In particular, the theorem asserts that for a given triangle ABC and points L, M, and N that lie on the sides AB, BC, and CA, respectively, a necessary and sufficient condition for the three lines from vertex to point opposite (AM, BN, CL) to intersect at a common point (be concurrent) is that the following relation hold between the line segments formed on the triangle:BM∙CN∙AL = MC∙NA∙LB.
The Laguerre-Samuelson inequality (see Jensen and Styan [31] and Samuelson [32]) asserts that for a sample of size n no observation lies more than (sqrt{n - 1}) standard deviation away from the arithmetic mean, that is, vert x_{i} - mu vert le sqrt{n - 1} cdot sigma, quad i = 1,ldots,n, mu = s_{1}, sigma = sqrt{s_{2} - s_{1}^{2}}.
The well-known Jensen inequality asserts that for a convex function (varphi colon Isubseteq mathbb{R}to mathbb{R}) we have varphi Biggl(frac{1}{P_{n}}sum_{i=1}^{n} p_{i} x_{i} Biggr)leq frac{1}{P _{n}}sum _{i=1}^{n} p_{i} varphi (x_{i}), (1.1) where (x_{i}in I) for (i=1,ldots,n), and (p_{i}) are nonnegative real numbers such that (P_{n}=sum_{i=1}^{n} p_{i}>0).
Economic theory asserts that for a creditor nation, "the outright cancellation of debt becomes... a necessary part of its foreign policy" if it hopes to promote economic growth as well as its own strategic interests in bankrupt states [ 47, 52].
Evans asserts that "for an evidence-based discipline, it is surprising that there has been little research or reflection on [the role of schools of public health in tackling global public health challenges] within the public health literature" [ 18] and Sadana and colleagues confirm that "there is almost no evaluation of the degree to which education practices match real-world dimensions" [ 19].
The celebrated Hardy's inequality (see [1, Theorem ] ) asserts that, for, (1.1).
The extensionality axiom asserts that for any A and B, ∀x(x ∈ A ≡ x ε B) → A = B. Notice that any object to which this axiom applies is regular.
Traditionally, this is done by a so-called comprehension principle which, intuitively, asserts that, for any well-formed formula ('wff') A, with n free variables, x1, …, xn, there is a corresponding n-adic property.
The main result asserts that, for any contraction T on an arbitrary Banach space X, ∥ Tn − Tn + 1 ∥ → 0 as n → ∞, if and only if the spectrum of T has no points on the unit circle except perhaps z = 1.
The Legitimacy theory asserts that for an organisation to be accepted by a society, it must engage in activities that are accepted by that society (Deephouse & Carter, 2005; Bebbington, et al., 2008).
The second Chen fundamental inequality, stated in [41], asserts that for any submanifold M n of a real space form M ¯ with constant sectional curvature c, we have δ ( n 1, …, n k ) ≤ d ( n 1, …, n k ) ∥ H ∥ 2 + b ( n 1, …, n k ) c (19). for any k-tuples ( n 1, …, n k ) ∈ S ( n ).
