Sheaves of spaces of generalized hyperfunctions $\mathcal{B}\mathcal{G}$ and algebras of megafunctions $\mathcal{M}\mathcal{G}$ are introduced. The first one is a flabby sheaf. Moreover, there exist injective sheaf homomorphisms $\mathcal{G} \rightarrow \mathcal{B}\mathcal{G}$ and $\mathcal{B}\mathcal{G} \rightarrow \mathcal{M}\mathcal{G},$ where $\mathcal{G}$ is the algebra of Colombeau generalized functions.

We study Noether's problem over $\mathbb{Q}$ for meta-cyclic groups. This paper is an extension of the previous work [2], which was concerned with the cyclic group ${C_n}$ of order $n$. We shall give a simple description of the action of the normalizer of $C_n$ in $S_n$ to the function field $\mathbb{Q}(x_1,\ldots,x_n)$, in terms of the generators of the fixed field of $C_n$ given in [2]. Using this, we settle Noether's problem for the dihedral group of order $2n$ $(n \leq 6)$ and the Frobenius group of order $20$ with explicit construction of independent generators of the fixed fields. We shall also reconstruct some simple one-parameter families of cyclic and dihedral polynomials.

We define a Nambu-Jacobi structure as a bracket of several functions which satisfies the Fundamental Identity. Then we express the Nambu-Jacobi structure in terms of two tensor fields and show the necessary and sufficient conditions that they should satisfy. We investigate the foliations associated with a Nambu-Jacobi structure. This allows us to give many examples of Nambu-Jacobi manifolds.

Let $(M,\omega)$ be a symplectic manifold and $L\subset M$ be a Lagrangian submanifold. In [Oh2], the cyclic condition of $L$ was defined. Y.-G. Oh proved that, in [Oh2], if $(M,\omega)$ is Kähler-Einstein with non-zero scalar curvature and $L$ is minimal, then $L$ is cyclic. In this article, first, we prove that $L$ is cyclic if and only if the ``mean cuvature cohomology class'' of $L$ is rational, when $(M,\omega)$ is Kähler-Einstein with non-zero scalar curvature. Secondly, we see that there are non-cyclic minimal Lagrangian submanifolds when $(M,\omega)$ is a prequantizable Ricci-flat Kähler manifold. Thirdly, if $(M,\omega)$ is Kähler-Einstein with non-zero scalar curvature, there are not minimal Lagrangian fibration structures on $M$ by a result of [Oh2]. Nevertheless we construct Hamiltonian minimal Lagrangian fibration.

We study a Kähler submanifold $M$ of a quaternionic Kähler manifold $\tilde {M}$. For such submanifold $M$ we construct a totally real and minimal submanifold ${\cal Z}$ in the twistor space $\tilde{\cal Z}$ of $\tilde {M}$.

The purpose of this paper is to show that the reduced Lefschetz module of the $G$-poset $\mathcal{B}_{p}^{cen}(G)$ consisting of all centric $p$-radical subgroups of a finite group $G$ is an $\mathcal{X}$-projective virtual $\mathbb{Z}_{p}[G]$-module where $\mathcal{X}$ is a family of $p$-subgroups of the normalizers of non-centric $p$-radical subgroups of $G$. As corollary, we have a lower bound of the $p$-power of the reduced Euler characteristic $\tilde{\chi}(\mathcal{B}_{p}^{cen}(G))$.

We give a correspondence among a Hamiltonian minimal Lagrangian cone in ${\mathbb C}^{m}$, a Legendrian minimal Legendrian submanifold in the unit sphere $S^{2m-1}(1)$ and a Hamiltonian minimal Lagrangian submanifold in the complex projective space ${\mathbb C}P^{m-1}$. As an application of this result, we prove that a Hamiltonian minimal Lagrangian cone in ${\mathbb C}^{m}$ such that the first Betti number of its link is 0 must be a special Lagrangian cone. Moreover, we construct Hamiltonian minimal (non-minimal) Lagrangian cones in ${\mathbb C}^{3}$ with a toroidal link, which are parametrized by a triple of relatively prime positive integers $(p,q,r)$, and discuss their Hamiltonian stabilities.

We calculate some invariants determined by the spectrum of the Jacobi operator $J$ of $n$-dimensional totally real submanifolds of the quaternionic projective space $QP^n$ and we use such invariants to characterize parallel submanifolds of $QP^n$.

We construct diffusion processes on disconnected selfsimilar fractal sets as limits of suitably scaled random walks. We also consider homogenization problems on disconnected fractal sets. We treat environments whose means are not only finite but infinite.

The continuity of the density function of the invariant probability measure for piecewise $C^2$ Bernoulli transformations is proved.

The $C^*$-algebras of continuous fields are enlarged and embedded into the associated direct products, and their stable rank and connected stable rank are estimated in terms of their base spaces and fibers. Using these estimates, we compute these ranks of $C^*$-algebras of continuous fields of elementary $C^*$-algebras, and those of group $C^*$-algebras of the discrete Heisenberg groups.

In this paper, we consider the partial survival and extinction of species in model governed by the following discrete model of nonautonomous Lotka-Volterra type: \[ \left\{ \begin{array}{@{}ll} N_i(p+1) = N_i(p) \exp\{ c_i(p)-\displaystyle \sum_{j=1}^n \displaystyle \sum_{l=0}^m a_{ij}^l(p) N_j(p-k_l) \}\,, \quad p \geq 0\,, \quad 1 \leq i \leq n\,, \\ N_i(p) = N_{ip} \geq 0\,, \quad p \leq 0\,, \quad \mbox{and} \quad N_{i0}>0\,, \quad 1 \leq i \leq n\,, \\ \end{array} \right. \] where each $ c_i(p) $ and $ a_{ij}^l(p) $ are bounded for $ p \geq 0 $ and \[ \displaystyle \sum_{l=0}^m (\displaystyle \inf_{p \geq 0} a_{ii}^l(p))>0\,,\quad \ a_{ij}^l(p) \geq 0\,, \quad i \leq j \leq n\,, \quad 1 \leq i \leq n\,, \quad k_l \geq 0\,, \quad 0 \leq l \leq m\,. \] To the above discrete system, we extend results on the {\it principle of competitive exclusion} in nonautonomous Lotka-Volterra differential systems which has been established by Shair Ahmad (1999, {\it Proceedings of the American Mathematical Society} {\bf 127}, 2905--2910), that is, if the coefficients satisfy certain inequalities, then any solution with positive components at some point will have all of its last $ n-1 $ components tend to zero, while the first one will stabilize at a certain solution of a discrete logistic equation.

In a symmetric space of non-compact type, we have recently defined the notion of a complex equifocal submanifold. In this paper, we introduce the notion of an infinite dimensional anti-Kaehlerian isoparametric submanifold. We show that the investigation of complete real analytic complex equifocal submanifolds is reduced to that of infinite dimensional anti-Kaehlerian isoparametric submanifolds. Also, we show that an infinite dimensional anti-Kaehlerian isoparametric submanifold is multi-foliated by complex spheres (or complex affine subspaces) and that the main part of the focal set of the submanifold at each point consists of some complex hyperplanes in the normal space.

Trees are constructed by one--dimensional piecewise linear transformations, and the Hausdorff dimension of trees are calculated.

We study the $\lambda$-invariant of the cyclotomic $\mathbf{Z}_2$-extension of $\mathbf{Q}(\sqrt{pq})$ with $p\equiv 3\pmod{8}$, $q\equiv 1\pmod{8}$ and $\bigl(\frac{q}{p}\bigr)=-1$. With further conditions on $q$, we show that $\lambda$-invariant is zero.

Let $G$ be a connected reductive algebraic group defined over $\mathbb{F}_q$ with Lie algebra $\mathcal{G}$. We define a Deligne-Lusztig induction for the $\overline{\mathbb{Q}}_{\ell}$-valued functions on $\mathcal{G}(\mathbb{F}_q)$ which are invariant under the adjoint action of $G(\mathbb{F}_q)$ on $\mathcal{G}(\mathbb{F}_q)$, by making use of the ``character formula'' where the ``two-variable Green functions'' are defined via a $G$-equivariant homeomorphism $\mathcal{G}_{nil}\rightarrow G_{uni}$. We verify that it satisfies properties analogous to the group case like transitivity, the Mackey formula or the commutation with duality. The interest of a Deligne-Lusztig induction for invariant functions comes from a conjecture on a commutation formula with Fourier transforms which has no counterpart in the group case. In a forthcoming paper, this conjecture will be proved in almost all cases.

It is introduced a split extension of groups $1\rightarrow P_2\rightarrow C_{1,2}\rightarrow \Theta\rightarrow 1$, where $P_2$ is the group of pure braids in 2 strings, $C_{1,2}$ is the group of cobordism classes of (pure) 2-string links and $\Theta$ is the group of cobordism classes of theta curves. The concept of boundary theta curve is introduced and it is proved that the group of boundary cobordism classes of boundary theta curves is isomorphic to the group of boundary cobordism classes of boundary string links in 2 strings.

For a non-trivial Kähler magnetic field on a Kähler manifold, we consider bow-shapes as substitutions for triangles on a Riemannian manifold. We give a comparison theorem for bow-shapes on a manifold whose sectional curvature is bounded from above.