Davenport constant with weights

For the cyclic group $G=\mathbb{Z}/n\mathbb{Z}$ and any non-empty $A\in\mathbb{Z}$. We define the Davenport constant of $G$ with weight $A$, denoted by $D_A(n)$, to be the least natural number $k$ such that for any sequence $(x_1, ..., x_k)$ with $x_i\in G$, there exists a non-empty subsequence $(x_{j_1}, ..., x_{j_l})$ and $a_1, ..., a_l\in A$ such that $\sum_{i=1}^l a_ix_{j_i} = 0$. Similarly, we define the constant $E_A(n)$ to be the least $t\in\mathbb{N}$ such that for all sequences $(x_1, >..., x_t)$ with $x_i \in G$, there exist indices $j_1, ..., j_n\in\mathbb{N}, 1\leq j_1 <... < j_n\leq t$, and $\vartheta_1, >..., \vartheta_n\in A$ with $\sum^{n}_{i=1} \vartheta_ix_{j_i} = 0$. In the present paper, we show that $E_A(n)=D_A(n)+n-1$. This solve the problem raised by Adhikari and Rath \cite{ar06}, Adhikari and Chen \cite{ac08}, Thangadurai \cite{th07} and Griffiths \cite{gr08}.Comment: 6page

Regulation of energy metabolism in skeletal muscle cells by PPARδ activation, in vitro exercise and perilipin 2 ablation

The prevalence of type 2 diabetes (T2D) has increased worldwide during the last decades. Obesity and accumulation of lipids in skeletal muscles are strongly associated with T2D. Much focus has been on the possibility of increasing lipid utilization by exercise or pharmacologically to avoid lipid accumulation in muscle. This thesis aimed to study regulation of energy metabolism related to obesity and T2D in cultured human myotubes by peroxisome proliferator-activated receptor δ (PPARδ) activation and in vitro exercise. The lipid droplet (LD)-associated protein, perilipin 2 (PLIN2), is one of the PPARδ target genes, and to study the functional role of PLIN2 and LDs in muscle energy metabolism we also examined myotube cultures established from PLIN2+/+ and PLIN2-/- mice. The results presented in this thesis suggest that myotubes to some extent retain the phenotype of their donors, and that responses to in vitro exercise reflected the in vivo characteristics of the donors. While both exercise in vitro, activation of PPARδ and lack of PLIN2 increased lipid oxidation, only in vitro exercise had any impact on the insulin-stimulated responses, whereas both PPARδ activation and PLIN2 ablation shifted the cells from glucose to lipid metabolism