Mathematization of nature: how it is done

Natural phenomena can be quantitatively described by means of mathematics, which is actually the only way of doing so. Physics is a convincing example of the mathematization of nature. This paper gives an answer to the question of how mathematization of nature is done and illustrates the answer. Here nature is to be taken in a wide sense, being a substantial object of study in, among others, large domains of biology, such as epidemiology and neurobiology, chemistry, and physics, the most outspoken example. It is argued that mathematization of natural phenomena needs appropriate core concepts that are intimately connected with the phenomena one wants to describe and explain mathematically. Second, there is a scale on and not beyond which a specific description holds. Different scales allow for different conceptual and mathematical descriptions. This is the scaling hypothesis, which has meanwhile been confirmed on many occasions. Furthermore, a mathematical description can, as in physics, but need not be universally valid, as in biology. Finally, the history of science shows that only an intensive gauging of theory, i.e., mathematical description, by experiment leads to progress. That is, appropriate core concepts and appropriate scales are a necessary condition for mathematizing nature, and so is its verification by experiment.

Dynamic Sampling Bias and Overdispersion Induced by Skewed Offspring Distributions

Natural populations often show enhanced genetic drift consistent with a strong skew in their offspring number distribution. The skew arises because the variability of family sizes is either inherently strong or amplified by population expansions. The resulting allele-frequency fluctuations are large and, therefore, challenge standard models of population genetics, which assume sufficiently narrow offspring distributions. While the neutral dynamics backward in time can be readily analyzed using coalescent approaches, we still know little about the effect of broad offspring distributions on the forward-in-time dynamics, especially with selection. Here, we employ an asymptotic analysis combined with a scaling hypothesis to demonstrate that over-dispersed frequency trajectories emerge from the competition of conventional forces, such as selection or mutations, with an emerging time-dependent sampling bias against the minor allele. The sampling bias arises from the characteristic time-dependence of the largest sampled family size within each allelic type. Using this insight, we establish simple scaling relations for allele-frequency fluctuations, fixation probabilities, extinction times, and the site frequency spectra that arise when offspring numbers are distributed according to a power law n−(1+α). To demonstrate that this coarse-grained model captures a wide variety of evolutionary dynamics, we validate our results in traveling waves, where the phenomenon of ’gene surfing’ can produce any exponent 1 < α < 2. We argue that the concept of a dynamic sampling bias is useful to develop both intuition and statistical tests for the unusual dynamics of populations with skewed offspring distributions, which can confound commonly used tests for selection or demographic history.

Noteworthy records of bats of the genus Eumops Miller, 1906 from Guatemala: first confirmed record of Underwood’s Bonneted Bat, Eumops underwoodi Goodwin, 1940 (Mammalia, Chiroptera, Molossidae), in the country

Two species of Eumops Miller, 1906 are reported with voucher specimens from Guatemala: E. auripendulus (Shaw, 1800) and E. ferox (Gundlach, 1861). Eumops underwoodi Goodwin, 1940 has been known only by recordings. We collected dead specimens of E. ferox and E. underwoodi in a wind farm. Additionally, we provide acoustic data, and an allometric scaling hypothesis supports the identification of E. underwoodi, as it reflects the largest body size for sympatric free-tailed bats. We increase the list of bats in Guatemala to 104 species.

Correlations of quantum curvature and variance of Chern numbers

We analyse the correlation function of the quantum curvature in complex quantum systems, using a random matrix model to provide an exemplar of a universal correlation function. We show that the correlation function diverges as the inverse of the distance at small separations. We also define and analyse a correlation function of mixed states, showing that it is finite but singular at small separations. A scaling hypothesis on a universal form for both types of correlations is supported by Monte-Carlo simulations. We relate the correlation function of the curvature to the variance of Chern integers which can describe quantised Hall conductance.

A mode I crack with multiple islands of ideal contact between the crack faces: An asymptotic model

The problem of a mode I crack having multiple contacts between the crack faces is considered. In the case of small contact islands of arbitrary shapes, which are arbitrarily located inside the crack, the first-order asymptotic model for the crack opening displacement is constructed using the method of matched asymptotic expansions. The case of a penny-shaped crack has been studied in detail. A scaling hypothesis for the compliance reduction factor is formulated.

Dynamic sampling bias and overdispersion induced by skewed offspring distributions

Natural populations often show enhanced genetic drift consistent with a strong skew in their offspring number distribution. The skew arises because the variability of family sizes is either inherently strong or amplified by population expansions, leading to so-called ‘jackpot’ events. The resulting allele frequency fluctuations are large and, therefore, challenge standard models of population genetics, which assume sufficiently narrow offspring distributions. While the neutral dynamics backward in time can be readily analyzed using coalescent approaches, we still know little about the effect of broad offspring distributions on the dynamics forward in time, especially with selection. Here, we employ an exact asymptotic analysis combined with a scaling hypothesis to demonstrate that over-dispersed frequency trajectories emerge from the competition of conventional forces, such as selection or mutations, with an emerging time-dependent sampling bias against the minor allele. The sampling bias arises from the characteristic time-dependence of the largest sampled family size within each allelic type. Using this insight, we establish simple scaling relations for allele frequency fluctuations, fixation probabilities, extinction times, and the site frequency spectra that arise when offspring numbers are distributed according to a power law n−(1+α). To demonstrate that this coarse-grained model captures a wide variety of non-equilibrium dynamics, we validate our results in traveling waves, where the phenomenon of ‘gene surfing’ can produce any exponent 1 < α < 2. We argue that the concept of a dynamic sampling bias is useful generally to develop both intuition and statistical tests for the unusual dynamics of populations with skewed offspring distributions, which can confound commonly used tests for selection or demographic history.

True scale-free networks hidden by finite size effects

We analyze about 200 naturally occurring networks with distinct dynamical origins to formally test whether the commonly assumed hypothesis of an underlying scale-free structure is generally viable. This has recently been questioned on the basis of statistical testing of the validity of power law distributions of network degrees. Specifically, we analyze by finite size scaling analysis the datasets of real networks to check whether the purported departures from power law behavior are due to the finiteness of sample size. We find that a large number of the networks follows a finite size scaling hypothesis without any self-tuning. This is the case of biological protein interaction networks, technological computer and hyperlink networks, and informational networks in general. Marked deviations appear in other cases, especially involving infrastructure and transportation but also in social networks. We conclude that underlying scale invariance properties of many naturally occurring networks are extant features often clouded by finite size effects due to the nature of the sample data.

Why Pandemics, Such as COVID-19, Require a Metropolitan Response

We introduce evidence from the COVID-19 pandemic in the United States that lends support to future political efforts to include multi-county metropolitan areas as an additional and critical institutional layer—over and above municipalities, countries, states, or the federal government—for the effective management of present and future pandemics. Multi-county metropolitan statistical areas (MSAs) accommodated 73% of the U.S. population and, as of 27 September 2020, they were home to 78% of reported cases of COVID-19 and 82% of reported deaths. The rationale for a renewed focus on these spatial units is that they are found to be densely interconnected yet easily identifiable locales for the spread of pandemics and, therefore, for their proper management as well. The paper uses available data on cases and deaths in U.S. counties as of 27 September 2020 to lend statistical support to four hypotheses: (1) The Onset Hypothesis: The onset of COVID-19 cases and deaths commenced earlier in multi-county metropolitan areas than in small-city counties or rural counties; (2) The Peak Hypothesis: The current peak of COVID-19 cases and deaths occurred earlier in multi-county metropolitan areas; (3) The Scaling Hypothesis: Multi-county metropolitan areas had more than their shares of COVID-19 cases and deaths than their shares in the population; and (4) The Neighbor Hypothesis: Levels of COVID-19 cases and deaths in counties within multi-county metropolitan areas were more strongly related to respective levels in their neighboring counties than small-city counties or rural counties. The reported statistical results demonstrate the value of adopting a metropolitan perspective on pandemics and working to empower effective institutional arrangements at the metropolitan level for managing the present and future pandemics.

Rural–urban scaling of age, mortality, crime and property reveals a loss of expected self-similar behaviour

The urban scaling hypothesis has improved our understanding of cities; however, rural areas have been neglected. We investigated rural–urban population density scaling in England and Wales using 67 indicators of crime, mortality, property, and age. Most indicators exhibited segmented scaling about a median critical density of 27 people per hectare. Above the critical density, urban regions preferentially attract young adults (25–40 years) and lose older people (> 45 years). Density scale adjusted metrics (DSAMs) were analysed using hierarchical clustering, networks, and self-organizing maps (SOMs) revealing regional differences and an inverse relationship between excess value of property transactions and a range of preventable mortality (e.g. diabetes, suicide, lung cancer). The most striking finding is that age demographics break the expected self-similarity underlying the urban scaling hypothesis. Urban dynamism is fuelled by preferential attraction of young adults and not a fundamental property of total urban population.