Terry Adams

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Rigidity Sequences of Power Rationally Weakly Mixing Transformations

arXiv (Cornell University), Mar 19, 2015

We prove that a class of infinite measure preserving transformations, satisfying a "strong" weak ... more

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Development of a Big Data Framework for Connectomic Research

arXiv (Cornell University), Jan 24, 2015

This paper outlines research and development of a new Hadoop-based architecture for distributed p... more 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.

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The Gap Dimension and Uniform Laws of Large Numbers for Ergodic Processes

arXiv (Cornell University), Jul 17, 2010

Let F be a family of Borel measurable functions on a complete separable metric space. The gap (or... more Let F be a family of Borel measurable functions on a complete separable metric space. The gap (or fat-shattering) dimension of F is a combinatorial quantity that measures the extent to which functions f ∈ F can separate finite sets of points at a predefined resolution γ > 0. We establish a connection between the gap dimension of F and the uniform convergence of its sample averages under ergodic sampling. In particular, we show that if the gap dimension of F at resolution γ > 0 is finite, then for every ergodic process the sample averages of functions in F are eventually within 10γ of their limiting expectations uniformly over the class F. If the gap dimension of F is finite for every resolution γ > 0 then the sample averages of functions in F converge uniformly to their limiting expectations. We assume only that F is uniformly bounded and countable (or countably approximable). No smoothness conditions are placed on F , and no assumptions beyond ergodicity are placed on the sampling processes. Our results extend existing work for i.i.d. processes.

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Over Recurrence for Mixing Transformations

arXiv (Cornell University), Jan 16, 2017

We show that every invertible strong mixing transformation on a Lebesgue space has strictly over-... more 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 the first question posed in [2]. We define ǫ-over-recurrence and show that given ǫ > 0, any ergodic measure preserving invertible transformation (including discrete spectrum) has ǫ-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 overrecurrent sets.

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Existence and non-existence of solutions to the coboundary equation for measure-preserving systems

Ergodic Theory and Dynamical Systems

A fundamental question in the field of cohomology of dynamical systems is to determine when there... more A fundamental question in the field of cohomology of dynamical systems is to determine when there are solutions to the coboundary equation: $$ \begin{align*} f = g - g \circ T. \end{align*} $$ In many cases, T is given to be an ergodic invertible measure-preserving transformation on a standard probability space $(X, {\mathcal B}, \mu )$ and is contained in $L^p$ for $p \geq 0$ . 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 as follows. Given measurable f, there exist an ergodic invertible measure-preserving transformation T and measurable function g such that $f(x) = g(x) - g(Tx)$ for almost every (a.e.) $x\in X$ , if and only if $\int _{f> 0} f\,d\mu...

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Description of a Tracking Metric Inspired by KL-divergence

ArXiv, 2018

A unified metric is given for the evaluation of tracking systems. The metric is inspired by KL-di... more A unified metric is given for the evaluation of 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). Preliminary scoring results are given on a standard tracking dataset (Oxford Town Centre Dataset). In the final section, prospective advantages of the metric are listed, along with ideas for improving the metric. We end with a couple of open questions concerning tracking metrics.

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AI-Powered Social Bots

ArXiv, 2017

This paper gives an overview of impersonation bots that generate output in one, or possibly, mult... more 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.

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Smorodinsky’s conjecture on rank-one mixing

Proceedings of the American Mathematical Society, 1998

We prove Smorodinsky’s conjecture: the rank-one transformation, obtained by adding staircases who... more

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Weak Mixing for Infinite Measure Invertible Transformations

Lecture Notes in Mathematics, 2018

The weak mixing property for finite measure-preserving transformations has several equivalent cha... more

Constructive symbolic presentations of rank one measure-preserving systems

Colloquium Mathematicum, 2017

Given a rank one measure-preserving system defined by cutting and stacking with spacers, we produ... more 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.

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Entropy and the uniform mean ergodic theorem for a family of sets

Transactions of the American Mathematical Society, 2016

We define a notion of entropy for an infinite family C of measurable sets in a probability space.... more We define a notion of entropy for an infinite family C of measurable sets in a probability space. We show that the mean ergodic theorem holds uniformly for C under every ergodic transformation if and only if C has zero entropy. When the entropy of C is positive, we establish a strong converse showing that the uniform mean ergodic theorem fails generically in every isomorphism class, including the isomorphism classes of Bernoulli transformations. As a corollary of these results, we establish that every strong mixing transformation is uniformly strong mixing on C if and only if the entropy of C is zero, and obtain a corresponding result for weak mixing transformations.

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On infinite transformations with maximal control of ergodic two-fold product powers

Israel Journal of Mathematics, 2015

We study the rich behavior of ergodicity and conservativity of Cartesian products of infinite mea... more We study the rich behavior of ergodicity and conservativity of Cartesian products of infinite measure preserving transformations. A class of transformations is constructed such that for any subset R ⊂ Q ∩ (0, 1) there exists T in this class such that T p × T q is ergodic if and only if p q ∈ R. This contrasts with the finite measure preserving case where T p × T q is ergodic for all nonzero p and q if and only if T × T is ergodic. We also show that our class is rich in the behavior of conservative products. For each positive integer k, a family of rank-one infinite measure preserving transformations is constructed which have ergodic index k, but infinite conservative index.

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ℤ D Staircase Actions

Ergodic Theory and Dynamical Systems

A counterexample concerning the extension of uniform strong laws to ergodic processes

Institute of Mathematical Statistics Collections, 2013

We present a construction showing that a class of sets C that is Glivenko-Cantelli for an i.i.d. ... more We present a construction showing that a class of sets C that is Glivenko-Cantelli for an i.i.d. process need not be Glivenko-Cantelli for every stationary ergodic process with the same one dimensional marginal distribution. This result provides a counterpoint to recent work extending uniform strong laws to ergodic processes, and a recent characterization of universal Glivenko Cantelli classes.

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Tower multiplexing and slow weak mixing

Colloquium Mathematicum, 2015

A technique is presented for multiplexing two ergodic measure preserving transformations together... more 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 sequences 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.

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Ergodicity and Mixing Properties

Encyclopedia of Complexity and Systems Science, 2009

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Binomial-coefficient multiples of irrationals

Monatshefte f�r Mathematik, 1998

Denote by x a random infinite path in the graph of Pascal's triangle (left and right turns are se... more Denote by x a random infinite path in the graph of Pascal's triangle (left and right turns are selected independently with fixed probabilities) and by d n (x) the binomial coefficient at the n'th level along the path x. Then for a dense G δ set of θ in the unit interval, {d n (x)θ} is almost surely dense but not uniformly distributed modulo 1.

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Finitary reconstruction of a measure preserving transformation

Israel Journal of Mathematics, 2001

This paper considers the finitary reconstruction of an ergodic measure preserving transformation ... more This paper considers the finitary reconstruction of an ergodic measure preserving transformation T of a complete separable metric space X from a single trajectory x, T x,. . ., or more generally, from a suitable reconstruction sequence x = x 1 , x 2 ,. .. with x i ∈ X. An n-sample reconstruction is a function T n : X n+1 → X; the map T n (•; x 1 ,. .. , x n) is treated as an estimate of T (•) based on the n initial elements of x. Given a reference probability measure µ 0 and constant M > 1, functions T 1 , T 2 ,. .. are defined, and it is shown that for every µ with 1/M ≤ dµ/dµ 0 ≤ M , every µpreserving transformation T , and every reconstruction sequence x for T , the estimateŝ T n (•; x 1 ,. .. , x n) converge to T in the weak topology. For the family of interval exchange transformations of [0, 1) a simple family of estimates is described and shown to be consistent both pointwise and in the strong topology. However, it is also shown that no finitary estimation scheme is consistent in the strong topology for the family of all ergodic Lebesgue measure preserving transformations of the unit interval, even if x is assumed to be a generic trajectory of T .

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Rank-one weak mixing for nonsingular transformations

Israel Journal of Mathematics, 1997

We construct rank-one infinite measure preserving transformations satisfying each of the followin... more We construct rank-one infinite measure preserving transformations satisfying each of the following dynamical properties: (1) Continuous L °° spectrum, conservative k-fold cartesian products but nonergodic cartesian square; (2) ergodic k-fold cartesian products; (3) nonconservative cartesian square. We show how to modify the construction of (1) to obtain type Ilia transformations with similar properties.

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Estimating a function from ergodic samples with additive noise

IEEE Transactions on Information Theory, 2001

We study the problem of estimating an unknown function from ergodic samples corrupted by additive... more We study the problem of estimating an unknown function from ergodic samples corrupted by additive noise. It is shown that one can consistently recover an unknown measurable function in this setting if the one dimensional distribution of the samples is comparable to a known reference distribution, and the noise is independent of the samples and has known mixing rates. The estimates are applied to deterministic sampling schemes, in which successive samples are obtained by repeatedly applying a fixed map to a given initial vector, and it is then shown how the estimates can be used to reconstruct an ergodic transformation from one of its trajectories.

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