A sufficiency class for global (in time) solutions to the 3D Navier-Stokes equations

Let Ω be an open domain of class C² contained in R³, let L²(Ω)³ be the Hilbert space of square integrable functions on Ω and let H[Ω]?H be the completion of the set, , with respect to the inner product of L²(Ω)³. A well-known unsolved problem is that of the construction of a sufficient class of functions in H which will allow global, in time, strong solutions to the three-dimensional Navier-Stokes equations. These equations describe the time evolution of the fluid velocity and pressure of an incompressible viscous homogeneous Newtonian fluid in terms of a given initial velocity and given external body forces. In this paper, we use the analytic nature of the Stokes semigroup to construct an equivalent norm for H, which provides strong bounds on the nonlinear term. This allows us to prove that, under appropriate conditions, there exists a number u₊, depending only on the domain, the viscosity, the body forces and the eigenvalues of the Stokes operator, such that, for all functions in a dense set D contained in the closed ball B(Ω)?B of radius in H, the Navier-Stokes equations have unique, strong, solutions in C¹((0,∞),H).     

Land cover change using an energy transition paradigm in a statistical mechanics approach

This paper explores a statistical mechanics approach as a means to better understand specific land cover changes on a continental scale. Integrated assessment models are used to calculate the impact of anthropogenic emissions via the coupling of technoeconomic and earth/atmospheric system models and they have often overlooked or oversimplified the evolution of land cover change. Different time scales and the uncertainties inherent in long term projections of land cover make their coupling to integrated assessment models difficult. The mainstream approach to land cover modelling is rule-based methodology and this necessarily implies that decision mechanisms are often removed from the physical geospatial realities, therefore a number of questions remain: How much of the predictive power of land cover change can be linked to the physical situation as opposed to social and policy realities? Can land cover change be understood using a statistical approach that includes only economic drivers and the availability of resources? In this paper, we use an energy transition paradigm as a means to predict this change. A cost function is applied to developed land covers for urban and agricultural areas. The counting of area is addressed using specific examples of a Pólya process involving Maxwell–Boltzmann and Bose–Einstein statistics. We apply an iterative counting method and compare the simulated statistics with fractional land cover data with a multi-national database. An energy level paradigm is used as a basis in a flow model for land cover change. The model is compared with tabulated land cover change in Europe for the period 1990–2000. The model post-predicts changes for each nation. When strong extraneous factors are absent, the model shows promise in reproducing data and can provide a means to test hypothesis for the standard rules-based algorithms.     

On Minimal Extensions of Rings

Given two rings R ? S, S is said to be a minimal ring extension of R, if R is a maximal subring of S. In this article, we study minimal extensions of an arbitrary ring R, with particular focus on those possessing nonzero ideals that intersect R trivially. We will also classify the minimal ring extensions of prime rings, generalizing results of Dobbs, Dobbs &; Shapiro, and Ferrand &; Olivier, on commutative minimal extensions.     

4-Chloro-2,6-bis(hydroxymethyl)pyridinium Chloride and 4-Dimethylamino-2,6-bis(hydroxymethyl)pyridinium Chloride Hemihydrate

Abstract  Two substituted pyridinium chloride salts, namely 4-chloro-2,6-bis(hydroxymethyl)pyridinium chloride (triclinic, P-1, a = 6.0651(8) ?, b = 8.4393(8) ?, c = 8.6554(9) ?, α = 78.845(1)°, β = 83.156(1)°, γ = 89.047(1)°; V = 431.55(8) ?³; Z = 2) and 4-dimethylamino-2,6-bis(hydroxymethyl)pyridinium chloride hemihydrate (orthorhombic, Aba2, a = 17.2179(14) ?, b = 17.6332(15) ?, c = 7.2068(6) ?; V = 2188.0(3) ?³; Z = 4), have been structurally characterized. Hydrogen bonding and van der Waals contacts are evident in both structures. Furthermore, the 4-dimethylamino derivative features a trigonal planar dimethylamino group with some π interaction between the nitrogen atom and the pyridine ring. Graphical Abstract  Two compounds, namely 4-chloro-2,6-bis(hydroxymethyl)pyridinium chloride and 4-dimethylamino-2,6-bis(hydroxymethyl)pyridinium chloride hemihydrate, have been spectroscopically and structurally characterized.      

Metal‐Mediated Functionalization of Natural Peptides and Proteins: Panning for Bioconjugation Gold

Selective modification of natural proteins is a daunting methodological challenge and a stringent test of selectivity and reaction scope. There is a continued need for new reactivity and new selectivity concepts. Transition metals exhibit a wealth of unique reactivity that is orthogonal to biological reactions and processes. As such, metal‐based methods play an increasingly important role in bioconjugation. This Review examines metal‐based methods as well as their reactivity and selectivity for the functionalization of natural proteins and peptides.     

Spin-dependent electronic states of the ferromagnetic semimetal EuB6

The spin polarization of EuB6 has been measured by using Andreev reflection spectroscopy. Analyses of the conductance spectra of the EuB6/Pb junctions yield a spin polarization of about 56%. The results demonstrate that the ferromagnetic EuB6 is not half-metallic. Combined with the Hall effect and magnetoresistivity data, the results indicate a semimetallic band structure with a fully spin-polarized hole band and an unpolarized electron band. The values and the spread of the measured spin polarization are quantitatively consistent with the experimentally determined Fermi surface and carrier densities.     

Luminescent Surface-Tethered Polymer Brush Materials

Surface-tethered polymers are unique molecular architectures that have been recently used in advanced sensors, electronics and biomedical applications. However, techniques for characterizing these materials in their surface-tethered form remain limited. The incorporation of luminescent functionality into these materials has enabled new characterization methods, while also unlocking new applications in optoelectronics, stenography and sensing. Micron-scale photolithography techniques have recently enabled the preparation of high-resolution patterns, as well as architectures with unique photophysical properties. Herein, we provide an overview of the techniques used to prepare luminescent polymer brush materials and their applications in stimuli-responsive sensors, cell adhesion materials, and optoelectronics. We also provide our perspective on the promising future uses of surface-tethered polymers, as well as the short-term challenges and opportunities in the field.