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Steven J. Miller

Publications and source records attributed to Steven J. Miller.

At least 235 records · Page 13Linked to original sources
arXiv2012

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

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.

stat.AP↗
arXiv2011

Generalized More Sums Than Differences Sets

A More Sums Than Differences (MSTD, or sum-dominant) set is a finite set $A\subset \mathbb{Z}$ 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 \cite{MO} proved 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 $|ε_1A+...+ε_kA|>|δ_1A+...+δ_kA|$ 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, notation as above we prove that for any $m$, $|ε_1A + ... + ε_kA| - |δ_1A + ... + δ_kA| = m$ a positive percentage of the time. We find the limiting behavior of $kA=A+...+A$ for an arbitrary set $A$ as $k\to\infty$ 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 the probability is at most $10^{-9}$, and almost surely significantly less). We prove the surprising result that for any $k$ a positive percentage of sets are $k$-generational, and no set can be $k$-generational for all $k$.

math.NT↗
arXiv2011

Low-lying Zeros of Cuspidal Maass Forms

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) agree 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 \cite{ILS} proved that the behavior of zeros near the central point of holomorphic cusp forms agree with the behavior of eigenvalues of orthogonal matrices for suitably restricted test functions. We prove a similar result for level 1 cuspidal Maass forms, the other natural family of ${\rm GL}_2$ $L$-functions. We use the explicit formula to relate sums of our test function at scaled zeros to sums of the Fourier transform at the primes weighted by the $L$-function coefficients, and then use the Kuznetsov trace formula to average the Fourier coefficients over the family. There are numerous technical obstructions in handling the terms in the trace formula, which are surmounted through the use of smooth weight functions for the Maass eigenvalues and results on Kloosterman sums and Bessel and hyperbolic functions.

math.NT↗
arXiv2011

A Random Matrix Model for Elliptic Curve L-Functions of Finite Conductor

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 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 tends 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.

math.NT↗
arXiv2011

Generalized Ramanujan Primes

In 1845, Bertrand conjectured that for all integers $x\ge2$, there exists at least one prime in $(x/2, x]$. This was proved by Chebyshev in 1860, and then generalized by Ramanujan in 1919. He showed that for any $n\ge1$, there is a (smallest) prime $R_n$ such that $π(x)- π(x/2) \ge n$ for all $x \ge R_n$. In 2009 Sondow called $R_n$ the $n$th Ramanujan prime and proved the asymptotic behavior $R_n \sim p_{2n}$ (where $p_m$ is the $m$th prime). In the present paper, we generalize the interval of interest by introducing a parameter $c \in (0,1)$ and defining the $n$th $c$-Ramanujan prime as the smallest integer $R_{c,n}$ such that for all $x\ge R_{c,n}$, there are at least $n$ primes in $(cx,x]$. Using consequences of strengthened versions of the Prime Number Theorem, we prove that $R_{c,n}$ exists for all $n$ and all $c$, that $R_{c,n} \sim p_{\frac{n}{1-c}}$ as $n\to\infty$, and that the fraction of primes which are $c$-Ramanujan converges to $1-c$. We then study finer questions related to their distribution among the primes, and see that the $c$-Ramanujan primes display striking behavior, deviating significantly from a probabilistic model based on biased coin flipping; this was first observed by Sondow, Nicholson, and Noe in the case $c = 1/2$. This model is related to the Cramer model, which correctly predicts many properties of primes on large scales, but has been shown to fail in some instances on smaller scales.

math.NT↗
arXiv2011

Virus Dynamics on Starlike Graphs

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.

math.DS↗
arXiv2011

From Fibonacci Numbers to Central Limit Type Theorems

A beautiful theorem of Zeckendorf states that every integer can be written uniquely as a sum of non-consecutive Fibonacci numbers $\{F_n\}_{n=1}^{\infty}$. Lekkerkerker proved that the average number of summands for integers in $[F_n, F_{n+1})$ is $n/(ϕ^2 + 1)$, with $ϕ$ the golden mean. This has been generalized to the following: given nonnegative integers $c_1,c_2,...,c_L$ with $c_1,c_L>0$ and recursive sequence $\{H_n\}_{n=1}^{\infty}$ with $H_1=1$, $H_{n+1} =c_1H_n+c_2H_{n-1}+...+c_nH_1+1$ $(1\le n< L)$ and $H_{n+1}=c_1H_n+c_2H_{n-1}+...+c_LH_{n+1-L}$ $(n\geq L)$, every positive integer can be written uniquely as $\sum a_iH_i$ under natural constraints on the $a_i$'s, the mean and the variance of the numbers of summands for integers in $[H_{n}, H_{n+1})$ are of size $n$, and the distribution of the numbers of summands converges to a Gaussian as $n$ goes to the infinity. 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 a multitude of other problems (in the sequel paper \cite{BM} we show how this perspective allows us to determine the distribution of gaps between summands in decompositions). For example, it is known that every integer can be written uniquely as a sum of the $\pm F_n$'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 $-(21-2ϕ)/(29+2ϕ) \approx -0.551058$.

math.NT↗
arXiv2011

Gaussian Behavior in Generalized Zeckendorf Decompositions

A beautiful theorem of Zeckendorf states that every integer can be written uniquely as a sum of non-consecutive Fibonacci numbers $\{F_n\}_{n=1}^{\infty}$; Lekkerkerker proved that the average number of summands for integers in $[F_n, F_{n+1})$ is $n/(ϕ^2 + 1)$, with $ϕ$ the golden mean. Interestingly, the higher moments seem to have been ignored. We discuss the proof that the distribution of the number of summands converges to a Gaussian as $n \to \infty$, and comment on generalizations to related decompositions. For example, every integer can be written uniquely as a sum of the $\pm F_n$'s, such that every two terms of the same (opposite) sign differ in index by at least 4 (3). The distribution of the numbers of positive and negative summands converges to a bivariate normal with computable, negative correlation, namely $-(21-2ϕ)/(29+2ϕ) \approx -0.551058$.

math.NT↗
arXiv2011

Finding and Counting MSTD sets

We review the basic theory of More Sums Than Differences (MSTD) sets, specifically their existence, simple constructions of infinite families, the proof that a positive percentage of sets under the uniform binomial model are MSTD but not if the probability that each element is chosen tends to zero, and 'explicit' constructions of large families of MSTD sets. We conclude with some 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 (meaning $A$, $A+A$, $...$, $kA = A + ...+A$ are each sum-dominant).

math.NT↗
arXiv2011

The Limiting Spectral Measure for Ensembles of Symmetric Block Circulant Matrices

Given an ensemble of NxN 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 --> oo. 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 f_m 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 f_m is the product of a Gaussian and a degree 2m-2 polynomial; the formula equals that of the m x m Gaussian Unitary Ensemble (GUE). The proof is by the moments. 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.

math.PR↗
arXiv2011

An elliptic curve test of the L-Functions Ratios Conjecture

We compare the L-Function Ratios Conjecture's 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 up to an error term of size X^{-(1-sigma)/2} for test functions supported in (-sigma, sigma); this gives us a power-savings for σ<1. This test of the Ratios Conjecture introduces complications not seen in previous cases (due to the level of the elliptic curve). Further, the results here are one of the key ingredients in the companion paper [DHKMS2], where they are used to determine the effective matrix size for modeling zeros near the central point for this family. The resulting model beautifully describes the behavior of these low lying zeros for finite conductors, explaining the data observed by Miller in [Mil3]. A key ingredient in our analysis is a generalization of Jutila's bound for sums of quadratic characters with the additional restriction that the fundamental discriminant be congruent to a non-zero square modulo a square-free integer M. This bound is needed for two purposes. The first is to analyze the terms in the explicit formula corresponding to characters raised to an odd power. The second is to determine the main term in the 1-level density of quadratic twists of a fixed form on GL_n. Such an analysis was performed by Rubinstein [Rub], who implicitly assumed that Jutila's bound held with the additional restriction on the fundamental discriminants; in this paper we show that assumption is justified.

math.NT↗
arXiv2010

If a prime divides a product

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.

math.HO↗
arXiv2010

The `Real' Schwarz Lemma

The purpose of this note is to discuss the real analogue of the Schwarz lemma from complex analysis.

math.HO↗
arXiv2010

Modeling Convolutions of $L$-Functions

A number of mathematical methods have been shown to model the zeroes of $L$-functions with remarkable success, including the Ratios Conjecture and Random Matrix Theory. In order to understand the structure of convolutions of families of $L$-functions, we investigate how well these methods model the zeros of such functions. Our primary focus is the convolution of the $L$-function associated to Ramanujan's tau function with the family of quadratic Dirichlet $L$-functions, for which J.B. Conrey and N.C. Snaith computed the Ratios Conjecture's prediction. Our main result is performing the number theory calculations and verifying these predictions for the one-level density for suitably restricted test functions up to square-root error term. Unlike Random Matrix Theory, which only predicts the main term, the Ratios Conjecture detects the arithmetic of the family and makes detailed predictions about their dependence in the lower order terms. Interestingly, while Random Matrix Theory is frequently used to model behavior of L-functions (or at least the main terms), there has been little if any work on the analogue of convolving families of L-functions by convolving random matrix ensembles. We explore one possibility by considering Kronecker products; unfortunately, it appears that this is not the correct random matrix analogue to convolving families.

math.NT↗
arXiv2010

The lowest eigenvalue of Jacobi random matrix ensembles and Painlevé VI

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. The first method solves an associated Painleve VI nonlinear differential equation numerically, with suitable initial conditions that we determine. The second method proceeds via constructing the power-series expansion of the Painleve VI function. Our results are applied in a forthcoming paper in which we model the distribution of the first zero above the central point of elliptic curve L-function families of finite conductor and of conjecturally orthogonal symmetry.

math.CA↗
arXiv2010

Distribution of Eigenvalues of Highly Palindromic Toeplitz Matrices

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 [BDJ,HM] showed that the limiting spectral measures (the density of normalized eigenvalues) converge weakly 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 (see [MMS]) 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 an universal distribution with a fatter tail than any previously seen limiting spectral measure.

math.PR↗
arXiv2010

On the number of summands in Zeckendorf decompositions

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 $n / (φ^2 + 1) + O(1)$, where $φ= \frac{1+\sqrt{5}}2$ 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 $n\to\infty$. 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; see [MW] for proofs in the general case.

math.NT↗