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Characterized Subgroups
Full text linkLet T = R / Z be the written additively circle group and u =
(un) be a sequence of integers. Many authors in various areas
of Mathematics gave their attention to the following subgroups
of T and their subsets
t u( T ) = { x ∈ T | unx → 0 } .
These subgroups are known with various names, here I refer
to these subgroups as topologically u-torsion subgroups, because
of their strong connection with torsion subgroups. Here, be-
sides these subgroups in the circle group, I consider their nat-
ural generalization for an arbitrary topological abelian group,
with particular attention to the compact case: for a topologi-
cal abelian group X and a sequence of characters v = (vn) the
following subgroup
s v(X) = { x ∈ X | vn(x) → 0 }
is called characterized subgroup.
Here I present some of my research results. In particular, I
give a complete description of the subgroups t u( T ) where u
is an arithmetic sequence, that is a strictly increasing sequence
where un | un+1 for every n ∈ N. I give also some new results
on the study of the Borel complexity of these subgroups, both
in the compact case and in the circle group. Moreover, I present
a structure theorem for the subgroups that admit a finer locally
compact Polish group topology. The latter is a sufficient condi-
tion for a subgroup to be characterized. Furthermore, I give a
complete description of closed characterized subgroups in arbi-
trary topological abelian groups and various useful reductions
to the metrizable case. Presenting these results, I take the op-
portunity to give an exhaustive description of the state of the
art in this topic and to show some applications to other areas of
Mathematics, with the aim of providing a useful handbook to
an expert audience and a starting point for potential research
purposes to non-expert users
Topologically torsion elements of the circle group
No full textLet (m_n) be a faithfully enumerated sequence of integers with m_n|m_{n+1} for every n in N. We describe the topologically (m_n)-torsion elements of the circle group T=R/Z (written additively), namely those elements x in T such that m_nx coverges to 0
Metrizability of some hereditarily normal compact like groups
No full textInspired by the fact that a compact topological group is
hereditarily normal if and only if it is metrizable, we prove that various levels of compactness-like properties imposed on a topological group G allow one to establish that G is hereditarily normal if and only if G is metrizable (among these properties are locally compactness, local minimality and \omega-boundedness). This extends recent results from [4] in the case of countable compactness
Visuospatial attention, motor intention, action affordance and brain plasticity: neurophysiology and network analysis
No full textThe project addresses sensorimotor integration for motor intention and execution within a unitary perspective, based on the
assumption that the underlying processes can only be understood if considered as different aspects of a unitary mental
construct, enabling non-human primates and humans to deploy attention and encode command functions to move and act
effortlessly in the extra-personal space.
The core objectives of the project concern: a) cortical encoding of kinetics and kinematics of grasping and reaching, their
coupling during coordinated action, and action recognition when performed by another agent; b) the analysis of the memory
reservoirs which help planning future action based on previous experience; c) how complex motor cortical circuits generate
ethologically relevant forms of behavior; d) brain encoding of action affordances; e) the mechanisms necessary to allocate
attention to a salient target, while resisting distractors, as a necessary prerequisite for successful action planning and
performance. The statistical analysis of the distributed cortical system responsible for these functions will unravel the
anatomical substrates of the above mentioned functions.
Our approach will combine behavioral neurophysiology and neuroanatomy in monkeys, and will also include
neurophysiological studies in humans. Neural activity (single unit, multi-unit, local field potentials) in different frontal,
parietal and extrastriate cortical areas, including some of their subcortical targets such as the putamen, will be studied while
monkeys perform ad hoc tasks assumed to recruit these areas, and inspired by their anatomical input and by the
consequences of their lesion on behavior. These studies will not only be of correlative nature, but will also include the
complex analysis of the causal relations between neural activity and behavior, through reversible inactivation of specific
cortical sites in behaving animals. Cutting-edge experimental techniques will also be adopted in humans as well, where
motor cortex will be electrically stimulated during neurosurgery in awake patients at behaviorally-relevant time scales, to
study how ethologically relevant actions are generated, how their repeated performance affects cortical circuits, and how the
wiring diagram of the cortical motor output can be reshaped by (hand) use.
The relevance of these studies of sensorimotor control for the rehabilitation of skilled hand action as consequence of brain
lesion is direct. Beyond the obvious statement that understanding brain function is essential to understand brain dysfunction,
our project aims at conveying the message that this can only be feasible at multi-scale level, i.e. the micro-scale of cell
function, the meso-scale level of cortical circuits, and the macro-scale level of behavioral analysis
Characterized subgroups of topological abelian groups
Full text linkA subgroup H of a topological abelian group X is said to be characterized by a sequence v of characters of X if H={x∈X:v_n(x)→0 in T}. We study the basic properties of characterized subgroups in the general setting, extending results known in the compact case. For a better description, we isolate various types of characterized subgroups. Moreover, we introduce the relevant class of auto-characterized groups (namely, the groups that are characterized subgroups of themselves by means of a sequence of non-null characters); in the case of locally compact abelian groups, these are proven to be exactly the non-compact ones. As a by-product of our results, we find a complete description of the characterized subgroups of discrete abelian groups
Claustral afferents of superior parietal areas PEc and PE in the macaque
Full text linkThe exposed surface of the primate superior parietal cortex
includes two cytoarchitectonically defined areas, the
PEc and PE. In the present study we describe the distribution
of neurons projecting from the claustrum to these
areas. Retrograde neuronal tracers were injected by
direct visualization of regions of interest, and the location
of injection sites was reconstructed relative to cytoarchitectural
borders. For comparison, the patterns of claustral
label that resulted from injections involving
neighboring cytoarchitectonic areas were analyzed. We
found that the claustral territories sending projections to
areas PE and PEc partially overlapped zones previously
shown to form projections to the posterior parietal, somatosensory,
visual, and motor cortex. The projection zones
to the PE and PEc overlapped extensively, and consisted
of multiple patches separated by label-free zones. Most
of the labeled neurons were located in the posterior–ventral
part of the claustrum. Area PE received additional
inputs from a posterior–dorsal part of the claustrum,
which has been previously reported to project to the
somatosensory cortex, while the PEc receives additional
input from an anterior–ventral region of the claustrum,
which has been reported to project to the visual association
cortex. These observations reflect the known functional
properties of the PE and PEc, with the former
containing neurons that are predominantly involved in
somatosensory processing, and the latter including both
somatosensory and visual neurons. The present results
suggest that the claustrum projections may help coordinate
the activity of an extensive neural circuit involved in
sensory and motor processing for movement execution
Thalamo-cortical projections to the macaque superior parietal lobule areas PEc and PE
Full text linkThe exposed surface of the superior parietal lobule in macaque brain contains two architectonically defined areas named PEc and PE. The aim of the present study is the characterization of thalamic afferents of these two areas. For this purpose, retrograde neuronal tracers were injected, or placed in crystal form, in areas PEc and PE. We found that the two areas show a similar pattern of thalamic inputs, mainly originating from Lateral Posterior (LP), Pulvinar (Pul), Ventral Posterior Lateral (VPL), and Ventral Lateral (VL) nuclei, all structures known to be involved in visual, somatosensory, and/or sensorimotor processing. Minor afferents were observed from the Centromedian/Parafascicular complex (CM/PF), Central Lateral (CL), Ventral Anterior (VA), and Medial Dorsal (MD) nuclei. LP and VL were more strongly connected to PEc than to PE, while the other main thalamic inputs to the two areas showed slight differences in strength. The part of the Pul mostly connected with areas PEc and PE was the Medial Pul. No labeled cells were found in the retinotopically organized Lateral and Inferior Pul. In the somatotopically organized VPL and VL nuclei, labeled neurons were mainly found in regions likely to correspond to the trunk and limb representations (in particular the legs). These findings are in line with the sensory-motor nature of areas PEc and PE, and with their putative functional roles, being them suggested to be involved in the preparation and control of limb interaction with the environment, and in locomotion
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