Math/Stat

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.

CSI: MSRI
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 p_k/q_k 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 q_k’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 + … + ε_k A|>|δ₁ A + … + δ_k A| a positive percent of the time for all nontrivial choices of ε_j,δ_j\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 + … + ε_k A| – |δ₁ A + … + δ_k 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/n^ε. 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 [F_n, F_{n+1}) is n / (j^{2 + 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 {F_n}_{n=1}^∞. Lekkerkerker \cite{Lek} proved the average number of summands for integers in [F_n, F_{n+1}) is n/(j² + 1), with phi the golden mean. This has been generalized: given nonnegative integers c₁,c₂,…,c_L with c₁,c_L>0 and recursive sequence {H_n}_{n=1}^∞ with H₁₌₁, H_{n+1} =c₁H_n + c₂H_{n-1} + … +c_nH₁₊₁ (1 ≤ ∞ n < L) and H_{n+1}=c₁H_n+c₂H_{n-1}+ … +c_LH_{n+1-L} (n ≥ L), every positive integer can be written uniquely as a sum of a_iH_i 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 ± F_n‘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 ℓ
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.

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