One of Berkeley’s arguments on compensating errors in the calculus

This paper addresses three questions related to George Berkeley’s theory of compensating errors in the calculus published in 1734. The first is how did Berkeley conceive of Leibnizian differentials? The second and most central question concerns Berkeley’s procedure which consisted in identifying two quantities as errors and proving that they are equal. The question is how was this possible? The answer is that this was not possible, because in his calculations Berkeley misguided himself by employing a result equivalent to what he wished to prove. In 1797 Lazare Carnot published the expression “a compensation of errors” in an attempt to explain why the calculus functions. The third question is: did Carnot by this expression mean the same as Berkeley?     

The Effects of the Broadband UVA Radiation on Myeloid Leukemia Cells: The Possible Role of Protein Kinase C in Mediation of UVA-lnduced Effects

Abstract— We examined the effects of broadband UVA radiation (320–400 nm) on a rat myeloid leukemia cell line–chlo-roma (ChL). A Phillips face tanner model HB 171/A was used as a light source. Chloroma were irradiated through a 5 mm thick glass Alter that cut off all of the UVB contamination. The irradiances were measured, from 250 to 400 nm, with a well-characterized and calibrated double-grating spectroradiometer Optronic 742. The overall uncertainty of dose evaluation was estimated to be <15% (2s?). The cells were irradiated with UVA doses of 4 and 8 J/cm² and cultured thereafter for 24 h. After this period of time, a marked decline up to 50% was observed in cell proliferation in UVA-irradiated ChL cultures. The cell proliferation decline was found to be caused by simultaneously occurring G2/M phase cell cycle arrest and apoptosis in part of the UVA-irradiated ChL population. Concomitantly, with the decline in cell proliferation, an increase was observed in the expression of the major histocompatibility (MHC) class I and II antigens. Because protein kinase C (PKC) is known to regulate cell proliferation, apoptosis and expression of MHC antigens, and because UVA was shown to regulate PKC activity/expression, we therefore examined whether UVA irradiation has any effect on the expression of isozymes of PKC. Western blots revealed that ChL express α, βI, δ, α, γ, and π isozymes of PKC and that expression of all isozymes declined 24 h after UVA irradiation (8 J/cm²). Finally, PKC activation in ChL by exposure to phorbol ester caused cell cycle arrest in G1 phase but did not induce apoptosis. This suggests that the previously shown UVA-induced PKC activation in ChL might be responsible for the induction of MHC antigens but the simultaneously observed ChL apoptosis is likely to be mediated by PKC down-regulation. All together, our results suggest that UVA, at irradiance levels that resemble the outdoor exposure, may have profound effects on the immune-related properties of leukocytes. Thus, we speculate that in vivo the immune functions of leukocytes passing through dermal capillaries might be altered by exposure to solar UVA radiation.     

Detection of microcystins with protein phosphatase inhibition assay, high-performance liquid chromatography-UV detection and enzyme-linked immunosorbent assay: Comparison of methods

A colorimetric protein phosphatase inhibition assay (PPI assay), a commercial enzyme-linked immunosorbent assay (ELISA) test and different HPLC methods using UV detection were compared for the detection of cyanobacterial hepatotoxins, microcystins (MCYST) and nodularin. The suitability of the methods to detect different toxin variants was evaluated by using pure toxins and laboratory cultures as well as water and bloom samples of toxic cyanobacteria. The emphasis of the study was on the analysis of polar demethyl microcystin variants that are common in nature but for which there exist no commercial standards. The IC₅₀ values of MCYST-LR for the PPI assay and the ELISA test were 2.2-2.5 and 0.26-0.38 μg l⁻¹, respectively. The most important factors that decreased toxin recovery in sample treatment were the use of C₁₈ cartridges and polypropylene containers. Good recoveries of toxins were obtained by using hydrophilic-lipophilic balanced (Oasis HLB, Waters) cartridges for concentrating the samples. The results obtained with the PPI assay, the ELISA test and HPLC correlated quantitatively well with the exception of [d-Asp³] microcystins. Concentrations of [d-Asp³]MCYST-RR measured with the PPI assay were only 5% of those obtained by the ELISA test and HPLC. Concentrations of hydrophobic microcystin variants were lower when analysed with ELISA than with the other methods. The World Health Organisation (WHO) has set a guideline value of 1 μg l⁻¹ for the world-wide most common microcystin variant, MCYST-LR in drinking water. Since the quantitative ranges of the PPI assay and the ELISA test are within microcystin concentrations in natural waters, and both tests are easy to perform, they show potential for routine use in the screening and monitoring of microcystins from drinking water supplies and from recreational waters.     

The State of Proof in Finnish and Swedish Mathematics Textbooks—Capturing Differences in Approaches to Upper-Secondary Integral Calculus

Students’ difficulties with proof, scholars’ calls for proof to be a consistent part of K-12 mathematics, and the extensive use of textbooks in mathematics classrooms motivate investigations on how proof-related items are addressed in mathematics textbooks. We contribute to textbook research by focusing on opportunities to learn proof-related reasoning in integral calculus, a key subject in transitioning from secondary to tertiary education. We analyze expository sections and nearly 2000 students’ exercises in the four most frequently used Finnish and Swedish textbook series. Results indicate that Finnish textbooks offer more opportunities for learning proof than do Swedish textbooks. Proofs are also more visible in Finnish textbooks than in Swedish materials, but the tasks in the latter reflect a higher variation in nature of proof-related reasoning. Our results are compared with methodologically similar U.S. studies. Consequences for learning and transition to university mathematics, as well as directions for future research, are discussed.     

Matchings,path covers and domination

We show that if $G$ is a graph with minimum degree at least three, then ${\gamma }_t\left(G\right)\le {\alpha }^{\prime }\left(G\right)+(\mathrm{pc}\left(G\right)?1)∕2$ and this bound is tight, where ${\gamma }_t\left(G\right)$ is the total domination number of $G$, ${\alpha }^{\prime }\left(G\right)$ the matching number of $G$ and $\mathrm{pc}\left(G\right)$ the path covering number of $G$ which is the minimum number of vertex disjoint paths such that every vertex belongs to a path in the cover. We show that if $G$ is a connected graph on at least six vertices, then ${\gamma }_{\mathrm{nt}}\left(G\right)\le {\alpha }^{\prime }\left(G\right)+\mathrm{pc}\left(G\right)∕2$ and this bound is tight, where ${\gamma }_{\mathrm{nt}}\left(G\right)$ denotes the neighborhood total domination number of $G$. We observe that every graph $G$ of order $n$ satisfies ${\alpha }^{\prime }\left(G\right)+\mathrm{pc}\left(G\right)∕2\ge n∕2$, and we characterize the trees achieving equality in this bound.     

Mass fingerprinting of toxic fractions from the venom of the Indian red scorpion, Mesobuthus tamulus: biotope-specific variation in the expression of venom peptides

The red scorpion, Mesobuthus tamulus, is found in two distinct biotopes within the Indian state of Maharastra-a tropical, sea-level biotope and a semi-arid biotope, up to 600 m. Scorpions from these two geographical areas show marked differences in toxicity. Using mass spectrometry, we have shown biotope-specific variation in the expression of peptides from scorpions collected from these two distinct areas. Matrix-assisted laser desorption/ionisation time-of-flight mass spectrometry (MALDI-TOFMS) and reversed-phase liquid chromatography/electrospray ionisation mass spectrometry (LC/ESI-MS) were assessed as techniques for obtaining mass fingerprint data. On line LC/ESI-MS was judged to be the method of choice and unique biotope-specific mass fingerprints, with key diagnostic markers, were obtained.     

Erythemally Weighted Radiometers in Solar UV Monitoring: Results from the WMO/STUK Intercomparison

The first international intercomparison of erythemally weighted (EW) broadband radiometers was arranged in 1995 to improve the accuracy and comparability of the measurements carried out by solar UV monitoring networks. The intercomparison was arranged at the Radiation and Nuclear Safety Authority in Helsinki, Finland, in cooperation with the University of Innsbruck and with support from the World Meteorological Organization. Altogether 20 EW meters of six different types from 16 countries were (1) tested in the laboratory by measuring the spectral and angular responsivities and (2) calibrated in solar radiation against two reference spectroradiometers. Calibration factors (CFs) for the EW meters were determined by using simultaneously measured EW solar UV spectra as a calibration reference. The CFs averaged over solar elevations higher than 35° varied from 0.87 to 1.75, with the estimated uncertainty being ±10%. As a result of this intercomparison, for the first time the calibrations of more than 100 EW radiometers around the world are possible to trace to the same origin. The present experience indicates that the accuracy of temperature-controlled EW radiometers is not significantly lower than the accuracy of spectroradiometers provided that strict quality assurance/quality control procedures are followed.     

Null-sets ofH-solutions

We considerC ²-solutionsf=u+iv+jw of the system