A Method for Weight Multiplicity Computation Based on Berezin Quantization

Let $G$ be a compact semisimple Lie group and $T$ be a maximal torus of $G$. We describe a method for weight multiplicity computation in unitary irreducible representations of $G$, based on the theory of Berezin quantization on $G/T$. Let $\Gamma_{\rm hol}(\mathcal{L}^{\lambda})$ be the reproducing kernel Hilbert space of holomorphic sections of the homogeneous line bundle $\mathcal{L}^{\lambda}$ over $G/T$ associated with the highest weight $\lambda$ of the irreducible representation $\pi_{\lambda}$ of $G$. The multiplicity of a weight $m$ in $\pi_{\lambda}$ is computed from functional analytical structure of the Berezin symbol of the projector in $\Gamma_{\rm hol}(\mathcal{L}^{\lambda})$ onto subspace of weight $m$. We describe a method of the construction of this symbol and the evaluation of the weight multiplicity as a rank of a Hermitian form. The application of this method is described in a number of examples

Cortical integration of contextual information across objects

Abstract Recognizing objects in the environment and understanding our surroundings often depends on context: the presence of other objects and knowledge about their relations with each other. Such contextual information activates a set of medial lobe brain regions, the parahippocampal cortex and the retrosplenial complex. Both regions are more activated by single objects with a unique contextual association than by objects not associated with any specific context. Similarly they are more activated by spatially coherent arrangements of objects when those are consistent with their known spatial relations. The current study tested how context in multiple-object displays is represented in these regions in the absence of relevant spatial information. Using an fMRI slow-event-related design, we show that the precuneus (a subpart of the retrosplenial complex) is more activated by simultaneously presented contextually related objects than by unrelated objects. This suggests that the representation of context in this region is cumulative, representing integrated information across objects in the display. We discuss these findings in relation to processing of visual information and relate them to previous findings of contextual effects in perception.</jats:p

Famous Faces Activate Contextual Associations in the Parahippocampal Cortex

The parahippocampal cortex (PHC) has been traditionally implicated both in place processing and in episodic memory. How could the same cortical region mediate these cognitive functions that seem quite different? We have recently proposed that the PHC should be seen as more generally mediating contextual associative processing, which is required for both navigation and memory. We therefore predicted that any associative objects should activate the PHC. To test this generalization, we investigated the extent to which common stimuli that are nonspatial by nature, namely faces, activate the PHC, although their perception is typically associated with other cortical structures. Specifically, we compared the activation elicited by famous faces, which are highly associated with rich pictorial and contextual information (e.g., Tom Cruise) and are not associated with a specific place, with activation elicited by unfamiliar faces. Consistent with our prediction, contrasting famous with unfamiliar faces revealed significant activation within the PHC. Taken collectively, these findings indicate that the PHC should be regarded as mediating contextual associations in general and not necessarily spatial or episodic informatio

The Cortical Underpinnings of Context-based Memory Distortion

Everyday contextual settings create associations that later afford generating predictions about what objects to expect in our environment. The cortical network that takes advantage of such contextual information is proposed to connect the representation of associated objects such that seeing one object (bed) will activate the visual representations of other objects sharing the same context (pillow). Given this proposal, we hypothesized that the cortical activity elicited by seeing a strong contextual object would predict the occurrence of false memories whereby one erroneously "remembers'' having seen a new object that is related to a previously presented object. To test this hypothesis, we used functional magnetic resonance imaging during encoding of contextually related objects, and later tested recognition memory. New objects that were contextually related to previously presented objects were more often falsely judged as "old'' compared with new objects that were contextually unrelated to old objects. This phenomenon was reflected by activity in the cortical network mediating contextual processing, which provides a better understanding of how the brain represents and processes context.Psycholog

Berry's phase for compact Lie groups

The Lie group adiabatic evolution determined by a Lie algebra parameter dependent Hamiltonian is considered. It is demonstrated that in the case when the parameter space of the Hamiltonian is a homogeneous K\"ahler manifold its fundamental K\"ahler potentials completely determine Berry geometrical phase factor. Explicit expressions for Berry vector potentials (Berry connections) and Berry curvatures are obtained using the complex parametrization of the Hamiltonian parameter space. A general approach is exemplified by the Lie algebra Hamiltonians corresponding to SU(2) and SU(3) evolution groups.Comment: 24 pages, no figure

Enabling global processing in simultanagnosia by psychophysical biasing of visual pathways

A fundamental aspect of visual cognition is our disposition to see the ‘forest before the trees'. However, damage to the posterior parietal cortex, a critical brain region along the dorsal visual pathway, can produce a neurological disorder called simultanagnosia, characterized by a debilitating inability to perceive the ‘forest' but not the ‘trees' (i.e. impaired global processing despite intact local processing). This impairment in perceiving the global shape persists even though the ventral visual pathway, the primary recognition pathway, is intact in these patients. Here, we enabled global processing in patients with simultanagnosia using a psychophysical technique, which allowed us to bias stimuli such that they are processed predominantly by the intact ventral visual pathway. Our findings reveal that the impairment in global processing that characterizes simultanagnosia stems from a disruption in the processing of low-spatial frequencies through the dorsal pathway. These findings advance our understanding of the relationship between visuospatial attention and perception and reveal the neural mechanism mediating the disposition to see the ‘forest before the trees

Realization of compact Lie algebras in K\"ahler manifolds

The Berezin quantization on a simply connected homogeneous K\"{a}hler manifold, which is considered as a phase space for a dynamical system, enables a description of the quantal system in a (finite-dimensional) Hilbert space of holomorphic functions corresponding to generalized coherent states. The Lie algebra associated with the manifold symmetry group is given in terms of first-order differential operators. In the classical theory, the Lie algebra is represented by the momentum maps which are functions on the manifold, and the Lie product is the Poisson bracket given by the K\"{a}hler structure. The K\"{a}hler potentials are constructed for the manifolds related to all compact semi-simple Lie groups. The complex coordinates are introduced by means of the Borel method. The K\"{a}hler structure is obtained explicitly for any unitary group representation. The cocycle functions for the Lie algebra and the Killing vector fields on the manifold are also obtained

Comments on the Morphology of Nouns in Mishnaic Hebrew: Nouns attested and unattested in Biblical Hebrew

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