We present an experimental study of pattern formation during the penetration of an aqueous surfactant solution into a liquid fatty acid in a Hele-Shaw cell. When a solution of the cationic surfactant cetylpyridinium chloride is injected into oleic acid, a wide variety of fingering patterns are observed as a function of surfactant concentration and flow rate, which are strikingly different than the classic Saffman-Taylor (ST) instability. We observe evidence of interfacial material forming between the two liquids, causing these instabilities. Moreover, the number of fingers decreases with increasing flow rate Q, while the average finger width increases with Q, both trends opposite to the ST case. Bulk rheology on related mixtures indicates a gel-like state. Comparison of experiments using other oils indicates the importance of pH and the carboxylic head group in the formation of the surfactant-fatty acid material.
We prove that the Möbius function is disjoint to all Lipschitz continuous skew product dynamical systems on the 3-dimensional Heisenberg nilmanifold over a minimal rotation of the 2-dimensional torus.
Let W ⊂ P 1 × P 1 × P 1 be a surface given by the vanishing of a (2, 2, 2)-form. These surfaces admit three involutions coming from the three projections W → P 1 × P 1 , so we call them tri-involutive K3 (TIK3) surfaces. By analogy with the classical Markoff equation, we say that W is of Markoff type (MK3) if it is symmetric in its three coordinates and invariant under double sign changes. An MK3 surface admits a group of automorphisms G generated by the three involutions, coordinate permutations, and sign changes. In this paper we study the G-orbit structure of points on TIK3 and MK3 surfaces. Over finite fields, we study fibral connectivity and the existence of large orbits, analogous to work of Bourgain, Gamburd, Sarnak and others for the classical Markoff equation. For a particular 1-parameter family of MK3 surfaces W k , we compute the full G-orbit structure of W k (F p ) for all primes p ≤ 79, and we use this data as a guide to find many finite G-orbits in W k (C), including a family of orbits of size 288 parameterized by a curve of genus 9.
Given n ∈ N, we study the conditions under which a finite field of prime order q will have adjacent elements of multiplicative order n. In particular, we analyze the resultant of the cyclotomic polynomial Φ n (x) with Φ n (x + 1), and exhibit Lucas and Mersenne divisors of this quantity. For each n = 1, 2, 3, 6, we prove the existence of a prime q n for which there is an element α ∈ Z qn where α and α + 1 both have multiplicative order n. Additionally, we use algebraic norms to set analytic upper bounds on the size and quantity of these primes.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.