This note serves to provide additional details for the proof of Lemma 3.6 in our paper [Liu, Zhang and Zhang, Comm. Math. Sci., 3(2005), pp.201-218]. Moreover, we will also present an alternative, yet simpler, proof based on arguments in [Wang, Zhang and Zhang, CPAM, 68(2015), no. 8, 1326-1398.In [1] we showed that in order to determine the number of solutions to the Doi-Onsager equation on the sphere with Maier-Saupe potential, it suffices to determine the number of zeros of B(η, α) in term of the intensity parameter α > 0, whereNote that, for η > 0, the mean value theorem gives 3e −η /

We analyse the structure of steady state solutions of the one-dimensional Doi model for rod-like molecules. We prove that if the interaction strength parameter U is less than 4, then the constant solution is the only possible steady state solution. If U is larger than 4, then there is a new solution that corresponds to the nematic phase. All other non-constant solutions are obtained from this solution by translation. We prove further that the nematic solutions have period π instead of 2π , a property that signifies the fact the rods are symmetric, i.e. they have orientations but no directions.

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Several herbivorous insects and plant-associated microorganisms control the phytohormonal balance, thus enabling them to successfully exploit the plant by inhibiting plant defenses and withdrawing plant resources for their own benefit. The leaf-mining moth Phyllonorycter blancardella modifies the cytokinin (CK) profile of mined leaf-tissues, and the insect symbiotic bacteria Wolbachia is involved in the plant manipulation to the benefit of the insect host. To gain a deeper understanding into the possible origin and dynamics of CKs, we conducted an extensive characterization of CKs in larvae and in infected apple leaves. Our results show the enhanced CK levels in mines, both on green and yellow leaves, allowing insects to control their nutritional supply under fluctuating environmental conditions. The spatial distribution of CKs within the mined leaves shows that hormone manipulation is strictly limited to the mine suggesting the absence of CK translocation from distant leaf areas toward the insect feeding site. Mass spectrometry analyses reveal that major CK types accumulating in mines and larvae are similar to what is observed for most gall-inducers, suggesting that strategies underlying the plant manipulation may be shared between herbivorous insects with distinct life histories. Results further show that CKs are detected in the highest levels in larvae, reinforcing our hypothesis that CKs accumulating in the mines originate from the insect itself. Presence of bacteriaspecific methylthio-CKs is consistent with previous results suggesting that insect bacterial symbionts contribute to the observed phenotype. Our study provides key findings toward the understanding of molecular mechanisms underlying this intricate plant-insect-microbe interaction.