Searcharxiv⌕ Search

arXiv subjects

Anna Haensch

Publications and source records attributed to Anna Haensch.

14 recordsLinked to original sources
arXiv2023

ODEs and Mandatory Voting

This paper presents mathematics relevant to the question whether voting should be mandatory. Assuming a static distribution of voters' political beliefs, we model how politicians might adjust their positions to raise their share of the vote. Various scenarios can be explored using our web-based app (see text for the link). Abstentions are found to have great impact on the dynamics of candidates, and in particular to introduce the possibility of discontinuous jumps in optimal candidate positions. This is a paper intended for undergraduate students. It is an unusual application of ODEs. We hope that it might help engage some students who may find it harder to connect with the more customary applications from the natural sciences.

physics.soc-ph↗
arXiv2023

A blue sky bifurcation in the dynamics of political candidates

Political candidates often shift their positions opportunistically in hopes of capturing more votes. When there are only two candidates, the best strategy for each of them is often to move towards the other. This eventually results in two centrists with coalescing views. However, the strategy of moving towards the other candidate ceases to be optimal when enough voters abstain instead of voting for a centrist who does not represent their views. These observations, formalized in various ways, have been made many times. Our own formalization is based on differential equations. The surprise and main result derived from these equations is that the final candidate positions can jump discontinuously as the voters' loyalty towards their candidate wanes. The underlying mathematical mechanism is a blue sky bifurcation.

math.DS↗
arXiv2022

A particle method for continuous Hegselmann-Krause opinion dynamics

We derive a differential-integral equation akin to the Hegselmann-Krause model of opinion dynamics, and propose a particle method for solving the equation. Numerical experiments demonstrate second-order convergence of the method in a weak sense. We also show that our differential-integral equation can equivalently be stated as a system of differential equations. An integration-by-parts argument that would typically yield an energy dissipation inequality in physical problems then yields a concentration inequality, showing that a natural measure of concentration increases monotonically.

math.NA↗
arXiv2022

A geospatial bounded confidence model including mega-influencers with an application to Covid-19 vaccine hesitancy

We introduce a geospatial bounded confidence model with mega-influencers, inspired by Hegselmann and Krause. The inclusion of geography gives rise to large-scale geospatial patterns evolving out of random initial data; that is, spatial clusters of like-minded agents emerge regardless of initialization. Mega-influencers and stochasticity amplify this effect, and soften local consensus. As an application, we consider national views on Covid-19 vaccines. For a certain set of parameters, our model yields results comparable to real survey results on vaccine hesitancy from late 2020.

cs.SI↗
arXiv2022

An Equity-Aware Recommender System for Curating Art Exhibits Based on Locally-Constrained Graph Matching

Public art shapes our shared spaces. Public art should speak to community and context, and yet, recent work has demonstrated numerous instances of art in prominent institutions favoring outdated cultural norms and legacy communities. Motivated by this, we develop a novel recommender system to curate public art exhibits with built-in equity objectives and a local value-based allocation of constrained resources. We develop a cost matrix by drawing on Schelling's model of segregation. Using the cost matrix as an input, the scoring function is optimized via a projected gradient descent to obtain a soft assignment matrix. Our optimization program allocates artwork to public spaces in a way that de-prioritizes "in-group" preferences, by satisfying minimum representation and exposure criteria. We draw on existing literature to develop a fairness metric for our algorithmic output, and we assess the effectiveness of our approach and discuss its potential pitfalls from both a curatorial and equity standpoint.

cs.IR↗
arXiv2022

Covid-19 vaccine hesitancy and mega-influencers

Covid-19 vaccines are widely available in the United States, yet our Covid-19 vaccination rates have remained far below 100%. Not only that, but CDC data shows that even in places where vaccine acceptance was proportionally high at the outset of the Covid-19 vaccination effort, that willingness has not necessarily translated into high rates of vaccination over the subsequent months. We model how such a shift could have arisen, using parameters in agreement with data from the state of Alabama. The simulations suggest that in Alabama, local interactions would have favored the emergence of tight consensus around the initial majority view, which was to accept the Covid-19 vaccine. Yet this is not what happened. We therefore add to our model the impact of mega-influencers such as mass media, the governor of the state, etc. Our simulations show that a single vaccine-hesitant mega-influencer, reaching a large fraction of the population, can indeed cause the consensus to shift radically, from acceptance to hesitancy. Surprisingly this is true even when the mega-influencer only reaches individuals who are already somewhat inclined to agree with them, and under the conservative assumption that individuals give no more weight to the mega-influencer than they would give to a single one of their friends or neighbors. Our simulations also suggest that a competing mega-influencer with the opposite view can shift the mean population opinion back, but cannot restore the tightness of consensus around that view. Our code and data are distributed in the ODyN (Opinion Dynamic Networks) library available at https://github.com/annahaensch/ODyN.

physics.soc-ph↗
arXiv2020

A Canonical Form for Positive Definite Matrices

We exhibit an explicit, deterministic algorithm for finding a canonical form for a positive definite matrix under unimodular integral transformations. We use characteristic sets of short vectors and partition-backtracking graph software. The algorithm runs in a number of arithmetic operations that is exponential in the dimension $n$, but it is practical and more efficient than canonical forms based on Minkowski reduction.

math.NT↗
arXiv2019

Classification of one-class spinor genera for quaternary quadratic forms

A quadratic form has a one-class spinor genus if its spinor genus consists of a single equivalence class. In this paper, we determine that there is only one primitive quaternary genus which has a one-class spinor genus but not a one-class genus. In all other cases, the genera of primitive quaternary lattices either have a genus and spinor genus which coincide, or the the genus splits into multiple spinor genera, which in turn split into multiple equivalence classes.

math.NT↗
arXiv2018

Completeness of the list of spinor regular ternary quadratic forms

Extending the notion of regularity introduced by Dickson in 1939, a positive definite ternary integral quadratic form is said to be spinor regular if it represents all the positive integers represented by its spinor genus (that is, all positive integers represented by any form in its spinor genus). Jagy conducted an extensive computer search for primitive ternary quadratic forms that are spinor regular, but not regular, resulting in a list of 29 such forms. In this paper, we will prove that there are no additional forms with this property.

math.NT↗
arXiv2017

Almost universal ternary sums of polygonal numbers

For a natural number $m$, generalized $m$-gonal numbers are those numbers of the form $p_m(x)=\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in \mathbb Z$. In this paper we establish conditions on $m$ for which the ternary sum $p_m(x)+p_m(y)+p_m(z)$ is almost universal.

math.NT↗
arXiv2011

Primitive prime divisors in zero orbits of polynomials

Let $(b_n) = (b_1, b_2, ...)$ be a sequence of integers. A primitive prime divisor of a term $b_k$ is a prime which divides $b_k$ but does not divide any of the previous terms of the sequence. A zero orbit of a polynomial $f(z)$ is a sequence of integers $(c_n)$ where the $n$-th term is the $n$-th iterate of $f$ at 0. We consider primitive prime divisors of zero orbits of polynomials. In this note, we show that for integers $c$ and $d$, where $d > 1$ and $c \neq \pm 1$, every iterate in the zero orbit of $f(z) = z^d + c$ contains a primitive prime whenever zero has an infinite orbit. If $c = \pm 1$, then every iterate after the first contains a primitive prime.

math.NT↗
arXiv2011

Ternary Sums of Squares and Triangular Numbers

For any integer $x$, let $T_x$ denote the triangular number $\frac{x(x+1)}{2}$. In this paper we give a complete characterization of all the triples of positive integers $(α, β, γ)$ for which the ternary sums $αx^2 +βT_y + γT_z$ represent all but finitely many positive integers. This resolves a conjecture of Kane and Sun \cite[Conjecture 1.19(i)]{KS08} and complete the characterization of all almost universal ternary mixed sums of squares and triangular numbers.

math.NT↗