We study the compressible Euler equations with spherical symmetry surrounding a solid ball. For the spherically symmetric flow, the global existence of $L^{\infty}$ entropy weak solutions has not yet obtained except a special case. In this paper, we prove the existence of global solutions in the more general case. We construct approximate solutions by using a modified Godunov scheme. The main point is to obtain an $L^{\infty}$ bound for the approximate solutions.

Let $C$ be a closed convex set on a complete simply connected surface $S$ whose Gaussian curvature is bounded above by a nonpositive constant $K$. For a relatively compact subset $\Omega \subset S \sim C$, we obtain the sharp relative isoperimeric inequality $2\pi \mathrm{Area}(\Omega )-K\mathrm{Area}(\Omega )^{2} \leq \mathrm{Length}(\partial \Omega \sim \partial C)^{2}$. And we also have a similar partial result with positive Gaussian curvature bound.

In this paper, we consider the Cauchy problem of Schrödinger-IMBq equations in $\mathbb{R}^{n}$, $n \geq 1$. We first show the global existence and blowup criterion of solutions in the energy space for the 3 and 4 dimensional system without power nonlinearity under suitable smallness assumption. Secondly the global existence is established to the system with $p$-powered nonlinearity in $H^{s}(\mathbb{R}^{n})$, $n = 1,2$ for some $\frac{n}{2} < s < \mathrm{min}(2, p)$ and some $p > \frac{n}{2}$ . We also provide a blowup criterion for $n = 3$ in Triebel-Lizorkin space containing BMO space naturally.

Let $A(M)$ be the automorphism group of the middle homology of a smooth 4-manifold $M$ and $D(M)$ be the subgroup induced by diffeomorphisms of $M$. In this paper we give explicit generators of $D(M)$ for rational and ruled 4-manifolds. We also prove the uniqueness of reduced forms for classes with minimal genus 0 and non-negative square.

In the second section of this paper, in analogy to modulation spaces, we define the space $M(p, q)(R^{d})$ to be the subspace of tempered distributions $f \in S'(R^{d})$ such that the Gabor transform $V_{g}(f)$ of $f$ is in the Lorentz space $L(p, q)(R^{2d})$, where the window function $g$ is a rapidly decreasing function. We endow this space with a suitable norm and show that the $M(p, q)(R^{d})$ becomes a Banach space and is invariant under time-frequency shifts for $1 \leq p,q \leq \infty$. We also discuss the dual space of $M(p, q)(R^{d})$ and the multipliers from $L^{1}(R^{d})$ into $M(p, q)(R^{d})$. In the third section we intend to study the intersection space $S (p, q)(R^{d}) = L^{1}(R^{d})\cap M (p, q)(R^{d})$ for $1 < p < \infty$, $1 \leq q \leq \infty$. We endow it with the sum norm and show that $S (p, q)(R^{d})$ becomes a Banach convolution algebra. Further we prove that it is also a Segal algebra. In the last section we discuss the multipliers of $S (p, q)(R^{d})$ and $M (p, q)(R^{d})$.

We give a necessary and sufficient condition for the stability and the freeness of the family of $A_{2}$-type arrangements. Moreover, we determine explicitly when the normalization of the sheafification of its module of reduced logarithmic vector fields is isomorphic to $T_{\mathrm{P}^{2}} (-2)$.

We present a method for estimating the measure of complexity of an unknown dynamical system by studying the time series observed from it. In particular, we propose a simple method for giving a lower bound to the topological entropy of the unknown dynamical system via a onedimensional multivalued map obtained from the time series. We illustrate this method using the H´enon map and the magnetoelastic ribbon data.

We prove $SL(2)-GL(n)$ strange duality conjecture for a general smooth projective irreducible curve.