808 results on '"Miller, Steven"'
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2. Geometric Proof of the Irrationality of Square-Roots for Select Integers
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Chen, Zongyun, Miller, Steven J., and Wu, Chenghan
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Mathematics - History and Overview - Abstract
This paper presents geometric proofs for the irrationality of square roots of select integers, extending classical approaches. Building on known geometric methods for proving the irrationality of sqrt(2), the authors explore whether similar techniques can be applied to other non-square integers. They begin by reviewing well-known results, such as Euclid's proof for the irrationality of sqrt(2), and discuss subsequent geometric extensions for sqrt(3), sqrt(5), and sqrt(6). The authors then introduce new geometric constructions, particularly using hexagons, to prove the irrationality of sqrt(6). Furthermore, the paper investigates the limitations and challenges of extending these geometric methods to triangular numbers. Through detailed geometric reasoning, the authors successfully generalize the approach to several square-free numbers and identify cases where the method breaks down. The paper concludes by inviting further exploration of geometric irrationality proofs for other integers, proposing potential avenues for future work., Comment: 11 pages, 8 figures
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- 2024
3. Lower Order Biases in Moment Expansions of One Parameter Families of Elliptic Curves
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Cheek, Timothy, Gilman, Pico, Jaber, Kareem, Miller, Steven J., Sharan, Vismay, and Tomé, Marie-Hélène
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Mathematics - Number Theory ,Mathematics - Algebraic Geometry ,11G05, 11G40 - Abstract
For a fixed elliptic curve $E$ without complex multiplication, $a_p := p+1 - \#E(\mathbb{F}_p)$ is $O(\sqrt{p})$ and $a_p/2\sqrt{p}$ converges to a semicircular distribution. Michel proved that for a one-parameter family of elliptic curves $y^2 = x^3 + A(T)x + B(T)$ with $A(T), B(T) \in \mathbb{Z}[T]$ and non-constant $j$-invariant, the second moment of $a_p(t)$ is $p^2 + O(p^{{3}/{2}})$. The size and sign of the lower order terms has applications to the distribution of zeros near the central point of Hasse-Weil $L$-functions and the Birch and Swinnerton-Dyer conjecture. S. J. Miller conjectured that the highest order term of the lower order terms of the second moment that does not average to zero is on average negative. Previous work on the conjecture has been restricted to a small set of highly nongeneric families. We create a database and a framework to quickly and systematically investigate biases in the second moment of any one-parameter family. When looking at families which have so far been beyond current theory, we find several potential violations of the conjecture for $p \leq 250,000$ and discuss new conjectures motivated by the data., Comment: 16 pages, 18 figures
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- 2024
4. Black Hole Zeckendorf Games
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Cashman, Caroline, Miller, Steven J., Shuffleton, Jenna, and Son, Daeyoung
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Mathematics - Number Theory - Abstract
Zeckendorf proved a remarkable fact that every positive integer can be written as a decomposition of non-adjacent Fibonacci numbers. Baird-Smith, Epstein, Flint, and Miller converted the process of decomposing a positive integer into its Zeckendorf decomposition into a game, using the moves of $F_i + F_{i-1} = F_{i+1}$ and $2F_i = F_{i+1} + F_{i-2}$, where $F_i$ is the $i$thFibonacci number. Players take turns applying these moves, beginning with $n$ pieces in the $F_1$ column. They showed that for $n \neq 2$, Player 2 has a winning strategy, though the proof is non-constructive, and a constructive solution is unknown. We expand on this by investigating "black hole'' variants of this game. The Black Hole Zeckendorf game on $F_m$ is played with any $n$ but solely in columns $F_i$ for $i < m$. Gameplay is similar to the original Zeckendorf game, except any piece that would be placed on $F_i$ for $i \geq m$ is locked out in a ``black hole'' and removed from play. With these constraints, we analyze the games with black holes on $F_3$ and $F_4$ and construct a solution for specific configurations, using a parity-stealing based non-constructive proof to lead to a constructive one. We also examine a pre-game in which players take turns placing down $n$ pieces in the outermost columns before the decomposition phase, and find constructive solutions for any $n$., Comment: 40 pages, 51 figures
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- 2024
5. A Spectral Representation of a Weighted Random Vectorial Field: Potential Applications to Turbulence and the Problem of Anomalous Dissipation in the Inviscid Limit
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Miller, Steven D
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Mathematics - Probability ,Mathematical Physics - Abstract
Let ${\mathfrak{G}}\subset\mathbb{R}^{3}$ with $vol(\mathfrak{G})\sim L^{3}$. Let ${\mathscr{T}}(x)$ be a Gaussian random field $\forall~x\in\mathfrak{G}$ with expectation $\mathbf{E}[{\mathscr{T}}(x)]=0$ and correlation $\mathbf{E}[{\mathscr{T}}(x)\otimes{\mathscr{T}}(y)]=K(x,y;\lambda)$, an isotropic and regulated kernel with correlation length $\lambda$. The field has a Karhunen-Loeve spectral representation ${\mathscr{T}}(x)=\sum_{I=1}^{\infty}\mathrm{Z}^{1/2}_{I}f_{I}(x)\otimes\mathscr{Z}_{I}$, with eigenvalues $\lbrace\mathrm{Z}_{I}\rbrace$, eigenfunctions $\lbrace f_{I}(x)\rbrace $ and Gaussian random variables $\mathscr{Z}_{I}$ with $\mathbf{E}[\mathscr{Z}_{I}]=0$ and $\mathbf{E}[\mathscr{Z}_{I}\otimes\mathscr{Z}_{J}]=\delta_{IJ}$. If $\mathfrak{G}$ contains incompressible fluid of viscosity $\nu$ with velocity $u_{a}(x,t)$ that evolves via the Navier-Stokes equations with a high 'Reynolds function' $\mathsf{RE}(x,t)=\tfrac{\|u_{a}(x,t)\|L}{\nu} $ then aspects of a turbulent flow with $\mathsf{RE}(x,t)\gg \mathsf{RE}_{*}$, a critical Reynolds number, might be represented by the 'weighted' random field $\mathscr{U}_{a}(x,t)= u_{a}(x,t)+\mathrm{A}u_{a}(x,t)\big(\mathsf{RE}(x,t)-\mathsf{RE}_{*}\big)^{\beta}\sum_{I=1}^{\infty} \mathrm{Z}^{1/2}_{I}f_{I}(x)\otimes\mathscr{Z}_{I}$ where random fluctuations and amplitude scale nonlinearly with $\mathsf{RE}(x,t)$, with mean $\mathbf{E}[{\mathscr{U}}_{a}(x,t)] =u_{a}(x,t)$. In the inviscid limit one can prove an anomalous dissipation-type law \begin{align} \lim_{\nu\rightarrow 0}\bigg(\lim_{u_{a}(x,t)\rightarrow {u}_{a}}\sup~\nu \int_{\mathfrak{G}}\int_{0}^{T}{\mathbf{E}}\bigg[\bigg|{\nabla}_{a}{\mathscr{U}}_{a}(x,s)\bigg|^{2}\bigg]d\mathcal{V}(x) ds\bigg)>0 \end{align} iff $\beta=\tfrac{1}{2}$ and $\sum_{I=1}^{\infty}\mathrm{Z}_{I}\int_{{\mathfrak{G}}}{\nabla}_{a}f_{I}(x){\nabla}^{a}f_{I}(x)d\mathcal{V}(x)>0$., Comment: 58 pages
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- 2024
6. Hyper-bishops, Hyper-rooks, and Hyper-queens: Percentage of Safe Squares on Higher Dimensional Chess Boards
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Cashman, Caroline, Cooper, Joseph, Marquez, Raul, Miller, Steven J., and Shuffelton, Jenna
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Mathematics - Combinatorics ,Mathematics - Probability ,60C05, 05A16 (primary) - Abstract
The $n$ queens problem considers the maximum number of safe squares on an $n \times n$ chess board when placing $n$ queens; the answer is only known for small $n$. Miller, Sheng and Turek considered instead $n$ randomly placed rooks, proving the proportion of safe squares converges to $1/e^2$. We generalize and solve when randomly placing $n$ hyper-rooks and $n^{k-1}$ line-rooks on a $k$-dimensional board, using combinatorial and probabilistic methods, with the proportion of safe squares converging to $1/e^k$. We prove that the proportion of safe squares on an $n \times n$ board with bishops in 2 dimensions converges to $2/e^2$. This problem is significantly more interesting and difficult; while a rook attacks the same number of squares wherever it's placed, this is not so for bishops. We expand to the $k$-dimensional chessboard, defining line-bishops to attack along $2$-dimensional diagonals and hyper-bishops to attack in the $k-1$ dimensional subspace defined by its diagonals in the $k-2$ dimensional subspace. We then combine the movement of rooks and bishops to consider the movement of queens in 2 dimensions, as well as line-queens and hyper-queens in $k$ dimensions., Comment: 19 pages, 6 figures
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- 2024
7. Stability of Matrix Recurrence Relations
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Bruda, Glenn, Fang, Bruce, Gilman, Pico, Marquez, Raul, Miller, Steven J., Prapashtica, Beni, Son, Daeyoung, Waheed, Saad, and Wang, Janine
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Mathematics - Combinatorics ,11B37, 11B39, 15A24 - Abstract
Motivated by the rich properties and various applications of recurrence relations, we consider the extension of traditional recurrence relations to matrices, where we use matrix multiplication and the Kronecker product to construct matrix sequences. We provide a sharp condition, which when satisfied, guarantees that any fixed-depth matrix recurrence relation defined over a product (with respect to matrix multiplication) will converge to the zero matrix. We also show that the same statement applies to matrix recurrence relations defined over a Kronecker product. Lastly, we show that the dual of this condition, which remains sharp, guarantees the divergence of matrix recurrence relations defined over a consecutive Kronecker product. These results completely determine the stability of nontrivial fixed-depth complex-valued recurrence relations defined over a consecutive product.
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- 2024
8. A Pair of Diophantine Equations Involving the Fibonacci Numbers
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Chen, Xuyuan, Chu, Hung Viet, Kesumajana, Fadhlannafis K., Kim, Dongho, Li, Liran, Miller, Steven J., Yang, Junchi, and Yao, Chris
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Mathematics - Number Theory ,11B39, 11D04 - Abstract
Let $a, b\in \mathbb{N}$ be relatively prime. Previous work showed that exactly one of the two equations $ax + by = (a-1)(b-1)/2$ and $ax + by + 1 = (a-1)(b-1)/2$ has a nonnegative, integral solution; furthermore, the solution is unique. Let $F_n$ be the $n$th Fibonacci number. When $(a,b) = (F_n, F_{n+1})$, it is known that there is an explicit formula for the unique solution $(x,y)$. We establish formulas to compute the solution when $(a,b) = (F_n^2, F_{n+1}^2)$ and $(F_n^3, F_{n+1}^3)$, giving rise to some intriguing identities involving Fibonacci numbers. Additionally, we construct a different pair of equations that admits a unique positive (instead of nonnegative), integral solution., Comment: 12 pages
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- 2024
9. On the Density of Low Lying Zeros of a Large Family of Automorphic $L$-functions
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Cheek, Timothy, Gilman, Pico, Jaber, Kareem, Miller, Steven J., and Tomé, Marie-Hélène
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Mathematics - Number Theory ,11M41 (Primary), 60B20 (Secondary) - Abstract
Under the generalized Riemann Hypothesis (GRH), Baluyot, Chandee, and Li nearly doubled the range in which the density of low lying zeros predicted by Katz and Sarnak is known to hold for a large family of automorphic $L$-functions with orthogonal symmetry. We generalize their main techniques to the study of higher centered moments of the one-level density of this family, leading to better results on the behavior near the central point. Numerous technical obstructions emerge that are not present in the one-level density. Averaging over the level of the forms and assuming GRH, we prove the density predicted by Katz and Sarnak holds for the $n$-th centered moments for test functions whose Fourier transform is compactly supported in $(-\sigma, \sigma)$ for $\sigma~=~\min\left\{3/2(n-1), 4/(2n-\mathbf{1}_{2\nmid n})\right\}$. For $n=3$, our results improve the previously best known $\sigma=2/3$ to $\sigma=3/4$. We also prove the two-level density agrees with the Katz-Sarnak density conjecture for test functions whose Fourier transform is compactly supported in $\sigma_1 = 3/2$ and $\sigma_2 = 5/6$, respectively, extending the previous best known sum of supports $\sigma_1 + \sigma_2 = 2$. This work is the first evidence of an interesting new phenomenon: by taking different test functions, we are able to extend the range in which the Katz-Sarnak density predictions hold. The techniques we develop can be applied to understanding quantities related to this family containing sums over multiple primes., Comment: 36 pages
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- 2024
10. Variants of Conway Checkers and k-nacci Jumping
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Bruda, Glenn, Cooper, Joseph, Jaber, Kareem, Marquez, Raul, and Miller, Steven J.
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Mathematics - Combinatorics - Abstract
Conway Checkers is a game played with a checker placed in each square of the lower half of an infinite checkerboard. Pieces move by jumping over an adjacent checker, removing the checker jumped over. Conway showed that it is not possible to reach row 5 in finitely many moves by weighting each cell in the board by powers of the golden ratio such that no move increases the total weight. Other authors have considered the game played on many different boards, including generalising the standard game to higher dimensions. We work on a board of arbitrary dimension, where we allow a cell to hold multiple checkers and begin with m checkers on each cell. We derive an upper bound and a constructive lower bound on the height that can be reached, such that the upper bound almost never fails to be equal to the lower bound. We also consider the more general case where instead of jumping over 1 checker, each checker moves by jumping over k checkers, and again show the maximum height reachable lies within bounds that are almost always equal., Comment: 17 pages, 7 figures
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- 2024
11. Sum of Consecutive Terms of Pell and Related Sequences
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Anand, Navvye, Basistha, Amit Kumar, Davenport, Kenny B., Gong, Alexander, Miller, Steven J., and Zhu, Alexander
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Mathematics - Number Theory ,Mathematics - Combinatorics ,11Bxx, 11B37, 11B39, 11B50 - Abstract
We study new identities related to the sums of adjacent terms in the Pell sequence, defined by $P_{n} := 2P_{n-1}+P_{n-2}$ for $ n\geq 2$ and $P_{0}=0, P_{1}=1$, and generalize these identities for many similar sequences. We prove that the sum of $N>1$ consecutive Pell numbers is a fixed integer multiple of another Pell number if and only if $4\mid N$. We consider the generalized Pell $(k,i)$-numbers defined by $p(n) :=\ 2p(n-1)+p(n-k-1) $ for $n\geq k+1$, with $p(0)=p(1)=\cdots =p(i)=0$ and $p(i+1)=\cdots = p(k)=1$ for $0\leq i\leq k-1$, and prove that the sum of $N=2k+2$ consecutive terms is a fixed integer multiple of another term in the sequence. We also prove that for the generalized Pell $(k,k-1)$-numbers such a relation does not exist when $N$ and $k$ are odd. We give analogous results for the Fibonacci and other related second-order recursive sequences., Comment: 33 Pages. Comments welcome!
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- 2024
12. A Random Matrix Model for a Family of Cusp Forms
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Barrett, Owen, Batterman, Zoë X., Jambhale, Aditya, Miller, Steven J., Narayanan, Akash L., Sharma, Kishan, and Yao, Chris
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Mathematics - Number Theory ,11M26, 11M50, 15B52, 15B10 - Abstract
The Katz-Sarnak philosophy states that statistics of zeros of $L$-function families near the central point as the conductors tend to infinity agree with those of eigenvalues of random matrix ensembles as the matrix size tends to infinity. While numerous results support this conjecture, S. J. Miller observed that for finite conductors, very different behavior can occur for zeros near the central point in elliptic curve $L$-function families. This led to the creation of the excised model of Due\~{n}ez, Huynh, Keating, Miller, and Snaith, whose predictions for quadratic twists of a given elliptic curve are well fit by the data. The key ingredients are relating the discretization of central values of the $L$-functions to excising matrices based on the value of the characteristic polynomials at 1 and using lower order terms (in statistics such as the one-level density and pair-correlation) to adjust the matrix size. We extended this model for a family of twists of an $L$-function associated to a given holomorphic cuspidal newform of odd prime level and arbitrary weight. We derive the corresponding "effective" matrix size for a given form by computing the one-level density and pair-correlation statistics for a chosen family of twists, and we show there is no repulsion for forms with weight greater than 2 and principal nebentype. We experimentally verify the accuracy of the model, and as expected, our model recovers the elliptic curve model., Comment: 55 pages, 13 figures. arXiv admin note: substantial text overlap with arXiv:2402.06641
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- 2024
13. On the Within-perfect Numbers
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Kwan, Chung-Hang and Miller, Steven J.
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Mathematics - Number Theory ,11A25 (primary), 11N25, 11B83 (secondary) - Abstract
Motivated by the works of Erd\"os, Pomerance, Wolke and Harman on the sum-of-divisor function $\sigma(n)$, we study the distribution of a special class of natural numbers closely related to (multiply) perfect numbers which we term `$(\ell;k)$-within-perfect numbers', where $\ell >1$ is a real number and $k: [1, \infty) \rightarrow (0, \infty)$ is an increasing and unbounded function., Comment: Minor changes, Accepted for publication. This article together with arXiv:1610.04253v4 (published in Acta Arithmetica 194 (2020), 341-366) supersede arXiv:1610.04253v2
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- 2024
14. Upper Bounds for the Lowest First Zero in Families of Cuspidal Newforms
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Tang, Xueyiming and Miller, Steven J.
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Mathematics - Number Theory ,11M41 (primary), 60B20 (secondary) - Abstract
Assuming the Generalized Riemann Hypothesis, the non-trivial zeros of $L$-functions lie on the critical line with the real part $1/2$. We find an upper bound of the lowest first zero in families of even cuspidal newforms of prime level tending to infinity. We obtain explicit bounds using the $n$-level densities and results towards the Katz-Sarnak density conjecture. We prove that as the level tends to infinity, there is at least one form with a normalized zero within $1/4$ of the average spacing. We also obtain the first-ever bounds on the percentage of forms in these families with a fixed number of zeros within a small distance near the central point., Comment: Version 1.0, 18 pages, 2 figures
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- 2024
15. $v$-Palindromes: An Analogy to the Palindromes
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Bispels, Chris, Boran, Muhammet, Miller, Steven J., Sosis, Eliel, and Tsai, Daniel
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Mathematics - History and Overview ,Mathematics - Number Theory ,(Primary) 11A63, (Secondary) 11A25, 11A51 - Abstract
Around the year 2007, one of the authors, Tsai, accidentally discovered a property of the number $198$ he saw on the license plate of a car. Namely, if we take $198$ and its reversal $891$, which have prime factorizations $198 = 2\cdot 3^2\cdot 11$ and $891 = 3^4\cdot 11$ respectively, and sum the numbers appearing in each factorization getting $2+3+2+11 = 18$ and $3+4+11 = 18$, both sums are $18$. Such numbers were later named $v$-palindromes because they can be viewed as an analogy to the usual palindromes. In this article, we introduce the concept of a $v$-palindrome in base $b$ and prove their existence for infinitely many bases. We also exhibit infinite families of $v$-palindromes in bases $p+1$ and $p^2+1$, for each odd prime $p$. Finally, we collect some conjectures and problems involving $v$-palindromes., Comment: 22 pages, 2 figures, 1 table. arXiv admin note: substantial text overlap with arXiv:2111.10211
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- 2024
16. The German Tank Problem with Multiple Factories
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Miller, Steven J., Sharma, Kishan, and Yang, Andrew K.
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Mathematics - Statistics Theory - Abstract
During the Second World War, estimates of the number of tanks deployed by Germany were critically needed. The Allies adopted two methods to estimate this information: espionage and statistical analysis. The latter approach was far more successful and is as follows: assuming that the tanks are sequentially numbered starting from 1, if we observe $k$ serial numbers from an unknown total of $N$ tanks, with the highest observed number being $M$, then the best linear unbiased estimator for $N$ is $M(1+1/k)-1$. This is now known as the German Tank Problem. Suppose one wishes to estimate the productivity of a rival by inspecting captured or destroyed tanks, each with a unique serial number. In many situations, the original German Tank Problem is insufficient, since typically there are $l>1$ factories, and tanks produced by different factories may have serial numbers in disjoint ranges that are often far separated, let alone sequentially numbered starting from 1. We wish to estimate the total tank production across all of the factories. We construct an efficient procedure to estimate the total productivity and prove that our procedure effectively estimates $N$ when $\log l/\log k$ is sufficiently small, and is robust against both large and small gaps between factories. In the final section, we show that given information about the gaps, we can make a far better estimator that is also effective when we have a small number of samples. When the number of samples is small compared to the number of gaps, the Mean Squared Error of this new estimator is several orders of magnitude smaller than the one that assumes no information. This quantifies the importance of hiding such information if one wishes to conceal their productivity from a rival.
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- 2024
17. Training in neonatal neurocritical care: a proposal for a hybrid model of competence by design and time-based methods
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Mohammad, Khorshid, Craig, Alexa K., Chang, Taeun, Tam, Emily W. Y., Ayed, Mariam, de Vries, Linda S., El-Dib, Mohamed A., Esser, Michael J., Ferriero, Donna M., Hellström-Westas, Lena, Miller, Steven P., Soul, Janet S., Vollmer, Brigitte, Glass, Hannah C., and Smyser, Christopher D.
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- 2024
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18. Limiting Behavior in Missing Sums of Sumsets
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Jambhale, Aditya, Kaldybayev, Rauan, Miller, Steven J., and Yao, Chris
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Mathematics - Number Theory ,11P99, 11B13 - Abstract
We study $|A + A|$ as a random variable, where $A \subseteq \{0, \dots, N\}$ is a random subset such that each $0 \le n \le N$ is included with probability $0 < p < 1$, and where $A + A$ is the set of sums $a + b$ for $a,b$ in $A$. Lazarev, Miller, and O'Bryant studied the distribution of $2N + 1 - |A + A|$, the number of summands not represented in $A + A$ when $p = 1/2$. A recent paper by Chu, King, Luntzlara, Martinez, Miller, Shao, Sun, and Xu generalizes this to all $p\in (0,1)$, calculating the first and second moments of the number of missing summands and establishing exponential upper and lower bounds on the probability of missing exactly $n$ summands, mostly working in the limit of large $N$. We provide exponential bounds on the probability of missing at least $n$ summands, find another expression for the second moment of the number of missing summands, extract its leading-order behavior in the limit of small $p$, and show that the variance grows asymptotically slower than the mean, proving that for small $p$, the number of missing summands is very likely to be near its expected value., Comment: 25 pages, 6 figures
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- 2024
19. A Survey of a Random Matrix Model for a Family of Cusp Forms
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Barrett, Owen, Batterman, Zoë X., Jambhale, Aditya, Miller, Steven J., Narayanan, Akash L., Sharma, Kishan, and Yao, Chris
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Mathematics - Number Theory ,11M26, 11M50 - Abstract
The Katz-Sarnak philosophy states that statistics of zeros of $L$-function families near the central point as the conductors tend to infinity agree with those of eigenvalues of random matrix ensembles as the matrix size tends to infinity. While numerous results support this conjecture, S. J. Miller observed that for finite conductors, very different behavior can occur for zeros near the central point in elliptic curve families. This led to the excised model of Due\~{n}ez, Huynh, Keating, Miller, and Snaith, whose predictions for quadratic twists of a given elliptic curve are beautifully fit by the data. The key ingredients are relating the discretization of central values of the $L$-functions to excising matrices based on the value of the characteristic polynomials at 1 and using lower order terms (in statistics such as the one-level density and pair-correlation) to adjust the matrix size. We discuss recent successes by the authors in extending this model to a family of quadratic twists of finite conductor of a given holomorphic cuspidal newform of level an odd prime level. In particular, we predict very little repulsion for forms with weight greater than 2., Comment: 28 pages, 7 figures
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- 2024
20. Applications of Moments of Dirichlet Coefficients in Elliptic Curve Families
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Batterman, Zoë, Jambhale, Aditya, Miller, Steven J., Narayanan, Akash L., Sharma, Kishan, Yang, Andrew, and Yao, Chris
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Mathematics - Number Theory ,Mathematics - Numerical Analysis ,11G05, 11G40 - Abstract
The moments of the coefficients of elliptic curve L-functions are related to numerous arithmetic problems. Rosen and Silverman proved a conjecture of Nagao relating the first moment of one-parameter families satisfying Tate's conjecture to the rank of the corresponding elliptic surface over Q(T); one can also construct families of moderate rank by finding families with large first moments. Michel proved that if j(T) is not constant, then the second moment of the family is of size p^2 + O(p^(3/2)); these two moments show that for suitably small support the behavior of zeros near the central point agree with that of eigenvalues from random matrix ensembles, with the higher moments impacting the rate of convergence. In his thesis, Miller noticed a negative bias in the second moment of every one-parameter family of elliptic curves over the rationals whose second moment had a calculable closed-form expression, specifically the first lower order term which does not average to zero is on average negative. This Bias Conjecture is confirmed for many families; however, these are highly non-generic families whose resulting Legendre sums can be determined. Inspired by the recent successes by Yang-Hui He, Kyu-Hwan Lee, Thomas Oliver, Alexey Pozdnyakov and others in investigations of murmurations of elliptic curve coefficients with machine learning techniques, we pose a similar problem for trying to understand the Bias Conjecture. As a start to this program, we numerically investigate the Bias Conjecture for a family whose bias is positive for half the primes. Since the numerics do not offer conclusive evidence that negative bias for the other half is enough to overwhelm the positive bias, the Bias Conjecture cannot be verified for the family.
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- 2023
21. Applications of Improvements to the Pythagorean Won-Loss Expectation in Optimizing Rosters
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Almeida, Alexander F., Dayaratna, Kevin, Miller, Steven J., and Yang, Andrew K.
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Statistics - Applications - Abstract
Bill James' Pythagorean formula has for decades done an excellent job estimating a baseball team's winning percentage from very little data: if the average runs scored and allowed are denoted respectively by ${\rm RS}$ and ${\rm RA}$, there is some $\gamma$ such that the winning percentage is approximately ${\rm RS}^\gamma / ({\rm RS}^\gamma + {\rm RA}^\gamma)$. One important consequence is to determine the value of different players to the team, as it allows us to estimate how many more wins we would have given a fixed increase in run production. We summarize earlier work on the subject, and extend the earlier theoretical model of Miller (who estimated the run distributions as arising from independent Weibull distributions with the same shape parameter; this has been observed to describe the observed run data well). We now model runs scored and allowed as being drawn from independent Weibull distributions where the shape parameter is not necessarily the same, and then use the Method of Moments to solve a system of four equations in four unknowns. Doing so yields a predicted winning percentage that is consistently better than earlier models over the last 30 MLB seasons (1994 to 2023). This comes at a small cost as we no longer have a closed form expression but must evaluate a two-dimensional integral of two Weibull distributions and numerically estimate the solutions to the system of equations; as these are trivial to do with simple computational programs it is well worth adopting this framework and avoiding the issues of implementing the Method of Least Squares or the Method of Maximum Likelihood.
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- 2023
22. Sums of Powers of Primes in Arithmetic Progression
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Boran, Muhammet, Byun, John, Li, Zhangze, Miller, Steven J., and Reyes, Stephanie
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Mathematics - Number Theory ,(Primary) 11N13, (Secondary) 11N05 - Abstract
Gerard and Washington proved that, for $k > -1$, the number of primes less than $x^{k+1}$ can be well approximated by summing the $k$-th powers of all primes up to $x$. We extend this result to primes in arithmetic progressions: we prove that the number of primes $p\equiv n \pmod m$ less than $x^{k+1}$ is asymptotic to the sum of $k$-th powers of all primes $p\equiv n \pmod m$ up to $x$. We prove that the prime power sum approximation tends to be an underestimate for positive $k$ and an overestimate for negative $k$, and quantify for different values of $k$ how well the approximation works for $x$ between $10^4$ and $10^8.$, Comment: 19 pages, 16 tables
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- 2023
23. Dynamics of the Fibonacci Order of Appearance Map
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FitzGibbons, Molly, Miller, Steven J., and Verga, Amanda
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Mathematics - Number Theory ,60B10, 11B39 (primary) 65Q30 (secondary) - Abstract
The \textit{order of appearance} $ z(n) $ of a positive integer $ n $ in the Fibonacci sequence is defined as the smallest positive integer $ j $ such that $ n $ divides the $ j $-th Fibonacci number. A \textit{fixed point} arises when, for a positive integer $ n $, we have that the $ n^{\text{th}} $ Fibonacci number is the smallest Fibonacci that $ n $ divides. In other words, $ z(n) = n $. In 2012, Marques proved that fixed points occur only when $ n $ is of the form $ 5^{k} $ or $ 12\cdot5^{k} $ for all non-negative integers $ k $. It immediately follows that there are infinitely many fixed points in the Fibonacci sequence. We prove that there are infinitely many integers that iterate to a fixed point in exactly $ k $ steps. In addition, we construct infinite families of integers that go to each fixed point of the form $12 \cdot 5^{k}$. We conclude by providing an alternate proof that all positive integers $n$ reach a fixed point after a finite number of iterations., Comment: 10 pages, 2 figures
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- 2023
24. The Reversed Zeckendorf Game
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Batterman, Zoë X., Jambhale, Aditya, Miller, Steven J., Narayanan, Akash L., Sharma, Kishan, Yang, Andrew K., and Yao, Chris
- Subjects
Mathematics - Combinatorics ,Mathematics - Number Theory ,11B39 (Primary), 05C57, 65Q30, 91A05, 91A46 (Secondary) - Abstract
Zeckendorf proved that every natural number $n$ can be expressed uniquely as a sum of non-consecutive Fibonacci numbers, called its Zeckendorf decomposition. Baird-Smith, Epstein, Flint, and Miller created the Zeckendorf game, a two-player game played on partitions of $n$ into Fibonacci numbers which always terminates at a Zeckendorf decomposition, and proved that Player 2 has a winning strategy for $n\geq 3$. Since their proof was non-constructive, other authors have studied the game to find a constructive winning strategy, and lacking success there turned to related problems. For example, Cheigh, Moura, Jeong, Duke, Milgrim, Miller, and Ngamlamai studied minimum and maximum game lengths and randomly played games. We explore a new direction and introduce the reversed Zeckendorf game, which starts at the ending state of the Zeckendorf game and flips all the moves, so the reversed game ends with all pieces in the first bin. We show that Player 1 has a winning strategy for $n = F_{i+1} + F_{i-2}$ and solve various modified games., Comment: 25 Pages, 4 figures
- Published
- 2023
25. Phase Transitions for Binomial Sets Under Linear Forms
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Jeong, Ryan and Miller, Steven J.
- Subjects
Mathematics - Number Theory ,Mathematics - Combinatorics ,Mathematics - Probability ,11P99, 11K99, 05D40 - Abstract
We generalize the study of the sum and difference sets of a subset of $\mathbb{N}$ drawn from a binomial model to the following setting. Given $A \subseteq \{0, 1, \dots, N\}$, an integer $h \geq 2$, and a linear form $L: \mathbb{Z}^h \to \mathbb{Z}$ given by $$L(x_1, \dots, x_h) = u_1x_1 + \cdots + u_hx_h, \quad u_i \in \mathbb{Z}_{\neq 0} \text{ for all } i \in [h],$$ we study the size of $$L(A) = \left\{u_1a_1 + \cdots + u_ha_h : a_i \in A \right\}$$ and its complement $L(A)^c$ when each element of $\{0, 1, \dots, N\}$ is independently included in $A$ with probability $p(N)$. We identify two phase transition phenomena. The first ``global" phase transition concerns the relative sizes of $L(A)$ and $L(A)^c$, with $p(N) = N^{-\frac{h-1}{h}}$ as the threshold. Asymptotically almost surely, it holds below the threshold that almost all sums generated in $L(A)$ are distinct and almost all possible sums are in $L(A)^c$, and above the threshold that almost all possible sums are in $L(A)$. Our asymptotic formulae substantially extend work of Hegarty and Miller and completely settle, with appropriate corrections made to its statement, their conjecture from 2009. The second ``local" phase transition concerns the asymptotic behavior of the number of distinct realizations in $L(A)$ of a given value, with $p(N) = N^{-\frac{h-2}{h-1}}$ as the threshold. Specifically, it identifies (in a sharp sense) when the number of such realizations obeys a Poisson limit. Our main tools are recent results concerning the asymptotic enumeration of partitions and weak compositions, classical theorems on Poisson approximation, and the martingale machinery of Kim and Vu., Comment: 35 pages, no figures
- Published
- 2023
26. On a Pair of Diophantine Equations
- Author
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Arachchi, Sujith Uthsara Kalansuriya, Chu, Hung Viet, Liu, Jiasen, Luan, Qitong, Marasinghe, Rukshan, and Miller, Steven J.
- Subjects
Mathematics - Number Theory ,11D04, 11B83 - Abstract
For relatively prime natural numbers $a$ and $b$, we study the two equations $ax+by = (a-1)(b-1)/2$ and $ax+by+1= (a-1)(b-1)/2$, which arise from the study of cyclotomic polynomials. Previous work showed that exactly one equation has a nonnegative solution, and the solution is unique. Our first result gives criteria to determine which equation is used for a given pair $(a,b)$. We then use the criteria to study the sequence of equations used by the pair $(a_n/\gcd{(a_n, a_{n+1})}, a_{n+1}/\gcd{(a_n, a_{n+1})})$ from several special sequences $(a_n)_{n\geq 1}$. Finally, fixing $k \in \mathbb{N}$, we investigate the periodicity of the sequence of equations used by the pair $(k/\gcd{(k, n)}, n/\gcd{(k, n)})$ as $n$ increases., Comment: 21 pages, 2 figures
- Published
- 2023
27. Generalized Continuous and Discrete Stick Fragmentation and Benford's Law
- Author
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Fang, Xinyu, Miller, Steven J., Sun, Maxwell, and Verga, Amanda
- Subjects
Mathematics - Probability ,60 - Abstract
Inspired by the basic stick fragmentation model proposed by Becker et al. in arXiv:1309.5603v4, we consider three new versions of such fragmentation models, namely, continuous with random number of parts, continuous with probabilistic stopping, and discrete with congruence stopping conditions. In all of these situations, we state and prove precise conditions for the ending stick lengths to obey Benford's law when taking the appropriate limits. We introduce the aggregated limit, necessary to guarantee convergence to Benford's law in the latter two models. We also show that resulting stick lengths are non-Benford when our conditions are not met. Moreover, we give a sufficient condition for a distribution to satisfy the Mellin transform condition introduced in arXiv:0805.4226v2, yielding a large family of examples., Comment: 43 pages, 2 figures
- Published
- 2023
28. Benford's Law under Zeckendorf expansion
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Chang, Sungkon and Miller, Steven J.
- Subjects
Mathematics - Number Theory - Abstract
In the literature, Benford's Law is considered for base-b expansions where b>1 is an integer. In this paper, we investigate the distribution of leading "digits" of a sequence of positive integers under other expansions such as Zeckendorf expansion, and declare what Benford's Law should be under generalized Zeckendorf expansion.
- Published
- 2023
29. Benford Behavior of a Higher-Dimensional Fragmentation Process
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Durmić, Irfan and Miller, Steven J.
- Subjects
Mathematics - Probability - Abstract
Nature and our world have a bias! Roughly $30\%$ of the time the number $1$ occurs as the leading digit in many datasets base $10$. This phenomenon is known as Benford's law and it arrises in diverse fields such as the stock market, optimizing computers, street addresses, Fibonacci numbers, and is often used to detect possible fraud. Based on previous work, we know that different forms of a one-dimensional stick fragmentation result in pieces whose lengths follow Benford's Law. We generalize this result and show that this can be extended to any finite-dimensional ``volume''. We further conjecture that even lower-dimensional volumes, under the unrestricted fragmentation process, follow Benford's Law.
- Published
- 2023
30. Towards Exascale Computation for Turbomachinery Flows
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Fu, Yuhang, Shen, Weiqi, Cui, Jiahuan, Zheng, Yao, Yang, Guangwen, Liu, Zhao, Zhang, Jifa, Ji, Tingwei, Xie, Fangfang, Lv, Xiaojing, Liu, Hanyue, Liu, Xu, Liu, Xiyang, Song, Xiaoyu, Tao, Guocheng, Yan, Yan, Tucker, Paul, Miller, Steven A. E., Luo, Shirui, Koric, Seid, and Zheng, Weimin
- Subjects
Computer Science - Distributed, Parallel, and Cluster Computing - Abstract
A state-of-the-art large eddy simulation code has been developed to solve compressible flows in turbomachinery. The code has been engineered with a high degree of scalability, enabling it to effectively leverage the many-core architecture of the new Sunway system. A consistent performance of 115.8 DP-PFLOPs has been achieved on a high-pressure turbine cascade consisting of over 1.69 billion mesh elements and 865 billion Degree of Freedoms (DOFs). By leveraging a high-order unstructured solver and its portability to large heterogeneous parallel systems, we have progressed towards solving the grand challenge problem outlined by NASA, which involves a time-dependent simulation of a complete engine, incorporating all the aerodynamic and heat transfer components., Comment: SC23, November, 2023, Denver, CO., USA
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- 2023
31. VC-Dimension of Hyperplanes over Finite Fields
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Ascoli, Ruben, Betti, Livia, Cheigh, Justin, Iosevich, Alex, Jeong, Ryan, Liu, Xuyan, McDonald, Brian, Milgrim, Wyatt, Miller, Steven J., Acosta, Francisco Romero, and Iannuzzelli, Santiago Velazquez
- Subjects
Mathematics - Combinatorics - Abstract
Let $\mathbb{F}_q^d$ be the $d$-dimensional vector space over the finite field with $q$ elements. For a subset $E\subseteq \mathbb{F}_q^d$ and a fixed nonzero $t\in \mathbb{F}_q$, let $\mathcal{H}_t(E)=\{h_y: y\in E\}$, where $h_y$ is the indicator function of the set $\{x\in E: x\cdot y=t\}$. Two of the authors, with Maxwell Sun, showed in the case $d=3$ that if $|E|\geq Cq^{\frac{11}{4}}$ and $q$ is sufficiently large, then the VC-dimension of $\mathcal{H}_t(E)$ is 3. In this paper, we generalize the result to arbitrary dimension and improve the exponent in the case $d=3$., Comment: 9 pages, 1 figure
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- 2023
32. A characterization of prime $v$-palindromes
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Boran, Muhammet, Choi, Garam, Miller, Steven J., Purice, Jesse, and Tsai, Daniel
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Mathematics - Number Theory ,(Primary) 11A45, 11A63 - Abstract
An integer $n\geq 1$ is a $v$-palindrome if it is not a multiple of $10$, nor a decimal palindrome, and such that the sum of the prime factors and corresponding exponents larger than $1$ in the prime factorization of $n$ is equal to that of the integer formed by reversing the decimal digits of $n$. For example, if we take 198 and its reversal 891, their prime factorizations are $198 = 2\cdot 3^2\cdot 11$ and $891 = 3^4\cdot 11$ respectively, and summing the numbers appearing in each factorization both give 18. This means that $198$ and $891$ are $v$-palindromes. We establish a characterization of prime $v$-palindromes: they are precisely the larger of twin prime pairs of the form $(5 \cdot 10^m - 3, 5 \cdot 10^m - 1)$, and thus standard conjectures on the distribution of twin primes imply that there are only finitely many prime $v$-palindromes., Comment: 16 pages
- Published
- 2023
33. Extending the support of $1$- and $2$-level densities for cusp form $L$-functions under square-root cancellation hypotheses
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Mauro, Annika, Miller, Jack B., and Miller, Steven J.
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Mathematics - Number Theory ,11M26 (primary), 11M41, 15A52 (secondary) - Abstract
The Katz-Sarnak philosophy predicts that the behavior of zeros near the central point in families of $L$-functions agrees with that of eigenvalues near 1 of random matrix ensembles. Under GRH, Iwaniec, Luo and Sarnak showed agreement in the one-level densities for cuspidal newforms with the support of the Fourier transform of the test function in $(-2, 2)$. They increased the support further under a square-root cancellation conjecture, showing that a ${\rm GL}(1)$ estimate led to additional agreement between number theory and random matrix theory. We formulate a two-dimensional analog and show it leads to improvements in the two-level density. Specifically, we show that a square-root cancellation of certain classical exponential sums over primes increases the support of the test functions such that the main terms in the $1$- and $2$-level densities of cuspidal newforms averaged over bounded weight $k$ (and fixed level $1$) converge to their random matrix theory predictions. We also conjecture a broad class of such exponential sums where we expect improvement in the case of arbitrary $n$-level densities, and note that the arguments in [ILS] yield larger support than claimed., Comment: 14 pages, to be submitted to Acta Arithmetica
- Published
- 2023
34. Benfordness of Measurements Resulting from Box Fragmentation
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Betti, Livia, Durmić, Irfan, McDonald, Zoe, Miller, Jack B., and Miller, Steven J.
- Subjects
Mathematics - Probability ,60A10, 11K06 (primary), 60E10 (secondary) - Abstract
We make progress on a conjecture made by [DM], which states that the $d$-dimensional frames of $m$-dimensional boxes resulting from a fragmentation process satisfy Benford's law for all $1 \leq d \leq m$. We provide a sufficient condition for Benford's law to be satisfied, namely that the maximum product of $d$ sides is itself a Benford random variable. Motivated to produce an example of such a fragmentation process, we show that processes constructed from log-uniform proportion cuts satisfy the maximum criterion for $d=1$., Comment: 13 pages, 3 figures, to be submitted to the Journal of Statistical Theory and Practice
- Published
- 2023
35. Schreier Multisets and the $s$-step Fibonacci Sequences
- Author
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Chu, Hung Viet, Irmak, Nurettin, Miller, Steven J., Szalay, Laszlo, and Zhang, Sindy Xin
- Subjects
Mathematics - Combinatorics ,11B39, 11B37, 11Y55 - Abstract
Inspired by the surprising relationship (due to A. Bird) between Schreier sets and the Fibonacci sequence, we introduce Schreier multisets and connect these multisets with the $s$-step Fibonacci sequences, defined, for each $s\geqslant 2$, as: $F^{(s)}_{2-s} = \cdots = F^{(s)}_0 = 0$, $F^{(s)}_1 = 1$, and $F^{(s)}_{n} = F^{(s)}_{n-1} + \cdots + F^{(s)}_{n-s}, \mbox{ for } n\geqslant 2$. Next, we use Schreier-type conditions on multisets to retrieve a family of sequences which satisfy a recurrence of the form $a(n) = a(n-1) + a(n-u)$, with $a(n) = 1$ for $n = 1,\ldots, u$. Finally, we study nonlinear Schreier conditions and show that these conditions are related to integer decompositions, each part of which is greater than the number of parts raised to some power., Comment: 11 pages. To appear in Proceedings of the Integers Conference 2023
- Published
- 2023
36. Second And Third-Order Structure Functions Of An 'Engineered' Random Field And Emergence Of The Kolmogorov 4/5 And 2/3-Scaling Laws Of Turbulence
- Author
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Miller, Steven D
- Subjects
Physics - Fluid Dynamics ,Mathematical Physics - Abstract
The 4/5 and 2/3 laws of turbulence can emerge from a theory of 'engineered' random vector fields $\mathcal{X}_{i}(x,t) =X_{i}(x,t)+\tfrac{\theta}{\sqrt{d(d+2)}} X_{i}(x,t)\psi(x)$ existing within $\mathbf{D}\subset\mathbf{R}^{d}$. Here, $X_{i}(x,t)$ is a smooth deterministic vector field obeying a nonlinear PDE for all $(x,t)\in\mathbf{D}\times\mathbf{R}^{+}$, and $\theta$ is a small parameter. The field $\psi(x)$ is a regulated and differentiable Gaussian random field with expectation $\mathbb{E}[\psi(x)]=0$, but having an antisymmetric covariance kernel $\mathscr{K}(x,y)=\mathbb{E}[\psi(x)\psi(y)]=f(x,y)K(\|x-y\|;\lambda)$ with $f(x,y)=-f(y,x)=1,f(x,x)=f(y,y)=0$ and with $K(\|x-y\|;\lambda)$ a standard stationary symmetric kernel. For $0\le\ell\le \lambda
- Published
- 2023
37. Critical congenital heart disease beyond HLHS and TGA: neonatal brain injury and early neurodevelopment
- Author
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Vassar, Rachel, Peyvandi, Shabnam, Gano, Dawn, Cox, Stephany, Zetino, Yensy, Miller, Steven, and McQuillen, Patrick
- Subjects
Biomedical and Clinical Sciences ,Cardiovascular Medicine and Haematology ,Clinical Sciences ,Infant Mortality ,Cardiovascular ,Clinical Research ,Brain Disorders ,Heart Disease ,Genetics ,Neurosciences ,Rare Diseases ,Pediatric ,Perinatal Period - Conditions Originating in Perinatal Period ,2.1 Biological and endogenous factors ,Aetiology ,Neurological ,Good Health and Well Being ,Infant ,Infant ,Newborn ,Child ,Humans ,Transposition of Great Vessels ,Hypoplastic Left Heart Syndrome ,Prospective Studies ,Heart Defects ,Congenital ,Brain Injuries ,Paediatrics and Reproductive Medicine ,Public Health and Health Services ,Pediatrics ,Paediatrics - Abstract
BackgroundCharacterization of brain injury and neurodevelopmental (ND) outcomes in critical congenital heart disease (cCHD) has primarily focused on hypoplastic left heart syndrome (HLHS) and transposition of the great arteries (TGA). This study reports brain injury and ND outcomes among patients with heterogeneous cCHD diagnoses beyond HLHS and TGA.MethodsThis prospective cohort study included infants with HLHS, TGA, or heterogenous "Other cCHD" including left- or right-sided obstructive lesions, anomalous pulmonary venous return, and truncus arteriosus. Brain injury on perioperative brain MRI and ND outcomes on the Bayley-II at 30 months were compared.ResultsA total of 218 participants were included (HLHS = 60; TGA = 118; "Other cCHD" = 40, including 8 with genetic syndromes). Pre-operative (n = 209) and post-operative (n = 189) MRI showed similarly high brain injury rates across groups, regardless of cardiopulmonary bypass exposure. At 30 months, participants with "Other cCHD" had lower cognitive scores (p = 0.035) compared to those with HLHS and TGA, though worse ND outcome in this group was driven by those with genetic disorders.ConclusionsFrequency of brain injury and neurodevelopmental delay among patients with "Other cCHD" is similar to those with HLHS or TGA. Patients with all cCHD lesions are at risk for impaired outcomes; developmental and genetic screening is indicated.ImpactThis study adds to literature on risk of brain injury in patients with critical congenital heart disease (cCHD) diagnoses other than hypoplastic left heart syndrome (HLHS) and transposition of the great arteries (TGA), a heterogenous cohort of patients that has often been excluded from imaging studies. Children with cCHD beyond HLHS and TGA have similarly high rates of acquired brain injury. The high rate of neurodevelopmental impairment in this heterogenous group of cCHD diagnoses beyond HLHS and TGA is primarily driven by patients with comorbid genetic syndromes such as 22q11.2 deletion syndrome.
- Published
- 2023
38. Determining optimal test functions for 2-level densities
- Author
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Bołdyriew, Elżbieta, Chen, Fangu, Devlin, Charles, Miller, Steven J, and Zhao, Jason
- Subjects
Applied Mathematics ,Mathematical Physics ,Pure Mathematics ,Mathematical Sciences ,Random matrix theory ,L-functions ,Low-lying zeros ,Optimal test functions ,Fredholm theory - Published
- 2023
39. Sums of Powers by L'Hopital's Rule
- Author
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Dueñez, Eduardo, Hamakiotes, Asimina S., and Miller, Steven J.
- Subjects
Mathematics - Number Theory - Abstract
For a positive integer $d$, let $p_d(n) := 0^d + 1^d + 2^d + \cdots + n^d$; i.e., $p_d(n)$ is the sum of the first $d^{\mathrm{th}}$-powers up to $n$. It's well known that $p_d(n)$ is a polynomial of degree $d+1$ in $n$. While this is usually proved by induction, once $d$ is not small it's a challenge as one needs to know the polynomial for the inductive step. We show how this difficulty can be bypassed by giving a simple proof that $p_d(n)$ is a polynomial of degree $d+1$ in $n$ by using L'Hopital's rule, and show how we can then determine the coefficients by Cramer's rule. This illustrates a general principle and the point of our paper: there's more than one path to a goal, different approaches have their advantages and disadvantages, and the more techniques one knows, the more likely one can successfully attack a problem.
- Published
- 2023
40. A Cluster Expansion Proof That The Stochastic Exponential Of A Brownian Motion Is A Martingale
- Author
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Miller, Steven D
- Subjects
Mathematics - Probability ,Mathematical Physics - Abstract
Let ${\psi}:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ be a smooth and continuous real function and $\psi\in\mathrm{L}^{2}(\mathbb{R}^{+})$. Let ${B}(t)$ be a standard Brownian motion defined with respect to a probability space $(\Omega,\mathscr{F},{\mathsf{P}})$ and where $d{B}(t)={\xi}(t)dt$ and $t\in\mathbb{R}^{+}$. The process $\xi(t)$ is a Gaussian white noise with expectation $\mathbf{\mathsf{E}}~\xi(t)=0$ and with covariance ${\mathsf{E}}~\xi(t)\xi(s)=\delta(t-s)$. The Dolean-Dades stochastic exponential ${Z}(t)$ is the solution to the linear stochastic differential equation describing a geometric Brownian motion such that $d{Z}(t)=\psi(t){Z}(t)d{B}(t)=\psi(t){Z}(t)\xi(t)dt$. Using a cluster expansion method, and the moment and cumulant generating functions for $\xi(t)$, it is shown that ${Z}(t)$ is a martingale. The original Novikov criteria for ${Z}(t)$ being a true martingale are reproduced and exactly satisfied, namely that \begin{align} {\mathsf{E}}\mathrm{Z}(t)={\mathsf{E}}\exp\left(\int_{o}^{t}{\psi}(u)d{B}(u) -\frac{1}{2}\int_{0}^{t}|{\psi}(u)|^{2}du\right)=1\nonumber \end{align} provided that $\exp\big(\int_{0}^{t}|{\psi}(u)|^{2}du\big)<\infty$ for all $t>0$. However, ${\mathsf{E}}\big[|{Z}(t)|^{p}\big] =\exp(\tfrac{1}{2}p(p-1)\phi(t))$, if $\phi(t)=\int_{0}^{t}|\psi(u)|^{2}du$ is monotone increasing and is a submartingale for all $p>1$., Comment: 8 Pages
- Published
- 2023
41. A Stochastic Differential Equation For Laser Propagation In Medias With Random Gaussian Absorption Coefficients: A Modified Beer's Law Solution Via A Van Kampen Cluster Expansion
- Author
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Miller, Steven D
- Subjects
Mathematical Physics ,Mathematics - Probability - Abstract
Let $\mathbb{I\!D}=[0,\mathrm{L}]\subset\mathbb{R}^{+}$ be a slab geometry with boundaries $z=0$ and $z=\mathrm{L}$. A laser beam with a flat incident intensity $\psi_{o}$ enters the slab along the z-axis or unit vector $\widehat{\mathbf{e}}_{3}$ at $z=0$. The slab contains matter with an absorption coefficient of $\mathsf{A}$ with respect to the wavelength. If $\mathsf{A}$ is constant and homogenous then the beam decays as Beer's law $\psi(z,\widehat{\mathbf{e}}_{3})=\psi_{o}\exp(-\mathsf{A} z)$. If the absorption coefficient is randomly fluctuating in space as $\mathbf{A}(z)=\mathsf{A}(1+\alpha \mathbf{G}(z))$--where $\alpha>0$ determines the magnitude of the fluctuations, and the Gaussian random function has expectation $\mathbb{E}\lbrace \mathbf{G}(z)\rbrace =0$ and a binary correlation $\mathbb{E}\lbrace\mathbf{G}(z_{1})\otimes\mathbf{G}(z_{2})\rbrace=\phi(z_{1},z_{2};\xi)={\mathsf{C}}\exp(-|z_{1}-z_{2}|^{2}\xi^{-2})$ for all $(z_{1},z_{2})\in\mathbb{I\!D}$ with correlation length $\xi$--then the beam propagation and attentuation within the medium is described by the stochastic differential equation \begin{equation} d\widehat{{\psi}(z,\widehat{\mathbf{e}}_{3})}=-\mathsf{A}\widehat{\psi(z,\mathbf{e}_{3})}dz-\alpha\mathsf{A}\widehat{\psi(z,\mathbf{e}_{3})}\mathbf{G}(z)dz \end{equation} The stochastically averaged solution is derived via a Van Kampen-type cluster expansion, truncated at 2nd order for Gaussianality, giving a modified Beer's law \begin{equation} \mathbb{I}(z,\widehat{\mathbf{e}}_{3})=\mathbb{E}\big\lbrace\widehat{\psi(z,\widehat{\mathbf{e}}_{3})}\big\rbrace=\psi_{o}\exp(-\mathsf{A}z)\exp\bigg(\frac{1}{4}\alpha^{2}\mathsf{A}^{2}{\mathsf{C}}\xi\bigg[\exp(-z^{2}/\xi^{2})\bigg(\sqrt{\pi}z Erf(\tfrac{z}{\xi})\exp\bigg(\frac{z^{2}}{\xi^{2}}\bigg)+\xi\bigg)-\xi\bigg]\bigg) \end{equation} The deterministic Beer's law is recovered as $\alpha\rightarrow 0 $., Comment: 21 pages, 3 figures. arXiv admin note: text overlap with arXiv:2211.14925
- Published
- 2023
42. Virus Dynamics on $k$-Level Starlike Graphs
- Author
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Takigawa, Akihiro and Miller, Steven J.
- Subjects
Mathematics - Dynamical Systems ,Mathematics - Probability - Abstract
Becker, Greaves-Tunnell, Kontorovich, Miller, Ravikumar, and Shen determined the long term evolution of virus propagation behavior on a hub-and-spoke graph of one central node and $n$ neighbors, with edges only from the neighbors to the hub (a $2$-level starlike graph), under a variant of the discrete-time SIS (Suspectible Infected Suspectible) model. The behavior of this model is governed by the interactions between the infection and cure probabilities, along with the number $n$ of $2$-level nodes. They proved that for any $n$, there is a critical threshold relating these rates, below which the virus dies out, and above which the probabilistic dynamical system converges to a non-trivial steady state (the probability of infection for each category of node stabilizes). For $a$, the probability at any time step that an infected node is not cured, and $b$, the probability at any time step that an infected node infects its neighbors, the threshold for the virus to die out is $b \leq (1-a)/\sqrt{n}$. We extend this analysis to $k$-level starlike graphs for $k \geq 3$ (each $(k-1)$-level node has exactly $n_k$ neighbors, and the only edges added are from the $k$-level nodes) for infection rates above and below the critical threshold of $(1-a)/\sqrt{n_1+n_2+\dots+n_{k-1}}$. We do this by first analyzing the dynamics of nodes on each level of a $3$-level starlike graph, then show that the dynamics of the nodes of a $k$-level starlike graph are similar, enabling us to reduce our analysis to just $3$ levels, using the same methodology as the $3$-level case., Comment: Version 1.2: Adjusted title
- Published
- 2022
43. Sums of Reciprocals of Recurrence Relations
- Author
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Cui, Hao, Cui, Xiaoyu, Davis, Sophia C., Durmić, Irfan, Hu, Qingcheng, Liu, Lisa, Miller, Steven J., Ren, Fengping, Reina, Alicia Smith, and Sosis, Eliel
- Subjects
Mathematics - Number Theory ,11B39 (primary), 33C05 (secondary) - Abstract
There is a growing literature on sums of reciprocals of polynomial functions of recurrence relations with constant coefficients and fixed depth, such as Fibonacci and Tribonacci numbers, products of such numbers, and balancing numbers (numbers $n$ such that the sum of the integers less than $n$ equals the sum of the $r$ integers immediately after, for some $r$ which is called the balancer of $n$; If $n$ is included in the summation, we have the cobalancing numbers, and $r$ is called the cobalancer of $n$). We generalize previous work to reciprocal sums of depth two recurrence sequences with arbitrary coefficients and the Tribonacci numbers, and show our method provides an alternative proof of some existing results. We define $(a,b)$ balancing and cobalancing numbers, where $a$ and $b$ are constants that multiply the left-hand side and right-hand side respectively, and derive recurrence relations describing these sequences. We show that for balancing numbers, the coefficients $(3,1)$ is unique such that every integer is a $(3,1)$ balancing number, and proved there does not exist an analogous set of coefficients for cobalancing numbers. We also found patterns for certain coefficients that have no balancing or cobalancing numbers., Comment: 31 pages, 2 figures, 3 tables
- Published
- 2022
44. Winning Strategies for Generalized Zeckendorf Game
- Author
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Miller, Steven J., Sosis, Eliel, and Ye, Jingkai
- Subjects
Mathematics - Number Theory ,91A05, 91A06 - Abstract
Zeckendorf proved that every positive integer $n$ can be written uniquely as the sum of non-adjacent Fibonacci numbers; a similar result holds for other positive linear recurrence sequences. These legal decompositions can be used to construct a game that starts with a fixed integer $n$, and players take turns using moves relating to a given recurrence relation. The game eventually terminates in a unique legal decomposition, and the player who makes the final move wins. For the Fibonacci game, Player $2$ has the winning strategy for all $n>2$. We give a non-constructive proof that for the two-player $(c, k)$-nacci game, for all $k$ and sufficiently large $n$, Player $1$ has a winning strategy when $c$ is even and Player $2$ has a winning strategy when $c$ is odd. Interestingly, the player with the winning strategy can make a mistake as early as the $c + 1$ turn, in which case the other player gains the winning strategy. Furthermore, we proved that for the $(c, k)$-nacci game with players $p \ge c + 2$, no player has a winning strategy for any $n \ge 3c^2 + 6c + 3$. We find a stricter lower boundary, $n \ge 7$, in the case of the three-player $(1, 2)$-nacci game. Then we extend the result from the multiplayer game to multialliance games, showing which alliance has a winning strategy or when no winning strategy exists for some special cases of multialliance games., Comment: 24 pages, 8 figures
- Published
- 2022
45. A Turbulent Fluid Mechanics via Nonlinear 'Mixing' of Smooth Velocity Flows With Weighted Random Fields: Stochastically Averaged Navier-Stokes Equations and Velocity Correlations
- Author
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Miller, Steven D
- Subjects
Mathematical Physics - Abstract
Let $\mathfrak{D}\subset\mathbb{R}^{3}$, with ${Vol}(\mathfrak{D})\sim L^{3}$, contain an incompressible fluid of viscosity $\nu$ and velocity $\mathrm{U}_{i}(x,t)$ with $(x,t)\in\mathfrak{D}\times[0,\infty)$, satisfying the Navier-Stokes equations with some boundary conditions on $\partial\mathfrak{D}$ and evolving from initial Cauchy data. Now let $\mathscr{B}(x)$ be a Gaussian random field defined for all $x\in\mathfrak{D}$ with expectation $\mathbb{I\!E}\langle\mathscr{B}(x)\rangle=0$, and a Bargmann-Fock binary correlation $\mathbb{I\!E}\big\langle\mathscr{B}(x)\otimes \mathscr{B}({y})\big\rangle=\mathsf{C}\exp(-\|{x}-{y}\|^{2}\lambda^{-2})$ with $\lambda\le {L}$. Define a volume-averaged Reynolds number $\mathbf{Re}(\mathfrak{D},t) =(|Vol(\mathfrak{D})|^{-1}\int_{\mathfrak{D}}\|\mathrm{U}_{i}(x,t)\|d\mu({x}){L}/\nu$. The critical Reynolds number is $\mathbf{Re}_{c}(\mathfrak{D})$ so that turbulence fully evolves within $\mathfrak{D}$ for $t$ such that $\mathbf{Re}(\mathfrak{D},t)>\mathbf{Re}_{c}(\mathfrak{D})$. Let $\psi(|\mathbf{Re}(\mathfrak{D},t)-\mathbf{Re}_{c}(\mathfrak{D})|)$ be an arbitrary monotone-increasing weighting functional. The turbulent flow evolving within $\mathfrak{D}$ is described by the random field $\mathscr{U}_{i}(x,t)$ via a 'mixing' ansatz $\mathscr{U}_{i}(x,t)=\mathrm{U}_{i}(x,t)+\beta\mathrm{U}_{i}(x,t) \big\lbrace\psi(|\mathbf{Re}(\mathfrak{D},t)-\mathbf{Re}_{c}(\mathfrak{D})|)\big\rbrace \mathbb{I}_{\mathcal{S}}[\mathbf{Re}(\mathfrak{D},t)\big]\mathscr{B}(x)$ where ${\beta}\ge 1$ is a constant and $\mathbb{I}_{\mathcal{S}}[\mathbf{Re}(\mathfrak{D},t)]$ an indicator function. The flow grows increasingly random if $\mathbf{Re}(\mathfrak{D},t)$ increases with $t$ so that this is a 'control parameter'. The turbulent flow $\mathscr{U}_{i}(x,t)$ is a solution of stochastically averaged N-S equations. Reynolds-type velocity correlations are estimated., Comment: 64 pages, 5 figures
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- 2022
46. Bounding the Order of Vanishing of Cuspidal Newforms via the nth Centered Moments
- Author
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Dutta, Sohom and Miller, Steven J.
- Subjects
Mathematics - Number Theory - Abstract
Building on the work of Iwaniec, Luo and Sarnak, we use the $n$-level density to bound the probability of vanishing to order at least $r$ at the central point for families of cuspidal newforms of prime level $N \to \infty$, split by sign. There are three methods to improve bounds on the order of vanishing: optimizing the test functions, increasing the support, and increasing the $n$-level density studied. Previous work determined the optimal test functions for the $1$ and $2$-level densities in certain support ranges, leading to marginal improvements in bounds and making it not a productive avenue for further research. Similarly the support has been increased as far as possible, and further progress is shown to be related to delicate and difficult conjectures in number theory. Thus we concentrate on the third method, and study the higher centered moments (which are similar to the $n$-level densities but combinatorially easier). We find the level at each rank for which the bounds on the order of vanishing is the best, thus producing world-record bounds on the order of vanishing to rank at least $r$ for every $r > 2$ (for example, our bounds for vanishing to order at least 5 or at least 6 are less than half the previous bounds, a significant improvement). Additionally, we explicitly calculate the optimal test function for the $1$-level density from previous work and compare it to the naive test functions for higher levels. In doing so, we find that the optimal test function for certain levels are not the optimal for other levels, and some test functions may outperform others for some levels but not in others. Finally, we explicitly calculate the integrals needed to determine the bounds, doing so by transforming an $n$-dimensional integral to a $1$-dimensional integral and greatly reducing the computation cost in the process.
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- 2022
47. Generalizing the German Tank Problem
- Author
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Lee, Anthony and Miller, Steven J.
- Subjects
Mathematics - Probability ,Mathematics - Statistics Theory ,62J05, 60C05 (primary), 05A10 (secondary) - Abstract
The German Tank Problem dates back to World War II when the Allies used a statistical approach to estimate the number of enemy tanks produced or on the field from observed serial numbers after battles. Assuming that the tanks are labeled consecutively starting from 1, if we observe $k$ tanks from a total of $N$ tanks with the maximum observed tank being $m$, then the best estimate for $N$ is $m(1 + 1/k) - 1$. We explore many generalizations. We looked at the discrete and continuous one dimensional case. We explored different estimators such as the $L$\textsuperscript{th} largest tank, and applied motivation from portfolio theory and studied a weighted average; however, the original formula was the best. We generalized the problem in two dimensions, with pairs instead of points, studying the discrete and continuous square and circle variants. There were complications from curvature issues and that not every number is representable as a sum of two squares. We often concentrated on the large $N$ limit. For the discrete and continuous square, we tested various statistics, finding the largest observed component did best; the scaling factor for both cases is $(2k+1)/2k$. The discrete case was especially involved because we had to use approximation formulas that gave us the number of lattice points inside the circle. Interestingly, the scaling factors were different for the cases. Lastly, we generalized the problem into $L$ dimensional squares and circles. The discrete and continuous square proved similar to the two dimensional square problem. However, for the $L$\textsuperscript{th} dimensional circle, we had to use formulas for the volume of the $L$-ball, and had to approximate the number of lattice points inside it. The formulas for the discrete circle were particularly interesting, as there was no $L$ dependence in the formula., Comment: Version 1.0, 47 pages
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- 2022
48. Towards the Gaussianity of Random Zeckendorf Games
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Cheigh, Justin, Moura, Guilherme Zeus Dantas e, Jeong, Ryan, Duke, Jacob Lehmann, Milgrim, Wyatt, Miller, Steven J., and Ngamlamai, Prakod
- Subjects
Mathematics - Combinatorics - Abstract
Zeckendorf proved that any positive integer has a unique decomposition as a sum of non-consecutive Fibonacci numbers, indexed by $F_1 = 1, F_2 = 2, F_{n+1} = F_n + F_{n-1}$. Motivated by this result, Baird, Epstein, Flint, and Miller defined the two-player Zeckendorf game, where two players take turns acting on a multiset of Fibonacci numbers that always sums to $N$. The game terminates when no possible moves remain, and the final player to perform a move wins. Notably, studied the setting of random games: the game proceeds by choosing an available move uniformly at random, and they conjecture that as the input $N \to \infty$, the distribution of random game lengths converges to a Gaussian. We prove that certain sums of move counts is constant, and find a lower bound on the number of shortest games on input $N$ involving the Catalan numbers. The works Baird et al. and Cuzensa et al. determined how to achieve a shortest and longest possible Zeckendorf game on a given input $N$, respectively: we establish that for any input $N$, the range of possible game lengths constitutes an interval of natural numbers: every game length between the shortest and longest game lengths can be achieved. We further the study of probabilistic aspects of random Zeckendorf games. We study two probability measures on the space of all Zeckendorf games on input $N$: the uniform measure, and the measure induced by choosing moves uniformly at random at any given position. Under both measures that in the limit $N \to \infty$, both players win with probability $1/2$. We also find natural partitions of the collection of all Zeckendorf games of a fixed input $N$, on which we observe weak convergence to a Gaussian in the limit $N \to \infty$. We conclude the work with many open problems., Comment: 29 pages
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- 2022
49. Recurrence Relations for $S$-Legal Index Difference Sequences
- Author
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Moura, Guilherme Zeus Dantas e, Keisling, Andrew, Lilly, Astrid, Mauro, Annika, Miller, Steven J., Phang, Matthew, and Iannuzzelli, Santiago Velazquez
- Subjects
Mathematics - Number Theory ,11B39, 65Q30 - Abstract
Zeckendorf's Theorem implies that the Fibonacci number $F_n$ is the smallest positive integer that cannot be written as a sum of non-consecutive previous Fibonacci numbers. Catral et al. studied a variation of the Fibonacci sequence, the Fibonacci Quilt sequence: the plane is tiled using the Fibonacci spiral, and integers are assigned to the squares of the spiral such that each square contains the smallest positive integer that cannot be expressed as the sum of non-adjacent previous terms. This adjacency is essentially captured in the differences of the indices of each square: the $i$-th and $j$-th squares are adjacent if and only if $|i - j| \in \{1, 3, 4\}$ or $\{i, j\} = \{1, 3\}$. We consider a generalization of this construction: given a set of positive integers $S$, the $S$-legal index difference ($S$-LID) sequence $(a_n)_{n=1}^\infty$ is defined by letting $a_n$ to be the smallest positive integer that cannot be written as $\sum_{\ell \in L} a_\ell$ for some set $L \subset [n-1]$ with $|i - j| \notin S$ for all $i, j \in L$. We discuss our results governing the growth of $S$-LID sequences, as well as results proving that many families of sets $S$ yield $S$-LID sequences which follow simple recurrence relations., Comment: 18 pages, 4 figures
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- 2022
50. VC-Dimension and Distance Chains in $\mathbb{F}_q^d$
- Author
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Ascoli, Ruben, Betti, Livia, Cheigh, Justin, Iosevich, Alex, Jeong, Ryan, Liu, Xuyan, McDonald, Brian, Milgrim, Wyatt, Miller, Steven J., Acosta, Francisco Romero, and Iannuzzelli, Santiago Velazquez
- Subjects
Mathematics - Combinatorics ,Mathematics - Classical Analysis and ODEs - Abstract
Given a domain $X$ and a collection $\mathcal{H}$ of functions $h:X\to \{0,1\}$, the Vapnik-Chervonenkis (VC) dimension of $\mathcal{H}$ measures its complexity in an appropriate sense. In particular, the fundamental theorem of statistical learning says that a hypothesis class with finite VC-dimension is PAC learnable. Recent work by Fitzpatrick, Wyman, the fourth and seventh named authors studied the VC-dimension of a natural family of functions $\mathcal{H}_t^{'2}(E): \mathbb{F}_q^2\to \{0,1\}$, corresponding to indicator functions of circles centered at points in a subset $E\subseteq \mathbb{F}_q^2$. They showed that when $|E|$ is large enough, the VC-dimension of $\mathcal{H}_t^{'2}(E)$ is the same as in the case that $E = \mathbb F_q^2$. We study a related hypothesis class, $\mathcal{H}_t^d(E)$, corresponding to intersections of spheres in $\mathbb{F}_q^d$, and ask how large $E\subseteq \mathbb{F}_q^d$ needs to be to ensure the maximum possible VC-dimension. We resolve this problem in all dimensions, proving that whenever $|E|\geq C_dq^{d-1/(d-1)}$ for $d\geq 3$, the VC-dimension of $\mathcal{H}_t^d(E)$ is as large as possible. We get a slightly stronger result if $d=3$: this result holds as long as $|E|\geq C_3 q^{7/3}$. Furthermore, when $d=2$ the result holds when $|E|\geq C_2 q^{7/4}$., Comment: 12 pages, 1 figure
- Published
- 2022
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