arXiv subjects

Elena Kim

Publications and source records attributed to Elena Kim.

12 recordsLinked to original sources

arXiv2026

Polynomial bounds for eigenfunctions and eigenvalues on random covers of hyperbolic surfaces

Let $X$ be a compact connected orientable hyperbolic surface and let $X_n$ be a degree $n$ cover, taken uniformly at random. We show that, with high probability, the $L^\infty$ norm of every Laplace eigenfunction on $X_n$ with bounded eigenvalue decays polynomially in $n$. This gives a polynomial decay analogue of the logarithmic bound of Gilmore--Le Masson--Sahlsten--Thomas [arXiv:1912.09961] in the Weil--Petersson model. Using similar methods, we also show that, with high probability, the distribution of eigenvalues of the Laplacian on $X_n$ converges to the spectral measure of the hyperbolic plane with polynomially decaying error. Our proof relies on the Selberg pre-trace formula and a variant of the polynomial method.

math.SP↗

arXiv2025

Rethinking Retrieval: From Traditional Retrieval Augmented Generation to Agentic and Non-Vector Reasoning Systems in the Financial Domain for Large Language Models

Recent advancements in Retrieval-Augmented Generation (RAG) have enabled Large Language Models to answer financial questions using external knowledge bases of U.S. SEC filings, earnings reports, and regulatory documents. However, existing work lacks systematic comparison of vector-based and non-vector RAG architectures for financial documents, and the empirical impact of advanced RAG techniques on retrieval accuracy, answer quality, latency, and cost remain unclear. We present the first systematic evaluation comparing vector-based agentic RAG using hybrid search and metadata filtering against hierarchical node-based systems that traverse document structure without embeddings. We evaluate two enhancement techniques applied to the vector-based architecture, i) cross-encoder reranking for retrieval precision, and ii) small-to-big chunk retrieval for context completeness. Across 1,200 SEC 10-K, 10-Q, and 8-K filings on a 150-question benchmark, we measure retrieval metrics (MRR, Recall@5), answer quality through LLM-as-a-judge pairwise comparisons, latency, and preprocessing costs. Vector-based agentic RAG achieves a 68% win rate over hierarchical node-based systems with comparable latency (5.2 compared to 5.98 seconds). Cross-encoder reranking achieves a 59% absolute improvement at optimal parameters (10, 5) for MRR@5. Small-to-big retrieval achieves a 65% win rate over baseline chunking with only 0.2 seconds additional latency. Our findings reveal that applying advanced RAG techniques to financial Q&A systems improves retrieval accuracy, answer quality, and has cost-performance tradeoffs to be considered in production.

cs.CL↗

arXiv2025

Semiclassical Measures on Hyperbolic Manifolds

We examine semiclassical measures for Laplace eigenfunctions on compact hyperbolic $(n+1)$-manifolds. We prove their support must contain the cosphere bundle of a compact immersed totally geodesic submanifold. Our proof adapts the argument of Dyatlov and Jin to higher dimensions and classifies the closures of horocyclic orbits using Ratner theory. An important step in the proof is a generalization of the higher-dimensional fractal uncertainty principle of Cohen to Fourier integral operators, which may be of independent interest.

math.AP↗

arXiv2024

Characterizing the support of semiclassical measures for higher-dimensional cat maps

Quantum cat maps are toy models in quantum chaos associated to hyperbolic symplectic matrices $A\in \operatorname{Sp}(2n,\mathbb{Z})$. The macroscopic limits of sequences of eigenfunctions of a quantum cat map are characterized by semiclassical measures on the torus $\mathbb{R}^{2n}/\mathbb{Z}^{2n}$. We show that if the characteristic polynomial of every power $A^k$ is irreducible over the rationals, then every semiclassical measure has full support. The proof uses an earlier strategy of Dyatlov-J\'ez\'equel [arXiv:2108.10463] and the higher-dimensional fractal uncertainty principle of Cohen [arXiv:2305.05022]. Our irreducibility condition is generically true, in fact we show that asymptotically for $100\%$ of matrices $A$, the Galois group of the characteristic polynomial of $A$ is $S_2 \wr S_n$. When the irreducibility condition does not hold, we show that a semiclassical measure cannot be supported on a finite union of parallel non-coisotropic subtori. On the other hand, we give examples of semiclassical measures supported on the union of two transversal symplectic subtori for $n=2$, inspired by the work of Faure-Nonnenmacher-De Bi\`evre [arXiv:nlin/0207060] in the case $n=1$. This is complementary to the examples by Kelmer [arXiv:math-ph/0510079] of semiclassical measures supported on a single coisotropic subtorus.

math.AP↗

arXiv2023

Bounds on Eigenfunctions of Quantum Cat Maps

We study $\ell^\infty$ norms of $\ell^2$-normalized eigenfunctions of quantum cat maps. For maps with short quantum periods (constructed by Bonechi and de Bi\`evre), we show that there exists a sequence of eigenfunctions $u$ with $\|u\|_{\infty}\gtrsim (\log N)^{-1/2}$. For general eigenfunctions we show the upper bound $\|u\|_\infty\lesssim (\log N)^{-1/2}$. Here the semiclassical parameter is $h=(2\pi N)^{-1}$. Our upper bound is analogous to the one proved by B\'{e}rard for compact Riemannian manifolds without conjugate points.

math.SP↗

arXiv2023

The strongly Leibniz property and the Gromov--Hausdorff propinquity

We construct a new version of the dual Gromov--Hausdorff propinquity that is sensitive to the strongly Leibniz property. In particular, this new distance is complete on the class of strongly Leibniz quantum compact metric spaces. Then, given an inductive limit of C*-algebras for which each C*-algebra of the inductive limit is equipped with a strongly Leibniz L-seminorm, we provide sufficient conditions for placing a strongly Leibniz L-seminorm on an inductive limit such that the inductive sequence converges to the inductive limit in this new Gromov--Hausdorff propinquity. As an application, we place new strongly Leibniz L-seminorms on AF-algebras using Frobenius--Rieffel norms, for which we have convergence of the Effros--Shen algebras in the Gromov--Hausdorff propinquity with respect to their irrational parameter.

math.OA↗

arXiv2022

Spectral Analysis of the Kohn Laplacian on Lens Spaces

We obtain an analog of Weyl's law for the Kohn Laplacian on lens spaces. We also show that two 3-dimensional lens spaces with fundamental groups of equal prime order are isospectral with respect to the Kohn Laplacian if and only if they are CR isometric.

math.DG↗

arXiv2021

Frobenius-Rieffel norms on finite-dimensional C*-algebras

In 2014, Rieffel introduced norms on certain unital C*-algebras built from conditional expectations onto unital C*-subalgebras. We begin by showing that these norms generalize the Frobenius norm, and we provide explicit formulas for certain conditional expectations onto unital C*-subalgebras of finite-dimensional C*-algebras. This allows us compare these norms to the operator norm by finding explicit equivalence constants. In particular, we find equivalence constants for the standard finite-dimensional C*-subalgebras of the Effros-Shen algebras that vary continuously with respect to their given irrational parameters.

math.OA↗

arXiv2021

A Tauberian Approach to an Analog of Weyl's law for the Kohn Laplacian on Compact Heisenberg Manifolds

Let $M= Γ\setminus \mathbb{H}_d$ be a compact quotient of the $d$-dimensional Heisenberg group $\mathbb{H}_d$ by a lattice subgroup $Γ$. We show that the eigenvalue counting function $N(λ)$ for any fixed element of a family of second order differential operators $\left\{\mathcal{L}_α\right\}$ on $M$ has asymptotic behavior $N\left(λ\right) \sim C_{d,α} \operatorname{vol}\left(M\right) λ^{d + 1}$, where $C_{d,α}$ is a constant that only depends on the dimension $d$ and the parameter $α$. As a consequence, we obtain an analog of Weyl's law (both on functions and forms) for the Kohn Laplacian on $M$. Our main tools are Folland's description of the spectrum of $\mathcal{L}_α$ and Karamata's Tauberian theorem.

math.CV↗

arXiv2021

Tinkering with Lattices: A New Take on the Erdős Distance Problem

The Erdős distance problem concerns the least number of distinct distances that can be determined by $N$ points in the plane. The integer lattice with $N$ points is known as \textit{near-optimal}, as it spans $Θ(N/\sqrt{\log(N)})$ distinct distances, the lower bound for a set of $N$ points (Erdős, 1946). The only previous non-asymptotic work related to the Erdős distance problem that has been done was for $N \leq 13$. We take a new non-asymptotic approach to this problem in a model case, studying the distance distribution, or in other words, the plot of frequencies of each distance of the $N\times N$ integer lattice. In order to fully characterize this distribution, we adapt previous number-theoretic results from Fermat and Erdős in order to relate the frequency of a given distance on the lattice to the sum-of-squares formula. We study the distance distributions of all the lattice's possible subsets; although this is a restricted case, the structure of the integer lattice allows for the existence of subsets which can be chosen so that their distance distributions have certain properties, such as emulating the distribution of randomly distributed sets of points for certain small subsets, or emulating that of the larger lattice itself. We define an error which compares the distance distribution of a subset with that of the full lattice. The structure of the integer lattice allows us to take subsets with certain geometric properties in order to maximize error; we show these geometric constructions explicitly. Further, we calculate explicit upper bounds for the error when the number of points in the subset is $4$, $5$, $9$ or $\left \lceil N^2/2\right\rceil$ and prove a lower bound in cases with a small number of points.

math.NT↗

arXiv2020

Constructions of Generalized MSTD Sets in Higher Dimensions

Let $A$ be a set of finite integers, define $$A+A \ = \ \{a_1+a_2: a_1,a_2 \in A\}, \ \ \ A-A \ = \ \{a_1-a_2: a_1,a_2 \in A\},$$ and for non-negative integers $s$ and $d$ define $$sA-dA\ =\ \underbrace{A+\cdots+A}_{s} -\underbrace{A-\cdots-A}_{d}.$$ A More Sums than Differences (MSTD) set is an $A$ where $|A+A| > |A-A|$. It was initially thought that the percentage of subsets of $[0,n]$ that are MSTD would go to zero as $n$ approaches infinity as addition is commutative and subtraction is not. However, in a surprising 2006 result, Martin and O'Bryant proved that a positive percentage of sets are MSTD, although this percentage is extremely small, about $10^{-4}$ percent. This result was extended by Iyer, Lazarev, Miller, ans Zhang [ILMZ] who showed that a positive percentage of sets are generalized MSTD sets, sets for $\{s_1,d_1\} \neq \{s_2, d_2\}$ and $s_1+d_1=s_2+d_2$ with $|s_1A-d_1A| > |s_2A-d_2A|$, and that in $d$-dimensions, a positive percentage of sets are MSTD. For many such results, establishing explicit MSTD sets in $1$-dimensions relies on the specific choice of the elements on the left and right fringes of the set to force certain differences to be missed while desired sums are attained. In higher dimensions, the geometry forces a more careful assessment of what elements have the same behavior as $1$-dimensional fringe elements. We study fringes in $d$-dimensions and use these to create new explicit constructions. We prove the existence of generalized MSTD sets in $d$-dimensions and the existence of $k$-generational sets, which are sets where $|cA+cA|>|cA-cA|$ for all $1\leq c \leq k$. We then prove that under certain conditions, there are no sets with $|kA+kA|>|kA-kA|$ for all $k \in \mathbb{N}.$

math.NT↗

arXiv2019

Sobolev and Schatten Estimates for the Complex Green Operator on Spheres

The complex Green operator $\mathcal{G}$ on CR manifolds is the inverse of the Kohn-Laplacian $\square_b$ on the orthogonal complement of its kernel. In this note, we prove Schatten and Sobolev estimates for $\mathcal{G}$ on the unit sphere $\mathbb{S}^{2n-1}\subset \mathbb{C}^n$. We obtain these estimates by using the spectrum of $\square_b$ and the asymptotics of the eigenvalues of the usual Laplace-Beltrami operator.

math.SP↗