Evolutionary branching of function-valued traits under constraints

Some evolutionary traits are described by scalars and vectors, while others are described by continuous functions on spaces (e.g., shapes of organisms, resource allocation strategies between growth and reproduction along time, and effort allocation strategies for continuous resource distributions along resource property axes). The latter are called function-valued traits. This study develops conditions for candidate evolutionary branching points, referred to as CBP conditions, for function-valued traits under simple equality constraints, in the framework of adaptive dynamics theory (i.e., asexual reproduction and rare mutation are assumed). CBP conditions are composed of conditions for evolutionary singularity, strong convergence stability, and evolutionary instability. The CBP conditions for function-valued traits are derived by transforming the CBP conditions for vector traits into those for infinite-dimensional vector traits.

Nonsmooth and level-resolved dynamics illustrated with a periodically driven tight-binding model

We point out that in the first-order time-dependent perturbation theory, the transition probability may behave nonsmoothly in time and have kinks periodically. Moreover, the detailed temporal evolution can be sensitive to the exact locations of the eigenvalues in the continuum spectrum, in contrast to coarse-graining ideas. Underlying this nonsmooth and level-resolved dynamics is a simple equality about the sinc function sinc x ≡ sin x/x. These physical effects appear in many systems with approximately equally spaced spectra, and are also robust for larger amplitude coupling beyond the domain of perturbation theory. We use a one-dimensional periodically driven tight-binding model to illustrate these effects, both within and outside the perturbative regime.

Nonsmooth and level-resolved dynamics illustrated with a periodically driven tight binding model

We point out that in the first order time-dependent perturbation theory, the transition probability may behave nonsmoothly in time and have kinks periodically. Moreover, the detailed temporal evolution can be sensitive to the exact locations of the eigenvalues in the continuum spectrum, in contrast to coarse-graining ideas. Underlying this nonsmooth and level-resolved dynamics is a simple equality about the sinc function $\sinc x \equiv \sin x / x$. These physical effects appear in many systems with approximately equally spaced spectra, and is also robust for larger-amplitude coupling beyond the domain of perturbation theory. We use a one-dimensional periodically driven tight-binding model to illustrate these effects, both within and outside the perturbative regime.

Living Standards and Capabilities: Equal Values or Equal Sets?

Inspired by Gaertner and Xu (2006), this paper examines the possibility to construct a social ordering over distributions of capability sets, and a measure of the value of individual capability sets, such that perfect equality of sets, across individuals, is preferable to a simple equality of the value of sets. It is shown that this is a rather demanding requirement.