Search results
Results from the WOW.Com Content Network
Given a homogeneous polynomial of degree with real coefficients that takes only positive values, one gets a positively homogeneous function of degree / by raising it to the power /. So for example, the following function is positively homogeneous of degree 1 but not homogeneous: ( x 2 + y 2 + z 2 ) 1 2 . {\displaystyle \left(x^{2}+y^{2}+z^{2 ...
A C k field, more generally, is one for which any homogeneous polynomial of degree d in N variables has a non-trivial zero, provided d k < N, for k ≥ 1. [11] The condition was first introduced and studied by Lang. [10] If a field is C i then so is a finite extension. [11] [12] The C 0 fields are precisely the algebraically closed fields. [13 ...
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.
Let H be the homogeneous ideal generated by the homogeneous parts of highest degree of the elements of I. If I is homogeneous, then H=I. Finally let B be a Gröbner basis of I for a monomial ordering refining the total degree partial ordering and G the (homogeneous) ideal generated by the leading monomials of the elements of B.
In thermodynamics, the phase rule is a general principle governing multi-component, multi-phase systems in thermodynamic equilibrium.For a system without chemical reactions, it relates the number of freely varying intensive properties (F) to the number of components (C), the number of phases (P), and number of ways of performing work on the system (N): [1] [2] [3]: 123–125
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 ...
Get AOL Mail for FREE! Manage your email like never before with travel, photo & document views. Personalize your inbox with themes & tabs. You've Got Mail!
The Ax–Kochen theorem, named for James Ax and Simon B. Kochen, states that for each positive integer d there is a finite set Y d of prime numbers, such that if p is any prime not in Y d then every homogeneous polynomial of degree d over the p-adic numbers in at least d 2 + 1 variables has a nontrivial zero.