OOCs, Partial Relative Difference Families and a Conjecture of Golomb

The cyclic difference sets constructed by Singer are also examples of perfect distinct difference sets (DDS). The Bose construction of distinct difference sets, leads to a relative difference set. In this paper we introduce the concept of partial relative DDS and prove that an optical orthogonal code (OOC) construction due to Moreno et. al., is a partial relative DDS. We generalize the concept of ideal matrices previously introduced by Kumar and relate it to the concepts of this paper. Another variation of ideal matrices is introduced in this paper: Welch ideal matrices of dimension n by (n - 1). We prove that Welch ideal matrices exist only for n prime. Finally, we recast an old conjecture of Golomb on the Welch construction of Costas arrays using the concepts of this paper. This connection suggests that our construction of partial relative difference sets is in a sense, unique

Thermal Weight Functions and Stress Intensity Factors for Bonded Dissimilar Media Using Body Analogy

In this study, an analytical method is presented for the computation of thermal weight functions in two dimensional bi-material elastic bodies containing a crack at the interface and subjected to thermal loads using body analogy method. The thermal weight functions are derived for two problems of infinite bonded dissimilar media, one with a semi-infinite crack and the other with a finite crack along the interface. The derived thermal weight functions are shown to reduce to the already known expressions of thermal weight functions available in the literature for the respective homogeneous elastic body. Using these thermal weight functions, the stress intensity factors are computed for the above interface crack problems when subjected to an instantaneous heat source.

Control of Molecular Weight and Polydispersity of Hyperbranched Polymers Using a Reactive B(3) Core: A Single-Step Route to Orthogonally Functionalizable Hyperbranched Polymers

Molecular weight and polydispersity are two structural features of hyperbranched polymers that are difficult to control because of the statistical nature of the step-growth polycondensation of AB(2) type monomers; the statistical growth also causes the polydispersity index to increase with percent conversion (or molecular weight). We demonstrate that using controlled amounts of a specifically designed B(3) core, containing B-type functionality that are more reactive than those present in the AB(2) monomer, both the molecular weight and the polydispersity can be readily controlled; the PDI was shown to improve with increasing mole-fraction of the B(3) core while the polymer molecular weight showed an expected decrease. Incorporation of a ``clickable'' propargyl group in the B(3) core unit permitted the generation of a core-functionalizable hyperbranched polymer. Importantly, this clickable core, in combination with a recently developed AB(2) monomer, wherein the B-type groups are allyl ethers and A is an hydroxyl group, led to the generation of a hyperbranched polymer carrying orthogonally functionalizable core and peripheral groups, via a single-step melt polycondensation. Selective functionalization of the core and periphery using two different types of chromophores was achieved, and the occurrence of fluorescence resonance energy transfer (FRET) between the donor and acceptor chromophores was demonstrated.

Utilizing an ionic liquid for synthesizing a soft matter polymer ``gel'' electrolyte for high rate capability lithium-ion batteries

A cross-linked polymer ``gel'' electrolyte obtained from free radical polymerization of a vinyl monomer (acrylonitrile; AN) in a room temperature ionic liquid electrolyte (N,N-methyl butyl pyrrolidinium-bis (trifluoromethanesulphonyl)imide-lithium bis(trifluoromethanesulphonyl) imide;LiTFSI-[Py(1,4)-TFSI]) for application in high rate capability rechargeable lithium-ion batteries is discussed here. This is a novel alternative compared to the often employed approach of using a molecular liquid as the medium for performing the polymerization reaction. The polymer ``gel'' electrolytes (AN:Py(1,4)-TFSI = 0.16-0.18, w/w) showed remarkable compliable mechanical strength and higher thermal stability compared to LiTFSI-[Py(1,4)-TFSI]. Despite two orders increase in magnitude of viscosity of polymer ``gels'', the room temperature ionic conductivity of the ``gels'' (1.1 x 10(-3)-1.7 x 10(-3) Omega(-1) cm(-1)) were nearly identical to that of the ionic liquid (1.8 x 10(-3) Omega(-1) cm(-1)). The present ``gel'' electrolytes did not exhibit any ageing effects on ionic conductivity similar to the conventional polymer gel electrolytes (e.g. high molecular weight polymer + salt + high dielectric constant molecular solvent). The disorder (ionic liquid) to a relative order (cross-linked polymer electrolyte) transformation does not at all influence the concentration of conducting species. The polymer framework is still able to provide efficient pathways for fast ion transport. Unlike the ionic liquid which is impossible to assemble without a conventional separator in a cell, the polymer ``gel'' electrolyte could be conveniently assembled without a separator in a Li vertical bar lithium iron phosphate (LiFePO(4)) cell. Compared to the ionic liquid, the ``gel'' electrolyte showed exceptional cyclability and rate capability (current density: 35-760 mA g(-1) with LiFePO(4) electronically wired with carbon (amorphous or multiwalled nanotube [MWCNT]).

Control of Molecular Weight and Polydispersity of Hyperbranched Polymers Using a Reactive B(3) Core: A Single-Step Route to Orthogonally Functionalizable Hyperbranched Polymers

Molecular weight and polydispersity are two structural features of hyperbranched polymers that are difficult to control because of the statistical nature of the step-growth polycondensation of AB(2) type monomers; the statistical growth also causes the polydispersity index to increase with percent conversion (or molecular weight). We demonstrate that using controlled amounts of a specifically designed B(3) core, containing B-type functionality that are more reactive than those present in the AB(2) monomer, both the molecular weight and the polydispersity can be readily controlled; the PDI was shown to improve with increasing mole-fraction of the B(3) core while the polymer molecular weight showed an expected decrease. Incorporation of a ``clickable'' propargyl group in the B(3) core unit permitted the generation of a core-functionalizable hyperbranched polymer. Importantly, this clickable core, in combination with a recently developed AB(2) monomer, wherein the B-type groups are allyl ethers and A is an hydroxyl group, led to the generation of a hyperbranched polymer carrying orthogonally functionalizable core and peripheral groups, via a single-step melt polycondensation. Selective functionalization of the core and periphery using two different types of chromophores was achieved, and the occurrence of fluorescence resonance energy transfer (FRET) between the donor and acceptor chromophores was demonstrated.

Constant weight codes based on ideals of commutative rings

We present a construction of constant weight codes based on the prime ideals of a Noetherian commutative ring. The coding scheme is based on the uniqueness of the primary decomposition of ideals in Noetherian rings. The source alphabet consists of a set of radical ideals constructed from a chosen subset of the prime spectrum of the ring. The distance function between two radical ideals is taken to be the Hamming metric based on the symmetric distance between sets. As an application we construct codes for random networks employing SAF routing.