A Mikhlin--H\"ormander multiplier theorem for the partial harmonic oscillator

We prove a Mikhlin--H\"ormander multiplier theorem for the partial harmonic oscillator H_{\textup{par}}=-\pa_\rho^2-\Delta_x+|x|^2 for $(\rho, x)\in\R\times\R^d$ by using the Littlewood--Paley $g$ and $g^\ast$ functions and the associated heat kernel estimate. The multiplier we have investigated is defined on $\mathbb R \times \mathbb N$.Comment: 14 pages, no figure. All comments are welcom

On the Discussion of Large Language Models: Symmetry of Agents and Interplay with Prompts

Two ways has been discussed to unlock the reasoning capability of a large language model. The first one is prompt engineering and the second one is to combine the multiple inferences of large language models, or the multi-agent discussion. Theoretically, this paper justifies the multi-agent discussion mechanisms from the symmetry of agents. Empirically, this paper reports the empirical results of the interplay of prompts and discussion mechanisms, revealing the empirical state-of-the-art performance of complex multi-agent mechanisms can be approached by carefully developed prompt engineering. This paper also proposes a scalable discussion mechanism based on conquer and merge, providing a simple multi-agent discussion solution with simple prompts but state-of-the-art performance.Comment: Working in progress, and code will be released soo

Symmetry breaking and criticality in tensor-product states

We discuss variationally optimized matrix-product states for the transverse-field Ising chain, using D*D matrices with small D=2-10. For finite system size N there are energy minimums for symmetric as well as symmetry-broken states, which cross each other at a field value hc(N,D); thus the transition is first-order. A continuous transition develops as N->infinity. The asymptotic critical behavior is then always of mean-field type (the magnetization exponent beta=1/2), but a window of field strengths where true Ising scaling holds (beta=1/8) emerges with increasing D. We also demonstrate asymptotic mean-field behavior for infinite-size two-dimensional tensor-product (iPEPS) states with small tensors.Comment: 4 pages, 5 figure