For a ramified covering $f$ : $Y\to X$ between compact complex manifolds, we establish a formula relating the Chern numbers of $Y$ and $X$. We obtain the formula by localizing characteristic classes via the Čech-de Rham cohomology theory. As corollaries, we deduce generalizations of such formulas as the Riemann-Hurwitz formula and a formula of Hirzebruch for the signature, as well as formulas, for other invariants such as the Todd genus.

For the real Hardy spaces, we shall establish Hardy's inequalities with respect to Laguerre expansions. The inequalities for the Hardy spaces with exponents smaller than one will be discussed.

Let $\alpha$ be a polygonal Jordan curve in $R^3$. We show that if $\alpha$ satisfies certain conditions, then the least-area Douglas-Radó disk in $R^3$ with boundary $\alpha$ is unique and is a smooth graph. As our conditions on $\alpha$ are not included amongst previously known conditions for embeddedness, we are enlarging the set of Jordan curves in $R^3$ which are known to be spanned by an embedded least-area disk. As an application, we consider the conjugate surface construction method for minimal surfaces. With our result we can apply this method to a wider range of complete catenoid-ended minimal surfaces in $R^3$.

Let $M^n\subset R^{n+2},$$n\ge 7$, be a conformally deformable submanifold of euclidean space in codimension two. In this paper we show that if the submanifold has sufficiently low conformal nullity, a generic conformal condition, then it can be realized as a hypersurface of a conformally deformable hypersurface. The latter have been classified by Cartan early this century. Furthermore, it turns out that all deformations of the former are induced by deformations of the latter.

Montiej Ros and Urbano [M3] showed a complete characterization of compact totally real minimal submanifold $M$ of $C{P}^n\left(c\right)$ with Ricci curvature $S$ of $M$ satisfying $S\ge 3(n-2)c/16$. The purpose of this paper is to answer Ogiue's conjecture which the above result remains true under the weaker condition of the scalar curvature $p$ of $M$ satisfying $p\ge 3n(n-2)c/16$.

At each prime number $p$, the homotopy groups ${\pi }_{*}\left(L_2{S}^0\right)$ of the $v_2^{-1}$BP-localized sphere spectrum play an crucial role to understand the category of $v_2^{-1}$BP-local spectra. For $p>3$, they are determined by using the Adams-Novikov spectral sequence (ANSS), which collapses in this case. At the prime 3, ${\pi }_{*}\left(L_{2}V\right(1\left)\right)$ is also determined by using the ANSS, in which $E_{\infty }=$$E_{10}$ in this case. Here $V\left(1\right)$ denotes the Toda-Smith 4-cells spectrum. In this paper, we determine the homotopy groups ${\pi }_{*}\left(L_{2}V\right(0\left)\right)$ of the mod 3 Moore spectrum from ${\pi }_{*}$$(L_{2}V$(1 )$)$ by the Bockstein spectral sequence (BSS). Actually, we first compute the $E_2$-term of the ANSS by the BSS and then study the Adams-Novikov differentials, and obtain $E_{\infty }=E_{10}$ as well.

As an application of the generalized Pontrjagin-Thom construction (see [5] $)$ here we prove some results on the orientability of singularity submanifolds. Our approach is based on the computation of symmetries of singularities and is different from the one based on the fundamental work of Boardman ([3]) which involves the high intrinsic derivatives. As an example we apply our method to all the ${\Sigma }^{1_r}$ and ${\Sigma }^{2,0}$ singularities.

For simple random walks $\left\{P_G^n\right\}$ on a homogeneous graph $G$ and $\left\{P_{L\left(G\right)}^n\right\}$ on its line graph $L\left(G\right)$, we obtain the relationship between the asymptotic behavior of the $n$-step transition probability $P_G^n(x,x)$ and that of $P_{L\left(G\right)}^n(x,x)$ as $n\to \infty$.

This paper is devoted to the computation of the index of a critical point for nonlinear operators with strong coefficient growth. These operators are associated with boundary value problems of the type \begin{center} $\displaystyle \sum_{|\alpha|=1}\mathscr{D}^{\alpha}\{\rho^{2}(u)\mathscr{D}^{\alpha}u+a_{\alpha}(x,\mathscr{D}^{1}u)\}=\lambda a_{0}(x,u,\mathscr{D}^{1}u),\ x\in\Omega$, $u(x)=0,\ x\in\partial\Omega$, \end{center} where $\Omega\subset\mathbf{R}^{n}$ is open, bounded and such that $\partial\Omega\in\mathbf{C}^{2}$, while $\rho: \mathbf{R} \to \mathbf{R}_{+}$ can have exponential growth. An index formula is given for such densely defined operators acting from the Sobolev space $W_0^{1,m}(\Omega)$ into its dual space. We consider different sets of assumptions for $m>2$ (the case of a real Banach space) and $m=2$ (the case of a real Hilbert space). The computation of the index is important for various problems concerning nonlinear equations: solvability, estimates for the number of solutions, branching of solutions, etc. The result of this paper are based upon recent results of the authors involving the computation of index of a critical point for densely defined abstract operators of type $(S_{+})$. The latter are based in turn upon a new degree theory for densely defined $(S_{+})$-mappings, which has also been developed by the authors in a recent paper. Applications of the index formula to the relevant bifurcation problems are also included.

We show that there exist on $4C{P}^2$, the connected sum of four complex projective planes, self-dual metrics with the following properties: (i) the sign of the scalar curvature is positive, (ii) the identity component of the isometry group is $U\left(1\right),$$\left(iii\right)$ the metrics are not conformally isometric to the self-dual metrics constructed by LeBrun [LB1]. These are the first examples of self-dual metrics with non semi-free $U\left(1\right)\text{-}$ isometries on simply connected manifolds. Our proof is based on the twistor theory: we use an equivariant orbifold version of the construction of Donaldson and Friedman [DF]. We also give a rough description of the structure of the algebraic reduction of the corresponding twistor spaces.

Riesz distributions are relatively invariant distributions supported by the closure $\overline{\Omega }$ of a homogeneous cone $\Omega$. In this paper, we clarify the positivity condition of Riesz distributions by relating it to the orbit structure of $\overline{\Omega }$. Moreover each of the positive Riesz distributions is described explicitly as a measure on an orbit in $\overline{\Omega }$.

By generalizing Erlijman's method, we construct a subfactor from a fusion rule algebra with quantum $6j$-symbols which produce periodic commuting squares. This construction produces the same subfactor as Ocneanu's asymptotic inclusion for the subfactor which is generated by the original periodic commuting square. This result can be applied to the quantum $SU(n{)}_k$ subfactors which is the same as Hecke algebra subfactors of type $A$ of Wenzl for example, which shows that Erlijman's construction gives the same subfactor as the asymptotic inclusion.

Using an algebraic analog of Cartan's method of moving frames and the complete classification of two-dimensional subalgebras in sI $(4,R)$, we describe all locally homogeneous surfaces in $R{P}^3$.