Mielnik and cantoni transition probabilities

It is shown that when a Mielnik transition probability space is given, Cantoni transition probabilities can also be defined on it. A condition is given under which these transition probabilities are equal.     

Packing a convex domain with similar convex domains

Given a convex domain C and a positive integer k, inscribe k nonoverlapping convex domains into C, all of them similar to C. Denote by f(k) the maximal sum of their circumferences. In this paper it is shown, that for C square, parallelogram or triangle (1) the first increase of f(k) after k = l² occurs not later than at k = l² + 2, (2) constructions can be given, where the following lower bounds are attained for f(k) = f(l² + j):

$\text{(}\text{1}\text{c}\text{) ? l +}\text{(j ? 1)}\text{2l}\text{j}\text{odd}\text{, l? 2}$

$\text{? l +}\text{j}\text{2}\text{(l + 1) j}\text{even}\text{, l?2}$

where c denotes the circumference of C.     

How many data clusters are in the Galaxy data set?

Advances in Data Analysis and Classification - In model-based clustering, the Galaxy data set is often used as a benchmark data set to study the performance of different modeling approaches. Aitkin...     

State on splitting subspaces and completeness of inner product spaces

We show that an inner product spaceV is complete iff the system of all splitting subspaces, i.e., of all subspacesM for whichM + M =V, possesses at least one completely additive state.     

Synthesis and conformational properties of 2,6-bis-anilino-3-nitropyridines

The synthesis of 2,6-bis-anilino-3-nitropyridines that are alkylated or acylated at the anilino nitrogen atoms is described. These derivatives show characteristic differences in the 1H-NMR spectra compared with the unsubstituted parent compound. These differences are used to determine structure-conformation relationships of this type of compounds. The conclusions drawn from the 1H-NMR spectra in this respect are supported by X-ray crystallographic data and by 1H-NMR data of conformationally restricted analogues. Preliminary investigations indicate that these relationships can in principle be extended to other diarylamines.     

Structure and electron density distribution of the nitrate ion and urea molecule upon protonation

Summary The changes in the structure and electron density distribution of the nitrate ion and urea molecule upon the presence of a point charge are discussed. Analyses of the Cambridge Structural Database are performed as well as Hartree-Fock calculations on the appropriate molecules in the presence of a point charge. The Hartree-Fock calculations confirm the correlations in structural parameters found in the database. A charge analysis of the molecules explains part of the structural changes caused by the presence of the point charge. The electrostatic potential and Laplacian of the electron density distribution explain the position of the point charge relative to the molecules.     

Reactions of π-cyclopentadienyltriphenylphosphinemetal diiodides and π-cyclopentadienylcarbonylmetal diiodides (M = Co,Rh, Ir) with 1,4-dilithio-1,2,3,4-tetraphenylbutadiene

Reactions between π-cyclopentadienyltriphenylphosphinemetàl diiodides (M = Rh and Ir) and 1,4-dilithio-l,2,3,4-tetraphenylbutadiene result in the formation of 1-(π-cyclopentadienyl)-l-triphenylphosphine-2,3,4,5-tetraphenylrhodole and 1-(π-cyclopentadienyl)-1-triphenylphosphine-2,3,4,5-tetraphenyliridole, respectively, in low yield. Reactions between π-cyclopentadienyltriphenylphosphinecobalt diiodide or π-cyclopentadienylcarbonylcobalt diiodide do not produce the expected cobaltacyclopentadiene complexes, but instead a low yield of π-cyclopentadienyl-(π-tetraphenylcyclobutadiene)cobalt. The trimeric rhodium complex (π-C₅H₅Rh)₃(CO)(PhCCPh) has been isolated from a reaction between 1,4-dilithio-l,2,3,4-tetraphenylbutadiene and π-cyclopentadienylcarbonylrhodium diiodide. The importance of metallocyclic intermediates in the formation of polynuclear complexes of this type is discussed.     

Fluorescence quenching studies on poly(methyl methacrylate) particles : Matching of the refraction index of the stabilizer phase to that of the solvent with added carbon disulphide

Detailed information about the interior morphology of sterically stabilized colloid particles ought to be accessible from light-scattering measurements when the index of refraction of the stabilizer layer is matched to that of the diluent. When carbon disulphide is added to dispersions of poly(methyl methacrylate) (PMMA) latex particlesstabilized with poly(12-hydroxystearic acid) (PHSA) is dodecane, the scattered light intensity varies continuously with time; a permanent match of indices of refraction in this system is not achieved. Fluorescence quenching experiments with CS₂ as quencher were performed on similar particles containing 0.02 mol% anthracene groups covalently bound to the PMMA chains in order to obtain information about CS₂ swelling of these particles. Time-dependent quenching experiments indicate rapid panetration of CS₂ (< 10 min) into the 150-nm diameter particles. Stern-Volmer analysis of steady-state CS₂ quenching data yielded results typical for polymer systems containing fluorophores in a distribution of environments; the particle-bound anthracenes are not all equally accessible to quencher. Nevertheless, all the anthracene labels are quenched to some extent. It is concluded that within 10 min after exposure, CS₂ penetrates throughout the entire volume of the particles.     

Construction of giant dendrimers using a tripodal building block

Giant pentane-soluble organo-silicon dendrimers have been synthesized using a triallylphenol brick according to a new divergent construction that uses a hydrosilylation-nucleophilic substitution sequence up to the ninth generation (G(9)). All the reactions were monitored by (1)H, (13)C, and (29)Si NMR until G(9), indicating that they were clean at the NMR accuracy until this last generation. MALDI TOF mass spectra were recorded for G(1) to G(4) and show the nature and amounts of defects that are intrinsic to the divergent construction. This technique and SEC (recorded up to G(5)) confirm the monodispersity (1.00 to 1.02) from G(1) to G(5). HRTEM and AFM images recorded for the high generations disclose the expected smooth dendrimer size progression and the globular shape. At G(9), the theoretical number of termini (TNT) is 177 407 branches (abbreviation: G(9)-177 047). It is estimated that more than 10(5) terminal branches are actually present in the G(9) dendrimer, far beyond the De Gennes "dense-packing" limit (6000 branches), and it is believed that the branch termini turn inside the dendrimer toward the core. This is corroborated by lower reaction rates and yields for the highest generation numbers presumably due to intradendritic reactions. It is probable that the dendritic construction is limited by the density of branches inside the dendrimer, i.e., far beyond the dense-packing limit.     

Study of blends of HDPE and different grades of CaCO3. Part I. Factorial experimental design

Calcium carbonate (CaCO₃)/high density polyethylene (HDPE) composites were prepared in a HAAKE twin screw extruder. The influence of the CaCO₃ particle size and content and the processing conditions were investigated by using a factorial experimental design. The dependent variables studied were tensile properties, such as Young's modulus, stress at break and toughness, and the rheological property, zero-shear viscosity, η₀. Preliminary results have shown that the most significant processing variables were content and mineral filler type. Mathematical modeling has given us a more efficient evaluation of the results obtained and has become an important tool in the development of this work.