Physical Sciences and Mathematics Commons^™

On Mountain Pass Type Algorithms, James Bisgard

All Faculty Scholarship for the College of the Sciences

We consider constructive proofs of the mountain pass lemma, the saddle point theorem and a linking type theorem. In each, an initial “path” is deformed by pushing it downhill using a (pseudo) gradient flow, and, at each step, a high point on the deformed path is selected. Using these high points, a Palais–Smale sequence is constructed, and the classical minimax theorems are recovered. Because the sequence of high points is more accessible from a numerical point of view, we investigate the behavior of this sequence in the final two sections. We show that if the functional satisfies the Palais–Smale condition …

New Bounds And Computations On Prime-Indexed Primes, Jonathan Bayless, Dominic Klyve, Tomas Oliveira E Silva

Mathematics Faculty Scholarship

In a 2009 article, Barnett and Broughan considered the set of prime-index primes. If the prime numbers are listed in increasing order (2, 3, 5, 7, 11, 13, 17, . . .), then the prime-index primes are those which occur in a prime-numbered position in the list (3, 5, 11, 17, . . .). Barnett and Broughan established a prime-indexed prime number theorem analogous to the standard prime number theorem and gave an asymptotic for the size of the n-th prime-indexed prime.

We give explicit upper and lower bounds for π²(x), the number of prime-indexed primes up to …