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A143148
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Decimal expansion of lower bound using Shannon entropy arising in randomly-projected hypercubes.
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2
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5, 1, 0, 6, 4, 6, 1, 1, 8, 9, 1, 3, 8, 3, 3, 8, 2, 7, 7, 6, 3, 1, 3, 0, 9, 5, 6, 7, 1, 6, 0, 0, 8, 7, 9, 1, 6, 4, 9, 0, 9, 9, 0, 4, 7, 8, 9, 1, 1, 1, 6, 3, 6, 1, 8, 1, 2, 3, 8, 5, 0, 6, 1, 9, 7, 8, 4, 4, 2, 1, 4, 5, 1, 5, 1, 1, 0, 6, 3, 5, 3, 2, 8, 1, 6, 0, 2, 9, 8, 7, 4, 0, 1, 3, 5, 2, 8, 7, 8, 6
(list; constant; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| Formula in Donoho, p.10. Upper bound in A143149.
Abstract: Let A be an n by N real valued random matrix and h denote the N-dimensional hypercube. For numerous random matrix ensembles, the expected number of k-dimensional faces of the random n-dimensional zonotope A\h obeys the formula E f_k(A\h) /f_k(\h) = 1-P_{N-n,N-k}, where P_{N-n,N-k} is a fair-coin-tossing probability.
The formula applies, for example, where the columns of A are drawn i.i.d. from an absolutely continuous symmetric distribution.
The formula exploits Wendel's Theorem. Let po denote the positive orthant; the expected number of k-faces of the random coneA\po obeys {E} f_k(A\po) /f_k(\po) = 1 - P_{N-n,N-k}.
The formula applies to numerous matrix ensembles, including those with iid random columns from an absolutely continuous, centrally symmetric distribution.
There is an asymptotically sharp threshold in the behavior of face counts of the projected hypercube; thresholds known for projecting the simplex and the cross-polytope, occur at very different locations.
We briefly consider face counts of the projected orthant when A does not have mean zero; these do behave similarly to those for the projected simplex. We consider non-random projectors of the orthant; the 'best possible' A is the one associated with the first n rows of the Fourier matrix.
These geometric face-counting results have implications for signal processing, information theory, inverse problems and optimization. Most of these flow in some way from the fact that face counting is related to conditions for uniqueness of solutions of underdetermined systems of linear equations.
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LINKS
| David L. Donoho and Jared Tanner, Counting the Faces of Randomly-Projected Hypercubes and Orthants, with Applications
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FORMULA
| (16/25)*(2/pi)^(1/2).
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EXAMPLE
| 0.5106461189138...
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CROSSREFS
| Cf. A143149.
Sequence in context: A019904 A002068 A021666 * A198541 A200420 A176324
Adjacent sequences: A143145 A143146 A143147 * A143149 A143150 A143151
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KEYWORD
| cons,easy,nonn
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AUTHOR
| Jonathan Vos Post (jvospost3(AT)gmail.com), Jul 27 2008
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EXTENSIONS
| Removed leading zero, adjusted offset- R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 05 2009
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