On the basis of Dupont's work, we exhibit a cocycle in the simplicial de Rham complex which represents the Chern character. We also prove the related conjecture due to Brylinski. This gives a way to construct a cocycle in a local truncated complex.

We study graph homomorphisms over finite graphs from a viewpoint of reproducing kernel Hilbert space theory. In particular, introducing de Branges-Rovnyak theory into graph theory, the relation between injective graph homomorphisms and de Branges-Rovnyak spaces is discussed in detail.

Let $M$ be a shift invariant subspace in the two variable Hardy space $H^2(\Gamma_z\times\Gamma_w)$. We study $\mathcal{M}(M_z)=\{\phi\in H^\infty(\Gamma_z\times \Gamma_w) : \phi M_z\subseteq M_z\}$ where $M_z=M\ominus zM$. We give several sufficient conditions for $\mathcal{M}(M_z)=H^\infty(\Gamma_w)$ where $H^\infty(\Gamma_w)$ is the one variable Hardy space.

We provide a mathematical model of $n$-membered straight-chain hydrocarbon molecules. The configuration space of the model is parametrized by chain lengths. By assuming the bond angle conditions required for hydrocarbon molecules, we determine the topological types of fibers of the configuration space of the model by chain lengths when $n = 5$.

Some new inequalities for the functional \begin{align*} & E_{T}\left( f,u\right) \\ & :=f\left( b\right) \left( u\left( b\right) -\frac{1}{b-a}% \int_{a}^{b}u\left( t\right) dt\right) +f\left( a\right) \left( \frac{1}{b-a}% \int_{a}^{b}u\left( t\right) dt-u\left( a\right) \right) \\ & -\int_{a}^{b}f\left( t\right) du\left( t\right) , \end{align*} under various assumptions for the functions $f$ and $u$ are given. Applications for functions of selfadjoint operators and unitary operators on complex Hilbert spaces are also provided.