$$(O( V \oplus F), O( V))$$ is a Gelfand pair for any quadratic space V over a local field F

Let V be a quadratic space with a form q over an arbitrary local field F of characteristic different from 2. Let with the form Q extending q with Q(e) = 1. Consider the standard embedding and the two-sided action of on . In this note we show that any -invariant distribution on is invariant with respect to transposition. This result was earlier proven in a bit different form in van Dijk (Math Z 193:581–593, 1986) for , in Aparicio and van Dijk (Complex generalized Gelfand pairs. Tambov University, 2006) for and in Bosman and van Dijk (Geometriae Dedicata 50:261–282, 1994) for p-adic fields. Here we give a different proof. Using results from Aizenbud et al. (arXiv:0709.1273 (math.RT), submitted), we show that this result on invariant distributions implies that the pair (O(V), O(W)) is a Gelfand pair. In the archimedean setting this means that for any irreducible admissible smooth Fréchet representation (π, E) of we have A stronger result for p-adic fields is obtained in Aizenbud et al. (arXiv:0709.4215 (math.RT), submitted).     

Many-to-many location-routing with inter-hub transport and multi-commodity pickup-and-delivery

In this paper, we consider a variant of the many-to-many location-routing problem, where hub facilities have to be located and customers with either pickup or delivery demands have to be combined in vehicle routes. In addition, several commodities and inter-hub transport processes are taken into account. A practical application of the problem can be found in the timber-trade industry, where companies provide their services using hub-and-spoke networks. We present a mixed-integer linear model for the problem and use CPLEX 12.4 to solve small-scale instances. Furthermore, a multi-start procedure based on a fix-and-optimize scheme and a genetic algorithm are introduced that efficiently construct promising solutions for medium- and large-scale instances. A computational performance analysis shows that the presented methods are suitable for practical application.     

Probabilistic analysis of geometric location problems

We analyze the behaviour of thek center and median problems forn points randomly distributed in an arbitrary regionA ofR ^d . Under a mild assumption on the regionA, we show that fork≦k(n)=o(n/logn), the objective function values of the discrete and continuous versions of these problems are equal to each otheralmost surely. For the two-dimensional case, both these problems can be solved by placing the centers or medians in an especially simple regular hexagonal pattern (the ‘honeycomb heuristic’ of Papadimitriou). This yields the exact asymptotic values for thek center and median problem, namely, α(|A|/k)^1/2 and β(|A|/k)^1/2, where |A| denotes the volume ofA, α and β are known constants, and the objective of the median problem is given in terms of the average, rather than the usual total, distance. For the 3- and 4-dimensional case, similar results can be obtained for the center problem to within an accuracy of roughly one percent. As a by-product, we also get asymptotically optimal algorithms for the 2-dimensionalp-normk median problem and for the twin problems of minimizing the maximum number of vertices served by any center and similarly for maximizing the minimum.     

Tetragonal low-temperature structure of LiAl

The cubic face-centered structure of LiAl (, at ) transforms into a tetragonal body-centered structure (I4₁/amd, , at ). This first-order phase transition at about during heating is probably the reason for the so-called “ anomalies” in some physical properties like specific heat, electrical resistivity and nuclear-spin lattice relaxation. This transition seems to be correlated with the composition Li:Al of the alloy and the amount of Li vacancies.     

Synthesis, crystal structure and magnetic properties of new indium rhenium and scandium rhenium oxides, In6ReO12 and Sc6ReO12

The new complex indium rhenium and scandium rhenium oxides, In₆ReO₁₂ and Sc₆ReO₁₂, have been synthesized as single phases in sealed silica tubes and by high-pressure high-temperature syntheses, and their crystal structures have been determined by single crystal X-ray diffraction.The compounds crystallize in a rhombohedral structure related to the distorted fluorite structure like Ln₆ReO₁₂ for some rare earth elements, S. G.: R-3, Z=3, a_H= 9.248(2) Å, c_H=8.720(2) Å for Sc₆ReO₁₂ and a_H=9.492(1) Å, c_H=8.933(1) Å for In₆ReO₁₂. A maximum in magnetization is observed for Sc₆ReO₁₂ at T(M_max)=1.89(2) K, whereas ferromagnetic ordering is found for In₆ReO₁₂ by a pronounced increase in the temperature dependence of magnetization at T_C=7.5(5) K. The magnetic moment per rhenium ion in In₆ReO₁₂ and Sc₆ReO₁₂ is 0.84(1) and 0.65(1) μ_B, respectively, derived from the paramagnetic regions.     

Ferromagnetism in a dinuclear nickel(II) complex containing triethylenetetramine and tricyanomethanide

Triethylenetetramine (L(4)) was used as a tetradentate blocking ligand that, after complexation with Ni(II), leaves two sites ready for ligation with tricyanomethanide. The formed binuclear complex [L(4)Ni(NCC(CN)CN)(2)NiL(4)](ClO(4))(2) exhibits a ferromagnetic coupling with J/hc = +0.15 cm(-1) and g(Ni) = 2.126; below 16 K, a ferromagnetic ordering is evidenced by ac magnetic susceptibility (both in-phase and out-of-phase), magnetization, field-cooled magnetization, and zero-field-cooled magnetization measurements.     

Dithiaporphyrin derivatives as photosensitizers in membranes and cells

We synthesized a series of analogues of 5,20-diphenyl-10,15-bis(4-carboxylatomethoxy)phenyl-21,23-dithiaporphyrin (I) as potential photosensitizers for photodynamic therapy (PDT). The photosensitizers differ in the length of the side chains that bind the carboxyl to the phenol at positions 10 and 15 of the thiaporphyrin. The spectroscopic, photophysical, and biophysical properties of these photosensitizers are reported. The structural changes have almost no effect on the excitation/emission spectra with respect to I's spectra or on singlet oxygen generation in MeOH. All of the photosensitizers have a very high, close to 1.00, singlet oxygen quantum yield in MeOH. On the contrary, singlet oxygen generation in liposomes was considerably affected by the structural change in the photosensitizers. The photosensitizers possessing short side chains (one and three carbons) showed high quantum yields of around 0.7, whereas the photosensitizers possessing longer side chains showed smaller quantum yield, down to 0.14 for compound X (possessing side-chain length of 10 carbons), all at 1 microM. Moreover a self-quenching process of singlet oxygen was observed, and the quantum yield decreased as the photosensitizer's concentration increased. We measured the binding constant of I to liposomes and found Kb = 23.3 +/- 1.6 (mg/mL)-1. All the other photosensitizers with longer side chains exhibited very slow binding to liposomes, which prevented us from assessing their Kb's. We carried out fluorescence resonance energy transfer (FRET) measurements to determine the relative depth in which each photosensitizer is intercalated in the liposome bilayer. We found that the longer the side chain the deeper the photosensitizer core is embedded in the bilayer. This finding suggests that the photosensitizers are bound to the bilayer with their acid ends close to the aqueous medium interface and their core inside the bilayer. We performed PDT with the dithiaporphyrins on U937 cells and R3230AC cells. We found that the dark toxicity of the photosensitizers with the longer side chain (X, VI, V) is significantly higher than the dark toxicity of sensitizers with shorter side chains (I, III, IV). Phototoxicity measurements showed the opposite direction; the photosensitizers with shorter side chains were found to be more phototoxic than those with longer side chains. These differences are attributed to the relationship between diffusion and endocytosis in each photosensitizer, which determines the location of the photosensitizer in the cell and hence its phototoxicity.     

Spectroscopy, binding to liposomes and production of singlet oxygen by porphyrazines with modularly variable water solubility

Three novel classes of porphyrazine-like structures were synthesized to form modular structures in which lipophilicity and water solubility can be tuned. Subtle modification of solubility is an important criterion in selecting a compound for biological photosensitization. The general structure takes the form H2[pz(AnB4-n)], where the core is a porphyrazine (pz) group, A is a pyrrole ring with two sulfide linkages (SR moieties) and B is a pyrrole fused with a 4,7-bis(isopropyloxy)benzo group, with n=4, 3 and 2. These molecules possess their longest wavelength absorption band between 700 and 810 nm, hence laser beams of higher tissue penetration depth could be used to illuminate them in photodynamic therapy (PDT). Armed with absorption bands in the far-red and near-infrared (near-IR), and a capability to tune the solubility, these molecules could make for better sensitizers because of optimized uptake by lipidic membranes and better optical properties. We tested several derivatives of the A4, A3B and A2B2 structures for their singlet oxygen quantum yields in methanol and in liposomes, using 9,10-dimethyl anthracene (DMA) as a singlet oxygen target. Singlet oxygen quantum yields in liposomes ranged from 0.01 to 0.44, with the A2B2 group showing the most promise. In the binding assay to find the equilibrium binding constant, Kb, we detected fluorescence changes due to a change in environment. Peripheral long-chain moieties (the R group in the SR moieties) dominate lipid binding. These moieties range in the hydrophobicity that they induce from C8H17 and benzene, which rendered the molecule totally insoluble in water, to polyethylene glycol (PEG) and carboxylate groups, which imparted water solubility. Each molecule had between 4 and 8 such identical chains. Chains bearing an ether or ester link resulted in measurable equilibrium constants, with a higher Kb for ether substituents. Results for Kb ranged from 0.23 to 26.52 (mg mL(-1))(-1). A delicate balance exists between water solubility and good partitioning to membranes. In general, a higher oxygen-to-carbon ratio in the chains improves binding. Fewer chains and a centrally coordinated zinc ion further improve binding and singlet oxygen production.