Consider the n×n matrix with (i, j)’th entry gcd (i, j). Its largest eigenvalue \lambda _{n} and sum of entries s_{n} satisfy \lambda _{n} > s_{n}/n. Because sn cannot be expressed algebraically as a function of n, we underestimate it in several ways. In examples, we compare the bounds so obtained with one another and with a bound from S.Hong, R.Loewy (2004). We also conjecture that \lambda _{n} > 6\Pi ^{-2}n log n for all n. If n is large enough, this follows from F.Balatoni (1969)., Jorma K. Merikoski., and Obsahuje seznam literatury
Consider $\mathcal T_n(F)$ - the ring of all $n\times n$ upper triangular matrices defined over some field $F$. A map $\phi$ is called a zero product preserver on ${\mathcal T}_n(F)$ in both directions if for all $x,y\in{\mathcal T}_n(F)$ the condition $xy=0$ is satisfied if and only if $\phi(x)\phi(y)=0$. In the present paper such maps are investigated. The full description of bijective zero product preservers is given. Namely, on the set of the matrices that are invertible, the map $\phi$ may act in any bijective way, whereas for the zero divisors and zero matrix one can write $\phi$ as a composition of three types of maps. The first of them is a conjugacy, the second one is an automorphism induced by some field automorphism, and the third one transforms every matrix $x$ into a matrix $x'$ such that $\{y\in\mathcal T_n(F) xy=0\}=\{y\in\mathcal T_n(F) x'y=0\}$, $\{y\in\mathcal T_n(F) yx=0\}=\{y\in\mathcal T_n(F) yx'=0\}$., Roksana Słowik., and Obsahuje bibliografii
The aim of the project is to study, develop and apply processes, methods and tools enabling the creation of an infrastructure and conditions for implementation of the Czech Mathematical Digital Library which will comprise a relevant part of the mathematical literature published in the Czech Republic and for its integration into the World Mathematical Digital Library (WMDL). The project includes the beginning of the digitization process, provision of the end users with an access to the digital staff, investigation of advanced technologies for search in mathematical documents and integration of born-digital documents.
Tento příspěvek je hrstkou poznámek a otázek, vztahujícících se k nekonečnu v matematice. Není v něm řeč o nekonečnech, jak je potkává fyzik (nekonečnost vesmíru, singularity, obecné teorie relativity, divergence v kvantové teorii polí atd.), ale o nekonečnech zaváděných v matematice. Množné číslo je na místě, matematika totiž zná nekonečen mnoho; na vkus fyzika asi až příliš mnoho. Příspěvek není matematickým textem, neusiluje o exaktní matematické formulace, je toliko pohledem uživatele matematiky., Josef Jelen., and Obsahuje seznam literatury
Přestože zahraniční výzkumy potvrzují silný vliv matematické self-efficacy na výkony a postoje v matematice, v českém prostředí dosud nebyla tomuto tématu věnována větší pozornost. Článek proto představuje koncept self-efficacy a specifickou oblast self-efficacy v matematice a dále prezentuje uskutečněný výzkum, který měl dva hlavní cíle: vyvinout a ověřit český dotazník matematické self-efficacy pro základní školy a zjistit souvislost mezi matematickou self-efficacy a výkonem v matematice. Výzkumný soubor tvořilo 436 žáků a žákyň ze 4. a 8. tříd, kteří vyplnili dotazník matematické self-efficacy, dotazník o matematice a matematický test složený z úloh uvolněných z projektu TIMSS 2007. Výsledný český dotazník matematické self-efficacy je tvořen 30 položkami, které jsou hodnoceny na pěti bodové škále, a vykazuje vysokou reliabilitu (Cronbachova alfa = 0,9). Matematická self-efficacy významně korelovala s výkonem v matema-tickém testu. Míra self-efficacy vykazovala významný rozdíl mezi mladšími a staršími dětmi. and Despite a rich research done abroad which has confirmed a strong influence of self-efficacy on performance and attitudes in relation to mathematics, there has been paid little attention to this concept in the Czech Republic. This article introduces the concept of self-efficacy and domain-specific self-efficacy in mathematics while also presents realized research with two main goals: 1) to develop and verify a Czech scale of mathematical self-efficacy for primary and secondary schools 2) to find out the relationship between mathematical self-efficacy and performance in mathematics. In the research group, there were 436 boys and girls from fourth and eighth grades. They completed a scale of mathematical self-efficacy, a mathematical questionnaire and a mathematical test with tasks provided by TIMSS 2007 project. The final Czech scale of mathematical self-efficacy consist of 30 items with a five-point evaluation scale and reports high reliability (Cronbach’s alpha = 0,9). Mathematical self-efficacy significantly correlated with performance in the mathematical test. There was also a significant difference in the level of self-efficacy between younger and older children.