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Zawad Chowdhury

Publications and source records attributed to Zawad Chowdhury.

5 recordsLinked to original sources

Obtaining the Chamanara Surface from the van der Corput sequence

We investigate a family of $4$-regular graphs constructed to test for the presence of combinatorial structure in a sequence of distinct real numbers. We show that the graphs constructed from the Kronecker sequence can be embedded into the torus, while the graphs constructed from the binary van der Corput sequence can be embedded into the Chamanara surface, in both cases with the possible removal of one edge. These results generalize to embeddings of sequence graphs coming from interval exchange transformations into associated translation surfaces.

math.CO↗

Uniformly Weighted Graphical Designs

A graphical design is a subset of vertices of a graph, along with a weight for each chosen vertex, that can perfectly average chosen subspaces of functions on the graph. A design is uniformly weighted if all the weights are equal, and several well-known combinatorial objects such as orthogonal arrays, combinatorial block designs and t-wise permutations are uniformly weighted graphical designs. While one might expect to see uniformly weighted designs in structured graphs, they do not always exist. In this paper we characterize the existence of uniformly weighted graphical designs, and use our result to provide several families of graphs that have, and do not have, such designs. Our results offer a polyhedral view of the structures that control the existence and cardinalities of these designs. In particular, we characterize all uniformly weighted designs of threshold graphs, and provide a geometric proof of the duality of linear codes and linear orthogonal arrays. We also provide a novel construction for graphs whose Laplacian characteristic polynomials are almost irreducible, to produce families without uniformly weighted designs.

math.CO↗

Graphical Designs find Combinatorial Structures

Graphical designs are subsets of vertices of a graph that perfectly average a selected set of eigenvectors of the Graph Laplacian. We show that in highly-structured graphs, graphical designs can coincide with highly structured and well-known combinatorial objects: orthogonal arrays in hypercube graphs, combinatorial block designs and extremizers of the Erdos-Ko-Rado theorem in Johnson graphs, and t-wise uniform sets of permutations and symmetric subgroups in normal Cayley graphs on the symmetric group. These connections allow tools from spectral graph theory to bear on these combinatorial objects. We also show that the central vertex in a Mycielskian is an extremely good design and certain designs of the Mycielskian coincide with designs of the original graph.

math.CO↗

Event knowledge in large language models: the gap between the impossible and the unlikely

Word co-occurrence patterns in language corpora contain a surprising amount of conceptual knowledge. Large language models (LLMs), trained to predict words in context, leverage these patterns to achieve impressive performance on diverse semantic tasks requiring world knowledge. An important but understudied question about LLMs' semantic abilities is whether they acquire generalized knowledge of common events. Here, we test whether five pre-trained LLMs (from 2018's BERT to 2023's MPT) assign higher likelihood to plausible descriptions of agent-patient interactions than to minimally different implausible versions of the same event. Using three curated sets of minimal sentence pairs (total n=1,215), we found that pre-trained LLMs possess substantial event knowledge, outperforming other distributional language models. In particular, they almost always assign higher likelihood to possible vs. impossible events (The teacher bought the laptop vs. The laptop bought the teacher). However, LLMs show less consistent preferences for likely vs. unlikely events (The nanny tutored the boy vs. The boy tutored the nanny). In follow-up analyses, we show that (i) LLM scores are driven by both plausibility and surface-level sentence features, (ii) LLM scores generalize well across syntactic variants (active vs. passive constructions) but less well across semantic variants (synonymous sentences), (iii) some LLM errors mirror human judgment ambiguity, and (iv) sentence plausibility serves as an organizing dimension in internal LLM representations. Overall, our results show that important aspects of event knowledge naturally emerge from distributional linguistic patterns, but also highlight a gap between representations of possible/impossible and likely/unlikely events.

cs.CL↗

Computing Periodic Points on Veech Surfaces

A non-square-tiled Veech surface has finitely many periodic points, i.e., points with finite orbit under the affine automorphism group. We present an algorithm that inputs a non-square-tiled Veech surface and outputs its set of periodic points. We apply our algorithm to Prym eigenforms in the minimal stratum in genus 3, proving that in low discriminant these surfaces do not have periodic points, except for the fixed points of the Prym involution.

math.DS↗