Location of Poles for the Hastings-McLeod Solution to the Second Painlev\'{e} Equation

We show that the well-known Hastings-McLeod solution to the second Painlev\'{e} equation is pole-free in the region $\arg x \in [-\frac{\pi}{3},\frac{\pi}{3}]\cup [\frac{2\pi}{3},\frac{ 4 \pi}{3}]$, which proves an important special case of a general conjecture concerning pole distributions of Painlev\'{e} transcedents proposed by Novokshenov. Our strategy is to construct explicit quasi-solutions approximating the Hastings-McLeod solution in different regions of the complex plane, and estimate the errors rigorously. The main idea is very similar to the one used to prove Dubrovin's conjecture for the first Painlev\'{e} equation, but there are various technical improvements.Comment: 31 pages, 2 figures. Minor revision, to appear in Constructive Approximatio

Multi-Pair Two-Way Relay Network with Harvest-Then-Transmit Users: Resolving Pairwise Uplink-Downlink Coupling

While two-way relaying is a promising way to enhance the spectral efficiency of wireless networks, the imbalance of relay-user distances may lead to excessive wireless power at the nearby-users. To exploit the excessive power, the recently proposed harvest-then-transmit technique can be applied. However, it is well-known that harvest-then-transmit introduces uplink-downlink coupling for a user. Together with the co-dependent relationship between paired users and interference among multiple user pairs, wirelessly powered two-way relay network suffers from the unique pairwise uplink-downlink coupling, and the joint uplink-downlink network design is nontrivial. To this end, for the one pair users case, we show that a global optimal solution can be obtained. For the general case of multi-pair users, based on the rank-constrained difference of convex program, a convergence guaranteed iterative algorithm with an efficient initialization is proposed. Furthermore, a lower bound to the performance of the optimal solution is derived by introducing virtual receivers at relay. Numerical results on total transmit power show that the proposed algorithm achieves a transmit power value close to the lower bound

RATIOMETRIC NEAR-INFRARED FLUORESCENT PROBES FOR THE SENSITIVE DETECTION OF INTRACELLULAR pH AND BIO-THIOLS IN LIVE CELLS

In the past twenty years, fluorescence sensing and imaging based on fluorescent probes has been developed as an imperative technique due to the merits including excellent sensitivity, operational simplicity, instant time effectiveness and outstanding selectivity in the research areas such as mineralogy, gemology, biological medicine, materials and environmental engineering. Protons act as a significant role in a variety of pathological and physiological processes, and there are obvious differences in the pH among organelles: the pH in lysosomes is acid within the range of 4.5–5.5, whereas mitochondrial pH is basic that can be as high as 8.0. Abnormal intracellular pH is always an indication of a disrupted pH homeostasis in the whole cell. Furthermore, intracellular bio-thiols are vital to cell metabolism, which by either elevated or deficiency levels of bio-thiols will lead to some diseases. Possessing the advantages of avoiding systematic errors and undesirable photophysical properties of certain fluorophores, novel near-infrared ratiometric fluorescent sensors for the accurately monitoring intracellular pH and biothiols have become the spotlight in research topics. Throughout this dissertation, we firstly have designed and synthesized two novel rhodamine-based dyes with high fluorescence quantum yield, good pH stability large Stokes shifts and excellent photostability by introducing an additional amino residue with fused rings into a classic rhodamine skeleton. We also have constructed a fluorescent sensor by incorporating a receptor to one of these dyes and applied it as an effective sensor for the quick and sensitive monitoring of lysosomal pH fluctuations. Then, we have prepared two sets of ratiometric fluorescent probes for the sensitive detection of lysosomal pH values. The former series were based on π-conjugation modulation strategy, which was accomplished by conjugating a visible coumarin motif to a classic near-infrared hemicyanine skeleton via a vinyl linker. The lysosome-targeting goal was reached by introducing a morpholine ligand or a o-phenylenediamine group to the hemicyanine acceptor. For the latter series, we have obtained three near-infrared ratiometric fluorescent sensors containing a TPE as a donor and a rhodamine as an acceptor for the quantitative, sensitive and comparative analysis of lysosomal pH alterations through FRET and TBET approaches. Furthermore, we have prepared two near-infrared hybrid rhodol dyes for the ratiometric and sensitive visualization of pH value alterations in mitochondria taking advantage of conjugating typical hemicyanine fluorophores into a classic rhodol motif. Upon pH changes, a rhodol hydroxyl group in the probe acts as a spiropyran switch, resulting in the change of π-conjugation and the appearance of a new fluorescent peak. Due to the positive charge, these two novel rhodol dyes possessed the mitochondria-targeting property. In the end, besides the ratiometric fluorescent pH probes, we have reported a FRET-based fluorescent sensor for the ratiometric, selective and accurate detection of cysteine (Cys), which was achieved by linking a visible coumarin skeleton and a near-infrared rhodamine motif through a piperazine spacer. This probe could be used to monitor the intracellular cysteine concentration ratiometrically and be further applied for imaging of Drosophila melanogaster larvae to detect cysteine concentration alterations in vivo