The ∞‐Besov capacity problem

A theory of ∞‐Besov capacities is developed and several applications are provided. In particular, we solve an open problem in the theory of limits of the ∞‐Besov semi‐norms, we obtain new restriction‐extension inequalities, and we characterize the pointwise multipliers acting on the ∞‐Besov spaces.     

An alternative proof of lower bounds for the first eigenvalue on manifolds

Recently, Andrews and Clutterbuck [1] gave a new proof of the optimal lower eigenvalue bound on manifolds via modulus of continuity for solutions of the heat equation. In this short note, we give an alternative proof of Theorem 2 in [1]. More precisely, following Ni's method (Section 6 of [5]), we give an elliptic proof of this theorem.     

Gold‐Catalyzed Suzuki Coupling of ortho‐Substituted Hindered Aryl Substrates

A method that allows hindered ortho ‐substituted aryl iodides to be efficiently coupled to phenylboronic acid using a gold‐catalyzed C−C bond formation is presented. The use of a molecularly‐defined dinuclear gold chloride catalytic precursor that is stabilized by a new tetradentate (N ,N′ )‐diamino‐(P,P′ )‐diphosphino ferrocene hybrid ligand in a Suzuki‐type reaction is described for the first time. Electron‐rich isopropyl groups on phosphorus were found essential for a superior activity, while the performances of a set of analogous gold dinuclear complexes that were fully characterized by multinuclear NMR spectroscopy and XRD analysis, were investigated. Therefore, arylation of para and ortho ‐substituted iodoarenes bearing electron‐rich, electron‐poor functional groups, and even hindered polycyclic aromatic compounds is described.     

N‐Benzylnicotinamide and N‐benzyl‐1,4‐dihydronicotinamide: useful models for NAD+ and NADH

3‐Aminocarbonyl‐1‐benzylpyridinium bromide (N‐benzylnicotinamide, BNA), C₁₃H₁₃N₂O⁺·Br⁻, (I), and 1‐benzyl‐1,4‐dihydropyridine‐3‐carboxamide (N‐benzyl‐1,4‐dihydronicotinamide, rBNA), C₁₃H₁₄N₂O, (II), are valuable model compounds used to study the enzymatic cofactors NAD(P)⁺ and NAD(P)H. BNA was crystallized successfully and its structure determined for the first time, while a low‐temperature high‐resolution structure of rBNA was obtained. Together, these structures provide the most detailed view of the reactive portions of NAD(P)⁺ and NAD(P)H. The amide group in BNA is rotated 8.4 (4)° out of the plane of the pyridine ring, while the two rings display a dihedral angle of 70.48 (17)°. In the rBNA structure, the dihydropyridine ring is essentially planar, indicating significant delocalization of the formal double bonds, and the amide group is coplanar with the ring [dihedral angle = 4.35 (9)°]. This rBNA conformation may lower the transition‐state energy of an ene reaction between a substrate double bond and the dihydropyridine ring. The transition state would involve one atom of the double bond binding to the carbon ortho to both the ring N atom and the amide substituent of the dihydropyridine ring, while the other end of the double bond accepts an H atom from the methylene group para to the N atom.     

Omega‐3 Fatty Acid Grafted PAMAM‐Paclitaxel Conjugate Exhibits Enhanced Anticancer Activity in Upper Gastrointestinal Cancer Cells

Upper Gastrointestinal Cancers (UGCs) are a leading cause of cancer‐related deaths worldwide. Paclitaxel (PTX) is frequently used for the treatment of UGCs; however, low bioavailability, reduced solubility, and dose‐dependent toxicity impede its therapeutic use. PAMAMG_4.0‐NH₂‐DHA is synthesized by linking amine‐terminated fourth‐generation poly(amidoamine) (PAMAMG_4.0‐NH₂) dendrimers with omega‐3 fatty acid docosahexaenoic acid (DHA). Next, PAMAMG_4.0‐NH₂‐DHA‐PTX (DHATX) and PAMAMG_4.0‐NH₂‐PTX (PAX) conjugates are synthesized by subsequent covalent binding of PTX with PAMAMG_4.0‐NH₂‐DHA and PAMAMG_4.0‐NH₂, respectively. ¹H‐NMR and MALDI‐TOF analyses are performed to confirm conjugation of DHA to PAMAMG_4.0‐NH₂ and PTX to PAMAMG_4.0‐NH₂‐DHA. The cell viability, clonogenic cell survival, and flow cytometry analyses are used to determine the anticancer activity of PTX, PAX, and DHATX in UGC cell lines. The in vitro data indicate that treatment with DHATX is significantly more potent than PTX or PAX at inhibiting cellular proliferation, suppressing long‐term survival, and inducing cell death in UGC cells.

     

A Sixteen‐Membered Au8P8 Macrocycle Based on Gold(I) and Diphospha(III)guanidine

The reaction of cyclo ‐P₄Mes₄C(NCy) ( 1 ) with two equivalents of [AuCl(tht)] (tht=tetrahydrothiophene) resulted in the formation of unusual sixteen‐membered Au–P macrocycle 2 . This macrocycle contains diphospha(III)guanidinate as a coordinating ligand, which is formed by P−P bond cleavage of 1 . Macrocycle 2 was characterized by multinuclear NMR spectroscopy, mass spectrometry and X‐ray crystallography.     

Graphs with No Induced Five‐Vertex Path or Antipath

We prove that a graph G contains no induced ‐vertex path and no induced complement of a ‐vertex path if and only if G is obtained from 5‐cycles and split graphs by repeatedly applying the following operations: substitution, split unification, and split unification in the complement, where split unification is a new class‐preserving operation introduced here.     

Asymptotic behavior of solutions to the compressible nematic liquid crystal system in ℝ3

In this paper, we study a nematic liquid crystals system in three-dimensional whole space ?³ and obtain the time decay rates of the higher-order spatial derivatives of the solution by the method of spectral analysis and energy estimates if the initial data belongs to L¹?³ additionally.     

Solitary wave solution of the generalized Hirota–Satsuma coupled KdV system

In this research, we find the exact traveling wave solutions involving parameters of the generalized Hirota–Satsuma couple KdV system according to the modified simple equation method with the aid of Maple 16. When these parameters are taken special values, the solitary wave solutions are derived from the exact traveling wave solutions. It is shown that the modified simple equation method provides an effective and a more powerful mathematical tool for solving nonlinear evolution equations in mathematical physics. Comparison between our results and the well-known results will be presented.