Umbilic hypersurfaces of constant sigma-k curvature in the Heisenberg group

We study immersed, connected, umbilic hypersurfaces in the Heisenberg group $H_{n}$ with $n$ $\geq$ $2.$ We show that such a hypersurface, if closed, must be rotationally invariant up to a Heisenberg translation. Moreover, we prove that, among others, Pansu spheres are the only such spheres with positive constant sigma-k curvature up to Heisenberg translations.Comment: 28 pages, 6 figure

Umbilicity and characterization of Pansu spheres in the Heisenberg group

For $n\geq 2$ we define a notion of umbilicity for hypersurfaces in the Heisenberg group $H_{n}$. We classify umbilic hypersurfaces in some cases, and prove that Pansu spheres are the only umbilic spheres with positive constant $p$(or horizontal)-mean curvature in $H_{n}$ up to Heisenberg translations.Comment: 32 pages, 2 figures; in Crelle's journal, 201

Effects of red and blue light ratio on the morphological traits and flower sex expression in Cucurbita moschata Duch.

Squash (Cucurbita moschata Duch.) is an important fruit vegetable that can be long-term transport and storage. Light-emitting diodes (LEDs) are commercially used light sources applied to improve the producing of leaf vegetables in plant factory. However, the influences of LEDs on the plant growth and flower development of fruit vegetables remain unknown. In this study, five effective light quality treatments, including white light, a 10:8 ratio of blue (B) to red (R) light, a 10:4 mixture of blue/red light, red light, and blue light, were used for growing squash and inducing female flowers to maximize production. Our results show that varying light quality influence morphological traits and flower appearance. Both blue and red light improved the development of first and second internodes and induced larger leaves and petiole lengths, whereas 10:4 mixture caused shorter plant heights and decreased internode and petiole lengths. Although 10:8 mixture treatment reduced chlorophyll content, this spectral regime increased leaf number and influenced flower sex development, inducing more female flowers and more fruits. Light quality manipulation thus beneficially influences the growth and flower sex proportion in squash plants. Squash plants under 10:8 mixture treatment exhibited increase in yield, and can be used as a supplementary light treatment in plant factory

Interpretations of Domain Adaptations via Layer Variational Analysis

Transfer learning is known to perform efficiently in many applications empirically, yet limited literature reports the mechanism behind the scene. This study establishes both formal derivations and heuristic analysis to formulate the theory of transfer learning in deep learning. Our framework utilizing layer variational analysis proves that the success of transfer learning can be guaranteed with corresponding data conditions. Moreover, our theoretical calculation yields intuitive interpretations towards the knowledge transfer process. Subsequently, an alternative method for network-based transfer learning is derived. The method shows an increase in efficiency and accuracy for domain adaptation. It is particularly advantageous when new domain data is sufficiently sparse during adaptation. Numerical experiments over diverse tasks validated our theory and verified that our analytic expression achieved better performance in domain adaptation than the gradient descent method.Comment: Published at ICLR 202