On coverings of plane convex sets by translates of strips

In the euclidean planeE ² letS ₁,S ₂, ... be a sequence of strips of widthsw ₁,w ₂, .... It is shown thatE ² can be covered by translates of the stripsS _i if w ₁ ^3/2 = . Further results concern conditions in order that a compact convex domain inE ² can be covered by translates ofS ₁,S ₂, ....This research was supported by National Science Foundation Research Grant MCS 76-06111.     

Sur la forme canonique des formes extérieures

Sans résumé

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The article is the text of a talk given at the Symposium on Differential Geometry in Debrecen, Hungary, on August 28–September 3, 1975.     

A central limit theorem for age- and density-dependent population processes

Consider a population consisting of one type of individual living in a fixed region with area A. In [8], we constructed a stochastic population model in which the death rate is affected by the age of the individual and the birth rate is affected by the population density P_A(t), i.e., the population size divided by the area A of the given region. In [8], we proposed a continuous deterministic model which in general is a nonlinear Volterra type integral equation and proved that under appropriate conditions the sequence P_A(t) would converge to the solution P(t) of our integral equation in the sense that

$\text{lim}\text{→∞}\text{P}\text{sup}\text{0?s?t}\text{|P}{}_{\mathrm{A}}\text{(s) ? P(s)|>ε}\text{=0}\text{for every}\text{ε > 0}$

.In this paper, we obtain a “central limit theorem” for the random element √A(P_A(t)?P(t)). We prove that under appropriate conditions √A(P_A(t)?P(t)) will converge to a Gaussian process. (See Theorem 3.4 for the explicit formula of this Gaussian process.)     

Stereochemicai study of 7-aryl-1,7,8,8a-tetrahydro-3(2H)indolizinones by1H and13C NMR

Stereochemistry of 7-aryl-1,7,8,8a-tetrahydro-3(2H)-indolizinones was studied by¹H and¹³C NMR. Complete assignment of¹H NMR signals and analysis of¹H-¹H coupling constants were performed using the iterative PANIC program. Values of³ J _6,7,³ J _7,8endo, and⁴ J _5,7 allow one to unambiguously identify the correspondingexo- andendo-stereoisomers. For stereoisomers with exo-orientation of H(7), complete assignment of¹³C NMR signals was performed on the basis of analysis of the¹³C-¹H coupling constants using two dimensional heteronuclear shift-correlating spectroscopy.Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 3, pp. 591–593, March, 1996.     

Infrared investigation of aluminum- and chromium-substituted magnetites and of the lacunar spinels resulting from their oxidation

When Fe³⁺ ions are substituted by aluminum or chromium on magnetite octahedral sites, the ir spectrum shows the conversion of an inverse spinel to a normal spinel. Both broad bands of magnetite are gradually replaced by the four characteristic bands of normal spinels II–III. They are also observed for solid solutions, FeCr₂O₄FeAl₂O₄, with, however, a further band at 780 cm^?1 which may be assigned to Al³⁺ ions on tetrahedral sites. Low-temperature (<400°C) oxidation of these compounds whose sizes are less than 2000 Å results in lacunar spinels III–III. The ir spectrum of these solids is characterized by two absorption bands (as for inverse spinels II–III) except for compounds close to pure γFe₂O₃ in which an order of vacancies could be put in evidence.     

Prescriptions and Ingredients for Controlled CC Bond Formation with Organometallic Reagents. New synthetic methods (5)

Merits and drawbacks of known carbon-carbon linking procedures are outlined. Two novel methods are discussed in some detail: the copper-catalyzed alkylation of Grignard reagents and reactions with allylpotassium compounds. Both methods provide a very efficient access to saturated, unsaturated, as well as functionally substituted hydrocarbons and moreover permit an astonishing degree of regio- and stereoselective control of olefin synthesis.     

The structure of 1-formyl-3-phenyl-Δ-pyrazoline in the gas phase (DFT calculations), in solution (NMR) and in the solid state (X-ray crystallography)

The molecular structure of 1-formyl-3-phenyl-Δ²-pyrazoline was determined by X-ray crystallography (triclinic, P-1). The geometry thus obtained was compared with that obtained by DFT calculations. The GIAO method was used to calculate absolute shieldings, which agree conveniently with those measured by ¹³C and ¹⁵N NMR. The title compound appears to be an essentially planar molecule.     

In-capillary micro solid-phase extraction and capillary electrophoresis separation of heterocyclic aromatic amines with nanospray mass spectrometric detection

A miniaturised technique to analyse and detect heterocyclic aromatic amines (HAs) using micro solid-phase extraction (SPE) coupled on-line (in-capillary) to capillary electrophoresis (CE) separation with nanospray (nESI) mass spectrometry (MS) detection has been developed. HAs are mutagenic and carcinogenic compounds formed at low levels in protein-rich food during cooking. Due to the low concentrations of HAs and the high complexity of the matrix in which they exist, sensitive and selective analytical methods are required for quantification. SPE was performed on a packed bed of C₁₈ particles inside the CE capillary, which minimised the dead volume. The on-line coupling of SPE, CE and nESI-MS reduced the time for extraction and identification to less than half an hour, which will allow for screening of several samples per day. The new technique provides short analysis time, low sample and solvent consumption, and HAs in standard solutions were easily detected at 12–17 fmol injections, and in spiked urine samples at 750–810 fmol injections.