Let $u$ be a $\lambda$-Hölder continuous function on the closure of a bounded domain $D$ with fractal boundary $\partial D$. We estimate the Besov norm of the restriction of $u$ to $\partial D$ by the $L^p\left(D\right)$-norm of $\left|\nabla u\right(y\left)\right|$ dist $(y,\partial D{)}^{\lambda }$ for an adequate $\lambda >0$. We apply it to the boundedness of operators related to the double layer potentials on the Besov spaces on $\partial D$.

We classify smooth complex projective algebraic curves $C$ of low genus $7\le g\le 10$ such that the variety of nets $W_{g-1}^2\left(C\right)$ has dimension $g-$$7$. We show that $\dim W_{g-1}^2\left(C\right)=g-$$7$ is equivalent to the following conditions according to the values of the genus $g$. $\left(i\right)C$ is either trigonal, a double covering of a curve of genus 2 or a smooth plane curve degree 6 for $g=10$. $\left(ii\right)C$ is either trigonal, a double covering of a curve of genus 2, a tetragonal curve with a smooth model of degree 8 in $P^3$ or a tetragonal curve with a plane model of degree 6 for $g=9$. $\left(iii\right)C$ is either trigonal or has a birationally very ample $g_6^2$ for $g=8$ or $g=7$.

In 1990, Gérard-Tahara [2] introduced the Briot-Bouquet type partial differential equation $t{\partial }_{t}u=F(t,x,u,{\partial }_{x}u)$, and they determined the structure of singular solutions provided that the characteristic exponent $\rho \left(x\right)$ satisfies $\rho \left(0\right)\notin \{1,2,\text{...}\}$. In this paper the author determines the structure of singular solutions in the case $\rho \left(0\right)\in \{1,2,\text{...}\}$.

Let $V$ be a complex analytic space and $x$ be an isolated singular point of $V$. We define the $q$-th punctured local holomorphic de Rham cohomology $H_h^q$$(V,x)$ to be the direct limit of $H_h^q(U-\{x\left\}\right)$ where $U$ runs over strongly pseudoconvex neighborhoods of $x$ in $V$, and $H_h^q(U-\{x\left\}\right)$ is the holomorphic de Rahm cohomology of the complex manifold $U-\left\{x\right\}$. We prove that punctured local holomorphic de Rham cohomology is an important local invariant which can be used to tell when the singularity $(V,x)$ is quasi-homogeneous. We also define and compute various Poincaré number ${\tilde{p}}_x^{\left(i\right)}$ and ${\overline{p}}_x^{\left(i\right)}$ of isolated hypersurface singularity $(V,x)$.

In this note, we will study Delta link homotopy, which is an equivalence relation of ordered and oriented link types. Previously, a necessary condition was given by a pair of numerical invariants derived from the Conway polynomials for two link types to be Delta link homotopic. In this note, we will show that, for two component links, if their pairs of numerical invariants coincide then the two links are Delta link homotopic.

We give an upper estimate for the Łojasiewicz exponent $\ell (J,I)$ of an ideal $J\subseteq A\left(K^n\right)$ with respect to another ideal I in the ring $A\left(K^n\right)$ of germs analytic functions $f$ : $(K^n,0)\to K$, where $K=C$ or $R$, using Newton polyhedrons. In particular, we give a method to estimate the Łojasiewicz exponent ${\alpha }_0\left(f\right)$ of a germ $f\in A\left(K^n\right)$ that can be applied when $f$ is Newton degenerate with respect to its Newton polyhedron.

The Goursat problem for certain types of second order linear equations is considered. The Goursat problem for those second order equations is not $\mathcal{E}$-wellposed in general. For a certain type homogeneous equations, the Goursat problem is $\mathcal{E}$-wellposed. Necessary or sufficient conditions on lower order terms for $\mathcal{E}$-wellposedness are given. Wellposedness in Gevrey class is discussed.

A well-known conjecture states that the kernel of representation associated to a modular fusion algebra is always a congruence subgroup. Assuming this conjecture, Eholzer studied modular fusion algebras such that the kernel of representation associated to each of them is a congruence subgroup using the fact that all irreducible representaions of $SL(2,Z/p^{\lambda }Z)$ are classified. He classified all strongly modular fusion algebras of dimension two, three, four and the nondegenerate ones with dimension $\le 24$. In this paper, we try to imitate Eholzer's work. We classify modular fusion algebras such that the kernel of representation associated to each of them is a noncongruence normal subgroup of $\Gamma :=PSL(2,Z)$ containing an element $\left(\begin{array}{cc}1& 6\\ 0& 1\end{array}\right)$. Among such normal subgroups, there exist infinitely many noncongruence subgroups. In a sense, they are the classes of near congruence subgroups. For such a normal subgroup $G$, we shall show that any irreducible representation of degree not equal to 1 of $\Gamma /G$ is not associated to a modular fusion algebra.

In this paper, we will study smoothability of a weak Fano 3-fold with only canonical singularities which is obtained as an image of a crepant primitive birational contraction from a smooth weak Fano 3-fold. Main part is on a contraction of type III.

If $M$ is a (separable) von Neumann algebra and $A$ is a Cartan subalgebra of $M$, then $M$ is determined by an equivalence relation and a 2-cocycle. By constructing an equivalence subrelation, we show that for any intermediate von Neumann subalgebra $N$ between $M$ and $A$, there exists a faithful normal conditional expectation from $M$ onto $N$.

In this paper, we estimate the degree of symmetry and the semi-simple degree of symmetry of certain fiber bundles by virtue of the rigidity theorem with respect to the harmonic map due to Schoen and Yau. As a corollary of this estimate, we compute the degree of symmetry and the semi-simple degree of symmetry of certain product manifolds. In addition, by Albanese map, we estimate the degree of symmetry and the semi-simple degree of symmetry of a compact smooth manifold under some topological assumptions.

Let $k$ be a number field and $\tilde{k}$ a fixed quadratic extension of $k$. In this paper and its companions, we find the mean value of the product of class numbers and regulators of two quadratic extensions $F,$$F^{*}\ne \tilde{k}$ contained in the biquadratic extensions of $k$ containing $\tilde{k}$.

We consider the initial-boundary value problem for the standard quasilinear wave equation:

$u_{\mathrm{tt}}-\text{div}\left\{\sigma \right(\left|{\nabla }_u{|}^2\right){\nabla }_u\}+a(x)u_t=0$ in $\Omega \times [0,\infty )$

$u(x,0)=u_0\left(x\right)$ and $u_t(x,0)=u_1\left(x\right)$ and $u{|}_{\partial \Omega }=0$

where $\Omega$ is an exterior domain in $R^N,\sigma \left(v\right)$ is a function like $\sigma \left(v\right)=1/\sqrt{1+v}$ and $a\left(x\right)$ is a nonnegative function. Under two types of hypotheses on $a\left(x\right)$ we prove existence theorems of global small amplitude solutions. We note that $a\left(x\right)u_t$ is required to be effective only in localized area and no geometrical condition is imposed on the boundary $\partial \Omega$.

We consider a compressible viscous fluid effected by general form external force in $R^3$. In part 1, an analysis of the linearized problem based on the weighted- $L_2$ method implies a condition on the external force for the existence and some regularities of the steady flow. In part 2, we study the stability of the steady flow with respect to the initial disturbance. What we proved is: if $H^3$-norm of the initial disturbance is small enough, then the solution to the non-stationary problem exists uniquely and globally in time.

We investigate the properties of ideals such that their corresponding partial orders preserve stationarity. We show that these ideals exhibit many large cardinal-like consequences. We also prove the existence of a certain non-reflecting stationary subset of ${\mathcal{P}}_{\kappa }\lambda$ under some hypotheses.

We prove a central limit theorem for the transition operator of the symmetric random walk on a covering graph with a covering transformation group of polynomial growth. As the limit, the continuous semigroup of the sub-Laplacian on a nilpotent Lie group is obtained.