arXiv subjects

Ni An

Publications and source records attributed to Ni An.

5 recordsLinked to original sources

arXiv2026

Length minimization of filling pairs on hyperbolic surfaces

A filling pair $(\alpha, \beta)$ of a surface $S_g$ is a pair of simple closed curves in minimal position such that the complement of $\alpha\cup\beta$ in $S_g$ is a disjoint union of topological disks. A filling pair is said to be minimally intersecting if the number of intersections between them, or equivalently, the number of complementary disks, is minimal among all filling pairs of $S_g$. For surfaces of genus $g \geq 3$, minimal filling pairs are well understood, whereas in genus two, such a pair divides the surface into exactly two disks. In this paper, we classify all minimal filling pairs up to the action of the mapping class group in genus two and determine the length of the shortest minimal filling pair.

math.GT↗

arXiv2025

Small genus, small index critical points of the systole function

In this paper the index of a family of critical points of the systole function on Teichm\"uller space is calculated. The members of this family are interesting in that their existence implies the existence of strata in the Thurston spine for which the systoles do not determine a basis for the homology of the surface. Previously, index calculations of critical points with this pathological feature were impossible, because the only known examples were in surfaces with huge genus. A related concept is that of a ``minimal filling subset'' of the systoles at the critical point. Such minimal filling sets are studied, as they relate to the dimension of the Thurston spine near the critical point. We find an example of a minimal filling set of simple closed geodesics in genus 5 with cardinality 8, that are presumably realised as systoles. More generally, we determine the smallest and largest cardinality of a minimal filling set related to a tesselation of a hyperbolic surface by regular, right-angled $m$-gons for $m \in \{ 5, 6, 7 \}$. For this, we use integer linear programming together with a hand-tailored symmetry breaking technique.

math.GT↗

arXiv2024

Algebraic aspects of holomorphic quantum modular forms

Matrix-valued holomorphic quantum modular forms are intricate objects that arise in successive refinements of the Volume Conjecture of knots and involve three holomorphic, asymptotic and arithmetic objects. It is expected that the algebraic properties of these objects can be deduced from the algebraic properties of descendant state integrals, and we illustrate this for the case of the $(-2,3,7)$-pretzel knot.

math.GT↗

arXiv2018

Efficiency and detectability of random reactive jamming in carrier sense wireless networks

A natural basis for the detection of a wireless random reactive jammer (RRJ) is the perceived violation by the detector (typically located at the access point (AP)) of the carrier sensing protocol underpinning many wireless random access protocols (e.g., WiFi). Specifically, when the wireless medium is perceived by a station to be busy, a carrier sensing compliant station will avoid transmission while a RRJ station will often initiate transmission. However, hidden terminals (HTs), i.e., activity detected by the AP but not by the sensing station, complicate the use of carrier sensing as the basis for RRJ detection since they provide plausible deniability to a station suspected of being an RRJ. The RRJ has the dual objectives of avoiding detection and effectively disrupting communication, but there is an inherent performance tradeoff between these two objectives. In this paper we capture the behavior of both the RRJ and the compliant stations via a parsimonious Markov chain model, and pose the detection problem using the framework of Markov chain hypothesis testing. Our analysis yields the receiver operating characteristic of the detector, and the optimized behavior of the RRJ. While there has been extensive work in the literature on jamming detection, our innovation lies in leveraging carrier sensing as a natural and effective basis for detection.

cs.IT↗

arXiv2015

On the performance overhead tradeoff of distributed principal component analysis via data partitioning

Principal component analysis (PCA) is not only a fundamental dimension reduction method, but is also a widely used network anomaly detection technique. Traditionally, PCA is performed in a centralized manner, which has poor scalability for large distributed systems, on account of the large network bandwidth cost required to gather the distributed state at a fusion center. Consequently, several recent works have proposed various distributed PCA algorithms aiming to reduce the communication overhead incurred by PCA without losing its inferential power. This paper evaluates the tradeoff between communication cost and solution quality of two distributed PCA algorithms on a real domain name system (DNS) query dataset from a large network. We also apply the distributed PCA algorithm in the area of network anomaly detection and demonstrate that the detection accuracy of both distributed PCA-based methods has little degradation in quality, yet achieves significant savings in communication bandwidth.

cs.DC↗