Cubature formulas of multivariate polynomials arising from symmetric orbit functions

The paper develops applications of symmetric orbit functions, known from irreducible representations of simple Lie groups, in numerical analysis. It is shown that these functions have remarkable properties which yield to cubature formulas, approximating a weighted integral of any function by a weighted finite sum of function values, in connection with any simple Lie group. The cubature formulas are specialized for simple Lie groups of rank two. An optimal approximation of any function by multivariate polynomials arising from symmetric orbit functions is discussed.Comment: 19 pages, 4 figure

Cubature formulae for orthogonal polynomials in terms of elements of finite order of compact simple Lie groups

AbstractThe paper contains a generalization of known properties of Chebyshev polynomials of the second kind in one variable to polynomials of n variables based on the root lattices of compact simple Lie groups G of any type and of any rank n. The results, inspired by work of H. Li and Y. Xu where they derived cubature formulae from A-type lattices, yield Gaussian cubature formulae for each simple Lie group G based on nodes (interpolation points) that arise from regular elements of finite order in G. The polynomials arise from the irreducible characters of G and the nodes as common zeros of certain finite subsets of these characters. The consistent use of Lie theoretical methods reveals the central ideas clearly and allows for a simple uniform development of the subject. Furthermore it points to genuine and perhaps far reaching Lie theoretical connections

Two dimensional symmetric and antisymmetric generalizations of sine functions

Properties of 2-dimensional generalizations of sine functions that are symmetric or antisymmetric with respect to permutation of their two variables are described. It is shown that the functions are orthogonal when integrated over a finite region $F$ of the real Euclidean space, and that they are discretely orthogonal when summed up over a lattice of any density in $F$. Decomposability of the products of functions into their sums is shown by explicitly decomposing products of all types. The formalism is set up for Fourier-like expansions of digital data over 2-dimensional lattices in $F$. Continuous interpolation of digital data is studied.Comment: 12 pages, 5 figure

Risks, experience and benefits of ski mountaineering

10 Abstract Title: Risks, experience and benefits of ski mountaineering Objectives: The aim of the study was to investigate feelings, risks and benefits during ski mountaineeering. Methods: In this study, we used a survey which is one of the questionnaire techniques. It is a quantitative research. Graphical representation was used to process the results. 337 respondents aged 18-72 years took part in this study. Results: We learned in the survey that for most ski mountaineers the greatest experience they get out of a hike is nature. Furthermore, most ski mountaineers don't know what green exercise means, 61% of ski mountaineers claim that the biggest subjective risk is not knowing the terrain, and the biggest objective risk are avalanches (66%). Almost all respondents agreed that the biggest benefit they perceived to be the improvement of their mood due to endorphin release. Conclusion: Thanks to the conducted survey, we have found out which benefits and risks skialpinists perceive as the most significant. According to our findings, the release of endorphins and the associated improvement in mood are among the greatest physiological benefits. Avalanche risks were identified by the respondents as the most feared objective risks, while lack of terrein knowledge was the most commonly selected subjective risk by...10 Abstrakt Název: Rizika, prožitek a benefity skialpinistické túry Cíl: Cílem práce je zjistit pocity, rizika a benefity během skialpinistické túry. Metody: V závěrečné části jsme použili jednu z dotazníkových technik a to anketu. Jedná se o kvantitativní výzkum. Ke zpracování výsledků bylo použito grafové znázornění. Výzkumný soubor se skládá ze 337 respondentů ve věku 18-72 let. Výsledky: V této práci jsme se dozvěděli, že většina skialpinistů považuje za největší prožitek z túry přírodu. Dále většina skialpinistů neví, co je green exercise, 61 % skialpinistů bere jako největší subjektivní nebezpečí riziko terénu a největší objektivní riziko vnímají laviny (66 %). Téměř všichni se shodli, že největším benefitem se jim jeví zlepšení nálady díky vyplavování endorfinu. Závěr: Díky vytvořené anketě jsme zjistili, jaké benefity a rizika vnímají skialpinisté jako nejpodstatnější. Dle našeho zjištění patří mezi největší fyziologické benefity vyplavování endorfinů a s tím spojené zlepšení nálady. Mezi nejobávanější objektivní rizika zařadili respondenti laviny, u subjektivních rizik nejčastěji skialpinisté zaškrtli neznalost terénu. Klíčová slova: objektivní rizika, subjektivní rizika, lavina, zážitek, green exerciseKatedra atletiky, sportů a pobytu v příroděDepartment of Athletics and Outdoor SportsFaculty of Physical Education and SportFakulta tělesné výchovy a sport

Six types of E−E-functions of the Lie groups O(5) and G(2)

New families of $E$-functions are described in the context of the compact simple Lie groups O(5) and G(2). These functions of two real variables generalize the common exponential functions and for each group, only one family is currently found in the literature. All the families are fully characterized, their most important properties are described, namely their continuous and discrete orthogonalities and decompositions of their products.Comment: 25 pages, 13 figure

On E-functions of Semisimple Lie Groups

We develop and describe continuous and discrete transforms of class functions on a compact semisimple, but not simple, Lie group $G$ as their expansions into series of special functions that are invariant under the action of the even subgroup of the Weyl group of $G$. We distinguish two cases of even Weyl groups -- one is the direct product of even Weyl groups of simple components of $G$, the second is the full even Weyl group of $G$. The problem is rather simple in two dimensions. It is much richer in dimensions greater than two -- we describe in detail $E-$transforms of semisimple Lie groups of rank 3.Comment: 17 pages, 2 figure