LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS IN ARROWHEAD FORM

This paper deals with different approaches for solving linear systems of the first order differential equations with the system matrix in the symmetric arrowhead form.Some needed algebraic properties of the symmetric arrowhead matrix are proposed.We investigate the form of invariant factors of the arrowhead matrix.Also the entries of the adjugate matrix of the characteristic matrix of the arrowhead matrix are considered. Some reductions techniques for linear systems of differential equations with the system matrix in the arrowhead form are presented.

Weighted Homology of Bi-Structures over Certain Discrete Valuation Rings

An RNA bi-structure is a pair of RNA secondary structures that are considered as arc-diagrams. We present a novel weighted homology theory for RNA bi-structures, which was obtained through the intersections of loops. The weighted homology of the intersection complex X features a new boundary operator and is formulated over a discrete valuation ring, R. We establish basic properties of the weighted complex and show how to deform it in order to eliminate any 3-simplices. We connect the simplicial homology, Hi(X), and weighted homology, Hi,R(X), in two ways: first, via chain maps, and second, via the relative homology. We compute H0,R(X) by means of a recursive contraction procedure on a weighted spanning tree and H1,R(X) via an inflation map, by which the simplicial homology of the 1-skeleton allows us to determine the weighted homology H1,R(X). The homology module H2,R(X) is naturally obtained from H2(X) via chain maps. Furthermore, we show that all weighted homology modules Hi,R(X) are trivial for i>2. The invariant factors of our structure theorems, as well as the weighted Whitehead moves facilitating the removal of filled tetrahedra, are given a combinatorial interpretation. The weighted homology of bi-structures augments the simplicial counterpart by introducing novel torsion submodules and preserving the free submodules that appear in the simplicial homology.

Red Wood Ants prefer mature Pine Forests in Variscan Granite Environments (Formica rufa-group)

0AbstractWe used presence/absence data of 5,160 red wood ant nests (RWA; Formica polyctena) acquired in a systematic large-scale area-wide survey in two study areas (≈350 ha) in the Oberpfalz, NE Bavaria, Germany to explore for the first time the influence of variable (e.g., forest type, tree age) and quasi-invariant factors (e.g., tectonics, geochemical composition of the bedrock) on nest size, spatial distribution and nest density for Variscan granites. A combination of the forest type (mature pine-dominated forests (≥80–140 years) as main variable factor and the geochemical property of the Variscan granites with their high natural Radon potential and moderate heat production as main quasi-invariant factor could explain the high nest numbers in both study areas. In addition, the spatially clustered distribution patterns of the observed nests suggest a strong interaction between nests and their quasi-invariant environment, especially the directionality of the present-day stress field and the direction of the tectonically formed “Erbendorfer Line”. In general, such a combination of variable and quasi-invariant factors can be addressed as particularly favorable RWA habitats.

A note on Dehn colorings and invariant factors

If [Formula: see text] is an abelian group and [Formula: see text] is an integer, let [Formula: see text] be the subgroup of [Formula: see text] consisting of elements [Formula: see text] such that [Formula: see text]. We prove that if [Formula: see text] is a diagram of a classical link [Formula: see text] and [Formula: see text] are the invariant factors of an adjusted Goeritz matrix of [Formula: see text], then the group [Formula: see text] of Dehn colorings of [Formula: see text] with values in [Formula: see text] is isomorphic to the direct product of [Formula: see text] and [Formula: see text]. It follows that the Dehn coloring groups of [Formula: see text] are isomorphic to those of a connected sum of torus links [Formula: see text].

A plea for realistic assumptions in economic modelling

The use of unrealistic assumptions in Economics is usually defended not only for pragmatic reasons, but also because of the intrinsic difficulties in determining the degree of realism of assumptions. Additionally, the criterion used for evaluating economic models is associated with their ability to provide accurate predictions. This mode of thought involves –at least implicitly– a commitment to the existence of unvarying invariant factors or regularities. Contrary to this, the present paper presents a critique to the use of invariant knowledge in economics. One reason for this analysis lies in the fact that economic phenomena are not compatible with the logic of invariance, but with the logic of "possibility trees" or "open-ended results". The other reason is that the use of invariant knowledge may entail both external validity problems and negative exposures to a "black swan". Alternatively, an approach where models are understood as possible scenarios is proposed. It is argued that the realism of (substantive) assumptions is crucial here, since it helps to ascertain the degree of resemblance between the different models and the target system.

The Consequences of Spatial Inequality for Adolescent Residential Mobility

A large body of literature suggests that neighborhood socioeconomic disadvantage is positively associated with out-mobility. However, prior research has been limited by (1) the inability to account for endogenous factors that both funnel families into deprived neighborhoods and increase their likelihood of moving out, and (2) the failure to consider how the spatial distribution of socioeconomic deprivation in the broader community conditions the effect of local deprivation on mobility. This paper attends to this gap in the literature by examining how changes in socioeconomic disadvantage between sending and receiving neighborhoods and the spatial patterning of deprivation in the areas surrounding destination neighborhoods influence future mobility among a representative sample of American adolescents. We employ a modeling strategy that allows us to examine the unique and separable effects of local and extralocal neighborhood disadvantage while simultaneously holding constant time-invariant factors that place some youth at a greater likelihood of experiencing a residential move. We find that moves to more impoverished neighborhoods decrease the likelihood of subsequent mobility and that this effect is most pronounced among respondents who move to neighborhoods surrounded by other similarly deprived neighborhoods. In this sense, geographical pockets of disadvantage strengthen the mobility-hampering effect of neighborhood deprivation on future mobility.