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1
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0001805648
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Scaling, universality and renormalization group theory
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Springer, Berlin
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e.g. M.E. Fisher, Scaling, Universality and Renormalization Group Theory, Lecture Notes in Physics, vol. 186, Springer, Berlin, 1986
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(1986)
Lecture Notes in Physics
, vol.186
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Fisher, M.E.1
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6
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19944400206
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note
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This point of view including the idea of pursuit of stability was presented in the Cherry Bud Workshop at Yokohama held on March 2004; the paper is an outgrowth of the workshop presentation.
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9
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32144435012
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N.S. Namachchivaya, Y.K. Lin (Eds.) Kluwer Academic Publishers
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See, also L. Arnold, in: N.S. Namachchivaya, Y.K. Lin (Eds.), IUTAM on Nonlinear Stochastic Dynamics, Kluwer Academic Publishers, 2003, 5.
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(2003)
IUTAM on Nonlinear Stochastic Dynamics
, pp. 5
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Arnold, L.1
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12
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19944420227
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note
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This is only formal. 〈 f 〉 A may not even be continuous, and the reduced equation may not have any solution in some cases. The meaning of the approximation of the true solution x (t) in terms of A (t) is also quite delicate. Mathematically, we expect the modulus of the discrepancy integrated over [ 0, t / ε ] converges to zero as ε → 0
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13
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19944392036
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note
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The interpretation of this stochastic equation is delicate. Our interpretation (conjecture) is, under the condition that the averaged equation (2.7) is well behaved, that ∫ d t (x ̇ - 〈 f 〉) 2 / 2 ε b (A) is the rate function for the large deviation of the true solution from, A (t). That is, we interpret the Langevin equation (2.11) as a shorthand notation of the large deviation, principle
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16
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0011229674
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Wiley, New York
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W. Feller, An Introduction to Probability Theory and Its Applications, vol. II, Wiley, New York, 1971, pp. 21-23.
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(1971)
An Introduction to Probability Theory and Its Applications
, vol.2
, pp. 21-23
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Feller, W.1
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17
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0001813888
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An invariance principle for certain probability limit theorems
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M.D. Donsker An invariance principle for certain probability limit theorems Mem. Am. Math. Soc. 6 1951
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(1951)
Mem. Am. Math. Soc.
, Issue.6
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Donsker, M.D.1
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21
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19944379907
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note
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This statement is equivalent to asserting that probability theory is used to describe the possible world consistent with the given empirical data, because objective probability must be consistent with empirical frequencies. One might say that Bayesian or subjective interpretation of probability is possible and free from frequencies, so the above assertion is incomplete.
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22
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19944368517
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note
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Here, the reader may wonder why we use only the inequality relations instead of more quantitative concepts such as distances to describe dissimilarities. It is our general belief that converting the dissimilarity information in terms of qualitative inequalities makes the analysis method much more robust against corrupted and/or missing data. Thus, our proposal is: converting all the quantitative data into a set of qualitative relations that could recover the quantitative relations if the original data is quantitative must be a general strategy in data analysis.
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23
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19944370211
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note
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We know that the requirement is enough to recover a geometrical object if the number of points in the object is not too small (e.g., not less than 20).
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27
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85039094755
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Advance Access published on February 22 doi:10.1093/bioinformatics/ bti067
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Y.-h. Taguchi, Y. Oono, Bioinformatics, Advance Access published on February 22, 2005; doi:10.1093/bioinformatics/ bti067.
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(2005)
Bioinformatics
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Taguchi, Y.-H.1
Oono, Y.2
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