Multi-term fractional integro-differential equations in power growth function spaces

In this paper we characterize the Laplace transform of functions with power growth square averages and study several multi-term Caputo and Riemann-Liouville fractional integro-differential equations in this space of functions.

The Clash of Security and Sovereignty in Quasi-Alliance

Which is needed more: sovereignty or security? The autonomy–security trade-off model sees this as a trade-off between a client and a patron. A client surrenders some measure of autonomy to a patron and, in turn, receives security. This paper explores whether the underlying logic is applicable to quasi-alliances between a state and a multinational regime, such as the United Nations Command (UNC). South Korea has maintained a quasi-alliance with the UNC since the outbreak of the Korean War in 1950. The level of trade-off between South Korea and the UNC, however, has been changing over time, particularly while being affected by the power growth of South Korea, a client, and the preference changes of the US, the most important actor of the UNC, a patron. This paper attempts to explain why South Korea is much more enthusiastic in seeking full sovereignty and more autonomy in the 2018–2019 détente era.

A universal power law for modelling the growth and form of teeth, claws, horns, thorns, beaks, and shells

Background A major goal of evolutionary developmental biology is to discover general models and mechanisms that create the phenotypes of organisms. However, universal models of such fundamental growth and form are rare, presumably due to the limited number of physical laws and biological processes that influence growth. One such model is the logarithmic spiral, which has been purported to explain the growth of biological structures such as teeth, claws, horns, and beaks. However, the logarithmic spiral only describes the path of the structure through space, and cannot generate these shapes. Results Here we show a new universal model based on a power law between the radius of the structure and its length, which generates a shape called a ‘power cone’. We describe the underlying ‘power cascade’ model that explains the extreme diversity of tooth shapes in vertebrates, including humans, mammoths, sabre-toothed cats, tyrannosaurs and giant megalodon sharks. This model can be used to predict the age of mammals with ever-growing teeth, including elephants and rodents. We view this as the third general model of tooth development, along with the patterning cascade model for cusp number and spacing, and the inhibitory cascade model that predicts relative tooth size. Beyond the dentition, this new model also describes the growth of claws, horns, antlers and beaks of vertebrates, as well as the fangs and shells of invertebrates, and thorns and prickles of plants. Conclusions The power cone is generated when the radial power growth rate is unequal to the length power growth rate. The power cascade model operates independently of the logarithmic spiral and is present throughout diverse biological systems. The power cascade provides a mechanistic basis for the generation of these pointed structures across the tree of life.