Densities of Hyperbolic Cusp Invariants

Colin Adams, Rose Kaplan-Kelly*, Michael Moore*, Brandon Shapiro*, Shruthi Sridhar*, Joshua Wakefield*

Proceedings of American Mathematical Society, Vol. 146, No. 9, 4073-4089, September 2018

This is an investigation into two new invariants associated to hyperbolic 3-manifolds.

Presidential Address

Colin C. Adams

Mathematical Intelligencer, Volume 39, Issue 2, 3-5, 2017

What if Donald Trump was the President of the American Mathematical Society?

Referee’s Report

Colin C. Adams

Mathematical Intelligencer, Volume 39, Issue 3, 24-26, 2017

The worst case scenario of a referee’s report.

The Seven Labors of Hercules

Colin C. Adams

Mathematical Intelligencer, Volume 39, Issue 4, 27-29, 2017

A mathematical take on the labors of Hercules.

Turning Knots into Flowers and Related Undergraduate Research

Colin C. Adams

American Mathematical Monthly, Vol. 124, No. 9, 791-806, 2017

An introduction to multi-crossing number and petal number, highlighting student research.

To the Fields Medal Committee

Colin C. Adams

Mathematical Intelligencer, Volume 40, Issue 1, 24-25 (Also appeared on Scientific American on-line), 2018

A self-nomination for the Fields medal.

What is the Greatest Number of all Time?

C. Adams and T. Garrity

Mother Jones on-line https://www.motherjones.com/media/2018/03/pi-day-is-a-great-day-for-a-math-argument/, March 2018

A debate on which is the greatest number of all time.

What is…a Hyperbolic 3-Manifold?

Colin C. Adams

AMS Notices, Vol. 65, No. 5, 544-546, May 2018

A brief introduction to hyperbolic 3-manifold theory

On Writing Math Fiction

Colin C. Adams

MAA Focus, Vol. 38, No. 3, 17-18, June/July 2018

A brief introduction to mathematical fiction and the writing thereof.

Zika Virus Dynamics: When Does Sexual Transmission Matter?

C Blackwood, O. Maxian, A. Neufeld ’18, EJ Talis and LM Childs

Epidemics, 21 48-55, 2017

The Zika virus (ZIKV) has captured worldwide attention with the ongoing epidemic in South America and its link to severe birth defects, most notably microcephaly. ZIKV is spread to humans through a combination of vector and sexual transmission, but the relative contribution of these transmission routes to the overall epidemic remains largely unknown. Furthermore, a disparity in the reported number of infections between males and females has been observed. We develop a mathematical model that describes the transmission dynamics of ZIKV to determine the processes driving the observed epidemic patterns. Our model reveals a 4.8% contribution of sexual transmission to the basic reproductive number, R0. This contribution is too minor to independently sustain an outbreak but suggests that vector transmission is the main driver of the ongoing epidemic. We also find a minor, yet statistically significant, difference in the mean number of cases in males and females, both at the peak of the epidemic and at equilibrium. While this suggests an intrinsic disparity between males and females, the differences do not account for the vastly greater number of reported cases for females, indicative of a large reporting bias. In addition, we identify conditions under which sexual transmission may play a key role in sparking an epidemic, including temperate areas where ZIKV mosquito vectors are less prevalent.

Unraveling Within-Host Signatures of Dengue Infection at the Population Level

R. Nikin-Beers, JC Blackwood, LM Childs, and SM Ciupe

Journal of Theoretical Biology, 446, 79-86, 2018

Dengue virus causes worldwide concern with nearly 100 million infected cases reported annually. The within-host dynamics differ between primary and secondary infections, where secondary infections with a different virus serotype typically last longer, produce higher viral loads, and induce more severe disease. We build upon the variable within-host virus dynamics during infections resulting in mild dengue fever and severe dengue hemorrhagic fever. We couple these within-host virus dynamics to a population-level model through a system of partial differential equations creating an immuno-epidemiological model. The resulting multiscale model examines the dynamics of between-host infections in the presence of two circulating virus strains that involves feedback from the within-host and between-hosts interactions, encompassing multiple scales. We analytically determine the relationship between the model parameters and the characteristics of the model’s solutions, and find an analytical threshold under which infections persist in the population. Furthermore, we develop and implement a full numerical scheme for our immuno-epidemiological model, allowing the simulation of population dynamics under variable parameter conditions.

A Cascade of Destabilizations: Combining Wolbachia and Allee Effects to Eradicate Insect Pests

JC Blackwood, R. Vargas, Jr. ’16. and X Fauvergue

Journal of Animal Ecology, 87, 59-72, 2018

The management of insect pests has long been dominated by the use of chemical insecticides, with the aim of instantaneously killing enough individuals to limit their damage. To minimize unwanted consequences, environmentally-friendly approaches have been proposed that utilize biological control and take advantage of intrinsic demographic processes to reduce pest populations. We address the feasibility of a novel pest management strategy based on the release of insects infected with Wolbachia, which causes cytoplasmic incompatibilities in its host population, into a population with a pre-existing Allee effect. We hypothesize that the transient decline in population size caused by a successful invasion of Wolbachia can bring the population below its Allee threshold and, consequently, trigger extinction. We develop a stochastic population model that accounts for Wolbachia-induced cytoplasmic incompatibilities in addition to an Allee effect arising from mating failures at low population densities. Using our model, we identify conditions under which cytoplasmic incompatibilities and Allee effects successfully interact to drive insect pest populations toward extinction. Based on our results, we delineate control strategies based on introductions of Wolbachia-infected insects. We extend this analysis to evaluate control strategies that implement successive introductions of two incompatible Wolbachia strains. Additionally, we consider methods that combine Wolbachia invasion with mating disruption tactics to enhance the pre-existing Allee effect. We demonstrate that Wolbachia-induced cytoplasmic incompatibility and the Allee effect act independently from one another: the Allee effect does not modify the Wolbachia-invasion threshold, and cytoplasmic incompatibilities only have a marginal effect on the Allee threshold. However, the interaction of these two processes can drive even large populations to extinction. The success of this method can be amplified by the introduction of multiple Wolbachia cytotypes as well as the addition of mating disruption. Our study extends the existing literature by proposing the use of Wolbachia introductions to capitalize on pre-existing Allee effects and consequently eradicate insect pests. More generally, it highlights the importance of transient dynamics, and the relevance of manipulating a cascade of destabilizations for pest management.

Modeling Alternative Stable States in Caribbean Coral Reefs

JC Blackwood, C. Okasaki, A. Archer, EW Matt ’18, E. Sherman, K. Montovan, R. Vargas, Jr. ’16

Natural Resource Modeling, 31, e12157, 2018

The resilience of Caribbean coral reefs, which are an important source of biodiversity and provide essential ecosystem services, is constantly challenged by many reef stressors including ocean acidification, hurricane damage, and overharvesting of herbivorous reef fish. The presence of two alternative stable states—a desirable state with high levels of coral cover and its coral‐depleted counterpart—has been widely documented in the literature. Increasing coral resilience to prevent phase shifts to the undesirable state is a critical research priority, and mathematical models can serve as an important tool to not only better understand the underlying dynamics of observed coral communities, but also to evaluate the potential impacts of stressors and the outcome of management strategies designed to promote coral persistence. Here, we review the existing literature of mathematical models designed to understand the processes that generate alternative stable states. We focus on models that are comprised of ordinary differential equations and, at their core, capture algal–coral dynamics.

Intro Stats, 5th Edition

Richard De Veaux with Paul Velleman and David Bock

Pearson, 2017

Stats: Data and Modeling the World, 5th Edition

Richard De Veaux with David Bock, Floyd Bullard and Paul Velleman

Pearson, 2017

The Distribution of the Maximum of Character Sums

Leo Goldmakher with J. Bober, A. Granville, D. Koukoulopoulos

J. Eur. Math. Soc. (JEMS) 20, no 7, 1759-1818, 2018

Refinements of Lagrange’s Four-Square Theorem

Leo Goldmakher with P. Pollack

Amer. Math. Monthly, 125, no. 3, 258-262, 2018

The q-Analog of Kostant’s Partition Function and the Highest Root of the Simple Lie Algebras

Pamela E. Harris, Erik Insko, and Mohamed Omar

Australasian Journal of Combinatorics, Volume 71, no. 1, 68-91, 2018

For a given weight of a complex simple Lie algebra the q-analog of Kostant’s partition function is a polynomial valued function in the variable q where the coefficient of qk is the number of ways the weight can be written as a nonnegative integral sum of exactly k positive roots. In this paper we determine generating functions for the q-analog of Kostant’s partition function when the weight in question is the highest root of the classical Lie algebras of types B, C, and D, and the exceptional Lie algebras of type G2, F4, E6, E7, and E8.

A Minimal-Preserving Crystal on Ordered Multiset Partitions

Georgia Benkart, Laura Colmenarejo, Pamela E. Harris, Rosa Orellana, Greta Panova, Anne Schilling, and Martha Yip

Advances in Applied Mathematics, 95, 96-115, 2018

We provide a crystal structure on the set of ordered multiset partitions, which recently arose in the pursuit of the Delta Conjecture. This conjecture was stated by Haglund, Remmel and Wilson as a generalization of the Shuffle Conjecture. Various statistics on ordered multiset partitions arise in the combinatorial analysis of the Delta Conjecture, one of them being the minimaj statistic, which is a variant of the major index statistic on words. Our crystal has the property that the minimaj statistic is constant on connected components of the crystal. In particular, this yields another proof of the Schur positivity of the graded Frobenius series of the generalization Rn,k due to Haglund, Rhoades and Shimozono of the coinvariant algebra Rn. The crystal structure also enables us to demonstrate the equidistributivity of the minimaj statistic with the major index statistic on ordered multiset partitions.

Computing Weight q-Multiplicities for the Representations of the Simple Lie Algebras

Pamela E. Harris, Erik Insko, and Anthony Simpson ‘19

Applicable Algebra in Engineering, Communication and Computing, 2017

The multiplicity of a weight μ in an irreducible representation of a simple Lie algebra g with highest weight λ can be computed via the use of Kostant’s weight multiplicity formula. This formula is an alternating sum over the Weyl group and involves the computation of a partition function. In this paper we consider a q- analog of Kostant’s weight multiplicity and present a SageMath program to compute q-multiplicities for the simple Lie algebras.

Computing Weight Multiplicities

Pamela E. Harris

A Primer for Undergraduate Research, Wootton A., Peterson V., Lee C. (editors)

Foundations for Undergraduate Research in Mathematics, Birkhauser, Cham, 193-222, 2017

Central to the study of the representation theory of Lie algebras is the computation of weight multiplicities, which are the dimensions of vector subspaces called weight spaces. The multiplicity of a weight can be computed using a well- known formula of Kostant that consists of an alternating sum over a finite group and involves a partition function. In this paper, we specialize to the Lie algebra slr+1(C) and focus on questions regarding the number of terms contributing non- trivially to Kostant’s weight multiplicity formula. Through this study, we show that these contributing sets, called Weyl alternation sets, show interesting combinatorial and geometric properties. We dedicate a section to detailed examples that illustrate accessible techniques students may use to begin investigating the open problems we present in this area.

How Gaussian Mixture Models Might Miss Detecting Factors That Impact Growth Patterns

Heggeseth, BC and Jewell NP

Annals of Applied Statistics, 12 (1) 222-245, 2018

Review of Handbook of Cluster Analysis by Christian Hennig, Marina Meila, Fionn Murtagh, and Roberto Rocci, editors
Heggeseth, BC and Jewell NP

Journal of American Statistical Association, 112 (518) 878-882, 2017

Let’s Join the conversation. Blog post about best practices for supporting inclusion and diversity in the field for the ASA Statistical Education Section

Heggeseth, BC and Jewell NP

Published at https://statisticseducatio.wixsite.com/mysite/single-post/2018/03/11/Lets-Join-the-Conversation.

Ovarian Cancer Survivors’ Acceptance of Treatment Side Effects Evolves as Goals of Care Change Over the Cancer Continuum

Fry, Ellis, Koontz, Shyne, Klingenberg, Fields, Chern and Blank

Gynecologic Oncology, Vol. 146 (2) 386–391, 2017

Women with ovarian cancer can have long overall survival and goals of treatment change over time from cure to remission to stable disease. We sought to determine whether survivors’ acceptance of treatment side effects also changes over the disease continuum.

Decreasing Recurrent Bowel Obstructions, Improving Quality of Life With Physiotherapy: Controlled Study

Rice, Patterson, Reed, Wurn, Robles, Klingenberg, Weinstock, Pratt, King and Wurn

World Journal of Gastroenterology, 24(19): 2108-2119, 2018

This paper compares over 30 quality of life measurements and rates of recurrent small bowel obstructions for patients treated with a novel manual physiotherapy versus no treatment.

Identification of Discrete, Intermingled Hypocretin Neuronal Populations

Iyer, Essner, Klingenberg and Carter

Journal of Comparative Neurology, doi: 10.1002/cne.24490, 2018

Neurons in the lateral hypothalamic area that express hypocretin (Hcrt) neuropeptides help regulate many behaviors including wakefulness and reward seeking. In the paper we test the hypotheses that Hcrt neurons can be classified based on their downstream projections and medial/lateral location within the hypothalamus.

The M & M Game: From Morsels to Modern Mathematics

Steven Miller (with Ivan Badinski, Christopher Huffaker, Nathan McCue ‘15, Cameron N. Miller, Kayla S. Miller and Michael Stone)

Mathematics Magazine, 90, no. 3, 197-207, 2017

To an adult, it’s obvious that the day of someone’s death is not precisely determined by the day of birth, but it’s a very different story for a child. When the third named author was four years old he asked his father, the fifth named author: If two people are born on the same day, do they die on the same day? While this could easily be demonstrated through murder, such a proof would greatly diminish the possibility of teaching additional lessons, and thus a different approach was taken. With the help of the fourth named author they invented what we’ll call the M&M Game: Given k people, each simultaneously flips a fair coin, with each eating an M&M on a head and not eating on a tail. The process then continues until all M&M’S are consumed, and two people are deemed to die at the same time if they run out of M&M’S together. This led to a great concrete demonstration of randomness appropriate for little kids; it also led to a host of math problems which have been used in probability classes and math competitions. There are many ways to determine the probability of a tie, which allow us in this article to use this problem as a springboard to a lot of great mathematics, including memoryless process, combinatorics, statistical inference, graph theory, and hypergeometric functions.

Constructing Families of Moderate-Rank Elliptic Curves Over Number Fields

Steven Miller (with David Mehrle, Tomer Reiter, Joseph Stahl and Dylan Yott)

Minnesota Journal of Undergraduate Mathematics, 2, 11 pages, 2016-2017

We generalize a construction of families of moderate rank elliptic curves over to number fields K/. The construction, originally due to Steven J. Miller, Alvaro Lozano-Robledo ´ and Scott Arms, invokes a theorem of Rosen and Silverman to show that computing the rank of these curves can be done by controlling the average of the traces of Frobenius; the construction for number fields proceeds in essentially the same way. One novelty of this method is that we can construct families of moderate rank without having to explicitly determine points and calculating determinants of height matrices.

Subsets of F_q[x] Free of 3-Term Geometric Progressions

Steven Miller (with Megumi Asada ’17, Eva Fourakis ’16, Sarah Manski, Gwyneth Moreland and Nathan McNew)

Finite Fields and Their Applications, 44, 135-147, 2017

Several recent papers have considered the Ramsey-theoretic problem of how large a subset of integers can be without containing any 3-term geometric progressions. This problem has also recently been generalized to number fields, determining bounds on the greatest possible density of ideals avoiding geometric progressions. We study the analogous problem over q[x], first constructing a set greedily which avoids these progressions and calculating its density, and then considering bounds on the upper density of subsets of q[x] which avoid 3-term geometric progressions. This new setting gives us a parameter q to vary and study how our bounds converge to 1 as it changes, and finite characteristic introduces some extra combinatorial structure that increases the tractibility of common questions in this area.

On Identities of Ruggles, Horadam, Howard, and Young

Steven Miller (with Curtis Cooper, Peter J.C. Moses, Murat Sahin and Thotsaporn Thanatipanonda

Fibonacci Quarterly, 55, No. 5, 52-65, 2017

Ruggles (1963) discovered that for integers n ≥ 0 and k ≥ 1

Fn+2k = Lk Fn+k + (−1)k+1Fn.

Horadam (1965), Howard (2001), and Young (2003) each expanded this identity to generalized linear recurrence relations of orders 2, 3, and integers r ≥ 2, respectively. In this paper we let r ≥ 2 be an integer and w0, w1, . . ., wr−1, and p1, p2, . . ., pr 0 be integers. For n ≥ r set

wn = p1 wn−1 + p2 wn−2 + · · · + pr wn−r.

We find identities like those of Ruggles, Horadam, Howard, and Young, of the form wn+rk = Rk(r − 1, r)wn+(r−1)k + Rk(r − 2, r)wn+(r−2)k + · · · + Rk(1, r)wn+k + Rk(0, r)wn, where, by a result of Young, Rk(i, r) is a linear recurrence relation of order for i = 0, 1, . . . , r − 1. Our proof uses the Cayley-Hamilton theorem. Next, we find the recurrences Rk(0, r) and Rk(r −1, r) for arbitrary r. Finally, we explicitly find identities for orders r = 3, r = 4 and r = 5.

A Collection of Central Limit Type Results in Generalized Zeckendorf Decompositions

Steven Miller (with Ray Li)

Fibonacci Quarterly, 55, No. 5, 105-114, 2017

Zeckendorf ‘s Theorem states that if the Fibonacci numbers are indexed as F1 = 1, F2 = 2, F3 = 3, F4 = 5, . . . , then every positive integer can be written uniquely as the sum of non-adjacent Fibonacci numbers. This result can be generalized to certain classes of linear recurrence relations {Gn} with appropriate notions of decompositions. For many decompositions, the distribution of the number of summands in the decomposition of an M ∈ [Gn, Gn+1) is known to converge to a Gaussian as n → ∞. This work discusses a more general approach to proving this kind of asymptotic Gaussian behavior that also bypasses technical obstructions in previous approaches. The approach is motivated by the binomials an,k = . The binomials satisfy the recursion an,k = an−1,k + an−1,k−1 and are well known to have the property that the random variables {Xn} given by Pr[Xn = k] = an,k / an,i converge to a Gaussian as n → ∞. This new approach proves that appropriate two-dimensional recurrences exhibit similar asymptotic Gaussian behavior. From this, we can reprove that the number of summands in decompositions given by many linear recurrence relations is asymptotic ally Gaussian and additionally prove that for any non-negative integer g, the number of gaps of size g in the decomposition of an M ∈ [Gn, Gn+1) also converges to a Gaussian as n → ∞.

On the Asymptomatic Behavior of Variance of PLRS Decompositions

Steven Miller (with Dawn Nelson, Zhao Pan, and Huanzhong Xu)

Fibonacci Quarterly, 55, No. 5, 135-143, 2017

A positive linear recurrence sequence is of the form Hn+1 = c1Hn + · · · + cLHn+1−L with each ci ≥ 0 and c1cL > 0, with appropriately chosen initial conditions. There is a notion of a legal decomposition (roughly, given a sum of terms in the sequence we cannot use the recurrence relation to reduce it) such that every positive integer has a unique legal decomposition using terms in the sequence; this generalizes the Zeckendorf decomposition, which states any positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers. Previous work proved not only that a decomposition exists, but that the number of summands Kn(m) in legal decompositions of m ∈ [Hn, Hn+1) converges to a Gaussian. Using partial fractions and generating functions it is easy to show the mean and variance grow linearly in n: an + b + o(1) and Cn + d + o(1), respectively; the difficulty is proving a and C are positive. Previous approaches relied on delicate analysis of polynomials related to the generating functions and characteristic polynomials, and is algebraically cumbersome. We introduce new, elementary techniques that bypass these issues. The key insight is to use induction and bootstrap bounds through conditional probability expansions to show the variance is unbounded, and hence C > 0 (the mean is handled easily through a simple counting argument).

Fringe Pairs in Generalized MSTD Sets

Steven Miller (with Sarah Manski and Hong Suh)

International Journal of Number Theory, 13, No. 10, 2653-2675, 2017

A More Sums Than Differences (MSTD) set is a set A for which |A + A| > |A – A|. Martin and O’Bryant proved that the proportion of MSTD sets in {0, 1, . . . , n} is bounded below by a positive number as n goes to infinity. Iyer, Lazarev, Miller and Zhang introduced the notion of a generalized MSTD set, a set A for which |sA – dA| > |σA − δA| for a prescribed s + d = σ + δ. We offer efficient constructions of k-generational MSTD sets, sets A where A, A + A, . . . , kA are all MSTD. We also offer an alternative proof that the proportion of sets A for which |sA−dA|−|σA−δA| = x is positive, for any x ∈ . We prove that for any > 0, Pr(1− < log |sA−dA|/ log |σA−δA| 1 a natural number) and exponential progressions (fn(x) = xn). Progression-free sets are often constructed “greedily,” including every number so long as it is not in progression with any of the previous elements. Rankin characterized the greedy geometric-progression-free set in terms of the greedy arithmetic set. We characterize the greedy exponential set and prove that it has asymptotic density 1, and then discuss how the optimality of the greedy set depends on the family of functions used to define progressions. Traditionally, the size of a progression-free set is measured using the (upper) asymptotic density, however we consider several different notions of density, including the uniform and exponential densities.

Benford’s Law and Continuous Dependent Random Variables

Steven Miller (with Thealexa Becker, David Burt ‘17, Taylor C. Corcoran, Alec Greaves-Tunnell ‘13, Joseph Iafrate ‘14, Joy Jing ‘13, Jaclyn Porfilio ‘15, Ryan Robab, Jirapat Samranvedhya ‘14, Frederick W. Strauch and Blaine Talbut)

Annals of Physics 388, 350-381, 2018

Many mathematical, man-made and natural systems exhibit a leading-digit bias, where a first digit (base 10) of 1 occurs not 11% of the time, as one would expect if all digits were equally likely, but rather 30%. This phenomenon is known as Benford’s Law. Analyzing which datasets adhere to Benford’s Law and how quickly Benford behavior sets in are the two most important problems in the field. Most previous work studied systems of independent random variables, and relied on the independence in their analyses. Inspired by natural processes such as particle decay, we study the dependent random variables that emerge from models of decomposition of conserved quantities. We prove that in many instances the distribution of lengths of the resulting pieces converges to Benford behavior as the number of divisions grow, and give several conjectures for other fragmentation processes. The main difficulty is that the resulting random variables are dependent. We handle this by using tools from Fourier analysis and irrationality exponents to obtain quantified convergence rates as well as introducing and developing techniques to measure and control the dependencies. The construction of these tools is one of the major motivations of this work, as our approach can be applied to many other dependent systems. As an example, we show that the n! entries in the determinant expansions of n × n matrices with entries independently drawn from nice random variables converges to Benford’s Law.

Irrationality Measure and Lower Bounds for \pi(x)

Steven Miller (with David Burt ’17, Sam Donow, Matthew Schiffman and Ben Wieland)

Pi Mu Epsilon Journal 14, no. 7, 421-429, 2017

In this note we show how the irrationality measure of ζ(s) = π2/6 can be used to obtain explicit lower bounds for π(x). We analyze the key ingredients of the proof of the finiteness of the irrationality measure, and show how to obtain good lower bounds for π(x) from these arguments as well. While versions of some of the results here have been carried out by other authors, our arguments are more elementary and yield a lower bound of order x/ log x as a natural boundary.

Random Matrix Ensembles With Split Limiting Behavior

Steven Miller (with Paula Burkhardt, Peter Cohen, Jonathan Dewitt, Max Hlavacek, Carsten Sprunger, Yen Nhi Truong Vu, Roger Van Peski, and an appendix joint with Manuel Fernandez and Nicholas Sieger)

Random Matrices: Theory and Applications, 7, no. 3, 30 pages, DOI: 10:1142/S2010326318500065, 2018

We introduce a new family of N × N random real symmetric matrix ensembles, the k-checkerboard matrices, whose limiting spectral measure has two components which can be determined explicitly. All but k eigenvalues are in the bulk, and their behavior, appropriately normalized, converges to the semi-circle as N tends to infinity; the remaining k are tightly constrained near N/k and their distribution converges to the k x k hollow GOE ensemble (this is the density arising by modifying the GOE ensemble by forcing all entries on the main diagonal to be zero). Similar results hold for complex and quaternionic analogues. We are able to isolate each regime separately through appropriate choices of weight functions for the eigenvalues and then an analysis of the resulting combinatorics.

Problem Session Notes from CANT Conferences

Steven Miller (currently 2009 through 2018, inclusive)

Distributed to interested parties and posted on the arXiv, 35 pages, 2018

These notes are a summary of the problem session discussions at various CANT (Combinatorial and Additive Number Theory Conferences). Currently they include all years from 2009 through 2018 (inclusive); the goal is to supplement this file each year. These additions will include the problem session notes from that year, and occasionally discussions on progress on previous problems. If you are interested in pursuing any of these problems and want additional information as to progress, please email the author. For more information, visit the conference homepage at http://www.theoryofnumbers.com/ or email either the typist at [email protected] or [email protected], or the organizer at [email protected]

Mathematics of Optimization: How To Do Things Faster

Steven Miller

AMS, Pure and Applied Undergraduate Texts, 30, 327 pages, 2017

Optimization Theory is an active area of research with numerous applications; many of the books are designed for engineering classes, and thus have an emphasis on problems from such fields. Covering much of the same material, there is less emphasis on coding and detailed applications as the intended audience is more mathematical. There are still several important problems discussed (especially scheduling problems), but there is more emphasis on theory and less on the nuts and bolts of coding. A constant theme of the text is the why and the how in the subject. Why are we able to do a calculation efficiently? How should we look at a problem? Extensive effort is made to motivate the mathematics and isolate how one can apply ideas/perspectives to a variety of problems. As many of the key algorithms in the subject require too much time or detail to analyze in a first course (such as the run-time of the Simplex Algorithm), there are numerous comparisons to simpler algorithms which students have either seen or can quickly learn (such as the Euclidean algorithm) to motivate the type of results on run-time savings.

The Isoperimetric Problem With Density

Frank Morgan

Math. Intelligencer, 39, 2-8, 2017

Enhanced write-up of talk at 2015 Lázló Fejes Tóth Centennial, including recent proofs of Log-Convex Density Theorem, the analog for perimeter and volume densities rk and rm, and the isoperimetric solution for density rp.

The Complex Body Isoperimetric Conjecture in R2

Frank Morgan, John Berry ’18, Eliot Bongiovanni, Wyatt Boyer ’17, Bryan Brown, Matthew Dannenberg, Paul Gallagher, David Hu, Jason Liang, Alyssa Loving, Zane Martin ’15, Maggie Miller, Byron Perpetua ’16, Sara Tammen, and Yingqi Zeng

Rose-Hulman Und. Math. J., 2018

Morgan’s 2012-2015 NSF SMALL undergraduate research Geometry Groups provide partial results on the convex body isoperimetric conjecture in R2.

The Isoperimetric Problem for the Sum of Two Gaussian Densities

Frank Morgan, John Berry ‘16, Matthew Dannenberg, Jason Liang, Yingyi Zeng

Involve, 11, 4, 549-567, 2018

Morgan’s 2015 NSF SMALL undergraduate research Geometry Group solves the isoperimetric problem for the sum of two Gaussian densities.

Strict Doubly Ergodic Infinite Transformations

Cesar E Silva with Isaac Loh

Dynamical Systems 32, no. 4, http://arxiv.org/abs/1512.09340, 519 – 543, 2017

We give examples of rank-one transformations that are weakly doubly ergodic and rigid (so all their Cartesian products are conservative), but with non-ergodic two-fold Cartesian product. We give conditions for rank-one infinite measure-preserving transformations to be weakly doubly ergodic and for their k-fold Cartesian product to be conservative. We also show that a weakly doubly ergodic nonsingular group action is ergodic with isometric coefficients, and that the latter strictly implies W-measurable sensitivity.

On Mixing-Like Notions in Infinite Measure

Cesar E. Silva

Amer. Math. Monthly, 124, no. 9, 807 – 825, 2017

Measurable dynamical systems are defined on a measure space, such as the unit interval or the real line, with a transformation or map acting on the space. After discussing dynamical properties for probability spaces such as ergodicity, weak mixing, and mixing, we consider analogues of mixing and weak mixing in infinite measure, and present related examples and definitions that are the result of research with undergraduates. Rank-one transformation are introduced and used to construct the main examples.

On Conservative Sequences and Their Application to Ergodic Multiplier Problems

Cesar E. Silva with Madeleine Elyze ‘18, Alexander Kastner ‘17, Juan Ortiz Rhoton, Vadim Semenov

Colloquium Mathematicum, 151, no. 1, https://arxiv.org/abs/1610.01438, 123 – 145, 2018

The conservative sequence of a set A under a transformation T is the set of all integers n such that T^n(A) intersects A. By studying these sequences, we prove that given any countable collection of nonsingular transformations with no sigma-finite invariant measure T_i, there exists a rank-one transformation S such that T_i x S is not ergodic for any i. Moreover, S can be chosen to be rigid or have infinite ergodic index. We establish similar results for actions of Z^d and for flows. We then find sufficient conditions on rank-one transformations T that guarantee the existence of a rank-one transformation S such that T x S is ergodic, or alternatively, conditions that guarantee that T x S is conservative but not ergodic.

Infinite Symmetric Ergodic Index and Related Examples in Infinite Measure

Cesar E. Silva with Isaac Loh, Ben Athiwaratkun

Studia Mathematica 243, no. 1, https://arxiv.org/abs/1702.01455, 101-115, 2018

For infinite-measure-preserving rank-one transformations, we give a condition guaranteeing that all finite Cartesian products of the transformation with its inverse are ergodic. We show that the infinite Chacon transformation satisfies this condition. We then explore the relationship between product conservativity and product ergodicity and answer a question of Danilenko. Finally, we define a class of infinite Chacon type transformations and show they do not have products of all powers conservative and are therefore not power weakly mixing.

Band Depth Clustering for Nonstationary Time Series and Wind Speed Behavior

Laura L. Tupper, David S. Matteson, C. Lindsay Anderson, and Luckny Zephyr

Technometrics 60(2), 245-254, 2018

We explore the behavior of wind speed over time, using a subset of the Eastern Wind Dataset published by the National Renewable Energy Laboratory. This dataset gives modeled wind speeds over three years at hundreds of potential wind farm sites. Wind speed analysis is necessary to the integration of wind energy into the power grid; short-term variability in wind speed affects decisions about usage of other power sources, so that the shape of the wind speed time series becomes as important as the overall level. To assess differences in intra-day time series, we propose a functional distance measure, the band distance, which extends the band depth of Lopez-Pintado and Romo (2009). This measure emphasizes the shape of time series or functional observations relative to other members of a dataset, and allows clustering of observations without reliance on pointwise Euclidean distance. We show a method for adjusting for seasonal effects in wind speed, and use these standardizations as input for the band distance. We demonstrate the utility of the new method in simulation studies and an application to the MOST power grid algorithm, where the band distance improves reliability over standard methods at a comparable cost.

Multi-Crossing Number for Knots and the Kauffman Bracket Polynomial

Colin Adams, Jesse Freeman ’15, J. Daniel Irvine, Samantha Petti ’15, Daniel Vitek, Ashley Weber, Sicong Zhang

Mathematical Proceedings of the Cambridge Philosophical Society, 1-32. Doi:10.1017/S0305004116000906, 2016.

We find a lower bound on the n-crossing number in terms of the span of the bracket polynomial, for any n. A multi-crossing (or n-crossing) is a singular point in a projection of a knot or link at which n strands cross so that each strand bisects the crossing. We generalize the classic result of Kauffman, Murasugi, and Thistlethwaite relating the span of the bracket polynomial to the double-crossing number of a link, span(<K>) ≤ 4c2, to the n-crossing number. We find a lower bound on the n-crossing number in terms of the span of the bracket polynomial for any n ≥ 3. Further, we present the first extensive list of calculations of n-crossing number for knots.

Volume and Determinant Density of Hyperbolic Rational Links

Colin Adams, Aaron Calderon, Xinyi Jiang, Alexander Kastner ’17, Gregory Kehne ’16, Nathaniel Mayer, Mia Smith ‘16

Journal of Knot Theory and its Ramifications, Vol. 26, 1750002, pages, 2016.

The volume density of a hyperbolic link is defined as the ratio of hyperbolic volume to crossing number. We study its properties and a closely-related invariant called the determinant density. It is known that the sets of volume densities and determinant densities of links are dense in the interval [0,voct]. We construct sequences of alternating knots whose volume and determinant densities both converge to any x in [0,voct]. We also investigate the distributions of volume and determinant densities for hyperbolic rational links, and establish upper bounds and density results for these invariants.

The Galactic Math Repository

Colin Adams

Mathematical Intelligencer, Vol. 38, No. 2, 1-3, 2016.

How will our mathematical achievements compare to the rest of the galaxy when we join the Galactic Union?


Colin Adams

Mathematical Intelligencer, Vol. 38, No. 3, 11-13, 2016.

What happens when students lead a “slave revolt” against mathematics?

The Math Museum

Colin Adams

Mathematical Intelligencer, Vol. 38, No. 4, 23-27, 2016.

A tale of a museum that is mathematical, but unfortunately, the exhibits do all they can to kill you.

The Topology Terrors

Colin Adams

Mathematical Intelligencer, Vol. 39, No. 1, 2017.

A dark tale of a murderous function that tears topological spaces asunder, and the detective who brings it to justice.

Predicting Bat Colony Survival Under Controls Targeting Multiple Transmission Routes of White-Nose Syndrome
Julie Blackwood,
A.D. Meyer, D.F. Stevens ’15,

Journal of Theoretical Biology, 409, 60-69, 2016.

White-nose syndrome (WNS) is a lethal infection of bats caused by the psychrophilic fungus Pseudo-gymnoascus destructans (Pd). Since the first cases of WNS were documented in 2006, it is estimated that as many as 5.5 million bats have succumbed in the United States — one of the fastest mammalian die-offs due to disease ever observed, and the first known sustained epizootic of bats. WNS is contagious between bats, and mounting evidence suggests that a persistent environmental reservoir of Pd plays a significant role in transmission as well. It is unclear, however, the relative contributions of bat-to-bat and environment-to-bat transmission to disease propagation within a colony. We analyze a mathematical model to investigate the consequences of both avenues of transmission on colony survival in addition to the efficacy of disease control strategies. Our model shows that selection of the most effective control strategies is highly dependent on the primary route of WNS transmission. Under all scenarios, however, generalized culling is ineffective and while targeted culling of infected bats may be effective under idealized conditions, it primarily has negative consequences. Thus, understanding the significance of environment-to-bat transmission is paramount to designing effective management plans.

The Role of Interconnectivity in Control of an Ebola Epidemic
Julie Blackwood,
and L.M. Childs

Scientific Reports, 6, 29262, 2016.

Several West African countries – Liberia, Sierra Leone and Guinea – experienced significant morbidity and mortality during the largest Ebola epidemic to date, from late 2013 through 2015. The extent of the epidemic was fueled by outbreaks in large urban population centers as well as movement of the pathogen between populations. During the epidemic there was no known vaccine or drug, so effective disease control required coordinated efforts that include both standard medical and community practices such as hospitalization, quarantine and safe burials. Due to the high connectivity of the region, control of the epidemic not only depended on internal strategies but also was impacted by neighboring countries. In this paper, we use a deterministic framework to examine the role of movement between two populations in the overall success of practices designed to minimize the extent of Ebola epidemics. We find that it is possible for even small amounts of intermixing between populations to positively impact the control of an epidemic on a more global scale.

Dynamics Done With Your Bare Hands
Diana Davis with Bryce Weaver, Roland K.W. Roeder, and Pablo Lessa

European Mathematical Society, 2017.

This book arose from four lectures given at the Undergraduate Summer School of the Thematic Program Dynamics and Boundaries, held at the University of Notre Dame.  It is intended to introduce (under)graduate students to the field of dynamical systems by emphasizing elementary examples, exercises, and bare hands constructions.

Negative Refraction and Tiling Billards
Diana Davis with Kelsey DiPietro, Jenny Rustad, and Alexander St. Laurent

Advances in Geometry, 2017.

We introduce a new dynamical system that we call tiling billiards, where trajectories refract through planar tilings. This system is motivated by a recent discovery of physical substances with negative indices of refraction.  We investigate several special cases where the planar tiling is created by dividing the plane by lines, and we describe the results of computer experiments.

Big Data and the Missing Links

Richard De Veaux with Ron Snee and R. W. Hoerl

Statistical Analysis and Data Mining, Volume 9, Issue 6, pp 411–416, December 2016

Curriculum Guidelines for Undergraduate Programs in Data Science

Richard De Veaux with Mahesh Agarwal, Maia Averett, Benjamin S. Baumer, Andrew Bray,5Thomas C. Bressoud, Lance Bryant, Lei Z. Cheng, Amanda Francis, Robert Gould, Albert Y. Kim, Matt Kretchmar, Qin Lu, Ann Moskol, Deborah Nolan, Roberto Pelayo, Sean Raleigh, Ricky J. Sethi, Mutiara Sondjaja,Neelesh Tiruviluamala, Paul X. Uhlig, Talitha M. Washington, Curtis L. Wesley, David White, and Ping Ye

The Annual Review of Statistics and Its Application, Vol. 4:15-30, Volume publication date March 2017

First published online as a Review in Advance on December 23, 2016  https://doi.org/10.1146/annurev-statistics-060116-053930

Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences

Thomas Garrity with Amburg ’14, Dasaratha, Flapan, Lee ’12, Mihaila, Neumann-Chun ’13, Peluse and Stoffregen

Journal of Integer Sequences, Article 17.1.7, 2017

The Stern diatomic sequence is closely linked to continued fractions via the Gauss map on the unit interval, which in turn can be understood via systematic subdivisions of the unit interval. Higher dimensional analogues of continued fractions, called multidimensional continued fractions, can be produced through various subdivisions of a triangle. We define triangle partition-Stern sequences (TRIP-Stern sequences for short), higher-dimensional generalizations of the Stern diatomic sequence, from the method of subdividing a triangle via various triangle partition algorithms. We then explore several combinatorial results about TRIP-Stern sequences, which may be used to give rise to certain well-known sequences. We finish by generalizing TRIP-Stern sequences and presenting analogous results for these generalizations.

A Proof of the Peak Polynomial Positivity Conjecture

Alexander Diaz-Lopez, Pamela E. Harris, Erik Insko, and Mohamed Omar

Journal of Combinatorial Theory, Series A, 149, 21-29, 2017.

We say that a permutation p = p1p2 … pÎ Sn has a peak at index i if pi-1 < pi > pi+1. Let P(p) denote the set of indices where p has a peak.  Given a set S of positive integers, we define PS(n) = {p Î Sn : P(p) = S}. In 2013 Billey, Burdzy, and Sagan showed that for subsets of positive integers S and sufficiently large

n, ½PS(n)½ = pS(n)2n-|S|-1 where pS(x) is a polynomial depending on S. They gave a recursive formula for pS(x) involving an alternating sum, and they conjectured that the coefficients of pS(x) expanded in a binomial coefficient basis centered at max(S) are all nonnegative. In this paper we introduce a new recursive formula for½PS(n)½without alternating sums and we use this recursion to prove that their conjecture is true.

Individual Gap Measures from Generalized Zeckendorf Decompositions

Robert Dorward, Pari Ford, Eva Fourakis ‘16, Pamela E. Harris, Steven J. Miller, Eyvi Palsson, and Hannah Paugh

Uniform Distribution Theory, vol. 12, No. 1, 27-36, 2017.

Zeckendorf’s theorem states that every positive integer can be uniquely decomposed as a sum of nonconsecutive Fibonacci numbers. The distribution of the number of summands converges to a Gaussian, and the individual measures on gaps between summands for m Î [Fn, Fn+1) converge to geometric decay for almost all m as n Ò ¥. While similar results are known for many other recurrences, previous work focused on proving Gaussianity for the number of summands or the average gap measure. We derive general conditions which are easily checked yield geometric decay in the individual gap measures of generalized Zeckendorf decompositions attached to many linear recurrence relations.

Peak Sets of Classical Coxeter Groups

Alexander Diaz-Lopez, Pamela E. Harris, Erik Insko, and Darleen Perez-Lavin

Involve, a Journal of Mathematics, Vol. 10, no. 2, pp. 263-290, 2017

We say a permutation p = p1p2… pn in the symmetric group Sn has a peak at index i if pi-1 < p1 > pi-1 and we let P(p) = {i Î {1, 2,…,ni is a peak of p}. Given a set S of positive integers, we let P(S;n) denote the subset of Sn consisting of all permutations p, where P(p) = S.  In 2013, Billey, Burdzy, and Sagan proved

½P(S;n)½= p(n)2n-|S|-1, where p(n) is a polynomial of degree max(S) – 1. In 2014, Castro-Velez et al. considered the Coxeter group of type Bn as the group of signed permutations on n letters and showed that ½PB(S;n)½= p(n)22n-|S|-1 where p(n) is the same polynomial of degree max(S) 1. In this paper we partition the sets P(S;n) Ì Sn studied by Billey, Burdzy, and Sagan into subsets P(S;n) of permutations with peak set S that end with an ascent to a fixed integer k or a descent and provide polynomial formulas for the cardinalities of these subsets. After embedding the Coxeter groups of Lie type Cn and Dn into S2n, we partition these groups into bundles of permutations p1p… pn | pn+1  … p2n  such that p1p2   pn  has the same relative order as some permutation s1s2…sn Î Sn. This allows us to count the number of permutations in types Cn and Dn with a given peak set S by reducing the enumeration to calculations in the symmetric group and sums across the rows of Pascal’s triangle.

A Generalization of Zeckendorf’s Theorem Via Circumscribed $m$-gons

Robert Dorward, Pari Ford, Eva Fourakis ‘16, Pamela E. Harris, Steven J. Miller, Eyvi Palsson, and Hannah Paugh

Involve, a Journal of Mathematics, 10-1, 125-150, 2017.

Zeckendorf’s theorem states that every positive integer can be uniquely decomposed as a sum of nonconsecutive Fibonacci numbers, where the Fibonacci numbers satisfy Fn = Fn -1 + Fn-2 for n ³ 3, F1 =1 and F2 =2. The distribution of the number of summands in such decomposition converges to a Gaussian, the gaps between summands converges to geometric decay, and the distribution of the longest gap is similar to that of the longest run of heads in a biased coin; these results also hold more generally, though for technical reasons previous work needed to assume the coefficients in the recurrence relation are non-negative and the first term is positive.

We extend these results by creating an infinite family of integer sequences called the m-gonal sequences arising from a geometric construction using circumscribed m-gons. They satisfy a recurrence where the first m+1leading terms vanish, and thus cannot be handled by existing techniques. We provide a notion of a legal decomposition, and prove that the decompositions exist and are unique. We then examine the distribution of the number of summands used in the decompositions and prove that it displays Gaussian behavior. There is geometric decay in the distribution of gaps, both for gaps taken from all integers in an interval and almost surely in distribution for the individual gap measures associated to each integer in the interval. We end by proving that the distribution of the longest gap between summands is strongly concentrated about its mean, behaving similarly as in the longest run of heads in tosses of a coin.

Legal Decompositions Arising from Non-positive Linear Recurrences

Minerva Catral, Pari Ford, Pamela E. Harris, Steven J. Miller, and Dawn Nelson

Fibonacci Quart. 54, no. 4, 348-365, 2016.

Zeckendorf’s theorem states that any positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers; this result has been generalized to many recurrence relations, especially those arising from linear recurrences with leading term positive. We investigate legal decompositions arising from two new sequences: the (s,b)-Generacci sequence and the Fibonacci Quilt sequence. Both satisfy recurrence relations with leading term zero, and thus previous results and techniques do not apply. These sequences exhibit drastically different behavior. We show that the (s,b)-Generacci sequence leads to unique legal decompositions, whereas not only do we have non-unique legal decompositions with the Fibonacci Quilt sequence, we also have that in this case the average number of legal decompositions grows exponentially. Another interesting difference is that while in the (s,b)-Generacci case the greedy algorithm always leads to a legal decomposition, in the Fibonacci Quilt setting the greedy algorithm leads to a legal decomposition (approximately) 93% of the time. In the (s,b)-Generacci case, we again have Gaussian behavior in the number of summands as well as for the Fibonacci Quilt sequence when we restrict to decompositions resulting from a modified greedy algorithm.

Multiple Regression Analysis: Understanding the Impact of Offensive and Defensive Contributions to Team Performance

Steven Miller (with Kevin D. Dayaratna)

The Hockey Research Journal, 41-43, 2014/2015.

From evaluating team performance to predicting outcomes of particular matchups to understanding individual player contributions, rigorous statistical analysis has become increasingly useful in hockey over the past several decades. In recent years, there has been a variety of research projects offering quantitative assessments from a variety of perspectives. This study adds to those efforts.

A Probabilistic Approach to Generalized Zeckendorf Decompositions

Steven Miller with Iddo Ben-Ari

SIAM Journal on Discrete Mathematics, 30, no. 2, 1302-1332, 2016.

Generalized Zeckendorf decompositions are expansions of integers as sums of elements of solutions to recurrence relations. The simplest cases are base-b expansions, and the standard Zeckendorf decomposition uses the Fibonacci sequence. The expansions are finite sequences of nonnegative integer coefficients (satisfying certain technical conditions to guarantee uniqueness of the decomposition) and which can be viewed as analogues of sequences of variable-length words made from some fixed alphabet. In this paper we present a new approach and construction for uniform measures on expansions, identifying them as the distribution of a Markov chain conditioned not to hit a set. This gives a unified approach that allows us to easily recover results on the expansions from analogous results for Markov chains, and in this paper we focus on laws of large numbers, central limit theorems for sums of digits, and statements on gaps (zeros) in expansions. We expect the approach to prove useful in other similar contexts.

Lower-Order Biases in Elliptic Curve Fourier Coefficients in Families

Steven Miller (with B. Mackall ‘16, C. Rapti and K. Winsor)

Frobenius Distributions: Lang-Trotter and Sato-Tate Conjectures

(David Kohel and Igor Shparlinsky, editors)

Contemporary Mathematics 663, AMS, Providence, RI, 2016.

Let E : y2 = x3 + A(T)x + B(T) be a nontrivial one-parameter family of elliptic curves over Q(T), with A(T), B(T) in Z[T], and consider the k-th moments Ak,E(p) := sumt mod p aEt(p)k of the Fourier coefficients aEt(p). Rosen and Silverman proved a conjecture of Nagao relating the first moment to the rank of the family over Q(T), and Michel proved that the second moment is equal to p2 + O (p3/2). Cohomological arguments show that the lower order terms are of sizes p3/2, p, p1/2, and 1. In every case we are able to analyze, the largest lower order term in the second moment expansion that does not average to zero is on average negative. We prove this “bias conjecture” for several large classes of families, including families with rank, complex multiplication, and unusual distributions of functional equation signs. We also identify all lower order terms in large classes of families, shedding light on the arithmetic objects controlling these terms. The negative bias in these lower order terms has implications toward the excess rank conjecture and the behavior of zeros near the central point of elliptic curve L-functions.

From Quantum Systems to L-Functions:  Pair Correlation Statistics and Beyond

Steven Miller (with Owen Barrett, Frank W.K. Firk, and Caroline Turnage-Butterbaugh)

Open Problems in Mathematics

(John Nash, Jr. and Michael Th. Rassias, editors)

Springer-Verlag, 2016.

The discovery of connections between the distribution of energy levels of heavy nuclei and spacings between prime numbers has been one of the most surprising and fruitful observations in the twentieth century. The connection between the two areas was first observed through Montgomery’s work on the pair correlation of zeros of the Riemann zeta function. As its generalizations and consequences have motivated much of the following work, and to this day remains one of the most important outstanding conjectures in the field, it occupies a central role in our discussion below. We describe some of the many techniques and results from the past sixty years, especially the important roles played by numerical and experimental investigations, that led to the discovery of the connections and progress towards understanding the behaviors. In our survey of these two areas, we describe the common mathematics that explains the remarkable universality. We conclude with some thoughts on what might lie ahead in the pair correlation of zeros of the zeta function, and other similar quantities.

Some Results in the Theory of Low-Lying Zeros

Steven Miller (with Blake Mackall ‘15, Christina Rapti, Caroline Turnage-Butterbaugh and Karl Winsor and an Appendix with Megumi Asada ’17, Eva Fourakis ‘16, Kevin Yang)

In Families of Automorphic Forms and the Trace Formula

(Werner Muller, Sug Woo Shin and Nicolas Templier, editors)

Simons Symposia Series, Springer-Verlag, 2016.

While Random Matrix Theory has successfully modeled the limiting behavior of many quantities of families of L-functions, especially the distributions of zeros and values, the theory frequently cannot see the arithmetic of the family. In some situations this requires an extended theory that inserts arithmetic factors that depend on the family, while in other cases these arithmetic factors result in contributions which vanish in the limit, and are thus not detected. In this chapter we review the general theory associated to one of the most important statistics, the n-level density of zeros near the central point. According to the Katz-Sarnak density conjecture, to each family of L-functions there is a a corresponding symmetry group (which is a subset of a classical compact group) such that the behavior of the zeros near the central point as the conductors tend to infinity agrees with the behavior of the eigenvalues near 1 as the matrix size tends to infinity. We show how these calculations are done, emphasizing the techniques, methods and obstructions to improving the results, by considering in full detail the family of Dirichlet characters with square-free conductors. We then move on and describe how we may associate a symmetry constant to each family, and how to determine the symmetry group of a compound family in terms of the symmetries of the constituents. These calculations allow us to explain the remarkable universality of behavior, where the main terms are independent of the arithmetic, as we see that only the first two moments of the Satake parameters survive to contribute in the limit. Similar to the Central Limit Theorem, the higher moments are only felt in the rate of convergence to the universal behavior. We end by exploring the effect of lower order terms in families of elliptic curves. We present evidence supporting a conjecture that the average second moment in one-parameter families without complex multiplication has, when appropriately viewed, a negative bias, and end with a discussion of the consequences of this bias on the distribution of low-lying zeros, in particular relations between such a bias and the observed excess rank in families.

Crescent Configurations

Steven Miller (with David Burt ‘17, Eli Goldstein, Sarah Manski, Eyvindur Ari Palsson and Hong Suh, editors)

Integers (Electronic Journal of Combinatorical Number Theory) 16, #A38, 2016

In 1989, Erdos conjectured that for a sufficiently large n it is impossible to place n points in general position in a plane such that for every 1 < = i < = n – 1 there is a distance that occurs exactly i times. For small n this is possible and in his paper he provided constructions for n = 8. The one for n = 5 was due to Pomerance while Palasti came up with the constructions for n = 7, 8. Constructions for n = 9 and above remain undiscovered, and little headway has been made toward a proof that for sufficiently large n no configuration exists. In this paper we consider a natural generalization to higher dimensions and provide a construction which shows that for any given n there exists a sufficiently large dimension d such that there is a configuration in d-dimensional space meeting Erdos’ criteria.

The Emergence of 4-Cycles in Polynomial Maps Over the Extended Integers

Steven Miller with Andrew Best ’15, Patrick Dynes, Jasmine Powell and Ben Weiss)

Minnesota Journal of Undergraduate Mathematics 2, no. 1, 14 pages, 2016-2017.

Let f(x) in Z[x]; for each integer alpha it is interesting to consider the number of iterates na, if possible, needed to satisfy fna(a) = a. The sets {a, f(a), …, fn-1(a)} generated by the iterates of f are called cycles. For Z[x] it is known that cycles of length 1 and 2 occur, and no others. While much is known for extensions to number fields, we concentrate on extending Z by adjoining reciprocals of primes. Let Z[1/p1, …, 1/pn] denote Z extended by adding in the reciprocals of the n primes p1, …, pn and all their products and powers with each other and the elements of Z. Interestingly, cycles of length 4, called 4-cycles, emerge under the appropriate conditions for polynomials in Z[1/p1, …, 1/pn][x]. The problem of finding criteria under which 4-cycles emerge is equivalent to determining how often a sum of four terms is zero, where the terms are pm 1 times a product of elements from the list of n primes. We investigate conditions on sets of primes under which 4-cycles emerge. We characterize when 4-cycles emerge if the set has one or two primes, and (assuming a generalization of the ABC conjecture) find conditions on sets of primes guaranteed not to cause 4-cycles to emerge.

The Probability Lifesaver

Steven Miller

Princeton University Press, 2017.

Welcome to “The Probability Lifesaver”. My goal is to write a book introducing students to the material through lots of worked out examples and code, and to have lots of conversations about not just why equations and theorems are true, but why they have the form they do. In a sense, this is a sequel to Adrian Banner’s successful “The Calculus Lifesaver”. In addition to many worked out problems, there are frequent explanations of proofs of theorems, with great emphasis placed on discussing why certain arguments are natural and why we should expect certain forms for the answers. Knowing why something is true, and how someone thought to prove it, makes it more likely for you to use it properly and discover new relations yourself. The book highlights at great lengths the methods and techniques behind proofs, as these will be useful for more than just a probability class. See, for example, the extensive entries in the index on proof techniques, or the discussion on Markov’s inequality in §17.1. There are also frequent examples of computer code to investigate probabilities. This is the 21st century; if you cannot write simple code you are at a competitive disadvantage. Writing short programs helps us check our math in situations where we can get a close form solution; more importantly, it allows us to estimate the answer in situations where the analysis is very involved and nice solutions may be hard to obtain (if possible at all!).

Isoperimetric Symmetry Breaking: A Counterexample to a Generalized Form of the Log-Convex Density Conjecture

Frank Morgan

Anal. Geom. Metr. Spaces 4, 314 – 316, 2016

We give an example of a smooth surface of revolution for which all circles about the origin are strictly stable for fixed area but small isoperimetric regions are nearly round discs away from the origin.

Isoperimetry with Density

Frank Morgan

CIRM Audiovisual Mathematics Library video


A video of my talk at the CIRM conference on “Shape Optimization and Isoperimetric and Functional Inequalities,” Luminy, November, 2016. The talk featured open questions and recent results on the isoperimetric problem in the presence of a density, including some by my students.

Symmetries of Cairo-Prismatic Tilings

Frank Morgan, John Berry ‘16, Matthew Dannenberg, Jason Liang, Yingyi Zeng

Rose-Hulman Und. Math. J. 17, http://scholar.rose-hulman.edu/rhumj/vol17/iss2/3,2016

Morgan’s 2014 NSF SMALL undergraduate research Geometry Group studies and catalogs isoperimetric, planar tilings by unit-area Cairo and Prismatic pentagons. In particular, in counterpoint to the five wallpaper symmetry groups known to occur in Cairo-Prismatic tilings, they show that the five with order three rotational symmetry cannot occur.

Isoperimetric Regions in Rn with Density rp

Frank Morgan, Wyatt Boyer ‘15, Bryan Brown, Gregory Chambers, Alyssa Loving, and Sarah Tammen

Anal. Geom. Metr. Spaces 4, 236–265, 2016

Morgan’s 2015 NSF SMALL undergraduate research Geometry Group proves the conjectured most efficient regions in Rn with density rp.

Math in Cuba

Frank Morgan

Huffington Post blog, March 6, 2017.

A report on the state of mathematics in Cuba, including undergraduate research, after my visit.

The Tropical Commuting Variety

Ralph Morrison and Ngoc M. Tran

Linear Algebra and its Applications 507, 300-321, 2016

We study tropical commuting matrices from two viewpoints: linear algebra and algebraic geometry. In classical linear algebra, there exist various criteria to test whether two square matrices commute. We ask for similar criteria in the realm of tropical linear algebra, giving conditions for two tropical matrices that are polytropes to commute. From the algebro-geometric perspective, we explicitly compute the tropicalization of the classical variety of commuting matrices in dimension 2 and 3.

Weak Rational Ergodicity Does Not Imply Rational Ergodicity

Cesar Silva with Terrence Adams

Israel Journal of Mathematics, 214, 491 – 506, 2016

We construct an uncountable family of rank-one infinite measure-preserving transformations that are weakly rationally ergodic, but are not rationally ergodic, thus answering an open question by showing that weak rational ergodicity does not imply rational ergodicity.

The Mathematical Work of John C. Oxtoby

Cesar Silva with Steve Alpern and Joseph Auslander

Contemporary Mathematics 678, 43—51, 2016

We describe the mathematical work of John C. Oxtoby.

Ergodic Theory, Dynamical Systems, and the Continuing Influence of John C. Oxtoby

Cesar Silva with Joseph Auslander and Aimee Johnson, Editors

Contemporary Mathematics, American Mathematical Society 678, 2016

Proceedings of the three conferences: Oxtoby Centennial Conference held at Bryn Mawr College, Bryn Mawr, PA, October 30–31, 2010; Williams Ergodic Theory Conference held at Williams College, Williamstown, MA, July 27–29, 2012; AMS Special Session on Ergodic Theory and Symbolic Dynamics held in Baltimore, MD, January 17–18, 2014.

If a Prime Divides a Product

Cesar Silva with Steven J. Miller

The College Mathematics Journal, 48, 123-128, 2017

One of the greatest difficulties encountered by all in their first proof-intensive class is subtly assuming an unproven fact in a proof. The purpose of this note is to describe a specific instance where this can occur, namely in results related to unique factorization and the concept of the greatest common divisor.

Orthogonal Polynomials on the Unit Circle, CMV Matrices, and the Distribution of Their Eigenvalues

Mihai Stoiciu, Associate Professor of Mathematics

Memoria – Seminario de Operadores y Fisica-Matematica, IIMAS-UNAM, Vol. 1, 41-62, 2016

Abstract: We give a brief introduction to the theory of orthogonal polynomials on the unit circle (OPUC) and the associated CMV matrices. From the point of view of orthogonal polynomials, the CMV matrices are unitary analogues of the Jacobi matrices. We consider various classes of random and deterministic CMV matrices and study the distribution of their eigenvalues. More precisely, we consider CMV matrices with random decaying coefficients and CMV matrices associated to hyperbolic reflection groups. As the spectral measures approach an absolutely continuous measure, the repulsion between the eigenvalues increases and the eigenvalue distribution converges to the “clock” (or “picket fence”) distribution.

Asymptotics for Scaled Kramers-Smoluchowski Equations

Peyam Tabrizian and Lawrence C. Evans

SIAM Journal of Mathematical Analysis, 84, no. 4, 2944-2961, 2016.

We offer fairly simple proofs of the asymptotics for the scaled Kramers-Smoluchowski equation in both one and higher dimensions.  For the latter, we invoke the sharp asymptotic capacity estimates of Bovier-Eckhoff-Gayrard-Klein.

Mixed Data and Classification of Transit Stop
Laura L. Tupper, David S. Matteson, and John C. Handle
Proceedings of the 2016 IEEE International Conference on Big Data: 2nd International Workshop on Big Data for Sustainable Development, Washington, DC, pp 2225-2232, December 5-8, 2016

An analysis of the characteristics and behavior of individual bus stops can reveal clusters of similar stops, which can be of use in making routing and scheduling decisions, as well as determining what facilities to provide at each stop. This paper provides an exploratory analysis, including several possible clustering results, of a dataset provided by the Regional Transit Service of Rochester, NY. The dataset describes ridership on public buses, recording the time, location, and number of entering and exiting passengers each time a bus stops.  A description

of the overall behavior of bus ridership is followed by a stop-level analysis.  We compare multiple measures of stop similarity, based on location, route information, and ridership volume over time.

Automated Parameter Blocking for Efficient Markov Chain Monte Carlo Sampling

Daniel Turek, Perry de Valpine, Christopher Paciorek and Clifford Anderson-Bergman

Bayesian Analysis, 12, 2, 465-490, 2017

We propose an automated procedure to determine an efficient MCMC block-sampling algorithm for a given model and computing platform. Our procedure dynamically determines blocks of parameters for joint sampling that result in efficient MCMC sampling of the entire model.

Programming with Models: Writing Statistical Algorithms for General Model Structures with NIMBLE

Perry de Valpine, Daniel Turek, Christopher Paciorek, Clifford Anderson-Bergman, Duncan Temple Lang and Rastislav Bodik

Journal of Computational and Graphical Statistics, 26, 2, 403-413, 2017

We describe NIMBLE, a system for programming statistical algorithms for general model structures within R. NIMBLE is designed to meet three challenges: flexible model specification, a language for programming algorithms that can use different models, and a balance between high-level programmability and execution efficiency.

A Pi Day Carol

Colin Adams

Mathematical Intelligencer, Vol. 37, No. 3, 22-25, 2015.


Colin Adams

Mathematical Intelligencer, Vol. 37, No. 4, 2015.

What is…A Laminated Deck Transformation?

Colin Adams

Mathematical Intelligencer, Vol. 38, No. 1, 32-33, 2016.

The Galactic Math Repository

Colin Adams

Mathematical Intelligencer, Vol. 38, No. 2, 1-3, 2016.

Bipyramids and Bounds on Volumes of Hyperbolic Links

Colin Adams

ArXiv, 1511.02372, 2015.

Volume and Determinant Density of Hyperbolic Rational Links

Colin Adams, Aaron Calderon, Xinyi Jiang, Alexander Kastner ’17, Gregory Kehne ’16, Nathaniel Mayer ’16, Mia Smith ‘16

ArXiv, 1510.06050, 2015.

Generalized Bipyramids and Hyperbolic Volumes of Tiling Links

Colin Adams, Aaron Calderon, Xinyi Jiang, Alexander Kastner ’17, Gregory Kehne ’16, Nathaniel Mayer ’16, Mia Smith ‘16

ArXiv, 1603.03715, 2016.

Using Age-Stratified Incidence Data to Examine the Transmission Consequences of Pertussis Vaccination
Julie Blackwood with DAT Cummings, S. Iamsirithaworn, and P. Rohani

Epidemics 16, pp. 1–7, 2016.

Business Statistics A First Course, 3rd Edition
Richard De Veaux with Norean Sharpe and Paul Velleman

Pearson, January 2016.

Teaching Statistics Algorithmically or Stochastically Misses the Point: Why Not Teach Holistically?
Richard De Veaux (with Paul F. Velleman, in response to George Cobb’s Mere Renovation is too Little Too Late: We Need to Rethink Our Undergraduate Curriculum from the Ground Up).

American Statistician, 69, 4, 262–282, 2015.

Review of Steven Weintraub’s Differential Forms: Theory and Practice

Thomas Garrity

American Mathematical Monthly, Vol. 121, No. 4, 407-412, 2016.

A review of Weintraub’s text and a discussion of how to teach differential forms.

Statistics: The Art and Science of Learning From Data

Bernhard Klingenberg with Agresti, A., Franklin, C.

Pearson, 2017.

Treating Small Bowell Obstruction With a Manual Physical Therapy: A Prospective Efficacy Study

Bernhard Klingenberg with Rice, Patterson, Reed, Wurn, King, Wurn

BioMed Research International, Article ID 7610387, dio:10.1155/2016/7610387, 2016.

Formal Fibers With Countably Many Maximal Elements

Susan Loepp (with D. Aiello ’09 and P. Vu ’11)

Rocky Mountain Journal of Mathematics, 45, no. 2, 371-388, 2015.

Let T be a complete local ring and G a set of prime ideals of T with countably many maximal elements. We find necessary and sufficient conditions for T to be the completion of a local integral domain whose generic formal fiber is exactly G. In addition, if n is a positive integer, we construct integral domains with a prime ideal of height n whose formal fiber has countably many maximal elements.

Controlling the Generic Formal Fibers of Local Domains and Their Polynomial Rings

Susan Loepp (with P. Jiang, A. Kirkpatrick, S. Mack-Crane, S. Tripp ‘14)

Commutative Algebra, 7, no. 2, 241-264, 2015.

Let T be a complete local ring, C a countable set of incomparable prime ideals of T, and B and D sets of prime ideals of the power series ring over T in n variables such that the cardinality of B and D is less than that of T. We find necessary and sufficient conditions for T to be the completion of an integral domain A such that the generic formal fiber of A has maximal elements equal to C and the generic formal fiber of the polynomial ring over A in n variables contains every element of B and no elements of D. If T has characteristic 0, we find necessary and sufficient conditions for the A above to be excellent.

Completions of Reduced Local Reduced With Prescribed Minimal Prime Ideals

Susan Loepp (with B. Perpetua ‘14)

Involve, 9, no. 1, 101-118, 2016.

Let T be a complete local ring of dimension at least one, and let C1,C2,…,Cm each be countable sets of prime ideals of T. We find necessary and sufficient conditions for T to be the completion of a reduced local ring A such that A has exactly m minimal prime ideals Q1,Q2,…,Qm, and such that, for every i = 1,2,…,m, the set of maximal elements of the formal fiber of Qi is the set Ci.

Limiting Spectral Measures for Random Matrix Ensembles With a Polynomial Link Function

Steven Miller (with Kirk Swanson, Kimsy Tor and Karl Winsor)

Random Matrices: Theory and Applications 4, no. 2, 1550004 (28 pages).

Consider the ensembles of real symmetric Toeplitz matrices and real symmetric Hankel matrices whose entries are i.i.d. random variables chosen from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous work on real symmetric Toeplitz matrices shows that the spectral measures, or densities of normalized eigenvalues, converge almost surely to a universal near-Gaussian distribution, while previous work on real symmetric Hankel matrices shows that the spectral measures converge almost surely to a universal non-unimodal distribution. Real symmetric Toeplitz matrices are constant along the diagonals, while real symmetric Hankel matrices are constant along the skew diagonals. We generalize the Toeplitz and Hankel matrices to study matrices that are constant along some curve described by a real-valued bivariate polynomial. Using the Method of Moments and an analysis of the resulting Diophantine equations, we show that the spectral measures associated with linear bivariate polynomials converge in probability and almost surely to universal non-semicircular distributions. We prove that these limiting distributions approach the semicircle in the limit of large values of the polynomial coefficients. We then prove that the spectral measures associated with the sum or difference of any two real-valued polynomials with different degrees converge in probability and almost surely to a universal semicircular distribution.

Relieving and Readjusting Pythagoras

Steven Miller with Victor Luo ‘15

By the Numbers – The Newsletter of the SABR Statistical Analysis Committee, 25, no. 1, 5-14, 2015.

Bill James invented the Pythagorean expectation in the late 70’s to predict a baseball team’s winning percentage knowing just their runs scored and allowed. His original formula estimates a winning percentage of RS^2 /(RS^2+RA^2), where RS stands for runs scored and RA for runs allowed; later versions found better agreement with data by replacing the exponent 2 with numbers near 1.83. Miller and his colleagues provided a theoretical justification by modeling runs scored and allowed by independent Weibull distributions. They showed that a single Weibull distribution did a very good job of describing runs scored and allowed, and led to a predicted won-loss percentage of (RSobs−1/2)^γ / ((RSobs−1/2)^γ + (RAobs−1/2)^γ), where RSobs and RAobs are the observed runs scored and allowed and γ is the shape parameter of the Weibull (typically close to 1.8). We show a linear combination of Weibulls more accurately determines a team’s run production and increases the prediction accuracy of a team’s winning percentage by an average of about 25% (thus while the currently used variants of the original predictor are accurate to about four games a season, the new combination is accurate to about three). The new formula is more involved computationally; however, it can be easily computed on a laptop in a matter of minutes from publicly available season data. It performs as well (or slightly better) than the related Pythagorean formulas in use, and has the additional advantage of having a theoretical justification for its parameter values (and not just an optimization of parameters to minimize prediction error).

Equipartitions and a Distribution for Numbers: A Statistical Model for Benford’s Law

Steven Miller (with Joe Iafrate ‘14 and Frederick Strauch)

Physical Review E91, no. 6, 062138 (6 pages), 2015.

A statistical model for the fragmentation of a conserved quantity is analyzed, using the principle of maximum entropy and the theory of partitions. Upper and lower bounds for the restricted partitioning problem are derived and applied to the distribution of fragments. The resulting power law directly leads to Benford’s law for the first digits of the parts.

Sets Characterized by Missing Sums and Differences in Dilating Polytopes

Steven Miller (with Thao Do, Archit Kulkarni, David Moon ’16, Jake Wellens and James Wilcox ’14)

Journal of Number Theory, 157, 123–153, 2015.

A sum-dominant set is a finite set A of integers such that |A+A| > |A-A|. As a typical pair of elements contributes one sum and two differences, we expect sum-dominant sets to be rare in some sense. In 2006, however, Martin and O’Bryant showed that the proportion of sum-dominant subsets of {0,…,n} is bounded below by a positive constant as ntoinfty. Hegarty then extended their work and showed that for any prescribed s,d in N_0, the proportion rho^{s,d}_n of subsets of {0,…,n} that are missing exactly s sums in {0,…,2n} and exactly 2d differences in {-n,…,n} also remains positive in the limit. We consider the following question: are such sets, characterized by their sums and differences, similarly ubiquitous in higher dimensional spaces? We generalize the integers in a growing interval to the lattice points in a dilating polytope. Specifically, let P be a polytope in R^D with vertices in Z^D, and let rho_n^{s,d} now denote the proportion of subsets of L(nP) that are missing exactly s sums in L(nP)+L(nP) and exactly 2d differences in L(nP)-L(nP). As it turns out, the geometry of P has a significant effect on the limiting behavior of rho_n^{s,d}. We define a geometric characteristic of polytopes called local point symmetry, and show that rho_n^{s,d} is bounded below by a positive constant as n -> infinity if and only if P is locally point symmetric. We further show that the proportion of subsets in L(nP) that are missing exactly s sums and at least 2d differences remains positive in the limit, independent of the geometry of P. A direct corollary of these results is that if P is additionally point symmetric, the proportion of sum-dominant subsets of L(nP) also remains positive in the limit.

Newman’s Conjecture in Function Fields

Steven Miller (with Alan Chang, David Mehrle, Tomer Reiter, Joseph Stahl ad Dylan Yott)

Journal of Number Theory, 157, pages 154–169, 2015.

De Bruijn and Newman introduced a deformation of the completed Riemann zeta function zeta, and proved there is a real constant Lambda which encodes the movement of the nontrivial zeros of zeta under the deformation. The Riemann hypothesis is equivalent to the assertion that Lambda <= 0. Newman, however, conjectured that Lambda >= 0, remarking, “the new conjecture is a quantitative version of the dictum that the Riemann hypothesis, if true, is only barely so.” Andrade, Chang and Miller extended the machinery developed by Newman and Polya to L-functions for function fields. In this setting we must consider a modified Newman’s conjecture: sup_{f in F} Lambda_f >= 0, for F a family of L-functions. We extend their results by proving this modified Newman’s conjecture for several families of L-functions. In contrast with previous work, we are able to exhibit specific L-functions for which Lambda_D = 0, and thereby prove a stronger statement: max_{L in F} Lambda_L = 0. Using geometric techniques, we show a certain deformed L-function must have a double root, which implies Lambda = 0. For a different family, we construct particular elliptic curves with p + 1 points over {F}_p. By the Weil conjectures, this has either the maximum or minimum possible number of points over F_{p^{2n}}. The fact that #E(F_{p^{2n}}) attains the bound tells us that the associated L-function satisfies Lambda = 0.

Distribution of Eigenvalues of Weighted, Structured Matrix Ensembles

Steven Miller (with Olivia Beckwith, Victor Luo ’15, Karen Shen and Nicholas Triantafillou)

Integers: Electronic Journal of Combinatorial Number Theory 15, paper A21, 28 pages, 2015.

The study of the limiting distribution of eigenvalues of N × N random matrices as N → ∞ has many applications, including nuclear physics, number theory and network theory. One of the most studied ensembles is that of real symmetric matrices with independent entries drawn from identically distributed nice random variables, where the limiting rescaled spectral measure is the semi-circle. Studies have also determined the limiting rescaled spectral measures for many structured ensembles, such as Toeplitz and circulant matrices. These systems have very different behavior; the limiting rescaled spectral measures for both have unbounded support. Given a structured ensemble such that (i) each random variable occurs o(N) times in each row of matrices in the ensemble and (ii) the limiting rescaled spectral measure µe exists, we introduce a parameter to continuously interpolate between these two behaviors. We fix a p ∈ [1/2, 1] and study the ensemble of signed structured matrices by multiplying the (i, j)-th and (j, i)-th entries of a matrix by a randomly chosen epsilon_ij ∈ {1, −1}, with Prob(epsilon_ij = 1) = p (i.e., the Hadamard product). For p = 1/2 we prove that the limiting signed rescaled spectral measure is the semi-circle. For all other p, we prove the limiting measure has bounded (resp., unbounded) support if µ has bounded (resp., unbounded) support, and converges to µ as p → 1. Notably, these results hold for Toeplitz and circulant matrix ensembles. The proofs are by Markov’s Method of Moments. The analysis of the 2k-th moment for such distributions involves the pairings of 2k vertices on a circle. The contribution of each pairing in the signed case is weighted by a factor depending on p and the number of vertices involved in at least one crossing. These numbers are of interest in their own right, appearing in problems in combinatorics and knot theory. The number of configurations with no vertices involved in a crossing is well-studied, and are the Catalan numbers. We discover and prove similar formulas for configurations with 4, 6, 8 and 10 vertices in at least one crossing. We derive a closed-form expression for the expected value and determine the asymptotics for the variance for the number of vertices in at least one crossing. As the variance converges to 4, these results allow us to deduce properties of the limiting measure.

The Weibull Distribution of Benford’s Law

Steven Miller (with Victoria Cuff and Allie Lewis)

Involve, A Journal of Mathematics 38 – 5, pages 859–874, DOI 10.2140/involve.2015.8.859.

Benford’s law states that many data sets have a bias towardslower leading digits (about 30% are 1s). There are numerous applications, from designing efficient computers to detecting tax, voter and image fraud. It’s important to know which common probability distributions are almost Benford. We show the Weibull distribution, for many values of its parameters, is close to Benford’s law, quantifying the deviations. As the Weibull distribution arises in many problems, especially survival analysis, our results provide additional arguments for the prevalence of Benford behavior. The proof is by Poisson summation, a powerful technique to attack such problems.

Leading Digit Laws on Linear Lie Groups

Steven Miller (with Corey Manack)

Research in Number Theory 1:22, DOI 10.1007/s40993-015-0024-4, 2015.

We study the leading digit laws for the matrix entries of a linear Lie group G. For non-compact G, these laws generalize the following observations: (1) the normalized Haar measure of the Lie group R + is dx/x and (2) the scale invariance of dx/x implies the distribution of the digits follow Benford’s law. Viewing this scale invariance as left invariance of Haar measure, we see either Benford or power law behavior in the significands from one matrix entry of various such G. When G is compact, the leading digit laws we obtain come as a consequence of digit laws for a fixed number of components of a unit sphere. The sequence of digit laws for the unit sphere exhibits periodic behavior as the dimension tends to infinity.

Maass Waveforms and Low-Lying Zeros

Steven Miller with Levent Alpoge, Nadime Amersi, Geoffrey Iyer, Oleg Lazarev and Liyang Zhang ’15

Analytic Number Theory: In Honor of Helmut Maier’s 60th Birthday, Springer-Verlag, 2015.

The Katz-Sarnak Density Conjecture states that the behavior of zeros of a family of L-functions near the central point (as the conductors tend to zero) agrees with the behavior of eigenvalues near 1 of a classical compact group (as the matrix size tends to infinity). Using the Petersson formula, Iwaniec, Luo and Sarnak proved that the behavior of zeros near the central point of holomorphic cusp forms agrees with the behavior of eigenvalues of orthogonal matrices for suitably restricted test functions φ. We prove similar results for families of cuspidal Maass forms, the other natural family of GL2/Q L-functions. For suitable weight functions on the space of Maass forms, the limiting behavior agrees with the expected orthogonal group. We prove this for supp(φ^) ⊆ (−3/2,3/2) when the level N tends to infinity through the square-free numbers; if the level is fixed the support decreases to being contained in (−1,1), though we still uniquely specify the symmetry type by computing the 2-level density.

Determining Optimal Test Functions for Bounding Average Rank in Families of L-Functions

Steven Miller with Jesse Freeman ’15

Proceedings for Ram Murty’s 60th Birthday, Contemporary Math. Series of AMS (jointly with CRM) 2015.

Given an L-function, one of the most important questions concerns its vanishing at the central point; for example, the Birch and Swinnerton-Dyer conjecture states that the order of vanishing there of an elliptic curve L-function equals the rank of the Mordell-Weil group. The Katz and Sarnak Density Conjecture states that this and other behavior is well-modeled by random matrix ensembles. This correspondence is known for many families when the test functions are suitably restricted. For appropriate choices, we obtain bounds on the average order of vanishing at the central point in families. In this note we report on progress in determining the optimal test functions for the various classical compact groups for different support restrictions, and discuss how this relates to improved rank bounds.

Senior Editor for Theory and Applications of Benford’s Law

Steven Miller

Princeton University Press, 2015.

One of the greatest beauties in mathematics is how the same equations can describe phenomena in widely different fields. Benfords Law of digit bias is an outstanding example of this. Briefly, it asserts that for many natural data sets we are more likely to see numbers with small leading digits than large ones. Our purposes here are to show students and researchers useful techniques from a variety of subjects, highlight the connections between the different areas and encourage research and cross-departmental collaboration on these problems. To do this, we develop much of the general theory in the first few chapters (concentrating on the methods which are applicable to a variety of problems), and then conclude with numerous chapters on applications written by world-experts in that field.

Gaussian Distribution of the Number of Summands in Generalized Zeckendorf Decompositions in Small Intervals

Steven Miller with Andrew Best ’15, Patrick Dynes, Xixi Edelsbrunner ’16, Brian McDonald ’15, Kimsy Tor, Caroline Turnage-Butterbaugh and Madeleine Weinstein

Integers 16, #A6, 2015.

Zeckendorf’s theorem states that every positive integer can be written uniquely as a sum of non-consecutive Fibonacci numbers {F_n}, with initial terms F_1 = 1, F_2 = 2. Previous work proved that as n \to \infty the distribution of the number of summands in the Zeckendorf decompositions of m in [F_n, F_{n+1}), appropriately normalized, converges to the standard normal. The proofs crucially used the fact that all integers in [F_n, F_{n+1}) share the same potential summands, and hold for more general positive linear recurrence sequences {G_n}. We generalize these results to subintervals of [G_n, G_{n+1}) as n \to \infty for certain sequences. The analysis is significantly more involved here as different integers have different sets of potential summands. Explicitly, fix an integer sequence alpha(n) to infinity. As n to infinity, for almost all m in [G_n, G_{n+1}) the distribution of the number of summands in the generalized Zeckendorf decompositions of integers in the subintervals [m, m + G_{alpha(n)}), appropriately normalized, converges to the standard normal. The proof follows by showing that, with probability tending to 1, m has at least one appropriately located large gap between indices in its decomposition. We then use a correspondence between this interval and [0, G_{alpha(n)}) to obtain the result, since the summands are known to have Gaussian behavior in the latter interval.

Are Circles Isoperimetric in the Plane With Density er?

Frank Morgan with Ping Ngai Chung, Miguel A. Fernandez, Niralee Shah ’14, and Luis Sordo Vieira

Rose-Hulman Und. Math. J., 16, No. 1, 2015.

Morgan’s 2011 NSF SMALL undergraduate research Geometry Group gives evidence that circles about the origin are isoperimetric in the plane with density er.

Isoperimetric Problem on the Plane with Density e-1/r

Frank Morgan with Paul Gallagher, David Hu, Zane Martin ’13, Maggie Miller and Byron Perpetua ‘15

Rose-Hulman Und. Math. J., 15, No. 2, 2015.

Morgan’s 2012/13 NSF SMALL undergraduate research Geometry Groups provide numerical and theoretical evidence that isoperimetric curves the plane with density e–1/r have an angle at the origin approaching 1 radian from above as area approaches zero and provide further estimates.

Perimeter-Minimizing Tilings by Convex and Nonconvex Pentagons

Frank Morgan with Whan Ghang, Zane Martin ’13 and Steven Warahui

Rose-Hulman Und. Math. J., 16, No. 1, 2015.

Morgan’s 2012 NSF SMALL undergraduate research Geometry Group studies the presumably unnecessary convexity hypothesis in the theorem of Chung et al. on perimeter-minimizing planar tilings by convex pentagons.

Geometric Measure Theory: A Beginner’s Guide, 5th Edition

Frank Morgan

Academic Press, 2016.

An easy-going, illustrated introduction for the newcomer to this somewhat technical field. The fifth edition provides comprehensive updates and a new chapter on the Log Convex Density Theorem, a major new result in an area of mathematics—manifolds with density—that has exploded since its appearance in Perelman’s proof of the Poincaré conjecture.

Six Milestones in Geometry

Frank Morgan, Stephen F. Kennedy, Editor

A Century of Advancing Mathematics, Math. Assn. Amer., 51–64, 2015.

My choices for the six biggest advances in geometry during the 100-year lifetime of the MAA.

Unsolved Mathematical Mysteries

Frank Morgan

Virginia Math. Teacher, 42, Vol. 1, 1–20, Fall 2015.

Write-up of talk on “Soap Bubbles and Mathematics” at the Connecting Mathematical Practices Conference at Radford University, May 8, 2015, including some open questions about soap bubble clusters.

Soap Bubbles and Mathematics

Frank Morgan

Eur. Math. Soc. Newsletter, 32–36, September 2015.

Write-up of Abel Science Lecture, May 20, 2015, Oslo.

The Inferiorities of MacMail

Frank Morgan

Huffington Post Blog, September 2015.

Problems with the Mac mail application.

Sphere Packing in Dimension 8

Frank Morgan

Huffington Post Blog, March 2016.

A brief account of this breakthrough in sphere packing by a woman Ukranian mathematician.

On v-Positive Type Transformations in Infinite Measure

Cesar E. Silva with Tudor Padurariu and Evangelie Zachoes

Mathematics Colloquium, 140, 149-170, 2015.

For each vector v we define the notion of a v-positive type for infinite measure-preserving transformations, a refinement of positive type as introduced by Hajian and Kakutani. We prove that a positive type transformation need not be (1,2)-positive type. We study this notion in the context of Markov shifts and multiple recurrence and give several examples.

On Infinite Transformations With Maximal Control of Ergodic Two-Fold Product Powers

Cesar E. Silva with T.A. Adams

Israel Journal of Mathematics, 209, 929-948, 2015.

We study the rich behavior of ergodicity and conservativity of Cartesian products of infinite measure preserving transformations. A class of transformations is constructed such that for any subset R of rationals in (0,1) there exists T in this class such that T^p x T^q is ergodic if and only if p/q is in R. This contrasts with the finite measure preserving case where T^p x T^q is ergodic for all nonzero p and q if and only if T x T is ergodic. We also show that our class is rich in the behavior of conservative products.

For each positive integer k, a family of rank-one infinite measure preserving transformations is constructed which have ergodic index k, but infinite conservative index.

Ergodicity and Conservativity of Products of Infinite Transformations and Their Inverses

Cesar E. Silva with Julien Clancy, Rina Friedberg, Isaac Loh ’15, Indraneel Kalsmarka, Sahana Vasudevan

Colloq. Math., 143, 271-291, 2016.

We construct a class of rank-one infinite measure-preserving transformations such that for each transformation T in the class, the cartesian product T x T of the transformation with itself is ergodic, but the product T x T^{-1} of the transformation with its inverse is not ergodic. We also prove that the product of any rank-one transformation with its inverse is conservative, while there are infinite measure-preserving conservative ergodic Markov shifts whose product with their inverse is not conservative.

Explicit Bounds for the Pseudospectra of Various Classes of Matrices and Operators

Mihai Stoiciu with Feixue Gong ’16, Olivia Meyerson ’16, Jeremy Meza, and Abigail Ward

Involve, A Journal of Mathematics, 9, no. 3, 517-540, 2016.

We study the Î-pseudospectra sÎ(A) of square matrices A Î CNxN. We give a complete characterization of the Î-pseudospectra of 2 x 2 matrices and describe the asymptotic behavior (as Î ® 0) of sÎ(A) for every square matrix A. We also present explicit upper and lower bounds for the Î-pseudospectra of bidiagonal matrices, as well as for finite rank operators.

Knot Projections with a Single Multi-Crossing

Colin Adams with Thomas Crawford, Benjamin Demeo ’15, Michael Landry, MurphyKate Montee, Seojung Park, Saraswathi Venkatesh, Farrah Yhee

Journal of Knot Theory and Its Ramifications, Vol. 24, No. 3, 1550011(30 pages), 2015.

Bounds on Ubercrossing and Petal Number for Knots

C. Adams, O. Capovilla-Searle, J. Freeman ’15, D. Irvine, S. Petti ’15, D. Vitek, A. Weber, S. Zhang

Journal of Knot Theory and Its Ramifications, Vol. 24, No. 2, 1550012 (16 pages), 2015.

The End of Mathematics

Colin Adams

Mathematical Intelligencer, Vol. 36, No. 3, 20-212, 2014.

Motivational Seminar

Colin Adams

Mathematical Intelligencer, Vol. 36, No. 4, 19-21, 2014.

Zombies and Calculus: An Excerpt

Colin Adams

Mathematical Intelligencer, Vol. 37, No. 1, 78-82, 2015.

Zombies and Calculus

Colin Adams

Princeton University Press, October 2014.

Calculus, 3rd Edition

Colin Adams with Jon Rogawski

W.H. Freeman, 36, December 2014.

Zombies and Calculus I and II

Colin Adams with Jon Rogawski

NOVA, produced by Ari Daniel, September 2014

Navigation in Tree Spaces

Satyan Devadoss with J. Morava

Advances in Applied Mathematics, 67, 75-95, 2015.

The orientable cover of the moduli space of real genus zero algebraic curves with marked points is a compact aspherical manifold tiled by associahedra, which resolves the singularities of the space of phylogenetic trees. The resolution maps planar metric trees to their underlying abstract representatives, collapsing and folding an explicit geometric decomposition of the moduli space into cubes. This decomposition endows the resolving space with an interesting canonical pseudometric.

Convex Polytopes From Nested Posets

Satyan Devadoss with S. Forcey, S. Reisdorf, P. Showers

European Journal of Combinatorics, 43, 229-248, 2015.

Motivated by the graph associahedron KG, a polytope whose face poset is based on connected subgraphs of G, we consider the notion of associativity and tubes on posets. This leads to a new family of simple convex polytopes obtained by iterated truncations. These generalize graph associahedra and nestohedra, even encompassing notions of nestings on CW-complexes, but fall in a different category altogether than generalized permutohedra.

Polyhedral Covers of Tree Space

Satyan Devadoss with D. Huang, D. Spadacene

SIAM Journal of Discrete Mathematics, 28, 1508-1514, 2014.

We construct the space of phylogenetic trees from local gluings of classical polytopes, the associahedron and the permutohedron. Its homotopy is also reinterpreted and calculated based on polytope data.

Stats: Data and Models, 4th Edition
Richard De Veaux with Paul Velleman and David Bock

Pearson, January 2015.

A Generalized Family of Multidimensional Continued Fractions: TRIP Maps

Thomas Garrity, with Krishna Dasaratha, Laure Flapan, Chansoo Lee ‘12, Cornelia Mihaila, Nicholas Neumann-Chun ’14, Sarah Peluse, Matt Stoffregen

International Journal of Number Theory, Vol. 10, 2151, 2014.

Most well-known multidimensional continued fractions, including the M\”{o}nkemeyer map and the triangle map, are generated by repeatedly subdividing triangles. This paper constructs a family of multidimensional continued fractions by permuting the vertices of these triangles before and after each subdivision. We obtain an even larger class of multidimensional continued fractions by composing the maps in the family. These include the algorithms of Brun, Parry-Daniels and G\”{u}ting. We give criteria for when multidimensional continued fractions associate sequences to unique points, which allows us to determine when periodicity of the corresponding multidimensional continued fraction corresponds to pairs of real numbers being cubic irrationals in the same number field.

Cubic Irrationals and Periodicity Via a Family of Multi-Dimensional Continued Fraction Algorithms

Thomas Garrity, with Krishna Dasaratha, Laure Flapan, Chansoo Lee ‘12, Cornelia Mihaila, Nicholas Neumann-Chun ’14, Sarah Peluse, Matt Stoffregen

Monatshefte für Mathematik, Vol. 174, Issue 4, 549-566, August 2014.

We construct a countable family of multi-dimensional continued fraction algorithms, built out of five specific multidimensional continued fractions, and show a real number is a cubic irrational precisely when its multidimensional continued fraction expansion with respect to at least one element of the countable family is eventually periodic. We interpret this result as the construction of a matrix with entries of non-negative integers such that at least one of the rows is eventually periodic if and only if the chosen real is a cubic irrational. This result is built on a careful technical analysis of certain units in cubic number fields and our family of multi-dimensional continued fractions.

Review of Spherical Tube Hypersurfaces, by Alexander Isaev

Thomas Garrity

Bulletin of the American Mathematical Society, Vol. 51, No. 4, 675-685, 2014.

A review of Isaev’s text and an overview of CR geometry.

Electricity and Magnetism for Mathematicians: A Guided Path from Maxwell’s Equations to Yang-Mills

Thomas Garrity

Cambridge University Press, 2015.

This text is an introduction to some of the mathematical wonders of Maxwell’s equations. These equations led to the prediction of radio waves, the realization that light is a type of electromagnetic wave, and the discovery of the special theory of relativity. In fact, almost all current descriptions of the fundamental laws of the universe can be viewed as deep generalizations of Maxwell’s equations. Even more surprising is that these equations and their generalizations have led to some of the most important mathematical discoveries of the past thirty years. It seems that the mathematics behind Maxwell’s equations is endless. The goal of this book is to explain to mathematicians the underlying physics behind electricity and magnetism and to show their connections to mathematics. Starting with Maxwell’s equations, the reader is led to such topics as the special theory of relativity, differential forms, quantum mechanics, manifolds, tangent bundles, connections, and curvature.

Multiple Comparisons of Marginal Probabilities Following GEE Estimation

Bernhard Klingenberg

Proceedings of the 30th International Workshop on Statistical Modelling, Linz 2015.

Newman’s conjecture in various settings

Steven Miller (with Julio Andrade and Alan Chang)

Journal of Number Theory 144, 70-91, 2014.

We explicitly determine the Newman constants in various function field settings, which has strong implications for Newman’s quantitative version of RH.

A Message From Professor Steven Miller, A SLICE OF Pi

Steven Miller

Palmdale High School Math Department News Letter, volume 2, issue 5, pages 1 and 7, November 2014.

Discussion of Math Riddles for High School students.

Generalized Ramanujan Primes

Steven Miller (with Nadine Amersi, Olivia Beckwith, Ryan Ronan and Jonathan Sondow)

Combinatorial and Additive Number Theory, CANT 2011 and 2012 (Melvyn B. Nathanson, editor), Springer Proceedings in Mathematics & Statistics, 1—13, 2014.

We generalize the notion of Ramanujan primes and prove many properties, and conjecture others.

Finding and Counting MSTD Sets

Steven Miller (with Geoffrey Iyer, Oleg Lazarev and Liyang Zhang)

Combinatorial and Additive Number Theory, CANT 2011 and 2012 (Melvyn B. Nathanson, editor), Springer Proceedings in Mathematics & Statistics, 79-98, 2014.

We give new constructions and results of generalized MSTD sets, including among other items results on a positive percentage of sets having a given linear combination greater than another linear combination, and a proof that a positive percentage of sets are k-generational sum-dominant.

Most Subsets are Balanced in Finite Groups

Steven Miller (with Kevin Vissuet)

Combinatorial and Additive Number Theory, CANT 2011 and 2012 (Melvyn B. Nathanson, editor), Springer Proceedings in Mathematics & Statistics, 147-157, 2014.

We show that if we take subsets of larger and larger finite groups uniformly at random, then not only does the probability of a set being sum-dominated tend to zero but the probability that |A + A| = |A – A| tends to one, and hence a typical set is balanced in this case.

Gaussian Behavior in Generalized Zeckendorf Decompositions

Steven Miller (with Yinghui Wang)

Combinatorial and Additive Number Theory, CANT 2011 and 2012 (Melvyn B. Nathanson, editor), Springer Proceedings in Mathematics & Statistics, 159-173, 2014.

We prove Gaussian behavior for generalized Zeckendorf decompositions arising from certain recurrences.

The Expected Eigenvalue Distribution of Large, Weighted d-regular Graphs

Steven Miller (with Leo Goldmakher, Cap Khoury and Kesinee Ninsuwan)

Random Matrices: Theory and Applications 3, no. 4, 1450015 (22 pages) 2014.

We compute the limiting spectral measure for weighted ensembles arising from d-regular graphs.

Pythagoras at the Bat

Steven Miller (with Taylor Corcoran, Jennifer Gossels ‘13, Victor Luo ‘14 and Jaclyn Porfilio ‘15)

Book chapter in Social Networks and the Economics of Sports (organized by Victor Zamaraev), Springer-Verlag, 2014.

This survey article describes some of the advances and successes in modeling and using the Pythagorean won-loss formula.

The n-Level Density of Dirichlet L-Functions over F_q[T]

Steven Miller (with Julio Andrade, Kyle Pratt and Minh-Tam Trinh)

Communications in Number Theory and Physics 8, no. 3, 1—29, 2014

We compute the n-level density of Dirichlet L-functions in the function field.

A Message From Professor Steven Miller, A SLICE OF Pi

Steven Miller

Palmdale High School Math Department News Letter, volume 2, issue 7, page 1 and 7 additional remarks online, December 2014

Discussion of Math Riddles for High School students.

Surpassing the Ratios Conjecture in the 1-level density of Dirichlet L-functions

Steven Miller (with Daniel Fiorilli)

Algebra & Number Theory Vol. 9, No. 1, 13—52, 2015

We compute the 1-level density for Dirichlet L-functions, computing lower order terms beyond square-root cancellation and showing how various conjectures influence the level of support computable.

The James Function

Steven Miller (with Christopher N. B. Hammond and Warren P. Johnson)

Mathematics Magazine 88, 54—71, 2015

We explore a set of natural conditions for a won-loss formula, and determine the class of functions which satisfy these.

Gaps Between Zeros of GL(2) L-Functions

Steven Miller (with Owen Barrett, Brian McDonald, Ryan Patrick, Caroline Turnage-Butterbaugh and Karl Winsor)

Journal of Mathematical Analysis and Applications 429, 204—232, 2015

We prove there are infinitely many zeros at least sqrt(3) times the average spacing for GL(2) L-functions, as well as similar results on gaps smaller than the average spacing.

Benford Behavior of Zeckendorf Decompositions

Steven Miller (with A. Best ‘15, P. Dynes, X. Edelsbrunner’16, B. McDonald, C. Turnage-Butterbaugh and M. Weinstein)

Fibonacci Quarterly 52 (2014), no. 5, 35—46, 2014.

We prove that the distribution of summands in Zeckendorf decompositions satisfy Benford’s law.

Gaussian Behavior of the Number of Summands in Zeckendorf Decompositions in Small Intervals

Steven Miller (with A. Best ‘15, P. Dynes, X. Edelsbrunner ‘16, B. McDonald C. Turnage-Butterbaugh and M. Weinstein)

Fibonacci Quarterly 52, no. 5, 47—53, 2014

We extend previous Gaussian results to smaller intervals.

Generalizing Zeckendorf’s Theorem: The Kentucky Sequence

Steven Miller (with M. Catral, P. Ford, P. Harris and D. Nelson)

Fibonacci Quarterly 52, no. 5, 68—90, 2014.

Previous studies on generalized Zeckendorf decompositions required the first coefficient in the recurrence relation to be a positive integer; we explore the consequences arising from a sequence with first term zero.

Benford’s Law: Theory and Applications

Steven Miller, Editor

Princeton University Press, 2015.

This is the first interdisciplinary book on Benford’s law, with 19 chapters on diverse areas such as accounting, auditing, economics, finance, gambling, imaging, medicine, natural sciences, psychology, statistics, and voting. I wrote the introductory chapter, a chapter on ‘Fourier Analysis and Benford’s Law’ (parts written with recent SMALL students Xixi Edelsbrunner ’16, Karen Huan’16, Blake Mackall ’16, Jasmine Powell and Madeleine Weinstein), and a chapter on ‘Application of Benfords Law to Images.’

Optimal City Hierarchy: A Dynamic Programming Approach to Central Place Theory

Thomas J. Holmes, Wen-Tai Hsu, and Frank Morgan

J. Econ. Theory 154, 245-273, 2014.

We place cities of various sizes on the line to minimize set-up and transportation costs, and we provide a dynamic programming solution. We show that there must be one and only one immediate smaller city between two neighboring larger-sized cities. Often the smaller city takes a “central place” halfway between the next larger cities, but not always.

Bubbles and Tilings: Art and Mathematics

Frank Morgan

Proc. Bridges, 2014.

The 2002 proof of the Double Bubble Conjecture on the ideal shape for a double soap bubble depended for its ideas and explanation on beautiful images of the multitudinous possibilities. Similarly recent results on ideal tilings depend on the artwork.

Academics Must Be Williams Top Priority

Frank Morgan

Williams Record, May 7, 2014.

Williams: Inclusive or Exclusive?

Frank Morgan

WilliamsAlternative.com, May 2014.

Arithmetic Properties of Generalized Rikuna Polynomials

Allison Pacelli with Z. Chonoles, H. Hausman ’12, S. Pegado ‘11, F. Wei

Mathematiques de Besancon: Algebre et Theorie des Nombres, 1, 19-33, 2014.

On Rationally Ergodic and Rationally Weakly Mixing Rank-One Transformations

Cesar E. Silva with Irving Dai, Xavier Garcia, and Tudor Padurariu

Ergodic Theory and Dynamical Systems, 35, No. 4, 1141-1154, 2015.

We study the notions of weak rational ergodicity and rational weak mixing as defined by Jon Aaronson in 1977 and 2012. We prove that various families of infinite measure-preserving rank-one transformations possess or do not posses these properties, and consider their relation to other notions of mixing in infinite measure.

On Li-Yorke Measurable Sensitivity

Cesar E. Silva with Lucas Manuelli and Jared Hallett ‘14

Amer. Math. Soc., 143, No. 6, 2411-2426, 2015.

The notion of Li-Yorke sensitivity has been studied extensively in the case of topological dynamical systems. We introduce a measurable version of Li-Yorke sensitivity, for nonsingular (and measure-preserving) dynamical systems, and compare it with various mixing notions. It is known that in the case of nonsingular dynamical systems, conservative ergodic Cartesian square implies double ergodicity, which in turn implies weak mixing, but the converses do not hold in general, though they are all equivalent in the finite measure-preserving case. We show that for nonsingular systems, ergodic Cartesian square implies Li-Yorke measurable sensitivity, which in turn implies weak mixing. As a consequence we obtain that, in the finite measure-preserving case, Li-Yorke measurable sensitivity is equivalent to weak mixing. We also show that with respect to totally bounded metrics, double ergodicity implies Li-Yorke measurable sensitivity.

Subsequence Bounded Rational Ergodicity of Rank-One Transformations

Cesar E. Silva with Francisc Bozgan, Anthony Sanchez, David Stevens ’14 and Jane Wang

Dynamical Systems, 30, No. 1, 70-84, 2015.

We show that all rank-one transformations are subsequence boundedly rationally ergodic and that there exist rank-one transformations that are not weakly rationally ergodic.

Spectral Properties of Random and Deterministic Matrices

Mihai Stoiciu

Math. Model. Nat. Phenom., Vol. 9, No. 5, 270-281, 2014.
The CMV matrices are unitary analogues of the discrete one-dimensional Schrödinger operators. We review spectral properties of a few classes of CMV matrices and describe families of random and deterministic CMV matrices which exhibit a transition in the distribution of their eigenvalues.

Improving Cross-Validated Bandwidth Selection Using Subsampling-Extrapolation Techniques

Qing Wang and Bruce Lindsay

Computational Statistica & Data Analysis 89, 51-71, 2015.
Cross-validation methodologies have been widely used as a means of selecting tuning parameters in nonparametric statistical problems. In this paper we focus on a new method for improving the reliability of cross-validation. We implement this method in the context of the kernel density estimator, where one needs to select the bandwidth parameter so as to minimize L2 risk. This method is a two-stage subsampling-extrapolation bandwidth selection procedure, which is realized by first evaluating the risk at a fictional sample size m (m <= sample size n) and then extrapolating the optimal bandwidth from m to n. This two-stage method can dramatically reduce the variability of the conventional unbiased cross-validation bandwidth selector. This simple first-order extrapolation estimator is equivalent to the rescaled “bagging-CV” bandwidth selector in Hall and Robinson (2009) if one sets the bootstrap size equal to the fictional sample size. However, our simplified expression for the risk estimator enables us to compute the aggregated risk without any bootstrapping. Furthermore, we developed a second-order extrapolation technique as an extension designed to improve the approximation of the true optimal bandwidth. To select the optimal choice of the fictional size m given a sample of size n, we propose a nested cross-validation methodology. Based on simulation study, the proposed new methods show promising performance across a wide selection of distributions. In addition, we also investigated the asymptotic properties of the proposed bandwidth selectors.

Robust Thresholding for Diffusion Index Forecast

Qing Wang and Vu Lee ‘14

Economics Letters 125, 52-56, 2014.
In this paper we propose a new methodology in improving the Diffusion Index forecasting model (Stock and Watson, 2002) using hard thresholding with robust KVB statistic for regression hypothesis tests (Kiefer, Vogelsang, and Bunzel, 2000). The new method yields promising results in the context of long forecasting horizons and existence of serial correlation. Numerical comparison indicates that the proposed methodology can improve upon the existing hard thresholding methods and outperform the soft thresholding methods (Bai and Ng, 2008) when applied to a real data set that forecasts eight macroeconomic variables in the United States.

A General Class of Linearly Extrapolated Variance Estimators

Qing Wang and Shiwen Chen ‘14

Statistics & Probability Letters 98, 29-38, 2015.
A general class of linearly extrapolated variance estimators was developed as an extension of the conventional leave-one-out jackknife variance estimator. In the context of U-statistic variance estimation, the proposed variance estimator is first-order unbiased. After showing the equivalence between the Hoeffding decomposition (Hoeffding, 1948) and the ANOVA decomposition (Efron and Stein, 1981), we study the bias property of the proposed variance estimator in comparison to the conventional jackknife method. Simulation studies indicate that the proposal has comparable performance to the jackknife method when assessing the variance of the sample variance in various distributions. An application to half-sampling cross-validation indicates that the proposal is more computationally efficient and shows better performance than its jackknife counterpart in the context of regression analysis.

Spanning Surfaces in Alternating Knot Complements

Colin Adams with Thomas Kindred ’07

Algebraic and Geometry Topology 13, 2967-3007, 2013.

Unknotting Tunnels, Bracelets and the Elder Sibling Property for Hyperbolic 3-Manifolds

Colin Adams with K. Knudson ’09

Journal of the Australian Mathematical Society 95, 1-19, 2013.

Quadruple Crossing Number of Knots and Link

Colin Adams

Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 156, No. 2, 241-253, 2014.

A Grader’s Dream

Colin Adams 

Mathematical Intelligencer, Vol. 35, No. 3, 34-35, 2013.

Math is Everywhere

Colin Adams 

Mathematical Intelligencer, Vol. 35, No. 4, 39-42, 2013.

The Pi Day Massacre

Colin Adams 

Mathematical Intelligencer, Vol. 36, No. 1, 61-63, 2014.

On-Line Edutainment

Colin Adams 

Mathematical Intelligencer, Vol. 36, No. 2, 15-17, 2014.

Resolving the Roles of Immunity, Pathogenesis and Immigration for Rabies Persistence in Vampire Bats

Julie C. Blackwood, DG Streicker, S. Altizer, and P. Rohani

Proceedings of the National Academy of Sciences 110, 9595-9600, 2013.

A More Sums Than Differences (MSTD) set is a set of integers A of {0, …, n-1} whose sumset A+A is larger than

Skeletal Configurations of Ribbon Trees

Satyan Devadoss with H. Cheng, B. Li, A. Risteski

Discrete Applied Mathematics 170, 46-54, 2014.

The straight skeleton construction creates a straight-line tree from a polygon. Motivated by moduli spaces from algebraic geometry, we consider the inverse problem of constructing a polygon whose straight skeleton is a given tree. We prove there exists only a finite set of planar embeddings of a tree appearing as straight skeletons of convex polygons. Computational issues are also considered, uncovering ties to a much older angle bisector problem.

Business Statistics, 3rd Edition

Richard De Veaux with Norean Sharpe and Paul Velleman

Pearson 22, 1350006-1 – 1350006–17, January 2014.

Follow the Fundamentals:  Four Data Analysis Basics Will Help You Do Big Data Projects the Right Way

Richard De Veaux with Ron Snee and R.W. Hoerl

Quality Progress 24-28, January 2014.

A Multi-dimensional Continued Fraction Generalization of Stern’s Diatomic SequenceFundamentals

Thomas Garrity

Journal of Integer Sequences Vol. 16, 2013.

Continued fractions are linked to Stern’s diatomic sequence 0,1,1,2,1,3,2,3,1,4,… (given by the recursion relation a{2n}=a(n)  and a(2n+1} = a(n) + a(n+1) where a(0)=0 and a(1)=1), which has long been known.  Using a particular multi-dimensional continued fraction algorithm (the Farey algorithm), we will generalize the diatomic sequence to a sequence of numbers that quite naturally should be called Stern’s triatomic sequence (or a two-dimensional Pascal’s sequence with memory).  As continued fractions and the diatomic sequence can be thought of as coming from a systematic subdivision of the unit interval, this new triatomic sequence will arise by a systematic subdivision of a triangle.  We will discuss some of the algebraic properties for the triatomic sequence.

Longitudinal Cluster Analysis with Applications to Growth Trajectories

Brianna Heggeseth

UC Berkeley Electronic Theses and Dissertations 1-130, 2013.

Longitudinal studies play a prominent role in health, social, and behavioral sciences as well as in the biological sciences, economics, and marketing. By following subjects over time, temporal changes in an outcome of interest can be directly observed and studied. An important question concerns the existence of distinct trajectory patterns. One way to discover potential patterns in the data is through cluster analysis, which seeks to separate objects (individuals, subjects, patients, observational units) into homogeneous groups. There are many ways to cluster multivariate data. Most methods can be categorized into one of two approaches: nonparametric and model-based methods. However, the bulk of the available clustering algorithms are not appropriate to be directly applied to repeated measures with inherent dependence and do not clustering according to change over time, the key feature measured by longitudinal data.

Multivariate Gaussian mixtures are a class of models that can be adapted to longitudinal data, but simplifying assumptions about the dependence structure are often made.  I study, through asymptotic bias calculations and simulation, the impact of covariance misspecification in multivariate Gaussian mixtures. Although maximum likelihood estimators of regression and prior probability parameters are not consistent under misspecification, they have little asymptotic bias when mixture components are well separated or if the assumed correlation is close to the truth even when the covariance is misspecified. I also present a robust standard error estimator and show that it outperforms conventional estimators in simulations and can provide evidence that the model is misspecified.

To fulfill the need for clustering based explicitly on shape, I propose three methods that are adaptations of available algorithms. One approach is to use a dissimilarity measure based on estimated derivatives of functions underlying the trajectories. One challenge for this approach is estimating the derivatives with minimal bias and variance. The second approach explicitly models the variability in the level within a group of similarly shaped trajectories using a mixture model resulting in a multilayer mixture model. One difficulty with this method comes in choosing the number of shape clusters. Lastly, vertically shifting the data by subtracting the subject-specific mean directly removes the level prior to modeling. This non-invertible transformation can result in singular covariance matrixes, which makes parameter estimation difficult. In theory, all of these methods should cluster based on shape, but each method has shortfalls. I compare these methods with existing clustering methods in a simulation study and apply them to a real data set of childhood growth trajectories from the Center for the Health Assessment of Mothers and Children of Salinas (CHAMACOS) study.

The Impact of Covariance Misspecification in Multivariate Gaussian Mixtures on Estimation and Inference:  An Application to Longitudinal Modeling

Brianna Heggeseth and Nicholas Jewell

Statistics in Medicine 32 (16) 2790-2803, 2013.

Multivariate Gaussian mixtures are a class of models that provide a flexible parametric approach for the representation of heterogeneous multivariate outcomes. When the outcome is a vector of repeated measurements taken on the same subject, there is often inherent dependence between observations. However, a common covariance assumption is conditional independence—that is, given the mixture component label, the outcomes for subjects are independent. In this paper, we study, through asymptotic bias calculations and simulation, the impact of covariance misspecification in multivariate Gaussian mixtures. Although maximum likelihood estimators of regression and mixing probability parameters are not consistent under misspecification, they have little asymptotic bias when mixture components are well separated or if the assumed correlation is close to the truth even when the covariance is misspecified. We also present a robust standard error estimator and show that it outperforms conventional estimators in simulations and can indicate that the model is misspecified. Body mass index data from a national longitudinal study are used to demonstrate the effects of misspecification on potential inferences made in practice.

Parity and Body Mass Index in U.S. Women:  A Prospective 25-Year Study

Brianna Heggeseth, Barbara Abrams, David Rehkopf, and Esa Davis

Obesity 21 (8), 1514-1518, 2013.

The objective of this paper was to investigate long-term body mass index (BMI) changes associated with childbearing. Adjusted mean BMI changes were estimated by race-ethnicity, baseline BMI, and parity using longitudinal regression models for 3,943 young females over 10 and 25 year follow-up from the ongoing 1979 National Longitudinal Survey of Youth cohort. Estimated BMI increases varied by group, ranging from a low of 2.1 BMI units for white, non-overweight nulliparas over the first 10 years to a high of 10.1 BMI units for black, overweight multiparas over the full 25-year follow-up. Impacts of parity were strongest among overweight multiparas and primaparas at 10 years, ranges 1.4-1.7 and 0.8-1.3 BMI units, respectively. Among non-overweight women, parity-related gain at 10 years varied by number of births among black and white but not Hispanic women. After 25 years, childbearing significantly increased BMI only among overweight multiparous black women. Childbearing is associated with permanent weight gain in some women, but the relationship differs by maternal BMI in young adulthood, number of births, race-ethnicity, and length of follow-up. Given that overweight black women may be at special risk for accumulation of permanent, long-term weight after childbearing, effective interventions for this group are particularly needed.

Adiponectin and Leptin Trajectories in Mexican-American Children from Birth to 9 Years of Age

Brianna Heggeseth, Vitaly Volberg, Kim Harley, Karen Huen, Paul Yousefi, Veronica Davé, Kristin Tyler, Michelle Vedar, Brenda Eskenazi and Nina Holland

PLoS ONE 8 (10):e77964, 2013.

The objective of this paper was to address molecular mechanisms underlying obesity development, we examined patterns of critical metabolism-related hormones, adiponectin and leptin (adipokines), over childhood. Plasma adiponectin and leptin were measured in 80 Mexican-American children at birth and again at 2, 5, and 9 years from the ongoing prospective cohort followed by the Center for the Health Assessment of Mothers and Children of Salinas (CHAMACOS). We used a mixture modeling approach to identify patterns in adipokine trajectories from birth to 9 years. Leptin was positively related to child body size within all ages, however adiponectin had inverse and weaker associations with BMI at 2, 5, and 9 years. Correlations between adipokine levels over the 0–2, 2–5, and 5–9-year periods increased for both leptin (r = 0.06, 0.31 and 0.62) and adiponectin (r = 0.25, 0.41 and 0.46). Our mixture modeling approach identified three trajectory clusters for both leptin (1L [slowly-rising], 2L [rapidly-rising], and 3L [stable]) and adiponectin (1A [steep-dropping and rebounding], 2A [moderately-dropping], and 3A [stable]). While leptin groups were most separated over the 2–9-year period, adiponectin trajectories displayed greatest heterogeneity from birth to 2 years. Children in the rapidly-rising 2L group had highest BMI and waist circumference at 9 years. Further, children with greater birth weight had increased odds of belonging to this high risk group (OR = 1.21 95% CI 1.03, 1.43, compared to stable group 3L). Children whose mothers consumed more sugar-sweetened beverages during pregnancy were at risk of being in the steep-dropping 1A group (OR = 1.08, 95% CI 1.01, 1.17, compared to stable group 3A). Our results highlight developmental differences in leptin and adiponectin over the childhood period. Leptin closely reflects child body size however factors affecting adiponectin and long-term consequences of its changes over infancy need to be further explored.

Development and Validation of a Questionnaire to Measure Serious and Common Quality of Life Issues for Patients Experiencing Small Bowel Obstructions

Bernhard Klingenberg, Amanda D. Rice, Leslie B. Wakefield, Kimberley Patterson, Evette D’Avy Reed, Belinda F. Wurn, C. Richard King, III and Lawrence J. Wurn

Healthcare 2, doi:10.3390/healthcare2010139, 139-149, 2014.

A validated questionnaire to assess the impact of small bowel obstructions (SBO) on patients’ quality of life was developed and validated. The questionnaire included measurements for the impact on the patients’ quality of life in respect to diet, pain, gastrointestinal symptoms and daily life. The questionnaire was validated using 149 normal subjects. Chronbach alpha was 0.86. Test retest reliability was evaluated with 72 normal subjects, the correlation coefficient was 0.93. Discriminate validity was determined to be significant using the normal subject questionnaires and 10 questionnaires from subjects with recurrent SBO. Normative and level of impact for each measured domain were established using one standard deviation from the mean in the normal population and clinical relevance. This questionnaire is a valid and reliable instrument to measure the impact of SBO on a patient’s quality of life related to recurrent SBOs; therefore establishing a mechanism to monitor and quantify changes in quality of life over time.

A New and Improved Confidence Interval for the Mantel-Haenszel Risk Difference

Bernhard Klingenberg

Statist. Med., doi:10.1002/sim.6122, 2014.

Writing the variance of the Mantel–Haenszel estimator under the null of homogeneity and inverting the corresponding test, we arrive at an improved confidence interval for the common risk difference in stratified 2 times 2 tables. This interval outperforms a variety of other intervals currently recommended in the literature and implemented in software. We also discuss a score-type confidence interval that allows to incorporate strata/study weights. Both of these intervals work very well under many scenarios common in stratified trials or in a metaanalysis, including situations with a mixture of both small and large strata sample sizes, unbalanced treatment allocation, or rare events. The new interval has the advantage that it is available in closed form with a simple formula. In addition, it applies to matched pairs data.We illustrate the methodology with various stratified clinical trials and a meta-analysis. R code to reproduce all analysis is provided in the Appendix.

Completions of Hypersurface Domains

Susan Loepp, J. Ahn ’12, E. Ferme, F. Jiang, and G. Tran

Communications in Algebra no. 12, 4491-4503, 2013.

Given a complete local ring T, the authors find necessary and sufficient conditions for there to exist a hypersurface domain whose completion is T.

The Low-Lying Zeros of Level 1 Maass Forms

Steven Miller with Levent Alpoge

Int Math Res Notices 24, pages doi:10.1093/imrn/rnu012, 2014.

The Katz-Sarnak density conjecture states that the scaling limits of the distributions of zeroes of families of automorphic L-functions agree with the scaling limits of eigenvalue distributions of classical subgroups of the unitary groups U(N). This conjecture is often tested by way of computing particular statistics, such as the one-level density, which evaluates a test function with compactly supported Fourier transform at normalized zeroes near the central point. Iwaniec, Luo, and Sarnak studied the one-level densities of cuspidal newforms of weight k and level N. They showed in the limit as kN tends to infinity that these families have one-level densities agreeing with orthogonal type for test functions with Fourier transform supported in (-2,2). Exceeding (-1,1) is important as the three orthogonal groups are indistinguishable for support up to (-1,1) but are distinguishable for any larger support. We study the other family of GL2 automorphic forms over Q: Maass forms. To facilitate the analysis, we use smooth weight functions in the Kuznetsov formula which, among other restrictions, vanish to order M at the origin. For test functions with Fourier transform supported inside (-2 + 3/(2(M+1)), 2 – 3/(2(M+1))), we unconditionally prove the one-level density of the low-lying zeros of level 1 Maass forms, as the eigenvalues tend to infinity, agrees only with that of the scaling limit of orthogonal matrices.

The n-Level Density of Zeros of Quadratic Dirichlet L-Functions

Steven Miller with Jake Levinson ‘11

Acta Arithmetica 161, 145-182, 2013.

Previous work by Rubinstein and Gao computed the n-level densities for families of quadratic Dirichlet L-functions for test functions where the sum of the supports is less than 2, and showed agreement with random matrix theory predictions in this range for n < 4 but only in a restricted range for larger n. We extend these results and show agreement for n < 8, and reduce higher n to a Fourier transform identity. The proof involves adopting a new combinatorial perspective to convert all terms to a canonical form, which facilitates the comparison of the two sides.

The Limiting Spectral Measure for Ensembles of Symmetric Block Circulant Matrices

Steven Miller with Murat Kologlu ‘12, Gene S. Kopp, Frederick W. Strauch, Associate Professor of Physics and Wentao Xiong ‘11

Journal of Theoretical Probability 26, no 4, 1020-1060, 2013.

Given an ensemble of N x N random matrices, a natural question to ask is whether or not the empirical spectral measures of typical matrices converge to a limiting spectral measure as N tends to infinity. While this has been proved for many thin patterned ensembles sitting inside all real symmetric matrices, frequently there is no nice closed form expression for the limiting measure. Further, current theorems provide few pictures of transitions between ensembles. We consider the ensemble of symmetric m-block circulant matrices with entries i.i.d.r.v. These matrices have toroidal diagonals periodic of period m. We view m as a “dial” we can “turn” from the thin ensemble of symmetric circulant matrices, whose limiting eigenvalue density is a Gaussian, to all real symmetric matrices, whose limiting eigenvalue density is a semi-circle. The limiting eigenvalue densities fm show a visually stunning convergence to the semi-circle as m tends to infinity, which we prove. In contrast to most studies of patterned matrix ensembles, our paper gives explicit closed form expressions for the densities. We prove that fm is the product of a Gaussian and a certain even polynomial of degree 2m-2; the formula is the same as that for the m x m Gaussian Unitary Ensemble (GUE). The proof is by derivation of the moments from the eigenvalue trace formula. The new feature, which allows us to obtain closed form expressions, is converting the central combinatorial problem in the moment calculation into an equivalent counting problem in algebraic topology. We end with a generalization of the m-block circulant pattern, dropping the assumption that the m random variables be distinct. We prove that the limiting spectral distribution exists and is determined by the pattern of the independent elements within an m-period, depending on not only the frequency at which each element appears, but also the way the elements are arranged.

Coordinate Sum and Difference Sets of d-Dimensional Modular Hyperbolas

Steven Miller with Amanda Bower, Victor Luo ’14 and Ron Evans

INTEGERS #A31, 16 pages, 2013.

Many problems in additive number theory, such as Fermat’s last theorem and the twin prime conjecture, can be understood by examining sums or differences of a set with itself. A finite set A of the integers is considered sum-dominant if |A+A|>|A-A|. If we consider all subsets of {0, 1, …, n-1}, as n tends to infinity it is natural to expect that almost all subsets should be difference-dominant, as addition is commutative but subtraction is not; however, Martin and O’Bryant in 2007 proved that a positive percentage are sum-dominant as n tends to infinity.

(WS) Generalizing Zeckendorf’s Theorem to f-Decompositions

Steven Miller with Philippe Demontigny ’14, Thao Do, Archit Kulkarni and Umang Varma

Journal of Number Theory 141, 136-158, 2014.

A beautiful theorem of Zeckendorf states that every positive integer can be uniquely decomposed as a sum of non-consecutive Fibonacci numbers {Fn}, where F1 = 1, F2 = 2 and F{n+1} = Fn + F{n-1}. For general recurrences {Gn} with non-negative coefficients, there is a notion of a legal decomposition which again leads to a unique representation, and the number of summands in the representations of uniformly randomly chosen m \in [Gn, G{n+1}) converges to a normal distribution as n tends to infinity. We consider the converse question: given a notion of legal decomposition, is it possible to construct a sequence {an} such that every positive integer can be decomposed as a sum of terms from the sequence? We encode a notion of legal decomposition as a function f from the non-negative integers to non-negative integers and say that if an is in an “f-decomposition”, then the decomposition cannot contain the f(n) terms immediately before a_n in the sequence; special choices of f yield many well known decompositions (including base-b, Zeckendorf and factorial). We prove that for any such f, there exists a sequence {an} such that every positive integer has a unique f-decomposition using {an}. Further, if f is periodic, then the unique increasing sequence {an} that corresponds to f satisfies a linear recurrence relation. Previous research only handled recurrence relations with no negative coefficients. We find a function f that yields a sequence that cannot be described by such a recurrence relation. Finally, for a class of functions f, we prove that the number of summands in the f-decomposition of integers between two consecutive terms of the sequence converges to a normal distribution.

(S) Sets of Special Primes in Function Fields

Steven Miller with Julio Andrade, Kyle Pratt, and Minh-Tam Trinh

INTEGERS 14, #A18, 2014.

When investigating the distribution of the Euler totient function, one encounters sets of primes P where if p is in P then r is in P for all r|(p-1). While it is easy to construct finite sets of such primes, the only infinite set known is the set of all primes. We translate this problem into the function field setting and construct an infinite such set in  Fp[x] whenever p is equivalent to 2 or 5 modulo 9.

The Pythagorean Won-Loss Formula and Hockey:  A Statistical Justification for Using the Classic Baseball Formula as an Evaluative Tool in Hockey

Steven Miller with Kevin Dayaratna

The Hockey Research Journal:  A Publication of the Society for International Hockey Research, 193-209, 2012/2013.

Originally devised for baseball, the Pythagorean Won-Loss formula estimates the percentage of games a team should have won at a particular point in a season. For decades, this formula had no mathematical justification. In 2006, Steven Miller provided a statistical derivation by making some heuristic assumptions about the distributions of runs scored and allowed by baseball teams. We make a similar set of assumptions about hockey teams and show that the formula is just as applicable to hockey as it is to baseball. We hope that this work spurs research in the use of the Pythagorean Won-Loss formula as an evaluative tool for sports outside baseball.

(WS) The Pi Mu Epsilon 100th Anniversary Problems:  Part 1

Steven Miller with James M. Andrews, Avery T. Carr and many students

The Pi Mu Epsilon Journal 13, no. 9, 513-534, 2013.

(WS) The Pi Mu Epsilon 100th Anniversary Problems:  Part 2

Steven Miller with James M. Andrews, Avery T. Carr and many students

The Pi Mu Epsilon Journal 13, no. 10, 577-608, 2014.

As 2014 marks the 100th anniversary of Pi Mu Epsilon, we thought it would be fun to celebrate with 100 problems related to important mathematics milestones of the past century. The problems and notes below are meant to provide a brief tour through some of the most exciting and influential moments in recent mathematics. No list can be complete, and of course there are far too many items to celebrate. This list must painfully miss many people’s favorites. As the goal is to introduce students to some of the history of mathematics, accessibility counted far more than importance in breaking ties, and thus the list below is populated with many problems that are more recreational. Many others are well known and extensively studied in the literature; however, as our goal is to introduce people to what can be done in and with mathematics, we’ve decided to include many of these as exercises since attacking them is a great way to learn. We have tried to include some background text before each problem framing it, and references for further reading. This has led to a very long document, so for space issues we split it into four parts (based on the congruence of the year modulo 4). That said: Enjoy!

Note: I also edited the problem section for these two issues and contributed original problems.

The Mathematics of Encryption:  An Elementary Introduction

Steven Miller with Midge Cozzens

AMS Mathematical World Series 29, Providence, RI, 332 pages, 2013.

This book is the outgrowth of introductory cryptography courses for nonmath majors taught at Rutgers University and Williams College. It is a pleasure to thank our colleagues and our students for many helpful conversations that have greatly improved the exposition and guided the emphasis, in particular Elliot Schrock ‘11 (who helped write the Enigma chapter) and Zane Martin ‘13 and Qiao Zhang (who were the TAs for the 2013 iteration at Williams College, and helped guide the class in writing the solutions manual for teachers).

The Isoperimetric Problem in Higher Codimension

Frank Morgan and Isabel M.C. Salavessa

Manuscripta Mathematica 142, 369-382, 2013.

We consider three generalizations of the isoperimetric problem to higher codimension and provide results on equilibrium, stability, and minimization.

Town Hall Meeting:  Minority Participation in Math

Alissa S. Crans, Frank Morgan and Talithia Williams

MAA Focus, December 2013/January 2014.

Report on the 2013 MathFest Town Hall Meeting on minority participation in mathematics.

Dark Matter and Worst Packings

Frank Morgan

Huffington Post Blog, 28 May 2013.

A report on the annual Geometry and Topology conference at Lehigh University.

Are Smaller College Classes Really Better?

Frank Morgan

Huffington Post Blog, 26 August 2013.

A study shows that larger classes are often just as good.

Adding Fractions

Frank Morgan

Huffington Post Blog, 14 March 2014.

A suggestion that we start by teaching that fractions cannot be added.

Measurable Time-Restricted Sensitivity

Cesar E Silva with Domenico Aiello ’11, Hansheng, Diao, Zhou Fan, Daniel O. King, Jessica Lin

Nonlinearity, 25, 3313-3325, 2012.

We develop two notions of sensitivity to initial conditions for measurable dynamical systems, where the time before divergence of a pair of paths is at most an asymptotically logarithmic function of a measure of their initial distance. In the context of probability measure-preserving transformations on a compact space, we relate these notions to the metric entropy of the system. We examine one of these notions for classes of non-measure-preserving, nonsingular transformations.


Edward Burger, Contributing Authors:  Sarah Flood-Ryland, Douglas Quinney, and Allison Pacelli

Thinkwell, 2013.

Variance Estimation of a General U-Statistic With Application to Cross-Validation

Qing Wang

Statistica Sinica,24(3), 2014.
This paper addresses the problem of variance estimation for a general U-statistic. U-statistics form a class of unbiased estimators for those parameters of interest that can be written as E(ϕ (X1,…,Xn)) where ϕ is a symmetric kernel function with k arguments. Although estimating the variance of a U-statistic is clearly of interest, asymptotic results for a general U-statistic are not necessarily reliable when the kernel size k is not negligible compared with the sample size n. Such situations arise in cross-validation and other nonparametric risk estimation problems. On the other hand, the exact closed form variance is complicated in form, especially when both k and n are large. We have devised an unbiased variance estimator for a general U-statistic. It can be written as a quadratic form of the kernel function ϕ and is applicable as long as k<=n/2. In addition, it can be represented in a familiar analysis of variance form as a contrast of between-class and within-class variation. As a further step to make the proposed variance estimator more practical, we developed a partition resampling scheme that can be used to realize the U-statistic and its variance estimator simultaneously with high computational efficiency. A data example in the context of model selection is provided. To study our estimator, we construct a U-statistic cross-validation tool, akin to the BIC criterion for model selection. With our variance estimator we can test which model has the smallest risk.

Triple Crossing Number of Knots and Links

Colin Adams

Journal of Knot Theory and Its Ramifications 22, 1350006-1 – 1350006–17, 2013.

The Silence of the Lemmas

Colin Adams

Mathematical Intelligencer, 34, no. 2, 27-28, 2012.

The Cabinet of Dr. Mobius

Colin Adams

Mathematical Intelligencer, 34, no. 3, 35-40, 2012.

From Doodles to Diagrams to Knots

Colin Adams with Noel MacNaughton and Charmaine Sia

Mathematical Magazine, 86, no. 2, 83-96, 2013.

Milestones in the Discovery of the Numbers

Colin Adams

Math Horizons, 11-13, 2012.

Letter or Recommendation

Colin Adams

Mathematical Intelligencer, 34, no. 4, 12-14, 2012.

The Tale of Paul Buniyan

Colin Adams

Mathematical Intelligencer, 35, no. 1, 25-27, 2013.

Intro Stats, 4th Edition
Richard De Veaux with Paul Velleman and David Bock

Pearson, January 2013.

Stats: Modeling the World, 3ed Edition
Richard De Veaux with Paul Velleman and David Bock

Pearson, January 2013.

Algebraic Geometry:  A Problem Solving Approach

Thomas Garrity with Richard Belshoff, Lynette Boos, Ryan Brown, Carl Lienert, David Murphy, Junalyn Navarra-Madsen, Pedro Poitevin, Shawn Robinson, Brian A. Snyder and Caryn Werner

American Mathematical Society, Student Mathematical Library, Number 66, 2013.

Simultaneous Confidence Intervals for Comparing Margins of Multivariate Binary Data

Bernhard Klingenberg and Ville Satopää ‘11

Computational Statistics & Data Analysis 64, 87-98, 2013.

In many applications two groups are compared simultaneously on several correlated binary variables for a more comprehensive assessment of group differences. Although the response is multivariate, the main interest is in comparing the marginal probabilities between the groups. Estimating the size of these differences under strong error control allows for a better evaluation of effects than can be provided by multiplicity adjusted P-values.

Simultaneous confidence intervals for the differences in marginal probabilities are developed through inverting the maximum of correlated Wald, score or quasi-score statistics. Taking advantage of the available correlation information leads to improvements in the joint coverage probability and power compared to straightforward Bonferroni adjustments. Estimating the correlation under the null is also explored. While computationally complex even in small dimensions, it does not result in marked improvements. Based on extensive simulation results, a simple approach that uses univariate score statistics together with their estimated correlation is proposed and recommended. All methods are illustrated using data from a vaccine trial that investigated the incidence of four pre-specified adverse events between two groups and with data from the General Social Survey.

The Effect of Short Formative Diagnostic Web Quizzes and Minimal Feedback

Bernhard Klingenberg, with Bälter, O. Enström, E.

Computers and Education 60, 234-242, 2013.

Explicit Constructions of Large Families of Generalized More Sums Than Differences Sets

Steven Miller with Sean Pegado ‘11 and Luc Robinson ‘12

Integers 12, #A30, 2012.

A More Sums Than Differences (MSTD) set is a set of integers A of {0, …, n-1} whose sumset A+A is larger than its difference set A-A. While it is known that as n\to\infty a positive percentage of subsets of {0, …, n-1} are MSTD sets, the methods to prove this are probabilistic and do not yield nice, explicit constructions. Recently Miller, Orosz and Scheinerman gave explicit constructions of a large family of MSTD sets; though their density is less than a positive percentage, their family’s density among subsets of {0, …, n-1} is at least C/n4 for some C>0, significantly larger than the previous constructions, which were on the order of 1/2n/2. We generalize their method and explicitly construct a large family of sets A with |A+A+A+A| > |(A+A)-(A+A)|. The additional sums and differences allow us greater freedom than in Miller, Orosz and Scheinerman, and we find that for any ε > 0 the density of such sets is at least C/nε. In the course of constructing such sets we find that for any integer k there is an A such that |A+A+A+A| – |A+A-A-A| = k, and show that the minimum span of such a set is 30.

Quadratic fields with cyclic 2-class groups

Steven Miller with Carlos Dominguez ‘13 and Siman Wong

Journal of Number Theory 133, no. 3, 926-939, 2013.

For any integer k ≥1, we show that there are infinitely many complex quadratic fields whose 2-class groups are cyclic of order 2k. The proof combines the circle method with an algebraic criterion for a complex quadratic ideal class to be a square.  In memory of David Hayes.

First Order Approximations of the Pythagorean Won-Loss Formula for Predicting MLB Teams Winning Percentages

Steven Miller with Kevin Dayaratna

By The Numbers, The Newsletter of the SABR Statistical Analysis Committee 22, no 1, 15-19, 2012.

We mathematically prove that an existing linear predictor of baseball teams’ winning percentages (Jones and Tappin 2005) is simply just a first-order approximation to Bill James’ Pythagorean Won-Loss formula and can thus be written in terms of the formula’s well-known exponent. We estimate the linear model on twenty seasons of Major League Baseball data and are able to verify that the resulting coefficient estimate, with 95% confidence, is virtually identical to the empirically accepted value of 1.82. Our work thus helps explain why this simple and elegant model is such a strong linear predictor.

Low-lying zeros of number field L-functions

Steven Miller with Ryan Peckner

Journal of Number Theory 132, 2866-2891, 2012.

One of the most important statistics in studying the zeros of L-functions is the 1-level density, which measures the concentration of zeros near the central point. Fouvry and Iwaniec proved that the 1-level density for L-functions attached to imaginary quadratic fields agrees with results predicted by random matrix theory. In this paper, we show a similar agreement with random matrix theory occurring in more general sequences of number fields. We first show that the main term agrees with random matrix theory, and similar to all other families studied to date, is independent of the arithmetic of the fields. We then derive the first lower order term of the 1-level density, and see the arithmetic enter.

Leading, learning and herding

Steven Miller with Daniel Stone

Mathematical Social Sciences 65, 222-231, DOI information: 10.1016/j.mathsocsci.2012.12.002, 2013.

We analyze a game theoretic model of social learning about a consumption good with endogenous timing and heterogeneous accuracy of private information. We show that if individuals value their reputation for the degree to which they are informed, this reduces the incentive to learn by observing others and exacerbates the incentive to consume the good before others, i.e., to attempt to be an “opinion leader.” Consequently, reputation concerns reduce the average delay of consumption of new goods, and increase (reduce) the probability of herding on consumption (non-consumption).

The Average Gap Distribution for Generalized Zeckendorf Decompositions

Steven Miller with O. Beckwith, A. Bower, L. Gaudet, R. Insoft, S. Li, and P. Tosteson ‘13

The Fibonacci Quarterly 51, 13-27, 2013.

An interesting characterization of the Fibonacci numbers is that, if we write them as F1 = 1, F2 = 2, F3 = 3, F4 = 5, …, then every positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers. This is now known as Zeckendorf’s theorem, and similar decompositions exist for many other sequences {Gn+1 = c1 Gn + … + cL Gn+1-L} arising from recurrence relations. Much more is known. Using continued fraction approaches, Lekkerkerker proved the average number of summands needed for integers in [G_n, Gn+1) is on the order of CLek n for a non-zero constant; this was improved by others to show the number of summands has Gaussian fluctuations about this mean.

Kologlu ‘12, Kopp, Miller and Wang recently recast the problem combinatorially, reproving and generalizing these results. We use this new perspective to investigate the distribution of gaps between summands. We explore the average behavior over all m in [Gn, Gn+1) for special choices of the ci‘s. Specifically, we study the case where each ci in {0, 1} and there is a g such that there are always exactly g-1 zeros between two non-zero ci‘s; note this includes the Fibonacci, Tribonacci and many other important special cases. We prove there are no gaps of length less than g, and the probability of a gap of length j > g decays geometrically, with the decay ratio equal to the largest root of the recurrence relation. These methods are combinatorial and apply to related problems; we end with a discussion of similar results for far-difference (i.e., signed) decompositions.

Closed Form Continued Fraction Expansions of Special Quadratic Irrationals

Steven Miller with Dan Fishman

ISRN Combinatorics, Article ID 414623, 5 pages, doi:10.1155/2013/414623, 2013.

We derive closed form expressions for the continued fractions of powers of certain quadratic surds. Specifically, consider the recurrence relation Gn+1 = m Gn+ l Gn-1 with G0 = 0, G1 = 1, m a positive integer and l = ±1 (note m = l =1 gives the Fibonacci numbers). Let φm,l = limn à  ∞ Gn / Gn-1. We find simple closed form continued fraction expansions for φm,lk for any integer k by exploiting elementary properties of the recurrence relation and continued fractions. This paper is dedicated to the memory of Alf van der Poorten.

Distribution of Missing Sums in Sumsets

Steven Miller with Oleg Lazarev and Kevin O’Bryant

Experimental Mathematics 22, no. 2, 132-156, 2013.

For any finite set of integers X, define its sumset X+X to be {x+y: x, y in X}. In a recent paper, Martin and O’Bryant investigated the distribution of |A+A| given the uniform distribution on subsets A of {0, 1, …, n-1}. They also conjectured the existence of a limiting distribution for |A+A| and showed that the expectation of |A+A| is 2n – 11 + O((3/4)n/2). Zhao proved that the limits m(k) := limn à Prob(2n-1-|A+A|=k) exist, and that the sum of the m(k) equals 1.

We continue this program and give exponentially decaying upper and lower bounds on m(k), and sharp bounds on m(k) for small k. Surprisingly, the distribution is at least bimodal; sumsets have an unexpected bias against missing exactly 7 sums. The proof of the latter is by reduction to questions on the distribution of related random variables, with large scale numerical computations a key ingredient in the analysis. We also derive an explicit formula for the variance of |A+A| in terms of Fibonacci numbers, finding Var(|A+A|) is approximately 35.9658. New difficulties arise in the form of weak dependence between events of the form {x in A+A}, {y in A+A}. We surmount these obstructions by translating the problem to graph theory. This approach also yields good bounds on the probability for A+A missing a consecutive block of length k.

Virus Dynamics on Spoke and Star Graphs

Steven Miller with Thealexa Becker, Alec Greaves-Tunnell ‘13, Leo Kontorovich and Karen Shen

The Journal of Nonlinear Systems and Applications 4, no. 1, 53-63, 2013.

The field of epidemiology has presented fascinating and relevant questions for mathematicians, primarily concerning the spread of viruses in a community. The importance of this research has greatly increased over time as its applications have expanded to also include studies of electronic and social networks and the spread of information and ideas. We study virus propagation on a non-linear hub and spoke graph (which models well many airline networks). We determine the long-term behavior as a function of the cure and infection rates, as well as the number of spokes n. For each n we prove the existence of a critical threshold relating the two rates. Below this threshold, the virus always dies out; above this threshold, all non-trivial initial conditions iterate to a unique non-trivial steady state. We end with some generalizations to other networks.

Existence of Isoperimetric Regions in Rn with Density

Frank Morgan and Aldo Pratelli

Ann. Global Anal. Geom., 2012.

We prove the existence of isoperimetric regions in Rn with density under various hypotheses on the growth of the density. Along the way we prove results on the boundedness of isoperimetric regions.

Are Large Perimeter-Minimizing Two-Dimensional Clusters of Equal-Area Bubbles Hexogonal or Circular?

S. J. Cox, Frank Morgan and F. Graner

Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., published online 24 October 2012.

A computer study of clusters of 10000 equal-area bubbles shows for the first time that rounding conjectured optimal hexagon planar soap bubble clusters reduces perimeter.

Math Now – Commencement Can Wait

Frank Morgan

Huffington Post Blog, 28 May 2012.

A week of mathematics up and down the East coast.

Why is Summer so Early

Frank Morgan

Huffington Post Blog, 17 June 2012.

Too many leap years make summer come earlier.

I Win Soap-Bubble-Cluster Controversy

Frank Morgan

Huffington Post Blog, 22 June 2012.

A report on recent work on the shape of large soap bubble clusters.

Spilled Orange Juice on May Way to a Math Conference in Spain

Frank Morgan

Huffington Post Blog, 30 June 2012.

Adventures of a traveling mathematician.

Why a Laptop is Not a Computer

Frank Morgan

Huffington Post Blog, 24 July 2012.

Computers should have more cut and paste registers.

U.S. Presidential Election Paradox

Frank Morgan

Huffington Post Blog, 16 October 2012.

You could win the U.S. Presidential election with under 22% of the popular vote.

Why I Don’t Like Energy Efficient Light Bulbs

Frank Morgan

Huffington Post Blog, 24 November 2012.

Heat-producing light bulbs let me set my thermostat lower.

How Often Should I Rebalance My Investments? 

Frank Morgan

Huffington Post Blog, 3 December 2012.

A report on joint work with student Walter Filkins ’12.

Measurable Time-Restricted Sensitivity

Cesar E Silva with Domenico Aiello ’11, Hansheng, Diao, Zhou Fan, Daniel O. King, Jessica Lin

Nonlinearity, 25, 3313-3325, 2012.

We develop two notions of sensitivity to initial conditions for measurable dynamical systems, where the time before divergence of a pair of paths is at most an asymptotically logarithmic function of a measure of their initial distance. In the context of probability measure-preserving transformations on a compact space, we relate these notions to the metric entropy of the system. We examine one of these notions for classes of non-measure-preserving, nonsingular transformations.

Investigation of Topics in U-statistics and Their Applications in Risk Estimation and Cross Validation

Qing Wang

Penn State University Library (electronic resource), Ph.D. Dissertation, 1-187, 2012.

The Complimentary Regions of Knot and Link Projections
Colin Adams, R. Shinjo and K. Tanaka

Annals of Combinatorics, Vol. 15, No. 4, 549-563 (2011)

An increasing sequence of integers is said to be universal for knots and links if every knot and link has a reduced projection on the sphere such that the number of edges of each complementary face of the projection comes from the given sequence. In this paper, it is proved that the following infinite sequences are each universal for knots and links:

(3,5,7, …), (2,n, n+1,n+2,…) for each n ≥ 3, (3,n,n+1,n+2, …) for each n ≥ 4. Moreover, the finite sequences  (2,4,5) and (3,4,n) for each n ≥ 5 are universal for all knots and links.

It is also shown that every knot has a projection with exactly two odd-sided faces, which can be taken to be triangles, and every link of n components has a projection with at most n odd-sided faces if n is even and n+1 odd-sided faces if n is odd.

Planar and Spherical Stick Indices of Knots
Colin Adams, D. Collins, K. Hawkins ‘10, C. Sia, R. Silversmith ’11, B. Tshishiku

Journal of Knot Theory and Its Ramifications, Vol. 20, No. 5, 721-739 (2011)

The stick index of a knot is the least number of line segments required to build the knot in space. We define two analogous 2-dimensional invariants, the planar stick index, which is the least number of line segments in the plane to build a projection, and the spherical stick index, which is the least number of great circle arcs to build a projection on the sphere. We find bounds on these quantities in terms of other knot invariants, and give planar stick and spherical stick constructions for torus knots and for compositions of trefoils. In particular, unlike most knot invariants, we show that the spherical stick index distinguishes between the granny and square knots, and that composing a nontrivial knot with a second nontrivial knot need not increase its spherical stick index.

Stick Index of Knots and Links in the Cubic Lattice
Colin Adams, M. Chu, T. Crawford ‘12, S. Jensen ’12, K. Siegel, L. Zhang ‘12

Journal of knot Theory and Its Ramifications, Vol. 21, No. 5 (2012)

The cubic lattice stick index of a knot type is the least number of sticks necessary to construct the knot type in the 3-dimensional cubic lattice. We present the cubic lattice stick index of various knots and links, including all (p, p+1)-torus knots, and show how composing and taking satellites can be used to obtain the cubic lattice stick index for a relatively large infinite class of knots. Additionally, we present several bounds relating cubic lattice stick index to other known invariants.

Duality Properties of Indicatrices of Knots
Colin Adams, D. Collins, K. Hawkins ‘10, C. Sia, R. Silversmith ’11, B. Tshishiku

Geometriae Dedicata (on-line publication), (September 1, 2011)

The bridge index and superbridge index of a knot are important invariants in knot theory.  We define the bridge map of a knot conformation, which is closely related to these two invariants, and interpret it in terms of the tangent indicatrix of the knot conformation.  Using the concepts of dual and derivative curves of spherical curves as introduced by Arnold, we show that the graph of the bridge map is the union of the binormal indicatrix, its antipodal curve, and some number of great circles.  Similarly, we define the inflection map of a knot conformation, interpret it in terms of the binormal indicatrix, and express its graph in terms of the tangent indicatrix.  This duality relationship is also studied for another dual pair of curves, the normal and Darboux indicatrices of a knot conformation.  The analogous concepts are defined and results are derived for stick knots.

Colin Adams

Mathematical Intelligencer, Vol. 33, No. 2, 18-21 (2011)

What happens when there is a crime committed at the Mathematical Sciences Research Institute?

The Book
Colin Adams

Mathematical Intelligencer, Vol. 33, No. 3, 107-109 (2011)

Paul Erdos hypothesized a book in God’s possession that contained all of the beautiful proofs ever discovered.  What happens if you have access to that book.

Leonhard Euler and Seven Bridges of Konigsberg
Colin Adams

Mathematical Intelligencer, Vol. 33, No. 4, 18-20 (2011)

Many people attribute the birth of topology to Euler’s solution of the Konigsberg Bridge Problem.  But what is the true story of what really happened?

The Dog Who Knew Calculus
Colin Adams

Mathematical Intelligencer, Vol. 34, No. 1, 16-17 (2012)

In a 2003 article, the author explained how his dog Elvis seemed to understand calculus, as he was so good at minimizing the time it took to get a ball thrown in the water.  So let’s give him a job teaching.

Derivative vs. Integral:  The Final Smackdown
Colin Adams, Thomas Garrity and Adam Falk

Mathematical Association of America (January 2012)

Which is better, the derivative or the integral?  Recorded at Williams Family Days, Fall 2011.

A Generalization of a Theorem of Lekkerkerker to Ostrowski’s Decomposition of Natural Numbers
Edward B. Burger, David C. Clyde, Cory H. Colbert, Gea Hyun Shin ’11, and Zhaoning Wang ‘11

Acta Arithmetica, 153, 217-249 (2012)

Let a be a fixed, irrational real number and pk/qk its associated kth convergent.  In 1921, Ostrowski proved that each natural number n can be expressed uniquely as a linear combination of the continuants of a, namely the qk’s, in which the integer coefficients satisfy certain natural diophantine conditions.  Here we analyze the asymptotic behavior of the average number of summands required in such decompositions relative to the size of the corresponding natural numbers in the case for which a is a quadratic irrational.  Our results generalize the work of Lekkerkerker, who in 1951 explicitly computed this asymptotic ratio for the particular case a = (1+√5)/2 and found it to equal (5–√5)/10 = 0.2763… .

The Shape of Associativity
Satyan Devadoss

Canadian Mathematical Society Notes, 44, 12-14 (2012)

Associativity is ubiquitous in mathematics.  Unlike commutativity, its more popular cousin, associativity has for the most part taken a backseat in importance.  But over the past few decades, this concept has blossomed and matured.  We show how to visualize the concept of associativity.

What Makes a Tree a Straight Skeleton?
Satyan Devadoss

Proceedings of the European Conference on Computational Geometry (2012)

Given any polygon, one can construct a geometric tree associated to it called its straight skeleton.  This appears in the construction of roof and origami folding designs.  We ask the inverse question:  For what tree does there exist polygons with the tree as its skeleton?

Triple Infinity
Satyan Devadoss, Associate Professor of Mathematics

Esopus Magazine (2011)

A conversation between a mathematician, a cosmologist, and an artist about the meaning and nature of infinity in these three fields.

A Robust Boosting Algorithm for Chemical Modeling
Richard DeVeaux and Ville Satopӓӓ ‘11

Current Analytical Chemistry, Vol. 8, No. 2, 254-265 (2012)

Baggins and boosting have become increasingly important ensemble methods for combining models in the data mining and machine learning literature. We review the basic ideas of these methods, propose a new robust boosting algorithm based on a non-convex loss function and compare the performance of these methods to both simulated and real data sets both with and without contamination.

Using Mathematical Maturity to Shape our Teaching, our Careers and our Departments
Thomas Garrity

Notices of the American Mathematical Society, 1592 – 1593 (2011)

Derivative vs. Integral:  The Final Smackdown
Thomas Garrity, Colin C. Adams and Adam Falk

Mathematical Association of America (January 2012)

Which is better, the derivative or the integral?  Recorded at Williams Family Days, Fall 2011.

Semi-Local Formal Fibers of Minimal Prime Ideals of Excellent Reduced Local Rings
Susan Loepp, Nicholas Arnosti ’11, Rachel Karpman, Caitlin Leverson, and Jake Levinson ’11

Journal of Commutative Algebra, No. 1, 29-56 (2012)

Given a complete local ring T containing the rationals, and a positive integer m, the authors find necessary and sufficient conditions for there to exist an excellent reduced local ring A, whose completion is T, such that A has exactly m minimal prime ideals.  In addition, the authors show that the formal fibers over the minimal prime ideals can be controlled.

Distribution of Eigenvalues for Highly Palindromic Real Symmetric Toeplitz Matrices
Steven J. Miller, Steven Jackson ’10 and Thuy Pham ‘11

Journal of Theoretical Probability, 25, 464-495 (2012)

Consider the ensemble of real symmetric Toeplitz matrices whose entries are i.i.d random variables chosen from a fixed probability distribution p of mean 0, variance 1 and finite higher moments.  Previous work showed that the limiting spectral measures (the density of normalized eigenvalues) converge in probability and almost surely to a universal distribution almost that of the Gaussian, independent of p.  The deficit from the Gaussian distribution is due to obstructions to solutions of Diophantine equations and can be removed by making the first row palindromic.  In this paper, we study the case where there is more than one palindrome in the first row of a real symmetric Toeplitz matrix.  Using the method of moments and an analysis of the resulting Diophantine equations, we show that the moments of this ensemble converge to a universal distribution with a fatter tail than any previously seen limiting spectral measure.

Rational Irrationality Proofs
Steven J. Miller and David Montague

Mathematics Magazine, 85, No. 2, 110-114 (2012)

Proving the irrationality of the square-root of 2 is a rite of passage for mathematicians.  The purpose of this note is to spread the word of a remarkable geometric proof, and to generalize it.  The proof was discovered by Stanley Tennenbaum in the 1950’s, and first appeared in print in John H. Conway’s article in Power. In the interest of space, we often leave out the algebra justifications for the lengths of the sides in our figures. The reader is encouraged to prove these expressions for themselves, or see the arxiv post for complete details.

Moments of the Rank of Elliptic Curves
Steven J. Miller and Siman Wong

Canadian Journal of Mathematics, 64, No. 1, 151-182 (2012)

Fix an elliptic curve E/Q, and assume the Riemann Hypothesis for the L-function L(E_D, s) for every quadratic twist E_D of E by D in Z.  We combine Weil’s explicit formula with techniques of Heath-Brown to derive an asymptotic upper bound for the weighted moments of the analytic rank of E_D.  We derive from this an upper bound for the density of low-lying zeros of L(E_D, s) which is compatible with the random matrix models of Katz and Sarnak.  We also show that for any unbounded increasing function f on R, the analytic rank and (assuming in addition the Birch and Swinnerton-Dyer conjecture) the number of integral points of E_D are less than f(D) for almost all D.

Generalized More Sums Than Differences Sets
Steven J. Miller, Geoffrey Iyer, Oleg Lazarev, Liyang Zhang ‘12

Journal of Number Theory, 132, No. 5, 1054-1073 (27 pp) (2012)

A More Sums Than Differences (MSTD, or sum-dominant) set is a finite set A of Z such that |A+A|<|A-A|.  Though it was believed that the percentage of subsets of {0,…,n} that are sum-dominant tends to zero, in 2006 Martin and O’Bryant proved that a positive percentage are sum-dominant. We generalize their result to the many different ways of taking sums and differences of a set.  We prove that |ε1 A + … + εk A|>|δ1 A + … + δk A| a positive percent of the time for all nontrivial choices of εjj\in {-1,1}. Previous approaches proved the existence of infinitely many such sets given the existence of one; however, no method existed to construct such a set. We develop a new, explicit construction for one such set, and then extend to a positive percentage of sets.

We extend these results further, finding sets that exhibit different behavior as more sums/differences are taken.  For example, we prove that for any m, |ε1 A + … + εk A| – |δ1 A + … + δk A| = m a positive percentage of the time.  We find the limiting behavior of kA = A+ … +A for an arbitrary set A as k goes to infinity and an upper bound of k for such behavior to settle down.  Finally, we say A is k-generational sum-dominant if A, A+A, …, kA are all sum-dominant.  Numerical searches were unable to find even a 2-generational set (heuristics indicate that the probability is at most 10{-9}, and quite likely significantly less).  We prove that for any k a positive percentage of sets are k-generational, and no set can be k-generational for all k.

Explicit Constructions of Large Families of Generalized More Sums Than Differences Sets
Steven J. Miller, Sidney Luc Robinson ’12 and Sean Pegado ‘11

Integers, 12, No. A30 (2012)

A More Sums Than Differences (MSTD) set is a set of integers A of {0, …, n-1} whose sumset A+A is larger than its difference set A-A.  While it is known that as n tends to infinity a positive percentage of subsets of {0, …,n-1} are MSTD sets, the methods to prove this are probabilistic and do not yield nice, explicit constructions.  Recently Miller, Orosz and Scheinerman gave explicit constructions of a large family of MSTD sets; though their density is less than a positive percentage, their family’s density among subsets of {0, …,n-1} is at least C/n4 for some C>0, significantly larger than the previous constructions, which were on the order of 1/2{n/2}.  We generalize their method and explicitly construct a large family of sets A with |A+A+A+A| > |(A+A)-(A+A)|. The additional sums and differences allow us greater freedom than in MOS, and we find that for any ε>0 the density of such sets is at least C/nε.  In the course of constructing such sets we find that for any integer k there is an A such that |A+A+A+A| – |A+A-A-A| = k, and show that the minimum span of such a set is 30.

Models for Zeros at the Central Point in Families of Elliptic Curves
Steven J. Miller, Eduardo Duenez, Duc Khiem Huynh, Jon Keating and Nina Snaith

J. Phys. A: Math. Theor., 45, 115207 (2012)

We propose a random matrix model for families of elliptic curve L-functions of finite conductor. A repulsion of the critical zeros of these L-functions away from the center of the critical strip was observed numerically by S. J. Miller in 2006; such behaviour deviates qualitatively from the conjectural limiting distribution of the zeros (for large conductors this distribution is expected to approach the one-level density of eigenvalues of orthogonal matrices after appropriate rescaling).  Our purpose here is to provide a random matrix model for Miller’s surprising discovery.  We consider the family of even quadratic twists of a given elliptic curve.  The main ingredient in our model is a calculation of the eigenvalue distribution of random orthogonal matrices whose characteristic polynomials are larger than some given value at the symmetry point in the spectra. We call this sub-ensemble of SO(2N) the excised orthogonal ensemble.  The sieving-off of matrices with small values of the characteristic polynomial is akin to the discretization of the central values of L-functions implied by the formula of Waldspurger and Kohnen-Zagier.  The cut-off scale appropriate to modeling elliptic curve L-functions is exponentially small relative to the matrix size on the order of N.  The one-level density of the excised ensemble can be expressed in terms of that of the well-known Jacobi ensemble, enabling the former to be explicitly calculated.  It exhibits an exponentially small (on the scale of the mean spacing) hard gap determined by the cut-off value, followed by soft repulsion on a much larger scale. Neither of these features is present in the one-level density of SO(2N).  When N goes to infinity we recover the limiting orthogonal behaviour.  Our results agree qualitatively with Miller’s discrepancy.  Choosing the cut-off appropriately gives a model in good quantitative agreement with the number-theoretical data.

On the Number of Summands in Zeckendorf Decompositions
Steven J. Miller, Murat Kologlu ’12, Gene S. Kopp, and Yinghui Wang

Fibonacci Quarterly, 49, No. 2, 116-130 (2011)

Zeckendorf proved that every positive integer has a unique representation as a sum of non-consecutive Fibonacci numbers. Once this has been shown, it’s natural to ask how many summands are needed.  Using a continued fraction approach, Lekkerkerker proved that the average number of such summands needed for integers in [Fn, F{n+1}) is n / (j2 + 1) + O(1), where j = (1+sqrt(5))/2 is the golden mean.  Surprisingly, no one appears to have investigated the distribution of the number of summands; our main result is that this converges to a Gaussian as n tends to infinity.  Moreover, such a result holds not just for the Fibonacci numbers but many other problems, such as linear recurrence relation with non-negative integer coefficients (which is a generalization of base B expansions of numbers) and far-difference representations.

In general the proofs involve adopting a combinatorial viewpoint and analyzing the resulting generating functions through partial fraction expansions and differentiating identities.  The resulting arguments become quite technical; the purpose of this paper is to concentrate on the special and most interesting case of the Fibonacci numbers, where the obstructions vanish and the proofs follow from some combinatorics and Stirling’s formula.

From Fibonacci Numbers to Central Limit Type Theorems
Steven J. Miller and Yinghui Wang

Journal of Combinatorial Theory, Series A, 119, No. 7, 1398-1413 (2012)

A beautiful theorem of Zeckendorf states that every integer can be written uniquely as a sum of non-consecutive Fibonacci numbers {Fn}_{n=1}.  Lekkerkerker \cite{Lek} proved the average number of summands for integers in [Fn, F{n+1}) is n/(j2 + 1), with phi the golden mean.  This has been generalized: given nonnegative integers c1,c2,…,cL with c1,cL>0 and recursive sequence {Hn}_{n=1} with H1=1, H{n+1} =c1Hn + c2H{n-1} + … +cnH1+1  (1 ≤ ∞  n < L) and H{n+1}=c1Hn+c2H{n-1}+ … +c_LH_{n+1-L} (n ≥ L), every positive integer can be written uniquely as a sum of aiHi under natural constraints on the a_i‘s, the mean and variance of the numbers of summands for integers in [H{n}, H{n+1}) are of size n, and as n tends to infinity the distribution of the number of summands converges to a Gaussian.  Previous approaches used number theory or ergodic theory.  We convert the problem to a combinatorial one.  In addition to re-deriving these results, our method generalizes to other problems (in the sequel paper we show how this perspective allows us to determine the distribution of gaps between summands).  For example, it is known that every integer can be written uniquely as a sum of the ± Fn‘s, such that every two terms of the same (opposite) sign differ in index by at least 4 (3).  The presence of negative summands introduces complications and features not seen in previous problems.  We prove that the distribution of the numbers of positive and negative summands converges to a bivariate normal with computable, negative correlation, namely -(21-2j)/(29+2j), which is approximately -0.551058.

Steiner and Schwarz Symmetrization in Warped Products and Fiber Bundles With Density
Frank Morgan, Sean Howe and Nate Harman

Revista Mat. Iberoamericana, 27, 909-918 (2011)

We provide very general symmetrization theorems in arbitrary dimension and codimension, in products, warped products, and certain fiber bundles such as lens spaces, including Steiner, Schwarz, and spherical symmetrization and admitting density.

Isoperimetric Pentagonal Tilings
Frank Morgan, Ping Ngai Chung, Miguel Fernandez, Yifei Li, Michael Mara ’12, Isamar Rosa Plata, Niralee Shah ’12, Luis Sordo Vieira, and Elena Wikner ‘11

Notices Amer. Math. Soc., 59, 632-640 (2012)

We generalize the isoperimetric problem from geometry to numbers.

Alan Alda’s Flame Challenge and Kids’ Five Most Popular Science Questions
Frank Morgan

Huffington Post Blog, (March 16, 2012)

Can Math Survive Without the Bees?
Frank Morgan

Huffington Post Blog, (March 6, 2012)

Recent and new results on perimeter-minimizing tilings.

Soap Bubbles in Scotland
Frank Morgan

Huffington Post Blog, (March 23, 2012)

The latest progress on the century-old search for the least-perimeter way to partition space into unit volumes.

Math Finds the Best Doughnut
Frank Morgan

Huffington Post Blog, (April 2, 2012)

A report on the proof of the Willmore Conjecture.

Geometry Festival
Frank Morgan

Huffington Post Blog, (April 30, 2012)

A mathematics progress report from this annual meeting of geometers.

Function Fields With Class Number Indivisible by A Prime
Allison Pacelli, Michael Daub ’08, J. Lang, M. Merling and Natee Pitiwan ‘09

Acta. Arith., 150, 339-359 (2011)

In this paper, we prove that there are infinitely many function fields of any degree over the rational function field with class number indivisible by an arbitrary prime number.

On Mu-Compatible Metrics and Measurable Sensitivity
Cesar E. Silva, Ilya Grigoriev, Nate Ince, Marius Catalin ’09, and Amos Lubin

Colloquium Math. 126, 53-72 (2012)

We introduce the notion of W-measurable sensitivity, which extends and strictly implies canonical measurable sensitivity, a measure-theoretic version of sensitive dependence on initial conditions.  This notion also implies pairwise sensitivity with respect to a large class of metrics.  We show that nonsingular ergodic and conservative dynamical systems on standard spaces must be either W-measurably sensitive, or isomorphic mod 0 to a minimal uniformly rigid isometry.  In the finite measure-preserving case they are W-measurably sensitive or measurably isomorphic to an ergodic isometry on a compact metric space.

The Spiral Index of Knots

Colin Adams, Thomas T. Read Professor of Mathematics with W. George, R. Hudson, R. Morrison, L. Starkston, S. Taylor, O. Turanova

Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 149, Issue 2, 297-315 (2010)

In this paper, we introduce two new invariants that are closely related to Milnor’s curvature-torsion invariant.  The first, a particularly natural invariant called the spiral index of a knot, captures the number of local maxima in a knot projection that is free of inflection points.  This invariant is sandwiched between the bridge and braid index of a knot, and captures more subtle properties.  The second invariant, the projective superbridge index, provides a method of counting the greatest number of local maxima that occur in a given projection. In addition to investigating the relationships among these invariants, we use them to classify all those knots for which Milnor’s curvature-torsion invariant is 6pi.

Looking Backward

Colin Adams, Thomas T. Read Professor of Mathematics

Mathematical Intelligencer, Vol. 32, No. 3 (2010)

What happens to someone who is hypnotized to sleep and reawakens 100 years later in their expectation of the present day world, and finds a mathematical paradise?

A Forgivably Flat Classic, review of Flatland

Colin Adams, Thomas T. Read Professor of Mathematics

American Scientist, Vol. 98, No. 6, 498-500 (Nov. – Dec. 2010)

A review of the classic book “Flatland” in a new edition.

Group Therapy

Colin Adams, Thomas T. Read Professor of Mathematics

Mathematical Intelligencer, Vol. 32, No. 4 (2010)

How does group therapy work for a group of mathematicians who know group theory?

Hardy and Ramanujan

Colin Adams, Thomas T. Read Professor of Mathematics

Mathematical Intelligencer, Vol. 33, No. 1 (2011)

A more detailed account of the enigmatic relationship between Hardy and Ramanujan.

Leonhard Euler and Seven Bridges of Konigsberg

Colin Adams, Thomas T. Read Professor of Mathematics

Mathematical Intelligencer, Vol. 33, No. 2 (2011)

The true story of how Leonhard Euler became the greatest bridge stroller problem solver in history.

Arithmetic From an Advanced Perspective:  An Introduction to the Adeles

Edward B. Burger, Professor of Mathematics and Lissack Professor for Social Responsibility and Personal Ethics

Pro Mathematica, 24, 9-54 (2010)

Here we offer an introduction to the adele ring over the field of rational numbers Q and highlight some of its beautiful algebraic and topological structure.  We then apply this rich structure to revisit some ancient results of number theory and place them within this modern context as well as make some new observations.  We conclude by indicating how this theory enables us to extend the basic arithmetic of Q to the more subtle, complicated, and interesting setting of an arbitrary number field k.

Deformations of Bordered Surfaces and Convex Polytopes

Satyan Devadoss, Associate Professor of Mathematics with T. Heath and C. Vipismakul

Notices of the American Mathematical Society, 58, 530-541 (2011)

We provide a combinatorial framework to understand how surfaces with boundary can deform, and then proceed to classify all such deformations which have polytopal structures.

Pseudograph Associahedra

Satyan Devadoss, Associate Professor of Mathematics with M. Carr and S. Forcey

Journal of Combinatorial Theory, Series A, 118, 2035-2055 (2011)

Given an arbitrary finite graph (with loops and multiple edges), we construct a polytope which captures the connectedness of the graph.

Discrete and Computational Geometry

Satyan Devadoss, Associate Professor of Mathematics with Joseph O’Rourke

Princeton University Press (2011)

This textbook bridges the gap between discrete geometry of pure mathematics and computational geometry of data-driven computer science, at an undergraduate level.  It includes traditional topics such as convex hulls, triangulations, and Voronoi diagrams, as well as advanced material such as curve reconstruction, quasigeodesics, and Dehn invariants.

Stats:  Data and Models, 3rd Edition

Richard DeVeaux, Professor of Statistics with Paul Velleman and David Bock

Pearson Education (2010)

Simultaneous Confidence Bounds for Relative Risks in Multiple Comparisons to Control

Bernhard Klingenberg, Associate Professor of Statistics

Statistics in Medicine, 29, 3232-3244 (2010)

We discuss the construction of asymptotic simultaneous upper confidence limits that jointly bound relative risks formed by comparing several treatments to a control. Motivated by a vaccine study, we investigate the performance of several methods under such settings. Inverting the minimum of score statistics, together with estimating the correlation matrix of these statistics under the null gives simultaneous coverage rates closest to the nominal level. In typical settings of vaccine studies, this method proves to be the most powerful of the ones considered, but computationally simpler alternatives are also worth exploring when the number of comparisons is large. Simultaneous lower and two-sided confidence intervals are also considered. All procedures can be implemented and evaluated using freely available and general R code.

Formal Fibers of Unique Factorization Domains

Susan Loepp, Professor of Mathematics with A. Boocher, M. Daub, R. Johnson, H. Lindo, and P. Woodard

Canadian Journal of Mathematics, 62, 721-736 (2010)

In this paper, the authors construct unique factorization domains such that most of the formal fibers of these integral domains are geometrically regular.  In addition, they construct unique factorization domains containing many ideals for which tight closure and completion do not commute.

A Unitary Test of the L-Functions Ratios Conjecture

Steven J. Miller, Assistant Professor of Mathematics with John Goes, Steven Jackson ‘10, David Montague, Kesinee Ninsuwan, Ryan Peckner and Thuy Pham ‘11

Journal of Number Theory, 130, 2238-2258 (2010)

We verify the L-function Ratios Conjecture’s predictions for the unitary family of all Dirichlet L-functions with prime conductor; we show square-root agreement between prediction and number theory if the support of the Fourier transform of the test function is in (-1, 1), and for support up to (-2, 2) we show agreement up to a power savings in the family’s cardinality.  The interesting feature in this family (which has not surfaced in previous investigations) is determining what is and what is not a diagonal term in the Ratios recipe.

Towards an Average Version of the Birch and Swinnerton-Dyer Conjecture

Steven J. Miller, Assistant Professor of Mathematics with John Goes

Journal of Number Theory, 130, 2341-2358 (2010)

We find non-trivial upper and lower bounds for the average number of normalized zeros in intervals on the order of 1/ log N_E (which is the expected scale). Our results may be interpreted as providing further evidence in support of the Birch and Swinnerton-Dyer conjecture, as well as the Katz-Sarnak density conjecture from random matrix theory (as the number of zeros predicted by random matrix theory lies between our upper and lower bounds).

Explicit Constructions of Infinite Families of MSTD Sets (with Dan Scheinerman) Additive Number Theory:  Festschrift in Honor of the Sixtieth Birthday of Melvyn B. Nathanson

Steven J. Miller, Assistant Professor of Mathematics, with David Chudnovsky and Gregory Chudnovsky, eds.

Springer-Verlag (2010)

We present a new construction that yields a family of sum-dominated sets in {1, 2, …, r} of size C 2r / r4 for a fixed, non-zero constant C; our family is significantly denser than previous constructions.

The Lowest Eigenvalue of Jacobi Random Matrix Ensembles and Painleve VI

Steven J. Miller, Assistant Professor of Mathematics, with Eduardo Duenez, Duc Khiem Huynh, Jon Keating and Nina Snaith

Journal of Physics A:  Mathematical and Theoretical, 43, 405204 (27 pp) (2010)

We present two complementary methods, each applicable in a different range, to evaluate the distribution of the lowest eigenvalue of random matrices in a Jacobi ensemble.

An Orthogonal Test of the L-Functions Ratios Conjecture, II

Steven J. Miller, Assistant Professor of Mathematics, with David Montague

Acta Arith., 146, 53-90 (2011)

We prove the accuracy of the Ratios Conjectures prediction for the 1-level density of families of cuspidal newforms of constant sign (up to square-root agreement for support in (-1, 1), and up to a power savings in (-2, 2)), and discuss the arithmetic significance of the lower order terms. This is the most involved test of the Ratios Conjectures predictions to date, as it is known that the error terms dropped in some of the steps do not cancel, but rather contribute a main term! Specifically, these are the non-diagonal terms in the Petersson formula, which lead to a Bessel-Kloosterman sum which contributes only when the support of the Fourier transform of the test function exceeds (-1, 1).

Effective Equidistribution and the Sato-Tate Law for Families of Elliptic Curves

Steven J. Miller, Assistant Professor of Mathematics, with Ram Murty

Journal of Number Theory, 131, No. 1, 25-44 (2011)

We provide effective bounds on the family of all elliptic curves and one-parameter families of elliptic curves modulo p (for p prime tending to infinity) obeying the Sato-Tate Law.

Isoperimetric Sets of Integers

Steven J. Miller, Assistant Professor of Mathematics, with Frank Morgan, Webster Atwell Class of 1921 Professor of Mathematics, Edward Newkirk, ’09, Lori Pedersen, Deividas Seferis ‘09

Mathematics Magazine, 84, 37-42 (2011)

The celebrated isoperimetric theorem says that the circle provides the least-perimeter way to enclose a given area. I n this note we discuss a generalization.

An Elliptic Curve Family Test of the Ratios Conjecture

Steven J. Miller, Assistant Professor of Mathematics, with Duc Khiem Huynh and Ralph Morrison ‘10

Journal of Number Theory, 131, 1117-1147 (2011)

We compare the L-Function Ratios Conjectures prediction with number theory for the family of quadratic twists of a fixed elliptic curve with prime conductor, and show agreement in the 1-level density.

Demand-Driven Scheduling of Movies in a Multiplex

Steven J. Miller, Assistant Professor of Mathematics, with Jehoshua Eliashberg and Charles B. Weinberg

Newsletter of the European Marketing Academy (October 2010)

Summary of Silver-Scheduler paper in honor of it receiving the IJRM Best Paper Award for 2009.

Stable Constant Constant Mean Curvature Hypersurfaces are Area Minimizing in Small L1 Neighborhoods

Frank Morgan, Webster Atwell Class of 1921 Professor of Mathematics and Antonio Ros

Interfaces Free Boundaries, 151-155 (2010)

We prove that a strictly stable constant-mean-curvature hypersurface in a smooth manifold of dimension less than or equal to 7 is uniquely homologically area minimizing for fixed volume in a small L1 neighborhood.

Isoperimetric Sequences

Frank Morgan, Webster Atwell Class of 1921 Professor of Mathematics with Steven J. Miller, Assistant Professor of Mathematics, Edward Newkirk ‘09, Lori Pedersen, and Deividas Seferis ‘09

Math Magazine, 84, 37-42 (2011)

We generalize the isoperimetric problem from geometry to numbers.

Rebalance Every (15000/V)1/3 Years

Frank Morgan, Webster Atwell Class of 1921 Professor of Mathematics and Walter Filkins

SSRN (2010)

An original formula for how often to rebalance investments.

The Log-Convex Density Conjecture

Frank Morgan, Webster Atwell Class of 1921 Professor of Mathematics with Christian Houdré, Michel Ledoux, Emanuel Milman, and Mario Milman, eds.

Concentration, Functional Inequalities and Isoperimetry (Proc. Intl. Wkshp., Florida Atlantic Univ., Oct./Nov. 2009) Contemporary Mathematics, 545, Amer. Math. Soc. (2011)

A short exposition of a conjecture on when balls about the origin are isoperimetric in Rn with density.

Mixing on Rank-One Transformations

Cesar E. Silva, Hagey Family Professor of Mathematics with D. Creutz ’03

Studia Mathematica, 199, No. 1, 43-72 (2010)

We prove mixing on rank-one transformations is equivalent to “the uniform convergence of ergodic averages (as in the mean ergodic theorem) over subsequences of partial sums.”  In particular, all polynomial staircase transformations are mixing.

Dynamics of the p-adic Shift and Applications

Cesar E. Silva, Hagey Family Professor of Mathematics with J. Kingsbery ’06, A. Levin, and A. Preygel

Discrete and Continuous Dynamical Systems, 30, No. 1, 209-218 (2011)

There is a natural continuous realization of the one-sided Bernoulli shift on the p-adic integers as the map that shifts the coefficients of the p-adic expansion to the left.  We study this map’s Mahler power series expansion.  We prove strong results on p-adic valuations of the coefficients in this expansion, and show that certain natural maps (including many polynomials) are in a sense small perturbations of the shift.  As a result, these polynomials share the shift map’s important dynamical   properties.  This provides a novel approach to an earlier result of the authors.

Digraph Representations of Rational Functions Over the p-adic Numbers

Cesar E. Silva, Hagey Family Professor of Mathematics with Hansheng Diao

P-adic Numbers, Ultrametric Analysis, and Applications, 3, No. 1, 23-38 (2011)

In this paper, we construct a digraph structure on p-adic dynamical systems defined by rational functions.  We study the conditions under which the functions are measure-preserving, invertible and isometric, ergodic, and minimal on invariant subsets, by means of graph theoretic properties.