arXiv subjects

Weiqi Zhou

Publications and source records attributed to Weiqi Zhou.

9 recordsLinked to original sources

arXiv2026

On number of cyclic $n$-roots and disjointness of Fourier supports

A cyclic $n$-root is an $n$-dimensional complex vector that solves a particular set of multivariate homogeneous polynomial equations. There is a one-to-one correspondence between unimodular cyclic $n$-roots and bi-unimodular vectors (CAZAC sequences) with leading entry one. It was conjectured by Björck and Saffari that the set of cyclic $n$-roots is finite if and only if $n$ is square free. It is known that such a set is infinite if $n$ is not square free, and finite if $n$ is prime. A critical reduction in Haagerup's proof for prime $n$ is to show that infinity of cyclic $n$-roots (for any $n$) implies existence of two vectors with disjoint supports in both the time domain and the frequency domain. In this paper we show that such pair of vectors always exist if $n$ is composite, indicating that the original reduction is not adequate for composite square free cases. A discussion on the existence of a single vector whose support is disjoint with its Fourier transform is also included.

math.CA↗

arXiv2026

Mutual Annihilation of Tiles

We prove that if the zero set of the Fourier transform of $A\subseteq\mathbb Z_n\times\mathbb Z_n$ contains an element of prime power order, then there is an equi-distribution relation in subsets of $A$ with respect to certain hyperplanes. With this we further show that if $A$ is a tiling complement of the subgroup generated by $(p,0)$ and $(0,p)$ in $\mathbb Z_{p^m}\times\mathbb Z_{p^m}$, then the zero set of its Fourier transform is disjoint with the orthogonal rotation of $A$. These results are motivated by a casual observation in $\mathbb Z_{p^2}\times\mathbb Z_{p^2}$.

math.CA↗

arXiv2026

On Tiling and Spectral Sets in $\mathbb Z_{p^2}\times\mathbb Z_{p^2}$

Let $p$ be a prime number, it is shown that tiling and spectral sets coincide in $\mathbb Z_{p^2}\times\mathbb Z_{p^2}$ by considering equivalently symplectic spectral pairs. The main approach is still to analyze the zero set of the Fourier transform. The zero set of the symplectic Fourier transform differs from the zero set of the usual Fourier transform by an orthogonal rotation, but using the symplectic Fourier transform allows more freedom when applying change of basis. Some auxiliary results concerning tiling sets and spectral sets of sizes $p$ and $p^{2m-1}$ in $\mathbb Z_{p^m}\times\mathbb Z_{p^m}$ are also presented.

math.CA↗

arXiv2026

Spectrality of Prime Size Tiles

We prove that if a tile in $\mathbb Z^d$ has prime size $p$, then it must be spectral. The proof is by contradiction, it is simply shown that the tiling complement of such a tile can not annihilate all $p$-subgroups. In addition, with a simple transformation we prove that any $p$ points in general linear positions in $\mathbb Z^d (d\ge p-1)$ must be both tiling and spectral.

math.CA↗

arXiv2026

Principal Non-singularity of Fourier Matrices on $\mathbb Z_p \times \mathbb Z_q$ and $\mathbb Z_2^k \times \mathbb Z_q$

Let $F_n$ be the $n\times n$ Fourier matrix on the cyclic group $\mathbb Z_n$, a renowned theorem of Chebotarëv asserts that all minors in $F_n$ for prime $n$ are non-zero. In this short note it is shown that (i) all principal minors in the Kronecker product $F_p\otimes F_q$ are non-vanishing (principal non-singularity) for distinct odd primes $p,q$ if $q$ is large enough and generates the multiplicative group $\mathbb Z_p^*$; (ii) the Fourier matrix on $\mathbb Z_2^k \times \mathbb Z_q$ is principally non-singular upon permutation (in particular, for $k=1$ the identity permutation suffices) for odd prime $q$ and $k=1,2,3$. The proof is just an exposition of existing techniques reorganized in a unified way. The result will have implications in combining Riesz bases of exponentials.

math.CA↗

arXiv2026

Frame Sets and Zeros of Zak transforms of Extended Gaussians

Let $a,b,c\in\mathbb C$ with $\re(a)<0$, we show that the extended Gaussian $e^{ax^2+bx+c}$ has maximal frame set (i.e., its frame set consists of precisely all positive pairs $(\alpha,\beta)$ with $\alpha\beta<1$), and its Zak transform has a unique simple zero in the unit square $[0,1)^2$ (in particular, the zero is at the center of the unit square if $b=0$). These statements extend the same results of the usual Gaussian (the cases when $a<0$ and $b,c\in\mathbb R$), and add more instances to the observation that if a continuous Wiener function has maximal frame set, then its Zak transform has a unique simple zero in the unit square. The proof of the maximality of the frame set combines metaplectic representation with a classical density result of the standard Gaussian. The proof of the uniqueness of the zero relies on properties of the theta function.

math.CA↗

arXiv2023

Global Inequality in Cooling from Urban Green Spaces and its Climate Change Adaptation Potential

Heat extremes are projected to severely impact humanity and with increasing geographic disparities. Global South countries are more exposed to heat extremes and have reduced adaptation capacity. One documented source of such adaptation inequality is a lack of resources to cool down indoor temperatures. Less is known about the capacity to ameliorate outdoor heat stress. Here, we assess global inequality in green infrastructure, on which urban residents critically rely to ameliorate lethal heat stress outdoors. We use satellite-derived indicators of land surface temperature and urban green space area to quantify the daytime cooling capacity of urban green spaces in the hottest months across ~500 cities with population size over 1 million per city globally. Our results show a striking contrast with an about two-fold lower cooling capacity in Global South cities compared to the Global North (2.1 degrees Celsius vs. 3.8 degrees Celsius). A similar gap occurs for the cooling adaptation benefits received by an average urban resident (Global South 1.9 degrees Celsius vs. North 3.6 degrees Celsius), i.e., accounting for relative spatial distributions of people and urban green spaces. This cooling adaptation inequality is attributed to the discrepancies in urban green space quantity and quality between Global North and South cities, jointly shaped by natural and socioeconomic factors. Our analyses suggest vast potential for enhancing outdoor cooling adaptation while reducing its global inequality through expanding and optimizing urban green infrastructure.

stat.AP↗

arXiv2023

Mega-cities dominate China's urban greening

Trees play a crucial role in urban environments, offering various ecosystem services that contribute to public health and human well-being. China has initiated a range of urban greening policies over the past decades, however, monitoring their impact on urban tree dynamics at a national scale has proven challenging. In this study, we deployed nano-satellites to quantify urban tree coverage in all major Chinese cities larger than 50 km2 in 2010 and 2019. Our findings indicate that approximately 6000 km2 (11%) of urban areas were covered by trees in 2019, and 76% of these cities experienced an increase in tree cover compared to 2010. Notably, the increase in tree cover in mega-cities such as Beijing, and Shanghai was approximately twice as large as in most other cities (7.69% vs 3.94%). The study employs a data-driven approach towards assessing urban tree cover changes in relation to greening policies, showing clear signs of tree cover increases but also suggesting an uneven implementation primarily benefiting a few mega-cities.

cs.CV↗

arXiv2016

Random Reordering in SOR-Type Methods

When iteratively solving linear systems By=b with Hermitian positive semi-definite $B$, and in particular when solving least-squares problems for $Ax=b$ by reformulating them as $AA^\ast y=b$, it is often observed that SOR-type methods (Gauss-Seidel, Kaczmarz) perform suboptimally for the given equation ordering, and that random reordering improves the situation on average.This paper is an attempt to provide some additional theoretical support for this phenomenon. We show eerror bounds for two randomized versions, called shuffled and preshuffled SOR, that improve asymptotically upon the best known bounds fro SOR with cyclic ordering. Our results are based on studying the behavior of the triangular truncation of Hermitian matrices with respect to their permutations.

math.NA↗