Volume 107, Issue 4
November 1997, pages 329-423
pp 329-361 November 1997
Positive values of non-homogeneous indefinite quadratic forms of type (1, 4)
Let Г_{r,n—r} denote the infimum of all number Г > 0 such that for any real indefinite quadratic form inn variables of type (r, n—r), determinantD ≠ 0 and real numbers c_{1}; c_{2},…, c_{n}, there exist integersx_{1},x_{2},…,x_{n} satisfying 0 < Q(x_{1}+c_{1},x_{2} + c_{2},…,x_{n} + c_{n}) ≤(Г¦Z > ¦)^{1/n}. All the values of Г_{r,n—r} are known except for г_{1,4}. Earlier it was shown that 8 ≤Г_{1,4} ≤16. Here we improve the upper bound to get Г_{1,4} < 12.
pp 363-376 November 1997
On asymptotic distribution on the a-adic integers
We show that the values of a polynomial with a-adic coefficients at integer and rational prime arguments are asymptotically distributed on the a-adic integers and that the integer parts of certain sequences known to be uniformly distributed modulo one, are uniformly distributed on the a-adic integers.
pp 377-385 November 1997
A new approximate functional equation for Hurwitz zeta function for rational parameter
For Hurwitz zeta function ζ(s, (a/k)) witha = 1,2,3,…,k, we obtain a new simple approximate functional equation (uniform ink andt) in critical strip. Our method should prove to be an alternative approach to Atkinson’s method in dealing with$$\sum\nolimits_{x(\bmod q)} {\int_0^T {|L(s,x)|^2 } } dt$$, whereL(s, x) is Dirichlet L-series moduloq and s = σ +it.
pp 387-389 November 1997
A weak Brun—Titchmarsh theorem for multiplicative functions
An important theorem of Shiu gives a (precise) bound for the average of values of multiplicative functions, of a certain class, over ‘short’ intervals. Here we obtain, by simple means, the above result of same qualitative order.
pp 391-403 November 1997
Abelian and Tauberian theorems for a new trigonometric method of summation
We first introduce a new trigonometric method of summation and then prove some Abelian and Tauberian theorems for this method.
pp 405-409 November 1997
Dirichlet problem for some hypoelliptic operators
In this paper, the Dirichlet problem for hypoelliptic operators verifying Hörmander condition and the maximum principle is considered.
pp 411-423 November 1997
Transformation of chaotic nonlinear polynomial difference systems through Newton iterations
Chaotic sequences generated by nonlinear difference systems or ‘maps’ where the defining nonlinearities are polynomials, have been examined from the point of view of the sequential points seeking zeroes of an unknown functionf following the rule of Newton iterations. Following such nonlinear transformation rule, alternative sequences have been constructed showing monotonie convergence. Evidently, these are maps of the original sequences. For second degree systems, another kind of possibly less chaotic sequences have been constructed by essentially the same method. Finally, it is shown that the original chaotic system can be decomposed into a fast monotonically convergent part and a principal oscillatory part showing sharp oscillations. The methods are exemplified by the well-known logistic map, delayed-logistic map and the Hénon map.
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