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  2. Asymptotic homogenization - Wikipedia

    en.wikipedia.org/wiki/Asymptotic_homogenization

    Of course, all matter is inhomogeneous at some scale, but frequently it is convenient to treat it as homogeneous. A good example is the continuum concept which is used in continuum mechanics . Under this assumption, materials such as fluids , solids , etc. can be treated as homogeneous materials and associated with these materials are material ...

  3. Product order - Wikipedia

    en.wikipedia.org/wiki/Product_order

    The product order generalizes to arbitrary (possibly infinitary) Cartesian products. Suppose A ≠ ∅ {\displaystyle A\neq \varnothing } is a set and for every a ∈ A , {\displaystyle a\in A,} ( I a , ≤ ) {\displaystyle \left(I_{a},\leq \right)} is a preordered set.

  4. Polynomial SOS - Wikipedia

    en.wikipedia.org/wiki/Polynomial_SOS

    To establish whether a form h(x) is SOS amounts to solving a convex optimization problem. Indeed, any h(x) can be written as = {} ′ (+ ()) {} where {} is a vector containing a base for the forms of degree m in x (such as all monomials of degree m in x), the prime ′ denotes the transpose, H is any symmetric matrix satisfying = {} ′ {} and () is a linear parameterization of the linear ...

  5. Homogeneous relation - Wikipedia

    en.wikipedia.org/wiki/Homogeneous_relation

    The set of all homogeneous relations () over a set X is the set 2 X×X, which is a Boolean algebra augmented with the involution of mapping of a relation to its converse relation. Considering composition of relations as a binary operation on B ( X ) {\displaystyle {\mathcal {B}}(X)} , it forms a monoid with involution where the identity element ...

  6. Homogeneous function - Wikipedia

    en.wikipedia.org/wiki/Homogeneous_function

    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.

  7. Homogeneity (disambiguation) - Wikipedia

    en.wikipedia.org/wiki/Homogeneity_(disambiguation)

    Homogeneous catalysis, a sequence of chemical reactions that involve a catalyst in the same phase as the reactants Homogeneous (chemistry) , a property of a mixture showing no variation in properties Homogenization (chemistry) , intensive mixing of mutually insoluble substance or groups of substance to obtain a soluble suspension or constant

  8. Hilbert's seventeenth problem - Wikipedia

    en.wikipedia.org/wiki/Hilbert's_seventeenth_problem

    Hilbert's question can be restricted to homogeneous polynomials of even degree, since a polynomial of odd degree changes sign, and the homogenization of a polynomial takes only nonnegative values if and only if the same is true for the polynomial.

  9. Opposite - Wikipedia

    en.wikipedia.org/wiki/Opposite

    The term antonym (and the related antonymy) is commonly taken to be synonymous with opposite, but antonym also has other more restricted meanings. Graded (or gradable) antonyms are word pairs whose meanings are opposite and which lie on a continuous spectrum (hot, cold).

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