Aspects of Number Theory in Terms of Potential Theory

We develop the Potential Theory on the set N^* of all natural numbers 2 associated with the kernel V (resp. V ^*) given by

Volume Contents

Volume Contents

Volume Contents     

Counting chemical compositions using Ehrhart quasi-polynomials

To count the number of chemical compositions of a particular mass, we consider an alphabet ${\mathop{\bf a}}$ with a mass function which assigns a mass to each letter in ${\mathop{\bf a}}$ . We then compute the mass of a word (an ordered sequence of letters) by adding the masses of the constituent letters. Our main interest is to count the number of words that have a particular mass, where we ignore the order of the letters within the word. We show first that counting the number of words of a given mass has a geometric interpretation, whose solutions are called Ehrhart quasi-polynomials, a class of functions defined on integers. These special functions are “periodic” in the sense that they use the same polynomial every λ steps. In addition to discovering the connection between counting compositions and Ehrhart quasi-polynomials, we also find number theoretic results that greatly reduce the number of candidates for the period, λ. Finally, we illustrate the usefulness of these results and the use of a software library named barvinok (by Verdoolaege et?al.) by applying them to eight different classes of chemical compositions, including organic molecules, peptides, DNA, and RNA.     

Analysis of a two-dimensional nonsmooth Poincaré-like map

A two-dimensional nonsmooth Poincaré-like map is investigated in the present work. The map generalizes in some sense the so-called Nordmark map, which is related to one-dimensional impacting oscillators near grazing points, and constitutes an intuitive basis for dynamics related to degenerate grazing in such oscillators.     

Bernstein-type Operators on a Triangle with One Curved Side

We construct Bernstein-type operators on a triangle with one curved side. We study univariate operators, their product and Boolean sum, as well as their interpolation properties, the order of accuracy (degree of exactness, precision set) and the remainder of the corresponding approximation formulas. We also give some illustrative examples.     

Synthesis of potential fungicides based on N-(3-furanyl)pyrrolecarboxamides and N-(3-furanyl)pyrazolecarboxamides

Abstract  

An efficient synthesis of novel N-(3-furanyl)pyrrolecarboxamides and N-(3-furanyl)pyrazolecarboxamides is presented starting from furan-3-carboxylic acid. Two complementary strategies to 2-aryl-3-furanamines with directed ortho lithiation, functionalization of the 2-position, and subsequent cross-coupling reaction as key steps were developed. Acylation of these intermediates with appropriate acid chlorides led finally to the target compounds.     

Comparative high-resolution spectroscopy and emission dynamics of Nd-doped GSGG crystals and transparent ceramics

The comparative spectroscopic and emission decay investigation of Nd:GSGG single crystals and transparent ceramics indicates that their properties are similar. The accurate characterization of the effect of energy transfer processes on emission decay enables a reliable calculation of the Nd concentration dependence of emission quantum efficiency.     

Cyclophanes or Cyclodextrins: What is the Best Host for Aromatic Volatile Organic Compounds?

The first results of the complexing ability of cyclobis-(paraquat-p-phenylene) as supramolecular host with different aromatic volatile organic compounds are presented. The formation constants of cyclobis(paraquat-p-phenylene) with toluene and halogenobenzenes were determined in aqueous solution by static headspace associated with gas chromatography and compared with the ones obtained by cyclodextrins. The data indicated the formation of 1:1 inclusion compounds in both cases. The results underlined a greater complexation ability for cyclobis(paraquat-p-phenylene) which was confirmed by a theoretical study.     

Synthesis and Antimicrobial Activity of Poly (Oxyethylene)S Functionalized with Quaternary Phosphonium End Groups

Abstract

Poly(oxyethylene)s functionalized with quaternary phosphonium end groups were obtained by converting terminal hydroxyl end groups into phosphoniurn ion groups in the polymer-analogous reactions with phosphines and hydrogmu chloride. The products were characterized by: IR, UV. H NMR spectroscopy, thin-layer chromatography.     

Synthesis of Some Mixed Dialkyl Phosphites and their Use as Forerunners for Potential Chiral Phosphates

Abstract

Organophosphorus compounds containing a c h i d ccntrc at the P atom are of considerable interest fiom a stcreochemical point of view. Accordingly, trialkyl phosphates (RO)(RIO)(R20)P(0) (4) were synthesized starting from dialkyl phosphites. Thus: direct hydrolysis of dialkyl phosphites (ROhP(0)H (1) with tetraethylammonium hydroxide (20% aqueous solution), followed by extraction with dichloromethane affords the corresponding tetraethylammonium alkyl hydrogen phosphitcs (2). Good yields (q) of mixed dialkyl phosphites (3) were obtained on heating stoichiometric amounts of (2) and alkyl iodides in acetonitrile solution at 6O°C, for 6 hrs. Mixed dialkyl phosphites (3) were used in synthesis of some trialkyl phosphates (4) by PTC (see scheme).