L2 regularity of measurable solutions of a finite-difference equation of the circle

We show that if $\varphi$ is a lacunary Fourier series and the equation $\psi (x) -\psi (x + \alpha) = \varphi(x), x \bmod 1$ has a measurable solution $\varphi$, then in fact the equation has a solution in L2. This work of Michel Herman (1942-2000) appeared only as a preprint of the Mathematics Institute, University of Warwick, dated May 1976. It was turned into TEX format by Claire Desescures. Minor editorial work was done by Albert Fathi

Interview with Stewart Herman, September 5, 2001

Steward Herman was interviewed on September 5, 2001 by Michael J. Birkner about his time at Gettysburg College in the 1920\u27s & 30\u27s. Herman discusses his classes at the time, as well as extra-curricular activities he participated in, including the Gettysburgian, the Mercury, Owl & Nightingale Players and Phi Sigma Kappa fraternity. He also describes his education at Gettysburg Seminary and working as an American pastor in Berlin during WWII. Length of Interview: 52 minutes Collection Note: This oral history was selected from the Oral History Collection maintained by Special Collections & College Archives. Transcripts are available for browsing in the Special Collections Reading Room, 4th floor, Musselman Library. GettDigital contains the complete listing of oral histories done from 1978 to the present. To view this list and to access selected digital versions please visit -- http://gettysburg.cdmhost.com/cdm/landingpage/collection/p16274coll

Hippopotamyrus ansorgii species complex in the Upper Zambezi River System with a description of a new species, H. szaboi (Mormyridae)

Specimens referable to Hippopotamyrus ansorgii sampled from the Upper Zambezi River System within Caprivi (Namibia) represent a complex of three species, two of which coexist in the Upper Zambezi River, and a third that inhabits a nearby river, the Kwando, with which the Zambezi has been connected during periods of flooding. All three are indistinguishable in terms of their general appearance, but differ consistently in electric organ discharges (EOD), morphology, and molecular genetic characters. All phenotypes display a monopolar, headpositive EOD pulse with specific post- or prepotentials. For H. ansorgii from the Zambezi River (HaZ), pulse duration is less than 0.5 ms (down to 0.205 ms; N = 34); for the syntopic H. szaboi sp. n., it is greater than 0.6 ms (up to 1.8 ms at 10% peak amplitude; N = 19). The parapatric phenotype of H. ansorgii from the Kwando River (HaK) has pulses shorter than 0.215 ms (down to 0.105 ms; N = 36). All three members of the species complex may be distinguished from each other by 7 − 9 anatomical characters, analysed by MANOVA. Based on 22 enzymes and proteins studied, the moderate to high Wright’s fixation index and the significant (P < 0.05) allele differentiation between EOD phenotypes provide additional evidence for incipient speciation. Pairwise analyses of the three different phenotypes showed the two parapatric species of H. ansorgii grouped together, and distinguishable from individuals of H. szaboi . Analyses of the mitochondrial cytochrome b gene revealed that all specimens which were attributed to H. szaboi form a well-supported monophyletic basal clade (bootstrap support 73% or 82%). The genetic distances (uncorrected p distances) between H. szaboi and the two species of H. ansorgii are between 0.6% and 1.7%. Within the derived H. ansorgii clade some phylogeographical differentiation can be seen for fishes from the Zambezi and Kwando Rivers, but the respective groups are not consistent or supported by significant bootstrap values. The question of which of the two parapatric morphological and EOD phenotypes of H. ansorgii recognized in the present paper represents H. ansorgii (Boulenger, 1905) cannot be resolved at present because of the paucity and unclear origin of the historical type material