33,517 research outputs found
On the Index and the Order of Quasi-regular Implicit Systems of Differential Equations
This paper is mainly devoted to the study of the differentiation index and
the order for quasi-regular implicit ordinary differential algebraic equation
(DAE) systems. We give an algebraic definition of the differentiation index and
prove a Jacobi-type upper bound for the sum of the order and the
differentiation index. Our techniques also enable us to obtain an alternative
proof of a combinatorial bound proposed by Jacobi for the order.
As a consequence of our approach we deduce an upper bound for the
Hilbert-Kolchin regularity and an effective ideal membership test for
quasi-regular implicit systems. Finally, we prove a theorem of existence and
uniqueness of solutions for implicit differential systems
On pairs of prime geodesics with fixed homology difference
We exhibit the analogy between prime geodesics on hyperbolic Riemann surfaces
and ordinary primes. We present new asymptotic counting results concerning
pairs of prime geodesics whose homology difference is fixed.Comment: 19 pages, Corrected typos, corrected MSC-clas
On finite groups whose Sylow subgroups have a bounded number of generators
Let G be a finite non-nilpotent group such that every Sylow subgroup of G is
generated by at most d elements, and such that p is the largest prime dividing
|G|. We show that G has a non-nilpotent image G/N, such that N is
characteristic and of index bounded by a function of d and p. This result will
be used to prove that the index of the Frattini subgroup of G is bounded in
terms of d and p. Upper bounds will be given explicitly for soluble groups.Comment: 7 page
Algorithms in algebraic number theory
In this paper we discuss the basic problems of algorithmic algebraic number
theory. The emphasis is on aspects that are of interest from a purely
mathematical point of view, and practical issues are largely disregarded. We
describe what has been done and, more importantly, what remains to be done in
the area. We hope to show that the study of algorithms not only increases our
understanding of algebraic number fields but also stimulates our curiosity
about them. The discussion is concentrated of three topics: the determination
of Galois groups, the determination of the ring of integers of an algebraic
number field, and the computation of the group of units and the class group of
that ring of integers.Comment: 34 page
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