By using the theory of Lie algebras associated with a standard quadruplet, we can embed an arbitrary reductive prehomogeneous vector space with completely reducible representation into some graded Lie algebra. The purpose of this paper is to study properties of graded Lie algebras which correspond to the prehomogeneity condition of triplets. Moreover, we give another proof of castling transformation as an application.

Let $\{G_{}\}^n_{i=1}$ be locally compact groups and $\mathscr{H}$ be Hilbert space. We define the n-variable Fourier-Stieltjes algebra $B(\Pi^{n}_{1} G_{i}, \mathrm{B}(\mathscr{H}))$ consists all functions \[ \phi : G_{1} \times \cdot \times G_{n} \rightarrow \mathrm{B}(\mathscr{H}) \] for which there exists unitary representations $\pi_{i} : \mathbb{G}_{i} \rightarrow \mathrm{B}(\mathscr{H}_i)$ and a diagram of bounded operators \[ \mathscr{H} \rightarrow^{V} \mathscr{H}_n \rightarrow^{T_{n-1}} \mathscr{H}_{n-1} \rightarrow^{T_{1}} \mathscr{H_1} \rightarrow^{U} \mathscr{H} \] with \[ \phi(s_{1},..., s_{n}) = U\pi_{1}(s_1)T_{1}\pi_{2})s_{2}) \cdot \pi_{n-1}(s_{n-1})T_{n}\pi_{n}(s_{n}V \] We extend the pointwise product on $B(\Pi^{n}_{1} G_{i}, \mathrm{B}(\mathscr{H}))$ under which it forms a completely contractive commutative unital Banach algebra. A diagram of its subalgebras will be introduced.

Let $X$ be a compact metrizable space and $Y$ be a nondegenerate dendrite with an end point 0. For each continuous function $f : X \rightarrow Y$, we define the hypo-graph $\downarrow f = \cup_{x \in X \{x} \times [0, f(x]$ of $f$, where $[0, f(x)]$ is the unique path from 0 to $f(x)$ in $Y$. Then we can regard $\downarrow \mathrm{C}(X,Y) = \{\downarrow f | f : X \rightarrow Y$ is continuous} as a subspace of the hyperspace consisting of non-empty closed sets in $X \times Y$ equipped with the Vietoris topology. In this paper, we prove that $\downarrow \mathrm{C}(X,Y)$ is a Baire space if and only if the set of isolated points of $X$ is dense.

Previously, several families of systems of differential equations that generalize the Heisenberg Laplacian equations were introduced. The study of one of these families is continued here. It is shown that the systems in this family are free of integrability conditions provided that a parameter appearing in the system avoids a certain set of bad values, which is explicitly determined. Properties of polynomial solutions to the systems are investigated and special polynomial solutions involving terminating Lauricella hypergeometric series are given in some cases.

We try to take a first step to a theory of the structural properties of ideals over $\mathscr P_{\kappa}\lambda$, that was studied in detail by Baumgartner, Taylor and Wagon [1] for $\kappa$;. In defining the basic notions, P-points, Q-points, and selective ideals, we put importance on the behavior of the function on $\mathscr P_{\kappa}\lambda$ to the bounded ideal and Rudin-Keisler ordering.

Several facts hold similarly as on $\kappa$;, for instance, the bounded ideal is a nowhere Q-point. However some differences exist such as the bounded ideal is isomorphic to another ideal. We state the sufficient condition for ideals to be Q-points and the weakly normal ideals selective.

The aim of this paper is to give a characterization of the tempered distributions supported by a (Whitney's) regular closed set in the Euclidean space and the Heisenberg group by means of the heat kernel method. The heat kernel method, introduced by T. Matsuzawa, is the method to characterize the generalized functions on the Euclidean space by the initial value of the solutions of the heat equation.

In this paper we investigate the Cauchy problem for hyperbolic operators with double characteristics and hyperbolic operators of third order whose coefficients depend only on the time variable. And we give sufficient conditions for $C^{\infty}$ well-posedness.

The review on [1] in Mathematical Reviews points out that the proof of its main result is incorrect. The aim of this paper is to correct the previous paper's argument and clarify the statement.