Non-analytic Smooth Functions

Related lectures (51)

Covers retractions, tangent bundles, and embedded submanifolds on manifolds with proofs and examples.

Covers the topology of Riemann surfaces, focusing on orientation and orientability.

Explores algebraic topology and differential geometry in understanding robot dynamics and global behavior.

Explores gradient descent methods for optimizing functions on manifolds, emphasizing small gradient guarantees and global convergence.

Explores smooth functions on manifolds, emphasizing continuity and atlas topologies.

Explores smooth maps, differentials, composition properties, linearity, and extensions on manifolds.

Explores the transition from embedded to general manifolds, upgrading foundational concepts and discussing mathematical reasons for both approaches.

Explores Riemannian metrics on manifolds and the concept of Riemannian submanifolds in Euclidean spaces.

Covers optimization on manifolds, smoothness, tools needed for optimization, and advanced algorithms using Hessians and Riemannian connections.

Explores smooth functions, n-dimensional charts, compatibility, and atlases for manifolds.

Explores upgrading foundations from embedded to general manifolds in optimization, discussing smooth sets and tangent vectors.

Covers the basic properties of holomorphic maps and Taylor series expansions in complex analysis.

Explores Taylor expansions and retractions on Riemannian manifolds, emphasizing second-order approximations and covariant derivatives.

Covers Riemannian connections, emphasizing their properties and significance in geometry.

Introduces differentiating vector fields along curves on manifolds with connections and the unique operator satisfying specific properties.

Explores necessary and sufficient optimality conditions for local minima on manifolds, focusing on second-order critical points.

Covers second-order retractions in optimization on manifolds, focusing on smooth curves and their relation to the gradient and Hessian of a function.

Covers the concept of local frames, their construction, and limitations.

Covers optimization on manifolds, focusing on smooth manifolds and functions, and the process of gradient descent.

Explores the definition of connections for smooth vector fields on manifolds.

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