We investigate a poly-modal logic $S{5}_{n}C$ which has $n$-modalities satisfying the axioms of $S5$ and has axioms expressing permutability of modalities. We show that the logic $S{5}_{n}C$ is complete concerning Kripke semantics, has the finite model property and is decidable, however we prove $S{5}_{n}C$ is not locally finite. A main result consists of an algorithmic criterion for recognizing the admissibility of inference rules in $S{5}_{n}C$, i.e. the logic $S{5}_{n}C$ is decidable with respect to the admissibility.

Boundary integral equations corresponding to the differential equations describing a transient flow of incompressible viscous fluid in three dimensions are considered. Emphasis is put on the treatment of edges and corners. The boundary $\text{Γ}$ is assumed piecewise Lyapunov surface and the interior solid angle $\text{Θ}(x)$ at the non-smooth boundary point $x$ must satisfy the inequality

Corresponding to the Dirichlet problem of the Navier-Stokes equations, the following series of Volterra integral equations of the first kind for unknown tractions ${\sigma }_j^{(n)}(j=1,2,3:n=0,1,2,\dots )$ is derived.

$$G{\sigma }_j^{(n)}(x,t)={\displaystyle {\int }_t^0{\displaystyle {\int }_{\mathrm{\Gamma }}{\sigma }_i^{(n)}(y,\tau )U_{ij}^{*}(y,\tau ;x,t)dS(y)d\tau =b_j^{(n)}(x,t)}},$$

where $U_{ij}^{*}$ are components of the Stokes fundamental solution tensor and $b_j^{(n)}$ can be regarded as given functions. The integral $G{\sigma }_j^{(n)}$ is the single layer potential. The integral involved in the definition of $b_j^{(n)}$ (see the text) is the double layer potential. Those integrals are shown to be weakly singular for the non-smooth domain under consideration. It is proved that, with ${\sum }^{\text{}}=\mathrm{\Gamma }\times [0,T]$, the operator

$$G:H^{-\frac{1}{2},-\frac{1}{4}}(\sum )\to H^{\frac{1}{2},\frac{1}{4}}(\sum )$$

is coercive;

$${((G\sigma ,\sigma ))}_{L^2(\sum )}\ge \beta |||\sigma ||{|}_{H^{-\frac{1}{2},-{\frac{1}{4}}_{(\sum )}}}^2$$

with a constant , $\sigma =({\sigma }_1,{\sigma }_2,{\sigma }_3)$.

An enhanced algorithm and the computational results on the classification of two-symbol orthogonal arrays of strength 2, size 24, 6 constraints and index 6 derivable from 130 representatives of saturated orthogonal arrays having maximal constraints are given. A table of 1,317 representative 2-OA(2,6,6)’s obtained is given by indicating the selected set of 6 columns of a representative saturated array of 23 constraints each.

Sufficient conditions are given for the fractional maximal operator to send a weighted Orlicz class into another one. As an application, an Orlicz version of the famous Fefferman-Stein inequality is obtained.

In this paper, we obtain a sufficient condition for a foliation $ℱ$ with an Ehresmann connection $\mathcal{D}={(\text{T}ℱ)}^{\perp }$ on a compact, connected manifold $M$ to have leaves whose universal covers have the same growth type. When such a foliation has a compact leaf $L$, we show that the growth type of all leaves is bounded from above by the growth types of ${\pi }_{1}L$ and $H_{\mathcal{D}}(L)$.

In this paper, we investigate sufficient conditions in order that a family $T\left(\epsilon \right)=T_0+\epsilon T_1$ of closable linear operators with domain $D\left(T\left(\epsilon \right)\right)=D\left(T_0\right){\cap }^{\text{}}D\left(T_1\right)$ converge to $T_0$ as $\epsilon \downarrow 0$ in the sense of uniform and strong resolvent convergence. The obtained abstract results are applied to selfadjoint and nonselfadjoint Schrödinger operators.

Let $V$ be an $n$-dimensional non-singular algebraic integral scheme over a perfect field $k$ of characteristic and $K$ its algebraic function field. In this paper, we will prove the following:

Theorem B. Let $\omega$ be a differential form in $Z_{\infty }(K/k)$. Then the following three conditions are equivalent:

The above theorem is a generalization of the main theorem in Nakakoshi[5]. He proved the theorem in case of degree$(\omega )=n$.

In this paper, we show that the regularity of solutions to wave equation with a non smooth coefficient propagates through the points at which the coefficient is singular.

The purpose of this paper is to show the existence of unique global $C^1$-solutions to the time-dependent complex Ginzburg-Landau equation. We regard the equation as a genuinely nonlinear equation and simultaneously as a semilinear equation.