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Geometry of music perception
(2022)
Prevalent neuroscientific theories are combined with acoustic observations from various studies to create a consistent geometric model for music perception in order to rationalize, explain and predict psycho-acoustic phenomena. The space of all chords is shown to be a Whitney stratified space. Each stratum is a Riemannian manifold which naturally yields a geodesic distance across strata. The resulting metric is compatible with voice-leading satisfying the triangle inequality. The geometric model allows for rigorous studies of psychoacoustic quantities such as roughness and harmonicity as height functions. In order to show how to use the geometric framework in psychoacoustic studies, concepts for the perception of chord resolutions are introduced and analyzed.
The aim of this work is to establish and generalize a relationship between fractional partial differential equations (fPDEs) and stochastic differential equations (SDEs) to a wider class of stochastic processes, including fractional Brownian motions and sub-fractional Brownian motions with Hurst parameter H ∈ (1/2,1). We start by establishing the connection between a fPDE and SDE via the Feynman-Kac Theorem, which provides a stochastic representation of a general Cauchy problem. In hindsight, we extend this connection by assuming SDEs with fractional and sub-fractional Brownian motions and prove the generalized Feynman-Kac formulas under a (sub-)fractional Brownian motion. An application of the theorem demonstrates, as a by-product, the solution of a fractional integral, which has relevance in probability theory.
This study empirically analyzes and compares return data from developed and emerging market data based on the Fama French five-factor model and compares it to previous results from the Fama French three-factor model by Kostin, Runge and Adams (2021). It researches whether the addition of the profitability and investment pattern factors show superior results in the assessment of emerging markets during the COVID-19 pandemic compared to developed markets. We use panel data covering eight indices of developed and emerging countries as well as a selection of eight companies from these markets, covering a period from 2000 to 2020. Our findings suggest that emerging markets do not generally outperform developed markets. The results underscore the need to reconsider the assumption that adding more factors to regression models automatically yields results that are more reliable. Our study contributes to the extant literature by broadening this research area. It is the first study to compare the performance of the Fama French three-factor model and the Fama French five-factor model in the cost of equity calculation for developed and emerging countries during the COVID-19 pandemic and other crisis events of the past two decades.