We consider a scaling limit for the Derezi\'nski-G\'erard model. We derive an effective potential by taking a scaling limit for the total Hamiltonian of the Derezi\'nski-G\'erard model. Our method to derive an effective potential is independent of whether or not the quantum field has a nonnegative mass. As an application of our theory developed in the present paper, we derive an effective potential of the Nelson model.

In this paper we will prove that there exists a time periodic solution of the Navier-Stokes equations with the inhomogeneous boundary condition for infinite symmetric channels in $\R^2$. In two and three dimensional more generalized infinite channels (than treated in this paper) H.~Beir\~ao~da Veiga \cite{Beirao} proved that there exists time periodic solutions of the Navier-Stokes equations with the homogeneous boundary condition under a small time periodic flux. G.~P.~Galdi and A.~M.~Robertson \cite{GalRob} obtained time-periodic Poiseuille flow in a straight channel with a smooth cross section. C.~J.~Amick \cite{Amick2} proved that in two and three dimensional unbounded channels there exists solutions of the stationary Navier-Stokes equations with the nonhomogenous boundary condition. H.~Morimoto and H.~Fujita \cite{Morimoto1} and H.~Morimoto \cite{Morimoto2} proved that in a two dimensional certain unbounded symmetric channel there exists symmetric solutions of the stationary Navier-Stokes equations with a special symmetric Dirichlet boundary condition. T-P.~Kobayashi \cite{Kobayashi3} demonstrated that for two and three dimensional infinite channels time periodic solutions of the Navier-Stokes equations exist under the same condition as C.~J.~Amick \cite{Amick2}. In this paper using the condition of H.~Morimoto and H.~Fujita \cite{Morimoto1} and H.~Morimoto \cite{Morimoto2}, we obtain time priodic solutions.

In this paper we consider generalized spin-boson models with singular perturbations. It is proven that under the infrared regularity condition Hamiltonians have the unique ground state for sufficiently small values of coupling constants. In addition it is shown that the asymptotic creation and annihilation operators of massless boson field exist.

First we give the new elementary proof of the structure equations of $G_2$ and the congruence theorem of hypersurfaces of the purely imaginary octonions $\ImO$ under the action of $G_2$. Next, we classify almost complex structures of homogeneous hypersurfaces of $\ImO$ into 4-types.

We prove that a transversely equicontinuous minimal lamination on a locally compact metric space $Z$ has a transversely invariant nontrivial Radon measure. Moreover if the space $Z$ is compact, then the tranversely invariant Radon measure is shown to be unique up to a scaling.