Optimal institutional advertising: Minimum-time problem

This paper considers the problem of optimizing the institutional advertising expenditure for a firm which produces two products. The problem is formulated as a minimum-time control problem for the dynamics of an extended Vidale-Wolfe advertising model, the optimal control being the rate of institutional advertising that minimizes the time to attain the specified target market shares for the two products. The attainable set and the optimal control are obtained by applying the recent theory developed by Hermes and Haynes extending the Green's theorem approach to higher dimensions. It is shown that the optimal control is a strict bang-bang control. An interesting side result is that the singular arc obtained by the Green's theorem application turns out to be a maximum-time solution over the set of all feasible controls. The result clarifies the connection between the Green's theorem approach and the maximum principle approach.     

Synthesis,characterisation, mesomorphic investigation and temperature-dependent Raman study of a novel calamitic liquid crystal: methyl 4-(4′-n-alkoxybenzylideneamino)benzoate

A new series of Schiff base calamitic liquid crystal; methyl 4-(4′-n-alkoxybenzylideneamino)benzoate (MABAB), H_2n+1C _n OC₆H₄C(H)=NC₆H₄COOCH₃ (n = 6, 8, 10, 12, 14, 16) has been synthesised and characterised by elemental analyses, Fourier transform infrared spectroscopy (FT-IR), ¹H and ¹³C Nuclear Magnetic Resonance (NMR) spectroscopy. The mesomorphic properties of these compounds were studied by differential scanning calorimetry (DSC) and polarising optical microscopy (POM). All members of the series exhibit enantiotropic smectic A (SmA) mesophase. Temperature-dependent micro-Raman study of one of the members, MABAB-10 has been employed to identify phase transitions and the molecular rearrangement therein. Analysis of Raman marker bands; C–H in-plane bending, C–C stretching of phenyl rings and –C(H)=N– linking group of core confirms the transitions clearly as observed through DSC and POM. An in situ Raman measurement of C–H in-plane bending mode has also been performed to visualise the molecular changes more clearly. The Raman study gives an evidence of induced co-planarity of rings at Cr→SmA phase transition. The density functional theoretical (DFT) optimisation of monomer, dimer and rotational conformer of MABAB-10 also support the induced co-planarity at Cr→SmA phase transition.