Search results for "Diophantine equation"
showing 7 items of 7 documents
On a linear diophantine problem of Frobenius
1993
Abstract In this paper, linear diophantine problem of Frobenius is discussed. A theorem concerning the largest integer g m (a1,a2) and the smallest integer G m (a1,a2) with m different representations with a1,a2 as basis is proved.
The linear diophantine problem of Frobenius for subsets of arithmetic sequences
1997
Let A k = {a 1,. . . , a k } $ \subset \Bbb N $ with gcd (a 1,. . . , a k ) = 1. We shall say that a natural number n has a representation by a 1,. . . , a k if $ n =\sum \limits_{i=1}^{k}a_ix_i,\; x_i\in \Bbb N_0 $ . Let g = g (A k ) be the largest integer with no such representation. We then study the set A k = {a,ha + d,ha + 2d,..., ha + (k - 1) d} h,d > 0, gcd (a,d) = 1). If l k denotes the greatest number of elements which can be omitted without altering g (A k ), we show that ¶¶ $ 1-{4 \over \sqrt k} \le {l_k\over k} \le 1 - {3\over k}, $ ¶¶ provided a > k, or a = k with $ d \ge 2 h \sqrt {k} $ . The lower bound can be improved to 1 - 4 / k if we choose a > (k - 4) k + 3. Moreover, we…
Appendix: Diophantine Approximation on Hyperbolic Surfaces
2002
In this (independent) appendix, we study the Diophantine approximation properties for the particular case of the cusped hyperbolic surfaces, in the spirit of Sect. 2 (or [11]), and the many still open questions that arise for them. We refer to [9], [10]for fundamental results and further developments. We study in particular the distance to a cusp of closed geodesics on a hyperbolic surface.
Linear Diophantine Problems
1996
The Frobenius number g(A k ) Let A k \({A_k} = \{ {a_1},...,{a_k}\}\subset\) IN with gcd(A k ) = 1, n\( \in I{N_0}.\) If $$n = \sum\limits_{i = 1}^k {{x_i}{a_i},{x_i}}\in I{N_0}$$ (1) we call this a representation or a g-representation of n by Ak (in order to distinguish between several types of representations that will be considered in the sequel). Then the Frobenius number g(A k ) is the greatest integer with no g-representation.
On a Linear Diophantine Problem of Frobenius: Extending the Basis
1998
LetXk={a1, a2, …, ak},k>1, be a subset of N such that gcd(Xk)=1. We shall say that a natural numbernisdependent(onXk) if there are nonnegative integersxisuch thatnhas a representationn=∑ki=1 xiai, elseindependent. The Frobenius numberg(Xk) ofXkis the greatest integer withnosuch representation. Selmer has raised the problem of extendingXkwithout changing the value ofg. He showed that under certain conditions it is possible to add an elementc=a+kdto the arithmetic sequencea,a+d,a+2d, …, a+(k−1) d, gcd(a, d)=1, without alteringg. In this paper, we give the setCof all independent numberscsatisfyingg(A, c)=g(A), whereAcontains the elements of the arithmetic sequence. Moreover, ifa>kthen we give …
Pietro Mengoli and the six-square problem
1994
The aim of this paper is to analyze a little known aspect of Pietro Mengoli's (1625-1686) mathematical activity: the difficulties he faced in trying to solve some problems in Diophantine analysis suggested by J. Ozanam. Mengoli's recently published correspondence reveals how he cherished his prestige as a scholar. At the same time, however, it also shows that his insufficient familiarity with algebraic methods prevented him, as well as other Italian mathematicians of his time, from solving the so-called “French” problems. Quite different was the approach used for the same problems by Leibniz, who, although likewise partially unsuccessful, demonstrated a deeper mathematical insight which led…
Quantum Nekhoroshev Theorem for Quasi-Periodic Floquet Hamiltonians
1998
A quantum version of Nekhoroshev estimates for Floquet Hamiltonians associated to quasi-periodic time dependent perturbations is developped. If the unperturbed energy operator has a discrete spectrum and under finite Diophantine conditions, an effective Floquet Hamiltonian with pure point spectrum is constructed. For analytic perturbations, the effective time evolution remains close to the original Floquet evolution up to exponentially long times. We also treat the case of differentiable perturbations.