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In mathematics, a homogeneous function is a function of several variables such that the following holds: If each of the function's arguments is multiplied by the same scalar, then the function's value is multiplied by some power of this scalar; the power is called the degree of homogeneity, or simply the degree.
A polynomial of degree 0 is always homogeneous; it is simply an element of the field or ring of the coefficients, usually called a constant or a scalar. A form of degree 1 is a linear form. [notes 2] A form of degree 2 is a quadratic form. In geometry, the Euclidean distance is the square root of a quadratic form.
Any distribution S on R homogeneous of degree α ≠ −1, −2, ... is of this form as well. As a result, every homogeneous distribution of degree α ≠ −1, −2, ... on R \ {0} extends to R. Finally, homogeneous distributions of degree −k, a negative integer, on R are all of the form:
A differential equation can be homogeneous in either of two respects.. A first order differential equation is said to be homogeneous if it may be written (,) = (,),where f and g are homogeneous functions of the same degree of x and y. [1]
Homogeneity of degree 1 / operation of scalar multiplication () = Thus, a linear map is said to be operation preserving. In other words, it does not matter whether the linear map is applied before (the right hand sides of the above examples) or after (the left hand sides of the examples) the operations of addition and scalar multiplication.
In consumer theory, a consumer's preferences are called homothetic if they can be represented by a utility function which is homogeneous of degree 1. [1]: 146 For example, in an economy with two goods ,, homothetic preferences can be represented by a utility function that has the following property: for every >:
From January 2008 to December 2012, if you bought shares in companies when Donald R. Keough joined the board, and sold them when he left, you would have a 9.0 percent return on your investment, compared to a -2.8 percent return from the S&P 500.
In mathematics, a linear map or linear function f(x) is a function that satisfies the two properties: [1] Additivity: f(x + y) = f(x) + f(y). Homogeneity of degree 1: f(αx) = α f(x) for all α. These properties are known as the superposition principle.