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This study is concerned with the existence of normalized solutions for the Choquard equations with critical nonlinearities −Δu+λu=f(u)+(Iα∗∣u∣2α*)∣u∣2α*−2u,inRN,∫RN∣u∣2dx=a2,\left\{\begin{array}{l}-\Delta u+\lambda u=f\left(u)+\left({I}_{\alpha }\ast {| u| }^{{2}_{\alpha }^{* }}){| u| }^{{2}_{\alpha }^{* }-2}u,\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\hspace{1.0em}\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{| u| }^{2}{\rm{d}}x={a}^{2},\hspace{1.0em}\end{array}\right. where N>2N\gt 2, α∈(0,N)\alpha \in \left(0,N), a>0a\gt 0, and Iα(x){I}_{\alpha }\left(x) is the Riesz potential given by Iα(x)=Aα∣x∣N−αwithAα=ΓN−α22απN2Γα2,{I}_{\alpha }\left(x)=\frac{{A}_{\alpha }}{{| x| }^{N-\alpha }}\hspace{1em}\hspace{0.1em}\text{with}\hspace{0.1em}\hspace{0.33em}{A}_{\alpha }=\frac{\Gamma \left(\phantom{\rule[-0.68em]{}{0ex}},\frac{N-\alpha }{2}\right)}{{2}^{\alpha }{\pi }^{\tfrac{N}{2}}\Gamma \left(\phantom{\rule[-0.68em]{}{0ex}},\frac{\alpha }{2}\right)}, and 2α*=N+αN−2{2}_{\alpha }^{* }=\frac{N+\alpha }{N-2} is the Hardy-Littlewood-Sobolev critical exponent and ff is a subcritical nonlinearity. In the case that ff is L2{L}^{2}-supercritical growth, by means of the Pohozaev manifold method and mountain pass theorem, we obtain a couple of the normalized solution; while in the case f(u)=μ∣u∣q−2uf\left(u)=\mu {| u| }^{q-2}u with 20\mu \gt 0 a parameter, we employ the truncation technique and the genus theory to prove the multiplicity of normalized solutions.

This article is concerned with the existence of multiple normalized solutions for a class of Choquard equations with a parametric perturbation −ε2Δu+V(x)u=λu+ε−α(Iα*F(u))f(u),x∈RN,∫RN∣u∣2dx=a2εN,\left\{\begin{array}{ll}-{\varepsilon }^{2}\Delta u+V\left(x)u=\lambda u+{\varepsilon }^{-\alpha }\left({I}_{\alpha }* F\left(u))f\left(u),\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{| u| }^{2}{\rm{d}}x={a}^{2}{\varepsilon }^{N},\hspace{1.0em}& \end{array}\right. where a>0a\gt 0 is a constant, ε>0\varepsilon \gt 0 is a parameter, N≥3N\ge 3, α∈(0,N)\alpha \in \left(0,N), λ∈R\lambda \in {\mathbb{R}} is unknown and appears as a Lagrange multiplier, ff is a continuous function with L2{L}^{2}-subcritical growth, and V:RN→[0,∞)V:{{\mathbb{R}}}^{N}\to \left[0,\infty ) is a continuous function, satisfying del Pino and Felmer’s local conditions. With the help of the penalization method, and Lusternik-Schnirelmann theory, we investigate the relationship between the number of positive normalized solutions and the topology of the set, where the potential VV attains its minimum value if the parameter ε>0\varepsilon \gt 0 is small.

In this article, we study the fractional critical Choquard equation with a nonlocal perturbation: (−Δ)su=λu+α(Iμ*∣u∣q)∣u∣q−2u+(Iμ*∣u∣2μ,s*)∣u∣2μ,s*−2u,inRN,{\left(-{\Delta })}^{s}u=\lambda u+\alpha \left({I}_{{\mu }^{* }}\hspace{-0.25em}{| u| }^{q}){| u| }^{q-2}u+\left({I}_{{\mu }^{* }}\hspace{-0.25em}{| u| }^{{2}_{\mu ,s}^{* }}){| u| }^{{2}_{\mu ,s}^{* }-2}u,\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}, having prescribed mass ∫RNu2dx=c2,\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{u}^{2}{\rm{d}}x={c}^{2}, where s∈(0,1),N>2s,00,c>0s\in \left(0,1),N\gt 2s,0\lt \mu \lt N,\alpha \gt 0,c\gt 0, and Iμ(x){I}_{\mu }\left(x) is the Riesz potential given by Iμ(x)=Aμ∣x∣μwithAμ=Γμ22N−μπN⁄2ΓN−μ2,{I}_{\mu }\left(x)=\frac{{A}_{\mu }}{{| x| }^{\mu }}\hspace{1em}\hspace{0.1em}\text{with}\hspace{0.1em}\hspace{0.33em}{A}_{\mu }=\frac{\Gamma \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{\mu }{2}\right)}{{2}^{N-\mu }{\pi }^{N/2}\Gamma \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{N-\mu }{2}\right)}, and 2N−μN

In this paper, we study the fractional Schrödinger-Poisson system

Abstract Background Cucumber, Cucumis sativus L. (2n = 2 × = 14) and melon, C. melo L. (2n = 2 × = 24) are two important vegetable species in the genus Cucumis (family Cucurbitaceae). Both species have an Asian origin that diverged approximately nine million years ago. Cucumber is believed to have evolved from melon through chromosome fusion, but the details of this process are largely unknown. In this study, comparative genetic mapping between cucumber and melon was conducted to examine syntenic relationships of their chromosomes. Results Using two melon mapping populations, 154 and 127 cucumber SSR markers were added onto previously reported F2- and RIL-based genetic maps, respectively. A consensus melon linkage map was developed through map integration, which contained 401 co-dominant markers in 12 linkage groups including 199 markers derived from the cucumber genome. Syntenic relationships between melon and cucumber chromosomes were inferred based on associations between markers on the consensus melon map and cucumber draft genome scaffolds. It was determined that cucumber Chromosome 7 was syntenic to melon Chromosome I. Cucumber Chromosomes 2 and 6 each contained genomic regions that were syntenic with melon chromosomes III+V+XI and III+VIII+XI, respectively. Likewise, cucumber Chromosomes 1, 3, 4, and 5 each was syntenic with genomic regions of two melon chromosomes previously designated as II+XII, IV+VI, VII+VIII, and IX+X, respectively. However, the marker orders in several syntenic blocks on these consensus linkage maps were not co-linear suggesting that more complicated structural changes beyond simple chromosome fusion events have occurred during the evolution of cucumber. Conclusions Comparative mapping conducted herein supported the hypothesis that cucumber chromosomes may be the result of chromosome fusion from a 24-chromosome progenitor species. Except for a possible inversion, cucumber Chromosome 7 has largely remained intact in the past nine million years since its divergence from melon. Meanwhile, many structural changes may have occurred during the evolution of the remaining six cucumber chromosomes. Further characterization of the genomic nature of Cucumis species closely related to cucumber and melon might provide a better understanding of the evolutionary history leading to modern cucumber.

The botanical family Cucurbitaceae includes a variety of fruit crops with global or local economic importance. How their genomes evolve and the genetic basis of diversity remain largely unexplored. In this study, we sequence the genome of the wax gourd (Benincasa hispida), which bears giant fruit up to 80 cm in length and weighing over 20 kg. Comparative analyses of six cucurbit genomes reveal that the wax gourd genome represents the most ancestral karyotype, with the predicted ancestral genome having 15 proto-chromosomes. We also resequence 146 lines of diverse germplasm and build a variation map consisting of 16 million variations. Combining population genetics and linkage mapping, we identify a number of regions/genes potentially selected during domestication and improvement, some of which likely contribute to the large fruit size in wax gourds. Our analyses of these data help to understand genome evolution and function in cucurbits.

In the context of the advancing construction engineering field in China, there has been a significant increase in the adoption of building information modeling (BIM) technology within engineering–procurement–construction (EPC) projects. This emerging technology is expected to significantly influence the decision‐making practices of professionals in the construction and engineering domain. Consequently, there is an urgent need for comprehensive research focused on construction management specifically pertaining to the effectiveness of construction developer management. This study examines seven representative EPC projects where BIM technology was implemented and it considers various factors that have the potential to influence project decision‐making while taking into account time and cost considerations. A comprehensive multidimensional evaluation method that combines the analytic hierarchy process (AHP) and the fuzzy comprehensive evaluation method is introduced in this study. The results of this study emphasize that the efficiency enhancement strategies derived from the judgment indicator method demonstrate favorable practical outcomes. This research highlights the importance of a comprehensive evaluation method combining the AHP and the fuzzy comprehensive evaluation method to enhance decision‐making efficiency for construction developer management staff. [ABSTRACT FROM AUTHOR]

Objective: The study aimed to investigate the clinical characteristics of patients with multi-canal benign paroxysmal positional vertigo (MC-BPPV). Methods: We performed a retrospective analysis of 927 patients with BPPV who were admitted to our hospital between January 1, 2016 and December 31, 2019. The clinical data of all patients were collected. The Dix-Hallpike, straight head-hanging, and supine Roll tests were performed in all patients. The nystagmus was recorded using videonystagmography. The clinical characteristics of patients with MC-BPPV and single canal BPPV (SC-BPPV) was analyzed and compared. Results: Among 927 patients included, 49 (5.29%) patients had MC-BPPV, 878 (94.71%) patients had SC-BPPV. There were significant differences in the male to female ratio (1:3.90 vs 1:1.81, P <.05), mean age (62.47±12.51 vs 59.04±13.72, P <.05), as well as the ratio of cupulolithiasis to canalithiasis (1:1.45 vs 1:4.78, P <.01) between patients with MC-BPPV and SC-BPPV. The frequency of involvement of PC, HC, and AC were involved for 66 (67.35%), 23 (23.47%), and 9 (9.18%) times, respectively, in patients with MC-BPPV, which were involved in 581 (66.17%), 281 (32.0%), and 16 (1.82%) patients, respectively, in patients with SC-BPPV. No significant difference was found in the frequency of involvement of PC and HC between patients with MC-BPPV and SC-BPPV, while there was significant difference in the frequency of AC involvement between 2 groups (P <.01). Ipsilateral PC-HC-BPPV (n = 18) and bilateral PC-BPPV (n=19) were the most common among patients with MC-BPPV. Twenty-six (53.06%) patients had ipsilateral MC-BPPV, 23 (46.94%) had bilateral MC-BPPV. Of the 26 patients with ipsilateral MC-BPPV, 7 patients combined with unilateral peripheral vestibular disorder. Conclusion: Patients with MC-BPPV had a significantly older mean age at disease onset and a higher proportion of females compared with patients with SC-BPPV. Cupulolithiasis was more common in MC-BPPV. AC involvement was also much more common in MC-BPPV than in SC-BPPV. [ABSTRACT FROM AUTHOR]

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