A problem on conjugacy of matrices

Let F be an infinite field and n?12. Then the number of conjugacy classes of the upper triangular nilpotent matrices in M_n(F) under action by the subgroup of GL_n(F) consisting of all the upper triangular matrices is infinite.     

Text Indexing and Dictionary Matching with One Error

The indexing problem is where a text is preprocessed and subsequent queries of the form “Find all occurrences of pattern P in the text” are answered in time proportional to the length of the query and the number of occurrences. In the dictionary matching problem a set of patterns is preprocessed and subsequent queries of the form “Find all occurrences of dictionary patterns in text T” are answered in time proportional to the length of the text and the number of occurrences.There exist efficient worst-case solutions for the indexing problem and the dictionary matching problem, but none that find approximate occurrences of the patterns, i.e., where the pattern is within a bound edit (or Hamming) distance from the appropriate text location.In this paper we present a uniform deterministic solution to both the indexing and the general dictionary matching problem with one error. We preprocess the data in time O(n log² n), where n is the text size in the indexing problem and the dictionary size in the dictionary matching problem. Our query time for the indexing problem is O(m log n log log n + tocc), where m is the query string size and tocc is the number of occurrences. Our query time for the dictionary matching problem is O(n log³ d log log d + tocc), where n is the text size and d the dictionary size. The time bounds above apply to both bounded and unbounded alphabets.     

Assembling Quasi‐enantiomeric Octahedral Complexes of Different Metals via Quasi‐racemate Crystallization

Aiming at a general methodology for binary co‐assembly of complexes of different metals through quasiracemate crystallization, the hexadentate ligand 1 comprised of the chiral bipyrrolidine core and two bipyridine peripheral arms is introduced. Ligand 1 was found to bind in a fully diastereoselective and uniform mode around Zn^II, Fe^II and Cd^II giving coordinatively inert octahedral “chiral‐at‐metal” complexes with the Δ₄Λ₂/Λ₄Δ₂ wrapping mode. Equimolar mixtures of quasienantiomeric pairs of these complexes exhibited a clear tendency to pack as quasiracemates as was revealed from the crystallographic structures of [(R,R)‐ 1 ‐Zn](PF₆)₂/[(S,S)‐ 1 ‐Fe](PF₆)₂ and [(R,R)‐ 1 ‐Zn](PF₆)₂/[(S,S)‐ 1 ‐Cd](PF₆)₂, in an isomorphous fashion to that of the racemic compound [rac‐ 1 ‐Zn](PF₆)₂ in space group C2/c.     

Ink‐Jet Printing of Metallic Nanoparticles and Microemulsions

Summary: Two types of ink‐jet inks are presented: ink containing an aqueous dispersion of silver nanoparticles and an oil‐in‐water microemulsion‐based ink. The metallic ink contains nanoparticles of silver, which are formed in the presence of an ionic polymeric stabilizer. Sintering of the printed image obtained with the use of such silver‐based inks at temperatures as low as 300 °C results in formation of patterns possessing noticeable conductivity. The microemulsion inks are based on a thermodynamically stable microemulsion, in which the dispersed oil phase is a volatile solvent containing a water‐insoluble colorant. After contact of the jetted ink droplets with a substrate, nanodroplets of the microemulsion are converted into nanoparticles of the solubilized colorant. In some cases, it was found that the evaporation of microemulsion ink droplets leads to formation of rings composed of ordered nanoparticles.

Scheme of ink‐jet printing of an oil‐in‐water microemulsion followed by conversion of the nanodroplets into nanoparticles, caused by quick evaporation of the solvent within the microemulsion droplets. Therefore, the ink behaves as a dye‐based ink prior to printing, but after printing it behaves like a pigment‐based ink.