arXiv subjects

Yun Gao

Publications and source records attributed to Yun Gao.

At least 19 recordsLinked to original sources

arXiv2026

A High-Rank Gap Theorem for Type-I Bounded Symmetric Domains

Let $Ω_{r,s}$ and $Ω_{r',s'}$ be Type-I bounded symmetric domains with $s>r$ and $s'>r'$. Let $F:U\to M_{r',s'}(\mathbb C)$ be holomorphic, where $\overlineΩ_{r,s}\subset U$, and suppose that $F(S_{r,s})\subset S_{r',s'}$. If \[ k(s-r)\leq s'-r'<(k+1)(s-r) \quad\text{and}\quad r'>kr, \] then, up to target coordinates, the map contains a fixed identity block of size $r'-kr$. The proof first turns the boundary equation into a matrix identity for the first derivative. A direct dimension count gives $g(s-r)\leq s'-r'$, where $g$ is the rank of a positive semidefinite matrix arising from this identity. The extremal property of the Shilov boundary and the scalar Hopf boundary lemma then show that certain rows of the map are constant. This gives the fixed identity block. We also give examples showing that its size is optimal and explain why the proof needs $F$ on a neighborhood of $\overlineΩ_{r,s}$.

math.CV↗

arXiv2026

Gap phenomenon of holomorphic maps between Shilov boundaries of type-I bounded symmetric domains

Motivated by the gap phenomenon for proper holomorphic maps between complex unit balls, this paper investigates smooth CR maps $f$ from an open piece $M$ of the Shilov boundary of the unit ball $B^s$ into the Shilov boundary $S_{r',s'}$ of a higher-rank Type I bounded symmetric domain $Ω_{r',s'}$. To the best of our knowledge, this is the first work to systematically establish a general gap phenomenon in the higher-rank setting. Our main result demonstrates that when the signature difference $s'-r'$ falls into the interval $k(s-1) \le s'-r' < (k+1)(s-1)$ for some integer $k$, the non-trivial component of $f$ is constrained to a much smaller Type I boundary. Up to automorphisms of the domain and target spaces, $f$ decomposes into the block-diagonal form $$f(z) = \begin{pmatrix} I_{r'-k} & 0 \\ 0 & ϕ(z) \end{pmatrix},$$ where the essential component $ϕ: M \to S_{k, s'-r'+k}$ is a smooth CR map into the corresponding Shilov boundary. We provide explicit constructions showing that these dimensional bounds are sharp. As an immediate corollary, in the initial gap regime $s-1 \le s' - r' < 2s-2$, the map $f$ reduces to the standard linear embedding.

math.CV↗

arXiv2025

Some results related to Macaulay's Theorem about Hilbert functions and applications

Let $I$ be a homogeneous ideal in the polynomial ring $R = k[z_1, \cdots, z_n]$ , where $k$ is an algebraically closed field of characteristic zero. Macaulay's Theorem provides constraints on the Hilbert function of $I$ or $R/I$ from one degree to the next. Nowadays, the standard quotation of Macaulay's theorem is $H_{R/I}(d + 1) \le H_{R/I}(d)^{\langle d\rangle}$, which is regarding the quotient $R/I$ and the combinatorial computation in the formula involves the number $d$ explicitly. However, the origin statement of Macaulay is in fact regarding the Hilbert function of $I$ itself and the relevant combinatorics explicitly involves the number of variables (i.e. $n$) and does not depend on $d$. In this paper, we provide an elementary proof of the equivalence between these two versions of Macaulay's theorem. The original degree-independent version is more suitable for problems such as those involving sums of polynomial squared norms. Motivated by the Hermitian analogue of Hilbert's 17th problem and proper holomorphic mappings between complex unit balls, some questions lead to the study of Hermitian polynomials $M(z, \bar{z}) \in \mathbb{C}[z_1, \ldots, z_n, \bar{z}_1, \ldots, \bar{z}_n]$ satisfying $M(z, \bar{z})\|z\|^{2l} = \|h\|^2$ for some $l$ and a holomorphic mapping $h = (h_1, \cdots, h_R)$ . Using Macaulay's Theorem, we derive new inequalities relating $n$, $l$, the signature $(p, q)$ of the coefficient matrix of $M(z, \bar{z})$ , and $R$ (the rank of $M(z, \bar{z})\|z\|^{2l}$ ) and extend these results to norms of arbitrary signatures, which hold uniformly for all bidegrees of $M(z, \bar{z})$.

math.CV↗

arXiv2025

Rigidity of holomorphic maps between Shilov boundaries of type-I bounded symmetric domains

We introduce the concept of orthogonal structure on complex Grassmannians. Based on this structure, we define the notion of orthogonal mappings. This class of maps generalizes holomorphic maps between the Shilov boundaries of type-I bounded symmetric domains. By analyzing the geometric properties of these orthogonal mappings, we obtain rigidity results for such mappings. As an application, we establish a rigidity result for the holomorphic maps from the Shilov boundary of rank $1$ type I bounded symmetric domain $Ω_{1,s}$ (i.e. unit spheres) to the Shilov boundary of a higher-rank type-I domain $ Ω_{r',s'}$, where $s\ge 2$ and $2\le r'\le s'$. More specifically, we show that: (1) such maps are constant when $s'-r' < s-1$; (2) for $s-1 \le s'-r' < 2s-2$, they reduce to the standard linear embeddings after normalization by applying automorphisms on both sides. Our results are optimal and generalize the well-known optimal bounds for proper mappings between rank $1$ cases and CR maps between Shilov boundaries of higher-rank type I bounded symmetric domains.

math.CV↗

arXiv2025

Fixed-Relative-Switched Threshold Strategies for Consensus Tracking Control of Nonlinear Multiagent Systems

This paper investigates event-triggered consensus tracking in nonlinear semi-strict-feedback multi-agent systems involving one leader and multiple followers. We first employ radial basis function neural networks and backstepping techniques to approximate the unknown nonlinear dynamics, facilitating the design of dual observers to measure the unknown states and disturbances. Then three adaptive event-triggered control schemes are proposed: fixed-threshold, relative-threshold, and switched-threshold configurations, each featuring specialized controller architectures and triggering mechanisms. Through Lyapunov stability analysis, we establish that the follower agents can asymptotically track the reference trajectory of the leader, meanwhile all error signals remain uniform bounded. Our proposed control strategies effectively prevent Zeno behaviors through stringent exclusion criteria. Finally, an illustrative example is presented, demonstrating the competitive performance of our control framework in achieving consensus tracking and optimizing triggering efficiency.

eess.SY↗

arXiv2024

Interface and spectrum multiplexing ptychographic reflection microscopy

Reflective ptychography is a promising lensless imaging technique with a wide field of view, offering significant potential for applications in semiconductor manufacturing and detection. However, many semiconductor materials are coated with different layers during processing, which leads to the reflected diffraction light being a coherent superposition of multiple light beams. Traditional phase recovery methods often overlook the multi-layered nature of these materials, resulting in artificial errors and, in some cases, failures in image reconstruction. This limitation has hindered the broader application and adoption of reflection ptychography. Here, we propose and experimentally demonstrate an innovative interface and spectrum multiplexing ptychographic reflection microscopy. By employing multi-wavelength light as the illumination source, our approach enables the accurate extraction of the wavelength-dependent surface structure imaging and topography mapping of materials in a single experiment. This advancement offers a reliable technique for element-specific detection of semiconductor materials, utilizing tabletop extreme ultraviolet light sources in the future.

physics.optics↗

arXiv2024

FGP: Feature-Gradient-Prune for Efficient Convolutional Layer Pruning

To reduce computational overhead while maintaining model performance, model pruning techniques have been proposed. Among these, structured pruning, which removes entire convolutional channels or layers, significantly enhances computational efficiency and is compatible with hardware acceleration. However, existing pruning methods that rely solely on image features or gradients often result in the retention of redundant channels, negatively impacting inference efficiency. To address this issue, this paper introduces a novel pruning method called Feature-Gradient Pruning (FGP). This approach integrates both feature-based and gradient-based information to more effectively evaluate the importance of channels across various target classes, enabling a more accurate identification of channels that are critical to model performance. Experimental results demonstrate that the proposed method improves both model compactness and practicality while maintaining stable performance. Experiments conducted across multiple tasks and datasets show that FGP significantly reduces computational costs and minimizes accuracy loss compared to existing methods, highlighting its effectiveness in optimizing pruning outcomes. The source code is available at: https://github.com/FGP-code/FGP.

cs.CV↗

arXiv2024

Spectral multiplexing based on multi-distance lensless imaging

We have demonstrated the capability of spectral multiplexing in multi-distance diffractive imaging, enabling the reconstruction of samples with diverse spectral responses. While previous methods like ptychography utilize redundancy in radial diffraction data to achieve information multiplexing, they typically require capturing a substantial amount of diffraction data. In contrast, our approach effectively harnesses the redundancy information in axial diffraction data. This significantly reduces the amount of diffraction data required and relaxes the stringent requirements on optical path stability.

physics.optics↗

arXiv2024

Hermitian Rank and Rigidity of Holomorphic Mappings

Huang's Lemma is an important tool in CR geometry to study rigidity problems. This paper introduces a generalization of Huang's Lemma based on the rigidity properties of holomorphic mappings preserving certain orthogonality on projective spaces, which is optimal for the case of partial linearity. By exploring the intricate relationship between rigidity and Huang's Lemma, we establish that the rigidity properties of Segre maps or proper holomorphic mappings between generalized balls with Levi-degenerate boundaries can be inferred from those between generalized balls with Levi-non-degenerate boundaries by in a coordinate-free and more geometric manner.

math.CV↗

arXiv2022

On some classification of finite-dimensional Hopf algebras over the Hopf algebra $H_{b:1}^*$ of Kashina

Let $H$ be the dual of $16$-dimensional nontrivial semisimple Hopf algebra $H_{b:1}$ in the classification work of Kashina \cite{K00}. We completely determine all finite-dimensional Nichols algebras satisfying $\mathcal{B}(N)\cong \bigotimes_{i\in I}\mathcal{B}(N_i)$, where $N=\bigoplus_{i\in I}N_i$, each $N_i$ is a simple object in $_H^H\mathcal{YD}$. Under this assumption, we classify all those Hopf algebras of finite-dimensional growth from the semisimple Hopf algebra $H$ via the relevant Nichols algebras $\mathcal B(N)$.

math.QA↗

arXiv2022

Quantum N-toroidal algebras and extended quantized GIM algebras of N-fold affinization

We introduce the notion of quantum $N$-toroidal algebras as natural generalization of the quantum toroidal algebras as well as extended quantized GIM algebras of $N$-fold affinization. We show that the quantum $N$-toroidal algebras are quotients of the extended quantized GIM algebras of $N$-fold affinization, which generalizes a well-known result of Berman and Moody for Lie algebras.

math.QA↗

arXiv2021

On the rank of Hermitian polynomials and the SOS Conjecture

Let $z\in\mathbb C^n$ and $\|z\|$ be its Euclidean norm. Ebenfelt proposed a conjecture regarding the possible ranks of the Hermitian polynomials in $z,\bar z$ of the form $A(z,\bar z)\|z\|^2$, known as the SOS Conjecture, where SOS stands for "sums of squares". In this article, we employed and extended the recent techniques developed for local orthogonal maps to study this conjecture and its generalizations to arbitrary signatures. We proved that for a given Hermitian form $\langle\cdot,\cdot\rangle_\star$ on $\mathbb C^n$ of any signature with at least two non-zero eigenvalues, the ranks of the Hermitian polynomials of the form $A(z,\bar z)\|z\|^2_\star$ do not lie in a finite number of non-trivial gaps on the real line. Not only is this consistent with the SOS Conjecture, but it also demonstrates that in fact the conjecture qualitatively holds in all signatures except the trivial ones. In addition, these gaps can be explicitly calculated in terms of $n$ and the signature of $\langle\cdot,\cdot\rangle_\star$, and the number of such gaps and their sizes are of the same order as those stated in the SOS Conjecture.

math.CV↗

arXiv2021

A hyperplane restriction theorem for holomorphic mappings and its application for the gap conjecture

We established a hyperplane restriction theorem for the local holomorphic mappings between projective spaces, which is inspired by the corresponding theorem of Green for homogeneous ideals in polynomial rings. Our theorem allows us to give the first proof for the existence of gaps (albeit smaller) at all level for the rational proper maps between complex unit balls, conjectured by Huang-Ji-Yin. In addition, our proof does not distinguish the unit balls from other generalized balls and thus it simultaneously demonstrates the same phenomenon for all generalized balls.

math.CV↗

arXiv2021

Orthogonal pair and a rigidity problem for Segre maps between hyperquadrics

Being motivated by the orthogonal maps studied in \cite{GN1}, orthogonal pairs between the projective spaces equipped with possibly degenerate Hermitian forms were introduced. In addition, orthogonal pairs are generalizations of holomorphic Segre maps between Segre families of real hyperquadrics. We showed that non-degenerate holomorphic orthogonal pairs also have certain rigidity properties under a necessary codimension restriction. As an application, we got a rigidity theorem for Segre maps related to Heisenberg hypersurfaces obtained by Zhang in \cite{Zh}.

math.CV↗

arXiv2021

Local orthogonal maps and rigidity of holomorphic mappings between real hyperquadrics

We introduced a new coordinate-free approach to study the Cauchy-Riemann (CR) maps between the real hyperquadrics in the complex projective space. The central theme is based on a notion of orthogonality on the projective space induced by the Hermitian structure defining the hyperquadrics. There are various kinds of special linear subspaces associated to this orthogonality which are well respected by the relevant CR maps and this is where the rigidities come from. Our method allows us to generalize a number of well-known rigidity theorems for the CR mappings between real hyperquadrics with much simpler arguments.

math.CV↗

arXiv2020

Vector Bundles on Rational Homogeneous Spaces

We consider a uniform $r$-bundle $E$ on a complex rational homogeneous space $X$ %over complex number field $\mathbb{C}$ and show that if $E$ is poly-uniform with respect to all the special families of lines and the rank $r$ is less than or equal to some number that depends only on $X$, then $E$ is either a direct sum of line bundles or $δ_i$-unstable for some $δ_i$. So we partially answer a problem posted by Muñoz-Occhetta-Solá Conde. In particular, if $X$ is a generalized Grassmannian $\mathcal{G}$ and the rank $r$ is less than or equal to some number that depends only on $X$, then $E$ splits as a direct sum of line bundles. We improve the main theorem of Muñoz-Occhetta-Solá Conde when $X$ is a generalized Grassmannian by considering the Chow rings. Moreover, by calculating the relative tangent bundles between two rational homogeneous spaces, we give explicit bounds for the generalized Grauert-Mülich-Barth theorem on rational homogeneous spaces.

math.AG↗

arXiv2020

Vector Bundles on Flag varieties

We study vector bundles on flag varieties over an algebraically closed field $k$. In the first part, we suppose $G=G_k(d,n)$ $(2\le d\leq n-d)$ to be the Grassmannian manifold parameterizing linear subspaces of dimension $d$ in $k^n$, where $k$ is an algebraically closed field of characteristic $p>0$. Let $E$ be a uniform vector bundle over $G$ of rank $r\le d$. We show that $E$ is either a direct sum of line bundles or a twist of a pull back of the universal bundle $H_d$ or its dual $H_d^{\vee}$ by a series of absolute Frobenius maps. In the second part, splitting properties of vector bundles on general flag varieties $F(d_1,\cdots,d_s)$ in characteristic zero are considered. We prove a structure theorem for bundles over flag varieties which are uniform with respect to the $i$-th component of the manifold of lines in $F(d_1,\cdots,d_s)$. Furthermore, we generalize the Grauert-M$\ddot{\text{u}}$lich-Barth theorem to flag varieties. As a corollary, we show that any strongly uniform $i$-semistable $(1\le i\le n-1)$ bundle over the complete flag variety splits as a direct sum of special line bundles.

math.AG↗

arXiv2019

Finite-dimensional Hopf algebras over the Hopf algebra $H_{b:1}$ of Kashina

Let $H$ be the 16-dimensional nontrivial (namely, noncommutative and noncocommutative) semisimple Hopf algebra $H_{b:1}$ appeared in Kashina's work \cite{K00}. We obtain all simple Yetter-Drinfeld modules over $H$ and then determine all finite-dimensional Nichols algebras satisfying $\mathcal{B}(V)\cong \bigotimes_{i\in I}\mathcal{B}(V_i)$, where $V=\bigoplus_{i\in I}V_i$, each $V_i$ is a simple object in $_H^H\mathcal{YD}$. Finally, we describe all liftings of those $\mathcal{B}(V)$ over $H$.

math.QA↗