What an infra-nilmanifold endomorphism really should be

Infra-nilmanifold endomorphisms were introduced in the late sixties. They play a very crucial role in dynamics, especially when studying expanding maps and Anosov diffeomorphisms. However, in this note we will explain that the two main results in this area are based on a false result and that although we can repair one of these two theorems, there remains doubt on the correctness of the other one. Moreover, we will also show that the notion of an infra-nilmanifold endomorphism itself has not always been interpreted in the same way. Finally, we define a slightly more general concept of the notion of an infra-nilmanifold endomorphism and explain why this is really the right concept to work with

Crystallographic actions on contractible algebraic manifolds

We study properly discontinuous and cocompact actions of a discrete subgroup $\Gamma$ of an algebraic group $G$ on a contractible algebraic manifold $X$. We suppose that this action comes from an algebraic action of $G$ on $X$ such that a maximal reductive subgroup of $G$ fixes a point. When the real rank of any simple subgroup of $G$ is at most one or the dimension of $X$ is at most three, we show that $\Gamma$ is virtually polycyclic. When $\Gamma$ is virtually polycyclic, we show that $\Gamma$ is virtually polycyclic. When $\Gamma$ is virtually polycyclic, we show that the action reduces to a NIL-affine crystallographic action. As applications, we prove that the generalized Auslander conjecture for NIL-affine actions holds up to dimension six and give a new proof of the fact that every virtually polycyclic group admits a NIL-affine crystallographic action.Comment: This final version has been accepted for publication in 2013. The statements of the main results are now more general as they cover algebraic groups G where the real rank of any simple subgroup of G is at most on

The power of Store brands: Intrinsic loyalty and conquesting power.

An important evolution in the retailing industry is the growing success of store brands. Still, their level of penetration varies widely across countries and industries. We provide an operational measure of quantify the power of store brands along two dimensions : the intrinsic loyalty of their customer base, and their conquesting power to attract potential switchers. Based on their position along these two dimensions, we classify store and national brands as 'giants, 'misers', 'fighters' or 'artisans'. We use the proposed operationalization to evaluate the absolute and relative strenght of Albert Heijn, the leading Dutch store brand, in 19 products categories.Brands; Loyalty;

Sustained spending and persistent response: a new look at long-term marketing profitability.

An intuitively appealing decision rule is to allocate a company's scarce marketing resources where they have the greatest long-term benefit. This principle, however, is easier to accept than it is to execute, because long-run effects of marketing spending are difficult to estimate. We address this problem by examining the over-time behavior of market response and marketing spending, and identify four commonly occurring strategic scenarios: business as usual, hysteresis in response, escalating expenditures and evolving-business practice. We explain and illustrate why each scenario can occur in practice, and describe its positive and negative consequences for long-term profitability.When good time-series data on revenue and marketing spending are available, it is possible to apply multivariate persistence measures to identify which of the four strategic scenarios is taking place. We apply these ideas to data from two major companies in the packaged-foods and pharmaceuticals industries. We observe several long-term marketing effect, some with profitable and some with unprofitable consequences, and offer recommendations for each case.We conclude that high-quality databases along with modern time-series methods can be instrumental in extracting vital long-term marketing-effectiveness information from readily available data. Therefore, managing marketing resources with long-run performance in mind need no longer be a pure act of faith on behalf of the executive. We hope that this and future work will contribute toward an improved allocation of scarce marketing resources in our companies.Marketing; Profitability;