arXiv subjects

Congzhou M Sha

Publications and source records attributed to Congzhou M Sha.

9 recordsLinked to original sources

arXiv2026

Vector alignment in matrix Lie groups

Two observers of the same physical system may differ in gauge by a group element acting on their common vector representations, and recovering that element from finite, noisy paired observations is useful in both theory and experiment. The Kabsch and Horn algorithms solve this problem for rotated frames in $\mathbb R^3$ (i.e. $SO(3)$); our earlier Lie algebra method extends it to the Lorentz group $SO(3,1)_+$. Here we report explicit formulae for the Lie algebra method on the classical matrix Lie groups ($GL(n)$, $SL(n)$, $SO(n)$, $U(n)$, $SO(p,q)$, $Sp(n)$, $Spin(n)$, and $SE(n)$) over the real and complex fields. The four steps (pseudoinverse, matrix logarithm, projection onto the Lie algebra, matrix exponential) are exact for noiseless data. Only the projection is group-dependent, and we show it yields the unique least squares-optimal element of the Lie algebra whenever its image lies in $\mathfrak g$ and its residual is orthogonal to $\mathfrak g$. For noisy data the method is optimal only to leading order, so we add a quasi-Newton correction whose accuracy interpolates between the uncorrected method and direct least squares optimization. The projections, their optimality, and the identity underlying the correction are formally proven in Lean 4.31.0 with Mathlib, and numerical experiments are benchmarked in Julia.

math.NA↗

arXiv2026

A putative, computationally stable structure of homotrimeric BP180/collagen XVII

Background: BP180, also known as collagen XVII and BPAG2 (bullous pemphigoid antigen 2), is a 180-kDa transmembrane protein within the hemidesmosomal plaque complex, and which is known to be a major antigen in bullous pemphigoid, gestational pemphigoid, cicatricial (mucous membrane) pemphigoid, and linear IgA bullous disease. Objective: At present, the 3D structure of BP180 is not known. The goal is to predict a reasonable structure for BP180 through machine learning and molecular dynamics. Methods: In this work, we use the recent Boltz-2 model to predict a putative structure for the intracellular, transmembrane, and proximal extracellular domains, including the NC16A antigenic region and a portion of its first extracellular collagenous domain, Col-15. We computationally embed BP180 in a simple phospholipid bilayer, demonstrate that the putative structure is stable using molecular dynamics, and analyze its allosteric properties. Results: The structures presented satisfy symmetry and secondary structure properties which are expected from homology modelling. Over three 500 ns trajectories, there is minor instability of the predicted globular head domain, but the homotrimer otherwise stays mostly folded. The putative NC16A domain is stiff, whereas the truncated Col-15 domain is highly flexible. There does not appear to be a nearby stable conformation distinct from the initial state. Conclusion: The structure presented is a useful starting point for targeting BP180 pharmacologically, for further experimental characterization of BP180, and for generating hypotheses regarding the relevant epitopes contributing to bullous disease. Diffusion models such as Boltz-2 and AlphaFold3 are useful, but their results must be evaluated carefully.

q-bio.BM↗

arXiv2026

Optimal alignment of Lorentz orientation and generalization to matrix Lie groups

There exist elegant methods of aligning point clouds in $\mathbb R^3$. Unfortunately, these methods fail to generalize to the case of Minkowski space, as we will show. Instead, we propose two solutions to the following problem: given inertial reference frames $A$ and $B$, and given (possibly noisy) measurements of a set of 4-vectors $\{v_i\}$ made in those reference frames with components $\{v_{A,i}\}$ and $\{v_{B,i}\}$, find the optimal Lorentz transformation $Λ$ such that $Λv_{A,i}=v_{B,i}$. The first method is direct least squares optimization through a parametrization of $SO(3,1)_+$ in terms of the familiar boost and rotation vectors. The second method takes a detour through the Lorentz algebra; in addition to being conceptually simple and possessing a computational advantage over the first method, it can easily be generalized to the alignment of vector representations in other matrix Lie groups.

math.NA↗

arXiv2025

tParton: Implementation of next-to-leading order evolution of transversity parton distribution functions

We provide code to solve the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations for the nucleon transversity parton distribution functions (PDFs), which encode nucleon transverse spin structure. Though codes are widely available for the evolution of unpolarized and polarized PDFs, there are few codes publicly available for the transversity PDF. Here, we present Python code which implements two methods of solving the leading order (LO) and next-to-leading order (NLO) approximations of the DGLAP equations for the transversity PDF, and we highlight the theoretical differences between the two.

hep-ph↗

arXiv2025

On the continuity of flows

Flow matching has emerged as a powerful framework for generative modeling through continuous normalizing flows. We investigate a potential topological constraint: when the prior distribution and target distribution have mismatched topology (e.g., unimodal to multimodal), the optimal velocity field under standard flow matching objectives may exhibit spatial discontinuities. We suggest that this discontinuity arises from the requirement that continuous flows must bifurcate to map a single mode to multiple modes, forcing particles to make discrete routing decisions at intermediate times. Through theoretical analysis on bimodal Gaussian mixtures, we demonstrate that the optimal velocity field exhibits jump discontinuities along decision boundaries, with magnitude approaching infinity as time approaches the target distribution. Our analysis suggests that this phenomenon is not specific to $L^2$ loss, but rather may be a consequence of topological mismatch between distributions. We validate our theory empirically and discuss potential implications for flow matching on manifolds, connecting our findings to recent work on Riemannian flow matching and the challenge of learning discontinuous representations in neural networks.

cs.LG↗

arXiv2025

Mitigating collusion in Gold Humanism Honor Society nominations

An anecdotally common complaint regarding induction into the Gold Humanism Honor Society is the bias toward close friends during the initial nomination process. In this work, we numerically simulate the nomination process under different assumptions, demonstrate that collusion can be detected, and propose a simple strategy to correct for bias in the nomination process.

physics.soc-ph↗

arXiv2024

Charting a finite element, mechanical atlas of dermatologic wound closure

Wound geometry and the mechanical properties of human skin govern the failure modes of partially healed or scarred tissue. Though dermatologists and surgeons develop an intuitive understanding of the mechanical characteristics of skin through clinical practice, finite element models of wounds can aid in formalizing intuition. In this work, we explore the effect of wound geometry and primary intention closure on the propagation of mechanical stresses through skin. We use a two-layer, orthotropic, hyperelastic model of the epidermis, dermis, and subcutis to accurately capture the mechanical and geometric effects at work. We highlight the key assumptions which must be made when modeling closure of wounds by primary intention, clearly delineating promising areas for model improvement. Models are implemented in DOLFINx, an open-source finite element framework, and reference code is provided for reproducible and extensible science.

physics.med-ph↗

arXiv2021

Simple exponential acceleration of the power iteration algorithm

Many real-world problems rely on finding eigenvalues and eigenvectors of a matrix. The power iteration algorithm is a simple method for determining the largest eigenvalue and associated eigenvector of a general matrix. This algorithm relies on the idea that repeated multiplication of a randomly chosen vector x by the matrix A gradually amplifies the component of the vector along the eigenvector of the largest eigenvalue of A while suppressing all other components. Unfortunately, the power iteration algorithm may demonstrate slow convergence. In this report, we demonstrate an exponential speed up in convergence of the power iteration algorithm with only a polynomial increase in computation by taking advantage of the commutativity of matrix multiplication.

math.NA↗

arXiv2021

Drug Discovery Approaches using Quantum Machine Learning

Traditional drug discovery pipeline takes several years and cost billions of dollars. Deep generative and predictive models are widely adopted to assist in drug development. Classical machines cannot efficiently produce atypical patterns of quantum computers which might improve the training quality of learning tasks. We propose a suite of quantum machine learning techniques e.g., generative adversarial network (GAN), convolutional neural network (CNN) and variational auto-encoder (VAE) to generate small drug molecules, classify binding pockets in proteins, and generate large drug molecules, respectively.

cs.ET↗