Symmetries of double ratios and an equation for Möbius structures

Orthogonal representations η n : S n ↷ R N \eta _n\colon S_n\curvearrowright \mathbb {R}^N of the symmetric groups S n S_n , n ≥ 4 n\ge 4 , with N = n ! / 8 N=n!/8 , emerging from symmetries of double ratios are treated. For n = 5 n=5 , the representation η 5 \eta _5 is decomposed into irreducible components and it is shown that a certain component yields a solution of the equations that describe the Möbius structures in the class of sub-Möbius structures. In this sense, a condition determining the Möbius structures is implicit already in symmetries of double ratios.

Topic Models Ensembles for AD-HOC Information Retrieval

Ad hoc information retrieval (ad hoc IR) is a challenging task consisting of ranking text documents for bag-of-words (BOW) queries. Classic approaches based on query and document text vectors use term-weighting functions to rank the documents. Some of these methods’ limitations consist of their inability to work with polysemic concepts. In addition, these methods introduce fake orthogonalities between semantically related words. To address these limitations, model-based IR approaches based on topics have been explored. Specifically, topic models based on Latent Dirichlet Allocation (LDA) allow building representations of text documents in the latent space of topics, the better modeling of polysemy and avoiding the generation of orthogonal representations between related terms. We extend LDA-based IR strategies using different ensemble strategies. Model selection obeys the ensemble learning paradigm, for which we test two successful approaches widely used in supervised learning. We study Boosting and Bagging techniques for topic models, using each model as a weak IR expert. Then, we merge the ranking lists obtained from each model using a simple but effective top-k list fusion approach. We show that our proposal strengthens the results in precision and recall, outperforming classic IR models and strong baselines based on topic models.

Ruelle Zeta Functions of Hyperbolic Manifolds and Reidemeister Torsion

This paper is concerned with the behavior of twisted Ruelle zeta functions of compact hyperbolic manifolds at the origin. Fried proved that for an orthogonal acyclic representation of the fundamental group of a compact hyperbolic manifold, the twisted Ruelle zeta function is holomorphic at $$s=0$$ s = 0 and its value at $$s=0$$ s = 0 equals the Reidemeister torsion. He also established a more general result for orthogonal representations, which are not acyclic. The purpose of the present paper is to extend Fried’s result to arbitrary finite dimensional representations of the fundamental group. The Reidemeister torsion is replaced by the complex-valued combinatorial torsion introduced by Cappell and Miller.

Maintained representations of the ipsilateral and contralateral limbs during bimanual control in primary motor cortex

Primary motor cortex (M1) almost exclusively controls the contralateral side of the body. However, M1 activity is also modulated during ipsilateral body movements. Previous work has shown that M1 activity related to the ipsilateral arm is independent of the M1 activity related to the contralateral arm. How do these patterns of activity interact when both arms move simultaneously? We explored this problem by training two monkeys (male, Macaca mulatta) in a postural perturbation task while recording from M1. Loads were applied to one arm at a time (unimanual) or both arms simultaneously (bimanual). We found 83% of neurons were responsive to both the unimanual and bimanual loads. We also observed a small reduction in activity magnitude during the bimanual loads for both limbs (25%). Across the unimanual and bimanual loads, neurons largely maintained their preferred load directions. However, there was a larger change in the preferred loads for the ipsilateral limb (~25%) than the contralateral limb (~9%). Lastly, we identified the contralateral and ipsilateral subspaces during the unimanual loads and found they captured a significant amount of the variance during the bimanual loads. However, the subspace captured more of the bimanual variance related to the contralateral limb (97%) than the ipsilateral limb (66%). Our results highlight that even during bimanual motor actions, M1 largely retains its representations of the contralateral and ipsilateral limbs.Significance StatementPrevious work has shown that primary motor cortex (M1) reflects information related to the contralateral limb, its downstream target, but also reflects information related to the ipsilateral limb. Can M1 still reflect both sources of information when performing simultaneous movements of the limbs? Here we use a postural perturbation task to show that M1 activity maintains a similar representation for the contralateral limb during bimanual motor actions, while there is only a modest change in the representation of the ipsilateral limb. Our results indicate that two orthogonal representations can be maintained and expressed simultaneously in M1.