A Cantor set Associated With Sierpinski's Algorithm

W. Sierpiski showed that each x in 0<x1 has a representation1/a₁...a_k is an increasing sequence of positive integers. Weshow that the subset of numbers x for which(a_k) is strictlyincreasing is a Cantor set.     

Organotransition-Metal Complexes of Multidentate Ligands 3. The Unexpected Derivatives of Bis(3,5-Dimethylpyrazol-1-YL)Methanetetracarbonylmolybdenlim(0) and-Tungsten(0) Complexes

Thermolysis of (H₂CPz′₂)M(CO)₄ (H₂CPz′₂ = bis(3,5-dimethylpyrazol-1-yl)methane; M=Mo, W) in 1,2-dimethoxyethane did not give the expected 16-electron complexes, (H₂CPz′₂)M(CO)₃, but gave dinuclear compounds, [(H₂CPz′₂)M(CO)₃]₂, probably containing two linear carbonyl bridges and no metal-metal interactions. The dimers reacted with CH₃CN to give mononuclear compounds, (H₂CPz′₂)M(CO)₃(NCCH₃), identical to the substitution products between (H₂CPz′₂)M(CO)₄ and CH₃CN.     

Synthesis of Lithiophorite in High Alkaline Conditions

Lithiophorite consists of alternatively stacked MnO₆ octahedral sheets and LiAl₂(OH)₆ octahedral sheets. Its applications in laboratories and industries have been hindered by sophisticated operation procedures, long reaction time, or impurities existing in the final product. We proposed a fast and simple method, mixing birnessite, aluminate and lithium hydroxide together (designated it as the BAL method) in high alkaline conditions (pH > 13), and treating it hydrothermally at 423 K for 6 hours to prepare pure lithiophorite. A specific reaction between lithium cations and aluminate anions plays as a key role in the BAL method. Due to this specific reaction, Li_xAl_n(OH)_m^+z complexed cations can form and penetrate into interlayers of birnessite to replace sodium cations. In high alkaline conditions (pH > 12), Li_xAl_n(OH)_m^+z complexed cations become smaller and are soluble. Thus, the higher alkaline Li_xAl_n(OH)_m^+z complexed cations can penetrate into interlayers of birnessite at a higher rate. Furthermore, impurities, such as lithium intercalated gibbsite (LIG), aluminum oxyhydroxides and aluminum hydroxides are not stable in high alkaline conditions. Consequently, pure lithiophorite can be easily obtained within 6 hours in high alkaline conditions.     

Syntheses, X-ray structures, and reactions of ruthenium carbonyl complexes containing 1,1-dithiolates

Treatment of [Ru₂(CO)₄(MeCN)₆][BF₄]₂ or [Ru₂(CO)₄(μ-O₂CMe)₂(MeCN)₂] with uni-negative 1,1-dithiolate anions via potassium dimethyldithiocarbamate, sodium diethyldithiocarbamate, potassium tert-butylthioxanthate, and ammonium O,O′-diethylthiophosphate gives both monomeric and dimeric products of cis-[Ru(CO)₂(η²-(SS))₂] ((SS)⁻=Me₂NCS₂⁻ (1), Et₂NCS₂⁻ (2), ^tBuSCS₂⁻ (3), (EtO)₂PS₂⁻ (4)) and [Ru(CO)(η²-(Me₂NCS₂))(μ,η²-Me₂NCS₂)]₂ (5). The lightly stabilized MeCN ligands of [Ru₂(CO)₄(MeCN)₆][BF₄]₂ are replaced more readily than the bound acetate ligands of [Ru₂(CO)₄(μ-O₂CMe)₂(MeCN)₂] by thiolates to produce cis-[Ru(CO)₂(η²-(SS))₂] with less selectivity. Structures 1 and 5 were determined by X-ray crystallography. Although the two chelating dithiolates are cis to each other in 1, the dithiolates are trans to each other in each of the {Ru(CO)(η²-Me₂NCS₂)₂} fragment of 5. The dimeric product 5 can be prepared alternatively from the decarbonylation reaction of 1 with a suitable amount of Me₃NO in MeCN. However, the dimer [Ru(CO)(η²-Et₂NCS₂)(μ,η²-Et₂NCS₂)]₂ (6), prepared from the reaction of 2 with Me₃NO, has a structure different from 5. The spectral data of 6 probably indicate that the two chelating dithiolates are cis to each other in one {Ru(CO)(η²-Et₂NCS₂)₂}fragment but trans in the other. Both 5 and 6 react readily at ambient temperature with benzyl isocyanide to yield cis-[Ru(CO)(CNCH₂Ph)(η²-(SS))₂] ((SS)=Me₂NCS₂⁻ (7) and Et₂NCS₂⁻ (8)). A dimerization pathway for cis-[Ru(CO)₂(η²-(SS))₂] via decabonylation and isomerization is proposed.     

Normal Components, Kekulé Patterns, and Clar Patterns in Plane Bipartite Graphs

As a general case of molecular graphs of polycyclic alternant hydrocarbons, we consider a plane bipartite graph G with a Kekulé pattern (perfect matching). An edge of G is called nonfixed if it belongs to some, but not all, perfect matchings of G. Several criteria in terms of resonant cells for determining whether G is elementary (i.e., without fixed edges) are reviewed. By applying perfect matching theory developed in plane bipartite graphs, in a unified and simpler way we study the decomposition of plane bipartite graphs with fixed edges into normal components, which is shown useful for resonance theory, in particular, cell and sextet polynomials. Further correspondence between the Kekulé patterns and Clar (resonant) patterns are revealed.     

Counting Sums of Three Squares

Let (x) be the error term associated with the asymptotic formulafor the counting function on the set of numbers that are sumsof three squares. We give a formula for (x) and use it to provethat (x) has average order and normal order 3 1og x/(16 1og2).     

Pyridyl Pyrrolide Boron Complexes: The Facile Generation of Thermally Activated Delayed Fluorescence and Preparation of Organic Light‐Emitting Diodes

The electron positive boron atom usually does not contribute to the frontier orbitals for several lower‐lying electronic transitions, and thus is ideal to serve as a hub for the spiro linker of light‐emitting molecules, such that the electron donor (HOMO) and acceptor (LUMO) moieties can be spatially separated with orthogonal orientation. On this basis, we prepared a series of novel boron complexes bearing electron deficient pyridyl pyrrolide and electron donating phenylcarbazolyl fragments or triphenylamine. The new boron complexes show strong solvent‐polarity dependent charge‐transfer emission accompanied by a small, non‐negligible normal emission. The slim orbital overlap between HOMO and LUMO and hence the lack of electron correlation lead to a significant reduction of the energy gap between the lowest lying singlet and triplet excited states (ΔE_T‐S) and thereby the generation of thermally activated delay fluorescence (TADF).     

Are the radical centers in peptide radical cations mobile? The generation, tautomerism, and dissociation of isomeric alpha-carbon-centered triglycine radical cations in the gas phase

The mobility of the radical center in three isomeric triglycine radical cations[G(*)GG](+), [GG(*)G](+), and [GGG(*)](+) has been investigated theoretically via density functional theory (DFT) and experimentally via tandem mass spectrometry. These radical cations were generated by collision-induced dissociations (CIDs) of Cu(II)-containing ternary complexes that contain the tripeptides YGG, GYG, and GGY, respectively (G and Y are the glycine and tyrosine residues, respectively). Dissociative electron transfer within the complexes led to observation of [Y(*)GG](+), [GY(*)G](+), and [GGY(*)](+); CID resulted in cleavage of the tyrosine side chain as p-quinomethide, yielding [G(*)GG](+), [GG(*)G](+), and [GGG(*)](+), respectively. Interconversions between these isomeric triglycine radical cations have relatively high barriers (> or = 44.7 kcal/mol), in support of the thesis that isomerically pure [G(*)GG](+), [GG(*)G](+), and [GGG(*)](+) can be experimentally produced. This is to be contrasted with barriers < 17 kcal/mol that were encountered in the tautomerism of protonated triglycine [Rodriquez C. F. et al. J. Am. Chem. Soc. 2001, 123, 3006-3012]. The CID spectra of [G(*)GG](+), [GG(*)G](+), and [GGG(*)](+) were substantially different, providing experimental proof that initially these ions have distinct structures. DFT calculations showed that direct dissociations are competitive with interconversions followed by dissociation.     

Metal‐Sensitive Hydrothermal Synthesis and Structural Characterization of a Mixed‐Metal and Porous Coordination Polymer Containing 2,3‐Pyridinedicarboxylate

The hydrothermal reaction of 2,3‐pyridinedicarboxylic acid (2,3‐H₂pda) with a mixture of Cd(NO₃)₂ and Ni(NO₃)₂ afforded a coordination polymer, [CdNi(2,3‐pda)₂(H₂O)₃]_∞ ( 1 ); in contrast, that with a mixture of Cd(NO₃)₂ and Zn(NO₃)₂ surprisingly produced a discrete molecule, trans‐[Cd(3‐pa)₂(H₂O)₄] ( 2 ) (3‐pa^? = 3‐pyridinecarboxylate). Since a direct reaction between a single metal salt, Cd(NO₃)₂ or Zn(NO₃)₂, and 3‐pyridinecarboxylic acid (3‐Hpa) under similar hydrothermal conditions yielded different coordination polymers containing 3‐pa^?, it appears that the apparently thermal decarboxylation from ligated 2,3‐pda^2? to 3‐pa^? occurs after complexation of both metal cations, Cd(II) and Zn(II). A new coordination mode, formed for 2,3‐pda^2? in structure 1 , appears to help formation of microporous channels by piling up the observed 2D hydrogen‐bonded heteropolynuclear layers. Each channel apparently consists of two interpenetrating 6³ Cd(II) and Ni(II) nets.