In this paper, we establish the necessary and sufficient conditions for solving a dual split quaternion matrix equation AXB=C, and present the general solution expression when solvability is achieved. As an application, we delve into the necessary and sufficient condition for the existence of Hermitian solution to this equation by using a newly defined real representation method. Furthermore, we obtain the solutions for the dual split quaternion matrix equations AX=C and XB=C. Finally, we provide a numerical example to demonstrate the findings of this paper.

Red djulis (Chenopodium formosanum) is a native cereal plant in Taiwan; it contains abundant polyphenols, betalian and dietary fiber. The appearance of red djulis is bright red. Therefore, it is also called the “ruby of cereals”. The antioxidative activity of red djulis extract is well-understood. However, the antiaging function still remains unclear. This study examined the potential of red djulis extract for enhancing collagen secretion and preventing cutaneous aging using red djulis extracts. The red djulis extracts are comprised of an abundant active component that can effectively enhance the ability of collagen secretion of dermal fibroblasts, prevent the glycation of collagen and resist the damage of ultraviolet light exposure. After fibroblast treatment with red djulis extracts, TGM1, KRT1, KRT10 and SOD2 genes were up-regulated significantly by 2.3, 4.3, 4.4 and 27.3 times, respectively, compared to those of the control group. Additionally, it can increase COL1A2 gene expression by 43% and decrease MMP9 gene expression 33%. Therefore, it was demonstrated that red djulis extracts affect gene expressions related to the skin barrier, antioxidation and collagen. Moreover, we found positive effects on skin barrier integrity, endogenous antioxidant activity and skin collagen-preservation. The preparation of the red djulis extracts is environmental friendly and can promote the economic value of Chenopodium formosanum; thus, the proposed extract is suitable for applications in the development of food products, especially beverages, skin care and cosmetic products.

Breast cancer is the most common malignancies in women and the second leading cause of cancer death in women. Triple negative breast cancer (TNBC) subtype is a breast cancer subset without ER, PR and HER2 expression, limiting treatment options and presenting a poorer survival rate. Thus, we investigated whether HDACi would be used as potential anti-cancer therapy on breast cancer cells. In this study, we found TNBC and HER2-enrich breast cancers are extremely sensitive to Panobinostat, Belinostat of HDACi via experiments of cell viability assay, apoptotic marker identification and flow cytometry measurement. On the other hand, we developed a bioluminescence based live cell non-invasive apoptosis detection sensor (IADS) detection system to evaluate the quantitative and kinetic analyses of apoptotic cell death by HDAC treatment on breast cancer cells. In addition, the use of HDACi may also be accompanied with chemotherapeutic agent such as doxorubicin to synergic drug sensitivity on TNBC cell (MDA-MB-231), but not in breast normal epithelia cells (MCF-10A), providing therapeutic benefits against breast tumor in clinic.