Some probabilistic and extremal results on subdivisions and odd subdivisions of graphs

A probabilistic result of Bollobás and Catlin concerning the largest integer p so that a subdivision of K_p is contained in a random graph is generalized to a result concerning the largest integer p so that a subdivision of A_p is contained in a random graph for some sequence A₁, A₂,… of graphs such that A_i+1 contains a subdivision of A_i. A similar result is proved for subdivisions with odd paths or cycles. The result is applied to disprove a conjecture of Chartrand, Geller, and Hedetniemi. The maximum number of edges in a graph without a subdivision of K_p, p = 4, 5, with odd paths or cycles is determined.     

On a diophantine problem of P. Erdös

Research supported in part by Grant 4055 from the Hungarian National Foundation for Scientific Research and by Kereskedelmi és Hitelbank Rt. Universitas Foundation.     

Rectangular convexity of convex domains of constant width

An E rc="/content/l848374432616290/xxlarge8834.gif" alt="sub" align="MIDDLE" BORDER="0"> R ² is r-convex if for every x, y rc="/content/l848374432616290/xxlarge8712.gif" alt="isin" align="MIDDLE" BORDER="0"> E there exists a closed rectangle R such that x, y rc="/content/l848374432616290/xxlarge8712.gif" alt="isin" align="MIDDLE" BORDER="0"> R and R rc="/content/l848374432616290/xxlarge8834.gif" alt="sub" align="MIDDLE" BORDER="0"> E. Several results about r-convexity appeared in [1]. Its authors formulated a conjecture about conditions for a compact, convex set in R ² to be r-convex. We prove this conjecture in the case of convex domains of constant width.     

A combinatorial property of points and ellipsoids

For eachdrc="/content/yjw4751758u4v63u/xxlarge8805.gif" alt="ge" align="MIDDLE" BORDER="0">1 there is a constantc _d>0 such that any finite setXrc="/content/yjw4751758u4v63u/xxlarge8834.gif" alt="sub" align="MIDDLE" BORDER="0">R ^d contains a subsetYrc="/content/yjw4751758u4v63u/xxlarge967.gif" alt="chi" align="MIDDLE" BORDER="0">X, |Y|rc="/content/yjw4751758u4v63u/xxlarge8804.gif" alt="le" align="MIDDLE" BORDER="0">[1/4d(d+3)]+1 having the following property: ifErc="/content/yjw4751758u4v63u/xxlarge8835.gif" alt="sup" align="MIDDLE" BORDER="0">Y is an ellipsoid, then |E rc="/content/yjw4751758u4v63u/xxlarge957.gif" alt="ngr" align="BASELINE" BORDER="0"> X|rc="/content/yjw4751758u4v63u/xxlarge8805.gif" alt="ge" align="MIDDLE" BORDER="0">c _d |X|.On leave from the Mathematical Institute of the Hungarian Academy of Sciences, 1364 Budapest, P.O. Box 127, Hungary. Supported by a research fellowship from the Science and Engineering Research Council, U.K., and by Hungarian National Foundation for Scientific Research Grant No. 1812.     

Investigation of 2-[(<Emphasis Type="Italic">E</Emphasis>)-2-(4-diethylaminophenyl)-1-ethenyl]-1,3,3-trimethyl-3H-indolium as a new highly sensitive reagent for the spectrophotometric determination of nitrophenols

A new, simple, rapid, and sensitive spectrophotometric method has been developed for the determination of nitrophenols [picric acid (PA); dinitrophenols (DNP)] in wastewater samples. The method is based on the reaction of nitrophenols with 2-[(E)-2-(4-diethylaminophenyl)-1-ethenyl]-1,3,3-trimethyl-3 H-indolium chloride reagent to form the colored ion associates, which are extracted by organic solvents. The molar absorptivity of the ion associates of PA with the investigated reagent ranges from 8.3×10⁴ to 11.3×10⁴ L mol^–1 cm^–1, depending on the extractant. Because only PA is extracted in an acidic medium with the investigated reagent, but both PA and DNP are extracted in an alkaline medium, it is possible to determine both substances in a mixture. Appropriate reaction conditions have been established. The absorbance of the colored extracts obeys Beerrc="/content/n912m14u71470217/xxlarge8217.gif" alt="rsquo" align="BASELINE" BORDER="0">s law in the range of 0.04–4.58 mg L^–1 PA, 1.0–18.4 mg L^–1 2,4-DNP and 1.2–14.7 mg L^–1 2,6-DNP, respectively. The limit of detections, calculated from a blank test (n=10; P=0.95), are 0.05 mg L^–1 PA, 0.9 mg L^–1 (2,4-DNP), and 1.1 mg L^–1 (2,6-DNP), respectively.     

Determination of selenoamino acids by gas chromatography-mass spectrometry

The application of the recently introduced ethylchloroformate derivatization method for the separation and determination of selenomethionine and selenocystein in selenium-enriched yeast and yeast-free tablets by means of a gas chromatography-mass spectrometry (GC-MS) system has been studied. The efficiency of three methods for the extraction of selenomethionine from the tablets were compared. Total selenium content of the same tablets were measured using inductively coupled plasma (ICP)-MS and it was found that in the selenized yeast tablets about 80% of the total selenium is present as selenomethionine. The results were in agreement with the values in the labels and with the literature. The accuracy of the total selenium analysis was controlled by the analysis of a reference material.     

Some problems concerning the use of automated radiochemical separation systems in destructive neutron activation analysis

The present state of a long term program is reviewed. It was started to elaborate a remote controlled automated radiochemical processing system for the neutron activation analysis of biological materials. The system is based on wet ashing of the sample, followed by reactive desorption of some volatile components. The distillation residue is passed through a series of columns filled with selective ion screening materials to remove the matrix activity. The solution is thus “stripped” from the interfering radioions, and it is processed to single-elements through group separations using ion-exchange chromatographic techniques. Some special problems concerning this system are treated. (a) General aspects of the construction of a (semi)automated radiochemical processing system are discussed; (b) Comparison is made between various technical realizations of the same basic concept; (c) Some problems concerning the “reconstruction” of an already published processing system are outlined.     

Synthesis, spectroscopic and thermal studies of the copper(II) aspartame chloride complex

Aspartame adduct of copper(II) chloride Cu(Asp)₂Cl₂·2H₂O (Asp=aspartame) is synthesized and characterized by elemental analysis, FT IR, UV/vis, ESR spectroscopies, TG, DTG, DTA measurements and molecular mechanics calculations. Aqueous solution of the green solid absorbs strongly at 774 and 367 nm. According to the FT IR spectra, the aspartame moiety coordinates to the copper(II) ion via its carboxylate ends, whereas the ammonium terminal groups give rise to hydrogen bonding network with the water, the chloride ions or neighboring carboxylate groups. The results suggest tetragonally distorted octahedral environment of the copper ions.