The chromatic number of random Borsuk graphs

We study a model of random graph where vertices are n i.i.d. uniform random points on the unit sphere S^d in , and a pair of vertices is connected if the Euclidean distance between them is at least 2??. We are interested in the chromatic number of this graph as n tends to infinity. It is not too hard to see that if ?>0 is small and fixed, then the chromatic number is d+2 with high probability. We show that this holds even if ?→0 slowly enough. We quantify the rate at which ? can tend to zero and still have the same chromatic number. The proof depends on combining topological methods (namely the Lyusternik–Schnirelman–Borsuk theorem) with geometric probability arguments. The rate we obtain is best possible, up to a constant factor—if ?→0 faster than this, we show that the graph is (d+1)‐colorable with high probability.25     

Solidifying framework nucleic acids

正Living organisms have developed their unique strategies during the natural evolution for building hard tissues with minerals, including silica, calcium carbonate, calcium phosphate, and ferric oxide [1]. Such biomineralized materials generally have complex hierarchical structures with excellent mechanical properties. Although bioinspired approaches have led to the creation of well-defined synthetic structural materials ranging from micro to macro scales, the rational design of discrete biomimetic structures at the nanoscale remains a grand challenge.     

Effects of multiple delays on dynamics of a five-neuron network model

A five-neuron network model with multiple delays is proposed. This paper presents the combined effect of different delays on the dynamics of the proposed network. Pitchfork bifurcation is discussed in detail with the variation of the value of coupled weight or attenuation rate of internal neurons. By analyzing the corresponding characteristic equation, some stable criteria on delay-dependence and delay-independence are derived including multiple delays and coupled weights and the periodic oscillation arises bifurcated from the trivial equilibrium after the network loses its stability. Stable regions on delay-dependence are displayed in the two delayed parameter plane. It is shown that multiple delays can produce stability switching between resting state and periodic activity. Finally, theoretical results are justified by providing two illustrative examples.     

Infinitely many reducts of homogeneous structures

It is shown that the countably infinite dimensional pointed vector space (the vector space equipped with a constant) over a finite field has infinitely many first order definable reducts. This implies that the countable homogeneous Boolean-algebra has infinitely many reducts.     

Benchtop NMR analysis of piperazine-based drugs hyperpolarised by SABRE

Piperazine-based drugs, such as N-benzylpiperazine (BZP), became attractive in the 2000s due to possessing effects similar to amphetamines. Herein, BZP, in addition to its pyridyl analogues, 2-, 3-, and 4-pyridylmethylpiperidine (2-PMP, 3-PMP, and 4-PMP respectively) was subjected to the hyperpolarisation technique Signal Amplification By Reversible Exchange (SABRE) in order to demonstrate the use of this technique to detect these piperazine-based drugs. Although BZP was not hyperpolarised via SABRE, 2-PMP, 3-PMP, and 4-PMP were, with the ortho- and meta-pyridyl protons of 4-PMP showing the largest enhancement of 313-fold and 267-fold, respectively, in a 1.4-T detection field, following polarisation transfer at Earth's magnetic field. In addition to the freebase, 4-PMP.3HCl was also appraised by SABRE and was found not to polarise, however, the addition of increasing equivalents of triethylamine (TEA) produced the freebase, with a maximum enhancement observed upon the addition of 3 equivalents of TEA. Further addition of TEA led to a reduction in the observed enhancement. SABRE was also employed to polarise 4-PMP.3HCl (~20% w/w) in a simulated tablet to demonstrate the forensic application of the technique (138-fold enhancement for the ortho-pyridyl protons). The amount of 4-PMP.3HCl present in the simulated tablet was quantified via NMR using D₂O as a solvent and compared well to complimentary gas chromatography–mass spectrometry data. Exchanging D₂O for CD₃OD as the solvent utilised for analysis resulted in a significantly lower amount of 4-PMP.3HCl being determined, thus highlighting safeguarding issues linked to drug abuse in relation to determining the amount of active pharmaceutical ingredient present.     

The Jordan property of Cremona groups and essential dimension

We use a recent advance in birational geometry to prove new lower bounds on the essential dimension of some finite groups.     

Application of ion-imprinted polymer synthesized by precipitation polymerization as an efficient and selective sorbent for separation and pre-concentration of chromium ions from some real samples

In this study, a new Cr(III)-imprinted polymer (Cr(III)-IIP) is prepared from CrCl₃·6H₂O, methacrylic acid functional monomer, ethyleneglycoldimethacrylate cross-linking agent, 2,2?-azobisisobutyronitrile radical initiator and 2,2-(azanediylbis (ethane-2,1-diyl))bis(isoindoline-1,3-dione) ligand. To obtain the maximum adsorption capacity, the optimum condition was studied through pH, type and concentration of eluent, IIP weight, sample volume as well as the adsorption and desorption times. The Cr(III) ion content was determined via flame atomic absorption spectrometer. In optimum conditions, the adsorption capacity of the IIP for Cr(III) was obtained to be 74.65 mg g^?1, using 50 mg of IIP and the initial pH solution of 3.0. Both the adsorption and desorption times for quantitative analyses of Cr(III) ions were 15 and 5 min; respectively. After elution of the adsorbed ions by 3 mL of 4 mol L^?1 HNO₃ aqueous solution, the established IIP-based SPE procedure provides a reasonable pre-concentration factor of 100. The IIP-based pre-concentration method provides a low detection limit of 1.7 µg L^?1 with good repeatability (RSD?=?3.22%). Reusability studies confirmed that synthesis IIP is reusable and recoverable up to six cycles. According to the selectivity experiments, it was concluded that the prepared sorbent possesses more affinity toward Cr(III) ions than other ions such as Al³⁺, Pb²⁺, Cu²⁺, Mn²⁺, Fe²⁺, Zn²⁺, and Ni²⁺ ions. To evaluate the potential applicability of the proposed separation method, the pre-concentration and determination of trace amounts of Cr(III) were performed successfully in food samples with complex matrices, a bestial sample (i.e. cow liver) and an herbal product (i.e., broccoli) as real samples.