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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.
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. [1] For example, x 5 + 2 x 3 y 2 + 9 x y 4 {\displaystyle x^{5}+2x^{3}y^{2}+9xy^{4}} is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5.
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 ...
Homogeneous coordinates are not uniquely determined by a point, so a function defined on the coordinates, say (,,), does not determine a function defined on points as with Cartesian coordinates. But a condition f ( x , y , z ) = 0 {\displaystyle f(x,y,z)=0} defined on the coordinates, as might be used to describe a curve, determines a condition ...
The standard torus is homogeneous under its diffeomorphism and homeomorphism groups, and the flat torus is homogeneous under its diffeomorphism, homeomorphism, and isometry groups. In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the action of a group.
In mathematics, a homogeneous relation (also called endorelation) on a set X is a binary relation between X and itself, i.e. it is a subset of the Cartesian product X × X. [ 1 ] [ 2 ] [ 3 ] This is commonly phrased as "a relation on X " [ 4 ] or "a (binary) relation over X ".
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In the case of polynomials in more than one indeterminate, a polynomial is called homogeneous of degree n if all of its non-zero terms have degree n. The zero polynomial is homogeneous, and, as a homogeneous polynomial, its degree is undefined. [c] For example, x 3 y 2 + 7x 2 y 3 − 3x 5 is homogeneous of degree 5. For more details, see ...