A Direct Attack on a Birthday Problem

1976 ◽  
Vol 49 (3) ◽  
pp. 130-131
Author(s):  
Samuel Goldberg
1976 ◽  
Vol 49 (3) ◽  
pp. 130 ◽  
Author(s):  
Samuel Goldberg

1985 ◽  
Vol 50 (4) ◽  
pp. 845-853 ◽  
Author(s):  
Miloslav Šorm ◽  
Miloslav Procházka ◽  
Jaroslav Kálal

The course of hydrolysis of an ester, 4-acetoxy-3-nitrobenzoic acid catalyzed with poly(1-methyl-3-allylimidazolium bromide) (IIa), poly[l-methyl-3-(2-propinyl)imidazolium chloride] (IIb) and poly[l-methyl-3-(2-methacryloyloxyethyl)imidazolium bromide] (IIc) in a 28.5% aqueous ethanol was investigated as a function of pH and compared with low-molecular weight models, viz., l-methyl-3-alkylimidazolium bromides (the alkyl group being methyl, propyl, and hexyl, resp). Polymers IIb, IIc possessed a higher activity at pH above 9, while the models were more active at a lower pH with a maximum at pH 7.67. The catalytic activity at the higher pH is attributed to an attack by the OH- group, while at the lower pH it is assigned to a direct attack of water on the substrate. The rate of hydrolysis of 4-acetoxy-3-nitrobenzoic acid is proportional to the catalyst concentration [IIc] and proceeds as a first-order reaction. The hydrolysis depends on the composition of the solvent and was highest at 28.5% (vol.) of ethanol in water. The hydrolysis of a neutral ester, 4-nitrophenyl acetate, was not accelerated by IIc.


1997 ◽  
Vol 34 (01) ◽  
pp. 192-207 ◽  
Author(s):  
Anyue Chen ◽  
Eric Renshaw

An M/M/1 queue is subject to mass exodus at rate β and mass immigration at rate when idle. A general resolvent approach is used to derive occupation probabilities and high-order moments. This powerful technique is not only considerably easier to apply than a standard direct attack on the forward p.g.f. equation, but it also implicitly yields necessary and sufficient conditions for recurrence, positive recurrence and transience.


2003 ◽  
Vol 2003 (60) ◽  
pp. 3827-3840 ◽  
Author(s):  
P. N. Rathie ◽  
P. Zörnig

We study the birthday problem and some possible extensions. We discuss the unimodality of the corresponding exact probability distribution and express the moments and generating functions by means of confluent hypergeometric functionsU(−;−;−)which are computable using the software Mathematica. The distribution is generalized in two possible directions, one of them consists in considering a random graph with a single attracting center. Possible applications are also indicated.


1976 ◽  
Vol 7 (4) ◽  
pp. 39 ◽  
Author(s):  
Joe Dan Austin
Keyword(s):  

2002 ◽  
Vol 34 (03) ◽  
pp. 609-625 ◽  
Author(s):  
N. Papadatos ◽  
V. Papathanasiou

The random variablesX1,X2, …,Xnare said to be totally negatively dependent (TND) if and only if the random variablesXiand ∑j≠iXjare negatively quadrant dependent for alli. Our main result provides, for TND 0-1 indicatorsX1,x2, …,Xnwith P[Xi= 1] =pi= 1 - P[Xi= 0], an upper bound for the total variation distance between ∑ni=1Xiand a Poisson random variable with mean λ ≥ ∑ni=1pi. An application to a generalized birthday problem is considered and, moreover, some related results concerning the existence of monotone couplings are discussed.


2008 ◽  
Vol 7 (2) ◽  
pp. 111-122 ◽  
Author(s):  
Michael P. McDonald ◽  
Justin Levitt
Keyword(s):  

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