Several improved asymptotic normality criteria and their applications to graph polynomials

As widely regarded, one of the most classical and remarkable tools to measure the asymptotic normality of combinatorial statistics is due to Harper's real-rooted method proposed in 1967. However, this classical theorem exists some obvious shortcomings, for example, it requests all the roots of the corresponding generating function, which is impossible in general.Aiming to overcome this shortcoming in some extent, in this paper we present an improved asymptotic normality criterion, along with several variant versions, which usually just ask for one coefficient of the generating function, without knowing any roots. In virtue of these new criteria, the asymptotic normality of some usual combinatorial statistics can be revealed and extended. Among which, we introduce the applications to matching numbers and Laplacian coefficients in detail. Some relevant conjectures, proposed by Godsil (Combinatorica, 1981) and Wang et al. (J. Math. Anal. Appl., 2017), are generalized and verified as corollaries.     

Theo’s reinvention of the logic of conditional statements’ proofs rooted in set-based reasoning

This report documents how one undergraduate student used set-based reasoning to reinvent logical principles related to conditional statements and their proofs. This learning occurred in a teaching experiment intended to foster abstraction of these logical relationships by comparing the relationships between predicates within the conditional statements and inference structures among various proofs (in number theory and geometry). We document the progression of Theo’s set-based emergent model (Gravemeijer, 1999) from a model-of the truth of statements to a model-for logical relationships. This constitutes some of the first evidence for how students can abstract such logical concepts in this way and provides evidence for the viability of the learning progression that guided the instructional design.     

Polyethylene glycol phospholipids encapsulated silicon 2,3-naphthalocyanine dihydroxide nanoparticles (SiNcOH-DSPEPEG(NH2) NPs) for single NIR laser induced cancer combination therapy

Currently, the combination of photothermal therapy (PTT) and photodynamic therapy (PDT) has emerged as a powerful technique for cancer treatment. However, most examples of combined PTT and PDT reported use multi-component nanocomposites under excitation of separate wavelength, resulting in complex treatment process. In this work, a novel theranostic nanoplatform (SiNcOH-DSPE-PEG(NH₂) NPs) has been successfully developed by coating silicon 2,3-naphthalocyanine dihydroxide (SiNcOH) with DSPE-PEG and DSPE-PEG-NH₂ for photoacoustic (PA) imaging-guided PTT and PDT tumor ablation for the first time. The as-prepared single-agent SiNcOH-DSPE-PEG(NH₂) NPs not only have good water solubility and biocompatibility, but also exhibit high photothermal conversion efficiency and singlet oxygen generation capability upon 808 nm NIR laser irradiation. In addition, owing to their high absorption at NIR region, the SiNcOH-DSPE-PEG(NH₂) NPs can also be employed as an effective diagnostic nanoagent for photoacoustic (PA) imaging. In vitro and in vivo experimental results clearly indicated that the simultaneously combined PTT and PDT under the guidance of PA imaging with single NIR laser excitation can effectively kill cancer cells or eradicate tumor tissues. Taking facile synthesis and high efficiency in cancer treatment by SiNcOH-DSPE-PEG(NH₂) NPs into consideration, our study provides a promising strategy to realize molecular imaging-guided combination therapy.     

Neighbor Distinguishing Edge Colorings Via the Combinatorial Nullstellensatz Revisited

Consider a simple graph and its proper edge coloring c with the elements of the set . We say that c is neighbor set distinguishing (or adjacent strong) if for every edge , the set of colors incident with u is distinct from the set of colors incident with v. Let us then consider a stronger requirement and suppose we wish to distinguishing adjacent vertices by sums of their incident colors. In both problems the challenging conjectures presume that such colorings exist for any graph G containing no isolated edges if only . We prove that in both problems is sufficient. The proof is based on the Combinatorial Nullstellensatz, applied in the “sum environment.” In fact the identical bound also holds if we use any set of k real numbers instead of as edge colors, and the same is true in list versions of the both concepts. In particular, we therefore obtain that lists of length ( in fact) are sufficient for planar graphs.     

A practical and inexpensive ‘convertible’ isonitrile for use in multicomponent reactions

N-tert-Butylamides are readily converted into the corresponding carboxylic acids by simple nitrosation. The process, which occurs under mild nonaqueous conditions, leaves carboxylic esters untouched and transforms multicomponent reaction products into useful building blocks for further synthetic elaboration.     

Synthesis of amide libraries with immobilized HOBt

Highly reactive N-acylating solid-phase reagents based on macroporous polystyrene-bound 1-hydroxybenzotriazole (P-HOBt) and silica-bound 1-hydroxybenzotriazole (Si-HOBt) were prepared and compared for reactivity by synthesis of small combinatorial libraries of acetamides and benzamides.     

Highly efficient synthesis of triazolo[1,2-a]indazole-triones and novel spiro triazolo[1,2-a]indazole-tetraones under solvent-free conditions

A new, convenient, and high yielding procedure for the synthesis of triazolo[1,2-a]indazole-triones by the condensation reaction between dimedone, aryl aldehydes, and ueazoles in the presence of a catalytic amount of sulfonated polyethylene glycol (PEG-SO₃H) as a highly stable and reusable eco-friendly degradable polymeric catalyst is described under solvent-free conditions. This procedure has also been applied successfully for the synthesis of novel spiro triazolo[1,2-a]indazole-tetraones.