see what's behind the horizon (never mind the consequences). How much time would Alice have to look around and see what's happening, before reaching the end of her worldline? Conventional wisdom, until relatively recently, was that she would have some amount of time, perhaps hours. Passing the event horizon of a supermassive black hole would not seem like any kind of a milestone to the infalling individual; it is only an outside observer who would notice something out of the ordinary.

Mathematics and computation are inextricably entangled3. We couldn't do one without the other. The need to calculate can be traced to early human history, and mathematics developed in large part to enable computation. And computation is necessary to propel mathematics. One often loses sight of the fact that the great mathematicians of the past were also prodigious computers: For example, Gauss, Kummer and the other great pioneers of number theory did vast amounts of computation to arrive at or reinforce many of their insights.

Computational geometry and topology are huge branches of mathematics. Focussing on concepts that lead to computation is one strategy to provide a concrete conceptual basis for ideas that hold in a more general context. Indeed, this short book gives an introduction to a surprisingly broad range of ideas that can serve as a good introduction to geometry and topology (even broadly conceived) for undergraduates.

In 1930, the mathematician Esther Klein observed that any five points in the plane in general position (i.e., no three points forming a line) contain four points forming a convex quadrilateral. This innocentsounding discovery led to major lines of research in discrete geometry. Klein's friends Paul Erdős and George Szekeres generalized this theorem, and also conjectured that 2k-2 + 1 points (again in general position) would be enough to force a convex k-gon to exist. The resolution of this conjecture became known as the "happy ending problem," because Klein and Szekeres ended up getting married. The unhappy side is that it has, to date, not been completely solved, although a recent breakthrough of Suk made significant progress. This both mathematically and personally charming little story is a great beginning for this elegant book about discrete geometry. It typifies the type of problems that are studied throughout, and also captures the spirit of curiosity that drives such studies. The book covers many problems that lie at the intersection of three fields: discrete geometry, algorithms and computational complexity.

The three books reviewed in this column are about central ideas in algorithms, complexity, and geometry. The third one brings together topics from the first two by applying techniques of both property testing (the subject of the first book) and parameterized complexity (including its more focused incarnation studied in the second book, kernelization) to geometric problems.

There are probably at least as many different approaches to number theory as there are books written about it. Some broad distinctions include those taking an historical versus (say) a purely modern approach, with many gradations in between, or those that are algebraically oriented (e.g., with an emphasis on reciprocity laws, or questions that relate to algebraic geometry), or still others that are more analytic. The book under review is definitely in the latter category. The "message" of the book is in the title, as primes and their density are the principle concern. In accordance with that theme, a highlight of the book is a complete proof of the prime number theorem. However, the theme and its variations are taken as springboards to other important fields, including aspects of algebraic number theory, as well as applications, such as primality testing and cryptography.

Some (admittedly controversial) models of physical reality propose that it is, at the fundamental level, based on a certain type of graph, called a spin-network (and this among other models of physical reality based on graphs). Recently, astronomers have verified that the otherwise empty reaches of intergalactic space are laced with wispy tendrils of hot gas that connect galaxies. On a more quotidian level, the machine on which I'm typing this document is constructed out of components that are essentially graphs, those components being connected by electrical edges forming an even larger graph, and this one device in turn being connected via various links to millions of other nodes in a huge global graph called the internet. The notion of a graph encompasses all levels of reality, from the subatomic through everyday life and reaching out to the cosmological scale.

The equation y2 = x3 + ax2 + bx + c might seem a little innocuous at first. However, studying the sets of rational points (x; y) obeying this equation has proven to be one of the most far-reaching and fruitful areas of mathematics. For example, it led, aided and abetted by much of the most powerful mathematics of the past century, to Wiles' proof of Fermat's Last Theorem. And furthermore, these so-called "elliptic curves" (the terminology having little to do with ellipses) are actually useful. You can factor numbers with them! And send secret messages!

review-article Share on Joint Review of Quadratic Residues and Non-Residues by Steve Wright and The Quadratic Reciprocity Law by Oswald Baumgart Edited and translated by Franz Lemmermeyer Reviewer: Frederic Green Department of Mathematics and Computer Science Clark University,Worcester, MA Department of Mathematics and Computer Science Clark University,Worcester, MAView Profile Authors Info & Claims ACM SIGACT NewsVolume 49Issue 1March 2018 pp 20–28https://doi.org/10.1145/3197406.3197411Published:14 March 2018Publication History 0citation30DownloadsMetricsTotal Citations0Total Downloads30Last 12 Months1Last 6 weeks1 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

4. Polyhedral and Algebraic Methods in Computational Geometry, by Michael Joswig and Thorsten Theobald. Reviewed by Brittany Terese Fasy and David L. Millman. This book treats a wide array of topics, beginning from the basics (e.g., convex hulls) and proceeding through the fundamental tools of computational algebraic geometry (e.g., Gröbner bases), and containing a section on applications as well.

For a number of years we have been investigating the geometric and algebraic properties of a family of discrete sets of points in Euclidean space generated by a simple binary operation: pairwise affine combination by a xed parameter, which we call xed-parameter extrapolation. By varying the parameter and the set of initial points, a large variety of point sets emerge. To our surprise, many of these sets display aperiodic order and share properties of so-called \quasicrystals" or \quasilattices." Such sets display some ordered crystal-like properties (e.g., generation by a regular set of local rules, such as a nite set of tiles, and possessing a kind of repetitivity), but are \aperiodic" in the sense that they have no translational symmetry. The most famous of such systems are Penrose's aperiodic tilings of the plane [16]. Mathematically, a widely accepted way of capturing the idea of aperiodic order is via the notion of Meyer sets, which we de ne later. Our goal is to classify the sets generated by xed-parameter extrapolation in terms of a number of these properties, including but not limited to aperiodicity, uniform discreteness, and relative density. We also seek to determine exactly which parameter values lead to which types of sets.

In this column we review these three books: 1. Compact Data Structures - A Practical Approach, by Gonzalo Navarro. An unusual approach focussing on data structures that use as little space as is theoretically possible, while maintaining practicality. Review by Laszlo Kozma. 2. Power Up: Unlocking the Hidden Mathematics in Video Games, by Matthew Lane. A revealing look at the mathematics, physics, and computer science that underlies (good) video games, with special attention to their potential valuable role in education. Review by S. V. Nagaraj. 3. Probability and Computing: Randomization and Probabilistic Techniques in Algorithms and Data Analysis (Second Edition), by Michael Mitzenmacher and Eli Upfal. The second edition2 of a distinguished text on a rich area that is central to computer and data science. Review by Aravind Srinivasan. Many thanks to these and all reviewers over the years who have graciously volunteered this valuable service to the community! Interested in reviewing? I hope so. Remember the perks: Learning (more?) about a field that interests you, a review on these pages, and a free book! The books listed on the next two pages are available, but the list is not exhaustive: I'm open to suggestions.

review-article Share on Review of Art in the Life of Mathematicians by Anna Kepes Szemerédi Reviewer: Frederic Green Clark University,Worcester, MA Clark University,Worcester, MAView Profile Authors Info & Claims ACM SIGACT NewsVolume 48Issue 1March 2017 pp 10–17https://doi.org/10.1145/3061640.3061643Published:10 March 2017Publication History 0citation54DownloadsMetricsTotal Citations0Total Downloads54Last 12 Months1Last 6 weeks1 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

It is a well-known phenomenon that if you take a random walk along a line, it’s more likely than not that you’ll end up back where you started. The probability distribution over position after a sufficient amount of time follows a Gaussian centered on the starting point, and with high probability you won’t get much more than √ t steps (the standard deviation) from that point in time t. At any rate, that is what holds for random walks following the laws of classical probability theory. Quantum mechanics, of course, is weird. If you take a quantum walk, it’s more likely than not that you won’t end up where you started! And the standard deviation is now t, i.e., linear. The walker is spread more thinly and uniformly in the quantum case. Now where have we seen this kind of “quadratic improvement” in going from classical to quantum before? In Grover’s algorithm, of course, in which we find that a quantum search of an unsorted database of N elements can be done in √ N time, as opposed to the best classical performance of N . This is no coincidence: Grover’s algorithm can be interpreted as a quantum walk on a complete graph. Roughly speaking, a quantum walker explores a search space much more widely, and in less time, than a classical walker. Note that in the phrase “quantum walk,” under current proper usage the word “quantum” really replaces the word “random” that is used in the classical case. One may be tempted to say “quantum random walk,” but the unitary evolution that describes the process is perfectly deterministic. The randomness only comes in after one makes a measurement, so it is prudent not to fall into the trap of thinking that each step of a quantum walk is in any sense “random.” It is “quantum.” These are the basic ideas introduced and extensively elaborated in Renato Portugal’s detailed, focused and clearly-written monograph “Quantum Walks and Search Algorithms” (“QWSA”). One valuable aspect of QWSA is that the bulk of it studies an area of quantum computing (“QC”) that is rarely covered in such depth anywhere else, and until now has been largely available only in the research literature. While Grover’s algorithm (a key concept in the book) dates to 1996, most of the theory of quantum walks was developed since around 2001, so “classic” QC texts such as Nielsen & Chuang ([NC00]) or Mermin ([M07], reviewed in SIGACT News 41 (3), 2010) are either too early or too elementary to cover them at all, and the more recent texts that do get into quantum walks do so in less detail (Moore & Mertens [MM11], reviewed in SIGACT News 47 (1), 2016; Lipton & Regan [LR14], reviewed in SIGACT News 47 (3), 2016). After the introduction to QC in Chapter 2, the book gets more serious in Chapter 3, where it first reviews classical random walks. It then proceeds to discrete-time quantum walks, which provides an elementary context for introducing the primary ingredients in a quantum walk, i.e., the coin operator (which “randomly,” or rather “quantumly,” determines the direction of movement) and the shift operator (which implements the actual step). Thus the whole theory of quantum walks revolves around the unitary operator U = S(C ⊗ I),

review-article Share on Review of Set Theory: A First Course by Daniel W. Cunningham Reviewer: Frederic Green Clark University, Worcester, MA Clark University, Worcester, MAView Profile Authors Info & Claims ACM SIGACT NewsVolume 48Issue 3September 2017 pp 7–9https://doi.org/10.1145/3138860.3138863Published:07 September 2017Publication History 0citation127DownloadsMetricsTotal Citations0Total Downloads127Last 12 Months14Last 6 weeks1 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access