Maximal rank

This is possible since P has maximal rank p. Now all roots of M are roots of A. Assume A to be nonsingular, and suppose that if A has any multiple roots occurring among the roots of M, then it has the same multiplicity as a root of M. Then it can be shown that... 

The generic rank of a structured matrix B is defined to be the maximal rank that B achieves as a function of its free parameters. 

The maximal rank of an (m x g) matrix having no specified structure is equal to min (m, g). The inclusion of the structure into the problem makes it possible for matrices to have less than full rank, independent of the values of the free parameters, as was shown by Schields and Pearson (1976). Therefore, a structured matrix B has full generic rank if, and only if, there exists an admissible matrix B with full rank. 

A closed subscheme X c Pr is of maximal rank in degree k if the restriction map... 

The a. C.M. schemes of positive dimension are of maximal rank because ali the maps are surjective. If X is of maximal rank then in particular n satisfies the hypothesis of theorem (9.1) because in degrees dv. .., d5 the restriction maps are not injective. 

The union of two disjoint lines in P3 is an example of a scheme of maximal rank which is not a. C M. 

Every X c Pr is obviously of maximal rank in every degree i 0, moreover it follows from theorem (1.10) that there is a k0, depending only on the Hilbert polynomial p(t) of X, such that X is of maximal rank in every degree k>k0 take k0 such that h1(Pr(tx(IO) 0 for all kik0. 

From the semicontinutty theorem it follows that condition (c) is open in HmrP(t)l in other words the locus of points that parametrize schemes of. maximal rank in degree k is an open ( and possibly empty ) subset Uk of HilbW Since any X with Hilbert polynomial p(t) is possibly not of maximal rank only in degrees 1 i k i k0, it follows that the locus of points of H1lbrpft) which parametrize schemes of maximal rank is open, being equal to u, n... n ... 

Maximal rank, scheme of, 9-9 (see also "curve of maximal rank") Monodromy, 8-8 Morphism closed, 5-4... 

Regular (quadratic) matrix means that its determinant is non-zero. Assertions in conditions 2,3 about ranks n —h (number of independent chemical reactions, see Sect. 4.2) foUow with the use of Lemma product of quadratic regular matrix with rectangultir matrix of maximal rank has also this maximal rank (this follows from Sylvester s inequeilities for rtmk of matrix product, see [134, 13.2.7]). 

From Table 1, it can be clearly seen that 01 is the preferred object (highest row sum) while 04 is the worst performing one (lowest row sum) 02 and 03 have the same row sums, which are derived from different variables. Hence, these two objects cannot be compared, and are therefore alternatives. The partial preorder of objects 1-4 may be represented as Ol, (02/03), 04 ( maximized ranking). If the criterion was reversed, i.e., the lowest row sum was preferred ( minimized ranking ), then the rank order would also be reversed, 04, (02/03), Ol. 

Note that if on an even-dimensional complex manifold Af there exists a holomorphic (or, more generally, meromorphic) 2-form, which has the maximal rank at the general point, then its restriction to an appropriate subset U G M open in Zariski topology sets on (7 is a symplecftic structure, and thus U is a symplectic manifold. 

This linear equation can be solved by standard methods, if G has maximal rank. Otherwise, the generalized constraint forces are no longer uniquely defined. 

If there are some regroupings, the minimum set of parameters becomes XRi. Its dimension is c. XRi is obtained by selecting the principal parameters which fit with the minor of maximal rank extracted from... 

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