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In mathematics, a characteristic class is a way of associating to each principal bundle of X a cohomology class of X. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classes are global invariants that measure the deviation of a local product structure from a global product structure. They are one of the unifying geometric concepts in algebraic topology, differential geometry, and algebraic geometry. The notion of characteristic class arose in 1935 in the work of Eduard Stiefel and Hassler Whitney about vector fields on manifolds. Let G be a topological group, and for a topological space , write for the set of isomorphism classes of principal G-bundles over . This is a contravariant functor from Top (the of topological spaces and continuous functions) to Set (the category of sets and functions), sending a map to the pullback operation . A characteristic class c of principal G-bundles is then a natural transformation from to a cohomology functor , regarded also as a functor to Set. In other words, a characteristic class associates to each principal G-bundle in an element c(P) in H*(X) such that, if f : Y → X is a continuous map, then c(fP) = fc(P). On the left is the class of the pullback of P to Y; on the right is the image of the class of P under the induced map in cohomology. Characteristic classes are elements of cohomology groups; one can obtain integers from characteristic classes, called characteristic numbers. Some important examples of characteristic numbers are Stiefel–Whitney numbers, Chern numbers, Pontryagin numbers, and the Euler characteristic. Given an oriented manifold M of dimension n with fundamental class , and a G-bundle with characteristic classes , one can pair a product of characteristic classes of total degree n with the fundamental class. The number of distinct characteristic numbers is the number of monomials of degree n in the characteristic classes, or equivalently the partitions of n into .

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This paper presents our approach to predicting future error-related events in a robot-mediated gamified phys- ical training activity for stroke patients. The ability to predict future error under such conditions suggests the existence of distinguishable features and separated class characteristics between the casual gameplay state and error prune state in the data. Identifying such features provides valuable insight to creating individually tailored, adaptive games as well as possible ways to increase rehabilitation success by patients. Considering the time-series nature of sensory data created by motor actions of patients we employed a predictive analysis strategy on carefully engineered features of sequenced data. We split the data into fixed time windows and explored logistic regression models, decision trees, and recurrent neural networks to predict the likelihood of a patient making an error based on the features from the time window before the error. We achieved an 84.4% F1-score with a 0.76 ROC value in our best model for predicting motion accuracy related errors. Moreover, we computed the permutation importance of the features to explain which ones are more indicative of future errors.

2022

Characteristic classes as complete obstructions

Martina Rovelli

In the first part of this paper, we propose a uniform interpretation of characteristic classes as obstructions to the reduction of the structure group and to the existence of an equivariant extension of a certain homomorphism defined a priori only on a single fiber of the bundle. Afterwards, we define a family of invariants of principal bundles that detect the number of group reductions that a principal bundle admits. We prove that they fit into a long exact sequence of abelian groups, together with the cohomology of the base space and the cohomology of the classifying space of the structure group.

SPRINGER HEIDELBERG2019

Towards new invariants for principal bundles

Martina Rovelli

We investigate the theory of principal bundles from a homotopical point of view. In the first part of the thesis, we prove a classification of principal bundles over a fixed base space, dual to the well-known classification of bundles with a fixed structure group. This leads to an adjointness property in a homotopical context between the classifying space and the loop space. We then focus on characteristic classes, which are invariants for principal bundles that take values in the cohomology of the base space. Each characteristic class captures different geo- metric features of principal bundles. We propose a uniform treatment to interpret most of known characteristic classes as obstructions to group reduction and to the extension of a universal cocycle. By plugging in the correct parameters, the method recovers several classical theorems. Afterwards, we construct a long exact sequence of abelian groups for any principal bundle. This sequence involves the cohomology of the base space and the group cohomology of the structure group. Moreover the connecting map is deeply related with the characteristic classes of the bundle.

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Characteristic class

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In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an -dimensional manifold, or -manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of -dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane.

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In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches of mathematics and physics, such as string theory, Chern–Simons theory, knot theory, Gromov-Witten invariants. Chern classes were introduced by . Chern classes are characteristic classes. They are topological invariants associated with vector bundles on a smooth manifold.

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