Sets with no Riesz bases of exponentials
We prove that sets in a certain class do not admit Riesz bases of exponentials. In particular, this class contains disks and triangles in the plane.
math.CA↗
arXiv subjects
Zhiqiang Wan
Publications and source records attributed to Zhiqiang Wan.
Heisenberg Uniqueness Pairs for a Hyperbola Branch: Supercritical Nonuniqueness for Shifted Lattice Crosses
We study Heisenberg uniqueness for the positive hyperbola branch and shifted lattice crosses in the supercritical regime $q=αγ>1$. We resolve the infinite-dimensionality clause of the arbitrary-shift problem posed by Giri and Manna: for arbitrary shifts on both arms, the normalized pre-annihilator is infinite-dimensional. More precisely, every $v\in BV((1,q))$ has a global $BV$ pre-annihilating extension; the extension is unique unless both twisting phases are trivial, in which case its ambiguity is one-dimensional. The proof reduces the annihilation conditions to a graph equation for a twisted Perron--Frobenius operator and combines a phase-uniform Lasota--Yorke estimate with peripheral spectral rigidity. We also give an exact operator-theoretic normal form for the entire $L^1$ pre-annihilator in terms of the maximal convergence domain of the associated Green series. Writing $Q$ for the twisted product and $A$ for the forcing operator, we show that $Q$ has the closed unit disk as its spectrum on $L^1((0,1))$, that $\Ran(I-Q)$ is not closed, and that, outside a countable set of algebraic values of $q>1$, the operator $\sum_{j=0}^{N-1}Q^jA:L^1((1,q))\to L^1((0,1))$ has norm $2N$ for every $N\ge1$.
math.CA↗
Observable sets for Schrödinger equations on combinatorial graphs
We study observable sets for Schrödinger equations on combinatorial graphs. For one-dimensional lattice Schrödinger operators \(H=-Δ_{\mathrm{disc}}+V\) with \(V(n)\to c\in\mathbb R\) as \(|n|\to\infty\), we prove that a set \(E\subset\mathbb Z\) is observable at some time, equivalently at any time, if and only if it satisfies a local arithmetic condition. This reveals an arithmetic obstruction absent from the Euclidean theory, where thickness is the decisive condition. The same criterion also characterizes observability for the corresponding heat equation on \(\mathbb Z\). In higher-dimensional lattices, we prove observability from the complement of any finite set. We further obtain arithmetic criteria on discrete tori, showing that positive density alone does not ensure observability.
math.AP↗
The Cauchy problem for the improved Boussinesq equation with spatially quasi-periodic initial data
We study the Cauchy problem for the improved Boussinesq equation \[ u_{tt}-u_{xx}-u_{xxtt}-(u^2)_{xx}=0 \] on the real line with spatially quasi-periodic initial data. For a non-resonant frequency vector $ω\in\mathbb R^ν$, we prove local existence and uniqueness of classical spatially quasi-periodic solutions with the same frequency vector $ω$ in two Fourier-side classes. First, for exponentially decaying initial Fourier coefficients, we obtain a spatially quasi-periodic solution whose Fourier coefficients remain exponentially decaying on an explicit time interval. Second, for initial Fourier coefficients $c(n)$ and $d(n)$ satisfying the polynomial decay $ |c(n)|+|d(n)|\lesssim (1+|n|)^{-r}, \; r>ν+2, $ we prove that the corresponding spatially quasi-periodic solution preserves the same polynomial decay rate as the initial data. We also extend these results to the nonlinearity $u^p$ with integer $p \geq 3$.
math.AP↗
Dispersive estimates for discrete Klein-Gordon equations on one-dimensional lattice with quasi-periodic potentials
We prove $\ell^{1}\!\to\!\ell^{\infty}$ dispersive estimates for the discrete Klein--Gordon equation on $\mathbb Z$ with small real-analytic quasi-periodic potentials, showing that the time-decay rate persists as $(\tfrac13)^{-}$. As applications, we derive the corresponding Strichartz estimates and establish small-data global well-posedness for the associated nonlinear discrete Klein--Gordon equation.
math.AP↗
Strichartz estimate for discrete Schrödinger equation on layered King's grid
We establish the sharp \( l^1 \to l^{\infty} \) decay estimate for the discrete Schrödinger equation (DS) on the Layered King's Grid (LKG), with a dispersive decay rate of \( \langle t \rangle^{-13/12} \), which is faster than that for $3$-dimensional lattice (\( \langle t \rangle^{-1} \), see \cite{SK05}). This decay estimate enables us to derive the corresponding Strichartz estimate via the standard Keel--Tao argument. Our approach relies on using techniques from Newton polyhedra to analyze singularities.
math.AP↗
iBARLE: imBalance-Aware Room Layout Estimation
Room layout estimation predicts layouts from a single panorama. It requires datasets with large-scale and diverse room shapes to train the models. However, there are significant imbalances in real-world datasets including the dimensions of layout complexity, camera locations, and variation in scene appearance. These issues considerably influence the model training performance. In this work, we propose the imBalance-Aware Room Layout Estimation (iBARLE) framework to address these issues. iBARLE consists of (1) Appearance Variation Generation (AVG) module, which promotes visual appearance domain generalization, (2) Complex Structure Mix-up (CSMix) module, which enhances generalizability w.r.t. room structure, and (3) a gradient-based layout objective function, which allows more effective accounting for occlusions in complex layouts. All modules are jointly trained and help each other to achieve the best performance. Experiments and ablation studies based on ZInD~\cite{cruz2021zillow} dataset illustrate that iBARLE has state-of-the-art performance compared with other layout estimation baselines.
cs.CV↗
U2RLE: Uncertainty-Guided 2-Stage Room Layout Estimation
While the existing deep learning-based room layout estimation techniques demonstrate good overall accuracy, they are less effective for distant floor-wall boundary. To tackle this problem, we propose a novel uncertainty-guided approach for layout boundary estimation introducing new two-stage CNN architecture termed U2RLE. The initial stage predicts both floor-wall boundary and its uncertainty and is followed by the refinement of boundaries with high positional uncertainty using a different, distance-aware loss. Finally, outputs from the two stages are merged to produce the room layout. Experiments using ZInD and Structure3D datasets show that U2RLE improves over current state-of-the-art, being able to handle both near and far walls better. In particular, U2RLE outperforms current state-of-the-art techniques for the most distant walls.
cs.CV↗
PSMNet: Position-aware Stereo Merging Network for Room Layout Estimation
In this paper, we propose a new deep learning-based method for estimating room layout given a pair of 360 panoramas. Our system, called Position-aware Stereo Merging Network or PSMNet, is an end-to-end joint layout-pose estimator. PSMNet consists of a Stereo Pano Pose (SP2) transformer and a novel Cross-Perspective Projection (CP2) layer. The stereo-view SP2 transformer is used to implicitly infer correspondences between views, and can handle noisy poses. The pose-aware CP2 layer is designed to render features from the adjacent view to the anchor (reference) view, in order to perform view fusion and estimate the visible layout. Our experiments and analysis validate our method, which significantly outperforms the state-of-the-art layout estimators, especially for large and complex room spaces.
cs.CV↗