arXiv subjects

Terrence Adams

Publications and source records attributed to Terrence Adams.

9 recordsLinked to original sources

arXiv2026

Perpetually Fair Assignments Via Balanced Sequences of Permutations

There is a set of $n$ indivisible items (goods or chores), and a set of $n$ players. Each day, a single item should be assigned to each player. Assignments based on latin squares guarantee fairness after every $n$ days; our goal is to ensure fairness after every single day. We present two 'balance' conditions on latin squares. Informally, a latin square is balanced if its top rows and leftmost columns contain all $n$ labels; this ensures that all $n$ players receive one of the top items in one of the early days. One such condition can always be satisfied, but is arguably too weak; a second condition is strong, and can be satisfied for all $n\leq 12$, but cannot be satisfied for some larger values of $n$, including all $n>108$. We show that the second balance condition guarantees that the cumulative assignment is always \emph{proportional up to one item (PROP1)}, where proportionality holds in a strong ordinal sense -- for every valuations that are consistent with the item ranking. Finally, we present a weaker balance condition on a sequence, that guarantees ordinal proportionality up to two items (PROP2). Whether or not this condition can be satisfied for all $n$ remains an open question.

math.CO↗

arXiv2023

Moving Averages

We consider the convergence of moving averages in the general setting of ergodic theory or stationary ergodic processes. We characterize when there is universal convergence of moving averages based on complete convergence to zero of the standard ergodic averages. Using a theorem of Hsu-Robbins (1947) for independent, identically distributed processes, we prove for any bounded measurable function $f$ on a standard probability space $(X,\mathcal{B},\mu)$, there exists a Bernoulli shift $T$, such that all moving averages $M(v_n, L_n)^T f = \frac{1}{L_n} \sum_{i=v_n+1}^{v_n+L_n} f \circ T^i$ with $L_n\geq n$ converge a.e. to $\int_X f d\mu$. We refresh the reader about the cone condition established by Bellow, Jones, Rosenblatt (1990) which guarantees convergence of certain moving averages for all $f \in L^1(\mu)$ and ergodic measure preserving maps $T$. We show given $f \in L^1(\mu)$ and ergodic measure preserving $T$, there exists a moving average $M(v_n,L_n)^T f$ with $L_n$ strictly increasing such that $(v_n,L_n)$ does not satisfy the cone condition, but pointwise convergence holds a.e. We show for any non-zero $f\in L^1(\mu)$, there is a generic class of ergodic maps $T$ such that each map has an associated moving average $M(v_n, L_n)^T f$ which does not converge pointwise. It is known if $f\in L^2(\mu)$ is mean-zero, then there exist solutions $T$ and $g\in L^1(\mu)$ to the coboundary equation: $f = g - g\circ T$. This implies $f$ and $T$ produce universal moving averages. We show this does not generalize to $L^p(\mu)$ for $p<2$ by explicitly defining functions $f\in \cap_{p<2}L^p(\mu)$ such that for each ergodic measure preserving $T$, there exists a moving average $M(v_n, L_n)^T f$ with $L_n\geq n$ such that these moving averages do not converge pointwise. Several of the results are generalized to the case of moving averages with polynomial growth.

math.DS↗

arXiv2019

Existence and Non-existence of Solutions to the Coboundary Equation for Measure Preserving Systems

Let $(X,\mathcal{B},\mu)$ be a standard probability space. We give new fundamental results determining solutions to the coboundary equation: \begin{eqnarray*} f = g - g \circ T \end{eqnarray*} where $f \in L^p$ and $T$ is ergodic invertible measure preserving on $(X, \mathcal{B}, \mu )$. We extend previous results by showing for any measurable $f$ that is non-zero on a set of positive measure, the class of measure preserving $T$ with a measurable solution $g$ is meager (including the case where $\int_X f d\mu = 0$). From this fact, a natural question arises: given $f$, does there always exist a solution pair $T$ and $g$? In regards to this question, our main results are: (i) Given measurable $f$, there exists an ergodic invertible measure preserving transformation $T$ and measurable function $g$ such that $f(x) = g(x) - g(Tx)$ for a.e. $x\in X$, if and only if $\int_{f > 0} f d\mu = - \int_{f < 0} f d\mu$ (whether finite or $\infty$). (ii) Given mean-zero $f \in L^p$ for $p \geq 1$, there exists an ergodic invertible measure preserving $T$ and $g \in L^{p-1}$ such that $f(x) = g(x) - g( Tx )$ for a.e. $x \in X$. (iii) In some sense, the previous existence result is the best possible. For $p \geq 1$, there exist mean-zero $f \in L^p$ such that for any ergodic invertible measure preserving $T$ and any measurable $g$ such that $f(x) = g(x) - g(Tx)$ a.e., then $g \notin L^q$ for $q > p - 1$. Also, we show this situation is generic for mean-zero $f \in L^p$. Finally, it is shown that we cannot expect finite moments for solutions $g$, when $f \in L^1$. In particular, given any $\phi : \mathbb{R} \to \mathbb{R}$ such that $\lim_{x\to \infty} \phi (x) = \infty$, there exist mean-zero $f \in L^1$ such that for any solutions $T$ and $g$, the transfer function $g$ satisfies: \begin{eqnarray*} \int_{X} \phi \big( | g(x) | \big) d\mu = \infty. \end{eqnarray*}

math.DS↗

arXiv2018

A Continuous, Full-scope, Spatio-temporal Tracking Metric based on KL-divergence

A unified metric is given for the evaluation of object tracking systems. The metric is inspired by KL-divergence or relative entropy, which is commonly used to evaluate clustering techniques. Since tracking problems are fundamentally different from clustering, the components of KL-divergence are recast to handle various types of tracking errors (i.e., false alarms, missed detections, merges, splits). Scoring results are given on a standard tracking dataset (Oxford Town Centre Dataset), as well as several simulated scenarios. Also, this new metric is compared with several other metrics including the commonly used Multiple Object Tracking Accuracy metric. In the final section, advantages of this metric are given including the fact that it is continuous, parameter-less and comprehensive.

cs.CV↗

arXiv2017

AI-Powered Social Bots

This paper gives an overview of impersonation bots that generate output in one, or possibly, multiple modalities. We also discuss rapidly advancing areas of machine learning and artificial intelligence that could lead to frighteningly powerful new multi-modal social bots. Our main conclusion is that most commonly known bots are one dimensional (i.e., chatterbot), and far from deceiving serious interrogators. However, using recent advances in machine learning, it is possible to unleash incredibly powerful, human-like armies of social bots, in potentially well coordinated campaigns of deception and influence.

cs.SI↗

arXiv2017

Over Recurrence for Mixing Transformations

We show that every invertible strong mixing transformation on a Lebesgue space has strictly over-recurrent sets. Also, we give an explicit procedure for constructing strong mixing transformations with no under-recurrent sets. This answers both parts of a question of V. Bergelson. We define $\epsilon$-over-recurrence and show that given $\epsilon > 0$, any ergodic measure preserving invertible transformation (including discrete spectrum) has $\epsilon$-over-recurrent sets of arbitrarily small measure. Discrete spectrum transformations and rotations do not have over-recurrent sets, but we construct a weak mixing rigid transformation with strictly over-recurrent sets.

math.DS↗

arXiv2016

Constructive symbolic presentations of rank one measure-preserving systems

Given a rank one measure-preserving system defined by cutting and stacking with spacers, we produce a rank one binary sequence such that its orbit closure under the shift transformation, with its unique {nonatomic} invariant probability, is isomorphic to the given system. In particular, the classical dyadic odometer is presented in terms of a recursive sequence of blocks on the two-symbol alphabet $\{0,1\}$. The construction is accomplished using a definition of rank one in the setting of adic, or Bratteli-Vershik, systems.

math.DS↗

arXiv2015

Development of a Big Data Framework for Connectomic Research

This paper outlines research and development of a new Hadoop-based architecture for distributed processing and analysis of electron microscopy of brains. We show development of a new C++ library for implementation of 3D image analysis techniques, and deployment in a distributed map/reduce framework. We demonstrate our new framework on a subset of the Kasthuri11 dataset from the Open Connectome Project.

cs.DC↗

arXiv2013

Tower multiplexing and slow weak mixing

A technique is presented for multiplexing two ergodic measure preserving transformations together to derive a third limiting transformation. This technique is used to settle a question regarding rigidity of weak mixing transformations. Namely, given any rigidity sequence for an ergodic measure preserving transformation, there exists a weak mixing transformation which is rigid along the same sequence. This establishes a wide range of rigidity sequences for weakly mixing dynamical systems.

math.DS↗