A useful property of Euclidean space originally shown by I. Benjamini and O. Schramm turns out to characterize doubling metric spaces.

James T. Gill

We examine the quantization of the motion of two charged vortices in a Ginzburg--Landau theory for the fractional quantum Hall effect recently proposed by the first two authors. The system has two second-class constraints which can be implemented either in the reduced phase space or Dirac-Gupta-Bleuler formalism. Using the intrinsic formulation of statistics, we show that these two ways of implementing the constraints are inequivalent unless the vortices are quantized with conventional statistics; either fermionic or bosonic.

Theodore J. Allen Andrew J. Bordner Dennis B. Crossley

We consider a 1+1 dimensional field theory which contains both a complex fermion field and a real scalar field. We then construct a unitary operator that, by a similarity transformation, gives a continuum of equivalent theories which smoothly interpolate between the massive Thirring model and the sine-Gordon model. This provides an implementation of smooth bosonization proposed by Damgaard et al. as well as an example of a quantum canonical transformation for a quantum field theory.

Andrew J. Bordner

One may obtain, using operator transformations, algebraic relations between the Fourier transforms of the causal propagators of different exactly solvable potentials. These relations are derived for the shape invariant potentials. Also, potentials related by real transformation functions are shown to have the same spectrum generating algebra with Hermitian generators related by this operator transformation.

Andrew J. Bordner

We show that various models of the elliptic Calogero-Moser systems are accompanied with an isomonodromic system on a torus. The isomonodromic partner is a non-autonomous Hamiltonian system defined by the same Hamiltonian. The role of the time variable is played by the modulus of the base torus. A suitably chosen Lax pair (with an elliptic spectral parameter) of the elliptic Calogero-Moser system turns out to give a Lax representation of the non-autonomous system as well. This Lax representation ensures that the non-autonomous system describes isomonodromic deformations of a linear ordinary differential equation on the torus on which the spectral parameter of the Lax pair is defined. A particularly interesting example is the ``extended twisted $BC_\ell$ model'' recently introduced along with some other models by Bordner and Sasaki, who remarked that this system is equivalent to Inozemtsev's generalized elliptic Calogero-Moser system. We use the ``root type'' Lax pair developed by Bordner et al. to formulate the associated isomonodromic system on the torus.

Kanehisa Takasaki

Despite the importance of a thermodynamically stable structure with a conserved fold for protein function, almost all evolutionary models neglect site-site correlations that arise from physical interactions between neighboring amino acid sites. This is mainly due to the difficulty in formulating a computationally tractable model since rate matrices can no longer be used. Here we introduce a general framework, based on factor graphs, for constructing probabilistic models of protein evolution with site interdependence. Conveniently, efficient approximate inference algorithms, like Belief Propagation, can be used to calculate likelihoods for these models. We fit an amino acid substitution model of this type that accounts for both solvent accessibility and site-site correlations. Comparisons of the new model with rate matrix models and a model accounting only for solvent accessibility demonstrate that it better fits the sequence data. We also examine evolution within a family of homohexameric enzymes and find that site-site correlations between most contacting subunits contribute to a higher likelihood. In addition, we show that the new substitution model has a similar mathematical form to the one introduced in (Rodrigue et al. 2005), although with different parameter interpretations and values. We also perform a statistical analysis of the effects of amino acids at neighboring sites on substitution probabilities and find a significant perturbation of most probabilities, further supporting the significant role of site-site interactions in protein evolution and motivating the development of new evolutionary models like the one described here. Finally, we discuss possible extensions and applications of the new substitution models.

Andrew J. Bordner Hans D. Mittelmann

Computational methods are needed to differentiate the small fraction of missense mutations that contribute to disease by disrupting protein function from neutral variants. We describe several complementary methods using large-scale homology modeling of human protein complexes to detect non-neutral mutations. Importantly, unlike sequence conservation-based methods, this structure-based approach provides experimentally testable biochemical mechanisms for mutations in disease. Specifically, we infer metal ion, small molecule, protein-protein, and nucleic acid binding sites by homology and find that disease-associated missense mutations are more prevalent in each class of binding site than are neutral mutations. Importantly, our approach identifies considerably more binding sites than those annotated in the RefSeq database. Furthermore, an analysis of metal ion and protein-protein binding sites predicted by machine learning shows a similar preponderance of disease-associated mutations in these sites. We also derive a statistical score for predicting how mutations affect metal ion binding and find many dbSNP mutations that likely disrupt ion binding but were not previously considered deleterious. We also cluster mutations in the protein structure to discover putative functional regions. Finally, we develop a machine learning predictor for detecting disease-associated missense mutations and show that it outperforms two other prediction methods on an independent test set.

Andrew J. Bordner Barry Zorman

We show that every n-point tree metric admits a (1+eps)-embedding into a C(eps) log n-dimensional L_1 space, for every eps > 0, where C(eps) = O((1/eps)^4 log(1/eps)). This matches the natural volume lower bound up to a factor depending only on eps. Previously, it was unknown whether even complete binary trees on n nodes could be embedded in O(log n) dimensions with O(1) distortion. For complete d-ary trees, our construction achieves C(eps) = O(1/eps^2).

James R. Lee Arnaud de Mesmay Mohammad Moharrami

We present and analyze two new algorithms for the package-exchange robot-routing problem (PERR): restriction to inidividual paths (RIP) and bubbletree. RIP provably produces a makespan that is $O(\text{SIC}+k^2)$, where SIC is the sum of the lengths of the individual paths and $k$ is the number of robots. Bubbletree produces a makespan that is $O(n)$, where $n$ is the number of nodes. With optimizations bubbletree can also achieve a makespan of $O((k+l)\text{log}k)$, where $l$ is the longest path from start to goal in the bubbletree subgraph.

James Drain

In a hypergraph on $n$ vertices where $D$ is the maximum size of a hyperedge, there is a weighted hypergraph spectral $\varepsilon$-sparsifier with at most $O(\varepsilon^{-2} \log(D) \cdot n \log n)$ hyperedges. This improves over the bound of Kapralov, Krauthgamer, Tardos and Yoshida (2021) who achieve $O(\varepsilon^{-4} n (\log n)^3)$, as well as the bound $O(\varepsilon^{-2} D^3 n \log n)$ obtained by Bansal, Svensson, and Trevisan (2019). The same sparsification result was obtained independently by Jambulapati, Liu, and Sidford (2022).

James R. Lee