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In robotics, Vector Field Histogram (VFH) is a real time motion planning algorithm proposed by Johann Borenstein and Yoram Koren in 1991. [1] The VFH utilizes a statistical representation of the robot's environment through the so-called histogram grid, and therefore places great emphasis on dealing with uncertainty from sensor and modeling errors.
Let V be a finite-dimensional vector space over some field K.The standard notion of a dual vector space V ∗ has the following property: for any K-vector spaces U and W there is an adjunction Hom K (U ⊗ V,W) = Hom K (U, V ∗ ⊗ W), and this characterizes V ∗ up to a unique isomorphism.
The dual of an infinite-dimensional space has greater dimension (this being a greater infinite cardinality) than the original space has, and thus these cannot have a basis with the same indexing set. However, a dual set of vectors exists, which defines a subspace of the dual isomorphic to the original space.
An example of a crinkled arc in the Hilbert space (,) is: [2]: [,] (,) [,] where [,]: (,) {,} is the indicator function defined by {[,] A crinkled arc can be found in every infinite−dimensional Hilbert space because any such space contains a closed vector subspace that is isomorphic to (,). [2] A crinkled arc : [,] is said to be normalized if ...
There is no “codimension of a vector space (in isolation)”, only the codimension of a vector subspace. If W is a linear subspace of a finite-dimensional vector space V, then the codimension of W in V is the difference between the dimensions: [1] = ().
In mathematics, the complexification of a vector space V over the field of real numbers (a "real vector space") yields a vector space V C over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include their scaling ("multiplication") by complex numbers.
A portion of the two dimensional grid used for Discretization is shown below: Graph of 2 dimensional plot. In addition to the east (E) and west (W) neighbors, a general grid node P, now also has north (N) and south (S) neighbors. The same notation is used here for all faces and cell dimensions as in one dimensional analysis.
Here, c[i:i+3] represents the four array elements from c[i] to c[i+3] and the vector processor can perform four operations for a single vector instruction. Since the four vector operations complete in roughly the same time as one scalar instruction, the vector approach can run up to four times faster than the original code.