Sentence examples for dual bundle from inspiring English sources

Definition of real and complex vector bundles, tangent and cotangent bundles, basic operations on bundles such as dual bundle, tensor products, exterior products, direct sums, pull-back bundles.

The dual bundle E* admits a linear Poisson structure, a 2-vector field, ^E* such that [^E*,^E*]=0 and ^E* df,d f') is a linear function for any linear functions f,f'on E*.

We also consider *π E * E M to be the vector bundle projection of the dual bundle E*to E. For trivial N-connection structures, we can write respectively E * and E (we shall also use 'not-boldfaced' symbols if the constructions will not depend on existence of the N-connection structure).

Linear algebra shows that, once we identify TM, TN with their dual bundles via the Riemannian metrics, (|Df|^2 = Tr ( (Df)^{vee } circ Df )), hence we integrate over M the sum of the eigenvalues of the endomorphism ((Df)^{vee } circ Df : TM rightarrow TM).

As an application, the duality breaking for group bundles vs. tensor C∗-categories with nonsimple unit is discussed in the setting of Nistor Troitsky gauge-equivariant K-theory: there is a map assigning a nonabelian gerbe to a tensor category, and "triviality" of the gerbe is equivalent to the existence of a dual group bundle.

During the Christmas season of 2008, Some retailers, such as Target, sold in-store a dual guitar bundle which included two identical wireless Les Paul styled guitars from previous game versions and did not include the new guitar with the tapping area.

Prolongations to Lie algebroids on dual vector bundles.

The aim of this section is to outline the main features of the Hamilton mechanics realized in terms of prolongations to Lie algebroids (with associated covector, dual and bundles) provided with N-connection structure, defined by a Lagrangian and/or Hamiltonian.

We solve the dual problem with a bundle method.

While describing complex structures as integrable almost complex structures (see Theorem 32), it is convenient to view an almost complex structure as a differentiable (0, 1 -form with values in the dual of the cotangent bundle ((TY^{1 -formvee }).

By contrast, Intel's quad-core chips merely bundle two dual-core chips together.