Let $\Gamma$ be a $C^\infty$ curve in $\Bbb{C}$ containing 0; it becomes $\Gamma_\theta$ after rotation by angle $\theta$ about 0. Suppose a $C^\infty$ function $f$ can be extended holomorphically to a neighborhood of each element of the family $\{\Gamma_\theta \}$. We prove that under some conditions on $\Gamma$ the function $f$ is necessarily holomorphic in a neighborhood of the origin. In case $\Gamma$ is a straight segment the well known Bochnak-Siciak Theorem gives such a proof for real analyticity. We also provide several other results related to testing holomorphy property on a family of certain subsets of a domain in $\Bbb{C}$.

We construct a Gibbs measure for the nonlinear Schrödinger equation (NLS) on the circle, conditioned on prescribed mass and momentum:

$$d \mu_{a,b} = Z^{-1} 1_{ \{\int_{\mathbb{T}} |u|^2 = a\}} 1_{\{i \int_{\mathbb{T}} u \overline{u}_x = b\}}e^{\pm 1/p \int_{\mathbb{T}} |u|^p-1/2\int_{\mathbb{T}} |u|^2 } d P$$

for $a \in \mathbb{R}^+$ and $b \in \mathbb{R}$, where $P$ is the complex-valued Wiener measure on the circle. We also show that $\mu_{a,b}$ is invariant under the flow of NLS. We note that $i \int_{\mathbb{T}} u \overline{u}_x$ is the Lévy stochastic area, and in particular that this is invariant under the flow of NLS.

A 2-dominating set of a graph $G = (V,E)$ is a set $D$ of vertices of $G$ such that every vertex of $V(G) \setminus D$ has at least two neighbors in $D$. The 2-domination number of a graph $G$, denoted by $\gamma_2(G)$, is the minimum cardinality of a 2-dominating set of $G$. The non-isolating 2-bondage number of $G$, denoted by $b_2'(G)$, is the minimum cardinality among all sets of edges $E' \subseteq E$ such that $\delta(G-E') \ge 1$ and $\gamma_2(G-E') > \gamma_2(G)$. If for every $E' \subseteq E$, either $\gamma_2(G-E') = \gamma_2(G)$ or $\delta(G-E') = 0$, then we define $b_2'(G) = 0$, and we say that $G$ is a $\gamma_2$-non-isolatingly strongly stable graph. First we discuss the basic properties of non-isolating 2-bondage in graphs. We find the non-isolating 2-bondage numbers for several classes of graphs. Next we show that for every non-negative integer there exists a tree having such non-isolating 2-bondage number. Finally, we characterize all $\gamma_2$-non-isolatingly strongly stable trees.

We shall show that any complex minimal surface of general type with $c_1^2 = 2\chi -1$ having non-trivial 2-torsion divisors, where $c_1^2$ and $\chi$ are the first Chern number of a surface and the Euler characteristic of the structure sheaf respectively, has the Euler characteristic $\chi$ not exceeding 4. Moreover, we shall give a complete description for the surfaces of the case $\chi =4$, and prove that the coarse moduli space for surfaces of this case is a unirational variety of dimension 29. Using the description, we shall also prove that our surfaces of the case $\chi = 4$ have non-birational bicanonical maps and no pencil of curves of genus 2, hence being of so called non-standard case for the non-birationality of the bicanonical maps.

In this paper, we give an isotopy classification of 3-bridge spheres of 3-bridge arborescent links, which are not Montesinos links. To this end, we prove a certain refinement of a theorem of J. S. Birman and H. M. Hilden [3] on the relation between bridge presentations of links and Heegaard splittings of 3-manifolds. In the proof of this result, we also give an answer to a question by K. Morimoto [23] on the classification of genus-2 Heegaard splittings of certain graph manifolds.

The present paper concerns the dynamics of surface diffeomorphisms. Given a diffeomorphism $f$ of a surface $S$, the torsion of the orbit of a point $z\in S$ is, roughly speaking, the average speed of rotation of the tangent vectors under the action of the derivative of $f$, along the orbit of $z$ under $f$. The purpose of the paper is to identify some situations where there exist measures and orbits with non-zero torsion. We prove that every area preserving diffeomorphism of the disc which coincides with the identity near the boundary has an orbit with non-zero torsion. We also prove that a diffeomorphism of the torus ${\mathbb T}^2$, isotopic to the identity, whose rotation set has non-empty interior, has an orbit with non-zero torsion.

We introduce the notion of an approximate Jacobian Newton diagram which is the Jacobian Newton diagram of the morphism $(f^{(k)},f)$, where $f$ is a branch and $f^{(k)}$ is a characteristic approximate root of $f$. We prove that the set of all approximate Jacobian Newton diagrams is a complete topological invariant. This generalizes theorems of Merle and Ephraim about the decomposition of the polar curve of a branch.

We study the long time behavior of solutions of the wave equation with a variable damping term $V(x)u_t$ in the case of critical decay $V(x)\geq V_0(1+|x|^2)^{-1/2}$ (see condition (A) below). The solutions manifest a new threshold effect with respect to the size of the coefficient $V_0$: for $1 < V_0 < N$ the energy decay rate is exactly $t^{-V_0}$, while for $V_0\geq N$ the energy decay rate coincides with the decay rate of the corresponding parabolic problem.

For two variable real analytic function germs we compare the blow-analytic equivalence in the sense of Kuo to other natural equivalence relations. Our main theorem states that $C^1$ equivalent germs are blow-analytically equivalent. This gives a negative answer to a conjecture of Kuo. In the proof we show that the Puiseux pairs of real Newton-Puiseux roots are preserved by the $C^1$ equivalence of function germs. The proof is achieved, being based on a combinatorial characterisation of blow-analytic equivalence, in terms of the real tree model.

We also give several examples of bi-Lipschitz equivalent germs that are not blow-analytically equivalent.

Let $p$ be an odd prime number. To ask the behavior of $\lambda$- and $\mu$-invariants is a basic problem in Iwasawa theory of $\mathbb{Z}_p$-extensions. Sands showed that if $p$ does not divide the class number of an imaginary quadratic field $k$ and if the $\lambda$-invariant of the cyclotomic $\mathbb{Z}_p$-extension of $k$ is 2, then $\mu$-invariants vanish for all $\mathbb{Z}_p$-extensions of $k$, and $\lambda$-invariants are less than or equal to 2 for $\mathbb{Z}_p$-extensions of $k$ in which all primes above $p$ are totally ramified. In this article, we show results similar to Sands' results without the assumption that $p$ does not divide the class number of $k$. When $\mu$-invariants vanish, we also give an explicit upper bound of $\lambda$-invariants of all $\mathbb{Z}_p$-extensions.

We generalize the notion of a tame automorphism to the context of an affine quadric threefold and we prove that there exist non-tame automorphisms.

We showed in [3] and [4] the variation formulas for Schiffer spans and harmonic spans of the moving domain $D(t)$ in $\mathbb{C}_z$ with parameter $t\in B=\{t\in\mathbb{C}_t : |t| < \rho\}$, respectively, such that each $\partial D(t)$ consists of a finite number of $C^{\omega}$ contours $C_j(t)$ $(j=1\ldots,\nu)$ in $\mathbb{C}_z$ and each $C_j(t)$ varies $C^{\omega}$smoothly with $t\in B$. This implied that, if the total space $\mathcal{D}=\bigcup_{t\in B}(t,D(t))$ is pseudoconvex in $B\times \mathbb{C}_z$, then the Schiffer span is logarithmically subharmonic and the harmonic span is subharmonic on $B$, respectively, so that we showed those applications. In this paper, we give the indispensable condition for generalizing these results to Stein manifolds. Precisely, we study the general variation under pseudoconvexity, i.e., the variation of domains $\mathcal{D}: t\in B \to D(t)$ is pseudoconvex in $B\times\mathbb{C}_z$ but neither each $\partial D(t)$ is smooth nor the variation is smooth for $t\in B$.