The vector space H k ( G ) of closed forms modulo exact forms is the cohomology group of dimension b k, the Betti number.
Theorem 5.9 Let X be a digital simplicial group of dimension 2 with 8-adjacency, then H q 8 ( X ) = { Z, q = 0, 0, q > 0. Proof.
Let G=U2m(2) be the unitary group of dimension 2m≥6 over the finite field of four elements GF(4), W=GF(4 2m the natural module of G. Then G acts transitively on the set Ω of maximal totally isotropic m-dimensional subspaces of W. This permutation representation over R contains an irreducible representation of dimension d=(4m+2)/3.
Furthermore, for (N=4) there are no proper actions of groups of dimensions (20, 21, 22).
Weights were assigned to the indicators describing a dimension, as well as to the four groups of dimensions introduced in the criteria tree to indicate their relative importance with respect to social vulnerability.
Usability defined in this study consists of the following two groups of dimensions: objective performance and subjective image/impression, which are considered equally important in designing and evaluating consumer electronic products.
Here the threefold groups of dimensions in Figure 2 are seen as independent variables, having a substantial influence on the REF competence (seen here as dependent variable) respectively explaining important variance parts of it.
We also note that Theorem 3.6 extends in the global setting a result of [28] (see p. 54 therein) on the non-existence of actions of groups of dimension (n(n+1)/2-1) for (nge 3).
While the above result includes only three domains, a complete description of general complex manifolds that admit transitive proper actions of groups of dimension (N^2) is probably much larger.
The second goal involved comparing trajectory groups on dimensions of school adjustment (social, academic, and emotional personal).
