Institute of Mathematics AS CR, v. v. i.
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On the meaning and applications of the concept of chance in the school teaching of probability
No full textsummary:Příspěvek se věnuje významovým rozdílům pojmů šance a pravděpodobnost. V teoretické části si čtenář připomene obecné poznatky o pravděpodobnosti a šancích a jejich základních vlastnostech. Článek obsahuje 12 jednoduchých úloh, v nichž je patrný rozdíl ve výpočtu i v hodnotě šancí a pravděpodobnosti jevu v nějakém náhodném pokusu. V závěru článku jsou uvedeny metodické poznámky ke správnému zařazení pojmu šance ve výuce matematiky na 2. stupni ZŠ.summary:The article introduces the reader to what "chances" are in Mathematics, how they are calculated, how they differ from the probability value, what is the relationship between them, and presents several probability problems that are suitable for teaching Mathematics at the 2nd grade of elementary schools in the Czech Republic. In the conclusion, there are methodological notes on the use of "chances" in teaching
Misconceptions about zero among primary school teacher students
No full textsummary:Vybudování správné představy o čísle nula u žáků 1. stupně se může v učivu matematiky jevit jako obtížný moment. Mnohé výzkumy v minulosti prokázaly, že nejen žáci a studenti, ale i někteří učitelé matematiky mají mylné představy o nule, jejichž důsledkem je chybné zavádění čísla nula v hodinách matematiky. Tento fenomén pak může přispět k výskytu potíží u některých žáků v aritmetice v případech, kdy provádějí operace s nulou nebo s čísly, v jejichž zápisu je nula zahrnuta. Cílem článku je prezentovat výsledky průzkumu, které odhalují obtíže, s jakými se mohou potýkat studenti čtvrtého ročníku učitelství 1. stupně při popisu postupu zavádění čísla nula ve výuce matematiky v prvním ročníku ZŠ. V návaznosti na tento průzkum autorka poukazuje na odkryté mezery, které tito studenti vykazují při formulaci pojmu nula jako početnosti prázdné množiny. Tato studie by se mohla stát inspirativní a užitečnou pro studenty učitelství matematiky a učitele na 1. stupni ZŠ.summary:Building the correct idea of the number zero in 1st grade students can appear as a difficult moment in the mathematics curriculum. Many researches in the past have shown that not only pupils and students, but also some mathematics teachers have misconceptions about zero, the consequence of which is the incorrect introduction of the number zero in mathematics lessons. This phenomenon can then contribute to the occurrence of difficulties for some students in arithmetic in cases where they perform operations with numbers whose notation includes zero. The aim of the article is to present the results of the survey, which reveal the difficulties that students of the fourth year of the elementary teaching may face when describing the procedure for introducing the number zero in the teaching of mathematics in the first year of elementary school. In connection with this investigation, the author points to the uncovered gaps that these students show when formulating the concept of zero as the number of the empty set. This study could become inspiring and useful for students of mathematics teaching as well as teachers at the elementary school
Exact solutions of generalized Lane-Emden equations of the second kind
No full textsummary:Contact and Lie point symmetries of a certain class of second order differential equations using the Lie symmetry theory are obtained. Generators of these symmetries are used to obtain first integrals and exact solutions of the equations. This class of equations is transformed into the so-called generalized Lane-Emden equations of the second kind Then we consider two types of functions and present first integrals and exact solutions of the Lane-Emden equation for them. One of the considered cases is new
Cauchy problem with Denjoy-Stieltjes integral
Get PDFsummary:This work is devoted to analyzing the existence of the Cauchy fractional-type problems considering the Riemann-Liouville derivative (in the distributional Denjoy integral sense) of real order . These kinds of equations are a generalization of the measure differential equations. Our results extend A. A. Kilbas, H. M. Srivastava, J. J. Trujillo (2006) and H. Zhou, G. Ye, W. Liu, O. Wang (2015)
Geometric approaches to establish the fundamentals of Lorentz spaces and
No full textsummary:The aim of this paper is to investigate the orthogonality of vectors to each other and the Gram-Schmidt method in the Minkowski space . Hyperbolic cosine formulas are given for all triangle types in the Minkowski plane . Moreover, the Pedoe inequality is explained for each type of triangle with the help of hyperbolic cosine formulas. Thus, the Pedoe inequality allowed us to establish a connection between two similar triangles in the Minkowski plane. In the continuation of the study, the rotation matrix that provides both point and axis rotation in the Minkowski plane is obtained by using the Lorentz matrix multiplication. Also, it is stated to be an orthogonal matrix. Moreover, the orthogonal projection formulas on the spacelike and timelike lines are given in the Minkowski plane. In addition, the distances of any point from the spacelike or timelike line \hbox {are formulated}
Totally contact umbilical screen-slant and screen-transversal lightlike submanifolds of indefinite Kenmotsu manifold
Get PDFsummary:We study totally contact umbilical screen-slant lightlike submanifolds and totally contact umbilical screen-transversal lightlike submanifolds of an indefinite Kenmotsu manifold. We prove a characterization theorem of totally contact umbilical screen-slant lightlike submanifolds of an indefinite Kenmotsu manifold. We further prove some results on a totally contact umbilical radical screen-transversal lightlike submanifold of an indefinite Kenmotsu manifold, such as the necessary and sufficient conditions for the screen distribution to be integrable and for the induced connection to be a metric connection
Characterization of automorphisms of Radford's biproduct of Hopf group-coalgebra
No full textsummary:We study certain subgroups of the Hopf group-coalgebra automorphism group of Radford's -biproduct. Firstly, we discuss the endomorphism monoid and the automorphism group of Radford's -biproduct , and prove that the automorphism has a factorization closely related to the factors and . What's more interesting is that a pair of maps can be used to describe a family of mappings . Secondly, we consider the relationship between the automorphism group and the automorphism group of , and a normal subgroup of the automorphism group . Finally, we specifically describe the automorphism group of an example
Linear preservers of rc-majorization on matrices
No full textsummary:Let , be matrices. The concept of matrix majorization means the th column of is majorized by the th column of and this is done for all by a doubly stochastic matrix . We define rc-majorization that extended matrix majorization to columns and rows of matrices. Also, the linear preservers of rc-majorization will be characterized
