We shall give two criterions of Wiener type which characterize minimally thin sets and rarefied sets in a cone. We shall also show that a positive superharmonic function on a cone behaves regularly outside a rarefied set in a cone. These facts are known for a half space which is a special cone.

In this paper we consider the following problem suggested by T.-C. Kuo. Given a convenient Newton polyhedron $\Gamma$ and a convergent power series $f$. Under what conditions the topological type of $f$ is not affected by perturbations by the functions whose Newton diagram lies above $\Gamma$? If $\Gamma$ consists of one face only (weighted homogeneous case) then the answer is given by theorems of Kuiper-Kuo and of Paunescu. In order to answer this problem we introduce a pseudo-metric adapted to the polyhedron $\Gamma$ which allows us to define the gradient of $f$ with respect to $\Gamma$. Using this construction we obtain versions relative to the Newton filtration of Łojasiewicz Inequality for $f$ and of Kuiper-Kuo-Paunescu theorem. We show that our result is optimal: if Łojasiewicz Inequality with exponent $r$ is not satisfied for $f$ then the $r$-jet of $f$ with respect to the Newton filtration is not $C^0$ sulficent. In homogeneous case this result is known as Bochnak-Łojasiewicz Theorem. Next we study one parameter families of germs $f_t$ : $(R^n,0)\to (R,0)$ of analytic functions under the assumption that the leading terms of $f_t$ with respect to the Newton filtration satisfy the uniform Łojasiewicz Inequality. We show that in this case there is a toric modification $\pi$ of $R^n$ such that the family $f_t\circ \pi$ is analytically trivial. Our result implies in particular the criteria for blow-analytic trivliality due to Kuo, Fukui-Paunescu, and Fukui-Yoshinaga. Our technique can be also used to improve the criteria on $C^k$-sufficiency of jets originally due to Takens.

A substantial proper submanifold $M$ of a Riemannian symmetric space $S$ is called a curved Lie triple if its tangent space at every point is invariant under the curvature tensor of $S$, i.e. a sub-Lie triple. E.g. any complex submanifold of complex projective space has this property. However, if the tangent Lie triple is irreducible and of higher rank, we show a certain rigidity using the holonomy theorem of Berger and Simons: $M$ must be intrinsically locally symmetric. In fact we conjecture that $M$ is an extrinsically symmetric isotropy orbit. We are able to prove this conjecture provided that a tangent space of $M$ is also a tangent space of such an orbit.

We consider the generalized Littlewood-Paley square functions arising from rough kernels and prove the $L^p$-boundedness for a certain range of $p$ depending on the kernel. We also study a class of singular integrals by similar methods.

The topology of a compact self-dual manifold whose twistor space has positive algebraic dimension is studied. When the algebraic dimension equals three, it is known by Campana [4] that the original self-dual manifold is homeomorphic to a connected sum of copies of a complex projecitve plane. In the remaining cases where the algebraic dimension is equal to two or one, we similarly determine the topology of the selfdual manifold except in a certain exceptional case where the algebraic dimension equals one.

Let $A$ be a tame algebra and $M$ a directing $A$-module (there exists no sequence $M_1\to \text{...}\to {\tau }_{A}X\to *\to X\to \text{...}\to M_2$ of nonzero maps between indecomposable $A$-modules for some indecomposable nonprojective $A$-module $X$ and indecomposable direct summands $M_1,$$M_2$ of $M$). Then the variety $mo{d}_A\left(\dim M\right)$ of Amodules with dimension vector $\dim M$ is a complete intersection. If, in addition, $M$ is a tilting $A$-module then $mo{d}_A\left(\dim M\right)$ is normal.

We investigate the endomorphism algebras $\Gamma$ of finite dimensional modules having the property that every indecomposable finite dimensional $\Gamma$-module is of projective dimension at most one or injective dimension at most one. In particular, we describe all matrix algebras $\left[\begin{array}{cc}A& 0\\ A& A\end{array}\right]$ with this homological property.

In the article, the Cauchy problem of the form

(*) $\partial_{2}u(x,t)=f(u(x,t),\partial_{1}^{p}u(x,\alpha(t)t),x,t),\ u(x,0)=0$

or of the form

$(\dagger)$ $\partial_{2}u(x,t)=f(u(x,t), \partial_{1}^{p}u(\alpha(x,t)x,t),x,t),\ u(x,0)=0$

is studied. In (*) and $(\dagger)$ $u(x,t)$ denotes a real valued unknown function of the real variables $x$ and $t$. $p$ denotes a fixed positive integer. It is assumed that $f(u,v,x,t)$ is continuous in $(u,v,x,t)$ and Gevrey in $(u,v,x)$. $\alpha \left(t\right)$ in (*) and $\alpha (x,t)$ in $(\dagger)$ are called shrinkings, since they satisfy the conditions sup$|\alpha (t\left)\right|<1$ and sup$|\alpha (x,t\left)\right|<1$, respectively.

Voiculescu's single variable free entropy is generalized in two different ways to the free relative entropy for compactly supported probability measures on the real line. The one is introduced by the integral expression and the other is based on matricial (or microstates) approximation; their equivalence is shown based on a large deviation result for the empirical eigenvalue distribution of a relevant random matrix. Next, the perturbation theory for compactly supported probability measures via free relative entropy is developed on the analogy of the perturbation theory via relative entropy. When the perturbed measure via relative entropy is suitably arranged on the space of selfadjoint matrices and the matrix size goes to infinity, it is proven that the perturbation via relative entropy on the matrix space approaches asymptotically to that via free relative entropy. The whole theory can be adapted to probability measures on the unit circle.