On the rate of convergence of solutions in free boundary problems via penalization

The rate of convergence of approximate solutions via penalization for free boundary problems are concerned. A key observation is to obtain global bounds of penalized terms which give necessary estimates on integrations by the nonlinear adjoint method by L.C. Evans.     

Synthetic Strategies of Carbon Nanobelts and Related Belt-Shaped Polycyclic Aromatic Hydrocarbons

The development of carbon nanobelts and related belt-shaped polycyclic aromatic hydrocarbons has gained momentum in recent years. This Minireview focuses on the synthetic strategies used in constructing these aesthetically appealing molecular nanocarbons. Examples of carbon nanobelts and related belt-shaped polycyclic aromatic hydrocarbons reported in recent years as well as some representative synthetic attempts in earlier times are discussed.     

Synthesis of Heptagon-Containing Polyarenes by Catalytic C−H Activation

Nanocarbons incorporating non-hexagonal aromatic rings - such as five-, seven-, and eight-membered rings - have various intriguing physical properties such as curved structures, unique one-dimensional packing, and promising magnetic, optical, and conductivity properties. Herein, we report an efficient synthetic approach to polycyclic aromatics containing seven-membered rings via a palladium-catalyzed intramolecular Ar−H/Ar−Br coupling. In addition to all-hydrocarbon scaffolds, heteroatom-embedded heptagon-containing polyarenes can be efficiently constructed with this method. Rhodium- and palladium-catalyzed sequential six- and seven-membered ring formations also afford complex heptagon-containing molecular nanocarbons from readily available arylacetylenes and biphenyl boronic acids. Detailed mechanistic analysis by DFT calculations showed the feasibility of seven-membered ring formation by a concerted metalation-deprotonation mechanism. This reaction can serve as a template for the synthesis of a wide range of seven-membered ring-containing molecular nanocarbons.     

Making Dimethylamino a Transformable Directing Group by Nickel‐Catalyzed CN Borylation

The dimethylamino (Me₂N) group is arguably the most versatile functional group capable of highly efficient and site‐selective directed aromatic functionalizations at the ortho‐, meta‐, and para‐positions depending on reaction conditions. While the repertoire of Me₂N‐directed reactions is growing at a rapid pace, the lack of a general method to transform this group to other functionalities hampers its wider application in organic synthesis. Here we report nickel‐catalyzed C?N borylations of aryl‐ and benzyl‐dimethylamines that permit the conversion of a huge library of largely underutilized Me₂N‐containing organic molecules into various functional molecules by taking advantage of the wealth of existing C?B functionalization methods.     

Synthesis of Extended π‐Systems through C–H Activation

Activation of aromatic C? H bonds by a transition metal catalyst has received significant attention in the synthetic chemistry community. In recent years, rapid and site‐selective extension of π‐electron systems by C–H activation has emerged as an ideal methodology for preparing organic materials with extended π‐systems. This Review focuses on recently reported π‐extending C–H activation reactions directed toward new optoelectronic conjugated materials.     

A novel approach for LC-MS/MS-based chiral metabolomics fingerprinting and chiral metabolomics extraction using a pair of enantiomers of chiral derivatization reagents

Chiral metabolites are found in a wide variety of living organisms and some of them are understood to be physiologically active compounds and biomarkers. However, the overall analysis of chiral metabolomics is quite difficult due to the high number of metabolites, the significant diversity in their physicochemical properties, and concentration range from metabolite-to-metabolite. To solve this difficulty, we developed a novel approach for chiral metabolomics fingerprinting and chiral metabolomics extraction, which is based on the labeling of a pair of enantiomers of chiral derivatization reagents (i.e., DMT-(S,R)-Pro-OSu and DMT-3(S,R)-Apy) and precursor ion scan chromatography of the derivatives. The multivariate statistics is also required for this strategy. The proposed procedures were evaluated by the detection of a diagnostic marker (i.e., d-lactic acid) using the saliva of diabetic patients. This method was used for the determination of biomarker candidates of chiral amines and carboxyls in Alzheimer's disease (AD) brain homogenates. As the results, l-phenylalanine (L-Phe) and l-lactic acid (L-LA) were identified as the decreased and increased biomarker candidates in the AD brain, respectively. Therefore, the proposed approach seems to be helpful for the determination of non-target chiral metabolomics possessing amines and carboxyls.     

An indefinite concave-convex equation under a Neumann boundary condition I

We investigate the problem (P _λ) ?Δu = λb(x)|u| ^q?2 u + a(x)|u| ^p?2 u in Ω, ?u/?n = 0 on ?Ω, where Ω is a bounded smooth domain in R ^N (N ≥ 2), 1 < q < 2 < p, λ ∈ R, and a, b ∈ \({C^\alpha }\left( {\overline \Omega } \right)\) with 0 < α < 1. Under certain indefinite type conditions on a and b, we prove the existence of two nontrivial nonnegative solutions for small |λ|. We then characterize the asymptotic profiles of these solutions as λ → 0, which in some cases implies the positivity and ordering of these solutions. In addition, this asymptotic analysis suggests the existence of a loop type component in the non-negative solutions set. We prove the existence of such a component in certain cases, via a bifurcation and a topological analysis of a regularized version of (P _λ).