Divergent trajectories in arithmetic homogeneous spaces of rational rank two

Let G be a semisimple real algebraic group defined over ${\mathbb {Q}}$ , $\Gamma $ be an arithmetic subgroup of G, and T be a maximal ${\mathbb {R}}$ -split torus. A trajectory in $G/\Gamma $ is divergent if eventually it leaves every compact subset. In some cases there is a finite collection of explicit algebraic data which accounts for the divergence. If this is the case, the divergent trajectory is called obvious. Given a closed cone in T, we study the existence of non-obvious divergent trajectories under its action in $G\kern-1pt{/}\kern-1pt\Gamma $ . We get a sufficient condition for the existence of a non-obvious divergence trajectory in the general case, and a full classification under the assumption that $\mathrm {rank}_{{\mathbb {Q}}}G=\mathrm {rank}_{{\mathbb {R}}}G=2$ .

On the regularity of Cauchy hypersurfaces and temporal functions in closed cone structures

We complement our work on the causality of upper semi-continuous distributions of cones with some results on Cauchy hypersurfaces. We prove that every locally stably acausal Cauchy hypersurface is stable. Then we prove that the signed distance [Formula: see text] from a spacelike hypersurface [Formula: see text] is, in a neighborhood of it, as regular as the hypersurface, and by using this fact we give a proof that every Cauchy hypersurface is the level set of a Cauchy temporal (and steep) function of the same regularity as the hypersurface. We also show that in a globally hyperbolic closed cone structure, compact spacelike hypersurfaces with boundary can be extended to Cauchy spacelike hypersurfaces of the same regularity. We end the work with a separation result and a density result.

Inclusion Complexes of Naringenin in Dimethylated and Permethylated β-Cyclodextrins: Crystal Structures and Molecular Dynamics Studies

The crystal structures of the inclusion complexes of naringenin in dimethylated and permethylated β-cyclodextrin (DM-β-CD and TM-β-CD) were determined and extensively analyzed. Naringenin is found with its 4-hydroxyphenyl residue fully immersed in the DM-β-CD cavity and its chromone group protruding from the narrow rim of the open-cone shaped host. The naringenin/DM-β-CD complex units are packed in a ‘herring bone’ fashion. In the case of naringenin/TM-β-CD, the complex units are arranged in a cage-type mode, the guest naringenin is partially encapsulated in the cavity of the closed-cone shaped host, with its chromone group laying equatorially and its 4-hydroxyphenyl protruding extensively from the wide rim of the host. Furthermore, the crystallographically-determined coordinates of both complexes were employed for Molecular Dynaimics simulations in explicit water solvent and in the absence of crystal contacts. The trajectories showed that naringenin rapidly penetrates the open narrow rim of DM-β-CD but not the closed narrow rim of TM-β-CD. Thus, in the latter case, the chromone group of naringenin is accommodated shallowly in the wide rim of the host, tethered via hydrogen bonds to the secondary methoxy groups of the host. Finally, a significantly higher binding affinity for naringenin in DM-β-CD than TM-β-CD was estimated by Molecular Mechanics/Generalized Born Surface Area calculations.

Causality theory for closed cone structures with applications

We develop causality theory for upper semi-continuous distributions of cones over manifolds generalizing results from mathematical relativity in two directions: non-round cones and non-regular differentiability assumptions. We prove the validity of most results of the regular Lorentzian causality theory including: causal ladder, Fermat’s principle, notable singularity theorems in their causal formulation, Avez–Seifert theorem, characterizations of stable causality and global hyperbolicity by means of (smooth) time functions. For instance, we give the first proof for these structures of the equivalence between stable causality, [Formula: see text]-causality and existence of a time function. The result implies that closed cone structures that admit continuous increasing functions also admit smooth ones. We also study proper cone structures, the fiber bundle analog of proper cones. For them, we obtain most results on domains of dependence. Moreover, we prove that horismos and Cauchy horizons are generated by lightlike geodesics, the latter being defined through the achronality property. Causal geodesics and steep temporal functions are obtained with a powerful product trick. The paper also contains a study of Lorentz–Minkowski spaces under very weak regularity conditions. Finally, we introduce the concepts of stable distance and stable spacetime solving two well-known problems (a) the characterization of Lorentzian manifolds embeddable in Minkowski spacetime, they turn out to be the stable spacetimes, (b) the proof that topology, order and distance (with a formula à la Connes) can be represented by the smooth steep temporal functions. The paper is self-contained, in fact we do not use any advanced result from mathematical relativity.

Solvability of Two Classes of Tensor Complementarity Problems

In this paper, we first introduce a class of tensors, called positive semidefinite plus tensors on a closed cone, and discuss its simple properties; and then, we focus on investigating properties of solution sets of two classes of tensor complementarity problems. We study the solvability of a generalized tensor complementarity problem with aD-strictly copositive tensor and a positive semidefinite plus tensor on a closed cone and show that the solution set of such a complementarity problem is bounded. Moreover, we prove that a related conic tensor complementarity problem is solvable if the involved tensor is positive semidefinite on a closed convex cone and is uniquely solvable if the involved tensor is strictly positive semidefinite on a closed convex cone. As an application, we also investigate a static traffic equilibrium problem which is reformulated as a concerned complementarity problem. A specific example is also given.

Guest-driven unusual conformations in two calix[6]arene solvates and a new calix[8]arene

Unusual conformations have been found in a new calix[8]arene and in new solvates of two known calix[6]arenes. The chair-like conformation with 2/m point group symmetry was found for the first time in the dimethylformamide (DMF) disolvate of the basic calix[6]arene (1) without substituents in the lower and upper rims. Such symmetry is driven by the guest geometry allowing for two equivalent hydrogen bonding patterns in the chair seat. This avoids cone distortion found in the other chair-like conformers, although they have energies lower than that of new symmetrical conformer. The molecular conformation of hexa(carboxymethoxy)calix[6]arene (2) is also described as a dimethylsulfoxide (DMSO) pentasolvate. Its conformation can be described as a 1,3,5-closed cone with three alternate phenyl rings inclined inwards to the cone, thereby closing the cone entrance. Such a conformation also suggests five acid groups are pointed towards the same side of the calyx base and are able to bind metal ions or basic compounds in the lower rim, while inclusion of guests into the cone cavity is hindered. Both inclusion and cooperative acid binding/coordination abilities are still more hindered in the lowest energy 1,2,3-alternate cone conformer of 2. The role of the solvent in avoiding cone distortion was highlighted by inspecting the conformations of 5,11,17,23,29,35,41,47-octanitro-49,50,51,52,53,54,55,56-octa-n-butoxycalix[8]arene (3) and the known nitro analogues having methyl instead of n-butyl groups. Cone distortion is found in the non-solvated crystal form of 3, while non-classical hydrogen bonds with tetrahydrofuran preclude this in the literature analogue.

Disturbance macroecology: integrating disturbance ecology and macroecology with different-age post-fire stands of a closed-cone pine forest

Macroecological studies have generally restricted their scope to relatively steady-state systems, and as a result, how biodiversity and abundance metrics are expected to scale in disturbance-dependent ecosystems is unknown. We examine macroecological patterns in a fire-dependent forest of Bishop pine (Pinus muricata). We target two different-aged stands in a stand-replacing fire regime, one a characteristically mature stand with a diverse understory, and one more recently disturbed by a stand-replacing fire (17 years prior to measurement). We compare the stands using macroecological metrics of species richness, abundance and spatial distributions that are predicted by the Maximum Entropy Theory of Ecology (METE), an information-entropy based theory that has proven highly successful in predicting macroecological metrics across a wide variety of systems and taxa. Ecological patterns in the mature stand more closely match METE predictions than do data from the recently disturbed stand. This suggests METE’s predictions are more robust in late-successional, slowly changing, or steady-state systems than those in rapid flux with respect to species composition, abundances, and organisms’ sizes. Our findings highlight the need for a macroecological theory that incorporates natural disturbance and other ecological perturbations into its predictive capabilities, because most natural systems are not in a steady state.