Counting lattice chains and Delannoy paths in higher dimensions

Lattice chains and Delannoy paths represent two different ways to progress through a lattice. We use elementary combinatorial arguments to derive new expressions for the number of chains and the number of Delannoy paths in a lattice of arbitrary finite dimension. Specifically, fix nonnegative integers n₁,…,n_d, and let L denote the lattice of points (a₁,…,a_d)∈Z^d that satisfy 0≤a_i≤n_i for 1≤i≤d. We prove that the number of chains in L is given by where . We also show that the number of Delannoy paths in L equals Setting n_i=n (for all i) in these expressions yields a new proof of a recent result of Duchi and Sulanke [9] relating the total number of chains to the central Delannoy numbers. We also give a novel derivation of the generating functions for these numbers in arbitrary dimension.     

Extension of the 2-p-opt and 1-shift algorithms to the heterogeneous probabilistic traveling salesman problem

The probabilistic traveling salesman problem is a well known problem that is quite challenging to solve. It involves finding the tour with the lowest expected cost for customers that will require a visit with a given probability. There are several proposed algorithms for the homogeneous version of the problem, where all customers have identical probability of being realized. From the literature, the most successful approaches involve local search procedures, with the most famous being the 2-p-opt and 1-shift procedures proposed by Bertsimas [D.J. Bertsimas, L. Howell, Further results on the probabilistic traveling salesman problem, European Journal of Operational Research 65 (1) (1993) 68–95]. Recently, however, evidence has emerged that indicates the equations offered for these procedures are not correct, and even when corrected, the translation to the heterogeneous version of the problem is not simple. In this paper we extend the analysis and correction to the heterogeneous case. We derive new expressions for computing the cost of 2-p-opt and 1-shift local search moves, and we show that the neighborhood of a solution may be explored in O(n²) time, the same as for the homogeneous case, instead of O(n³) as first reported in the literature.     

The circular chromatic numbers of planar digraphs

The circular chromatic number is a refinement of the chromatic number of a graph. It has been established in [3,6,7] that there exists planar graphs with circular chromatic number r if and only if r is a rational in the set {1} ∪ [2,4]. Recently, Mohar, in [1,2] has extended the concept of the circular chromatic number to digraphs and it is interesting to ask what the corresponding result is for digraphs. In this article, we shall prove the new result that there exist planar digraphs with circular chromatic number r if and only if r is a rational in the interval [1,4]. © 2006 Wiley Periodicals, Inc. J Graph Theory 55: 14–26, 2007     

Evaluation of long‐term performance of antioxidants using prooxidants instead of thermal acceleration

Evaluation of stabilization efficiencies of different antioxidants in polymers at low temperature and relatively short time was performed using incorporation of a prooxidant for catalytic oxidation. Comparisons were made between polypropylene films stabilized with primary antioxidants (Irganox 1076, Irganox 1010, and α‐tocopherol), with or without the prooxidant manganese stearate at different temperatures. A faster degradation was obtained in the presence of a prooxidant than without it. The relative efficiency of the antioxidants at prooxidant acceleration correlated better to low temperature long‐term test than at the thermal acceleration. The results were affected by initial differences in the amounts of the antioxidants present after the processing of the films. These differences were corrected for by a recalculation using microwave‐assisted extraction (MAE) and high performance liquid chromatography analysis from the exponential decrease in the amount of antioxidant in the films. The fastest comparison of the antioxidants efficiency was obtained from oxidation induction times, using total luminescence intensity measurements, but reliable results could also be obtained from the time to apparent failure. © 2005 Wiley Periodicals,Inc. J Polym Sci Part A: Polym Chem 43: 4537–4546, 2005     

Polyester hydrogels with swelling properties controlled by the polymer architecture,molecular weight,and crosslinking agent

Hydrogels of poly(1,5‐dioxepan‐2‐one) and hydrogels of block copolymers of poly(1,5‐dioxepan‐2‐one) and poly(L ‐lactide) were synthesized. Both star‐shaped polymers and linear polymers were polymerized with ring‐opening polymerization and crosslinked in situ with a tetrafunctional acid chloride (1,2,3,4‐cyclopentane tetracarboxylic acid chloride) or a difunctional acid chloride (succinyl chloride). Different network architectures were synthesized in this way. The initial monomer concentrations and the molecular weights of the macromonomers were also altered. The networks were characterized with ¹H NMR and differential scanning calorimetry, and the swelling abilities of the different hydrogels were investigated in water and dichloromethane. The ratio of the monomer to the crosslinking agent was assessed by the quantification of the signal intensities in the ¹H NMR spectra of the swelled network and agreed with the theoretical crosslinking density. Both the homopolymers of 1,5‐dioxepan‐2‐one and the copolymers of 1,5‐dioxepan‐2‐one and L ‐lactide swelled to a high degree in water. The swelling properties of the materials could be varied over a broad range by changes in the architecture, molecular weight, and content of the precursor in the network. Star‐shaped poly(1,5‐dioxepan‐2‐one) crosslinked with a difunctional acid chloride had the highest degree of swelling among the different homopolymer hydrogels. This network also had the lowest glass‐transition temperature because of the flexible units in the structure. The same trends found for the homopolymer hydrogels were also seen in the hydrogels with block copolymers. The hydrogels swelled enormously in dichloromethane, and as in water, the star‐shaped copolymer crosslinked with a difunctional acid chloride had the highest degree of swelling. © 2003 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 41: 1296–1305, 2003     

Spontaneous crosslinking of poly(1,5‐dioxepan‐2‐one) originating from ether bond fragmentation

The spontaneous reaction of unsaturated double bonds induced by the fragmentation of ether bonds is presented as a method to obtain a crosslinked polymer material. Poly(1,5‐dioxepan‐2‐one) (PDXO) was synthesized using three different polymerization techniques to investigate the influence of the synthesis conditions on the ether bond fragmentation. It was found that thermal fragmentation of the ether bonds in the polymer main chain occurred when the synthesis temperature was 140 °C or higher. The double bonds produced reacted spontaneously to form crosslinks between the polymer chains. The formation of a network structure was confirmed by Fourier transform infrared spectrometry and differential scanning calorimetry. In addition, the low molar mass species released during hydrolysis of the DXO polymers were monitored by ESI‐MS and MALDI‐TOF‐MS. Ether bond fragmentation also occurred during the ionization in the electrospray instrument, but predominantly in the lower mass region. No fragmentation took place during MALDI ionization, but it was possible to detect water‐soluble DXO oligomers with a molar mass up to approximately 5000 g/mol. The results show that ether bond fragmentation can be used to form a network structure of PDXO. © 2008 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 46: 7258–7267, 2008     

Selective flourinations with 1-flouro-2-pyridone

1-Fluoro-2-pyridone has been used to fluorinata selectively enamines and Grignard reagents under mild conditions.