This paper solves the open problem of the simplicity of the second Dirichlet eigenvalue for nearly equilateral triangles, offering a complete solution to Conjecture 6.47 posed by R. Laugesen and B. Siudeja in A. Henrot’s book “Shape Optimization and Spectral Theory.” Our proof is achieved by introducing a new difference quotient formula for the behavior of nearly degenerate eigenvalues resulting from domain perturbations, and a novel numerical algorithm that rigorously estimates the formula using verified computation.
Abstract We propose a new scenario to realize the Coleman-Weinberg (CW) type chiral phase transition in the QCD thermal history. This scenario predicts a heavy axionlike particle (ALP) with mass ~ 5 MeV, consistently with the current experimental and cosmological bounds. The chiral phase transition is evaluated by monitoring ordinary QCD setup in a view of a two-flavor Nambu-Jona-Lasinio model including a simplified meson fluctuation contribution. The present work thus can open a new window to search for the ALP associated with the QCD phase transition epoch of the thermal history. The new QCD cosmological scenario potentially predicts rich epochs around the QCD scale: a mini-inflation; a nonperturbative preheating and/or reheating, which can provide characteristic gravitational wave and primordial black hole productions. This proposal is based on a generic classification of the order of the chiral phase transition at the level of the mean field approximation in view of the scale violation classes: the soft-scale breaking term and the CW-type scale anomaly term, in or off the medium with or without chemical potentials. On this theoretical ground, we also revisit existing scenarios which undergo the supercooling chiral phase transition, such as nearly scale-invariant QCD and QCD with a large baryon chemical potential.
Bilinear Fourier multipliers of the form e(i(vertical bar xi vertical bar+vertical bar eta vertical bar+vertical bar xi+eta vertical bar))sigma((xi, eta)) are considered. It is proved that if sigma(xi, eta) is in the Hormander class S-1,0(m)(R-2n) with m = -(n + 1)/2 then the corresponding bilinear operator is bounded in L-infinity x L-infinity -> bmo, h(1) x L-infinity -> L-1, and L-infinity x h(1) -> L-1. This improves a result given by Rodriguez-Lopez, Rule and Staubach. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Gauge fixing is an essential step in lattice QCD calculations, particularly for studying gauge-dependent observables. Traditional iterative algorithms are computationally expensive and often suffer from critical slowing down and scaling bottlenecks on large lattices. We present a novel machine learning framework for lattice gauge fixing, where Wilson lines are utilized to construct gauge transformation matrices within a convolutional neural network. The model parameters are optimized via backpropagation, and we introduce a hybrid strategy that combines a neural-network-based transformation with subsequent iterative methods. Preliminary tests on SU(3) gauge theory ensembles for Coulomb gauge demonstrate the potential of this approach to improve the efficiency of lattice gauge fixing. Furthermore, we show that the model exhibits lattice size transferability, where parameters optimized on smaller lattices remain effective for larger volumes without additional training. This framework provides a scalable path toward mitigating critical slowing down in high-precision gauge fixing.
We study a parameter optimization of domain-wall fermions to improve chiral symmetry based on machine learning. Domain-wall fermions involve coefficients along the fifth dimension, which can be treated as trainable parameters to reduce the chiral symmetry violation caused by the finite extent of the fifth dimension. As the loss function, we use the residual mass estimated stochastically on a single gauge configuration. Numerical tests on a L^3× T× L_5=4^3×8×8 lattice demonstrate the feasibility of this framework.
