The vertex-cover polynomial of a graph

In this paper we define the vertex-cover polynomial Ψ(G,τ) for a graph G. The coefficient of τ^r in this polynomial is the number of vertex covers V′ of G with |V′|=r. We develop a method to calculate Ψ(G,τ). Motivated by a problem in biological systematics, we also consider the mappings f from {1, 2,…,m} into the vertex set V(G) of a graph G, subject to f⁻¹(x)f⁻¹(y)≠ for every edge xy in G. Let F(G,m) be the number of such mappings f. We show that F(G,m) can be determined from Ψ(G,τ).     

Repurposing of Omarigliptin as a Neuroprotective Agent Based on Docking with A2A Adenosine and AChE Receptors,Brain GLP-1 Response and Its Brain/Plasma Concentration Ratio after 28 Days Multiple Doses in Rats Using LC-MS/MS

The authors in the current work suggested the potential repurposing of omarigliptin (OMR) for neurodegenerative diseases based on three new findings that support the preliminary finding of crossing BBB after a single dose study in the literature. The first finding is the positive results of the docking study with the crystal structures of A_2A adenosine (A2AAR) and acetylcholine esterase (AChE) receptors. A2AAR is a member of non-dopaminergic GPCR superfamily receptor proteins and has essential role in regulation of glutamate and dopamine release in Parkinson’s disease while AChE plays a major role in Alzheimer’s disease as the primary enzyme responsible for the hydrolytic metabolism of the neurotransmitter acetylcholine into choline and acetate. Docking showed that OMR perfectly fits into A2AAR binding pocket forming a distinctive hydrogen bond with Threonine 256. Besides other non-polar interactions inside the pocket suggesting the future of the marketed anti-diabetic drug (that cross BBB) as a potential antiparkinsonian agent while OMR showed perfect fit inside AChE receptor binding site smoothly because of its optimum length and the two fluorine atoms that enables quite lean fitting. Moreover, a computational comparative study of OMR docking, other 12 DPP-4 inhibitors and 11 SGLT-2 inhibitors was carried out. Secondly, glucagon-like peptide-1 (GLP-1) concentration in rats’ brain tissue was determined by the authors using sandwich GLP-1 ELISA kit bio-analysis to ensure the effect of OMR after the multiple doses’ study. Brain GLP-1 concentration was elevated by 1.9-fold following oral multiple doses of OMR (5 mg/kg/day, p.o. for 28 days) as compared to the control group. The third finding is the enhanced BBB crossing of OMR after 28 days of multiple doses that had been studied using LC-MS/MS method with enhanced liquid–liquid extraction. A modified LC-MS/MS method was established for bioassay of OMR in rats’ plasma (10–3100 ng/mL) and rats’ brain tissue (15–2900 ng/mL) using liquid–liquid extraction. Alogliptin (ALP) was chosen as an internal standard (IS) due to its LogP value of 1.1, which is very close to the LogP of OMR. Extraction of OMR from samples of both rats’ plasma and rats’ brain tissue was effectively achieved with ethyl acetate as the extracting solvent after adding 1N sodium carbonate to enhance the drug migration, while choosing acetonitrile to be the diluent solvent for the IS to effectively decrease any emulsion between the layers in the stated method of extraction. Validation results were all pleasing including good stability studies with bias of value below 20%. Concentration of OMR in rats’ plasma were determined after 2 h of the latest dose from 28 days multiple doses, p.o, 5 mg/kg/day. It was found to be 1295.66 ± 684.63 ng/mL estimated from the bio-analysis regression equation. OMR passed through the BBB following oral administration and exhibited concentration of 543.56 ± 344.15 ng/g in brain tissue, taking in consideration the dilution factor of 10. The brain/plasma concentration ratio of 0.42 (543.56/1295.66) was used to illustrate the penetration power through the BBB after the multiple doses for 28 days. Results showed that OMR passed through the BBB more effectively in the multiple dose study as compared to the previously published single dose study by the authors. Thus, the present study suggests potential repositioning of OMR as antiparkinsonian agent that will be of interest for researchers interested in neurodegenerative diseases.     

Self-injection length in La0.7Ca0.3MnO3-YBa2Cu3O7-δ ferromagnet-superconductor multilayer thin films

We have carried out extensive studies on the self-injection problem in barrierless heterojunctions between La_0.7Ca_0.3MnO₃ (LCMO) and YBa₂Cu₃O_7-δ (YBCO) thin films. The heterojunctions were formed in situ by sequentially growing LCMO and YBCO films on 〈100〉 LaAlO₃ (LAO) substrate using a pulsed laser deposition (PLD) system. YBCO micro-bridges with 64 μm width were patterned both on the LAO (control) and LCMO side of the substrate. Critical current, I _c, was measured at 77 K on both the control side as well as the LCMO side for different YBCO film thickness. It was observed that while the control side showed a J _c of ∼ 2 × 10⁶ A/cm², the LCMO side showed about half the value for the same thickness (1800 ?). The difference in J _c indicates that a certain thickness of YBCO has become ‘effectively’ normal due to self-injection. From the measurement of J _c at two different thicknesses (1800 ? and 1500 ?) of YBCO films both on the LAO as well as the LCMO side, the value of self-injection length (at 77 K) was estimated to be ∼ 900 ?. To the authors’ best knowledge, this is the first time that self-injection length has been quantified. A control experiment carried out with LaNiO₃ deposited by PLD on YBCO did not show any evidence of self-injection.     

A semi-linear delayed diffusion-wave system with distributed order in time

A numerical scheme for a class of non-linear distributed order fractional diffusion-wave equations with fixed time-delay is considered. The focus lies on the derivation of a linearized compact difference scheme as well as on quantitatively analyzing it. We prove unique solvability, convergence, and stability of the resulted numerical solution in \(L_{\infty }\)-norm by means of the discrete energy method. Numerical examples are introduced to illustrate the accuracy and efficiency of the proposed method.     

Superheating and solid-liquid phase coexistence in nanoparticles with nonmelting surfaces

We present a phenomenological model of melting in nanoparticles with facets that are only partially wet by their liquid phase. We show that in this model, as the solid nanoparticle seeks to avoid coexistence with the liquid, the microcanonical melting temperature can exceed the bulk melting point and that the onset of coexistence is a first-order transition. We show that these results are consistent with molecular dynamics simulations of aluminum nanoparticles which remain solid above the bulk melting temperature.     

Static, transient, and dynamic phase coexistence in metal nanoclusters

Molecular dynamics simulations are used to examine static and dynamic coexistence between solid and liquid phases in nanoscale silver, copper, and nickel clusters. We find static coexistence in the 561-atom copper icosahedron, the 561-atom silver icosahedron, and the 923-atom nickel icosahedron, and in cluster sizes above these thresholds, but not in smaller clusters. Nonetheless, in smaller clusters we typically observe either dynamic coexistence between fully solid and liquid states or transient coexistence which is essentially dynamic coexistence between a fully solid state and a solid-liquid state.