Exact solutions and critical behaviour for a linear growth-diffusion equation on a time-dependent domain

A linear growth-diffusion equation is studied in a time-dependent interval whose location and length both vary. We prove conditions on the boundary motion for which the solution can be found in exact form and derive the explicit expression in each case. Next, we prove the precise behaviour near the boundary in a ‘critical’ case: when the endpoints of the interval move in such a way that near the boundary there is neither exponential growth nor decay, but the solution behaves like a power law with respect to time. The proof uses a subsolution based on the Airy function with argument depending on both space and time. Interesting links are observed between this result and Bramson's logarithmic term in the nonlinear FKPP equation on the real line. Each of the main theorems is extended to higher dimensions, with a corresponding result on a ball with a time-dependent radius.

The story of one diamond: the heterogeneous distribution of the optical centres within a diamond crystal from the Ichetju placer, northern Urals

We have investigated a diamond crystal that consists of several misorientated subgrains. The main feature of the crystal is the dark areas in the cathodoluminescent core that has ‘estuary-like’ boundaries extending along the subgrain interfaces. The core has >3100 ppm of nitrogen, and the share of the B form is >95%; the absorbance of the centre N3VH at 3107 cm–1 reaches 75 cm–1. The N3 centre absorbance, as well as N3 luminescence, is absent in the core. In the outer part of the crystal, bright blue luminescence of the N3 centre is apparent, and the N3 absorbance reaches 5.3 cm–1. These observations could be explained by the conversion of N3 centres to N3VH after attaching a hydrogen atom. After the full conversion of the N3 centres, the diamond becomes darker under CL. We hypothesise the dark core has a specific shape due to the post-growth diffusion of the hydrogen.

Innovation diffusion as a catalyst for industrial company’s economic growth

The goal of the paper concerns essential issue of economic growth diffusion and innovation diffusion in the context of its spatial dissemination forming framework of the electro technical industry of Russia and the Netherlands main actors micro-level growth comparison. Comparison of two companies economic growth and innovation diffusion proves that Russian high-tech LOMO4 having inormous innnovation portfolio but lack of financial capital could proceed only organic growth and hierarchical innovation diffusion compared to financial and innovational world giant Philips which shows growth scenario of organic and inorganic growth assimilation that implies hierarchical innovation diffusion priority and higher competitiveness than LOMO.