Solving Fr\'echet Distance Problems by Algebraic Geometric Methods

We study several polygonal curve problems under the Fr\'{e}chet distance via algebraic geometric methods. Let $\mathbb{X}_m^d$ and $\mathbb{X}_k^d$ be the spaces of all polygonal curves of $m$ and $k$ vertices in $\mathbb{R}^d$, respectively. We assume that $k \leq m$. Let $\mathcal{R}^d_{k,m}$ be the set of ranges in $\mathbb{X}_m^d$ for all possible metric balls of polygonal curves in $\mathbb{X}_k^d$ under the Fr\'{e}chet distance. We prove a nearly optimal bound of $O(dk\log (km))$ on the VC dimension of the range space $(\mathbb{X}_m^d,\mathcal{R}_{k,m}^d)$, improving on the previous $O(d^2k^2\log(dkm))$ upper bound and approaching the current $\Omega(dk\log k)$ lower bound. Our upper bound also holds for the weak Fr\'{e}chet distance. We also obtain exact solutions that are hitherto unknown for curve simplification, range searching, nearest neighbor search, and distance oracle.Comment: To appear at SODA24, correct some reference

Curve Simplification and Clustering under Fr\'echet Distance

We present new approximation results on curve simplification and clustering under Fr\'echet distance. Let $T = \{\tau_i : i \in [n] \}$ be polygonal curves in $R^d$ of $m$ vertices each. Let $l$ be any integer from $[m]$. We study a generalized curve simplification problem: given error bounds $\delta_i > 0$ for $i \in [n]$, find a curve $\sigma$ of at most $l$ vertices such that $d_F(\sigma,\tau_i) \le \delta_i$ for $i \in [n]$. We present an algorithm that returns a null output or a curve $\sigma$ of at most $l$ vertices such that $d_F(\sigma,\tau_i) \le \delta_i + \epsilon\delta_{\max}$ for $i \in [n]$, where $\delta_{\max} = \max_{i \in [n]} \delta_i$. If the output is null, there is no curve of at most $l$ vertices within a Fr\'echet distance of $\delta_i$ from $\tau_i$ for $i \in [n]$. The running time is $\tilde{O}\bigl(n^{O(l)} m^{O(l^2)} (dl/\epsilon)^{O(dl)}\bigr)$. This algorithm yields the first polynomial-time bicriteria approximation scheme to simplify a curve $\tau$ to another curve $\sigma$, where the vertices of $\sigma$ can be anywhere in $R^d$, so that $d_F(\sigma,\tau) \le (1+\epsilon)\delta$ and $|\sigma| \le (1+\alpha) \min\{|c| : d_F(c,\tau) \le \delta\}$ for any given $\delta > 0$ and any fixed $\alpha, \epsilon \in (0,1)$. The running time is $\tilde{O}\bigl(m^{O(1/\alpha)} (d/(\alpha\epsilon))^{O(d/\alpha)}\bigr)$. By combining our technique with some previous results in the literature, we obtain an approximation algorithm for $(k,l)$-median clustering. Given $T$, it computes a set $\Sigma$ of $k$ curves, each of $l$ vertices, such that $\sum_{i \in [n]} \min_{\sigma \in \Sigma} d_F(\sigma,\tau_i)$ is within a factor $1+\epsilon$ of the optimum with probability at least $1-\mu$ for any given $\mu, \epsilon \in (0,1)$. The running time is $\tilde{O}\bigl(n m^{O(kl^2)} \mu^{-O(kl)} (dkl/\epsilon)^{O((dkl/\epsilon)\log(1/\mu))}\bigr)$.Comment: 28 pages; Corrected some wrong descriptions concerning related wor

Green production of a novel sorbent from kaolin for capturing gaseous PbCl2 in a furnace

The pollution of semi-volatile heavy metals is one of the key environmental risks for municipal solid waste incineration, and in-situ adsorption of metals within the furnace by mineral sorbents such as kaolin has been demonstrated as a promising emission control method. To lessen the consumption of sorbent, a novel material of amorphous silicate was produced from kaolin through pressurised hydrothermal treatment. Its performance of gaseous PbCl2 capture was tested in a fixed bed furnace and compared with unmodified kaolin and metakaolin. With increasing temperature, the adsorption rates for all sorbents declined due to higher saturated vapour pressure, while the partitions of residual form lead increased which indicated higher stability of heavy metals in the sorbent because of melting effect. The new sorbent with a larger surface area and reformed structure presented 26% more adsorption efficiency than raw kaolin at 900 °C, and increasing the modification pressure improved these properties. Additionally, the production of this high-temperature sorbent was relatively inexpensive, required little thermal energy and no chemicals to produce and no waste effluent was generated, thus being much cleaner than other modification methods

Identification and analysis of differential miRNA–mRNA interactions in coronary heart disease: an experimental screening approach

ObjectiveThis aim of this study is to screen the differential molecules of kidney deficiency and blood stasis (KDBS) syndrome in coronary heart disease by high-throughput sequencing. In addition, the study aims to verify the alterations in the expression levels of miR-4685-3p and its regulated downstream, namely, C1QC, C4, and C5, using quantitative polymerase chain reaction (qPCR) and enzyme-linked immunosorbent assay (ELISA), and to determine whether the complement and coagulation cascade pathway is the specific pathogenic pathway.MethodsPatients diagnosed with unstable angina pectoris with KDBS syndrome, patients with non-kidney deficiency blood stasis (NKDBS) syndrome, and a Normal group were recruited. The clinical symptoms of each group were further analyzed. Illumina's NextSeq 2000 sequencing platform and FastQC software were used for RNA sequencing and quality control. DESeq software was used for differential gene expression (DGE) analysis. qPCR and ELISA verification were performed on DGE analysis.ResultsThe DGE profiles of 77 miRNA and 331 mRNA were selected. The GO enrichment analysis comprised 43 biological processes, 49 cell components, and 42 molecular functions. The KEGG enrichment results included 40 KEGG pathways. The PCR results showed that, compared with the Normal group, the miR-4685-3p levels decreased in the CHD_KDBS group (P = 0.001), and were found to be lower than those observed in the CHD_NKDBS group. The downstream mRNA C1 regulated by miR-4685-3p showed an increasing trend in the CHD_KDBS group, which was higher than that in the Normal group (P = 0.0019). The mRNA C4 and C5 in the CHD_KDBS group showed an upward trend, but the difference was not statistically significant. ELISA was utilized for the detection of proteins associated with the complement and coagulation cascade pathway. It was found that the expression level of C1 was significantly upregulated in the CHD_KDBS group compared with the Normal group (P < 0.0001), which was seen to be higher than that in the CHD_NKDBS group (P < 0.0001). The expression levels of C4 and C5 in the CHD_KDBS group were significantly lower than the Normal group, and were lower than that in the CHD_NKDBS group (P < 0.0001).ConclusionThe occurrence of CHD_KDBS might be related to the activation of the complement and coagulation cascade pathway, which is demonstrated by the observed decrease in miR-4685-3p and the subsequent upregulation of its downstream C1QC. In addition, the expression levels of complement C4 and C5 were found to be decreased, which provided a research basis for the prevention and treatment of this disease

Symbiosis of Arbuscular Mycorrhizal Fungi and <i>Lycium barbarum</i> L. Prefers NO₃⁻ over NH₄⁺

Nitrogen (N) is an essential nutrient that plants require and is, most of the time, limited in different terrestrial ecosystems. Forming symbioses with plants, arbuscular mycorrhizal (AM) fungi improve mineral element uptake and the net primary production of plants. Recent reports have suggested that AM fungi mediate N uptake in plants. However, there are fewer studies on the influence of AM fungi on the response of Lycium barbarum, a medicinal plant in northwest China, under different N-addition conditions. In this study, the effect of Rhizophagus irregularis, N forms (NO3− and NH4+), and N levels (1.5, 7.5, 15, 30 mM) on the performance of L. barbarum was evaluated through a pot experiment. The application of R. irregularis significantly improved L. barbarum biomass, net photosynthetic rate, and root tissue viability under adequate NO3− and NH4+ supplies, and mycorrhizal plants showed better performance under NO3− supply. AM colonization enhanced N acquisition under adequate NO3− supply and strongly induced the expression of LbAMT3-1 in L. barbarum roots. Based on these results, we propose that NO3−-dominated N supply favors mycorrhizal symbiosis to a greater extent than NH4+; this study provides a basis for maintaining beneficial AM symbiosis during nitrogen fertilizer use in arable land

Dynamic transformations of metals in the burning solid matter during combustion of heavy metal-contaminated biomass

Combustion as an efficient and reliable method is widely used for metal-enriched biomass to achieve energy and metal recoveries, but there are emission risks of heavy metals in the flue gas and bottom ash that can give rise to secondary pollutions. To optimize such combustion processes, this work investigated the combustion characteristics of a kind of hyperaccumulator biomass and focused on the intermediate states and dynamic transformations of metals for the first time. A pseudo-in situ sampling method was used to collect the burning solid residues at different time intervals before further analysis. The conversions between elemental forms were revealed, and their conversion rates were also calculated. It was found that the transformation of metals was determined by their elemental natures, species distributions, and combustion progress where there was not a consecutive process but separated by several stages, which were related to (1) the release of volatile matters, (2) the formation and consumption of the char, and (3) the fixation by silicates. Based on the information of dynamic metal characteristics, a new strategy was proposed to optimize metal distribution by adjusting the combustion time of operations. The methodology introduced in this work will also help emission control and metal recovery for other metal-rich fuels

Comprehensive Identification of the <i>Pum</i> Gene Family and Its Involvement in Kernel Development in Maize

The Pumilio (Pum) RNA-binding protein family regulates post-transcription and plays crucial roles in stress response and growth. However, little is known about Pum in plants. In this study, a total of 19 ZmPum genes were identified and classified into two groups in maize. Although each ZmPum contains the conserved Pum domain, the ZmPum members show diversity in the gene and protein architectures, physicochemical properties, chromosomal location, collinearity, cis-elements, and expression patterns. The typical ZmPum proteins have eight α-helices repeats, except for ZmPum2, 3, 5, 7, and 14, which have fewer α-helices. Moreover, we examined the expression profiles of ZmPum genes and found their involvement in kernel development. Except for ZmPum2, ZmPum genes are expressed in maize embryos, endosperms, or whole seeds. Notably, ZmPum4, 7, and 13 exhibited dramatically high expression levels during seed development. The study not only contributes valuable information for further validating the functions of ZmPum genes but also provides insights for improvement and enhancing maize yield