We prove $L^p-L^q$ maximal regularity estimates for the Stokes equations in spatial regions with moving boundary. Our result includes bounded and unbounded regions. The method relies on a reduction of the problem to an equivalent nonautonomous system on a cylindrical space-time domain. By applying suitable abstract results for nonautonomous Cauchy problems we show maximal regularity of the associated propagator which yields the result. The abstract results, also proved in this note, are a modified version of a nonautonomous maximal regularity result of Y. Giga, M. Giga, and H. Sohr and a suitable perturbation result. Finally we describe briefly the application to the special case of rotating regions.

In this paper, we prove that the holomorphic automorphism groups of the spaces ${\mathbf{C}}^k\times ({\mathbf{C}}^{*}{)}^{n-k}$ and $({\mathbf{C}}^k-\{0\left\}\right)\times ({\mathbf{C}}^{*}{)}^{n-k}$ are not isomorphic as topological groups. By making use of this fact, we establish the following characterization of the space ${\mathbf{C}}^k\times ({\mathbf{C}}^{*}{)}^{n-k}$: Let $M$ be a connected complex manifold of dimension $n$ that is holomorphically separable and admits a smooth envelope of holomorphy. Assume that the holomorphic automorphism group of $M$ is isomorphic to the holomorphic automorphism group of ${\mathbf{C}}^k\times ({\mathbf{C}}^{*}{)}^{n-k}$ as topological groups. Then $M$ itself is biholomorphically equivalent to ${\mathbf{C}}^k\times ({\mathbf{C}}^{*}{)}^{n-k}$. This was first proved by us in [5] under the stronger assumption that $M$ is a Stein manifold.

In this paper we classify all configurations of singular fibers of elliptic fibrations on the double cover of ${\mathbf{P}}^2$ ramified along six lines in general position.

An integral representation of the $p$-series of odd $p$ is shown;$$\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^{2p+1}}=(-1{)}^p\frac{(2\pi {)}^{2p}}{\left(2p\right)!}{\int }_0^1{B}_{2p}(t\left)\log \right(\sin \pi t\left)\mathrm{d}t\right(p=1,2,\dots ),$$where $B_{2p}\left(t\right)$ is a Bernoulli polynomial of degree $2p$. As a consequence of this we have $$\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^{2p+1}}=(-1{)}^p\frac{(2\pi {)}^{2p}}{\left(2p\right)!}2\left[\sum _{k=0}^p\left(\begin{array}{c}2p\\ 2k\end{array}\right)B_{2p-2k}\left(\frac{1}{2}\right)b_{2k}\right],$$where $b_{2k}={\int }_0^{\frac{1}{2}}{t}^{2k}\log \left(\cos \pi t\right)\mathrm{d}t$, $k=0,1,\dots ,p$.

Let $a$ be a positive integer which is not a perfect $b$-th power with $b\ge 2$, $q$ be a prime number and $Q_a(x;q^i,j)$ be the set of primes $p\le x$ such that the residual order of $a\left(\mathrm{m}\mathrm{o}\mathrm{d}p\right)$ in $(\mathbf{Z}/p\mathbf{Z}{)}^{\times }$ is congruent to $j$ modulo $q^i$. In this paper, which is a sequel of our previous papers [1] and [6], under the assumption of the Generalized Riemann Hypothesis, we determine the natural densities of $Q_a(x;q^i,j)$ for $i\ge 3$ if $q=2$, $i\ge 1$ if $q$ is an odd prime, and for an arbitrary nonzero integer $j$ (the main results of this paper are announced without proof in [3], [7] and [2]).

In this paper we classify the singular fibres of stable maps of closed (possibly non-orientable) 4-manifolds into 3-manifolds up to the $C^{\mathrm{\infty }}$ equivalence. Furthermore, we obtain several results on the co-existence of the singular fibres of such maps. As a consequence, we show that under certain conditions, the Euler number of the source 4-manifold has the same parity as the total number of certain singular fibres. This generalises Saeki's result in the orientable case.

We establish the existence, uniqueness and $L^q$ estimates of weak solutions to the stationary Stokes equations with rotation effect both in the whole space and in exterior domains. The equation arises from the study of viscous incompressible flows around a body that is rotating with a constant angular velocity, and it involves an important drift operator with unbounded variable coefficient that causes some difficulties.

We present a very simple proof of Lou's fixed point theorem in a space of continuous mappings[Proc. Amer. Math. Soc., 127 (1999), 2259--2264]. We also discuss another similar fixed point theorem.

A spherical $t$-design is a finite subset $X$ in the unit sphere $S^{n-1}\subset {\mathbf{R}}^n$which replaces the value of the integral on the sphere of any polynomial of degree at most $t$ by the average of the values of the polynomial on the finite subset $X$. Generalizing the concept of spherical designs, Neumaier and Seidel (1988) defined the concept of Euclidean $t$-design in ${\mathbf{R}}^n$ as a finite set $X$ in ${\mathbf{R}}^n$ for which ${\sum }_{i=1}^p\left(w\right(X_i)/(\left|S_i\right|\left)\right){\int }_{S_i}f\left(x\right)d{\sigma }_i\left(x\right)={\sum }_{x\in X}w\left(x\right)f\left(x\right)$ holds for any polynomial $f\left(x\right)$ of $\mathrm{deg}\left(f\right)\le t$, where $\{S_i,1\le i\le p\}$ is the set of all the concentric spheres centered at the origin and intersect with $X$, $X_i=X\cap S_i$, and $w:X\to {\mathbf{R}}_{>0}$ is a weight function of $X$. (The case of $X\subset S^{n-1}$ and with a constant weight corresponds to a spherical $t$-design.) Neumaier and Seidel (1988), Delsarte and Seidel (1989) proved the (Fisher type) lower bound for the cardinality of a Euclidean $2e$-design. Let $Y$ be a subset of ${\mathbf{R}}^n$ and let ${𝒫}_e\left(Y\right)$ be the vector space consisting of all the polynomials restricted to $Y$ whose degrees are at most $e$. Then from the arguments given by Neumaier-Seidel and Delsarte-Seidel, it is easy to see that $\left|X\right|\ge \dim \left({𝒫}_e\right(S\left)\right)$ holds, where $S={\cup }_{i=1}^p{S}_i$. The actual lower bounds proved by Delsarte and Seidel are better than this in some special cases. However as designs on $S$, the bound $\dim \left({𝒫}_e\right(S\left)\right)$ is natural and universal. In this point of view, we call a Euclidean $2e$-design $X$ with $\left|X\right|=\dim \left({𝒫}_e\right(S\left)\right)$ a tight $2e$-design on $p$ concentric spheres. Moreover if $\dim \left({𝒫}_e\right(S\left)\right)=\dim \left({𝒫}_e\right({\mathbf{R}}^n\left)\right)(=\left(\begin{array}{c}n+e\\ e\end{array}\right))$ holds, then we call $X$ a Euclidean tight $2e$-design. We study the properties of tight Euclidean $2e$-designs by applying the addition formula on the Euclidean space. Furthermore, we give the classification of Euclidean tight 4-designs with constant weight. It is possible to regard our main result as giving the classification of rotatable designs of degree 2 in ${\mathbf{R}}^n$in the sense of Box and Hunter (1957) with the possible minimum size $\left(\begin{array}{c}n+2\\ 2\end{array}\right)$. We also give examples of nontrivial Euclidean tight 4-designs in ${\mathbf{R}}^2$ with nonconstant weight,which give a counterexample to the conjecture of Neumaier and Seidel (1988) that there are no nontrivial Euclidean tight $2e$-designs even for the nonconstant weight case for $2e\ge 4$.

We consider the Cauchy problem for the damped wave equation with absorption$$u_{tt}-\mathrm{\Delta }u+u_t+|u{|}^{\rho -1}u=0,(t,x)\in {\mathbf{R}}_{+}\times {\mathbf{R}}^N,(*)$$with $N=3,4$. The behavior of $u$ as $t\to \mathrm{\infty }$ is expected to be the Gauss kernel in the supercritical case $\rho >{\rho }_c\left(N\right):=1+2/N$. In fact, this has been shown by Karch [12] (Studia Math., 143 (2000), 175--197) for $\rho >1+\frac{4}{N}(N=1,2,3)$, Hayashi, Kaikina and Naumkin [8] (preprint (2004)) for $\rho >{\rho }_c\left(N\right)(N=1)$ and by Ikehata, Nishihara and Zhao [11] (J. Math. Anal. Appl., 313 (2006), 598--610) for and . Developing their result, we will show the behavior of solutions for , . For the proof, both the weighted $L^2$-energy method with an improved weight developed in Todorova and Yordanov [22] (J. Differential Equations, 174 (2001), 464--489) and the explicit formula of solutions are still usefully used. This method seems to be not applicable for $N=5$, because the semilinear term is not in $C^2$ and the second derivatives are necessary when the explicit formula of solutions is estimated.

Recently N. Kumano-go [15] succeeded in proving that piecewise linear time slicing approximation to Feynman path integral $$\int F\left(\gamma \right)e^{i\nu S\left(\gamma \right)}𝒟\left[\gamma \right]$$actually converges to the limit as the mesh of division of time goes to $0$ if the functional $F\left(\gamma \right)$ of paths $\gamma$ belongs to a certain class of functionals, which includes, as a typical example, Stieltjes integral of the following form; $$F\left(\gamma \right)={\int }_0^{T}f(t,\gamma (t\left)\right)\rho \left(dt\right),\left(1\right)$$ where $\rho \left(t\right)$ is a function of bounded variation and $f(t,x)$ is a sufficiently smooth function with polynomial growth as $\left|x\right|\to \mathrm{\infty }$. Moreover, he rigorously showed that the limit, which we call the Feynman path integral, has rich properties (see also [10]).

The present paper has two aims. The first aim is to show that a large part of discussion in [15] becomes much simpler and clearer if one uses piecewise classical paths in place of piecewise linear paths.

The second aim is to explain that the use of piecewise classical paths naturally leads us to an analytic formula for the second term of the semi-classical asymptotic expansion of the Feynman path integrals under a little stronger assumptions than that in [15]. If $F\left(\gamma \right)\equiv 1$, this second term coincides with the one given by G. D. Birkhoff [1].

In this paper, we study a new class of affine minimal hypersurfaces as higher dimensional analogues of affine minimal ruled surfaces.

Let $p$ be an odd prime number and $F$ a number field. Let $K=F\left({\zeta }_p\right)$ and $\mathrm{\Delta }=\mathrm{G}\mathrm{a}\mathrm{l}(K/F)$. Let ${𝒮}_{\mathrm{\Delta }}$ be the Stickelberger ideal of the group ring $\mathbf{Z}\left[\mathrm{\Delta }\right]$ defined in the previous paper [8]. As a consequence of a $p$-integer version of a theorem of McCulloh [15], [16], it follows that $F$ has the Hilbert-Speiser type property for the rings of $p$-integers of elementary abelian extensions over $F$ of exponent $p$ if and only if the ideal ${𝒮}_{\mathrm{\Delta }}$ annihilates the $p$-ideal class group of $K$. In this paper, we study some properties of the ideal ${𝒮}_{\mathrm{\Delta }}$,and check whether or not a subfield of $\mathbf{Q}\left({\zeta }_p\right)$ satisfies the above property.

In this paper, we study Möbius characterizations of submanifolds without umbilical points in a unit sphere $S^{n+p}\left(1\right)$. First of all, we proved that, for an $n$-dimensional $(n\ge 2)$ submanifold $\mathbf{x}:M\mapsto S^{n+p}\left(1\right)$ without umbilical points and with vanishing Möbius form $\mathrm{\Phi }$, if $(n-2)\left|\right|\tilde{\mathbf{A}}\left|\right|\le \sqrt{\frac{n-1}{n}}\left\{nR-\frac{1}{n}\left[\right(n-1)\left(2-\frac{1}{p}\right)-1]\right\}$ is satisfied, then, $\mathbf{x}$ is Möbius equivalent to an open part of either the Riemannian product $S^{n-1}\left(r\right)\times S^1\left(\sqrt{1-r^2}\right)$ in $S^{n+1}\left(1\right)$, or the image of the conformal diffeomorphism $\sigma$ of the standard cylinder $S^{n-1}\left(1\right)\times \mathbf{R}$ in ${\mathbf{R}}^{n+1}$, or the image of the conformal diffeomorphism $\tau$ of the Riemannian product $S^{n-1}\left(r\right)\times {\mathbf{H}}^1\left(\sqrt{1+r^2}\right)$in ${\mathbf{H}}^{n+1}$, or $\mathbf{x}$ is locally Möbius equivalent to the Veronese surface in $S^4\left(1\right)$. When $p=1$, our pinching condition is the same as in Main Theorem of Hu and Li [6], in which they assumed that $M$ is compact and the Möbius scalar curvature $n(n-1)R$ is constant. Secondly, we consider the Möbius sectional curvature of the immersion $\mathbf{x}$. We obtained that, for an $n$-dimensional compact submanifold $\mathbf{x}:M\mapsto S^{n+p}\left(1\right)$ without umbilical points and with vanishing form $\mathrm{\Phi }$, if the Möbius scalar curvature $n(n-1)R$ of the immersion $\mathbf{x}$ is constant and the Möbius sectional curvature $K$ of the immersion $\mathbf{x}$ satisfies $K\ge 0$ when $p=1$ and $K>0$ when $p>1$. Then, $\mathbf{x}$ is Möbius equivalent to either the Riemannian product $S^k\left(r\right)\times S^{n-k}\left(\sqrt{1-r^2}\right)$, for $k=1,2,\cdots ,n-1$, in $S^{n+1}\left(1\right)$; or $\mathbf{x}$ is Möbius equivalent to a compact minimal submanifold with constant scalar curvature in $S^{n+p}\left(1\right)$.