We demonstrate a suspended graphene-(poly(methyl methacrylate) (PMMA) polymer angular displacement actuator enabled by variable elastic modulus of the perforated stacked structure. Azimuthal flexures support a central disc-shaped membrane, and compression of the membrane can be used to control the rotation of the entire structure. Irradiating the PMMA on graphene stack with 5 kV electrons in a convention scanning electron microscope reduces the elastic modulus of the PMMA and allows graphene’s built in strain to dominate and compress the flexures, thus rotating the actuator.

In this report, we demonstrate the use of helium ion milling for the controllable fabrication of nanostructures in few-layer hexagonal boron nitride (h-BN). Using the direct-write lithographic capabilities of a scanning helium ion microscope (HIM), nanopores with diameters as small as 4 nm and nanoribbons with widths of 3 – 10 nm are etched from suspended h-BN sheets. This ability to pattern h-BN sheets with high-throughput and sub-10 nm precision paves the way for future studies that make use of atomically-thin, nanostructured insulators such as those needed for nanopore sequencing and patterned van der Waals heterostructures.

The Green function of singular limit-circle problems is constructed directly for the problem, not as a limit of sequences of regular Green's functions. This construction is used to obtain adjointness and self-adjointness conditions which are entirely analogous to the regular case. As an application, a new and explicit formula for the Green function of the classical Legendre problem is found.

Over the last decades the steel industries worldwide haveevolved more or less independently from each other, inthe EU and in North America in particular. A competitivecomparison of two different Basic Oxygen Furnace (BOF)maintenance operations is presented, discussing various costfigures in a theoretical case study.

We investigate the behaviour of the eigenvalues of a self-adjoint Sturm–Liouville problem with a separated boundary condition when the interval of the problem shrinks to an end point. It is shown that all the eigenvalues, except possibly the first, approach $+\infty$. The choices of the boundary condition are found for which the first eigenvalue tends to $+\infty$, independent of the coefficient functions, and the same is done for the $-\infty$ limit. For the remaining choices of the boundary condition, several types of condition on the coefficient functions are given, so that the first eigenvalue has a finite or infinite limit and, when the limit is finite, an explicit expression for the limit is obtained. Moreover, numerous examples are presented to illustrate these results, and a construction is given to perturb the finite-limit case to the no-limit case.

A transmission electron microscope (TEM) is much more than just a tool for imaging the static state of materials. To demonstrate this, we present work on studying the mechanical and electrical properties of carbon nanotube devices. Multiwall carbon nanotubes are concentrically stacked tubular sheets of graphite, where the spacing between each cylinder is simply the natural spacing of graphite. Using a custom-built in-situ nanomanipulation probe, we have shown that it is possible to slide the nanotube layers in a telescopic extension mode that exhibits low friction, demonstrating the potential of nanotubes as the ultimate synthetic nanobearing. During this telescopic extension, the electrical resistance of the nanotube devices increases, opening the possibility that these devices can also be used as nanoscale rheostats. We also briefly describe work on using electron holography inside a TEM to study the electric field distribution in nanotube field-emission devices and on using a nanotube itself as a biprism for electron holography. These measurements together demonstrate the wealth of information that can be obtained and frontiers that can be opened by putting operational nanodevices inside an electron microscope.

For a given Sturm-Liouville equation whose leading coefficient function changes sign, we establish inequalities among the eigenvalues for any coupled self-adjoint boundary condition and those for two corresponding separated self-adjoint boundary conditions. By a recent result of Binding and Volkmer, the eigenvalues (unbounded from both below and above) for a separated self-adjoint boundary condition can be numbered in terms of the Prüfer angle; and our inequalities can then be used to index the eigenvalues for any coupled self-adjoint boundary condition. Under this indexing scheme, we determine the discontinuities of each eigenvalue as a function on the space of such Sturm-Liouville problems, and its range as a function on the space of self-adjoint boundary conditions. We also relate this indexing scheme to the number of zeros of eigenfunctions. In addition, we characterize the discontinuities of each eigenvalue under a different indexing scheme.

We study the spectrum of regular and singular Sturm–Liouville problems with real-valued coefficients and a weight function that changes sign. The self-adjoint boundary conditions may be regular or singular, separated or coupled. Sufficient conditions are found for (i) the spectrum to be real and unbounded below as well as above and (ii) the essential spectrum to be empty. Also found is an upper bound for the number of non-real eigenvalues. These results are achieved by studying the interplay between the indefinite problems (with weight function which changes sign) and the corresponding definite problems. Our approach relies heavily on operator theory of Krein space.

We consider some geometric aspects of regular Sturm—Liouville problems. First, we clarify a natural geometric structure on the space of boundary conditions. This structure is the base for studying the dependence of Sturm—Liouville eigenvalues on the boundary condition, and reveals many new properties of these eigenvalues. In particular, the eigenvalues for separated boundary conditions and those for coupled boundary conditions, or the eigenvalues for self-adjoint boundary conditions and those for non-self-adjoint boundary conditions, are closely related under this structure. Then we give complete characterizations of several subsets of boundary conditions such as the set of self-adjoint boundary conditions that have a given real number as an eigenvalue, and determine their shapes. The shapes are shown to be independent of the differential equation in question. Moreover, we investigate the differentiability of continuous eigenvalue branches under this structure, and discuss the relationships between the algebraic and geometric multiplicities of an eigenvalue.

If a Sturm—Liouville problem is given in an open interval of the real line, then regular boundary value problems can be considered on compact sub-intervals. For these regular problems, all with necessarily discrete spectra, the eigenvalues depend on both the end-points of the compact intervals, and upon the choice of the real separated boundary conditions at these end-points. These eigenvalues are not, in general, continuous functionsof the end-points and boundary conditions. The paper shows the surprising form of these discontinuities. The results have applications to the approximations of singular Sturm—Liouville problems by regular problems, and to the theoretical aspects of the Sleign2 Computer program.

Fatigue lifetime measurements have been performed on foamed Al-Mg-Si wrought alloys and Al-Si cast alloys in the high cycle range using an ultrasonic resonance testing method. The porous structure of the material is described by quantitative image analysis of optical micrographs and non destructively by X-ray computer tomography. The static mechanical properties as determined by tensile, compression and bending tests in earlier studies are used for material characterisation in this paper. The evaluation of the stress strain curves is specified for porous structures to obtain the stiffness and the plateau strength. The influence of the surface skin on the mechanical properties as well as on oscillation behaviour during lifetime measurements was studied.

Eigenvalues of both regular and singular Sturm–Liouville (S–L) problems with general coupled self-adjoint boundary conditions are characterised. This characterisation, although elementary, appears to be new even in the regular case. The singular characterisation is an exact parallel of the regular one and reduces to it. One application yields inequalities among the eigenvalues of different coupled boundary conditions. This is a far-reaching extension, even in the regular case, of the well-known relationship among the periodic and semiperiodic eigenvalues.

We consider the question: when do two ordinary, linear, quasi-differential expressions commute? For classical differential expressions, answers to this question are well known. The set of all expressions which commute with a given such expression form a commutative ring. For quasi-differential expressions less is known and such an algebraiastructure can no longer be exploited. Using the theory of very general quasi-differential expressions with matrix-valued coefficients, we prove some general results concerning commutativity of such expressions. We show how, when specialised to scalar expressions, these results include a proof of the conjecture that if a 2nth-order scalar, J-symmetric (or real symmetric) quasi-differential expression commutes with a second order expression having the same properties, then the former must be an nth-order polynomial in the latter. This result was conjectured in a paper by Race and Zettl, to which this paper is a sequel.

A necessary and sufficient condition for a general, scalar, quasi-differential expression of order n to be factorisable into a product of expressions of order n − k and k, for any 0 < k < n, is given. The factors are characterised completely in terms of elements of the null space of the expression and its adjoint. The results obtained extend existing results due to both Polya and Zettl from the case of classical linear differential expressions to quasi-differential expressions.

For regular symmetric ordinary differential expressions we show that (i) the minimal operator is bounded below and (ii) the Friedrichs extension is determined by Dirichlet boundary conditions. Both proofs are based on elementary inequalities.