N. G. de Bruijn, Asymptotic Methods in Analysis, North-Holland Publishing Co., Amsterdam, 1958. Home/Search Document Not in Database Summary Related Articles Check |
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Deformations of Coxeter Hyperplane Arrangements - Postnikov, Stanley (2000) (14 citations) (Correct)
....Thus, for large n, approximately 7:75425 of the regions of the Linial arrangement Ln 1 are bounded. Note that by (9.9) the portion of the bounded regions in the Shi arrangement Sn 1 is equal to (n 1) n 1 and tends to e 0:1353353. In the proof of Theorem 11.1 we use methods described in [9]. The general outline of the proof is the following: a) use the Stirling formula for the function to approximate the summands in (11.1) b) approximate the summation by integration; c) use the Laplace method to approximate the integral. The Laplace method amounts to the following statement; see ....
N. G. de Bruijn, Asymptotic Methods in Analysis, North-Holland Publishing Co., Amsterdam, 1958.
On The Taylor Coefficients Of The Composition Of Two Analytic.. - Hilberdink (1996) (Correct)
....are bounded away from perfect squares and hence l k Gamma2 l k Gamma3 l k from which it will be seen that l n Gamma2 l n Gamma3 as required. Note that the converse holds trivially, so that we have an equivalence. Slowly varying functions have been studied in great detail (see Bingham et al. [3] or Feller [5] and it is well known that they can be represented as follows: If s(x) is slowly varying, then s(x) c(x) exp aeZ x (t) t dt oe where c(x) c 0 and (x) 0 as x 1 . Using this, we easily obtain Lemma 2.2. If l n [ p n ] l n as n 1 for all real , then l n = c n exp ....
de Bruijn, N.G.: Asymptotic Methods in Analysis. - North-Holland Publishing Co., 1970 (3rd edition).
Empirical Risk Approximation: An Induction Principle for.. - Buhmann (1998) (4 citations) (Correct)
....axis through the stationarity point. The stationarity point is located on the real axis which renders the temperature parameter 1=x a real variable at the saddle point. Approximately, the cardinality is given by jH ffl j 1 x sp exp ( nS(x sp ; y sp ; p sp ) O ( n) 40) See [Bruijn, 1981] for details of the saddle point stationary phase method) To capture the O ( n) terms correctly, we have to calculate the Hessian of S at the stationary J. M. Buhmann: Empirical Risk Approximation; Technical Report IAI TR98 3 17 point. The second derivatives are given by the following ....
Bruijn, N. G. D. (1958, (1981)). Asymptotic Methods in Analysis. North-Holland Publishing Co., (repr. Dover), Amsterdam.
Empirical Risk Approximation: An Induction Principle for.. - Buhmann (1998) (4 citations) (Correct)
.... cardinality of the function ball centered at the minimal loss function h (x) i.e. jHR j jHj jB(h (x) j : 8) The cardinality fi fi B(h (x) fi fi of a function ball with radius R can be approximated by adopting techniques from statistical physics and asymptotic analysis [Bruijn, 1981]: fi fi fiB(h (x) fi fi fi = X ff2 Theta R Gamma Z Omega jh(x; ff) Gamma h (x)j d (x) 1 Z Gamma 1 dx 2 x exp (nx(R Gamma F) 9) F : Gamma 1 nx log X ff2 exp Gamma xn Z Omega jh(x; ff) Gamma h (x)j d (x) 10) J. M. ....
Bruijn, N. G. D. (1958, (1981)). Asymptotic Methods in Analysis. North-Holland Publishing Co., (repr. Dover), Amsterdam.
On Explicit Bounds For The Solutions Of A Class Of Parametrized.. - Heuberger (Correct)
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N. G. de Bruijn, Asymptotic methods in analysis, North-Holland Publishing Co., Amsterdam, 1958, Bibliotheca Mathematica. Vol. 4.
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