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Furthermore, for a smooth, bounded domain Ω, since the Rayleigh quotient for the Laplace operator in the space , is minimized by the eigenfunction corresponding to the minimal eigenvalue λ 1 of the (negative) Laplacian, it is a simple consequence that, for any , (), ‖ ‖ ‖ ‖ and furthermore, that the constant λ 1 is optimal.
Hoeffding's inequality is a special case of the Azuma–Hoeffding inequality and McDiarmid's inequality. It is similar to the Chernoff bound, but tends to be less sharp, in particular when the variance of the random variables is small. [2] It is similar to, but incomparable with, one of Bernstein's inequalities.
Intuitively, the graph of a bounded function stays within a horizontal band, while the graph of an unbounded function does not. In mathematics , a function f {\displaystyle f} defined on some set X {\displaystyle X} with real or complex values is called bounded if the set of its values is bounded .
gives the inequality. In the special case of n = 1, the Nash inequality can be extended to the L p case, in which case it is a generalization of the Gagliardo-Nirenberg-Sobolev inequality (Brezis 2011, Comments on Chapter 8). In fact, if I is a bounded interval, then for all 1 ≤ r < ∞ and all 1 ≤ q ≤ p < ∞ the following inequality holds
Be careful, in this application, Garding's Inequality seems useless here as the final result is a direct consequence of Poincaré's Inequality, or Friedrich Inequality. (See talk on the article). As a simple example, consider the Laplace operator Δ. More specifically, suppose that one wishes to solve, for f ∈ L 2 (Ω) the Poisson equation
For example, the feasible set defined by the constraint set {x ≥ 0, y ≥ 0} is unbounded because in some directions there is no limit on how far one can go and still be in the feasible region. In contrast, the feasible set formed by the constraint set { x ≥ 0, y ≥ 0, x + 2 y ≤ 4} is bounded because the extent of movement in any ...
Thus the image of a bounded set under a continuous operator is also bounded. Because of this property, the continuous linear operators are also known as bounded operators . In order to "measure the size" of A , {\displaystyle A,} one can take the infimum of the numbers c {\displaystyle c} such that the above inequality holds for all v ∈ V ...
Such inequalities are of importance in several fields, including communication complexity (e.g., in proofs of the gap Hamming problem [13]) and graph theory. [14] An interesting anti-concentration inequality for weighted sums of independent Rademacher random variables can be obtained using the Paley–Zygmund and the Khintchine inequalities. [15]