A compatible analytic manifold structure for Lie pseudogroups of infinite type

Author(s):  
Niky Kamran ◽  
Thierry Robart
2018 ◽  
Vol 30 (10) ◽  
pp. 1810
Author(s):  
Lei Hua ◽  
Chongcheng Chen ◽  
Liyu Tang ◽  
Ying Jiang

2021 ◽  
Vol 58 (2) ◽  
pp. 335-346
Author(s):  
Mackenzie Simper

AbstractConsider an urn containing balls labeled with integer values. Define a discrete-time random process by drawing two balls, one at a time and with replacement, and noting the labels. Add a new ball labeled with the sum of the two drawn labels. This model was introduced by Siegmund and Yakir (2005) Ann. Prob.33, 2036 for labels taking values in a finite group, in which case the distribution defined by the urn converges to the uniform distribution on the group. For the urn of integers, the main result of this paper is an exponential limit law. The mean of the exponential is a random variable with distribution depending on the starting configuration. This is a novel urn model which combines multi-drawing and an infinite type of balls. The proof of convergence uses the contraction method for recursive distributional equations.


2006 ◽  
Vol 18 (10) ◽  
pp. 2509-2528 ◽  
Author(s):  
Yoshua Bengio ◽  
Martin Monperrus ◽  
Hugo Larochelle

We claim and present arguments to the effect that a large class of manifold learning algorithms that are essentially local and can be framed as kernel learning algorithms will suffer from the curse of dimensionality, at the dimension of the true underlying manifold. This observation invites an exploration of nonlocal manifold learning algorithms that attempt to discover shared structure in the tangent planes at different positions. A training criterion for such an algorithm is proposed, and experiments estimating a tangent plane prediction function are presented, showing its advantages with respect to local manifold learning algorithms: it is able to generalize very far from training data (on learning handwritten character image rotations), where local nonparametric methods fail.


2020 ◽  
Vol 32 (5) ◽  
pp. 1289-1296
Author(s):  
Junbin Dong

AbstractWe generalize the Alvis–Curtis duality to the abstract representations of reductive groups with Frobenius maps. Similar to the case of representations of finite reductive groups, we show that the Alvis–Curtis duality of infinite type, which we define in this paper, also interchanges the irreducible representations in the principal representation category.


Author(s):  
Arye Juhász

It is conjectured that an irreducible Artin group which is of infinite type has trivial center. The conjecture is known to be true for two-dimensional Artin groups and for a few other types of Artin groups. In this work, we show that the conjecture holds true for Artin groups which satisfy a condition stronger than being of infinite type. We use small cancellation theory of relative presentations.


2021 ◽  
Author(s):  
Corson N Areshenkoff ◽  
Daniel J Gale ◽  
Joe Y Nashed ◽  
Dominic Standage ◽  
John Randall Flanagan ◽  
...  

Humans vary greatly in their motor learning abilities, yet little is known about the neural mechanisms that underlie this variability. Recent neuroimaging and electrophysiological studies demonstrate that large-scale neural dynamics inhabit a low-dimensional subspace or manifold, and that learning is constrained by this intrinsic manifold architecture. Here we asked, using functional MRI, whether subject-level differences in neural excursion from manifold structure can explain differences in learning across participants. We had subjects perform a sensorimotor adaptation task in the MRI scanner on two consecutive days, allowing us to assess their learning performance across days, as well as continuously measure brain activity. We find that the overall neural excursion from manifold activity in both cognitive and sensorimotor brain networks is associated with differences in subjects' patterns of learning and relearning across days. These findings suggest that off-manifold activity provides an index of the relative engagement of different neural systems during learning, and that intersubject differences in patterns of learning and relearning across days are related to reconfiguration processes in cognitive and sensorimotor networks during learning.


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