Zestaw zadań dotyczących liczb całkowitych

Publikacja bezpłatna - rozdział z książki „Matematyka. Materiały metodyczne”, red. R. J. Pawlak, Z. Walczak, Wydawnictwo Uniwersytetu Łódzkiego, Łódź 2014Publikacja przygotowana w ramach realizacji projektu „Nowoczesny nauczyciel Matematyki. Wzmocnienie kompetencji nauczycieli matematyki z województwa łódzkiego”, współfinansowanego ze środków Unii Europejskiej w ramach Europejskiego Funduszu Społecznego

Definiowanie pojęć matematycznych

Publikacja bezpłatna - rozdział z książki „Matematyka. Materiały metodyczne”, red. R. J. Pawlak, Z. Walczak, Wydawnictwo Uniwersytetu Łódzkiego, Łódź 2014Publikacja przygotowana w ramach realizacji projektu „Nowoczesny nauczyciel Matematyki. Wzmocnienie kompetencji nauczycieli matematyki z województwa łódzkiego”, współfinansowanego ze środków Unii Europejskiej w ramach Europejskiego Funduszu Społecznego

The mathematical background of the first grade technical university students

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Complementary Use of Magnetometric Measurements for Geochemical Investigation of Light REE Concentration in Anthropogenically Polluted Soils

The purpose of this study was to use fast geophysical measurements of soil magnetic susceptibility (κ) as supplementary data for chemical measurements of selected light rare earth elements (REEs) in soil. In order to ensure diversity in soil conditions, anthropogenic conditions and types of land use, seven areas were selected, all located in regions subjected to past or present industrial pollution. Magnetometric parameters were measured using a selected magnetic sensor that was specially designed for measurements of soil cores and were used to classify collected soil cores into six distinctive types. The analysis of REEs concentrations in soil was carried out taking into account the grouping of collected soil samples based on the type of study area (open, forested and mountain), and additionally on the measured magnetometric parameters of collected soil cores. A use of magnetometric measurements provided different, but complementary to chemical measurements information, which allowed to obtain deeper insight on REEs concentrations in soils in studied areas

Comparative analysis of soil magnetic susceptibility and concentration of rare earth elements in soil of problematic areas

Soil contamination with rare earth metals (REE) can have both direct and indirect connection with industrial activity and ore-exploration. In the studies conducted so far, the presence of rare earth elements in coal seams, coal combustion waste as well as fly ash was found. It is important that detailed studies of the REE content in soil were not carried out in Poland. Until now, in a few studies, a high content of cerium and lanthanum was found in relation to the average content of the torn ones in the world. This work focuses on the areas under the influence of the industry associated with the extraction and combustion of hard coal, but also with other types of industry. Analyzes of REE content in soil were conducted in selected areas of the Upper Silesian Industrial Region and Norway, located near the Bjornevatn mine. In study areas, soil samples were collected and used for chemical and magnetometric measurements. Firstly, concentrations of REE were determined, and after that soil samples were used to measure soil magnetic susceptibility. Finally, statistical analyses were performed in order to check the correlation between REE concentrations in soil and soil magnetic susceptibility

Redukcja niepewności oceny zanieczyszczenia gleb przy wykorzystaniu pomiarów podatności magnetycznej i metody Co_Est

The deposition of anthropogenic dusts originating mostly from industrial processes and solid fuel combustion causes the substantial changes in magnetic susceptibility of topsoil layers. Consequently, the magnetic susceptibil-ity is one of the soil properties, which makes it possible to assess the soil contamination with heavy metals. Moreover, in situ field measurements of magnetic susceptibility are significantly less expensive and time-consuming than laboratory chemical analyses. However, the geochemical measurements are usually sparse and precise, whereas the magnetometric ones are numerous or even excessive, but not very exact. For these reasons, in order to assess the extent of soil contamination it should be performed both geochemical and magnetometric measure-ments at the same time, and then integrate them using some geostatistical methods such as cokriging. Because of usually too small set of geochemical measurements, this integration may be highly difficult task. In most eases, the modeling of auto-semivariance of primary variable and especially cross-semivariance in ordinary cokriging procedure become almost impossible. This study presents the Co_Est procedure potential to reduce soil contamination uncertainty when geochemical measurements are too sparse, and magnetic susceptibility measurements serve as secondary data. For this purpose, topsoil pollution at few different size areas placed in forests or parks, located within the Upper Silesian Industrial Area was measured and analyzed. Then the maps of contaminations obtained using cokriging and Co_Est methods were compared. In particular, reduction of uncertainty in soil contamination was quantified and discussed.Depozycja pyłów pochodzenia antropogenicznego, pochodzących głównie ze źródeł przemysłowych i spalania paliw stałych, powoduje znaczne zmiany podatności magnetycznej górnych warstw gleby. Z tego względu podatność magnetyczna jest jedną z właściwości gleby, które mogą być wykorzystane w celu określenia stopnia zanieczyszczenia gleby metalami ciężkimi. Terenowe pomiary podatności magnetycznej są znacznie tańsze i mniej skomplikowane w porównaniu do analiz chemicznych. Ponadto pomiary chemiczne są zazwyczaj dokładniejsze i wykonywane z mniejszą gęstością powierzchniową niż pomiary magnetometryczne. Z powyższych powodów, celowe jest wykonywanie pomiarów obu rodzajów, a następnie zintegrowanie ich za pomocą metod geostaty-stycznych, takich jak np. cokriging. Jednak zazwyczaj ze względu na małą ilość danych chemicznych tego typu integracja jest bardzo trudna. Dokładne wymodelowanie wariogramów i wariogramów krzyżowych dla metody cokrigingu może być niemożliwe. Niniejsza praca prezentuje możliwości zastosowania metody Co_Est w celu redukcji niepewności oceny zanieczyszczenia gleb, w przypadku gdy dostępna jest mała ilość pomiarów geochemicznych, a pomiary magnetometryczne służą jako dane uzupełniające. W tym celu wykonane zostały pomiary zanieczyszczenia gleb oraz pomiary magnetometryczne na kilku obszarach leśnych zlokalizowanych na terenie Górnośląskiego Okręgu Przemysłowego. Następnie wykonane i przeanalizowane zostały mapy rozkładów przestrzennych metodami krigingu i Co_Est

Using Geostatistical Gaussian Simulation for Designing and Interpreting Soil Surface Magnetic Susceptibility Measurements

This paper presents a new approach to the assessment of the uncertainty of using geostatistical Gaussian simulation in soil magnetometry. In the study area, numerous measurements of soil magnetic susceptibility were made, and spatial distributions of soil magnetic susceptibility were simulated. The parameters of variograms of soil magnetic susceptibility measured in the study area were determined and compared with those of simulated soil magnetic susceptibility. Regardless of the measurement scheme used, reproducibility of the original semivariograms of soil magnetic susceptibility was satisfactorily achieved when applying simulated values. A nugget effect, a sill, and a range of correlations of variograms of simulated values of soil magnetic susceptibility were similar to those of measured values. When the input data for the geostatistical simulation were averaged, the measured values of soil magnetic susceptibility and simulated spatial distributions were characterized by slightly lower standard deviations in comparison with the result of simulations based on the non-averaged, measured ones. At the same time, however, local variability of soil magnetic susceptibility was reproduced less. The accuracy of the calculations of point parameters and spatial distributions—based on the averaged values of soil magnetic susceptibility—were satisfactory, but when using geostatistical methods, it is recommended to use non-averaged magnetic susceptibility measurements

The student's work on a mathematical text

The present paper presents the results of the investigations carried out among the students of mathematics of Łódź University, concerning individual techniques of reading and analysing a mathematical text. These results point out that young people taking up their studies do not bring away from the secondary school any skill in reading a scientific text. It is only in the course of the studies that they labour their own methods of work. The techniques of work of students of higher years and very good students are considerably closer to the model techniques presented in the paper. Some of the failures in the learning of students of lower years result from the ignorance of effective ways of learning. It is therefore worth while for the academic teachers to devote already during the first years of studies, more attention to acquainting their students with the methods of working on a mathematical text and to bring them into the arcana of the effective studying.Zadanie pt. „Digitalizacja i udostępnienie w Cyfrowym Repozytorium Uniwersytetu Łódzkiego kolekcji czasopism naukowych wydawanych przez Uniwersytet Łódzki” nr 885/P-DUN/2014 zostało dofinansowane ze środków MNiSW w ramach działalności upowszechniającej naukę