Almost sure global well posedness of the defocusing energy-critical wave equation with a Cubic Convolution

We consider the defocusing energy-critical wave equation with a cubic convolution nonlinearity $\partial_t^2u-\Delta u+(|\cdot|^{-4}\ast|u|^2)u=0$ in spatial dimension $d=5$ and establish almost sure global existence for randomized initial data in $H_x^s\times H_x^{s-1}$ for $s\in(\frac{1}{2},1)$. The nonlocal nonlinearity term brings challenges in controlling the energy, which is a fundamental property to show global well-posedness. We are inspired to consider this problem by the work of Oh-Pocovnicu \cite{Oh-Pocovnicu-1}, but unlike the proof in \cite{Oh-Pocovnicu-1}, the novelty of our article lies in fully leveraging the potential energy

Fractional Leibniz rule on the torus

We discuss the fractional Leibniz rule for periodic functions on the $d$-dimensional torus, including the endpoint cases.Comment: 10 page

The Development and Validation of an EFL Learner Attitude Scale for Phonetic Symbol Learning in A Chinese University Context

Research has established that learner attitude significantly impacts the outcomes of learning English as a foreign language (EFL). However, previous studies have rarely considered a validated attitude scale for English phonetic symbol learning (EPSL). This study aims to develop and validate a scale to measure students\u27 attitudes toward EPSL by integrating findings from learning attitude research in education and psychology. The results demonstrate that the attitude scale can predict attitudes towards EPSL in the context of a Chinese university and has good reliability and validity (KMO=0.892, α=0.749). Pedagogical suggestions are made to help students develop a more positive attitude toward EPSL