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Fuhai Zhu

Publications and source records attributed to Fuhai Zhu.

10 recordsLinked to original sources
arXiv2026

Reduction of integer tiles via CRT and base-p digits

Coven and Meyerowitz gave two cyclotomic conditions, (T1) and (T2), which characterize integer tiles whose cardinalities have at most two distinct prime factors. We prove that the same characterization holds without this restriction. The proof uses a reduction in Chinese remainder coordinates: slicing by the lowest base-p digit produces sets with a common tiling complement in a group of order smaller by a factor of p. This reduction preserves the cyclotomic data needed for an induction on the exponents in (T2).

math.NT↗
arXiv2026

Harmonicity and Existence of Algebraic Generalized Ricci Solitons

In this paper, we study the harmonicity and existence of algebraic generalized Ricci solitons. Firstly, we characterize harmonic torsion of algebraic generalized Ricci solitons on arbitrary metric Lie algebras by an identity involving the Killing form. In particular, positive semidefiniteness of the Killing form implies harmonicity without a unimodularity assumption. Then, we construct generalized nilsolitons with nonzero torsion on indecomposable three-step nilpotent Lie algebras admitting no classical nilsoliton, in dimension seven and in dimensions $6m+d$ for $m>d\geq1$. Furthermore, we provide a spectral obstruction for nilpotent Lie algebras with abelian derived algebra and an obstruction based on the action of derivations on the quotient by the center. Combining these obstructions with explicit constructions, we prove that the filiform Lie algebra $\mathfrak m_2(n)$, $n\geq5$, admits a generalized nilsoliton if and only if $5\leq n\leq8$.

math.DG↗
arXiv2026

Error Analysis on a Novel Class of Exponential Integrators with Local Linear Extension Techniques for Highly Oscillatory ODEs

This paper investigates a class of non-autonomous highly oscillatory ordinary differential equations characterized by a linear component inversely proportional to a small parameter $\varepsilon$, with purely imaginary eigenvalues, and an $\varepsilon$-independent nonlinear part. When $0<\varepsilon\ll 1$, the rapidly oscillatory nature of the solution imposes severe constraints on step size selection and numerical accuracy, leading to considerable computational difficulties. Inspired by a linearization technique that introduces auxiliary polynomial variables, a new family of explicit exponential integrators has recently been proposed. These methods do not require the linear part to be diagonal or to have eigenvalues that are integer multiples of a fixed value - a common assumption in multiscale approaches - and they achieve arbitrarily high orders of convergence without imposing order conditions. The main contribution of this work is to provide a rigorous error analysis for this new class of methods under a bounded oscillatory energy condition. To this end, we first establish the equivalence between the high-dimensional system and the original problem using algebraic techniques. Building on these foundational results, we prove that the numerical schemes, when employing auxiliary polynomial variables of degree $k$, achieve a uniform convergence order of $O(h^{k+1})$. In particular, an improved order of $O(\varepsilon h^k)$ is attained when $h$ is larger than the scale of $\varepsilon$. These theoretical findings are further applied to second-order oscillatory systems, leading to improved uniform accuracy with respect to $\varepsilon$. Finally, numerical experiments confirm the optimality of the derived error estimates.

math.NA↗
arXiv2022

Multiplicative maps on matrix algebras

In this paper, we use elementary method to give a classification of the multiplicative maps on matrix algebra $M_{n}(\mF)$ over a field $\mF$ of characteristic $0$. All the multiplicative maps are classified into three classes: the trivial ones, the degenerate ones and the non-degenerate ones.

math.RT↗
arXiv2022

Flat Lie groups, Frobenius Lie algebras and étale prehomogeneous vector spaces for reductive Lie groups

In this paper, we established the relationship among left-invariant flat connections on Lie groups, left-symmetric algebras, Frobenius Lie algebras and étale prehomogeneous vector spaces, gave a one-to-one correspondence between the left-symmetric Lie algebras with a right identity and the étale prehomogeneous vector spaces for a Lie group, and proved that, in essence, any left-symmetric structure on a reductive Lie algebra has a right identity, which implies that the classification of flat connections on a reductive Lie group $G$ amounts to that of étale prehomogeneous vector spaces for $G$. We classified the étale prehomogeneous vector spaces for $G$ with simple Levi factors.

math.RT↗
arXiv2022

An improved condition for a graph to be determined by its generalized spectrum

A fundamental and challenging problem in spectral graph theory is to characterize which graphs are uniquely determined by their spectra. In Wang [J. Combin. Theory, Ser. B, 122 (2017): 438-451], the author proved that an $n$-vertex graph $G$ is uniquely determined by its generalized spectrum (DGS) whenever $2^{-\lfloor\frac{n}{2}\rfloor}\det W$ is odd and square-free. Here, $W$ is the walk matrix of $G$, namely, $W=[e,Ae,\ldots,A^{n-1}e]$ with $e$ all-one vector and $A$ the adjacency matrix of $G$. In this paper, we focus on a larger family of graphs with $d_n$ square-free, where $d_n$ refers to the last invariant factor of $W$. We introduce a new kind of polynomials for a graph $G$ associated with a prime $p$. Such a polynomial is invariant under generalized cospectrality. Using the newly defined polynomials, we obtain a sufficient condition for a graph in the larger family to be DGS. The main result of this paper improves upon the aforementioned result of Wang while the proof for the main result gives a new way to attack the problem of generalized spectral characterization of graphs.

math.CO↗
arXiv2014

Non-trivial $m$-quasi-Einstein metrics on simple Lie groups

We call a metric $m$-quasi-Einstein if $Ric_X^m$, which replaces a gradient of a smooth function $f$ by a vector field $X$ in $m$-Bakry-Emery Ricci tensor, is a constant multiple of the metric tensor. It is a generalization of Einstein metrics which contains Ricci solitons. In this paper, we focus on left-invariant metrics on simple Lie groups. First, we prove that $X$ is a left-invariant Killing vector field if the metric on a compact simple Lie group is $m$-quasi-Einstein. Then we show that every compact simple Lie group admits non-trivial $m$-quasi-Einstein metrics except $SU(3)$, $E_8$ and $G_2$, and most of them admit infinitely many metrics. Naturally, the study on $m$-quasi-Einstein metrics can be extended to pseudo-Riemannian case. And we prove that every compact simple Lie group admits non-trivial $m$-quasi-Einstein Lorentzian metrics and most of them admit infinitely many metrics. Finally, we prove that some non-compact simple Lie groups admit infinitely many non-trivial $m$-quasi-Einstein Lorentzian metrics.

math.DG↗
arXiv2008

On pseudo-Riemannian Lie algebras: a class of new Lie-admissible algebras

M. Boucetta introduced the notion of pseudo-Riemannian Lie algebra in [2] when he studied the line Poisson structure on the dual of a Lie algebra. In this paper, we redefine pseudo-Riemannian Lie algebra, which, in essence, is a class of new Lie admissible algebras and prove that all pseudo-Riemannian Lie algebras are solvable. Using our main result and method, we prove some of M. Boucetta's results in [2, 3] in a simple and new way, then we give an explicit construction of Riemann-Lie algebras and a classification of pseudo-Riemannian Lie algebras of dimension 2 and 3.

math.RA↗