Microlocal properties of basic operations in Colombeau algebras

The Colombeau algebra of generalized functions allows to unrestrictedly carry out products of distributions. We analyze this operation from a microlocal point of view, deriving a general inclusion relation for wave front sets of products in the algebra. Furthermore, we give explicit examples showing that the given result is optimal, i.e. its assumptions cannot be weakened. Finally, we discuss the interrelation of these results with the concept of pullback under smooth maps.Comment: LaTeX, 18 page

Elliptic regularity and solvability for partial differential equations with Colombeau coefficients

The paper addresses questions of existence and regularity of solutions to linear partial differential equations whose coefficients are generalized functions or generalized constants in the sense of Colombeau. We introduce various new notions of ellipticity and hypoellipticity, study their interrelation, and give a number of new examples and counterexamples. Using the concept of \G^\infty-regularity of generalized functions, we derive a general global regularity result in the case of operators with constant generalized coefficients, a more specialized result for second order operators, and a microlocal regularity result for certain first order operators with variable generalized coefficients. We also prove a global solvability result for operators with constant generalized coefficients and compactly supported Colombeau generalized functions as right hand sides

Changes in the McGurk Effect across Phonetic Contexts. I. Fusions

The McGurk effect has generally been studied within a limited range of phonetic contexts. With the goal of characterizing the McGurk effect through a wider range of contexts, a parametric investigation across three different vowel contexts, /i/, /α/, and /u/, and two different syllable types, consonant-vowel (CV) and vowel-consonant (VC), was conducted. This paper discusses context-dependent changes found specifically in the McGurk fusion phenomenon (Part II addresses changes found in combination percepts). After normalizing for differences in the magnitude of the McGurk effect in different contexts, a large qualitative change in the effect across vowel contexts became apparent. In particular, the frequency of illusory /g/ percepts increased relative to the frequency of illusory /d/ percepts as vowel context was shifted from /i/ to /α/ to /u/. This trend was seen in both syllable sets, and held regardless of whether the visual stimulus used was a /g/ or /d/ articulation. This qualitative change in the McGurk fusion effect across vowel environments corresponded systematically with changes in the typical second formant frequency patterns of the syllables presented. The findings are therefore consistent with sensory-based theories of speech perception which emphasize the importance of second formant patterns as cues in multimodal speech perception.National Institue on Deafness and other Communication Disorders (R29 02852); Alfred P. Sloan Foundation and National Institute on Deafness and other Communication Disorders (R29 02852