Composite quasianalytic functions

We prove two main results on Denjoy–Carleman classes: (1) a composite function theorem which asserts that a function $f(x)$ in a quasianalytic Denjoy–Carleman class ${\mathcal{Q}}_{M}$, which is formally composite with a generically submersive mapping $y=\unicode[STIX]{x1D711}(x)$ of class ${\mathcal{Q}}_{M}$, at a single given point in the source (or in the target) of $\unicode[STIX]{x1D711}$ can be written locally as $f=g\circ \unicode[STIX]{x1D711}$, where $g(y)$ belongs to a shifted Denjoy–Carleman class ${\mathcal{Q}}_{M^{(p)}}$; (2) a statement on a similar loss of regularity for functions definable in the $o$-minimal structure given by expansion of the real field by restricted functions of quasianalytic class ${\mathcal{Q}}_{M}$. Both results depend on an estimate for the regularity of a ${\mathcal{C}}^{\infty }$ solution $g$ of the equation $f=g\circ \unicode[STIX]{x1D711}$, with $f$ and $\unicode[STIX]{x1D711}$ as above. The composite function result depends also on a quasianalytic continuation theorem, which shows that the formal assumption at a given point in (1) propagates to a formal composition condition at every point in a neighbourhood.

A counterexample to the composition condition conjecture for polynomial Abel differential equations

Polynomial Abel differential equations are considered a model problem for the classical Poincaré center–focus problem for planar polynomial systems of ordinary differential equations. In the last few decades, several works pointed out that all centers of the polynomial Abel differential equations satisfied the composition conditions (also called universal centers). In this work we provide a simple counterexample to this conjecture.

The contributions of compositional structure and performance expression to the communication of emotion in music

In this investigation, eight highly-trained musicians communicated emotions through composition, performance expression, or the combination of the two. In the performance condition, they performed melodies with the intention of expressing six target emotions: anger, fear, happiness, neutral, sadness, and tenderness. In the composition condition, they composed melodies to express the same six emotions. The notated compositions were then played digitally without performance expression. In the combined condition, musicians performed the melodies they composed to convey the target emotions. Forty-two listeners heard the stimuli and attempted to decode the emotions in a forced-choice paradigm. Decoding accuracy varied significantly as a function of the channel of communication. Fear was comparatively well-decoded in the composition condition whereas anger was comparatively well decoded in the performance condition. Happiness and sadness were comparatively well-decoded in all three channels of communication. A principal component analysis of cues used by musicians clarified the distinct approaches adopted in composition and performance to differentiate emotional intentions. The results confirm that composition and performance involve the manipulation of distinct cues and have different emotional capabilities.