On the simple connectedness of hyperplane complements in dual polar spaces

Let $\Delta$ be a dual polar space of rank \geq 4$,$ be a hyperplane of $\Delta$ and $\Gamma: = \Delta\setminus H$ be the complement of $in$\Delta$. We shall prove that, if all lines of$\Delta$have more than$ points, then $\Gamma$ is simply connected. Then we show how this theorem can be exploited to prove that certain families of hyperplanes of dual polar spaces, or all hyperplanes of certain dual polar spaces, arise from embeddings

Hyperplanes of symplectic dual polar spaces : a survey

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On extensions of hyperplanes of dual polar spaces

AbstractLet Δ be a thick dual polar space and F a convex subspace of diameter at least 2 of Δ. Every hyperplane G of the subgeometry F˜ of Δ induced on F will give rise to a hyperplane H of Δ, the so-called extension of G. We show that F and G are in some sense uniquely determined by H. We also consider the following problem: if e is a full projective embedding of Δ and if eF is the full embedding of F˜ induced by e, does the fact that G arises from the embedding eF imply that H arises from the embedding e? We will study this problem in the cases that e is an absolutely universal embedding, a minimal full polarized embedding or a Grassmann embedding of a symplectic dual polar space. Our study will allow us to prove that if e is absolutely universal, then also eF is absolutely universal

Intercellular Ca2+ waves: mechanisms and functions

Intercellular calcium (Ca2+) waves (ICWs) represent the propagation of increases in intracellular Ca2+ through a syncytium of cells and appear to be a fundamental mechanism for coordinating multicellular responses. ICWs occur in a wide diversity of cells and have been extensively studied in vitro. More recent studies focus on ICWs in vivo. ICWs are triggered by a variety of stimuli and involve the release of Ca2+ from internal stores. The propagation of ICWs predominately involves cell communication with internal messengers moving via gap junctions or extracellular messengers mediating paracrine signaling. ICWs appear to be important in both normal physiology as well as pathophysiological processes in a variety of organs and tissues including brain, liver, retina, cochlea, and vascular tissue. We review here the mechanisms of initiation and propagation of ICWs, the key intra-and extracellular messengers (inositol 1,4,5-trisphosphate and ATP) mediating ICWs, and the proposed physiological functions of ICWs

The hyperplanes of finite symplectic dual polar spaces which arise from projective embeddings

AbstractWe characterize the hyperplanes of the dual polar space DW(2n−1,q) which arise from projective embeddings as those hyperplanes H of DW(2n−1,q) which satisfy the following property: if Q is an ovoidal quad, then Q∩H is a classical ovoid of Q. A consequence of this is that all hyperplanes of the dual polar spaces DW(2n−1,4), DW(2n−1,16) and DW(2n−1,p) (p prime) arise from projective embeddings

Direct constructions of hyperplanes of dual polar spaces arising from embeddings

Let e be one of the following full projective embeddings of a finite dual polar space Delta of rank n >= 2: (i) The Grassmann-embedding of the symplectic dual polar space Delta congruent to DW(2n 1,q); (ii) the Grassmann-embedding of the Hermitian dual polar space Delta congruent to DH(2n-1, q(2)); (iii) the spin-embedding of the orthogonal dual polar space Delta congruent to DQ(2n, q); (iv) the spin-embedding of the orthogonal dual polar space Delta congruent to DQ(-)(2n+ 1, q). Let H-e denote the set of all hyperplanes of Delta arising from the embedding e. We give a method for constructing the hyperplanes of H-e without implementing the embedding e and discuss (possible) applications of the given construction

Is there a Jordan geometry underlying quantum physics?

There have been several propositions for a geometric and essentially non-linear formulation of quantum mechanics. From a purely mathematical point of view, the point of view of Jordan algebra theory might give new strength to such approaches: there is a ``Jordan geometry'' belonging to the Jordan part of the algebra of observables, in the same way as Lie groups belong to the Lie part. Both the Lie geometry and the Jordan geometry are well-adapted to describe certain features of quantum theory. We concentrate here on the mathematical description of the Jordan geometry and raise some questions concerning possible relations with foundational issues of quantum theory.Comment: 30 page