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A norm over a real vector space is an example of a positively homogeneous function that is not homogeneous. A special case is the absolute value of real numbers. The quotient of two homogeneous polynomials of the same degree gives an example of a homogeneous function of degree zero. This example is fundamental in the definition of projective ...
A linear differential equation is homogeneous if it is a homogeneous linear equation in the unknown function and its derivatives. It follows that, if φ(x) is a solution, so is cφ(x), for any (non-zero) constant c. In order for this condition to hold, each nonzero term of the linear differential equation must depend on the unknown function or ...
The function defined by a homogeneous polynomial is always a homogeneous function. An algebraic form, or simply form, is a function defined by a homogeneous polynomial. [notes 1] A binary form is a form in two variables. A form is also a function defined on a vector space, which may be expressed as a homogeneous function of the coordinates over ...
2.1 Example: homogeneous case. 2.2 Example: nonhomogeneous case. 2.3 Example: mixed derivatives. ... The function u-v then satisfies homogeneous boundary condition, ...
In the above example, one can take a nonzero from the kernel of . Consequently, the ... A nonnegative sublinear function is a nonnegative homogeneous function : ...
Multiplying this by the generating function for the complete homogeneous symmetric polynomials, one obtains the constant series 1 (equivalently, plethystic exponentials satisfy the usual properties of an exponential), and the relation between the elementary and complete homogeneous polynomials follows from comparing coefficients of t m.
A first-order homogeneous function of two positive variables and , (that is, a ... Examples of functions that are monotonically increasing but not convex include () = ...
An example is given by the three-dimensional Euclidean space and the square of the Euclidean norm expressing the ... The map Q is a homogeneous function of degree ...