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

On Orbits of the Ring Zmn under Action of the Group SL(m, Zn)

We consider the action of the finite matrix group SL(m,Zn ) on the ring Zmn. We determine orbits of this action for n arbitrary natural number. It is a generalization of the task which was studied by A. A. Kirillov for m = 2 and n prime number

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

Influence of Moisture Content on the Application of ND and MD Tests to Various Species of Timber Elements

Variations in environmental conditions can affect the structural performance of timber components and constructions. Especially in case of historic masonry buildings, where timber elements can be found either in roof or floor structures, intrinsic defects (e.g., knots, cracks, degradation), as well as the surrounding environmental conditions (e.g., temperature and moisture) can play an important role in the assessment of the mechanical response of the elements. In this connection, non-destructive (ND) and minor-destructive (MD) procedures are used to investigate the onsite conditions of wood, which are commonly based on either tool penetration resistance or wave transmission devices. In the paper, the influence of moisture content (MC) measured with various devices on a series of both old and new timber elements was compared. The elements made of various wood species (mainly red fir, but also white fir, pine and larch) were subjected to laboratory ND and MD testing by three research groups. Results showed the low influence of the MC on MD tests (i.e., resistance drilling and pin penetration) in comparison to the ND (i.e., sonic/ultrasonic) ones. Correlation relationships are provided, which illustrate and approximatively quantify the effect of MC for the experimentally investigated conditions and test equipment

Effect of environmental conditions on species composition of macrophytes – study from two distinct biogeographical regions of Central Europe

The relationship between macrophytes and environmental conditions of lotic and lentic habitats was studied within two distinct biogeographical regions (Pannonian and Carpathian) represented by two model areas (Borská nížina lowland and Turčianska kotlina basin) in 2011. Altogether, 72 taxa of macrophytes were found within both studied biogeographical regions, while almost a third of them grew only in one from the regions. Species-environmental variables relationship was studied by Canonical Correspondence Analysis (forward selection), which revealed that three and seven environmental variables explaining 10.35% and 24.45% of variance of species data had significant effect on species composition of macrophytes in the Pannonian and Carpathian region, respectively. Flow regime and the portion of fine substrate on the bottom are the main drivers of species composition-environmental condition relationships and explained 3.57% and 6.21% of variance in the Pannonian and Carpathian region, respectively. However, the highest values of pure effect on species composition was detected in case of Altitude (Pannonian region; 3.81%) and Connectivity (Carpathian region; 3.19). Based on the variation partitioning, landscape variables (including geographical variables; Pannonian region; 6.8%) and hydrological variables (including morphological variables of the bottom; Carpathian region; 11.3%) explained a bigger portion of the variability of macrophytes in regions than the other groups of environmental variables