Proofs by Descent
نویسنده
چکیده
The method of descent is a technique developed by Fermat for proving certain equations have no (or few) integral solutions. The idea is to show that if there is an integral solution to an equation then there is another integral solution which is smaller in some way. Repeating this process and comparing the sizes of the successive solutions leads to an infinitely decreasing sequence a1 > a2 > a3 > · · · of positive integers, and that is impossible. Let’s take a look at two examples.
منابع مشابه
Potential-Function Proofs for First-Order Methods
This note discusses proofs for convergence of first-order methods based on simple potentialfunction arguments. We cover methods like gradient descent (for both smooth and non-smooth settings), mirror descent, and some accelerated variants.
متن کاملClassifying Descents According to Equivalence mod k
In [5] the authors refine the well-known permutation statistic “descent” by fixing parity of (exactly) one of the descent’s numbers. In this paper, we generalize the results of [5] by studying descents according to whether the first or the second element in a descent pair is equivalent to k mod k ≥ 2. We provide either an explicit or an inclusion-exclusion type formula for the distribution of t...
متن کامل6 Classifying Descents According to equivalence mod k
In [5] the authors refine the well-known permutation statistic “descent” by fixing parity of (exactly) one of the descent’s numbers. In this paper, we generalize the results of [5] by studying descents according to whether the first or the second element in a descent pair is equivalent to k mod k ≥ 2. We provide either an explicit or an inclusion-exclusion type formula for the distribution of t...
متن کاملSequent calculi for induction and infinite descent
This paper formalises and compares two different styles of reasoning with inductively defined predicates, each style being encapsulated by a corresponding sequent calculus proof system. The first system, LKID, supports traditional proof by induction, with induction rules formulated as rules for introducing inductively defined predicates on the left of sequents. We show LKID to be cut-free compl...
متن کاملTowards stability and optimality in stochastic gradient descent
Lemmas 1, 2, 3 and 4, and Corollary 1, were originally derived by Toulis and Airoldi (2014). These intermediate results (and Theorem 1) provide the necessary foundation to derive Lemma 5 (only in this supplement) and Theorem 2 on the asymptotic optimality of θ̄n, which is the key result of the main paper. We fully state these intermediate results here for convenience but we point the reader to t...
متن کامل