On concentration of solution to a Schrödinger logarithmic equation with deepening potential well

In this work, we prove the existence of positive solution for the following class of problems where λ>0 and is a potential satisfying some conditions. Using the variational method developed by Szulkin for functionals, which are the sum of a C¹ functional with a convex lower semicontinuous functional, we prove that for each large enough λ>0, there exists a positive solution for the problem, and that, as λ→+∞, such solutions converge to a positive solution of the limit problem defined on the domain Ω=int(V^?1({0})).     

求解矩阵方程AXB+CXD=F参数迭代法的最优参数分析

本文针对求矩阵方程AXB+CXD=F唯一解的参数迭代法，分析当矩阵A，B，C，D均是Hermite正（负）定矩阵时，迭代矩阵的特征值表达式，给出了最优参数的确定方法，并提出了相应的加速算法.     

Cationic para‐phenyl‐substituted α‐diimine nickel catalyzed ethylene and 4‐methyl‐1‐pentene (co)polymerizations via living/controlled chain‐walking

Upon activation with diethylaluminium chloride (Et₂AlCl), a series of phenyl‐substituted α‐diimine nickel precatalysts conducted 4‐methyl‐1‐pentene (4MP) and ethylene (E) (co)polymerizations via controlled chain‐walking to generate branched amorphous polymers with high molecular weight and narrow molecular weight distribution (M_w/M_n < 1.6). The obtained poly(4MP)s were amorphous elastomers with glass transition temperature (T_g) of ?10 ~ ?24 °C, which are higher than that of E‐4MP copolymer ( ? 63.0 °C). At room temperature (25 °C), 4MP polymerization proceeds in a living manner. The microstructures of the produced poly(4MP)s indicated the 2,1‐ and 1,2‐insertion followed by chain‐walking, the latter being predominant. The NMR analyses of the polymers showed that the obtained poly(4MP) possessed methyl, isobutyl, 2,4‐dimethylpentyl and 2‐methylhexyl groups, while the isobutyl and 2,4‐dimethylalkyl branches derived from 4MP were observed in the E‐4MP copolymer. The branch structures and the insertion‐type of monomer were depended on the polymerization temperature, and the content of methyl branch increased with an increase in the polymerization temperature.     

Single‐armed salamo‐like dioxime and its multinuclear Cu (II), Zn (II) and Cd (II) complexes: Syntheses,structural characterizations,Hirshfeld analyses and fluorescence properties

Three multinuclear Cu (II), Zn (II) and Cd (II) complexes, [Cu₂(L)(μ‐OAc)]·CHCl₂ ( 1 ), [Zn₂(L)(μ‐OAc)(H₂O)]·3CHCl₃ ( 2 ) and [{Cd₂(L)(OAc)(CH₃CH₂OH)}₂]·2CH₃CH₂OH ( 3 ) with a single‐armed salamo‐like dioxime ligand H₃L have been synthesized, and characterized by FT‐IR, UV–vis, X‐ray crystallography and Hirshfeld surfaces analyses. The ligand H₃L has a linear structure and C‐H···π interactions between the two molecules. The complex 1 is a dinuclear Cu (II) complex, Cu1 and Cu2 are all five‐coordinate possessing distorted square pyramidal geometries. The complex 2 also forms a dinuclear Zn (II) structure, and Zn1 and Zn2 are all five‐coordinate bearing distorted trigonal bipyramidal geometries. The complex 3 is a symmetrical tetranuclear Cd (II) complex, and Cd1 is a hexa‐coordinate having octahedral configuration and Cd2 is hepta‐coordinate with a pentagonal bipyramidal geometry, and it has π···π interactions inside the molecule. In addition, fluorescence properties of the ligand and its complexes 1 – 3 have also been discussed.     

On the minimal sum of Betti numbers of an almost complex manifold

We show that the only rational homology spheres which can admit almost complex structures occur in dimensions two and six. Moreover, we provide infinitely many examples of six-dimensional rational homology spheres which admit almost complex structures, and infinitely many which do not. We then show that if a closed almost complex manifold has sum of Betti numbers three, then its dimension must be a power of two.     

Classical Backfitting for Smooth-Backfitting Additive Models

Smooth backfitting has been shown to have better theoretical properties than classical backfitting for fitting additive models based on local linear regression. In this article, we show that the smooth backfitting procedure in the local linear case can be alternatively performed as a classical backfitting procedure with a different type of smoother matrices. These smoother matrices are symmetric and shrinking and some established results in the literature are readily applicable. The connections allow the smooth backfitting algorithm to be implemented in a much simplified way, give new insights on the differences between the two approaches in the literature, and provide an extension to local polynomial regression. The connections also give rise to a new estimator at data points. Asymptotic properties of general local polynomial smooth backfitting estimates are investigated, allowing for different orders of local polynomials and different bandwidths. Cases of oracle efficiency are discussed. Computer-generated simulations are conducted to demonstrate finite sample behaviors of the methodology and a real data example is given for illustration. Supplementary materials for this article are available online.     

New palladium (II) complexes containing phosphine‐nitrogen ligands and their use as catalysts in aminocarbonylation reaction

Aminocarbonylation of aryl halides, homogeneously catalysed by palladium, is an efficient method that can be employed for obtaining amides for pharmaceutical and synthetic applications. In this work, palladium (II) complexes containing P^N ligands were studied as catalysts in the aminocarbonylation of iodobenzene in the presence of diethylamine. Two types of systems were used: a palladium (II) complex formed in situ; and one prepared prior to the catalytic reaction. In general, the palladium complexes studied achieved high conversions in an average reaction time of less than 2 hr, which is less than that for the standard system (Pd (II)/PPh₃) used. The pre‐synthesized complexes were faster than their in situ counterparts, as the latter require an induction time to form the Pd/P^N species. The structure and electronic properties of the ligand P^N can influence both the activity and the selectivity of the reaction, stabilizing the acyl‐palladium intermediates formed in a better manner.     

Fe3O4‐supported Schiff‐base copper (II) complex: A valuable heterogeneous nanocatalyst for one‐pot synthesis of new pyrano[2,3‐b]pyridine‐3‐carboxamide derivatives

A novel Cu (II) Schiff‐base complex immobilized on core‐shell magnetic Fe₃O₄ nanoparticles (Fe₃O₄@SPNC) was successfully designed and synthesized. The structural features of these nanoparticles were studied and confirmed by using various techniques including FT‐IR spectroscopy, scanning electron microscopy (SEM), transmission electron microscopy (TEM), energy‐dispersive X‐ray spectroscopy (EDS), vibrating sample magnetometer (VSM), X‐Ray diffraction (XRD), wavelength dispersive X‐ray spectroscopy (WDX), and inductively coupled plasma (ICP). These newly synthesized nanoparticles have been used as efficient heterogeneous catalytic system for one‐pot multicomponent synthesis of new pyrano[2,3‐b]pyridine‐3‐carboxamide derivatives. Notably, the catalyst could be easily separated from the reaction mixture by using an external magnet and reused for several successive reaction runs with no significant loss of activity or copper leaching. The present protocol benefits from a hitherto unreported MNPs‐immobilized Cu (II) Schiff‐base complex as an efficient nanocatalyst for the synthesis of newly reported derivatives of pyrano[2,3‐b]pyridine‐3‐carboxamide from one‐pot multicomponent reactions.     

Organometallic titanocene complex as highly efficient bifunctional catalyst for intramolecular Mannich reaction

Bifunctional catalysts bearing two catalytic sites, Lewis acidic organometallic titanocene and Brønsted acidic COOH, have been assembled in situ from Cp₂TiCl₂ with carboxylic acid ligands, showing high catalytic activity over an intramolecular Mannich reaction towards synthesis of 2‐aryl‐2,3‐dihydroquinolin‐4(1H)‐ones. The determination of the bifunctional catalyst Cp₂Ti(C₈H₄NO₆)₂ was elucidated by single X‐ray HR‐MS and investigation of catalytic behavior. In particular, masking the Brønsted acidic COOH catalytic site with dormant COOMe lowered the reaction yield greatly, indicating that two catalytic sites work together to maintain high catalytic efficiency.     

Synthesis,X‐ray characterization and density functional theory studies of N6‐benzyl‐N6‐methyladenine–M(II) complexes (M = Zn,Cd): The prominent role of π–π, C–H···π and anion–π interactions

We report the synthesis and X‐ray characterization of the N⁶‐benzyl‐N⁶‐methyladenine ligand (L) and three metal complexes, namely [Zn(HL)Cl₃]·H₂O ( 1 ), [Cd(HL)₂Cl₄] ( 2 ) and [H₂L]₂[Cd₃(μ‐L)₂(μ‐Cl)₄Cl₆]·3H₂O ( 3 ). Complex 1 consists of the 7H‐adenine tautomer protonated at N3 and coordinated to a tetrahedral Zn(II) metal centre through N9. The octahedral Cd(II) in complex 2 is N⁹‐coordinated to two N⁶‐benzyl‐N⁶‐methyladeninium ligands (7H‐tautomer protonated at N3) that occupy apical positions and four chlorido ligands form the basal plane. Compound 3 corresponds to a trinuclear Cd(II) complex, where the central Cd atom is six‐coordinated to two bridging μ‐L and four bridging μ‐Cl ligands. The other two Cd atoms are six‐coordinated to three terminal chlorido ligands, to two bridging μ‐Cl ligands and to the bridging μ‐L through N3. Essentially, the coordination patterns, degree of protonation and tautomeric forms of the nucleobase dominate the solid‐state architectures of 1 – 3 . Additionally, the hydrogen‐bonding interactions produced by the endocyclic N atoms and NH groups stabilize high‐dimensional‐order supramolecular assemblies. Moreover, energetically strong anion–π and lone pair (lp)–π interactions are important in constructing the final solid‐state architectures in 1 – 3 . We have studied the non‐covalent interactions energetically using density functional theory calculations and rationalized the interactions using molecular electrostatic potential surfaces and Bader's theory of atoms in molecules. We have particularly analysed cooperative lp–π and anion–π interactions in 1 and π⁺–π⁺ interactions in 3 .