This is the final part of the paper on the dimensions of Jordan blocks in the images of regular unipotent elements from subsystem subgroups of type (C_2 ) in (p)-restricted irreducible representations of groups of type (C_n) in characteristic (pgeq 11 ) with locally small highest weights. Here the case where (n>3 ) and the restriction of a representation considered to a canonical subgroup of type (A_1) containing such element has a weight not less than (p), is investigated.
We find conditions such that every (lambda )-homogeneous model with small ( lambda ) satisfying these conditions is homogeneous. As a corollary, we obtain conditions guaranteeing that the following implication holds: If (T ) is a theory, (mu >|T| ), and every model of (T ) of cardinality ( mu ) is (omega _1) -homogeneous then every model of (T) of sufficiently large cardinality is homogeneous.
We study links between generative classes and their generative restrictions with respect to semantic and syntactic properties of corresponding generic structures. Generations and specificity of generative classes are investigated. Spectra for generative classes with respect to generic structures for subclasses and their theories are introduced. Values for these spectra are described for generative classes with complete diagrams and in general cases for finitely, countably and uncountably generated generative classes.
We solve the problem of finding all ((n+2))-dimensional geometries defined by a nondegenerate analytic function
$$ varphi (varepsilon _1x^1_Ax^1_B+ cdots +varepsilon _{n+1}x^{n+1}_Ax^{n+1}_B,w_A,w_B),$$which is an invariant of a motion group of dimension ((n+1)(n+2)/2). As a result, we have two solutions: the expected scalar product (varepsilon _1x^1_Ax^1_B+ cdots +varepsilon _{n+1}x^{n+1}_Ax^{n+1}_B+varepsilon w_Aw_B ) and the unexpected scalar product (varepsilon _1x^1_Ax^1_B+ cdots +varepsilon _{n+1}x^{n+1}_Ax^{n+1}_B+w_A+w_B ). The solution of the problem is reduced to the analytic solution of a functional equation of a special kind.
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