Pizza again? On the division of polygons into sections with a common origin

The paper explores the division of a polygon into equal-area pieces using line segments originating at a common point. The mathematical background of the proposed method is very simple and belongs to secondary school geometry. Simple examples dividing a square into two, four or eight congruent pieces provide a starting point to discovering how to divide a regular polygon into any number of equal-area pieces using line segments originating from the centre. Moreover, it turns out that there are infinite ways to do the division. Discovering the basic invariant involved allows application of the same procedure to divide any tangential polygon, as after suitable adjustment, it can be used also for rectangles and parallelograms. Further generalization offers many additional solutions of the problem, and some of them are presented for the case of an arbitrary triangle and a square. Links to dynamic demonstrations in GeoGebra serve to illustrate the main results.     

Foundation, analysis, and numerical investigation of a variational network-based model for rubber

Since the pioneering work by Treloar, many models based on polymer chain statistics have been proposed to describe rubber elasticity. Recently, Alicandro, Cicalese, and the first author rigorously derived a continuum theory of rubber elasticity from a discrete model by variational convergence. The aim of this paper is twofold. First, we further physically motivate this model and complete the analysis by numerical simulations. Second, in order to compare this model to the literature, we present in a common language two other representative types of models, specify their underlying assumptions, check their mathematical properties, and compare them to Treloar’s experiments.     

Ultralow concentrations of ibuprofen activate cell prostaglandin synthesis

he interest in the prostaglandin (PG) synthesis by animal cells today grows steadily because of the difficulties in obtaining them by any other way. Murine peritoneal macrophages can under certain con ditions synthesize large amounts of PGs. The effect of well-known nonsteroidal anti-inflammatory drug ibuprofen on PG synthesis by the cells using a high-performance liquid chromatography (HPLC) method with fluorescence detection of 4-bromomethyl-7-methoxy coumarin (BrMMC) derivatives was studied. In our case, the main metabolites were PGE₂ and PGF_2a. The PG synthesis activation effect was shown by ibuprofen concentrations in the 10^-10-10^-14M range with the maximum effect at the 10^-12M. In this case, the ibupro fen effect was comparable in value with the effect of the well-known cell PG synthesis activator—calcium ionophore A₂₃₁₈₇. Although the exact mechanism of such an effect is not clear at the moment, at low concentration, ibuprofen itself is able to activiate PG synthesis in murine peritoneal macrophages.     

Voltammetric Study of Reduction Processes in Rhodium(+3)–Selenium(+4) Acidic Mixtures

Using square‐wave voltammetry and linear scan voltammetry, some reduction processes, that take place on the hanging mercury drop electrode (HMDE) in rhodium(+3)–selenium(+4) mixtures, were studied in highly acidic sulfate, chloride and perchlorate solutions. The special attention was paid to the peak located at ?1.1 V, that reflects catalytic hydrogen evolution on the rhodium islets, and its changes under the influence of added selenium(+4), or selenium(?2) produced at the electrode. It can be concluded that addition of selenite to a dilute rhodium solution has the similar effect as the increase of rhodium(+3) concentration, making the catalytic hydrogen evolution more pronounced, probably by changing the properties of metal clusters on the mercury drop. If so, the effect could be of some practical importance in production of rhodium deposits on other surfaces, too.     

Binding of some first-row transition metal ions by a poly(iminoethylene)dithiocarbamate copolymer

The binding of the transition metal ions VO²⁺, Fe²⁺, Fe³⁺, CO²⁺, Co³⁺, Ni²⁺ and Cu²⁺ by a poly(iminoethylene) dithiocarbamate copolymer has been investigated by uptake studies and physical measurements (electronic, IR, and ESR spectra and magnetic susceptibility). Metal ions may be bound by both the dithiocarbamato and amino groups of the co-polymer. Binding to nitrogen (in addition to binding to sulphur) increases in the order FE(II)<Ni(II)<Cu(II) and accounted for increasing metal ion uptake by the copolymer in the same order. Factors which determine the relative uptake of the metal ions by the copolymer are discussed.