We study one parameter deformations of a pair consisting of an analytic singular space $X_0$ and a function $f_0$ on it, in case this defines an isolated singularity. We prove, under general conditions, a bouquet decomposition of the Milnor fibre when the isolated singularity splits in the deformation and the invariance of the Milnor fibration if there is no splitting.

We prove that a countable regular space has a continuous selection if and only if it is scattered. Further we prove that a paracompact scattered space admits a continuous selection if each of its points has a countable pseudo-base. We also provide two examples to show that: (1) paracompactness can not be replaced by countable compactness even together with (collectionwise) normality, and (2) having countable pseudo-base at each of its points can not be omitted even in the class of regular Lindelöf linearly ordered spaces.

Definition and fundamentals of tilings generated by Pisot numbers are shown by arithmetic consideration. Results include the case that a Pisot number does not have a finitely expansible property, i.e. a sofic Pisot case. Especially we show that the boundary of each tile has Lebesgue measure zero under some weak condition.

We give a detailed proof of convergence of a normalized solution of the Maurer-Cartan equation in the Barannikov-Kontsevich construction.

Smooth complex surfaces polarized with an ample and globally generated line bundle of degree three and four, such that the adjoint bundle is not globally generated, are considered. Scrolls of a vector bundle over a smooth curve are shown to be the only examples in degree three. Two classes of examples in degree four are presented, one of which is shown to characterize regular such pairs. A Reider-type theorem is obtained in which the assumption on the degree of $L$ is removed.

Let $B$ denote the unit ball in $C^n$, and $v$ the normalized Lebesgue measure on $B$. For $\alpha >-1$, define $d{v}_{\alpha }\left(z\right)=\Gamma (n+\alpha +1)/\left\{\Gamma \right(n+1\left)\Gamma \right(\alpha +1\left)\right\}(1-|z{|}^2{)}^{\alpha }dv\left(z\right)$, $z\in B$. Let $H\left(B\right)$ denote the space of holomorphic functions in $B$. For $p\ge 1$, define

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${\left(\mathrm{AN}\right)}^p\left({\nu }_{\alpha }\right)$ is an $F$-space with respect to the metric $\rho \left(f,g\right)\equiv ∥f-g∥$. In this paper we prove that every linear isometry $T$ of ${\left(\mathrm{AN}\right)}^p\left({\nu }_{\alpha }\right)$ into itself is of the form $Tf=c\left(f\circ \psi \right)$ for all $f\in {\left(\mathrm{AN}\right)}^p\left({\nu }_{\alpha }\right)$ , where $c$ is a complex number with $\left|c\right|=1$ and $\psi$ is a holomorphic self-map of $B$ which is measure-preserving with respect to the measure ${\nu }_{\alpha }$.

Let $\mathcal{N}$ be the moduli space of sextics with 3 $(3,4)$-cusps. The quotient moduli space $\mathcal{N}/G$ is one-dimensional and consists of two components, ${\mathcal{N}}_{\mathrm{torus}}/G$ and ${\mathcal{N}}_{\mathrm{gen}}/G$. By quadratic transformations, they are transformed into one-parameter families $C_s$ and $D_s$ of cubic curves respectively. First we study the geometry of ${\mathcal{N}}_{\epsilon }/G$, torus, gen and their structure of elliptic fibration. Then we study the Mordell-Weil torsion groups of cubic curves $C_s$ over $Q$ and $D_s$ over $Q\left(\sqrt{-3}\right)$ respectively. We show that $C_s$ has the torsion group $Z/3Z$ for a generic $s\in Q$ and it also contains subfamilies which coincide with the universal families given by Kubert [Ku] with the torsion groups $Z/6Z,$$Z/6Z+Z/2Z,$$Z/9Z$, or $Z/12Z$. The cubic curves $D_s$ has torsion $Z/3Z+Z/3Z$ generically but also $Z/3Z+Z/6Z$ for a subfamily which is parametrized by $Q\left(\sqrt{-3}\right)$.

We investigate the distribution of length of closed geodesics on geodesic spheres and tubes around complex hyperplane in a non-flat complex space form. The feature of the length spectrum of a geodesic sphere of radius $r$ in a complex projective space of holomorphic sectional curvature 4 is quite different according as ${\tan }^{2}r$ is rational or irrational. Each length spectrum is simple when ${\tan }^{2}r$ is irrationaj but when ${\tan }^{2}r$ is rationaj it is not necessarily simple and moreover the multiplicity is not uniformly bounded.

The spectrum of the Milnor-Gromoll-Meyer sphere is given. For this purpose the system of orthogonal functions on the symplectic group $Sp\left(2\right)$(isomorphic to the covering group of SO(5)$)$ is studied.

This paper is concerned with the convergence rates to viscous shock profile for general scalar viscous conservation laws. Compared with former results in this direction, the main novelty in this paper lies in the fact that the initial disturbance can be chosen arbitrarily large. This answers positively an open problem proposed by A. Matsumura in [12] and K. Nishihara in [16]. Our analysis is based on the $L^1\text{-}$ stability results obtained by H. Freistiihler and D. Serre in [1].

In this work we study the essential sets for some Kato measure in $(R^d-\{0\left\}\right),$$d\ge 2$. Using the caracterisation of Picard principle via the Green kernel associated to the Schrödinger operator $\Delta -\mu$; we give a new caracterisation of such sets when $\mu =\left(f\right(.)/||.||{)}^2\lambda$ where $f$ is assumed to be rotation free nonnegative, decreasing and locally Hölder continuous on . In particular we obtain results given by T. Tada in the case where $d=2$ and $f\left(r\right)=-\log r$.