Ind--varieties of generalized flags as homogeneous spaces for classical ind--groups

The purpose of the present paper is twofold: to introduce the notion of a generalized flag in an infinite dimensional vector space $V$ (extending the notion of a flag of subspaces in a vector space), and to give a geometric realization of homogeneous spaces of the ind--groups $SL(\infty)$, $SO(\infty)$ and $Sp(\infty)$ in terms of generalized flags. Generalized flags in $V$ are chains of subspaces which in general cannot be enumerated by integers. Given a basis $E$ of $V$, we define a notion of $E$--commensurability for generalized flags, and prove that the set \cFl (\cF, E) of generalized flags E$--commensurable with a fixed generalized flag$\cF$in$V$has a natural structure of an ind--variety. In the case when$V$is the standard representation of$G = SL(\infty)$, all homogeneous ind--spaces$G/P$for parabolic subgroups$P$containing a fixed splitting Cartan subgroup of$G$, are of the form$\cFl (\cF, E)$. We also consider isotropic generalized flags. The corresponding ind--spaces are homogeneous spaces for$SO(\infty)$and$Sp(\infty)$. As an application of the construction, we compute the Picard group of$\cFl (\cF, E)$(and of its isotropic analogs) and show that$\cFl (\cF, E)$is a projective ind--variety if and only if$\cF$is a usual, possibly infinite, flag of subspaces in$V$

Metabolic syndrome and cerebrovascular disease: a crucial relationship

Stroke is without any doubt among the top reasons for disability and mortality in neurology. This is especially true in Bulgaria, where vascular diseases in general, and cerebrovascular disease (CVD) in particular, are significantly more prevalent than in other European countries. More than 80`000 cases of CVD are registered yearly in Bulgaria, over 35`000 having stroke (1). Speaking of stroke, we should not underestimate the other manifestations of CVD, such as transient ischemic attacks and vascular cognitive impairment. The disease may develop silently or cause non-specific complaints for years, and only then become clearly symptomatic. For this reason the diagnosis of CVD can be delayed, which usually leads to less successful prophylaxis and treatment

Borel subalgebras of gl(oo)

The paper is aself-contained statement of results and examples. Proofs willappear elsewhere