198 research outputs found
A System for Unsteady Pressure Measurements Revisited
An overview is presented of some recent developments in the field of the design of effective sound absorbers. The first part deals with the application of socalled coupled tubes. For this purpose use is made of a system originally applied for unsteady pressure measurements on oscillating wind tunnel models. The second part deals with an extension of the theory of tubing systems to thin air layers, trapped between flexible walls
Multivariate Diophantine equations with many solutions
We show that for each n-tuple of positive rational integers (a_1,..,a_n)
there are sets of primes S of arbitrarily large cardinality s such that the
solutions of the equation a_1x_1+...+a_nx_n=1 with the x_i all S-units are not
contained in fewer than exp((4+o(1))s^{1/2}(log s)^{-1/2}) proper linear
subspaces of C^n. This generalizes a result of Erdos, Stewart and Tijdeman for
m=2 [Compositio 36 (1988), 37-56].
Furthermore we prove that for any algebraic number field K of degree n, any
integer m with 1<=m<n, and any sufficiently large s there are integers
b_0,...,b_m in a number field which are linearly independent over the
rationals, and prime numbers p_1,...,p_s, such that the norm polynomial
equation |N_{K/Q}(b_0+b_1x_1+...+b_mx_m)|=p_1^{z_1}...p_s^{z_s} has at least
exp{(1+o(1)){n/m}s^{m/n}(log s)^{-1+m/n}) solutions in integers
x_1,..,x_m,z_1,..,z_s. This generalizes a result of Moree and Stewart [Indag.
Math. 1 (1990), 465-472].
Our main tool, also established in this paper, is an effective lower bound
for the number of ideals in a number field K of norm <=X composed of prime
ideals which lie outside a given finite set of prime ideals T and which have
norm <=Y. This generalizes a result of Canfield, Erdos and Pomerance [J. Number
Th. 17 (1983), 1-28], and of Moree and Stewart (see above).Comment: 29 page
On conjectures and problems of Ruzsa concerning difference graphs of S-units
Given a finite nonempty set of primes S, we build a graph with
vertex set by connecting x and y if the prime divisors of both the
numerator and denominator of x-y are from S. In this paper we resolve two
conjectures posed by Ruzsa concerning the possible sizes of induced
nondegenerate cycles of , and also a problem of Ruzsa concerning
the existence of subgraphs of which are not induced subgraphs.Comment: 15 page
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