Contractions of Low-Dimensional Lie Algebras

Theoretical background of continuous contractions of finite-dimensional Lie algebras is rigorously formulated and developed. In particular, known necessary criteria of contractions are collected and new criteria are proposed. A number of requisite invariant and semi-invariant quantities are calculated for wide classes of Lie algebras including all low-dimensional Lie algebras. An algorithm that allows one to handle one-parametric contractions is presented and applied to low-dimensional Lie algebras. As a result, all one-parametric continuous contractions for the both complex and real Lie algebras of dimensions not greater than four are constructed with intensive usage of necessary criteria of contractions and with studying correspondence between real and complex cases. Levels and co-levels of low-dimensional Lie algebras are discussed in detail. Properties of multi-parametric and repeated contractions are also investigated.Comment: 47 pages, 4 figures, revised versio

Alternativity and reciprocity in the Cayley-Dickson algebra

We calculate the eigenvalue \rho of the multiplication mapping R on the Cayley-Dickson algebra A_n. If the element in A_n is composed of a pair of alternative elements in A_{n-1}, half the eigenvectors of R in A_n are still eigenvectors in the subspace which is isomorphic to A_{n-1}. The invariant under the reciprocal transformation A_n \times A_{n} \ni (x,y) -> (-y,x) plays a fundamental role in simplifying the functional form of \rho. If some physical field can be identified with the eigenspace of R, with an injective map from the field to a scalar quantity (such as a mass) m, then there is a one-to-one map \pi: m \mapsto \rho. As an example, the electro-weak gauge field can be regarded as the eigenspace of R, where \pi implies that the W-boson mass is less than the Z-boson mass, as in the standard model.Comment: To be published in J. Phys. A: Mathematical and Genera

Non-semisimple Lie algebras with Levi factor \frak{so}(3), \frak{sl}(2,R) and their invariants

We analyze the number N of functionally independent generalized Casimir invariants for non-semisimple Lie algebras \frak{s}\overrightarrow{% oplus}_{R}\frak{r} with Levi factors isomorphic to \frak{so}(3) and \frak{sl}(2,R) in dependence of the pair (R,\frak{r}) formed by a representation R of \frak{s} and a solvable Lie algebra \frak{r}. We show that for any dimension n >= 6 there exist Lie algebras \frak{s}\overrightarrow{\oplus}_{R}\frak{r} with non-trivial Levi decomposition such that N(\frak{s}% \overrightarrow{oplus}_{R}\frak{r}) = 0.Comment: 16 page

Quantifying the Detrimental Impacts of Land-Use and Management Change on European Forest Bird Populations

The ecological impacts of changing forest management practices in Europe are poorly understood despite European forests being highly managed. Furthermore, the effects of potential drivers of forest biodiversity decline are rarely considered in concert, thus limiting effective conservation or sustainable forest management. We present a trait-based framework that we use to assess the detrimental impact of multiple land-use and management changes in forests on bird populations across Europe. Major changes to forest habitats occurring in recent decades, and their impact on resource availability for birds were identified. Risk associated with these changes for 52 species of forest birds, defined as the proportion of each species' key resources detrimentally affected through changes in abundance and/or availability, was quantified and compared to their pan-European population growth rates between 1980 and 2009. Relationships between risk and population growth were found to be significantly negative, indicating that resource loss in European forests is an important driver of decline for both resident and migrant birds. Our results demonstrate that coarse quantification of resource use and ecological change can be valuable in understanding causes of biodiversity decline, and thus in informing conservation strategy and policy. Such an approach has good potential to be extended for predictive use in assessing the impact of possible future changes to forest management and to develop more precise indicators of forest health

Variable strength of forest stand attributes and weather conditions on the questing activity of Ixodes ricinus ticks over years in managed forests

Given the ever-increasing human impact through land use and climate change on the environment, we crucially need to achieve a better understanding of those factors that influence the questing activity of ixodid ticks, a major disease-transmitting vector in temperate forests. We investigated variation in the relative questing nymph densities of Ixodes ricinus in differently managed forest types for three years (2008–2010) in SW Germany by drag sampling. We used a hierarchical Bayesian modeling approach to examine the relative effects of habitat and weather and to consider possible nested structures of habitat and climate forces. The questing activity of nymphs was considerably larger in young forest successional stages of thicket compared with pole wood and timber stages. Questing nymph density increased markedly with milder winter temperatures. Generally, the relative strength of the various environmental forces on questing nymph density differed across years. In particular, winter temperature had a negative effect on tick activity across sites in 2008 in contrast to the overall effect of temperature across years. Our results suggest that forest management practices have important impacts on questing nymph density. Variable weather conditions, however, might override the effects of forest management practices on the fluctuations and dynamics of tick populations and activity over years, in particular, the preceding winter temperatures. Therefore, robust predictions and the detection of possible interactions and nested structures of habitat and climate forces can only be quantified through the collection of long-term data. Such data are particularly important with regard to future scenarios of forest management and climate warming