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A league is a unit of length. It was common in Europe and Latin America, but is no longer an official unit in any nation. Derived from an ancient Celtic unit and adopted by the Romans as the leuga, the league became a common unit of measurement throughout western Europe. Since the Middle Ages, many values have been specified in several ...
1.6.1 Linear algebra. 1.6.1.1 Eigenvalue inequalities. 1.7 Number theory. 1.8 Probability theory and statistics. ... Bernstein's inequality (mathematical analysis)
where , is the inner product.Examples of inner products include the real and complex dot product; see the examples in inner product.Every inner product gives rise to a Euclidean norm, called the canonical or induced norm, where the norm of a vector is denoted and defined by ‖ ‖:= , , where , is always a non-negative real number (even if the inner product is complex-valued).
For instance, to solve the inequality 4x < 2x + 1 ≤ 3x + 2, it is not possible to isolate x in any one part of the inequality through addition or subtraction. Instead, the inequalities must be solved independently, yielding x < 1 / 2 and x ≥ −1 respectively, which can be combined into the final solution −1 ≤ x < 1 / 2 .
A sailor and a man on shore, both sounding the depth with a line. Depth sounding, often simply called sounding, is measuring the depth of a body of water.Data taken from soundings are used in bathymetry to make maps of the floor of a body of water, such as the seabed topography.
The Loomis–Whitney inequality can be used to relate the Lebesgue measure of a subset of Euclidean space to its "average widths" in the coordinate directions. This is in fact the original version published by Loomis and Whitney in 1949 (the above is a generalization). [1]
For instance, if the one solving the math word problem has a limited understanding of the language (English, Spanish, etc.) they are more likely to not understand what the problem is even asking. In Example 1 (above), if one does not comprehend the definition of the word "spent," they will misunderstand the entire purpose of the word problem.
The reverse inequality follows from the same argument as the standard Minkowski, but uses that Holder's inequality is also reversed in this range. Using the Reverse Minkowski, we may prove that power means with p ≤ 1 , {\textstyle p\leq 1,} such as the harmonic mean and the geometric mean are concave.
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