Math seminar, 26 April

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From: math at ictp.it

M A T H E M A T I C S   S E M I N A R S   2006



Series of Lectures in "Experimental Discoveries of Mathematical Facts"


Wednesday, 26 April, at 14.30 hrs.


Professor V.I. Arnold
(Steklov Institute, Moscow, Russia)

Frobenius numbers, geometry and statistics
of additive semigroups of integers.
(Lecture 3)

Abstract:

The Frobenius number N(a_1, ..., a_n), where the a_i are natural
numbers (with no common divisor greater than 1) is the minimal integer,
such that itself and all greater integers are representable as linear
combinations x_1 a_1 + ... + x_n a_n with nonnegative integral
coefficients  x_i . For instance, N(a,b)=(a-1)(b-1). But for n > 2 there
is no  explicit formula for N, and even its growth rate for growing
a=(a_1, ..., a_n) is unknown. The talk proves that it grows
at least as sigma^(1+(1/n-1)) and
at most  like (sigma)^2, where  sigma = a_1 + ... + a_n.

Both boundary cases are attained for some directions of the vector a, 
but
the growth rate depends peculiarly on this direction.  The average 
growth
rate has been studied experimentally and the talk will present the
empirical mean values ( for sigma = 7, 19, 41, 97 and 199). The observed
rate is (sigma)^P with  p ~ 2  at the beginning, declining to  p ~ 1,6 
for sigma between 100 and 200.  This confirms the  author's conjecture 
of 1999 that p tends to  1+1/(n-1) = 3/2  for large sigma.



VENUE:  SEMINAR ROOM
(Main Building, first floor)



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