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The finite form of Jensen's inequality is a special case of this result. Consider the real numbers x 1, …, x n ∈ I and let := + + + denote their arithmetic mean.Then (x 1, …, x n) majorizes the n-tuple (a, a, …, a), since the arithmetic mean of the i largest numbers of (x 1, …, x n) is at least as large as the arithmetic mean a of all the n numbers, for every i ∈ {1, …, n − 1}.
The system + =, + = has exactly one solution: x = 1, y = 2 The nonlinear system + =, + = has the two solutions (x, y) = (1, 0) and (x, y) = (0, 1), while + + =, + + =, + + = has an infinite number of solutions because the third equation is the first equation plus twice the second one and hence contains no independent information; thus any value of z can be chosen and values of x and y can be ...
It is therefore not decreasing and not increasing, but it is neither non-decreasing nor non-increasing. A function f {\displaystyle f} is said to be absolutely monotonic over an interval ( a , b ) {\displaystyle \left(a,b\right)} if the derivatives of all orders of f {\displaystyle f} are nonnegative or all nonpositive at all points on the ...
The definition of an A p weight and the reverse Hölder inequality indicate that such a weight cannot degenerate or grow too quickly. This property can be phrased equivalently in terms of how much the logarithm of the weight oscillates: (a) If w ∈ A p, (p ≥ 1), then log(w) ∈ BMO (i.e. log(w) has bounded mean oscillation).
A weight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set. The result of this application of a weight function is a weighted sum or weighted average .
Mathematics addresses only a part of human experience. Much of human experience does not fall under science or mathematics but under the philosophy of value, including ethics, aesthetics, and political philosophy. To assert that the world can be explained via mathematics amounts to an act of faith. 4. Evolution has primed humans to think ...
Dilution is the process of decreasing the concentration of a solute in a solution, usually simply by mixing with more solvent like adding more water to the solution. To dilute a solution means to add more solvent without the addition of more solute.
The following notation will be used throughout this article: is a fixed positive integer and is a fixed non-empty open subset of Euclidean space. = {,,, …} denotes the natural numbers.