We obtain approximation bounds for products of quasimodes for the Laplace–Beltrami operator on compact Riemannian manifolds of all dimensions without boundary. We approximate the products of quasimodes uv by a low-degree vector space , and we prove that the size of the space is small. In this paper, we first study bilinear quasimode estimates of all dimensions , and , respectively, to make the highest frequency disappear from the right-hand side. Furthermore, the result of the case of bilinear quasimode estimates improves quasimodes estimates of Sogge and Zelditch in [C. D. Sogge and S. Zelditch,
A note on -norms of quasi-modes,
Some Topics in Harmonic Analysis and Applications,
Adv. Lect. Math. (ALM) 34,
International Press, Somerville 2016, 385–397] when . And on this basis, we give approximation bounds in -norm. We also prove approximation bounds for the products of quasimodes in -norm using the results of -estimates for quasimodes in [M. Blair, Y. Sire and C. D. Sogge,
Quasimode, eigenfunction and spectral projection bounds for Schrodinger operators on manifolds with critically singular potentials,
preprint 2019, https://arxiv.org/abs/1904.09665]. We extend the results of Lu and Steinerberger in [J. F. Lu and S. Steinerberger,
On pointwise products of elliptic eigenfunctions,
preprint 2018, https://arxiv.org/abs/1810.01024v2] to quasimodes.
In this paper, a generalized (G'/G)-expansion method, combined with suitable transformations, is used to construct exact solutions of the nonlinear Schrödinger equation with variable coefficients. As a result, hyperbolic function solutions, trigonometric function solutions, and rational solutions with parameters are obtained. When the parameters are taken as special values, some solutions including the known kink-type solitary wave solution and the singular travelling wave solution are derived from these obtained solutions. It is shown that the generalized (G'/G)-expansion method is direct, effective, and can be used for many other nonlinear evolution equations with variable coefficients in mathematical physics.
A modified (Gʹ/G)-expansion method is proposed to construct exact solutions of nonlinear evolution equations. To illustrate the validity and advantages of the method, the (3+1)-dimensional potential Yu-Toda-Sasa-Fukuyama (YTSF) equation is considered and more general travelling wave solutions are obtained. Some of the obtained solutions, namely hyperbolic function solutions, trigonometric function solutions, and rational solutions contain an explicit linear function of the variables in the considered equation. It is shown that the proposed method provides a more powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.
Phytochemical investigation of the n-BuOH extract of the leaves of processed Epimedium koreanum led to the isolation of three new prenylflavonol glycosides, epimedkoresides A–C (1–3), along with the 15 known ones (4–18). The structures of the three new compounds were established on the basis of chemical and spectroscopic methods as 8-γ-hydroxy-γ,γ-dimethylpropyl-5,7,4′-trihydroxy-flavonol-3- O-β-d-glucopyranosyl(1 → 2)-α-l-rhamnopyranoside (1), 8-γ-hydroxy-γ,γ-dimethylpropyl-5,7,4′-trihydroxy-flavonol-3-O-α-l-rhamnopyranosyl(1 → 2)-α-l-rhamnopyranoside (2), and anhydroicaritin 3-O-α-l-rhamnopyranosyl(1 → 2)- α-l-rhamnopyranosyl-7-O-α-l-rhamnopyranosyl(1 → 2)-β-d-glucopyranoside (3), respectively. Additionally, partial structures were evaluated for their anti-inflammatory activity.
Aim: To investigate the association of serum resistin levels with metabolic syndrome (MS) and early atherosclerosis in obese children.
Methods: A total of 176 obese children and 88 healthy children were enrolled in this study, and were gender and age matched. Obesity was defined as a body mass index (BMI) of ≥ the 95th percentile for age and sex. All children had a physical examination and routine hematology testing for fasting blood glucose, insulin, and lipids profile. Homeostasis model of assessment of insulin resistance (HOMA-IR) was calculated, as insulin resistance has a central role in the pathophysiology of MS. Non-invasive ultrasound measurement was obtained to investigate carotid intima-media thickness (IMT) as the markers of early atherosclerosis. Path analysis was used to evaluate the value of resistin levels to early atherosclerosis.
Results: The resistin levels were higher in obese children compared to healthy children (23.14±7.35 vs. 17.1±5.7 ng/mL, p<0.05), and it is positively correlated with BMI, waist circumference, systolic blood pressure, fasting insulin, HOMA-IR, IMT and high sensitive CRP (Hs-CRP), but not related to diastolic blood pressure, blood lipids and fasting glucose. A positive linear correlation was observed between resistin and the number of MS components. Path analysis indicated serum resistin can directly (β=0.304, p=0.001), and indirectly via HOMA-IR (β=0.085, p=0.008) and Hs-CRP (β=0.047, p=0.029), contribute to early atherosclerosis.
Conclusion: Resistin not only play a certain role in the presence of MS, but also indirectly via insulin resistance and Hs-CRP to contribute to early atherosclerosis in obese children.
An effective and regioselective synthesis of aminomethylated daidzein derivatives using a Mannich reaction of daidzein, formaldehyde with a variety of amines under ultrasound irradiation is described. The structures of the products were characterized by HRMS, 1H NMR, 13C NMR and 2D NMR spectra. Parallel reactions showed that ultrasound irradiation accelerated the reaction rate and improved the yields.
In the current study a facile synthetic route for preparing antineoplastic drug GDC-0449 is investigated. Starting with pyridine-1-oxide and 1-iodo-3-nitrobenzene, the intermediate product 2-(2-chloro-5-nitrophenyl) pyridine was prepared by cross-coupling, deoxidation and halogenation. The final compound was then synthesised by reduction of the nitro group followed by amidation. This synthetic route avoids the use of unstable organometallic or organic boride compounds; it employs relatively inexpensive and bench-stable reagents, involves readily controllable reaction conditions, and achieves a relatively high yield.
ADP-glucose pyrophosphorylase (AGP) consists of two large (AGP-L) and two small (AGP-S) subunits; it regulates the limiting step in the biosynthesis of starch. Here, we isolated the full-length cDNA sequence of the AGP-S gene (designed as Agp1) from wheat and whole genomic sequences from relatives of wheat. The coding region of Agp1 ranged from 5948 bp to 7666 bp and contained 9 exons and 8 introns. The full-length cDNA sequence of Agp1 encoded a polypeptide containing 473 amino acids without transit peptides. The sequence alignment and phylogenetic tree analyses indicated that the Agp1 cDNA sequence was highly conserved (99.7%) across the diploid Triticeae species examined in this study. Variations in introns caused the Agp1 genomic sequences from wheat relatives to be less similar (79.5%), and Agp1 sequences from the B genome donor species were divergent. In developing grains, rapid increase in Agp1 expression at prophase was closely followed by the accumulation of starch, and a significant (P < 0.001) correlation was observed between them. The heterogeneous mixtures of small and large subunits from different wheat species restarted glycogen synthesis in the Escherichia coli AC70R1-504 cells, which are deficient in endogenous AGP activity (glg C−/−). Our data provide useful information to evaluate and utilize Agp1 sequences in Triticeae species.