Boxicity and maximum degree

A d-dimensional box is a Cartesian product of d closed intervals on the real line. The boxicity of a graph is the minimum dimension d such that it is representable as the intersection graph of d-dimensional boxes. We give a short constructive proof that every graph with maximum degree D has boxicity at most 2D2. We also conjecture that the best upper bound is linear in D.

Design Optimization of Field Emission From A Stacked Carbon Nanotube Array

Fluctuation of field emission in carbon nanotubes (CNTs) is riot desirable in many applications and the design of biomedical x-ray devices is one of them. In these applications, it is of great importance to have precise control of electron beams over multiple spatio-temporal scales. In this paper, a new design is proposed in order to optimize the field emission performance of CNT arrays. A diode configuration is used for analysis, where arrays of CNTs act as cathode. The results indicate that the linear height distribution of CNTs, as proposed in this study, shows more stable performance than the conventionally used unifrom distribution.

d-Regular Graphs of Acyclic Chromatic Index at Least d+2

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a'(G). It was conjectured by Alon, Suclakov and Zaks (and earlier by Fiamcik) that a'(G) <= Delta+2, where Delta = Delta(G) denotes the maximum degree of the graph. Alon et al. also raised the question whether the complete graphs of even order are the only regular graphs which require Delta+2 colors to be acyclically edge colored. In this article, using a simple counting argument we observe not only that this is not true, but in fact all d-regular graphs with 2n vertices and d>n, requires at least d+2 colors. We also show that a'(K-n,K-n) >= n+2, when n is odd using a more non-trivial argument. (Here K-n,K-n denotes the complete bipartite graph with n vertices on each side.) This lower bound for Kn,n can be shown to be tight for some families of complete bipartite graphs and for small values of n. We also infer that for every d, n such that d >= 5, n >= 2d+3 and dn even, there exist d-regular graphs which require at least d+2-colors to be acyclically edge colored. (C) 2009 Wiley Periodicals, Inc. J Graph Theory 63: 226-230, 2010.

Bone health and vitamin D status in children with motor disability and adults with intellectual disability

Osteoporosis is not only a disease of the elderly, but is increasingly diagnosed in chronically ill children. Children with severe motor disabilities, such as cerebral palsy (CP), have many risk factors for osteoporosis. Adults with intellectual disability (ID) are also prone to low bone mineral density (BMD) and increased fractures. This study was carried out to identify risk factors for low BMD and osteoporosis in children with severe motor disability and in adults with ID. In this study 59 children with severe motor disability, ranging in age from 5 to 16 years were evaluated. Lumbar spine BMD was measured with dual-energy x-ray absorptiometry. BMD values were corrected for bone size by calculating bone mineral apparent density (BMAD), and for bone age. The values were transformed into Z-scores by comparison with normative data. Spinal radiographs were assessed for vertebral morphology. Blood samples were obtained for biochemical parameters. Parents were requested to keep a food diary for three days. The median daily energy and nutrient intakes were calculated. Fractures were common; 17% of the children had sustained peripheral fractures and 25% had compression fractures. BMD was low in children; the median spinal BMAD Z-score was -1.0 (range -5.0 – +2.0) and the BMAD Z-score <-2.0 in 20% of the children. Low BMAD Z-score and hypercalciuria were significant risk factors for fractures. In children with motor disability, calcium intakes were sufficient, while total energy and vitamin D intakes were not. In the vitamin D intervention studies, 44 children and adolescents with severe motor disability and 138 adults with ID were studied. After baseline blood samples, the children were divided into two groups; those in the treatment group received 1000 IU peroral vitamin D3 five days a week for 10 weeks, and subjects in the control group continued with their normal diet. Adults with ID were allocated to receive either 800 IU peroral vitamin D3 daily for six months or a single intramuscular injection of 150 000 IU D3. Blood samples were obtained at baseline and after treatment. Serum concentrations of 25-OH-vitamin D (S-25-OHD) were low in all subgroups before vitamin D intervention: in almost 60% of children and in 77% of adults the S-25-OHD concentration was below 50 nmol/L, indicating vitamin D insufficiency. After vitamin D intervention, 19% of children and 42% adults who received vitamin D perorally and 12% of adults who received vitamin D intramuscularly had optimal S-25-OHD (>80 nmol/L). This study demonstrated that low BMD and peripheral and spinal fractures are common in children with severe motor disabilities. Vitamin D status was suboptimal in the majority of children with motor disability and adults with ID. Vitamin D insufficiency can be corrected with vitamin D supplements; the peroral dose should be at least 800 IU per day.

Possible high-temperature superconducting state with a d plus id pairing symmetry in doped graphene

Motivated by a suggestion in our earlier work [G. Baskaran, Phys. Rev. B 65, 212505 (2002)], we study electron correlation driven superconductivity in doped graphene where on-site correlations are believed to be of intermediate strength. Using an extensive variational Monte Carlo study of the repulsive Hubbard model and a correlated ground state wave function, we show that doped graphene supports a superconducting ground state with a d+id pairing symmetry. We estimate superconductivity reaching room temperatures at an optimal doping of about 15%-20%. Our work suggests that correlations can stabilize superconductivity even in systems with intermediate coupling.