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The word problem is a well-known example of an undecidable problem. If A {\displaystyle A} is a finite set of generators for G {\displaystyle G} , then the word problem is the membership problem for the formal language of all words in A {\displaystyle A} and a formal set of inverses that map to the identity under the natural map from the free ...
Bennett's inequality, an upper bound on the probability that the sum of independent random variables deviates from its expected value by more than any specified amount Bhatia–Davis inequality , an upper bound on the variance of any bounded probability distribution
In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. [1] It is used most often to compare two numbers on the number line by their size. The main types of inequality are less than (<) and greater than (>).
Many important inequalities can be proved by the rearrangement inequality, such as the arithmetic mean – geometric mean inequality, the Cauchy–Schwarz inequality, and Chebyshev's sum inequality. As a simple example, consider real numbers : By applying with := for all =, …,, it follows that + + + + + + for every permutation of , …,.
Proof [2]. Since + =, =. A graph = on the -plane is thus also a graph =. From sketching a visual representation of the integrals of the area between this curve and the axes, and the area in the rectangle bounded by the lines =, =, =, =, and the fact that is always increasing for increasing and vice versa, we can see that upper bounds the area of the rectangle below the curve (with equality ...
The uses of this inequality are not limited to applications in probability theory.One example of its use in analysis is the following: if we let be a linear operator between two L p spaces (,) and (,), < <, with bounded norm ‖ ‖ <, then one can use Khintchine's inequality to show that
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]
An example is boron carbide, whose formula of CB n is a variable non-whole number ratio, with n ranging from over 4 to more than 6.5. When the chemical compound of the formula consists of simple molecules , chemical formulas often employ ways to suggest the structure of the molecule.