On the Chow ring of certain rational cohomology tori

Let $f: X \rightarrow A$ be an abelian cover from a complex algebraic variety with quotient singularities to an abelian variety. We show that $f^*$ induces an isomorphism between the rational cohomology rings $H^\bullet(A, \mathbb{Q})$ and $H^\bullet(X, \mathbb{Q})$ if and only if $f^*$ induces an isomorphism between the Chow rings with rational coefficients $\mathrm{CH}^\bullet(A)_{\mathbb{Q}}$ and $\mathrm{CH}^\bullet(X)_{\mathbb{Q}}$.Comment: 6 page

Endogenous Business Cycles with Consumption Externalities

Empirical evidences tell us that in the recent years the expansion period is increased with reduction of the contraction period in the U.S. business cycles. Moreover, the business cycles in the United States also show the trend to be moderated with recent economic growth induced and supported by high technologies and their industries. We study endogenous business cycles by a modified synthesized endogenous business cycles model “in which expansions are neoclassical growth periods driven by productivity improvements and capital accumulation, while downturns are the results of Keynesian contractions in aggregate demand†(Francois and Lloyd-Ellis, 2002), with consumption externalities. By considering consumption externalities, the endogenized business cycles will be more likely to happen, the optimal consumption level will be higher, the technology growth rate will be bigger, the length of expansion will be longer and the length of contraction will be shorter. All of these results will lead to a faster and longer economic growth and smoother cycles. These theoretical results are significantly different from those in circumstances without the consumption externalities Francois and Lloyd-Ellis (2002) obtained, and are strongly supported by the data from the United States in the different periods.Endogenous Business Cycle, Consumption Externality, Endogenous Growth

Sobolev, Besov and Triebel-Lizorkin spaces on quantum tori

This paper gives a systematic study of Sobolev, Besov and Triebel-Lizorkin spaces on a noncommutative $d$-torus $\mathbb{T}^d_\theta$ (with $\theta$ a skew symmetric real $d\times d$-matrix). These spaces share many properties with their classical counterparts. We prove, among other basic properties, the lifting theorem for all these spaces and a Poincar\'e type inequality for Sobolev spaces. We also show that the Sobolev space $W^k_\infty(\mathbb{T}^d_\theta)$ coincides with the Lipschitz space of order $k$, already studied by Weaver in the case $k=1$. We establish the embedding inequalities of all these spaces, including the Besov and Sobolev embedding theorems. We obtain Littlewood-Paley type characterizations for Besov and Triebel-Lizorkin spaces in a general way, as well as the concrete ones in terms of the Poisson, heat semigroups and differences. Some of them are new even in the commutative case, for instance, our Poisson semigroup characterizations improve the classical ones. As a consequence of the characterization of the Besov spaces by differences, we extend to the quantum setting the recent results of Bourgain-Br\'ezis -Mironescu and Maz'ya-Shaposhnikova on the limits of Besov norms. The same characterization implies that the Besov space $B^\alpha_{\infty,\infty}(\mathbb{T}^d_\theta)$ for $\alpha>0$ is the quantum analogue of the usual Zygmund class of order $\alpha$. We investigate the interpolation of all these spaces, in particular, determine explicitly the K-functional of the couple $(L_p(\mathbb{T}^d_\theta), \, W^k_p(\mathbb{T}^d_\theta))$, which is the quantum analogue of a classical result due to Johnen and Scherer. Finally, we show that the completely bounded Fourier multipliers on all these spaces do not depend on the matrix $\theta$, so coincide with those on the corresponding spaces on the usual $d$-torus