Neuroanatomical shifts mirror patterns of ecological divergence in three diverse clades of mimetic butterflies

Microhabitat partitioning in heterogenous environments can support more diverse communities but may expose partitioned species to distinct perceptual challenges. Divergence across microhabitats could therefore lead to local adaptation to contrasting sensory conditions across small spatial scales, but this aspect of community structuring is rarely explored. Diverse communities of ithomiine butterflies provide an example where closely related species partition tropical forests, where shifts in mimetic coloration are tightly associated with shifts in habitat preference. We test the hypothesis that these mimetic and ecological shifts are associated with distinct patterns of sensory neural investment by comparing brain structure across 164 individuals of 16 species from three ithomiine clades. We find distinct brain morphologies between Oleriina and Hypothyris, which are mimetically homogenous and occupy a single microhabitat. Oleriina, which occurs in low‐light microhabitats, invests less in visual brain regions than Hypothyris, with one notable exception, Hyposcada anchiala, the only Oleriina sampled to have converged on mimicry rings found in Hypothyris. We also find that Napeogenes, which has diversified into a range of mimicry rings, shows intermediate patterns of sensory investment. We identify flight height as a critical factor shaping neuroanatomical diversity, with species that fly higher in the canopy investing more in visual structures. Our work suggests that the sensory ecology of species may be impacted by, and interact with, the ways in which communities of closely related organisms are adaptively assembled

A transference theorem for ergodic H1

The final version of this paper appears in: "Quarterly Journal of Mathematics" 48 (1997) 417-430. Print.In this paper, we extend the basic transference theorem for convolution operators on Lp spaces of Coifman and Weiss to H1 spaces

On a weak type (1, 1) inequality for a maximal conjugate function

The final version of this paper appears in: "Studia Mathematica" 125 (1997): 13-21. Print.In a celebrated paper, Burkholder, Gundy, and Silverstein used Brownian motion to derive a maximal function characterization of Hp spaces for 0 < p < ∞. In this paper, we show that their method extends to higher dimensions and yields a dimension-free weak type (1,1) estimate for a conjugate function on the N-dimensional torus

Analytic measures and Bochner measurability

Many authors have made great strides in extending the celebrated F. and M. Riesz Theorem to various abstract settings. Most notably, we have, in chronological order, the work of Bochner, Helson and Lowdenslager, de Leeuw and Glicksberg, and Forelli. These formidable papers build on each other's ideas and provide broader extensions of the F. and M. Riesz Theorem. Our goal in this paper is to use the analytic Radon-Nikodym property and prove a representation theorem (Main Lemma 2.2 below) for a certain class of measure-valued mappings on the real line

Hardy martingales and Jensen's inequality

The final version of this paper appears in: "Bulletin of the Australian Mathematical Society" 55 (1997): 185-195. Print.Hardy martingales were introduced by Garling and used to study analytic functions on the N-dimensional torus TN, where analyticity is defined using a lexicographic order on the dual group ZN. We show how, by using basic properties of orders on ZN, we can apply Garling's method in the study of analytic functions on an arbitrary compact abelian group with an arbitrary order on its dual group. We illustrate our approach by giving a new and simple proof of a famous generalized Jensen's Inequality due to Helson and Lowdenslager

Decomposition of analytic measures on groups and measure spaces

The final version of this paper appears in: "Studia Mathematica" 146 (2001): 261-284. Print.This paper provides a new approach to proving generalizations of the F.&M. Riesz Theorem, for example, the result of Helson and Lowdenslager, the result of Forelli (and de Leeuw and Glicksberg), and more recent results of Yamagushi. We study actions of a locally compact abelian group with ordered dual onto a space of measures, and consider those measures that are analytic, that is, the spectrum of the action on the measure is contained within the positive elements of the dual of the group. The classical results tell us that the singular and absolutely continuous parts of the measure (with respect to a suitable measure) are also analytic. The approach taken in this paper is to adopt the transference principle developed by the authors and Saeki in another paper, and apply it to martingale inequalities of Burkholder and Garling. In this way, we obtain a decomposition of the measures, and obtain the above mentioned results as corollaries