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Grant Molnar

Publications and source records attributed to Grant Molnar.

18 recordsLinked to original sources
arXiv2026

The Tropical Algebra of Binary-Tree Height

The binary-tree height recursion defines an algebra $\mathcal{H}$ on $\mathbb{N}\cup\{-\infty\}$, with join given by $\max$ and product \[ a\star b=\max\{a,b\}+1. \] We show that weighted evaluation of a labelled tree depends only on the greatest depth of each label. Single-tree profiles are exactly the vectors satisfying the binary Kraft inequality, while finite joins realize every vector in $\bigl(\mathbb{N}\cup\{-\infty\}\bigr)^n$; hence the $n$-variable term operations form the free algebra $\mathcal{H}^n$. We also classify $\mathcal{H}$'s compatible semilattice operation, subalgebras, endomorphisms, congruences, and finite quotients, and recover the dyadic-composition spectrum at the full-linear boundary.

math.CO↗
arXiv2026

The Gregory function and its completed Gregory transform

We study the entire interpolation \[ \mathcal{G}(z)=\int_0^1 \binom{x}{z}\,dx \] of the Gregory coefficients. Its completion satisfies the positive Markov-transform identity \[ \frac{\pi z}{\sin(\pi z)}\mathcal{G}(z) =\sum_{n=1}^{\infty}\frac{n\left|G_n\right|}{n-z}. \] Consequently, every zero is real and simple; the negative zeros are the integers $-1,-2,\ldots$, and one zero $\rho_n$ lies in each $(n,n+1)$. We derive complete logarithmic asymptotics for $\rho_n-n$, determine the Cartwright growth and canonical products of $\mathcal{G}$, and realize $1/\rho_n$ spectrally. The resulting relative determinant yields \[ \gamma=\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{\rho_n}\right). \]

math.GM↗
arXiv2026

A weighted angle distance on strings

We define a multi-scale metric $d_\rho$ on strings by aggregating angle distances between all $n$-gram count vectors with exponential weights $\rho^n$. We benchmark $d_\rho$ in DBSCAN clustering against edit and $n$-gram baselines, give a linear-time suffix-tree algorithm for evaluation, prove metric and stability properties (including robustness under tandem-repeat stutters), and characterize isometries.

math.MG↗
arXiv2026

Integral-Differential Calculus

We give an exposition of the Newton-Leibniz calculus. We begin by defining the integral as a limit of Riemann sums, verify the integrals of the standard catalog of functions by direct manipulation, prove the substitution lemmas as theorems about Riemann sums, cross the Fundamental Theorem of Calculus, and harvest the differential calculus on the other side.

math.HO↗
arXiv2026

On the symmetry of evidential support

For events $A$ and $B$, we have \[ \mathbb{P}(A\mid B) > \mathbb{P}(A\mid \neg B) \qquad\Longleftrightarrow\qquad \mathbb{P}(B\mid A) > \mathbb{P}(B\mid \neg A) \] whenever all four quantities are defined. In other words, $B$ is evidence for $A$ if and only if $A$ is evidence for $B$. This note gives seven different proofs of this fact -- by cross-multiplication, covariance, coupling parameters, odds ratios, pointwise mutual information, combinatorial double counting, and mixed discrete derivatives -- and develops a surrounding web of interpretations. Once the marginals $\mathbb{P}(A)$ and $\mathbb{P}(B)$ are fixed, a $2\times 2$ table has only one degree of freedom, so every scalar notion of positive association must be governed by the same signed parameter.

stat.OT↗
arXiv2025

A family of analogues to the Robin criterion

The Robin criterion states that the Riemann hypothesis is equivalent to the inequality $\sigma(n) < e^\gamma n \log \log n$ for all $n>5040$, where $\sigma(n)$ is the sum of divisors of $n$, and $\gamma$ is the Euler--Mascheroni constant. Define the family of functions \[ \sigma^{[k]} (n):=\sum_{[d_1,\dots,d_k]=n}d_1\dots d_k \] where $[d_1, \dots, d_k]$ is the least common multiple of $d_1, \dots, d_k$. These functions behave asymptotically like $\sigma(n)^k$ as $k\to\infty$. We prove the following analogue of the Robin criterion: for any $k \geq 2$, the Riemann hypothesis holds if and only if $\sigma^{[k]} (n) < \frac{(e^\gamma n \log \log n)^k}{\zeta(k)}$ for all $n > 2162160$, where $\zeta$ is the Riemann zeta function.

math.NT↗
arXiv2024

Positive spoof Lehmer factorizations

We investigate the integer solutions of Diophantine equations related to Lehmer's totient conjecture. We give an algorithm that computes all nontrivial spoof Lehmer factorizations with a fixed number of factors, and enumerate all nontrivial spoof Lehmer factorizations with 6 or fewer factors.

math.NT↗
arXiv2024

Counting elliptic curves with a cyclic $m$-isogeny over $\mathbb{Q}$

Using methods from analytic number theory, for $m > 5$ and for $m = 4$, we obtain asymptotics with power-saving error terms for counts of elliptic curves with a cyclic $m$-isogeny up to quadratic twist over the rational numbers. For $m > 5$, we then apply a Tauberian theorem to achieve asymptotics with power saving error for counts of elliptic curves with a cyclic $m$-isogeny up to isomorphism over the rational numbers.

math.NT↗
arXiv2023

Reactive means in the Iterated Prisoner's Dilemma

The Iterated Prisoner's Dilemma (IPD) is a well studied framework for understanding direct reciprocity and cooperation in pairwise encounters. However, measuring the morality of various IPD strategies is still largely lacking. Here, we partially address this issue by proposing a suit of plausible morality metrics to quantify four aspects of justice. We focus our closed-form calculation on the class of reactive strategies because of their mathematical tractability and expressive power. We define reactive means as a tool for studying how actors in the IPD and Iterated Snowdrift Game (ISG) behave under typical circumstances. We compute reactive means for four functions intended to capture human intuitions about ``goodness'' and ``fair play''. Two of these functions are strongly anticorrelated with success in the IPD and ISG, and the other two are weakly anticorrelated with success. Our results will aid in evaluating and comparing powerful IPD strategies based on machine learning algorithms, using simple and intuitive morality metrics.

physics.soc-ph↗
arXiv2021

Multiplicative summations into algebraically closed fields

In this paper, extending our earlier program, we derive maximal canonical extensions for multiplicative summations into algebraically closed fields. We show that there is a well-defined analogue to minimal polynomials for a series algebraic over a ring of series, the "scalar polynomial". When that ring is the domain of a summation $\mathfrak{S}$, we derive the related concepts of the $\mathfrak{S}$-minimal polynomial for a series, which is mapped by $\mathfrak{S}$ to a scalar polynomial. When the scalar polynomial for a series has the form $(t-a)^n$, $a$ is the unique value to which the series can be mapped by an extension of the original summation.

math.AC↗
arXiv2021

Telescopic, Multiplicative, and Rational Extensions of Summations

A summation is a shift-invariant ${\rm R}$-module homomorphism from a submodule of ${\rm R}[[\sigma]]$ to ${\rm R}$ or another ring. [11] formalized a method for extending a summation to a larger domain by telescoping. In this paper, we revisit telescoping, we study multiplicative closures of summations (such as the usual summation on convergent series) that are not themselves multiplicatively closed, and we study rational extensions as a generalization of telescoping.

math.AC↗
arXiv2021

Fast-growing series are transcendental

Let $R$ be a subring of $\mathbb{C}[[z]]$, and let $X \in \mathbb{C}[[z]]$. The Newton-Puiseux Theorem implies that if the coefficients of $X$ grow sufficiently rapidly relative to the coefficients of the series in $R$, then $X$ is transcendental over $R$. We prove an alternative proof of this result by establishing a relationship between the coefficients of $A(X)$ and $A^\prime(X)$, where $A(T)$ is a polynomial over $\mathbb{C}[[z]]$.

math.NT↗
arXiv2020

The arithmetic of modular grids

A modular grid is a pair of sequences $(f_m)_m$ and $(g_n)_n$ of weakly holomorphic modular forms such that for almost all $m$ and $n$, the coefficient of $q^n$ in $f_m$ is the negative of the coefficient of $q^m$ in $g_n$. Zagier proved this coefficient duality in weights $1/2$ and $3/2$ in the Kohnen plus space, and such grids have appeared for Poincar\'{e} series, for modular forms of integral weight, and in many other situations. We give a general proof of coefficient duality for canonical row-reduced bases of spaces of weakly holomorphic modular forms of integral or half-integral weight for every group $\Gamma \subseteq {\text{SL}}_2(\mathbb{R})$ commensurable with ${\text{SL}}_2(\mathbb{Z})$. We construct bivariate generate functions that encode these modular forms, and study linear operations on the resulting modular grids.

math.NT↗
arXiv2017

Zagier duality for level $p$ weakly holomorphic modular forms

We prove Zagier duality between the Fourier coefficients of canonical bases for spaces of weakly holomorphic modular forms of prime level $p$ with $11 \leq p \leq 37$ with poles only at the cusp at $\infty$, and special cases of duality for an infinite class of prime levels. We derive generating functions for the bases for genus 1 levels.

math.NT↗
arXiv2017

A Savage-Like Axiomatization for Nonstandard Expected Utility

Since Leonard Savage's epoch-making "Foundations of Statistics", Subjective Expected Utility Theory has been the presumptive model for decision-making. Savage provided an act-based axiomatization of standard expected utility theory. In this article, we provide a Savage-like axiomatization of nonstandard expected utility theory. It corresponds to a weakening of Savage's 6th axiom.

cs.AI↗
arXiv2015

Graphs with the strong Havel-Hakimi property

The Havel-Hakimi algorithm iteratively reduces the degree sequence of a graph to a list of zeroes. As shown by Favaron, Mah\'eo, and Sacl\'e, the number of zeroes produced, known as the residue, is a lower bound on the independence number of the graph. We say that a graph has the strong Havel-Hakimi property if in each of its induced subgraphs, deleting any vertex of maximum degree reduces the degree sequence in the same way that the Havel-Hakimi algorithm does. We characterize graphs having this property (which include all threshold and matrogenic graphs) in terms of minimal forbidden induced subgraphs. We further show that for these graphs the residue equals the independence number, and a natural greedy algorithm always produces a maximum independent set.

math.CO↗