In this paper, we prove the functional equations for the zeta functions in two variables associated with prehomogeneous vector spaces acted on by maximal parabolic subgroups of orthogonal groups. Moreover, applying the converse theorem of Weil type, we show that elliptic modular forms of integral or half integral weight can be obtained from the zeta functions.

We present in this paper sufficient conditions for the topological triviality of families of germs of functions defined on an analytic variety $V$. The main result is an infinitesimal criterion based on a convenient weighted inequality, similar to that introduced by T. Fukui and L. Paunescu. When $V$ is a weighted homogeneous variety, we obtain as a corollary, the topological triviality of deformations by terms of non negative weights of a weighted homogeneous germ consistent with $V$. Application of the results to deformations of Newton non-degenerate germs with respect to a given variety is also given.

We prove that every symplectic filling of the link of a rational surface singularity with reduced fundamental cycle admits a rational compactification, possibly after a modification of the filling in a collar neighbourhood of the link.

The test ideal $\tau(R)$ of a ring $R$ of prime characteristic is an important object in the theory of tight closure. In this paper, we study a generalization of the test ideal, which is the ideal $\tau({\frak a}^t)$ associated to a given ideal $\frak a$ with rational exponent $t \ge 0$. We first prove a key lemma of this paper (Lemma \ref{key lemma}), which gives a characterization of the ideal $\tau({\frak a}^t)$. As applications of this key lemma, we generalize the preceding results on the behavior of the test ideal $\tau(R)$. Moreover, we prove an analogue of so-called Skoda's theorem, which is formulated algebraically via adjoint ideals by Lipman in his proof of the "modified Briançon-Skoda theorem."

Let $f(z)$ be an entire function of order less than $1/2.$ We consider an analogue of the Wiman-Valiron theory rewriting power series of $f(z)$ into binomial series. As an application, it is shown that if a transcendental entire solution $f(z)$ of a linear difference equation is of order $\chi < 1/2,$ then we have %$\chi$ is obtained from the Newton polygon for the equation, and $\log M(r,f) = Lr^{\chi}(1 + o(1))$ with a constant $L > 0.$

We propose certain obstructions for the existence of hyperelliptic splitting families of degenerations of curves. Moreover we determine the complete system of hyperelliptic atomic fibers of genus 3.

In this paper, the problem on formation and construction of a shock wave for three dimensional compressible Euler equations with the small perturbed spherical initial data is studied. If the given smooth initial data satisfy certain nondegeneracy conditions, then from the results in [22], we know that there exists a unique blowup point at the blowup time such that the first order derivatives of a smooth solution blow up, while the solution itself is still continuous at the blowup point. From the blowup point, we construct a weak entropy solution which is not uniformly Lipschitz continuous on two sides of a shock curve. Moreover the strength of the constructed shock is zero at the blowup point and then gradually increases. Additionally, some detailed and precise estimates on the solution are obtained in a neighbourhood of the blowup point.

We proved that any hyperconvex manifold has a complete Bergman metric.

We compute the Hochschild homology groups of the adiabatic algebra $\Psi_\alpha(X)$, a deformation of the algebra of pseudodifferential operators $\Psi(X)$ when $X$ is the total space of a fibration of closed manifolds. We deduce the existence and uniqueness of traces on $\Psi_\alpha(X)$ and some of its ideals and quotients, in the spirit of the noncommutative residue of Wodzicki and Guillemin. We introduce certain higher homological versions of the residue trace. When the base of the fibration is $S^1$, these functionals are related to the $\eta$ function of Atiyah-Patodi-Singer.