We consider the Navier-Stokes equations in half-space and a perturbed half-space in $L^p$ space with Muckenhoupt weight. As the first step, we shall describe the Helmholtz decompostion of $L^p$ space with Muckenhoupt weight and the weighted resolvent estimates for the Stokes equations. Next we shall show the $L^p - L^q$ estimates of Stokes semigroup with $\langle{x}\rangle^s$ weight. Finally as the application of the weighted $L^p - L^q$ estimates, we shall obtain the weighted asymptotic behavior of the solution to the Navier-Stokes equations.

We study slicing functions, which are called direct connections in the smooth category, and parallel displacements along sequences in a topological connection theory. We define holonomy groups for such parallel displacements, and prove a holonomy reduction theorem and related results. In particular, we study a category of principal bundles with parallel displacements over a fixed base space. Assuming the existence of an initial object of a category of principal $G$-bundles, we obtain a classification theorem of topological principal $G$-bundles in terms of topological group homomorphisms. It is shown that a certain object is an initial object if it is the holonomy reduction of itself with respect to the identification topology. The result is applied to the universal bundle over a countable simplicial complex constructed by Milnor.

We explicitly describe the Cartier dual of the $l$-th Frobenius kernel $N_1$ of the group scheme $\mathscr{G}^\lambda$, which deforms $\mathbf{G}_a$ to $\mathbf{G}_m$. Then the Cartier dual of $N_1$ is given by a certain Frobenius type kernel of the Witt scheme. Here we assume that the base ring $A$ is a $\mathbf{Z}_{(p)} / (p^n)$-algebra, where $p$ is a prime number. The obtained result generalizes a previous result by the author [1] which assumes that $A$ is an $\mathbf{F}_p$--algebra.

In [4], the author introduced the system $\lambda\rho$-calculus and stated without proof that the strong normalization theorem holds. Here we introduce a lemma (Lemma 4.10) and use it to prove the strong normalization theorem. While a typed $\lambda$-term itself is a derivation of the natural deduction for intuitionistic implicational logic (cf. [2]), a typed $\lambda\rho$-term itself is a derivation of the natural deduction for classical implicational logic. Our system is simpler than the implicational fragment of Parigot's $\lambda\mu$-calculus (cf. [5]).

Compactly supported orthonormal wavelets are often used in numerical analysis. However, since these functions have not an explicit formula in the time domain, the difficulty of integrations often occurs. In this paper, we introduce the Galerkin method with Deslauriers-Dubuc interpolating scaling functions, and we use the biorthogonality of the wavelets to overcome the difficulties of integration. We present numerical results that show the efficiency and accuracy of this method.

In this paper the helicoidal surfaces in the 3-dimensional Lorentz-Minkowski space are classified under the condition $\Delta^{III}r=Ar$, where $Ar$, is a real $3 \times 3$ matrix and $\Delta^{III}$ is the Laplace operator with respect to the third fundamental form.