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David Handelman

Publications and source records attributed to David Handelman.

At least 19 recordsLinked to original sources
arXiv2025

Abstract fractional linear transformations

We begin with (densely-defined) fractional linear transformations (FLT) on (some) Banach algebras and their relatives. This leads to Wedderburn's continued fractions (recursively-defined noncommutative polynomials) for any ring. Along the way, we discover a one-parameter family of (noncommutative) polynomials \st if one of them is invertible, then read in the opposite order, the corresponding polynomial is also invertible (extending the well known $1+ab$ is invertible if $1+ ba$ is, and the not-so-well-known, $a + abc + c$ and $a + cba + c$). This in turn leads to a definition of FLT for general rings $R$, which turns out to be PE$(2,R)$ (the projective elementary group). Using Wedderburn's polynomials, this permits us to define a length function on PE$(2,R)$, which suggests a stable range type condition (for $n =1$, it {\it is\/} stable range one, but higher values do not correspond. Again using the length results, we prove the expected results for PE$(2,R)$: under very modest conditions on $R$, the commutator subgroup of PE$(2,R)$ is perfect and of index one or two. Along the same lines, we also prove results on simplicity of the commutator subgroup: we require the usual generative properties on the simple ring $R$, as well either the very restrictive $1$ in the range, or a mild condition about invertibles, involving intersections of three translates of GL$(1,R)$. This last property is explored in the appendices, which give examples (and non-examples). Numerous questions suggest themselves throughout.

math.FA↗
arXiv2025

FLEET: Formal Language-Grounded Scheduling for Heterogeneous Robot Teams

Coordinating heterogeneous robot teams from free-form natural-language instructions is hard. Language-only planners struggle with long-horizon coordination and hallucination, while purely formal methods require closed-world models. We present FLEET, a hybrid decentralized framework that turns language into optimized multi-robot schedules. An LLM front-end produces (i) a task graph with durations and precedence and (ii) a capability-aware robot--task fitness matrix; a formal back-end solves a makespan-minimization problem while the underlying robots execute their free-form subtasks with agentic closed-loop control. Across multiple free-form language-guided autonomy coordination benchmarks, FLEET improves success over state of the art generative planners on two-agent teams across heterogeneous tasks. Ablations show that mixed integer linear programming (MILP) primarily improves temporal structure, while LLM-derived fitness is decisive for capability-coupled tasks; together they deliver the highest overall performance. We demonstrate the translation to real world challenges with hardware trials using a pair of quadruped robots with disjoint capabilities.

cs.RO↗
arXiv2024

ConceptAgent: LLM-Driven Precondition Grounding and Tree Search for Robust Task Planning and Execution

Robotic planning and execution in open-world environments is a complex problem due to the vast state spaces and high variability of task embodiment. Recent advances in perception algorithms, combined with Large Language Models (LLMs) for planning, offer promising solutions to these challenges, as the common sense reasoning capabilities of LLMs provide a strong heuristic for efficiently searching the action space. However, prior work fails to address the possibility of hallucinations from LLMs, which results in failures to execute the planned actions largely due to logical fallacies at high- or low-levels. To contend with automation failure due to such hallucinations, we introduce ConceptAgent, a natural language-driven robotic platform designed for task execution in unstructured environments. With a focus on scalability and reliability of LLM-based planning in complex state and action spaces, we present innovations designed to limit these shortcomings, including 1) Predicate Grounding to prevent and recover from infeasible actions, and 2) an embodied version of LLM-guided Monte Carlo Tree Search with self reflection. In simulation experiments, ConceptAgent achieved a 19% task completion rate across three room layouts and 30 easy level embodied tasks outperforming other state-of-the-art LLM-driven reasoning baselines that scored 10.26% and 8.11% on the same benchmark. Additionally, ablation studies on moderate to hard embodied tasks revealed a 20% increase in task completion from the baseline agent to the fully enhanced ConceptAgent, highlighting the individual and combined contributions of Predicate Grounding and LLM-guided Tree Search to enable more robust automation in complex state and action spaces.

cs.AI↗
arXiv2021

Another invariant for AT actions

We construct a collection of numerical invariants for approximately transitive (AT) actions (of $\Z$). We use them (sometimes supplemented by other invariants to show that members of various one-parameter families of AT actions are mutually non-isomorphic.

math.DS↗
arXiv2021

Boundary of the boundary for random walks on groups

We study fine structure related to finitely supported random walks on infinite finitely generated discrete groups, largely motivated by dimension group techniques. The unfaithful extreme harmonic functions (defined only on proper space-time cones), aka unfaithful pure traces, can be represented on systems of finite support, avoiding dead ends. This motivates properties of the random walk (WC) and of the group (SWC) which become of interest in their own right. While all abelian groups satisfy WC, the do not satisfy SWC; however some abelian by finite groups do satisfy the latter, and we characterize when this occurs. In general, we determine the maximal order ideals, aka, maximal proper space-time subcones of that generated by the group element $1$ at time zero), and show that the corresponding quotients are stationary simple dimension groups, and that all such can occur for the free group on two generators. We conclude with a case study of the discrete Heisenberg group, determining among other things, the pure traces (these are the unfaithful ones, not arising from characters).

math.FA↗
arXiv2020

Asymptotic hollowness of lattice simplices

An $(n-1)$-tuple $a = (a(1), \dots, a(n-1))$ consisting of positive integers is said to be asymptotically hollow if there exist infinitely many positive integers $N$ such that the convex hull, $K(a(n))$, in $n$-dimensional Euclidean space of $\{ 0,e_1, \dots, e_{n-1}, \alpha(N)^T\}$ is hollow (has no lattice points in its interior), where $e_i$ run over all but the last standard basis elements, and $\alpha(N) $ is the row $(a(1), \dots, a(N-1), N)$. The tuple is trivial if $\min a(i) = 1$. Nontrivial asymptotically hollow tuples are characterized in terms of modular inequalities, and turn out to be rare. We show that for a tuple $a$, there exists an effectively computable constant $C$ (depending on $a$) such that if for some $N > C$, $K(\alpha(N))$ is (not) hollow, then for all $M > C$, $K(\alpha(M))$ is (not) hollow (respectively). When $n = 4$, the nontrivial asymptotically hollow triples are completely determined; there are eleven of them, together with a one-parameter family.

math.NT↗
arXiv2016

Random sets and intersections

The following class of problems arose out of vain attempts to show that the Pascal's triangle adic transformation has trivial spectrum. Partition a set of size $N$ into sets of size $S \equiv S(N)$ (ignoring leftovers). What is the likelihood that a set of size $K \equiv K(N)$ will intersect each set in the partition in at least $R \equiv R(N)$ members (as $N$ increases)? Via elementary techniques and under reasonable hypotheses, we obtain an easy-to-use formula. Although different from the corresponding minimum problem for balls and bins (with $m = K$ balls and $n = N/S$ bins), under modest constraints, the asymptotic probabilities are the same.

math.PR↗
arXiv2016

Orbit equivalence of Cantor minimal systems and their continuous spectra

To any continuous eigenvalue of a Cantor minimal system $(X,\,T)$, we associate an element of the dimension group $K^0(X,\,T)$ associated to $(X,\,T)$. We introduce and study the concept of irrational miscibility of a dimension group. The main property of these dimension groups is the absence of irrational values in the additive group of continuous spectrum of their realizations by Cantor minimal systems. The strong orbit equivalence (respectively orbit equivalence) class of a Cantor minimal system associated to an irrationally miscible dimension group $(G,\,u)$ (resp. with trivial infinitesimal subgroup) with trivial rational subgroup, have no non-trivial continuous eigenvalues.

math.DS↗
arXiv2015

Nonsingular transformations and dimension spaces

For any adic transformation $T$ defined on the path space $X$ of an ordered Bratteli diagram, endowed with a Markov measure $\mu$, we construct an explicit dimension space (which corresponds to a matrix values random walk on $\mathbb{Z}$) whose Poisson boundary can be identified as a $\mathbb{Z}$-space with the dynamical system $(X,\mu,T)$. We give a couple of examples to show how dimension spaces can be used in the study of nonsingular transformations.

math.DS↗
arXiv2015

One-sided approximation in affine function spaces

Let $H$ be a subgroup of a partially ordered abelian group $G$ with order unit $u$, and let $S(G,u)$ denote the convex subset of $\bR^G$ consisting of all traces (states) $\tau$ on $G$ with $\tau(u)=1$. We say that $H$ has property $(B)$ if, for any integer $m\ge 2$, any $h\in H$ and any $\epsilon>0$, there exists $h'\in H$ such that $\tau(h)-m\tau(h')\ge -\epsilon$ for each $\tau\in S(G,u)$. We show that, if $S(G,u)$ is finite-dimensional, this condition is equivalent to asking that $\tau(H)$ is $\{0\}$ or dense in $\bR$ for all $\tau$ in the smallest face of $S(G,u)$ containing all traces that vanish identically on $H$. When $G$ is a simple dimension group and $H$ is a convex subgroup of $G$, we show that $G/H$ is unperforated if and only if $H$ has property $(B)$. We apply both results to provide a criterion for a trace of $G$ to be refinable when $G$ is a simple dimension group with finitely many pure traces.

math.NT↗
arXiv2014

Nearly approximate transitivity (AT) for circulant matrices

By previous work of Giordano and the author, ergodic actions of $\Z$ (and other discrete groups) are completely classified measure-theoretically by their dimension space, a construction analogous to the dimension group used in C*-algebras and topological dynamics. Here we investigate how far from AT (approximately transitive) can actions be which derive from circulant (and related) matrices. It turns out not very: although non-AT actions can arise from this method of construction, under very modest additional conditions, ATness arises; in addition, if we drop the positivity requirement in the isomorphism of dimension spaces, then all these ergodic actions satisfy an analogue of AT. Many examples are provided

math.DS↗
arXiv2013

Invariants for critical dimension groups and permutation-Hermite equivalence

Motivated by classification, up to order isomorphism, of some dense subgroups of Euclidean space that are free of minimal rank, we obtain apparently new invariants for an equivalence relation (intermediate between Hermite and Smith) on integer matrices. These then participate in the classification of the dense subgroups. The same equivalence relation has appeared before, in the classification of lattice simplices. We discuss this equivalence relation (called {\it permutation-Hermite}), obtain fairly fine invariants for it, and have density results, and some formulas counting the numbers of equivalence classes for fixed determinant.

math.AC↗
arXiv2013

Good measures for non-simple dimension groups

Akin's notion of good measure, introduced to classify measures on Cantor sets has been translated to dimension groups and corresponding traces by Bezuglyi and the author, but emphasizing the simple (minimal dynamical system) case. Here we deal with non-simple (non-minimal) dimension groups. In particular, goodness of tensor products of large classes of non-good traces (measures) is established. We also determine the pure faithful traces on the dimension groups associated to xerox type actions on AF C*-algebras; the criteria turn out to involve algebraic geometry and number theory. We also deal with a coproduct of dimension groups, wherein, despite expectations, goodness of direct sums is nontrivial. In addition, we verify a conjecture of [BeH] concerning good subsets of Choquet simplices, in the finite-dimensional case.

math.FA↗
arXiv2013

Non-direct limit of simple dimension groups with finitely many pure traces

There exist simple dimension groups which cannot be expressed as a direct limit of simple, or even approximately divisible dimension groups, each with finitely many pure traces, and we can specify its infinite-dimensional Choquet simplex of traces; a more drastic property is noted. On the other hand, a very easy argument shows that if $G$ is a $p$-divisible simple dimension group (for some integer $p>1$), then it can be expressed as such a direct limit. We also enlarge the class of initial objects for AF (and slightly more general) C*-algebras.

math.FA↗
arXiv2013

Equal column sum and equal row sum dimension group realizations

Motivated by connections between minimal actions, especially T\"oplitz, on Cantor sets, and dimension groups, we find realizations of classes of dimension groups as limits of primitive matrices all of which have equal column sums, or equal row sums.

math.FA↗
arXiv2012

Measures on Cantor sets: the good, the ugly, the bad

We translate Akin's notion of {\it good} (and related concepts) from measures on Cantor sets to traces on dimension groups, and particularly for invariant measures of minimal homeomorphisms (and their corresponding simple dimension groups), this yields characterizations and examples, which translate back to the original context. Good traces on a simple dimension group are characterized by their kernel having dense image in their annihilating set of affine functions on the trace space; this makes it possible to construct many examples with seemingly paradoxical properties. In order to study the related property of {\it refinability,} we consider goodness for sets of measures (traces on dimension groups), and obtain partial characterizations in terms of (special) convex subsets of Choquet simplices. These notions also very closely related to unperforation of quotients of dimension groups by convex subgroups (that are not order ideals), and we give partial characterizations. Numerous examples illustrate the results.

math.DS↗
arXiv2011

Log concavity of $(1+x)^m (1+ x^k)$

Let $m$ and $k \geq 2$ be positive integers. We show that polynomial $P = (1+x)^m(1+x^k)$ is strongly unimodal (frequently known as {\it log concave\/}) if and only if $m \geq k^2 -3$; this is also the criterion for $P$ to be merely unimodal (that is, for $P$ of this form, unimodality implies strong unimodality).{ }In section 2, we investigate an analogous question, concerning the property $\EE$ of functions $f$ analytic on a neighbourhood of the unit circle [H2], and show that the corresponding minimal $m$ is rather surprisingly of order $k^4$.

math.CO↗