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关键词： Knot   bridge decomposition   mapping class group  

摘要： A (g, n)-decomposition of a link L in a closed orientable 3-manifold M is a decomposition of M by a closed orientable surface of genus g into two handebodies each intersecting the link L in n trivial arcs. The Goeritz group of that decomposition is then defined to be the group of isotopy classes of orientation-preserving homeomorphisms of the pair (M,L) that preserve the decomposition. We compute the Goeritz groups of all (1, 1)-decompositions.

关键词： TiO2 nanoparticles   green catalysis   sol-gel method   4-(((8-hydroxyquinolin-7-yl)(phenyl) methyl)amino)-1,5-dimethyl-2-phenyl-1,2-dihydro-3H-pyrazol-3-one derivatives   one-pot synthesis  

摘要： A series of 4-(((8-hydroxyquinolin-7-yl)(phenyl)methyl)amino)-1,5-dimethyl-2-phenyl-1,2-dihydro-3H-pyrazol-3-one derivatives (4a-j) were synthesized via a one-pot, three-component reaction. The reaction employed benzaldehyde derivatives (1a-j), 4-aminoantipyrine (2), and 8-hydroxyquinoline (3), using titanium dioxide nanoparticles (TiO2 NPs) as a catalyst. The TiO2 NPs, synthesized through a sol-gel method, efficiently catalyzed the transformation under mild conditions, delivering high yields in just 9 minutes at room temperature. Optimization revealed that 0.010 g of catalyst in methanol was optimal, with protic solvents outperforming aprotic and non-polar ones. The reaction progress was monitored using thin-layer chromatography (TLC), and the final products were isolated via recrystallization. A systematic study of reaction parameters confirmed TiO2 NPs as an efficient, reusable, and environmentally friendly catalyst for multicomponent organic synthesis.

关键词： GLP-1 receptors agonists   insulin   physicochemical   pharmacokinetic   type 2 diabetes mellitus   glucagon  

摘要： Introdcution Glucagon-like peptide-1 receptor agonists (GLP-1 RAs) are among the most effective treatments for type 2 diabetes mellitus (T2DM). GLP-1 RAs stimulate pancreatic receptors, improving glycemia by boosting insulin secretion while decreasing glucagon secretion. GLP-1 receptors are present in pancreatic tissue. They are also found in extra-pancreatic tissue and have been shown to reduce body weight while also protecting the heart and endothelial cells. The most prevalent types of GLP-1 RAs can be injected twice daily (exenatide), once daily (lixisenatide and liraglutide), or once weekly (albiglutide, dulaglutide, exenatide once, semaglutide, tirzepatide). GLP-1 receptor agonists also reduce gastric emptying, preventing substantial post-meal glycaemic increases. Many publications have been written regarding GLP-1 RAs, covering various features of this family. However, the purpose of this study is to investigate the pharmacological design models and pharmacokinetic characteristics of the most regularly used members of this class, as well as to highlight contemporary developments in GLP-1 RAs. It also describes the physicochemical features, techniques of manufacture, the effects of molecular structure, and structural modifications on pharmacological *** The literature review was completed using a structured approach to identify and integrate relevant literature. It involved a broad search of reputable medical databases using inclusion and exclusion *** They are classified as short-acting or long-acting based on the length of their action. Short-acting GLP-1 RAs and long-acting GLP-1 RAs have differing efficacy profiles. Furthermore, the methods of administration, mode of action, and side effects of these medications are relevant to their pharmacological design and pharmacokinetic *** The treatment of type 2 diabetes and obesity has evolved with the advent of GLP-1 RAs. These drugs have a multifaceted approach, emphas

关键词： Graph   Coloring   S-packing coloring   S   Packing chromatic number   Cubic graph   Subcubic graph  

摘要： For a non-decreasing sequence of integers $S=(s_1,s_2,\dots ,s_k)$, an $S$-packing coloring of $G$ is a partition of $V\left(G\right)$ into $k$ subsets $V_1,V_2,\dots ,V_k$ such that the distance between any two distinct vertices $x,y\in V_i$ is at least

摘要： Photoelectron spectra of LiP₄ˉ and Li_nP₅ˉ (n = 1–2) clusters were measured using size-selected anion photoelectron spectroscopy. The vertical detachment energies of LiP₄ˉ, LiP₅ˉ and Li₂P₅ˉ were determined to be 2.94 ± 0.08, 1.30 ± 0.08, and 1.74 ± 0.08 eV, respectively. Complementary theoretical calculations were performed for both the anionic clusters and their corresponding neutrals, and the calculated results were compared with the experimental spectra. The results indicate that LiP₄ˉ adopts a C_4v symmetric pyramidal structure consisting of a planar P₄ unit coordinated to a lithium atom. LiP₅ˉ favors a C_s symmetric pentagonal pyramidal geometry derived from a nonplanar P₅ framework coordinated to a Li atom, while neutral LiP₅ has similar structure with C_5v symmetry. Li₂P₅ˉ and neutral Li₂P₅ both adopt the distorted pentagonal bipyramidal geometries. Notably, the photoelectron spectrum of LiP₅ˉ reveals that neutral LiP₅ has a large gap between the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO), implying that the LiP₅ neutral is highly stable. Further anisotropy of the induced current density (AICD) analysis reveals a distinct diatropic ring current delocalized over the P₅ framework, providing clear magnetic evidence for aromatic character in the LiP₅ cluster.

关键词： Gaussian estimates   Parabolic equation   Corrector   Generalized eigenfunctions   Nash type estimates  

摘要： We derive in this paper Gaussian estimates for a general parabolic equation $u_t-{(a(x)u_x)}_x=r(x)u$ over $\mathbb{R}$. Here a and r are only assumed to be bounded, measurable and ${\mathrm{essinf}}_{\mathbb{R}}a>0$. We first consider a canonical equation