arXiv subjects

Jaeseong Heo

Publications and source records attributed to Jaeseong Heo.

4 recordsLinked to original sources

arXiv2026

Integral zeros of Krawtchouk polynomials and quantum entanglements

In Heo and Kiem (2019), the authors raised a problem relating integral zeros of binary Krawtchouk polynomials, parametrized by quadruples $(m,n,k,\ell)$ with $m\le n$ and $k\le \ell$, to the existence of pairs of subspaces satisfying the range criterion in quantum information theory, and settled the symmetric case $k=\ell$. In this paper, we solve the problem for the following three asymmetric families: $(m,n,k,\ell)=(3,t^2-1,t(t-1)/2,t(t+1)/2)$, $t\ge2$; $(m,n,k,\ell)=(4,3t^2+2t-1,(3t^2-t)/2,(3t^2+5t+2)/2)$, $t\ge2$; $(m,n,k,\ell)=(4,3s^2+4s,(3s^2+s)/2,(3s^2+7s+4)/2)$, $s\ge1$. The remaining cases are sporadic zeros, with only finitely many for each $m\ge5$.

quant-ph↗

arXiv2026

Wavenumber-Aware Deep Learning for Robust Analysis of Raman Spectra

Machine-learning models for spectroscopy usually treat a spectrum as a fixed-grid intensity vector, leaving the physical coordinate implicit in the array index. Such models are tied to a single sampling grid, spectral range and resolution, which limits their use when measurement conditions change. Here we represent each spectral point as a pair of intensity and wavenumber and compute Transformer positional encodings from the wavenumber itself. On Raman spectra of three bacterial species and their synthetic mixtures, this representation remained robust to noise, sparser sampling and partial spectral coverage across convolutional, residual and Transformer networks. In controlled comparisons, wavenumber-based encoding outperformed index-based and learned encodings under partial observation and retained its performance on sampling grids never seen during training, where index-based encoding degraded. These findings were reproduced on a public 30-isolate bacterial benchmark. Explicit physical coordinates offer a simple route to robust spectral analysis under variable measurement conditions.

cs.LG↗

arXiv2026

Lean-GAP: A Dataset of Formalized Graduate Algebra Problems

We present Lean-GAP (Lean-Graduate Agebra Problems), 430 formalized graduate-level algebra problems from the textbook Abstract Algebra by Dummit and Foote. We develop a scalable pipeline consisting of PDF-to-LaTeX preprocessing, autoformalization into Lean 4, and verification of informal-formal correspondence. While the preprocessing and autoformalization stages can be largely automated, we find that verification remains the most subtle and labor-intensive component, requiring careful human oversight. Our contributions include (i) the construction of a structured dataset of formalized exercises, (ii) a systematic methodology for formalizing textbook mathematics, and (iii) an analysis of recurring challenges in the formalization process. We also compare the performance of different autoformalization models and highlight key bottlenecks in translating informal statements into formal language.

cs.LO↗

arXiv2019

Dirichlet forms and ultrametric Cantor sets associated to higher-rank graphs

The aim of this paper is to study the heat kernel and jump kernel of the Dirichlet form associated to ultrametric Cantor sets $\partial\BB_Λ$ that is the infinite path space of the stationary $k$-Bratteli diagram $\BB_Λ$, where $Λ$ is a finite strongly connected $k$-graph. The Dirichlet form which we are interested in is induced by an even spectral triple $(C_{\operatorname{Lip}}(\PB_Λ), π_ϕ, \mathcal{H}, D, Γ)$ and is given by \[ Q_s(f,g)=\frac{1}{2} \int_Ξ \operatorname{Tr}\big(\vert D\vert^{-s} [D,π_ϕ(f)]^{\ast} [D,π_ϕ(g)] \big) \, dν(ϕ), \] where $Ξ$ is the space of choice functions on $\partial \BB_Λ\times \partial \BB_Λ$. There are two ultrametrics, $d^{(s)}$ and $d_{w_δ}$, on $\partial \BB_Λ$ which make the infinite path space $\PB_Λ$ an ultrametric Cantor set. The former $d^{(s)}$ is associated to the eigenvalues of Laplace-Beltrami operator $Δ_s$ associated to $Q_s$, and the latter $d_{w_δ}$ is associated to a weight function $w_δ$ on $\BB_Λ$, where $δ\in (0,1)$. We show that the Perron-Frobenius measure $μ$ on $\partial \BB_Λ$ has the volume doubling property with respect to both $d^{(s)}$ and $d_{w_δ}$ and we study the asymptotic behaviors of the heat kernel associated to $Q_s$. Moreover, we show that the Dirichlet form $Q_s$ coincides with a Dirichlet form $\mathcal{Q}_{J_s, μ}$ which is associated to a jump kernel $J_s$ and the measure $μ$ on $\partial \BB_Λ$, and we investigate the asymptotic behavior and moments of displacements of the process.

math.PR↗