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We present a construction that enables one to find Banach spaces
of norm attaining functionals do not contain two-dimensional subspaces and such that, consequently,
does not contain proximinal subspaces of finite codimension greater than one, extending the results recently provided by Read [Banach spaces with no proximinal subspaces of codimension 2, Israel J. Math. (to appear)] and Rmoutil [Norm-attaining functionals need not contain 2-dimensional subspaces, J. Funct. Anal.272 (2017), 918–928]. Roughly speaking, we construct an equivalent renorming with the requested properties for every Banach space
where the set
for the original norm is not “too large”. The construction can be applied to every Banach space containing
and having a countable system of norming functionals, in particular, to separable Banach spaces containing
. We also provide some geometric properties of the norms we have constructed.
By themselves, mesencephalic subcortical mechanisms provide a preattentive kind of consciousness, related to stimulus-related, short-latency dopamine release triggered by collicular input. Elaborate forms of consciousness, containing identifiable objects (visual, auditory, tactile, or chemical), imply longer-lasting phenomena that depend on the activation of prosencephalic networks. Nevertheless, the maintenance of these higher-level networks strongly depends on long-lasting mesencephalic dopamine release.
in the target article, sagvolden and collaborators propose that attentional-deficit/hyperactivity disorder (adhd) is the result of a general behavioral deficit which is mainly caused by a hypofunctioning mesolimbic dopaminergic system. although we partly agree with this view, we think that it tends to overlook the dysfunction of prefrontal and frontostriatal executive functions by considering them to be a consequence of alterations in reward and extinction mechanisms. rather, we believe that adhd is the result of an overall cognitive and behavioral condition, associated to a generalized dopaminergic network dysfunction, and may not be easily attributable to a single basic behavioral function.
We commend Arbib for his original proposal that a mirror neuron system may have participated in language origins. However, in our view he proposes a complex evolutionary scenario that could be more parsimonious. We see no necessity to propose a hand-based signing stage as ancestral to vocal communication. The prefrontal system involved in human speech may have its precursors in the monkey's inferior frontal cortical domain, which is responsive to vocalizations and is related to laryngeal control.
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