Global solutions describing the collapse of a spherical or cylindrical cavity

The collapse of a spherical (cylindrical) cavity in air is studied analytically. The global solution for the entire domain between the sound front, separating the undisturbed and the disturbed gas, and the vacuum front is constructed in the form of infinite series in time with coefficients depending on an ldquoappropriaterdquo similarity variable. At timet=0+, the exact planar solution for a uniformly moving cavity is assumed to hold. The global analytic solution of this initial boundary value problem is found until the collapse time (=(gamma–1)/2) for gamma le 1+(2/(1+v)), wherev=1 for cylindrical geometry, andv=2 for spherical geometry. For higher values of gamma, the solution series diverge at timet — 2(beta–1)/ (v(1+beta)+(1–beta)2) where beta=2/(gamma–1). A close agreement is found in the prediction of qualitative features of analytic solution and numerical results of Thomaset al. [1].

Solution of a hypersingular integral equation of the second kind

A straightforward analysis involving the complex function-theoretic method is employed to determine the closed-form solution of a special hypersingular integral equation of the second kind, and its known solution is recovered.

Transformation of a laterally diverging boundary layer flow to a two-dimensional boundary layer flow

Laterally diverging boundary layer flow over a plate is shown to be reducible to a two-dimensional flow by modelling the diverging streamlines by a source flow.

Unitary invariants for Hilbert modules of finite rank

We associate a sheaf model to a class of Hilbert modules satisfying a natural finiteness condition. It is obtained as the dual to a linear system of Hermitian vector spaces (in the sense of Grothendieck). A refined notion of curvature is derived from this construction leading to a new unitary invariant for the Hilbert module. A division problem with bounds, originating in Douady's privilege, is related to this framework. A series of concrete computations illustrate the abstract concepts of the paper.

Structural characterization of backbone-expanded helices in hybrid peptides: (αγ) n and (αβ) n sequences with unconstrained β and γ homologues of L-Val

Learning your αβγ's: The diversity of hydrogen-bonding patterns in backbone-expanded hybrid helices is shown by crystal-structure determination of several oligomeric peptides (see scheme; C=gray; H=white; O=red; N=blue). C 12 helices were observed in the αγ peptide series for n=2-8. In comparison, the αα peptide and αβ peptide sequences show C 10 and mixed C 14/C 15 helices, respectively. Copyright © 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Proton Conduction in Metal-Organic Frameworks and Related Modularly Built Porous Solids

Proton-conducting materials are an important component of fuel cells. Development of new types of proton-conducting materials is one of the most important issues in fuel-cell technology. Herein, we present newly developed proton-conducting materials, modularly built porous solids, including coordination polymers (CPs) or metalorganic frameworks (MOFs). The designable and tunable nature of the porous materials allows for fast development in this research field. Design and synthesis of the new types of proton-conducting materials and their unique proton-conduction properties are discussed.

Nanoindentation in Crystal Engineering: Quantifying Mechanical Properties of Molecular Crystals

Nanoindentation is a technique for measuring the elastic modulus and hardness of small amounts of materials. This method, which has been used extensively for characterizing metallic and inorganic solids, is now being applied to organic and metalorganic crystals, and has also become relevant to the subject of crystal engineering, which is concerned with the design of molecular solids with desired properties and functions. Through nanoindentation it is possible to correlate molecular-level properties such as crystal packing, interaction characteristics, and the inherent anisotropy with micro/macroscopic events such as desolvation, domain coexistence, layer migration, polymorphism, and solid-state reactivity. Recent developments and exciting opportunities in this area are highlighted in this Minireview.