Harmonic mean | EPFL Graph Search

In mathematics, the harmonic mean is one of several kinds of average, and in particular, one of the Pythagorean means. It is sometimes appropriate for situations when the average rate is desired. The harmonic mean can be expressed as the reciprocal of the arithmetic mean of the reciprocals of the given set of observations. As a simple example, the harmonic mean of 1, 4, and 4 is : \left(\frac{1^{-1} + 4^{-1} + 4^{-1}}{3}\right)^{-1} = \frac{3}{\frac{1}{1} + \frac{1}{4} + \frac{1}{4}} = \frac{3}{1.5} = 2,. Definition The harmonic mean H of the positive real numbers x_1, x_2, \ldots, x_n is defined to be :H(x_1, x_2, \ldots, x_n) = \frac{n}{\displaystyle \frac1{x_1} + \frac1{x_2} + \cdots + \frac1{x_n}} = \frac{n}{\displaystyle \sum_{i=1}^n \frac1{x_i}}. It is the reciprocal of the arithmetic mean of the reciprocals, and vice versa: :\begin{align} H(x_1, x_2, \ldots, x_n) &= \frac{1}{\displaystyle A\left(\frac1{x_1}, \frac1{x_

Second harmonic scattering of water in a biological context

Tereza Schönfeldová

Water is one of the most abundant molecules in the universe. It forms a hydrogen bond network with unique structure and dynamics. Hydrogen bonding is highly relevant to many biological processes including membrane formation, protein folding, adsorption, and lubrication. However, the underlying molecular interactions within water and between water and membranes are difficult to study and hence not fully understood. In this thesis, we utilize second harmonic scattering (SHS) to investigate inhomogeneities in bulk water, probe structural changes in lipid bilayer membranes, and detect and characterize protein-membrane interactions. Firstly, we explore the structure of bulk water in comparison to a common model liquid, carbon tetrachloride. Combining SHS measurements and molecular dynamics simulations, we report on the nanoscale inhomogeneities in bulk water that occur on femtosecond timescales. We attribute the rise in the coherent second harmonic (SH) intensity to charge density fluctuations with enhanced nanoscale polarizabilities around transient voids. Even though the transient voids also exist in carbon tetrachloride, they do not show polarizability fluctuations and the SH response of carbon tetrachloride thus corresponds to that of a model isotropic liquid. Then, we study the structure of water around large unilamellar vesicles (LUVs), which serve as a model for cell membranes. We focus on the role of water during the lipid melting phase transition. Using SHS, differential scanning calorimetry, and temperature-dependent zeta-potential measurements, we investigate the structural changes of hydration shells around the vesicle membrane. We use lipids with phosphoserine, phosphocholine, and phosphate headgroups and observe different hydration behavior upon the lipid melting transition depending on the membrane composition. Next, we characterize the interface of membranes composed of lipids containing phosphocholine and phosphate headgroups in solutions with different CaCl2 concentrations. We observe that the SH intensity is decreasing with the addition of CaCl2, which we attribute to the formation of a condensed layer of Na+ counterions on the inner leaflet of the LUVs. Additionally, we extract the vesicle surface potential and second-order surface susceptibility from SHS data for membranes with different content of lipids with phosphate headgroups, and observe composition dependent behaviour which suggests structural membrane reorganization. We continue by exploring protein-membrane interactions using water as a contrast agent. We study a pore-forming toxin, perfringolysin O (PFO), known for selective insertion into cholesterol-rich membranes. Combining SHS measurements and cryo-electron microscopy, we compare the interaction between PFO and LUV membranes with a high and low cholesterol content. By obtaining a sub-picomolar insertion constant of PFO, we show that SHS can be used to determine protein-membrane binding constants in concentration ranges not accessible by conventional methods.Finally, we investigate the interaction of protein aerolysin and its mutants with lipid membranes. We observe changes in surface charge in the cap region of membrane-bound aerolysin at different pH of the surrounding solution. By combining SHS with single-channel measurements, we obtain the membrane binding constant of wild-type aerolysin and its mutants, and we dissect their pore-forming mechanism.

EPFL2022

Device and method for measuring and imaging second harmonic and multi-photon generation scattered radiation

Carlos Alberto Macias Romero, Sylvie Roke

Embodiments of the subject invention relate to a method and apparatus for performing measurements using multiphoton or second harmonic generation (SHG) scattered radiation from a sample including a turbid (scattering) medium includes providing a beam of laser pulses from a laser source having high pulse energies and a repetition rate; splitting the beam of laser pulses into two or more partial beams and focussing and overlaying the partial beams on a sample including the turbid medium; and detecting multiphoton and second harmonic radiation scattered from the sample.

2015

The Annotated Mozart Sonatas: Score, Harmony, and Cadence

Johannes Hentschel, Markus Franz Josef Neuwirth, Martin Alois Rohrmeier

This article describes a new expert-labelled dataset featuring harmonic, phrase, and cadence analyses of all piano sonatas by W.A. Mozart. The dataset draws on the DCML standard for harmonic annotation and is being published adopting the FAIR principles of Open Science. The annotations have been verified using a data triangulation procedure which is presented as an alternative approach to handling annotator subjectivity. This procedure is suited for ensuring consistency, within the dataset and beyond, despite the high level of analytical detail afforded by the employed harmonic annotation syntax. The harmony labels also encode contextual information and are therefore suited for investigating music theoretical questions related to tonal harmony and the harmonic makeup of cadences in the classical style. Apart from providing basic statistical analyses characterizing the dataset, its music theoretical potential is illustrated by two preliminary experiments, one on the terminal harmonies of cadences and the other on the relation between performance durations and harmonic density. Furthermore, particular features can be selected to produce more coarse-grained training data, for example for chord detection algorithms that require less analytical detail. Facilitating the dataset’s reusability, it comes with a Python script that allows researchers to easily access various representations of the data tailored to their particular needs.

2021

Second harmonic scattering of water in a biological context

Tereza Schönfeldová

EPFL2022