Generalized Ricci bounds and convergence of metric measure spaces

We introduce and analyze curvature bounds $\underset{?}{\mathbb{C}\mathrm{urv}}(M\text{,}\mathsf{d}\text{,}m)?K$ for metric measure spaces $(M\text{,}\mathsf{d}\text{,}m)$, based on convexity properties of the relative entropy $\mathrm{Ent}(?|m)$. For Riemannian manifolds, $\underset{?}{\mathbb{C}\mathrm{urv}}(M\text{,}\mathsf{d}\text{,}m)?K$ if and only if ${\mathrm{Ric}}_M(\xi \text{,}\xi )?K?{\left|\xi \right|}^2$ for all $\xi \in TM$. We define a complete separable metric $\mathbb{D}$ on the family of all isomorphism classes of normalized metric measure spaces. It has a natural interpretation in terms of mass transportation. Our lower curvature bounds are stable under $\mathbb{D}$-convergence. We also prove that the family of normalized metric measure spaces with doubling constant $?C$ is closed under $\mathbb{D}$-convergence. Moreover, the subfamily of spaces with diameter $?R$ is compact. To cite this article: K.-T. Sturm, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Contracting an element from a cocircuit

We consider the situation that M and N are 3-connected matroids such that $|E(N)|⩾4$ and $C^1$ is a cocircuit of M with the property that $M/x_0$ has an N-minor for some $x_0\in C^1$. We show that either there is an element $x\in C^1$ such that $\mathrm{si}(M/x)$ or $\mathrm{co}(\mathrm{si}(M/x))$ is 3-connected with an N-minor, or there is a four-element fan of M that contains two elements of $C^1$ and an element x such that $\mathrm{si}(M/x)$ is 3-connected with an N-minor.     

The division problem for tempered distributions of one variable

We give a characterization, in one variable case, of those $C^{\infty }$ multipliers F such that the division problem is solvable in $S^{\prime }(\mathbb{R})$. For these functions $F\in O_M(\mathbb{R})$ we even prove that the multiplication operator $M_F(G)=FG$ has a continuous linear right inverse on $S^{\prime }(\mathbb{R})$, in contrast to what happens in the several variables case, as was shown by Langenbruch.     

On the positive radial solutions of a class of singular semilinear elliptic equations

In this paper, we consider the following elliptic equation(0.1)$\mathit{div}\left(A\right(|x|\left)\mathrm{?}u\right)+B(|x|)u^p=0\text{in}{\mathbb{R}}^n,$ where $p>1$, $n?3$, $A(|x|)>0$ is differentiable in ${\mathbb{R}}^n?\{0\}$ and $B(|x|)$ is a given nonnegative Hölder continuous function in ${\mathbb{R}}^n?\{0\}$. The asymptotic behavior at infinity and structure of separation property of positive radial solutions with different initial data for (0.1) are discussed. Moreover, the existence and separation property of infinitely many positive solutions for Hardy equation and an equation related to Caffarelli–Kohn–Nirenberg inequality are obtained respectively, as special cases.