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In particular, a C k-atlas that is C 0-compatible with a C 0-atlas that defines a topological manifold is said to determine a C k differential structure on the topological manifold. The C k equivalence classes of such atlases are the distinct C k differential structures of the manifold. Each distinct differential structure is determined by a ...
The study of dynamic equations on time scales reveals such discrepancies, and helps avoid proving results twice—once for differential equations and once again for difference equations. The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so-called time scale (also known as a time-set ...
Most differentiable programming frameworks work by constructing a graph containing the control flow and data structures in the program. [7] Attempts generally fall into two groups: Static, compiled graph-based approaches such as TensorFlow, [note 1] Theano, and MXNet.
The ability to define such a local differential structure on an abstract space allows one to extend the definition of differentiability to spaces without global coordinate systems. A locally differential structure allows one to define the globally differentiable tangent space, differentiable functions, and differentiable tensor and vector fields.
More precisely, what is the structure of the set of additional structures? In more general categories, this structure set has more structure: in Diff it is simply a set, but in Top it is a group, and functorially so. Many of these structures are G-structures, and the question is reduction of the structure group. The most familiar example is ...
In type theory, many abstract data types can be described as the algebra generated by a transformation that maps structures based on the type back into the type. For example, the type T of binary trees containing values of type A can be represented as the algebra generated by the transformation 1+A×T 2 →T.
Meanwhile, retail sales in October were revised up to a 0.5% increase from a prior reading that showed a 0.4% increase in the month, according to Census Bureau data. A 2.4% month-over-month ...
In multivariable calculus, the directional derivative measures the rate at which a function changes in a particular direction at a given point. [citation needed]The directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a direction ...