By Ben Ayed M., El Mehdi K., Grossi M.

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**Extra info for Asymptotic behavior of least energy solutions of a biharmonic equation in dimension four**

**Sample text**

For each x is S, let Sx = {y|y is related to x}. Show that two such sets Sx and Sy are either disjoint or identical. Explain why this means that our relationship is an equivalence relation (as deﬁned in this section of the notes, not as deﬁned in the footnote). c Parts b and c of this problem prove that a relationship is an equivalence relation if and only if it is symmetric, reﬂexive, and transitive. Explain why. ) 15. Consider the following C++ function to compute n k . int pascal(int n, int k) { if (n < k) { cout << "error: n

G. interchanging two variables doesn’t change the result), then a proof that explains this symmetry is likely to give us additional insight into the formula. The proof above does not account for the symmetry of the k! term and the (n − k)! term in the n! (n−k)! This symmetry arises because choosing a k element subset is equivalent to choosing the (n − k)-element subset of elements we don’t want. ” For each labelling, the k objects that get the label “in” are in our subset. This explains the symmetry in our formula, but it doesn’t prove the formula.

We will denote Alice’s public and secret keys as KPA and KSA and Bob’s as KPB and KSB . They each keep their secret keys to themselves, but can publish their public keys and make them available to anyone, including the adversary. While the key published is likely to be a symbol string of some sort, the key is used in some standardized way (we shall see examples soon) to create a function from the set D of possible messages onto itself. (In complicated cases, the key might be the actual function).