Yuzvinsky [7] has shown that free arrangements are formal. In this note we define a more general class of arrangements which we call k-formal, and we show that free arrangements are k-formal. We close with an example which distinguishes k-formal arrangements from formal arrangements. Note that this is reverse inclusion. Thus V is the unique minimal element.

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Abstract. Manin and Schechtman defined the discriminantal arrangement of a generic hyperplane arrangement as a generalization of the braid arrangement. This paper shows their construction is dual to the fiber zonotope construction of Billera and Sturmfels, and thus makes sense even when the base arrangement is not generic. The hyperplanes, face lattices and intersection lattices of discriminantal arrangements are studied. The discriminantal arrangement over a generic arrangement is shown to be formal (and in some cases 3-formal), though it is in general not free. An example of a free discriminantal arrangement over a generic arrangement is given.

has been postulated that RAR antagonists may prevent or reverse retinoid-mediated cartilage destruction and a RAR pan antagonist was previously shown to improve clinical scores in the collagen-induced arthritis (CIA) model, albeit with unacceptable adverse effects on testes. We have postulated that the primary beneficial joint effects of RAR antagonists are associated with RARgamma, while the adverse effects on testes are associated with RARalpha. Thus, we have identified a highly selective RARgamma antagonist (LY2813631) to test this hypothesis. Methods: The RAR antagonist LY2813631 demonstrated in vitro selective affinity for RARgamma in a SPA-based binding assay using full length RAR alpha, beta and gamma protein and the synthetic pan agonist 3H-TTNPB. Functional antagonism and selectivity was demonstrated using a GAL4-RAR alpha, beta and gamma and Gal4 response element/Luciferase constructs, co-transfected into HEK 293 cells. Functional activity in chondrocytes was demonstrated using agonist TTNPB and primary bovine chondrocytes, looking at the ability of LY2813631 to regulate OA-relevant genes, such as MMP13, ADAMTS-5 and Type 2 Collagen. In vivo studies utilized Lewis rats to show reversal of RAR gamma agonist induced changes in OA relevant genes in the articular cartilage and reduction in OA-related neoepitopes in the synovial fluid. The collagen induced arthritis (CIA) model was used to determine joint efficacy in rats. Results: In vitro, LY2813631 binds RARgamma (Ki ¼ 0.74 nM) with significantly higher affinity than RARalpha (Ki ¼ 400 nM) and RAR beta (Ki ¼ 25 nM) and shows selective functional antagonism at RAR gamma (Kb ¼ 42 nM) in HEK 293 cells, compared to RARalpha (Kb ¼ 2010 nM) and RAR beta (Kb ¼ 359 nM). Additionally, LY2813631 normalized RAR agonist-induced increases in the catabolic enzyme ADAMTS-5 in primary bovine chondrocytes. In vivo, a selective RARgamma agonist was shown to increase mRNA levels of ADAMTS-5 and decrease mRNA levels of type 2 collagen in the articular cartilage of rats after 3 days of oral dosing. The animals also showed a 5-fold increase in the type 2 collagen neoepitopes 9A4 and CTX-II in synovial fluid. These effects could be blocked by co-dosing RARgamma antagonist LY2813631, demonstrating specific RARgamma-mediated target engagement in the joint space. LY2813631 also improved ankle and knee histopathology scores in the rat CIA model at doses that did not cause testicular degeneration. Conclusions: Our findings support a role for endogenous retinoids in the destruction of articular cartilage in OA and suggest that these effects are primarily mediated through RARgamma signaling. We have also shown that the beneficial effects of RARgamma antagonist LY2813631 can be demonstrated without concomitant adverse effects on the testes, supporting RARgamma as a potentially safe target for disease modification in arthritic diseases.Purpose: Modulation of the BMP pathway remains an attractive target for development of novel anabolic treatments for osteoarthritis. We are aiming to ide...

This note presents an example showing the relevance of apportionment to weighted voting systems. The example is appropriate for class presentations or student projects. Also, some of the basic notions of apportionment are presented in a way different from that of standard textbooks. This approach may be useful as a supplement to the more typical treatment.