• Volume 107, Issue 4

November 1997,   pages  329-423

• Positive values of non-homogeneous indefinite quadratic forms of type (1, 4)

Let Гr,n—r denote the infimum of all number Г &gt; 0 such that for any real indefinite quadratic form inn variables of type (r, n—r), determinantD ≠ 0 and real numbers c1; c2,…, cn, there exist integersx1,x2,…,xn satisfying 0 &lt; Q(x1+c1,x2 + c2,…,xn + cn) ≤(Г¦Z &gt; ¦)1/n. All the values of Гr,n—r are known except for г1,4. Earlier it was shown that 8 ≤Г1,4 ≤16. Here we improve the upper bound to get Г1,4 &lt; 12.

• 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.

• 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.

• 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.

• 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.

• Dirichlet problem for some hypoelliptic operators

In this paper, the Dirichlet problem for hypoelliptic operators verifying Hörmander condition and the maximum principle is considered.

• 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.