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Yanhui Su

Publications and source records attributed to Yanhui Su.

10 recordsLinked to original sources
arXiv2026

Interlacing for zeros of the Serre derivative of Eisenstein series

In 1970, Rankin and Swinnerton-Dyer showed that the non-elliptic zeros of Eisenstein series $E_k$ in the fundamental domain all lie on the lower arc $\{ e^{i\theta}: \frac{\pi}{2} < \theta < \frac{2\pi}{3}\}$. Very recently, Sugibayashi showed that the same property also holds for the Serre derivative $\vartheta_k(E_k)$ of Eisenstein series. In this paper, we first give very precise estimates for where exactly these zeros are located on the lower arc. These location estimates then allow us to prove four main results. First, we show that the zeros of $\vartheta_\ell(E_\ell)$ Stieltjes interlace with the zeros of $\vartheta_k(E_k)$ on the lower arc for all $\ell > k$. Second, we classify precisely when the zeros of $\vartheta_\ell(E_\ell)$ (standard) interlace with the zeros of $\vartheta_k(E_k)$ on the lower arc. Third, we show that the zeros of $\vartheta_k(E_k)$ always (standard) interlace with the zeros of $E_{k+2}$ on the lower arc. Fourth, as an application of the third main result, we show that the zeros of the cuspidal projection of $\vartheta_k(E_k)$ all lie on the lower arc, extending a result of Xue and Zhu.

math.NT↗
arXiv2026

Explicit generators of the space of modular forms

Let $S_{\kappa}$ be the space of cusp forms of weight $\kappa$ and level one, and let $S_{\kappa}^{\ast}$ denote its dual space. In this paper, we find explicit spanning subsets of $S_{\kappa}$ consisting of Rankin-Cohen brackets of Eisenstein series and explicit subsets of periods that span $S_{\kappa}^{\ast}$.

math.NT↗
arXiv2020

Graphs on surfaces with positive Forman curvature or corner curvature

On one hand, we study the class of graphs on surfaces, satisfying tessellation properties, with positive Forman curvature on each edge. Via medial graphs, we provide a new proof for the finiteness of the class, and give a complete classification. On the other hand, we classify the class of graphs on surfaces with positive corner curvature.

math.CO↗
arXiv2019

A curvature notion for planar graphs stable under planar duality

Woess \cite{Woess98} introduced a curvature notion on the set of edges of a planar graph, called $Ψ$-curvature in our paper, which is stable under the planar duality. We study geometric and combinatorial properties for the class of infinite planar graphs with non-negative $Ψ$-curvature. By using the discharging method, we prove that for such an infinite graph the number of vertices (resp. faces) of degree $k,$ except $k=3,4$ or $6,$ is finite. As a main result, we prove that for an infinite planar graph with non-negative $Ψ$-curvature the sum of the number of vertices of degree at least $8$ and the number of faces of degree at least $8$ is at most one.

math.CO↗
arXiv2018

The set of vertices with positive curvature in a planar graph with nonnegative curvature

In this paper, we give the sharp upper bound for the number of vertices with positive curvature in a planar graph with nonnegative combinatorial curvature. Based on this, we show that the automorphism group of a planar---possibly infinite---graph with nonnegative combinatorial curvature and positive total curvature is a finite group, and give an upper bound estimate for the order of the group.

math.DG↗
arXiv2018

Collapse of Deep and Narrow Neural Nets

Recent theoretical work has demonstrated that deep neural networks have superior performance over shallow networks, but their training is more difficult, e.g., they suffer from the vanishing gradient problem. This problem can be typically resolved by the rectified linear unit (ReLU) activation. However, here we show that even for such activation, deep and narrow neural networks (NNs) will converge to erroneous mean or median states of the target function depending on the loss with high probability. Deep and narrow NNs are encountered in solving partial differential equations with high-order derivatives. We demonstrate this collapse of such NNs both numerically and theoretically, and provide estimates of the probability of collapse. We also construct a diagram of a safe region for designing NNs that avoid the collapse to erroneous states. Finally, we examine different ways of initialization and normalization that may avoid the collapse problem. Asymmetric initializations may reduce the probability of collapse but do not totally eliminate it.

stat.ML↗
arXiv2018

Areas of spherical polyhedral surfaces with regular faces

For a finite planar graph, it associates with some metric spaces, called (regular) spherical polyhedral surfaces, by replacing faces with regular spherical polygons in the unit sphere and gluing them edge-to-edge. We consider the class of planar graphs which admit spherical polyhedral surfaces with the curvature bounded below by 1 in the sense of Alexandrov, i.e. the total angle at each vertex is at most $2π$. We classify all spherical tilings with regular spherical polygons, i.e. total angles at vertices are exactly $2π$. We prove that for any graph in this class which does not admit a spherical tiling, the area of the associated spherical polyhedral surface with the curvature bounded below by 1 is at most $4π- ε_0$ for some $ε_0 > 0$. That is, we obtain a definite gap between the area of such a surface and that of the unit sphere.

math.MG↗