E1E_1-degeneration and d′d′′d'd''-lemma

For a double complex $(A, d', d'')$, we show that if it satisfies the $d'd''$-lemma and the spectral sequence $\{E^{p, q}_r\}$ induced by $A$ does not degenerate at $E_0$, then it degenerates at $E_1$. We apply this result to prove the degeneration at $E_1$ of a Hodge-de Rham spectral sequence on compact bi-generalized Hermitian manifolds that satisfy a version of $d'd''$-lemma

Attractive Factors of Environment-Friendly Daily Necessities

Going green is increasingly important to the market. The present research indicates that functional and emotional factors can achieve the best-perceived effects when choosing an environment-friendly product. Therefore, this study aims to gather these attractive factors from high-involvement groups by using Miryoku engineering. First, we capture the factors through the Evaluation Grid Method and use Quantification Theory Type I for quantitative analysis. Then, generalize four feelings about environment-friendly products, namely “Assured,” “Responsible,” “Safe,” and “Comfortable.” We also define a linear dimension with short-, normal-, and far-sight for locating attractive factors and feelings. The result shows that high-involvement groups are more concerned about the long-term impacts of “Responsible” feelings, while low-involvement groups focus more on the obvious benefits of “Responsible” and “Safe” feelings. Moreover, the emphasis on natural ingredients is necessary for achieving “Assured” and “Comfortable” feelings for both the high- and low-involvement groups

Learning to Predict the Cosmological Structure Formation

Matter evolved under influence of gravity from minuscule density fluctuations. Non-perturbative structure formed hierarchically over all scales, and developed non-Gaussian features in the Universe, known as the Cosmic Web. To fully understand the structure formation of the Universe is one of the holy grails of modern astrophysics. Astrophysicists survey large volumes of the Universe and employ a large ensemble of computer simulations to compare with the observed data in order to extract the full information of our own Universe. However, to evolve trillions of galaxies over billions of years even with the simplest physics is a daunting task. We build a deep neural network, the Deep Density Displacement Model (hereafter D$^3$M), to predict the non-linear structure formation of the Universe from simple linear perturbation theory. Our extensive analysis, demonstrates that D$^3$M outperforms the second order perturbation theory (hereafter 2LPT), the commonly used fast approximate simulation method, in point-wise comparison, 2-point correlation, and 3-point correlation. We also show that D$^3$M is able to accurately extrapolate far beyond its training data, and predict structure formation for significantly different cosmological parameters. Our study proves, for the first time, that deep learning is a practical and accurate alternative to approximate simulations of the gravitational structure formation of the Universe.Comment: 8 pages, 5 figures, 1 tabl