China
Africa
Explore chapters and articles related to this topic
Similarity Principle—The Fundamental Principle of All Sciences
Published in Mark Chang, Artificial Intelligence for Drug Development, Precision Medicine, and Healthcare, 2020
A metric on a set X is a function (called the distance function or simply distance) d : X × X → [0, ∞), where [0, ∞) is the set of non-negative real numbers and for all x, y, z ∈ X, the following conditions are satisfied: non-negativity or separation axiom, d(x, y) ≥ 0identity of indiscernibility, d(x, y) = 0 ⇔ x = ysymmetry, d (x, y) = d (y, x)subadditivity or triangle inequality, d (x, z) ≤ d (x, y) + d (y, z)Transitional invariance, d (x, y) = d (x + z, y + z) = d (y − x, 0) = d (x − y, 0)
Measurable Sets
Published in Hugo D. Junghenn, Principles of Analysis, 2018
The definition asserts that E “splits” the outer measure of each subset C of X, a property that may be seen as a precursor to finite additivity. Note that by subadditivity the inequality ≤ $ \le $ in (1.8) always holds. Thus the measurability criterion singles out precisely those sets E for which the inequality ≥ $ \ge $ in (1.8) is satisfied. The collection of all μ∗ $ \mu ^* $ -measurable subsets of X is denoted by M(μ∗) $ \boldsymbol{{ \fancyscript {M}}}(\mu ^*) $ . Here is the main result regarding outer measure.
Uniqueness for Spherically Convergent Multiple Trigonometric Series
Published in George Anastassiou, Handbook of Analytic-Computational Methods in Applied Mathematics, 2019
The first line follows from inequality (7.34); the second line follows from inequality (7.32); the third line follows from subadditivity of harmonic measure in the second coordinate; and the last line follows from combining the trivial estimate ω(·, ·, ·) ≤ 1 and inequality (7.37) with η = 2–k.
On deferred-statistical convergence of uncertain fuzzy sequences
Published in International Journal of General Systems, 2022
L. Nayak, B. C. Tripathy, P. Baliarsingh
By , we denote a σ-algebra on a non-empty set Γ. The set function is said be an uncertain measure if it satisfies these four axioms such as the normality (i.e.), the duality (i.e. for any ), the subadditivity i.e. for every countable sequence of , and the product axiom, i.e. where are arbitrarily chosen events from for and stands for the minimum over the given set.
Constant along primal rays conjugacies and the l 0 pseudonorm
Published in Optimization, 2022
Jean-Philippe Chancelier, Michel De Lara
For any vector , is the support of . The so-called pseudonorm is the function defined by where denotes the cardinal of a subset . The pseudonorm shares three out of the four axioms of a norm: nonnegativity, positivity except for , subadditivity. The axiom of 1-homogeneity does not hold true; in contrast to norms, the pseudonorm is 0-homogeneous: The level sets of the pseudonorm
Supply chain network design under the risk of uncertain disruptions
Published in International Journal of Production Research, 2020
Let Γ be a nonempty set, and be a σ-algebra over Γ. Every element Λ in is called an event. If a set function : satisfies the following conditions: (Normality Axiom) for the universal set Γ.(Duality Axiom) for any event Λ.(Subadditivity Axiom) For every countable sequence of , we have