arXiv subjects

Shi-Zhuo Looi

Publications and source records attributed to Shi-Zhuo Looi.

9 recordsLinked to original sources

arXiv2026

Asymptotics in all regimes for the Schrödinger equation with time-independent coefficients

Using the recent analysis of the output of the low-energy resolvent of Schrödinger operators on asymptotically conic manifolds (including Euclidean space) when the potential is short-range, we produce detailed asymptotic expansions for the solutions of the initial-value problem for the Schrödinger equation (assuming Schwartz initial data). Asymptotics are calculated in all joint large-radii large-time regimes, these corresponding to the boundary hypersurfaces of a particular compactification of spacetime.

math.AP↗

arXiv2026

Global existence and pointwise decay for nonlinear waves under the null condition

This paper proves global existence and sharp pointwise decay for solutions to nonlinear wave equations satisfying the semilinear null condition, on a class of three-dimensional, asymptotically flat, and notably, non-stationary spacetimes. We consider nonlinearities satisfying a generalized null condition which does not necessarily retain its structure when commuted with vector fields. For sufficiently small initial data, and under the assumption that the underlying linear operator satisfies an integrated local energy decay estimate, we prove that solutions exist for all time and we establish sharp pointwise decay estimates for the solution $ϕ$ and its vector-fields. The solution itself decays as $|ϕ(t,x)| \lesssim \langle t+r \rangle^{-1} \langle t-r \rangle^{-1}$. This rate matches that of the nonlinear equation on a flat background. This rate is sharp, as this behavior holds already for certain time-dependent perturbations of the classical null form on Minkowski space, which we specify.

math.AP↗

arXiv2025

A proof of Onsager's Conjecture for the SQG equation

We construct solutions to the SQG equation that fail to conserve the Hamiltonian while having the maximal allowable regularity for this property to hold. This result solves the generalized Onsager conjecture on the threshold regularity for Hamiltonian conservation for SQG.

math.AP↗

arXiv2024

What's the Magic Word? A Control Theory of LLM Prompting

Prompt engineering is crucial for deploying LLMs but is poorly understood mathematically. We formalize LLM systems as a class of discrete stochastic dynamical systems to explore prompt engineering through the lens of control theory. We offer a mathematical analysis of the limitations on the controllability of self-attention as a function of the singular values of the parameter matrices. We present complementary empirical results on the controllability of a panel of LLMs, including Falcon-7b, Llama-7b, and Falcon-40b. Given initial state $\mathbf x_0$ from Wikitext and prompts of length $k \leq 10$ tokens, we find that the "correct" next token is reachable at least 97% of the time, and that the top 75 most likely next tokens are reachable at least 85% of the time. Intriguingly, short prompt sequences can dramatically alter the likelihood of specific outputs, even making the least likely tokens become the most likely ones. This control-theoretic analysis of LLMs demonstrates the significant and poorly understood role of input sequences in steering output probabilities, offering a foundational perspective for enhancing language model system capabilities.

cs.CL↗

arXiv2022

Improved decay for quasilinear wave equations close to asymptotically flat spacetimes including black hole spacetimes

We study the quasilinear wave equation $\Box_{g}ϕ=0$ where the metric $g = g(ϕ,t,x)$ is close to and asymptotically approaches $g(0,t,x)$, which equals the Schwarzschild metric or a Kerr metric with small angular momentum, as time tends to infinity. Under only weak assumptions on the metric coefficients, we prove an improved pointwise decay rate for the solution $ϕ$. One consequence of this rate is that for bounded $|x|$, we have the integrable decay rate $|ϕ(t,x)| \le Ct^{-1-\min(δ,1)}$ where $δ>0$ is a parameter governing the decay, near the light cone, of the coefficient of the slowest-decaying term in the quasilinearity. We also obtain the same aforementioned pointwise decay rates for the quasilinear wave equation $(\Box_{\tilde g} + B^α(t,x)\partial_α+ V(t,x))ϕ=0$ with a more general asymptotically flat metric $\tilde g = \tilde g(ϕ,t,x)$ and with other time-dependent asymptotically flat lower order terms.

math.AP↗

arXiv2022

Decay rates for cubic and higher order nonlinear wave equations on asymptotically flat spacetimes

In this paper, we prove pointwise decay rates for cubic and higher order nonlinear wave equations, including quasilinear wave equations, on asymptotically flat and time-dependent spacetimes. We assume that the solution to the linear equation (rather than the nonlinear equation) satisfies a weaker form of the standard integrated local energy decay, or Morawetz, estimate. For nonlinearities with a total derivative structure, we prove better pointwise decay rates.

math.AP↗

arXiv2022

Pointwise decay for the wave equation on nonstationary spacetimes

The first article in a two-part series (the second article being [arXiv:2205.13197]) assumes a weak local energy decay estimate holds and proves that solutions to the linear wave equation with variable coefficients in $\mathbb R^{1+3}$, first-order terms, and a potential decay at a rate depending on how rapidly the vector fields of the metric, first-order terms, and potential decay at spatial infinity. We prove results for both stationary and nonstationary metrics. The proof uses local energy decay to prove an initial decay rate, and then uses the one-dimensional reduction repeatedly to achieve the full decay rate.

math.AP↗

arXiv2022

Pointwise decay for the energy-critical nonlinear wave equation

This second article in a two-part series (following [arXiv:2105.02865], listed here as \cite{L}) proves optimal pointwise decay rates for the quintic defocusing wave equation with large initial data on nonstationary spacetimes, and both the quintic defocusing and quintic focusing wave equations with small initial data on nonstationary spacetimes. We prove a weighted local energy decay estimate, and use local energy decay and Strichartz estimates on these variable-coefficient backgrounds. By using an iteration scheme, we obtain the optimal pointwise bounds. In addition, we explain how the iteration scheme reaches analogous pointwise bounds for other integral power nonlinearities that are either higher or lower than the quintic power, given the assumption of global existence for those powers (and in the case of the lower powers, given certain initial decay rates).

math.AP↗

arXiv2019

Scattering for critical wave equations with variable coefficients

We prove that solutions to the quintic semilinear wave equation with variable coefficients in $\mathbb R^{1+3}$ scatter to a solution to the corresponding linear wave equation. The coefficients are small and decay as $|x|\to\infty$, but are allowed to be time dependent. The proof uses local energy decay estimates to establish the decay of the $L^6$ norm of the solution as $t\to\infty$.

math.AP↗