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A crucial ingredient in the recent discovery by Ablowitz, Halburd, Herbst and Korhonen (2000, 2007) that a connection exists between discrete Painlevé equations and (finite order) Nevanlinna theory is an estimate of the integrated average of $log ^+|f(z+1)/f(z)|$ on $|z|=r$. We obtained essentially the same estimate in our previous paper (2008) independent of Halburd et al. (2006). We continue our study in this paper by establishing complete asymptotic relations amongst the logarithmic differences, difference quotients and logarithmic derivatives for finite order meromorphic functions. In addition to the potential applications of our new estimates in integrable systems, they are also of independent interest. In particular, our findings show that there are marked differences between the growth of meromorphic functions with Nevanlinna order less than and greater than one. We have established a ``difference'' analogue of the classical Wiman-Valiron type estimates for meromorphic functions with order less than one, which allow us to prove that all entire solutions of linear difference equations (with polynomial coefficients) of order less than one must have positive rational order of growth. We have also established that any entire solution to a first order algebraic difference equation (with polynomial coefficients) must have a positive order of growth, which is a ``difference'' analogue of a classical result of Pólya. [ABSTRACT FROM AUTHOR]

Abstract  We investigate the growth of the Nevanlinna characteristic of f(z+η) for a fixed η∈C in this paper. In particular, we obtain a precise asymptotic relation between T(r,f(z+η)) and T(r,f), which is only true for finite order meromorphic functions. We have also obtained the proximity function and pointwise estimates of f(z+η)/f(z) which is a discrete version of the classical logarithmic derivative estimates of f(z). We apply these results to give new growth estimates of meromorphic solutions to higher order linear difference equations. This also allows us to solve an old problem of Whittaker (Interpolatory Function Theory, Cambridge University Press, Cambridge, 1935) concerning a first order difference equation. We show by giving a number of examples that all of our results are best possible in certain senses. Finally, we give a direct proof of a result in Ablowitz, Halburd and Herbst (Nonlinearity 13:889–905, 2000) concerning integrable difference equations. [ABSTRACT FROM AUTHOR]

For wave equations with variable coefficients on regions which are not necessarily smooth, we study the energy decay rate when a nonlinear damping is applied on a suitable subrigion. [ABSTRACT FROM AUTHOR]

Hundreds of bamboo shoots have been reported to be edible, but the accumulation of trace elements and hazardous elements in bamboo shoots is poorly understood. Here, 100 bamboo species have been evaluated by screening elements including B, Fe, Mn, Cu, Zn, Cd, Pb and As in bamboo shoots using different assessment systems. Bamboo shoots displayed different morphological characteristics, and large differences were found in the concentration of elements. Most bamboo shoots were rich in Fe and Zn and low concentrations of hazardous elements, but the concentration of Cd and Pb exceeded the maximum permissible limits of tuber vegetables in some bamboo species. Different bamboo shoots were ranked differently in the four assessment systems, and the comprehensive evaluation assigned final scores to all 100 bamboo shoots. This study provides valuable recommendations for selecting high-quality bamboo shoots that are rich in trace elements nutrition while minimizing the potential for hazardous element accumulation., Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (© 2023 The Author(s).)

Uptake of boron (B) in rice (Oryza sativa) is mediated by the Low silicon rice 1 (OsLsi1) channel, belonging to the NOD26-like intrinsic protein III subgroup, and the efflux transporter B transporter 1 (OsBOR1). However, it is unknown how these transporters cooperate for B uptake and how they are regulated in response to B fluctuations. Here, we examined the response of these two transporters to environmental B changes at the transcriptional and posttranslational level. OsBOR1 showed polar localization at the proximal side of both the exodermis and endodermis of mature root region, forming an efficient uptake system with OsLsi1 polarly localized at the distal side of the same cell layers. Expression of OsBOR1 and OsLsi1 was unaffected by B deficiency and excess. However, although OsLsi1 protein did not respond to high B at the protein level, OsBOR1 was degraded in response to high B within hours, which was accompanied with a significant decrease of total B uptake. The high B-induced degradation of OsBOR1 was inhibited in the presence of MG-132, a proteasome inhibitor, without disturbance of the polar localization. In contrast, neither the high B-induced degradation of OsBOR1 nor its polarity was affected by induced expression of dominant-negative mutated dynamin-related protein 1A (OsDRP1AK47A) or knockout of the mu subunit (AP2M) of adaptor protein-2 complex, suggesting that clathrin-mediated endocytosis is not involved in OsBOR1 degradation and polar localization. These results indicate that, in contrast to Arabidopsis thaliana, rice has a distinct regulatory mechanism for B uptake through clathrin-independent degradation of OsBOR1 in response to high B., (© American Society of Plant Biologists 2021. All rights reserved. For permissions, please email: journals.permissions@oup.com.)