Stability of a quadratic Jensen type functional equation in the spaces of generalized functions

AbstractMaking use of the fundamental solution of the heat equation we find the solution and prove the stability theorem of the quadratic Jensen type functional equation9f(x+y+z3)+f(x)+f(y)+f(z)=4[f(x+y2)+f(y+z2)+f(z+x2)] in the spaces of Schwartz tempered distributions and Fourier hyperfunctions

Vav1 inhibits RANKL-induced osteoclast differentiation and bone resorption

Vav1 is a Rho/Rac guanine nucleotide exchange factor primarily expressed in hematopoietic cells. In this study, we investigated the potential role of Vav1 in osteoclast (OC) differentiation by comparing the ability of bone marrow mononuclear cells (BMMCs) obtained from Vav1-deficient (Vav1−/−) and wild-type (WT) mice to differentiate into mature OCs upon stimulation with macrophage colony stimulating factor and receptor activator of nuclear kappa B ligand in vitro. Our results suggested that Vav1 deficiency promoted the differentiation of BMMCs into OCs, as indicated by the increased expression of tartrate-resistant acid phosphatase, cathepsin K, and calcitonin receptor. Therefore, Vav1 may play a negative role in OC differentiation. This hypothesis was supported by the observation of more OCs in the femurs of Vav1−/− mice than in WT mice. Furthermore, the bone status of Vav1−/− mice was analyzed in situ and the femurs of Vav1−/− mice appeared abnormal, with poor bone density and fewer number of trabeculae. In addition, Vav1-deficient OCs showed stronger adhesion to vitronectin, an αvβ3 integrin ligand important in bone resorption. Thus, Vav1 may inhibit OC differentiation and protect against bone resorption

Effect of Magnolol on the Function of Osteoblastic MC3T3-E1 Cells

Objectives. In the present study, the ability of magnolol, a hydroxylated biphenyl compound isolated from Magnolia officinalis, to stimulate osteoblast function and inhibit the release of bone-resorbing mediators was investigated in osteoblastic MC3T3-E1 cells. Methods. Osteoblast function was measured by cell growth, alkaline phosphatase activity, collagen synthesis, and mineralization. Glutathione content was also measured in the cells. Bone-resorbing cytokines, receptor activator of nuclear factor-κB ligand (RANKL), TNF-α, and IL-6 were measured with an enzyme immunoassay system. Results. Magnolol caused a significant elevation of cell growth, alkaline phosphatase activity, collagen synthesis, mineralization, and glutathione content in the cells (P < 0.05). Skeletal turnover is orchestrated by a complex network of regulatory factors. Among cytokines, RANKL, TNF-α, and IL-6 were found to be key osteoclastogenetic molecules produced by osteoblasts. Magnolol significantly (P < 0.05) decreased the production of osteoclast differentiation inducing factors such as RANKL, TNF-α, and IL-6 in the presence of antimycin A, which inhibits mitochondrial electron transport and has been used as an ROS generator. Conclusion. Magnolol might be a candidate as an agent for the prevention of bone disorders such as osteoporosis