Language-processing functions follow heterogeneous developmental trajectories. The human embryo can already distinguish vowels in utero, but grammatical complexity is usually not fully mastered until at least 7 years of age. Examining the current literature, we propose that the ontogeny of the cortical language network can be roughly subdivided into two main developmental stages. In the first stage extending over the first 3 years of life, the infant rapidly acquires bottom-up processing capacities, which are primarily implemented bilaterally in the temporal cortices. In the second stage continuing into adolescence, top-down processes emerge gradually with the increasing functional selectivity and structural connectivity of the left inferior frontal cortex.

In this paper we study the problem of dilating unital completely positive (CP) semigroups (quantum dynamical semigroups) to weak Markov flows and then to semigroups of endomorphisms (E0-semigroups) using the language of Hilbert modules. This is a very effective, representation free approach to dilation. In this way we are able to identify the right algebra (maximal in some sense) for endomorphisms to act. We are led inevitably to the notion of tensor product systems of Hilbert modules and units for them, generalizing Arveson's notions for Hilbert spaces. In the course of our investigations we are not only able to give new natural and transparent proofs of well-known facts for semigroups on [Formula: see text], but also extend the results immediately to much more general setups. For instance, Arveson classifies E0-semigroups on [Formula: see text] up to cocycle conjugacy by product systems of Hilbert spaces.5 We find that conservative CP-semigroups on arbitrary unital C*-algebras are classified up to cocycle conjugacy by product systems of Hilbert modules. Looking at other generalizations, it turns out that the role played by E0-semigroups on [Formula: see text] in dilation theory for CP-semigroups on [Formula: see text] is now played by E0-semigroups on [Formula: see text], the full algebra of adjointable operators on a Hilbert module E. We have CP-semigroup versions of many results proved by Paschke27 for CP maps.

The relation between brain function and behavior on the one hand and the relation between structural changes and behavior on the other as well as the link between the 2 aspects are core issues in cognitive neuroscience. It is an open question, however, whether brain function or brain structure is the better predictor for age-specific cognitive performance. Here, in a comprehensive set of analyses, we investigated the direct relation between hemodynamic activity in 2 pairs of frontal and temporal cortical areas, 2 long-distance white matter fiber tracts connecting each pair and sentence comprehension performance of 4 age groups, including 3 groups of children between 3 and 10 years as well as young adults. We show that the increasing accuracy of processing complex sentences throughout development is correlated with the blood-oxygen-level-dependent activation of 2 core language processing regions in Broca's area and the posterior portion of the superior temporal gyrus. Moreover, both accuracy and speed of processing are correlated with the maturational status of the arcuate fasciculus, that is, the dorsal white matter fiber bundle connecting these 2 regions. The present data provide compelling evidence for the view that brain function and white matter structure together best predict developing cognitive performance.

Christensen and Evans showed that, in the language of Hilbert modules, a bounded derivation on a von Neumann algebra with values in a two-sided von Neumann module (i.e. a sufficiently closed two-sided Hilbert module) are inner. Then they use this result to show that the generator of a normal uniformly continuous completely positive (CP-) semigroup on a von Neumann algebra decomposes into a (suitably normalized) CP-part and a derivation like part. The backwards implication is left open.In these notes we show that both statements are equivalent among themselves and equivalent to a third one, namely, that any type I tensor product system of von Neumann modules has a unital central unit. From existence of a central unit we deduce that each such product system is isomorphic to a product system of time ordered Fock modules. We, thus, find the analogue of Arveson's result that type I product systems of Hilbert spaces are symmetric Fock spaces.On the way to our results we have to develop a number of tools interesting on their own right. Inspired by a very similar notion due to Accardi and Kozyrev, we introduce the notion of semigroups of completely positive definite kernels (CPD-semigroups), being a generalization of both CP-semigroups and Schur semigroups of positive definite C-valued kernels. The structure of a type I system is determined completely by its associated CPD-semigroup ARTICLE IN PRESS$ This work has been supported by a DAAD-DST Project based Personnel exchange Programme. and the generator of the CPD-semigroup replaces Arveson's covariance function. As another tool we give a complete characterization of morphisms among product systems of time ordered Fock modules. In particular, the concrete form of the projection endomorphisms allows us to show that subsystems of time ordered systems are again time ordered systems and to find a necessary and sufficient criterion when a given set of units generates the whole system. As a byproduct we find a couple of characterizations of other subclasses of morphisms. We show that the set of contractive positive endomorphisms are order isomorphic to the set of CPD-semigroups dominated by the CPD-semigroup associated with type I system. r