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New PDF release: A new approach to linear filtering and prediction problems

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L. ˜ The Q ˜ − -orbit Q ˜ − P˜ in G/P˜ is open in its closure, Proof (i) Write P˜ − = Q ˜ − P˜ is open dense in G. and by dimension considerations, the closure is G. Hence Q We give the proof for x = e; the unipotent case is essentially the same. Suppose − ˜ − and p ∈ P˜ . Then eq− = ep−1 . But eq− ∈ e + L(Q ˜ − ), ep−1 ∈ eq p = e with q − ∈ Q 34 2. PRELIMINARIES ˜ − )∩L(P˜ ) = 0. It follows that C ˜ − ˜ (e) = C ˜ − (e)C ˜ (e). A similar e+L(P˜ ), and L(Q Q P Q P argument shows that CP˜ (e) = CQ˜ (e)CL˜ (e), so that CQ˜ − P˜ (e) = CQ˜ − (e)CQ˜ (e)CL˜ (e) and the dimension of CQ˜ − P˜ (e) is the sum of the dimensions of the individual centralizers.

Part (ii) is entirely similar, noting that eQ ⊆ e + L(Q)(≥2) . 7. Centralizers of nilpotent elements Continue to assume that G is a simple algebraic group over the algebraically closed field K. In this section we establish a key result (and some refinements of it) on the structure of the centralizers of nilpotent elements in L(G). The result holds in arbitrary characteristic. The section concludes with a result concerning the dimensions of centralizers of nilpotent elements. 25. Let L ¯ . Suppose there exists e be a distinguished nilpotent element of the Lie algebra of L ¯ , such that eT (c) = c2 e for all a 1-dimensional torus T = {T (c) : c ∈ K ∗ } in L c ∈ K ∗ .

Therefore, dim CP (l) = dim P − dim Q = dim L(P ) − dim L(Q). 4 of [26] yields dim CP (l) = dim CL(P ) (l). This implies that [L(P ), l] = L(Q). Now we use an argument of Jantzen to show that all weights of h are even. Write l = i>0 li for li ∈ L(Q)i . Intersect both sides of the equality [L(P ), l] = L(Q) of the last paragraph with L(P )1 + L(P )2 . 3) [L(P )0 , l1 + l2 ] + [L(P )1 , l1 ] = L(P )1 + L(P )2 . Suppose L(P )1 = 0. Then as l1 ∈ L(P )1 we have dim([L(P )1 , l1 ]) < dim(L(P )1 ). 37.

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A new approach to linear filtering and prediction problems by Kallenrode


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