Let $X$ be a closed oriented smooth 4-manifold of simple type with $b_{1}(X)=0$ and $b_{+}(X)\geq 2$, and let $\tau : X \longrightarrow X$ generate an involution preserving a spin^c structure $c$. Under certain topological conditions we show in this paper that the Seiberg-Witten invariant $SW(X, c)$ is zero modulo 2. This then enables us to investigate the mod 2 Seiberg-Witten invariants of real algebraic surfaces with the opposite orientation, which is motivated by the Kotschick’s conjecture. The basic strategy is to use the new interpretation of the Seiberg-Witten invariants as a certain equivariant degree of a map constructed from the Seiberg-Witten equa- tions and the generalization of the results of Fang.

In this paper, we show that, for some birational mapping $F$ of $\mathbb{P} ^{2}$ with an indeterminate point $I_{1}$, there exists a partial horseshoe structure at $I_{1}$ and periodic points of $F$ accumulate at $I_{1}$. This is a new dynamical model that gives a chaotic phenomenon in a neighbourhood of the indeterminate point $I_{1}$ at which $F$ is not continuous.

The family of harmonic Hilbert spaces is a natural enlargement of those classical $L^{2}$-Sobolev spaces on $\mathbb{R}^{d}$ which consist of continuous functions. In the present paper we demonstrate that the use of basic results from the theory of Wiener amalgam spaces allows to establish fundamental properties of harmonic Hilbert spaces even if they are defined over an arbitrary locally compact abelian group $\mathcal{G}$. Even for $\mathcal{G}=\mathbb{R}^{d}$ this new approach improves previously known results. In this paper we present results on minimal norm interpolators over lattices and show that the infinite minimal norm interpolations are the limits of finite minimal norm interpolations. In addition, the new approach paves the way for the study of stability problems and error analysis for norm interpolations in harmonic Hilbert and Banach spaces on locally compact abelian groups.

This article is a continuation of the paper [6]. Smooth complex projective 3-folds with nonnegative Kodaira dimension admitting nontrivial surjective endomorphisms are completely determined. Especially, it is proved that, for such a 3-fold $X$, there exist a finite étale Galois covering $\Tilde{X} \longrightarrow X$ and an abelian scheme structure $\Tilde{X} \longrightarrow T$ over a smooth variety $T$ of dimension $\leq 2$.

For an effective divisor $A$ with support $B$ in a compact Kähler manifold $M$ of dimension $\geq 3$, the following are antinomic.

a) $M\backslash B$ has a $C^{\infty}$ plurisubharmonic exhaustion function whose Levi form has pointwise at least 3 positive eigenvalues outside a compact subset of $M\backslash B$.

b) $[A]|B$, the normal bundle of $A$, is topologically trivial.

In this work, we study a class of operators coming from the linearization of some kinetic equations such as Boltzmann equations and Vlasov-Fokker-Planck equations. Since it is not a standard class of pseudodifferential operators, we obtain hypoelliptic estimates in some weight functions space and show the regularity of weak solutions for linear and semilinear equations.

We investigate a limit theorem on traversable length inside semicylinder in the 2-dimensional supercritical Bernoulli bond percolation, which gives an extension of Theorem 2 in [5]. This type of limit theorems was originally studied for the extinction time for the 1-dimensional contact process on a finite interval in [10]. Actually, our main result Theorem 2.1 is stated under a rather general 2-dimensional bond percolation setting.

We will show that the theory of ordered divisible vector spaces over an ordered field satisfies the bounded condition treated in [5].