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In the TE framework, the entailing and entailed texts are termed text (t) and hypothesis (h), respectively.Textual entailment is not the same as pure logical entailment – it has a more relaxed definition: "t entails h" (t ⇒ h) if, typically, a human reading t would infer that h is most likely true. [1]
Directional statistics (also circular statistics or spherical statistics) is the subdiscipline of statistics that deals with directions (unit vectors in Euclidean space, R n), axes (lines through the origin in R n) or rotations in R n. More generally, directional statistics deals with observations on compact Riemannian manifolds including the ...
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
For n > 1, take as the induction hypothesis that the generalization is true for n − 1. We want to prove it for n. Assume the function f satisfies the hypotheses of the theorem. By the standard version of Rolle's theorem, for every integer k from 1 to n, there exists a c k in the open interval (a k, b k) such that f ′(c k) = 0.
Hypothesis testing remains a subject of controversy for some users, but the most widely accepted alternative method, confidence intervals, is based on the same mathematical principles. Due to the historical development of testing, there is no single authoritative source that fully encompasses the hybrid theory as it is commonly practiced in ...
The question of whether this ideal is the sum of two properly smaller ideals is independent of ZFC, as was proved by Andreas Blass and Saharon Shelah in 1987. [ 22 ] Charles Akemann and Nik Weaver showed in 2003 that the statement "there exists a counterexample to Naimark's problem which is generated by ℵ 1 , elements" is independent of ZFC.
In mathematics, the term "trivial" is often used to refer to objects (e.g., groups, topological spaces) with a very simple structure. These include, among others: Empty set: the set containing no or null members; Trivial group: the mathematical group containing only the identity element; Trivial ring: a ring defined on a singleton set
In mathematics, a metric space is a set where a notion of distance (called a metric) between elements of the set is defined. Much of analysis happens in some metric space; the most commonly used are the real line , the complex plane , Euclidean space , other vector spaces , and the integers .