arXiv subjects

Qiaoling Wei

Publications and source records attributed to Qiaoling Wei.

11 recordsLinked to original sources

arXiv2026

Generic Spectral Determination of Semiclassical Schr\"odinger Operators with $\mathbb Z_2$-Symmetry

Consider the two-dimensional semiclassical Schr\"odinger operator $P_\hbar=-\frac{\hbar^2}{2}\Delta+V$ on $\mathbb R^2$, where $V$ has a nondegenerate well at the origin, its harmonic frequencies are rationally independent, and $V$ is $\mathbb Z_2$-symmetric. We prove that, generically, the first three layers of the quantum Birkhoff normal form (QBNF) determine the full Taylor series of $V$ at the origin, up to the unavoidable spatial inversion $V(x)\mapsto V(-x)$. The generic condition depends only on the jet of $V$ through order ten. Consequently, for real analytic potentials with a unique isolated global well at the origin, the low-lying semiclassical spectrum generically determines $V$ up to spatial inversion.

math.DS↗

arXiv2025

Rotation Numbers and Geometric Invariants in Bicycle Dynamics

We study planar bicycle dynamics via the rotation number function associated with a closed front track and bicycle length R. We prove that mode-locking plateaus occur only at integer rotation numbers and that the rotation number function is real-analytic off resonance. From the rotation number function we introduce two new geometric invariants: the critical B-length (right end of the first plateau) and the turning B-length (left end of the maximal monotone interval). We prove that, for a star-shaped curve, these invariants coincide, yielding a sharp transition of the bicycle monodromy: hyperbolic for R below the critical B-length and elliptic for R above it. The proofs combine projectivized SU(1,1) dynamics with Riccati equations and rotation-number theory.

math.DS↗

arXiv2025

From i-boxes to signed words

The combinatorics of i-boxes has recently been introduced by Kashiwara--Kim--Oh--Park in the study of cluster algebras arising from the representation theory of quantum affine algebras. In this article, we associate to each chain of i-boxes a signed word, which canonically determines a cluster seed following Berenstein--Fomin--Zelevinsky. By bridging these two different languages, we are able to provide a quick solution to the problem of explicit determining the exchange matrices associated with chains of i-boxes.

math.RT↗

arXiv2025

Geometric normalization

For a local analytic diffeomorphism of the plane with an irrational elliptic fixed point at 0, we introduce the notion of ``geometric normalization'', which includes the classical formal normalizations as a special case: it is a formal conjugacy to a formal diffeomorphism which preserves the foliation by circles centered at 0. We show that geometric normalizations, despite of non-uniqueness, correspond in a natural way to a unique formal invariant foliation. We show, in various contexts, generic results of divergence for the geometric normalizations, which amount to the generic non-existence of any analytic invariant foliation.

math.DS↗

arXiv2024

EFCNet: Every Feature Counts for Small Medical Object Segmentation

This paper explores the segmentation of very small medical objects with significant clinical value. While Convolutional Neural Networks (CNNs), particularly UNet-like models, and recent Transformers have shown substantial progress in image segmentation, our empirical findings reveal their poor performance in segmenting the small medical objects and lesions concerned in this paper. This limitation may be attributed to information loss during their encoding and decoding process. In response to this challenge, we propose a novel model named EFCNet for small object segmentation in medical images. Our model incorporates two modules: the Cross-Stage Axial Attention Module (CSAA) and the Multi-Precision Supervision Module (MPS). These modules address information loss during encoding and decoding procedures, respectively. Specifically, CSAA integrates features from all stages of the encoder to adaptively learn suitable information needed in different decoding stages, thereby reducing information loss in the encoder. On the other hand, MPS introduces a novel multi-precision supervision mechanism to the decoder. This mechanism prioritizes attention to low-resolution features in the initial stages of the decoder, mitigating information loss caused by subsequent convolution and sampling processes and enhancing the model's global perception. We evaluate our model on two benchmark medical image datasets. The results demonstrate that EFCNet significantly outperforms previous segmentation methods designed for both medical and normal images.

eess.IV↗

arXiv2023

Twist automorphisms and Poisson structures

We introduce (quantum) twist automorphisms for upper cluster algebras and cluster Poisson algebras with coefficients. Our constructions generalize the twist automorphisms for quantum unipotent cells. We study their existence and their compatibility with Poisson structures and quantization. The twist automorphisms always permute well-behaved bases for cluster algebras. We explicitly construct (quantum) twist automorphisms of Donaldson-Thomas type and for principal coefficients.

math.QA↗

arXiv2022

On Planar Shadowing Curves to Closed Escaping Curves

We introduce a new dynamical system model called the shadowing problem, where a shadower chases after an escaper by always staring at and keeping the distance from him. When the escaper runs along a planar closed curve, we associate to the reduced shadowing equations the rotation number, and show that it depends only on the geometry of the escaping curve. Two notions called the critical shadowing distance and turning shadowing distance are introduced to characterize different dynamical behaviors. We show that a planar closed escaping curve could have shadowing curves of different types including periodic, subharmonic and ergodic ones, depending on the shadowing distance. Singularities of cusp type are found when the shadowing distance is large. Shadowing curves to an escaping circle are examined in details analytically and numerically. Finally, we conjecture that the critical shadowing distance and turning shadowing distance are coincident for typical escaping curves.

math.DS↗

arXiv2021

Elliptic fixed points with an invariant foliation: Some facts and more questions

We address the following question: let F:(R^2,0)->(R^2,0) be an analytic local diffeomorphism defined in the neighborhood of the non resonant elliptic fixed point 0 and let Φbe a formal conjugacy to a normal form N. Supposing F leaves invariant the foliation by circles centered at 0, what is the analytic nature of Φand N?

math.DS↗

arXiv2017

Dynamical Spectral rigidity among $\mathbb Z_2$-symmetric strictly convex domains close to a circle

We show that any sufficiently (finitely) smooth $\mathbb Z_2$-symmetric strictly convex domain sufficiently close to a circle is dynamically spectrally rigid, i.e. all deformations among domains in the same class which preserve the length of all periodic orbits of the associated billiard flow must necessarily be isometric deformations. This gives a partial answer to a question of P. Sarnak.

math.DS↗

arXiv2013

Viscosity solution of Hamilton-Jacobi equation by a limiting minmax method

For non convex Hamiltonians, the viscosity solution and the more geometric minimax solution of the Hamilton-Jacobi equation do not coincide in general. They are nevertheless related: we show that iterating the minimax procedure during shorter and shorter time intervals one recovers the viscosity solution.

math.AP↗

arXiv2013

Front tracking and iterated minmax for Hamilton-Jacobi equation in one space variable

The viscosity solution of the Hamilton-Jacobi equation was constructed by an "iterated minimax" procedure. Using Dafermos' front tracking method, we give another proof of this construction in the case of Hamilton-Jacobi equations in one space dimension. This allows us to get a better understanding in this case of the singularities of the viscosity solution.

math.AP↗