Second, when G is coamenable, complete positive definiteness may be replaced with the weaker notion of positive definiteness, which models the classical notion.
And in the Metaphysics: "The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree" (Aristotle, volume 2 1705 [1078a36]).
Doing so would threaten the clarity and definiteness which the notion of identity brings to theory; local gains from postulating propositions are not worth this global loss.
Such change can become very large when the noise destroys the Hessian matrix's positive definiteness, which implies that a global optimal solution (like the one for the log-likelihood function of a traditional logistic regression model) may not be attained.
Proposed are generalizations and refinements of a well-known result on robust matrix sign-definiteness, which is extensively exploited in quadratic stability, design of robust quadratically stabilizing controllers, robust LQR-problem, etc.
This system, together with inequalities which guarantee positive definiteness of the matrix M, results in a one parameter family of inner products which give exact gradients for linear functions.
(E^{-1}AE^{-1}) being positive definite implies that (B(E^{-1}AE^{-1})B^{T}) is positive semi-definite, from which one derives (H+tau B(E^{-1}AE^{-1})B^{T}) to be positive definite for (forall tau>0) by combining with the positive definiteness of H.
In summary, this study has identified referentiality as linked to grammar and particularly definiteness, along with specific measures of grammatical complexity, as linguistic markers that distinguish FTD (but not SZ at large) from non-clinical controls.
Therefore, the absolute stability investigation problem is reduced to the task of checking of positive definiteness for two matrices, one of which is 2n-dimensional and the other is (n+1 -dimensional.
