# separation theorem

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## separation theorem

[‚sep·ə′rā·shən ‚thir·əm]
(control systems)
A theorem in optimal control theory which states that the solution to the linear quadratic Gaussian problem separates into the optimal deterministic controller (that is, the optimal controller for the corresponding problem without noise) in which the state used is obtained as the output of an optimal state estimator.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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(b) Let A;B[member of] WCC (X) with A [not equal to] B: We may assume without loss of generality that there exists some a [member of] A such that a [member of] B: Since B is a [T.sub.w]-compact subset of the locally convex topological vector space (X;[T.sub.w]) ; it follows from the Hahn-Banach Separation Theorem that there exists some x* [member of] X* such that sup Rex* (b) < Rex* (a): Let [delta] = Rex* (a) [mathematical expression not reproducible] (b) > 0: We have Rex* (a) Rex* (b) [greater than or equal to] [delta] for all b [member of] B which in turn implies that
In fact, if z [not member of] C, then there exists [y.sup.*] [member of] [E.sup.*] by the separation theorem  such that <z, [y.sup.*]> < [inf.sub.y [member of] C] <y,[y.sup.*]>.
Hence by the separation theorem of convex sets, there exists ([phi], [OMEGA], [xi]) [member of] ([R.sup.m] x [S.sup.p] x [R.sup.q])\ {0, 0,0}, such that
On the other hand, as far as [mathematical expression not reproducible] in Theorem 1 is concerned, the conclusion is also directly derived from Guo et al.'s separation theorem between a point and a [T.sub.c]-closed [L.sup.0]-convex subset in .
Then by the Hahn-Banach Separation Theorem, there is [x.sup.*] [member of] [X.sup.*], [gamma] [member of] R, and [epsilon] > 0, such that
For this we need a "separation theorem" for a finite set of points inside a [C.sup.1]-convex body.
By the classical separation theorem, there exists a nonzero functional [z.sup.*] [member of] [Z.sup.*], such that [z.sup.*] x [greater than or equal to] [z.sup.*] y for all x [member of] K and y [member of] W.
By the Sturm type separation theorem, one solution of (6.1) is (non)oscillatory iff every solution of (6.1) is (non)oscillatory.
The separation theorem says, for instance, that there is a vector orthogonal to the point b' on the arm such that for any h in H, ph' [less than or equal to] ph, and for any b in B, pb' [greater than or equal to] pb.
This paper extends the Fisher Separation Theorem of finance and microeconomic theory to include the Keynesian model of macroeconomics.
The Fisher separation theorem of finance theory is an application of pure microeconomics and has appeared in standard general finance textbooks for years [Brealy, Myers, Sick, and Whaley, 1986].
These papers derive two major theorems: One is the "separation theorem" which states that, when futures markets exist, the firm's export production decision is determined solely by technology and input-output prices, including the futures prices.

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