By Zhao W., Liu P.
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Extra resources for A Biplurisubharmonic Characterization of AUMD Spaces
As mentioned in the introduction, the natural state space has the essential state properties. The natural state space is minimal as set. This is different than minimizing the dimension of the state vector. Also, a state vector in  and  is contained in an open set; however, a natural state may not be contained in an open set (although Remark 13 gives conditions when the natural state space is connected). No coordinate chart can be defined on the natural state space since its dimension is not specified.
As we have seen each of the lower integral types is zero; hence, the last integral type is zero. Since the kernels f and g are continuous, h D 0, so we have f . 1 ; : : : ; N / D g. 1 ; : : : ; N / 8 1 ; : : : ; N . N th Order Scalar Polynomial Case Following, we consider the N th order nonhomogeneous case. v0;1 /. / Z 1 f1 . /Œu0 C! v0;1 . D f0 C Z 1 C 0 C Z 0 1 0 /d f2 . 1 ; 2 /Œu0 C! v0;1 . Z C 1 Z 1 fN . 1 ; : : : ; 0 1 /Œu0 C! N /Œu0 C! 0 Œu0 C! v0;1 . N /d 1 d N v0;1 . v0;1 . 2 /d 1 d 2 1/ B Proof of Proposition 20 49 Â Ã Â ÃZ 1 Â ÃZ 0 1 1 f0 C D f1 .
We provide a definition of a polynomial integral operator. Definition 19 (). By a time-invariant N th degree causal, polynomial integral operator with M -dimensional input space u D Œu1 ; ; uM 2 U with norm kuks;t D maxi kui ks;t and kui ks;t D sups< Ät jui . s;t b/< Ät jyi . t 1 0 1/ M X fpi1 ; ;in . t n /d 1 ; ;d n: The above sum is taken over all combinations without repeating; hence, there are M n terms. Such an operator is unchanged if the kernels fpi1 ; ;in , all p and n are symmetrized.