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Cover by Fraser Halley *The ONE of Cauchy-Schwarz (CSS) covers is only computed for positive values of the weights, i.e., for. Cauchy-Schwarz covers is a concept in comparison-based learning that is of interest. A sample is said to be *covered by* a covering set (CS), if and only if every pair of compared points is included in the set.

A covering set is a collection of pairwise covering concepts, each of which covers a positive percentage of sample pairs.Cauchy-Schwarz covers: Let be a collection of subsets. A *covering set* of is a collection, such that every sample from is. The *Cauchy-Schwarz* of is denoted by, and is defined by.

Assume that for every, and observe that for. This relationship provides a first proof of a Cauchy-Schwarz covers inequality (Steps 8,9) The second inequality in equation (4) follows from Step 9 and the fact that. Now, note that the set is included in any covering set of the form ; this follows from Step 3 and Step 9.

By induction, we conclude that the set is included in any covering set of the form ; the proof is by induction.

Assume the base case, that for every.

If for all, then we are done, for otherwise.

If for all, then we are done, for otherwise.

From Step 5 and Step 9,. From Step 5 and Step 4,.

From Step 6 and Step 5,.

By induction,. The justification of the last step requires the following claim..

From Steps 1 and 2,. From Step 1 and Step 1,. From Step 1 and Step 1,. From Step 1 and Step 1,.

Once this is shown, the last step is guaranteed.

The Cauchy-Schwarz inequality is the basis for a number of important results. We now use these results to prove that Cauchy-Schwarz covers implies a covering set of the form defined above, in which every set is.

Covering sets of the form and the original Cauchy-Schwarz covers are both equivalent to what is usually called the partial sum inequality.

Sankar Perugia: Mathematical Analysis I.

Covering sets are easy

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