Commuting self-adjoint extensions of symmetric operators defined from the partial derivatives

We consider the problem of finding commuting self-adjoint extensions of the partial derivatives {(1/i)(\partial/\partial x_j):j=1,...,d} with domain C_c^\infty(\Omega) where the self-adjointness is defined relative to L^2(\Omega), and \Omega is a given open subset of R^d. The measure on \Omega is Lebesgue measure on R^d restricted to \Omega. The problem originates with I.E. Segal and B. Fuglede, and is difficult in general. In this paper, we provide a representation-theoretic answer in the special case when \Omega=I\times\Omega_2 and I is an open interval. We then apply the results to the case when \Omega is a d-cube, I^d, and we describe possible subsets \Lambda of R^d such that {e^(i2\pi\lambda \dot x) restricted to I^d:\lambda\in\Lambda} is an orthonormal basis in L^2(I^d).Comment: LaTeX2e amsart class, 18 pages, 2 figures; PACS numbers 02.20.Km, 02.30.Nw, 02.30.Tb, 02.60.-x, 03.65.-w, 03.65.Bz, 03.65.Db, 61.12.Bt, 61.44.B

Clar Sextet Analysis of Triangular, Rectangular and Honeycomb Graphene Antidot Lattices

Pristine graphene is a semimetal and thus does not have a band gap. By making a nanometer scale periodic array of holes in the graphene sheet a band gap may form; the size of the gap is controllable by adjusting the parameters of the lattice. The hole diameter, hole geometry, lattice geometry and the separation of the holes are parameters that all play an important role in determining the size of the band gap, which, for technological applications, should be at least of the order of tenths of an eV. We investigate four different hole configurations: the rectangular, the triangular, the rotated triangular and the honeycomb lattice. It is found that the lattice geometry plays a crucial role for size of the band gap: the triangular arrangement displays always a sizable gap, while for the other types only particular hole separations lead to a large gap. This observation is explained using Clar sextet theory, and we find that a sufficient condition for a large gap is that the number of sextets exceeds one third of the total number of hexagons in the unit cell. Furthermore, we investigate non-isosceles triangular structures to probe the sensitivity of the gap in triangular lattices to small changes in geometry

Isomorphs in model molecular liquids

Isomorphs are curves in the phase diagram along which a number of static and dynamic quantities are invariant in reduced units. A liquid has good isomorphs if and only if it is strongly correlating, i.e., the equilibrium virial/potential energy fluctuations are more than 90% correlated in the NVT ensemble. This paper generalizes isomorphs to liquids composed of rigid molecules and study the isomorphs of two systems of small rigid molecules, the asymmetric dumbbell model and the Lewis-Wahnstrom OTP model. In particular, for both systems we find that the isochoric heat capacity, the excess entropy, the reduced molecular center-of-mass self part of the intermediate scattering function, the reduced molecular center-of-mass radial distribution function to a good approximation are invariant along an isomorph. In agreement with theory, we also find that an instantaneous change of temperature and density from an equilibrated state point to another isomorphic state point leads to no relaxation. The isomorphs of the Lewis-Wahnstrom OTP model were found to be more approximative than those of the asymmetric dumbbell model, which is consistent with the OTP model being less strongly correlating. For both models we find "master isomorphs", i.e., isomorphs have identical shape in the virial/potential energy phase diagram.Comment: 20 page

Intra-articular treatment of hip osteoarthritis: a randomized trial of hyaluronic acid, corticosteroid, and isotonic saline

SummaryObjectiveHyaluronic acid (HA) and corticosteroids are both widely used for intra-articular treatment of knee osteoarthritis (OA). We examined the effect of both drugs in intra-articular treatment for hip OA.MethodsOne hundred and one patients with hip OA were included in a prospective double blind study, using a randomized controlled trial with a three-armed parallel-group design. Three ultrasound-guided, intra-articular injections were given at 14 days interval. The primary outcome measure was ‘pain on walking’, registered on a visual analogue scale (VAS). Evaluation was performed at baseline and after 14, 28 and 90 days. The study adhered to the Consolidated Standards of Reporting Trials. All analyses were based on intention-to-treat analyses, and used ‘mixed-procedures’ with the baseline-observation as covariate.ResultsThere were no significant interactions with respect to Treatment×Time for any of the analyzed outcome measures. There was a significant treatment effect for ‘pain on walking’ (P=0.044) due to a significant improvement following corticosteroid compared to saline with an effect-size of 0.6 (95% confidence interval: 0.1–1.1, P=0.021). By contrast, HA compared to saline had an effect size of 0.4 (−0.1 to 0.9; P=0.13). The peak-effect was obtained after 2 weeks. There was no difference between the treatment groups at endpoint. No significant side effects of the injections were observed.ConclusionsPatients treated with corticosteroids experienced significant improvement during the 3 months of intervention, with an effect size indicating a moderate clinical effect. Although a similar significant result following treatment with HA could not be shown, the effect size indicated a small clinical improvement. A higher number of patients in future HA studies would serve to clarify this point

Ptychographic X-ray computed tomography of extended colloidal networks in food emulsions

As a main structural level in colloidal food materials, extended colloidal networks are important for texture and rheology. By obtaining the 3D microstructure of the network, macroscopic mechanical properties of the material can be inferred. However, this approach is hampered by the lack of suitable non-destructive 3D imaging techniques with submicron resolution. We present results of quantitative ptychographic X-ray computed tomography applied to a palm kernel oil based oil-in-water emulsion. The measurements were carried out at ambient pressure and temperature. The 3D structure of the extended colloidal network of fat globules was obtained with a resolution of around 300 nm. Through image analysis of the network structure, the fat globule size distribution was computed and compared to previous findings. In further support, the reconstructed electron density values were within 4% of reference values.Comment: 19 pages, 4 figures, to be published in Food Structur