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Hertz vectors, or the Hertz vector potentials, are an alternative formulation of the electromagnetic potentials. They are most often introduced in electromagnetic theory textbooks as practice problems for students to solve. [1] There are multiple cases where they have a practical use, including antennas [2] and waveguides. [3]
The block Wiedemann algorithm can be used to calculate the leading invariant factors of the matrix, ie, the largest blocks of the Frobenius normal form.Given and , where is a finite field of size , the probability that the leading < invariant factors of are preserved in = is
Relevant is the distinction between polar and axial vectors in vector algebra, which is natural in geometric algebra as the distinction between vectors and bivectors (elements of grade two). The I {\displaystyle I} here is a unit pseudoscalar of Euclidean 3-space, which establishes a duality between the vectors and the bivectors, and is named ...
Several methods for solving Wahba's problem are discussed by Markley and Mortari. This is an alternative formulation of the Orthogonal Procrustes problem (consider all the vectors multiplied by the square-roots of the corresponding weights as columns of two matrices with N columns to obtain the
It is then determined by the coordinates of that bound vector's terminal point. Thus the free vector represented by (1, 0, 0) is a vector of unit length—pointing along the direction of the positive x-axis. This coordinate representation of free vectors allows their algebraic features to be expressed in a convenient numerical fashion.
This is an illustration of the shortest vector problem (basis vectors in blue, shortest vector in red). In the SVP, a basis of a vector space V and a norm N (often L 2) are given for a lattice L and one must find the shortest non-zero vector in V, as measured by N, in L.
Vectorization is used in matrix calculus and its applications in establishing e.g., moments of random vectors and matrices, asymptotics, as well as Jacobian and Hessian matrices. [5] It is also used in local sensitivity and statistical diagnostics.
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
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