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.
The nature of forgetting has long been a critical issue in the study of serial order memory, with output interference recognized as a significant factor in shaping serial position curves. However, it remains unclear whether this interference arises from active representations following retrieval (such as capacity consumption, overwriting of other items, blocking retrieval of other information, increased competitions) or if it stems directly from the act of retrieval itself. The present study sought to clarify these possibilities by utilizing retrieval-induced forgetting paradigms, which have been primarily developed within the domain of single-item memory. Through four preregistered experiments, we found that engaging in retrieval practice for part of an ordered list led to reduced order accuracy for the remaining list items. However, the effect size of this retrieval practice was comparable to that of merely re-presenting order information in working memory without engaging in retrieval, suggesting that the observed impairment was not retrieval-specific. This retrieval unspecificity was replicated in an additional experiment. This indicates that output interference in serial order memory is more likely attributable to active representations maintained in working memory, rather than being a direct consequence of retrieval. These findings are discussed within the framework of the two-factor account of retrieval-induced forgetting in the domain of item memory, which posits that both inhibition and competition play roles in the forgetting process.
