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

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

Two-dimensional symmetric and antisymmetric generalizations of exponential and cosine functions

Properties of the four families of recently introduced special functions of two real variables, denoted here by $E^\pm$, and $\cos^\pm$, are studied. The superscripts $^+$ and $^-$ refer to the symmetric and antisymmetric functions respectively. The functions are considered in all details required for their exploitation in Fourier expansions of digital data, sampled on square grids of any density and for general position of the grid in the real plane relative to the lattice defined by the underlying group theory. Quality of continuous interpolation, resulting from the discrete expansions, is studied, exemplified and compared for some model functions.Comment: 22 pages, 10 figure

Three variable exponential functions of the alternating group

New class of special functions of three real variables, based on the alternating subgroup of the permutation group $S_3$, is studied. These functions are used for Fourier-like expansion of digital data given on lattice of any density and general position. Such functions have only trivial analogs in one and two variables; a connection to the $E-$functions of $C_3$ is shown. Continuous interpolation of the three dimensional data is studied and exemplified.Comment: 10 pages, 3 figure

Gaussian cubature arising from hybrid characters of simple Lie groups

Lie groups with two different root lengths allow two mixed sign homomorphisms on their corresponding Weyl groups, which in turn give rise to two families of hybrid Weyl group orbit functions and characters. In this paper we extend the ideas leading to the Gaussian cubature formulas for families of polynomials arising from the characters of irreducible representations of any simple Lie group, to new cubature formulas based on the corresponding hybrid characters. These formulas are new forms of Gaussian cubature in the short root length case and new forms of Radau cubature in the long root case. The nodes for the cubature arise quite naturally from the (computationally efficient) elements of finite order of the Lie group.Comment: 23 pages, 3 figure

Bryophytes and macro-algal growths as a part of macrophyte monitoring in rivers used for ecological assessment

Altogether, 62 taxa of macrophytes including 18 bryophytes and 16 macro-algal growths were determined at 87 survey sites (73 rivers) representing the both ecoregions in Slovakia (Pannonian and Carpathian) during the years 2010–2013. Bryophytes represented the dominant community in the Carpathians, while the occurrence of macro-algal growths was relatively balanced in both ecoregions. Ordination analyses (DCA) showed an obvious shift within studied survey sites from vascular plants to bryophytes, while macro-algal growths were more or less uniform distributed in the whole ordination space. Based on stepwise (forward) selection in CCA, altitude and water surfaces as a land use type were the main environmental factors responsible for this pattern and explained 13.7% of the variability. Variation partitioning showed that the shares of environmental variables on the total variation decreased in the following order: both groups together 8.3% (landscape and geographical variables, physicochemical variables), followed by landscape and geographical variables (5.8%) and purely physicochemical variables which had an insignificant effect on macrophyte composition. The importance of both groups (bryophytes and macro-algal growths) in ecological assessment was also confirmed by their contribution to the mean IBMR value determined for each water body type. Anyway, our study showed that their contribution to ecological assessment is not focused only on small mountain streams where they are dominant. They may obviously affect ecological assessment also in many water body types in lowland rivers and large upland rivers as well

The stands with the Phalaroides arundinacea dominance in the Ipeľ River catchment area (Slovakia and Hungary)

The Phalaridetum arundinaceae Libbert 1931 and the Rorippo-Phalaridetum arundinaceae Kopecký 1961 associations were recorded in the Ipeľ River catchment area from 1997 to 2000. Four other plant communities not ranked into hierarchical system were detected, too. The synmorphology, synecology and synchorology characteristics of these communities are given in the paper. The characteristics and syntaxonomic note are added for the Carici gracilis-Phalaridetum (Kovács et Máthé 1967) Soó 1971 corr. Borhidi 1996 association, which was described from the studied area several decades ago. The floristical and ecological differentiation of these communities is included. The tabular synthesis, polythetic divisive classification (Twinspan) and indirect gradient analysis (DCA) methods were used for classification and ordination of the communities

The Charetum contrariae association, new to Hungary

The Charetum contrariae, stoneworth plant community, was found during the research of aquatic biotopes in the catchment area of the Ipeľ river in the summer of 2000. It has not been detected in Hungary. Up to now, the community is known from Slovakia and Poland at the territory of Central Europe