We are concerned with the splittability problem of degenerations with stellar singular fibers. In this paper we give an interesting splitting criterion for such degenerations: if a stellar singular fiber has exactly three branches, and its central component (core) is the projective line, then this degeneration admits a splitting deformation.

Existence of a time-periodic solution to a non-linear wave equation with resonance is established by a variational method. We consider the $2\pi$-periodic weak solution to a wave equation $\Box u(x,t)=h(x,t,u(x,t))$ of space dimension 1, where $h(x,t,\xi)$ is asymptotically linear in $\xi$ both as $\xi\to0$ or $\xi\to\infty$, with the co-efficient as $\xi\to\infty$ belonging to $\sigma(\Box)$. It is proved that there are some cases, where the difference of $h(t,x,\xi)$ from its linear approximation is not bounded, that guarantee the existence of a non-trivial weak solutions. The proof is based on local linking theory and $({\it WPS})^*$ condition for the existence of a non-trivial critical point of a functional.

In this paper we consider a projective model for the time- and spacelike ultrahyperbolic unit balls in the orthogonal space $\mathbf{R}^{m,m}$. By means of an associated principal fibre bundle, a Dirac operator on these mani\-folds is defined and its fundamental solution is constructed (in case $m \in 2\mathbf{N} + 1$) with the aid of generalized Riesz distributions. Using the method of descent, we then construct fundamental solutions for the Dirac operator on time- or spacelike ultrahyperbolic unit balls in spaces of signature $(m,q)$ and $(p,m)$ respectively (with $p,q < m$).

We investigate the multiple Hurwitz zeta function $\zeta_n(s_1, \ldots, s_n;a)$, in particular those values at non-positive integers. Then, as an application, we give a generalization of Lerch's formula.

We construct dual objects for quantum complex projective spaces as quantum homogeneous spaces of quantum unitary groups, in which the deformation parameters are antisymmetric matrices. We prove the splitting formula and the nondegeneracy of the Hochschild dimensions for the quantum complex projective spaces.

Let $G$ be a $C^{\infty }$-mapping from a connected Riemann surface $M$ into the complex quadric $Q_{n-1}$ in the $n$-dimensional complex projective space. We give a condition for the existence of a surface in the $n$-dimensional Euclidean unit sphere $S^{n}$ such that the Gauss map is $G$. Under this condition, if $M$ is a torus, there exists a surface in $S^{n}$ such that the Gauss map is $G$. We also show that for a connected Riemann surface $M$ there exists an immersion $X:M\rightarrow RP^{n}$ such that a neighborhood of each point of $X(M)$ is covered by a surface in $S^{n}$ with prescribed Gauss map $G$ where $RP^{n}$ is the $n$-dimensional real projective space.

For an operator $T$ with some trace class condition, let $g_{T^n}$ and $g_{{T^n}}^P$ be the principal functions related to the Cartesian decomposition $T^n=X_n+iY_n$ and the polar decomposition $T^n = U_n|T^n|$ for a positive integer $n$, respectively. In this paper, we study properties of $g_{T^n}$ and $g_{T^n}^P$ and invariant subspaces of $T^n.$

The block number $Bl(M)$ introduced in our previous paper is a new topological invariant of a closed orientable 3-manifold $M$ which estimates a combinatorial complexity of $M$ just like the Heegaard genus $HG(M)$. In our previous paper, we have shown an inequality $HG(M) \leq Bl(M)$ for any $M \ne S^2 \times S^1$. In this paper, we will show that $Bl(M)=HG(M)$ for any $M$ with $HG(M)=2$ and moreover that $Bl(M) \leq 4$ for any $M$ with $HG(M)=3$.

Let $\phi$ be a bounded function on $[0,\infty)$ continuous except on a null set, and $\phi_{\epsilon}(\xi)=\phi(\epsilon\xi)\ (\epsilon>0).$ Also let $\tilde{T}_{\epsilon}$ be the operator on Jacobi series such that $(\tilde{T}_{\epsilon}f)^{\wedge}(n)=\phi_{\epsilon}(n)\hat{f}(n)\ (n\in{\bf Z})$, where $\hat{f}(n)$ is the coefficient of Jacobi expanstion of $f$, and ${\cal H}_{\alpha}(Tf)(\xi)$ be defined by $\phi(\xi){\cal H}_{\alpha}f(\xi)\ (\xi\in(0,\infty))$, where ${\cal H}_{\alpha}f$ is the modified Hankel transform of $f$ with order $\alpha$. Then the author [7] proved that if the operator norm of $\tilde{T}_{\epsilon}$ is uniformly bounded for all $\epsilon>0$, $T$ is a bounded operator on the modified Hankel transforms in the Lorentz spaces, and we have the maximal type theorem in the Lorentz spaces, respectively. In this paper, we give a generalized definition of the modified Hankel transform and the Hankel transform, and prove a generalization of the results in [7].

A mathematical model of environmental fluid is presented to describe fluid flow motions with large density variations. Moreover the associated numerical methods are discussed. The model of environmental fluid is formulated as an unsteady low-Mach number flow based on the compressible Navier-Stokes equations. For low-Mach number flows, the acoustic effects are assumed to be weak relative to the advection effects. Under this assumption, detailed acoustic effects can be removed from governing equations. The low-Mach number formulation thus enables numerical flow analysis with a projection methodology that uses high-order accurate upwind difference of the convection terms with a time step restricted solely by an advection Courant-Friedrichs-Lewy (CFL) condition. The algorithm presented here is based on an iterative implicit time evolution of second order accuracy and a high-accurate spatial discretization with TVD properties for unsteady low-Mach number flows. It is seen from the results on the verification for test cases of flows with a wide range of density variations that our numerical method is validated.

In a complex projective 3-space, we consider a domain with a projective line. If there is a compact non-singular quotient of the domain and the quotient manifold admits a non-constant meromorphic function, then the domain is dense in the projective 3-space and its complement is properly contained in a finite union of complex hypersurfaces and a set with Hausdorff dimension not more than two. Further, if the complement admits a certain fiber space structure, then it is either a disjoint union of two projective lines, a projective line, or an empty set.

In this article, we study strong triviality of two-bridge links. We prove that every (non-trivial) two-bridge link can not be strongly $n$-trivial for $n\geq 1$.

We study some conditions that the Iwasawa $\lambda$-, $\mu$-invariants of the the cyclotomic $\mathbb{Z}_{2}$-extension of $k = \mathbb{Q}(\sqrt{pq})$ with $p \equiv 7 \pmod{8}, q \equiv 1 \pmod{8}, (\frac{p}{q}) = -1$ are zero.

Let $(M^{n},g)$ be a closed, connected, oriented, $C^{\infty}$, Riemannian, $n$-manifold with a transversely oriented foliation $\mathbb{F}$. We show that if $\lbrace X,Y \rbrace$ are basic vector fields, the leaf component of $[X,Y]$, $\mathcal{V}[X,Y]$, has vanishing leaf divergence whenever ${\kappa}\wedge \chi_{\mathbb{F}}$ is a closed (possibly zero) de Rham cohomology $(p+1)$-form. Here ${\kappa}$ is the mean curvature one-form of the foliation $\mathbb{F}$ and $\chi_{\mathbb{F}}$ is its characteristic form. In the codimension-$2$ case, ${\kappa}\wedge \chi_{\mathbb{F}}$ is closed if and only if ${\kappa}$ is horizontally closed. In certain restricted cases, we give necessary and sufficient conditions for ${\kappa}\wedge{\chi_{\mathbb{F}}}$ to be harmonic. As an application, we give a characterization of when certain closed $3$-manifolds are locally Riemannian products. We show that bundle-like foliations with totally umbilical leaves with leaf dimension greater than or equal to two on a constant curvature manifold, with non-integrable transversal distribution, and with Einstein-like transversal geometry are totally geodesic