The cyclic and simplicial cohomology of the Cuntz semigroup algebra

The main objective of this paper is to determine the simplicial and cyclic cohomology groups of the Cuntz semigroup algebra ${?}^1({\mathbf{S}}_m)$. We also determine the simplicial and cyclic cohomology of the tensor algebra of a Banach space, a class which includes the algebra on the free semigroup on m-generators ${?}^1({\mathbf{FS}}_m)$. In order to do so, we first establish some general results which can be used when studying simplicial and cyclic cohomology of Banach algebras in general. We then turn our attention to ${?}^1({\mathbf{S}}_m)$, showing that the cyclic cohomology groups of degree n vanish when n is odd and are one-dimensional when n is even ($n?2$). Using the Connes–Tzygan exact sequence, these results are used to show that the simplicial cohomology groups of degree n vanish for $n?1$. A similar strategy is used for the tensor algebra of a Banach space.     

On a p-Laplace equation with multiple critical nonlinearities

Using the Mountain-Pass Theorem of Ambrosetti and Rabinowitz we prove that $?{\mathrm{\Delta }}_{p}u?\mu {|x|}^{?p}{u}^{p?1}={|x|}^{?s}{u}^{p^{?}(s)?1}+u^{p^{?}?1}$ admits a positive weak solution in ${\mathbb{R}}^n$ of class $D_1^p({\mathbb{R}}^n)\cap C^1({\mathbb{R}}^n?\{0\})$, whenever $\mu <{\mu }_1$, and ${\mu }_1={[(n?p)/p]}^p$. The technique is based on the existence of extremals of some Hardy–Sobolev type embeddings of independent interest. We also show that if $u\in D_1^p({\mathbb{R}}^n)$ is a weak solution in ${\mathbb{R}}^n$ of $?{\mathrm{\Delta }}_{p}u?\mu {|x|}^{?p}{|u|}^{p?2}u={|x|}^{?s}{|u|}^{p^{?}(s)?2}u+{|u|}^{q?2}u$, then $u\equiv 0$ when either $1<q<p^{?}$, or $q>p^{?}$ and u is also of class $L_{\mathrm{loc}}^{\infty }({\mathbb{R}}^n?\{0\})$.     

Powerful numbers in (1ℓ + qℓ)(2ℓ + qℓ)⋯(nℓ + qℓ)

Let q be a positive integer. Recently, Niu and Liu proved that, if $n\ge \max ?\{q,1198?q\}$, then the product $(1^3+q^3)(2^3+q^3)?(n^3+q^3)$ is not a powerful number. In this note, we prove (1) that, for any odd prime power ? and $n\ge \max ?\{q,11?q\}$, the product $(1^{?}+q^{?})(2^{?}+q^{?})?(n^{?}+q^{?})$ is not a powerful number, and (2) that, for any positive odd integer ?, there exists an integer $N_{q,?}$ such that, for any positive integer $n\ge N_{q,?}$, the product $(1^{?}+q^{?})(2^{?}+q^{?})?(n^{?}+q^{?})$ is not a powerful number.     

On the sub poly-harmonic property for solutions to (−Δ)pu < 0 in Rn

In this note, we mainly study the relation between the sign of ${(?\mathrm{\Delta })}^{p}u$ and ${(?\mathrm{\Delta })}^{p?i}u$ in ${\mathbb{R}}^n$ with $p?2$ and $n?2$ for $1?i?p?1$. Given the differential inequality ${(?\mathrm{\Delta })}^{p}u<0$, first we provide several sufficient conditions so that ${(?\mathrm{\Delta })}^{p?1}u<0$ holds. Then we provide conditions such that ${(?\mathrm{\Delta })}^{i}u<0$ for all $i=1,2,\dots ,p?1$, which is known as the sub poly-harmonic property for u. In the last part of the note, we revisit the super poly-harmonic property for solutions to ${(?\mathrm{\Delta })}^{p}u={\mathrm{e}}^{2pu}$ and ${(?\mathrm{\Delta })}^{p}u=u^q$ with $q>0$ in ${\mathbb{R}}^n$.     

On constructing normal and non-normal Cayley graphs

Bamberg and Giudici (2011) showed that the point graphs of certain generalised quadrangles of order $(q?1,q+1)$, where $q=p^k$ is a prime power with $p\ge 5$, are both normal and non-normal Cayley graphs for two isomorphic groups. We call these graphs BG-graphs. In this paper, we show that the Cayley graphs obtained from a finite number of BG-graphs by Cartesian product, direct product, and strong product also possess the property of being normal and non-normal Cayley graphs for two isomorphic groups.     

On the geography of threefolds of general type

Let X be a complex nonsingular projective 3-fold of general type. We show that there are positive constants c, $c^{\prime }$ and $m_1$ such that $\chi ({\omega }_X)??c\mathrm{Vol}(X)$ and $P_m(X)?c^{\prime }{m}^3\mathrm{Vol}(X)$ for all $m?m_1$.     

On a Hölder covariant version of mean dimension

Let Γ be a infinite countable group which acts naturally on ${?}^p(\Gamma )$. We introduce a modification of mean dimension which is an obstruction for ${?}^p(\Gamma )$ and ${?}^q(\Gamma )$ to be Hölder conjugates. To cite this article: A. Gournay, C. R. Acad. Sci. Paris, Ser. I 347 (2009).