By Bart de Bruyn

Devoted to the Russian mathematician Albert Shiryaev on his seventieth birthday, this can be a choice of papers written via his former scholars, co-authors and co-workers. The booklet represents the cutting-edge of a quick maturing idea and may be an important resource for researchers during this zone. the range of subject matters and complete variety of the papers make the e-book beautiful for Ph.D. scholars and younger researchers.

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**Sample text**

If α is an (n − 1 − i)-dimensional subspace of a polar space Γ of rank n (0 ≤ i ≤ n), then the set of all maximal subspaces through α deﬁnes a convex subspace of diameter i of the dual polar space S associated with Γ. Conversely, every convex subspace of S is obtained in this way. e. convex subspaces of diameter 2) are thick generalized quadrangles. Given a number of polar spaces, many others can be constructed. Let (Γi )i∈I be a family of polar spaces deﬁned on the sets (Pi )i∈I and with (Ai )i∈I as a collection of subspaces.

Suppose that any common neighbour of a and c lies at distance i − 1 from x. 21, there exists a path of length 3 in Γ2 (πQ (x)) ∩ Q connecting a and c. This path is completely contained in Γi (x). Deﬁnitions. If γ = (y0 , . . , yk ) denotes a path of S, then we deﬁne b(γ) := y0 and e(γ) := yk . For every point x of S, let Ωx denote the set of all paths (y0 , . . , yk ) in S for which (S(x, yi ) \ S(x, yi−1 )) ∩ S(x, y0 ) = ∅ for every i ∈ {1, . . , k} such that k d(x, yi ) > d(x, yi−1 ). For each such path γ, we deﬁne i(γ) := i=0 3d(x,y0 )−d(x,yi ) .

Let x be a given point of Q. For every point y ∈ Γ2 (x), we deﬁne A(y) := Γ1 (x) ∩ Γ1 (y) and A(y) := Γ1 (x) \ A(y). If y and y are two collinear points of Γ2 (x), then A(y) ∩ A(y ) consists of the unique point of yy collinear with x. 21, the diameter of Γ2 (x) is at most 3. So, if y and y are two points of Γ2 (x), we have one of the following possibilities: • y = y . Then |A(y) ∩ A(y )| = 0. • y ∼ y . Then |A(y) ∩ A(y )| = 1. • y and y have distance 2 in Γ2 (x). Let y denote a point of Γ2 (x) collinear with y and y .