arXiv subjects

Jingyan Li

Publications and source records attributed to Jingyan Li.

15 recordsLinked to original sources

arXiv2026

Quantum Steering Geometry at High Energy Particle Colliders

We formulate collider observables based on quantum steering ellipsoids (QSEs) for reconstructed bipartite systems of spin-$1/2$ particles. A collider spin density matrix defines a two-qubit state, while its QSE gives the geometry of conditional states accessible through local measurements. This makes the ellipsoid a direct probe of the quantum properties of fundamental particles, encoding polarization, spin correlation anisotropy, accessible-state volume, and the orientation of the dominant correlation axes. Using top-quark pair production as a benchmark process, we show how QSE observables organize the Standard Model spin state, probe entanglement, steerability, and Bell-nonlocality criteria, and use an expected precision metric to assess sensitivity to non-local correlations in the boosted central region. We show that different dimension-six operators generate distinctive QSE deformations, and provide a geometric interpretation of quantum information observables in high-energy particle physics. Quantum steering geometry therefore provides a unified framework for particle collider tomography, quantum information diagnostics, and precision searches for physics beyond the Standard Model with applications spanning the HL--LHC and future lepton, muon, flavor, and electron-ion collider programs.

hep-ph↗

arXiv2026

A Sharp Curvature Threshold for GLMY Path Homology

Let $G$ be a finite simple graph with at least one edge. We prove the sharp vanishing theorem \[ \kappa_{\min}^{\mathrm{LLY}}(G)>\frac12 \quad\Longrightarrow\quad \PathH_1(G;\R)=0. \] Equivalently, nonzero first GLMY path homology forces an edge of Lin--Lu--Yau curvature at most $1/2$. The threshold $1/2$ is sharp and is attained by $C_5$. The proof combines the cycle-space description of first GLMY path homology with the limit-free Laplacian characterization of Lin--Lu--Yau curvature. As a secondary consequence of the curvature-preserving universal-cover method, we prove that if $G$ is connected and $\kappa_{\min}^{\mathrm{LLY}}(G)>0$, then $\pi_1(\Xshort{5}(G),o)$ is finite, where $\Xshort{5}(G)$ is obtained by filling every simple cycle of length at most five. Equivalently, the normal subgroup generated by based simple $5$-cycle loops has finite index in $\pi_1^{\mathrm{GLMY}}(G,o)$. In higher degrees the situation is different: for each integer $r\geq1$, the Cartesian product $T_r=C_5^{\square r}$ has curvature $1/(2r)$ on every edge and, for every field $\F$, \[ \PathH_p(T_r;\F)\cong\F^{\binom rp}\qquad(0\leq p\leq r), \] so strict positivity of Lin--Lu--Yau curvature does not force higher-dimensional GLMY path homology to vanish.

math.AT↗

arXiv2026

Predicting Inference-Time Scaling Gains from Labeled Validation-Set Output Statistics

Best-of-$N$ inference scaling (drawing $N$ candidate answers from a language model and returning the one a reward model ranks highest) improves accuracy by an amount that varies across models, but predicting that amount in advance currently requires running the procedure end-to-end. Prior work links cheap statistics of a model's sampled outputs and validation-set correctness (how often samples agree, how diverse they are, how confident the model is, and where correct samples appear) to model behavior, but does not isolate which of these form a stable, compact predictor of best-of-$N$ gain. We fit ridge predictors on features computed from a single labeled validation-set sampling pass, use bootstrap-Lasso as a stability analysis of the candidate feature set, and give a concentration analysis with an explicit linear-approximation residual. Across three base-model families, six post-training methods, and math and reasoning task domains, the stability analysis identifies a strict three-feature core spanning prompt-level agreement spread, label-assisted first-correct-sample position, and completion-length variance; a compact ridge predictor built from this core plus an entropy add-on reaches Spearman $\rho = 0.90$ with actual best-of-$N$ gain under a reward-model verifier. The intended use is labeled validation-set screening of candidate configurations before paying the full reward-model scoring cost.

cs.CL↗

arXiv2026

Persistent magnitude homology on finite metric space

Magnitude homology is an emerging approach that captures the intrinsic topological and geometric features of metric spaces. It offers a distinct theoretical lens for interpreting structural information within topological and geometric data analysis. This work introduces persistent magnitude homology, an extension of magnitude homology that captures multi-scale geometric and topological features of metric spaces. We construct the category of finite metric spaces with isometric embeddings and show that magnitude homology defines a functor to the category of abelian groups, naturally leading to the definition of persistent magnitude homology. We also introduce weighted persistent modules and weighted barcodes to offer both an algebraic and visual description of persistent magnitude homology. Additionally, we present an isometry theorem that relates interleaving distances and bottleneck distances, and establish stability results for persistent magnitude homology and magnitude profile. These results establish the stability of magnitude-based descriptors, bridging the gap between theory and practical application.

math.AT↗

arXiv2025

A "Periodicity" Phenomenon of the Attaching Map of the Suspended Two-Cell Complex

In this paper, we determine the 3-cell skeleton of $F$, where $F$ is the homotopy fiber of the canonical pinch map from a suspension of a simply-connected 2-cell complex onto a sphere. The main result is stated $p$-locally: for $p=2$, and for $p\geq5$ under an additional assumption. The proof is based on Selick-Wu's $\mathrm{A}^{\mathrm{min}}$-theory and the machinery of the Eilenberg-Moore spectral sequence. As an application, we compute the 2-primary component of $\pi_{18} (\Sigma^{3}\mathbb{C}P^{2})$, a homotopy group outside the metastable range.

math.AT↗

arXiv2024

IntComplex for high-order interactions

Graphs serve as powerful tools for modeling pairwise interactions in diverse fields such as biology, material science, and social networks. However, they inherently overlook interactions involving more than two entities. Simplicial complexes and hypergraphs have emerged as prominent frameworks for modeling many-body interactions; nevertheless, they exhibit limitations in capturing specific high-order interactions, particularly those involving transitions from $n$-interactions to $m$-interactions. Addressing this gap, we propose IntComplex as an innovative framework to characterize such high-order interactions comprehensively. Our framework leverages homology theory to provide a quantitative representation of the topological structure inherent in such interactions. IntComplex is defined as a collection of interactions, each of which can be equivalently represented by a binary tree. Drawing inspiration from GLMY homology, we introduce homology for the detailed analysis of structural patterns formed by interactions across adjacent dimensions, $p$-layer homology to elucidate loop structures within $p$-interactions in specific dimensions, and multilayer homology to analyze loop structures of interactions across multiple dimensions. Furthermore, we introduce persistent homology through a filtration process and establish its stability to ensure robust quantitative analysis of these complex interactions. The proposed IntComplex framework establishes a foundational paradigm for the analysis of topological properties in high-order interactions, presenting significant potential to drive forward the advancements in the domain of complex network analysis.

math.AT↗

arXiv2024

Primitive path homology

In this paper we introduce a primitive path homology theory on the category of simple digraphs. On the subcategory of asymmetric digraphs, this theory coincides with the path homology theory which was introduced by Grigor'yan, Lin, Muranov, and Yau, but these theories are different in general case. We study properties of the primitive path homology and describe relations between the primitive path homology and the path homology. Let $a,b$ two different vertices of a digraph. Our approach gives a possibility to construct primitive homology theories of paths which have a given tail vertex $a$ or (and) a given head vertex $b$. We study these theories and describe also relationships between them and the path homology theory.

math.AT↗

arXiv2024

New Aspects of Analyzing Amyloid Fibrils

This is a summary of mathematical tools we used in research of analyzing the structure of proteins with amyloid form \cite{xi2024Top}. We defined several geometry indicators on the discrete curve namely the hop distance, the discrete curvature and the discrete torsion. Then, we used these indicators to analyze the structure of amyloid fibrils by regarding its peptide chains as discrete curves in $\Rds^3$. We gave examples to show that these indicators give novel insights in the characterization analysis of the structure of amyloid fibrils, for example the discrete torsion can detect the hydrogen bonds interactions between layers of amyloid fibril. {Moreover,} the topological tool performs better than the root mean square deviation (RMSD) in quantifying the difference of the structure of amyloid fibrils, etc.

math.AT↗

arXiv2024

Homotopy Groups and Puppe Sequence of Digraphs

We introduce homotopy groups of digraphs that admit an intuitive description of grid structures, which is a variation of the GLMY homotopy groups introduced by Grigor'yan, Lin, Muranov and Yau in 2014. This direct approach enables a descriptive interpretation of GLMY theory in applications such as network science. Furthermore, we prove that there exists a long exact sequence of homotopy groups of digraphs associated to any based digraph map, that is, there exists a digraph version of the Puppe sequence.

math.AT↗

arXiv2023

Search for charged-lepton flavor violation in the production and decay of top quarks using trilepton final states

This document describes a search for charged-lepton flavor violation (CLFV) in the production and decay of top quarks using 138 fb$^{-1}$ of data collected by the CMS experiment at a center-of-mass energy of 13 TeV. Events are selected for analysis if they contain an opposite-sign electron-muon pair, a third charged lepton (electron or muon), at least one jet, and at most one jet associated with a bottom quark. The analysis utilizes boosted decision trees to separate background processes from a possible signal. The data were found to be consistent with the standard model expectation. Exclusion limits were placed on different CLFV interactions, constituting the most stringent limits to date on these processes.

hep-ex↗

arXiv2023

The magnitude homology of a hypergraph

The magnitude homology, introduced by R. Hepworth and S. Willerton, offers a topological invariant that enables the study of graph properties. Hypergraphs, being a generalization of graphs, serve as popular mathematical models for data with higher-order structures. In this paper, we focus on describing the topological characteristics of hypergraphs by considering their magnitude homology. We begin by examining the distances between hyperedges in a hypergraph and establish the magnitude homology of hypergraphs. Additionally, we explore the relationship between the magnitude and the magnitude homology of hypergraphs. Furthermore, we derive several functorial properties of the magnitude homology for hypergraphs. Lastly, we present the K\"{u}nneth theorem for the simple magnitude homology of hypergraphs.

math.AT↗

arXiv2023

The algebraic stability for persistent Laplacians

The stability of topological persistence is one of the fundamental issues in topological data analysis. Numerous methods have been proposed to address the stability of persistent modules or persistence diagrams. Recently, the concept of persistent Laplacians has emerged as a novel approach to topological persistence, attracting significant attention and finding applications in various fields. In this paper, we investigate the stability of persistent Laplacians. We introduce the notion of ``Laplacian trees'', which captures the collection of persistent Laplacians that persist from a given parameter. To formalize our study, we construct the category of Laplacian trees and establish an algebraic stability theorem for persistent Laplacian trees. Notably, our stability theorem is applied to the real-valued functions on simplicial complexes and digraphs.

math.AT↗

arXiv2022

On the Cayley-persistence algebra

In this paper, we introduce a persistent (co)homology theory for Cayley digraph grading. We give the algebraic structures of Cayley-persistence object. Specifically, we consider the module structure of persistent (co)homology and show the decomposition of a finitely generated Cayley-persistence module. Moreover, we introduce the persistence-cup product on the Cayley-persistence module and study the twisted structure with respect to the persistence-cup product. As an application on manifolds, we show that the persistent (co)homology is closely related to the persistent map of fundamental classes.

math.AT↗

arXiv2018

The Embedded Homology of Hypergraphs and Applications

Hypergraphs are mathematical models for many problems in data sciences. In recent decades, the topological properties of hypergraphs have been studied and various kinds of (co)homologies have been constructed (cf. [3, 4, 12]). In this paper, generalising the usual homology of simplicial complexes, we define the embedded homology of hypergraphs as well as the persistent embedded homology of sequences of hypergraphs. As a generalisation of the Mayer-Vietoris sequence for the homology of simplicial complexes, we give a Mayer-Vietoris sequence for the embedded homology of hypergraphs. Moreover, as applications of the embedded homology, we study acyclic hypergraphs and construct some indices for the data analysis of hyper-networks.

math.AT↗

arXiv2017

Quantum correlations across two octaves from combined up and down conversion

We propose and analyse a cascaded optical parametric system which involves three interacting modes across two octaves of frequency difference. Our system, combining degenerate optical parametric oscillation (OPO) with second harmonic generation (SHG), promises to be a useful source of squeezed and entangled light at three differing frequencies. We show how changes in damping rates and the ratio of the two concurrent nonlinearities affect the quantum correlations in the output fields. We analyse the threshold behaviour, showing how the normal OPO threshold is changed by the addition of the SHG interactions. We also find that the inclusion of the OPO interaction removes the self-pulsing behaviour found in normal SHG. Finally, we show how the Einstein-Podolsky-Rosen correlations can be controlled by the injection of a coherent seed field at the lower frequency.

quant-ph↗