Distance-regular Cayley graphs with least eigenvalue $$$$

We classify the distance-regular Cayley graphs with least eigenvalue \(-2\) and diameter at most three. Besides sporadic examples, these comprise of the lattice graphs, certain triangular graphs, and line graphs of incidence graphs of certain projective planes. In addition, we classify the possible connection sets for the lattice graphs and obtain some results on the structure of distance-regular Cayley line graphs of incidence graphs of generalized polygons.     

An efficient anti-aliasing spectral continuous window shifting technique for PIV

A new sub-pixel correlation peak locating algorithm for PIV analysis is introduced. The method is theoretically consistent with the method of continuously shifting interrogation sub-windows by fractional displacements, which has proven to be an effective way to reduce the bias error associated with integer pixel aliasing, or peak-locking. However the proposed algorithm performs continuous window shifting in the spatial frequency domain using the shift property of the Fourier transform, thus it is equivalent to interpolating the original digital image with the Fourier transform reconstruction. Synthetic and real PIV images are used to test the new algorithms performance relative to that of traditional (non-iterative) peak-finding methods and other peak-locking reduction algorithms, such as the continuous window shifting technique. The resultant bias error of the proposed algorithm is smaller (by an order of magnitude in some cases), and importantly, the periodic nature of the bias error, the characteristic signature of peak-locking, is eliminated as long as the discrete particle images have been sampled at a rate greater than the Nyquist sampling frequency. Moreover, this new algorithm is shown to be computationally efficient and it converges faster than the competing algorithms.     

Order acceptance and scheduling on two identical parallel machines

We study the order acceptance and scheduling problem on two identical parallel machines. At the beginning of the planning horizon, a firm receives a set of customer orders, each of which has a revenue value, processing time, due date, and tardiness weight. The firm needs to select orders to accept and schedule the accepted orders on two identical parallel machines so as to maximize the total profit. The problem is intractable, so we develop two heuristics and an exact algorithm based on some optimal properties and the Lagrangian relaxation technique. We evaluate the performance of the proposed solution methods via computational experiments. The computational results show that the heuristics are efficient and effective in approximately solving large-sized instances of the problem, while the exact algorithm can only solve small-sized instances.     

Organic fluorine compounds: a great opportunity for enhanced materials properties

Interactions of "organic fluorine" have gained great interest not only in the context of crystal engineering, but also in the systematic design of functional materials. The first part of this tutorial review presents an overview on interactions known by organic fluorine. This involves π-π(F), C-F···H, F···F, C-F···π(F), C-F···π, C-F···M(+), C-F···C=O and anion-π(F) interactions, as well as other halogen bonds. The effect of the exchange of H vs. F is discussed by means of several examples and a short introduction to the young field of "fluorous" chemistry is given. The second part is dedicated to numerous applications of fluorine and fluorous interactions. It is shown how application of fluorination is used to enable a number of reactions, to improve materials properties and even open up new fields of research.     

Bridge-substituted calix[4]arenes: syntheses, conformations and application

The bridge-substituted calix[4]arene carboxylic acid, 5,11,17,23-tetra-tert-butyl-25,26,27,28-tetramethoxy-calix[4]arene-2-carboxylic acid (1), can be readily converted to various esters 2-4 and reduced to the alcohol 5, which reacts with methyl iodide to give the ether 6. The alcohol can be dansylated to give 7, the fluorescence of which is selectively quenched by Cu(II) in acetonitrile. An attempt to convert the acid 1 to an amide resulted unexpectedly in the formation of a lactone 8. The conformational characteristics of 1-8 have been studied in solution and, in the cases of 2 and 4, in the solid state by determination of their single-crystal X-ray structures. With the exception of 8, in all these compounds the bridge substituent adopts an equatorial (lateral) orientation.     

When is a metallopolymer not a metallopolymer? When it is a metallomacrocycle

We report reactions of cobalt(II) acetate with a series of ditopic bis(tpy) ligands (tpy = 2,2':6',2'-terpyridine) containing flexible polyethyleneoxy spacers (tpy-4'-O{(CH(2))(2)O}(n)-4'-tpy, n = 2, 3, 4 or 6 ligands 1-4, respectively) which result in the formation of complicated mixtures of species, presumed to be both open chain and cyclic species. Well resolved paramagnetically shifted (1)H NMR spectra are a powerful tool for the analysis of these solution systems. Upon equilibration, [n + n] metallomacrocycles are isolated as the dominant (thermodynamic) species in some cases, and the single crystal X-ray structures of [Co(2)(3)(2)][PF(6)](4)·6MeCN and [Co(2)(4)(2)][PF(6)](5)·2MeCN (a mixed cobalt(ii)/cobalt(iii) species) are presented. Oxidation of the equilibrated cobalt(II) mixtures to kinetically inert cobalt(III) species provides additional evidence for the formation of metallomacrocycles as the thermodynamic products. Single crystal structural data for [Co(3)(2)(3)][PF(6)](9)·2MeCN·3.5H(2)O, [Co(2)(3)(2)][PF(6)](6)·10MeCN and [Co(2)(4)(2)][PF(6)](6)·6MeCN confirm the assembly of [2 + 2] and [3 + 3] metallomacrocycles. PGSE NMR spectroscopy has been used to determine the hydrodynamic radii of the solution species.     

An odd characterization of the generalized odd graphs

We show that any connected regular graph with d+1 distinct eigenvalues and odd-girth 2d+1 is distance-regular, and in particular that it is a generalized odd graph.