High n-gram occurrence probability in baroque, classical and romantic melodies

An [Formula: see text]-gram in music is defined as an ordered sequence of [Formula: see text] notes of a melodic sequence [Formula: see text]. [Formula: see text] is calculated as the average of the occurrence probability without self-matches of all [Formula: see text]-grams in [Formula: see text]. Then, [Formula: see text] is compared to the averages Shuff[Formula: see text] and Equip[Formula: see text], calculated from random sequences constructed with the same length and set of symbols in [Formula: see text] either by shuffling a given sequence or by distributing the set of symbols equiprobably. For all [Formula: see text], both [Formula: see text], [Formula: see text], and this differences increases with [Formula: see text] and the number of notes, which proves that notes in musical melodic sequences are chosen and arranged in very repetitive ways, in contrast to random music. For instance, for [Formula: see text] and for all analyzed genres we found that [Formula: see text], while [Formula: see text] and [Formula: see text]. [Formula: see text] of baroque and classical genres are larger than the romantic genre one. [Formula: see text] vs [Formula: see text] is very well fitted to stretched exponentials for all songs. This simple method can be applied to any musical genre and generalized to polyphonic sequences.

Representation of Kinetics Models in Batch Flotation as Distributed First-Order Reactions

Four kinetic models are studied as first-order reactions with flotation rate distribution f(k): (i) deterministic nth-order reaction, (ii) second-order with Rectangular f(k), (iii) Rosin–Rammler, and (iv) Fractional kinetics. These models are studied because they are considered as alternatives to the first-order reactions. The first-order representation leads to the same recovery R(t) as in the original domain. The first-order R∞-f(k) are obtained by inspection of the R(t) formulae or by inverse Laplace Transforms. The reaction orders of model (i) are related to the shape parameters of first-order Gamma f(k)s. Higher reaction orders imply rate concentrations at k ≈ 0 in the first-order domain. Model (ii) shows reverse J-shaped first-order f(k)s. Model (iii) under stretched exponentials presents mounded first-order f(k)s, whereas model (iv) with derivative orders lower than 1 shows from reverse J-shaped to mounded first-order f(k)s. Kinetic descriptions that lead to the same R(t) cannot be differentiated between each other. However, the first-order f(k)s can be studied in a comparable domain.

Approximation of and by completely monotone functions

We investigate convergence in the cone of completely monotone fu nctions. Particular attention is paid to the approximation of and by exponentials and stretched exponentials. The need for such an analysis is a consequence of the fact that although stretched exponentials can be approximated by sums of exponentials, exponentials cannot in general be approximated by sums of stretched exponentials. doi:10.1017/S1446181120000012

The persistent and informative distribution of returns on capital

This Letter draws on series of very large national samples to show that cross-sectional distributions of realized returns on capital (RoC) are persistently well described by the same functional form: double stretched-exponentials. The Letter shows how the tails of these distributions can be understood as entropy maxima, suggesting complex patterns of competitive interactions across decentralized, market economies sustain formally persistent statistical equilibria in markets for capital. Such equilibria and their characteristics set the explanatory burden for successful economic accounts of the competitive regulation of profitability. They also point toward interesting new lines of inquiry on the systemic consequences of market competition in those economies and on the price structures it conditions.

APPROXIMATION OF AND BY COMPLETELY MONOTONE FUNCTIONS

We investigate convergence in the cone of completely monotone functions. Particular attention is paid to the approximation of and by exponentials and stretched exponentials. The need for such an analysis is a consequence of the fact that although stretched exponentials can be approximated by sums of exponentials, exponentials cannot in general be approximated by sums of stretched exponentials.

Black Soldier Fly Larvae Rearrange under Compression

Thousands of black soldier larvae hatch simultaneously from eggs laid within rotting vegetation or animal carcasses. Over the next few weeks, they grow while compressed by both their surroundings and each other. When compressed, these larvae rearrange to reduce the forces upon them. How quickly can larvae rearrange, and what final state do they choose? In this experimental study, we use a universal testing machine to conduct creep tests on larvae, squeezing them to set volume fractions and measuring the time course of their reaction force. Live larvae come to equilibrium at a rate 10 times faster than dead larvae, indicating that their small movements can rearrange them faster than just settling. The relaxation of dead larvae is well described by stretched exponentials, which also characterize hierarchical self-avoiding materials such as polymers or balls of crumpled aluminum foil. The equilibrium pressures of live larvae are comparable to those of dead larvae, suggesting that such pressures are dictated by the physics of their bodies rather than their behavior. Live larvae perform fluctuations to actively maintain this equilibrium pressure. This ability to survive large pressures might have applications in the larvae-rearing industry, where both live and dead larvae are packed in containers for shipping.

The Mittag-Leffler Fitting of the Phillips Curve

In this paper, a mathematical model based on the one-parameter Mittag-Leffler function is proposed to be used for the first time to describe the relation between the unemployment rate and the inflation rate, also known as the Phillips curve. The Phillips curve is in the literature often represented by an exponential-like shape. On the other hand, Phillips in his fundamental paper used a power function in the model definition. Considering that the ordinary as well as generalised Mittag-Leffler function behave between a purely exponential function and a power function it is natural to implement it in the definition of the model used to describe the relation between the data representing the Phillips curve. For the modelling purposes the data of two different European economies, France and Switzerland, were used and an “out-of-sample” forecast was done to compare the performance of the Mittag-Leffler model to the performance of the power-type and exponential-type model. The results demonstrate that the ability of the Mittag-Leffler function to fit data that manifest signs of stretched exponentials, oscillations or even damped oscillations can be of use when describing economic relations and phenomenons, such as the Phillips curve.