Sums and differences of four k-th powers

We prove an upper bound for the number of representations of a positive integer $N$ as the sum of four $k$-th powers of integers of size at most $B$, using a new version of the Determinant method developed by Heath-Brown, along with recent results by Salberger on the density of integral points on affine surfaces. More generally we consider representations by any integral diagonal form. The upper bound has the form $O_{N}(B^{c/\sqrt{k}})$, whereas earlier versions of the Determinant method would produce an exponent for $B$ of order $k^{-1/3}$ in this case. Furthermore, we prove that the number of representations of a positive integer $N$ as a sum of four $k$-th powers of non-negative integers is at most $O_{\epsilon}(N^{1/k+2/k^{3/2}+\epsilon})$ for $k \geq 3$, improving upon bounds by Wisdom.Comment: 18 pages. Mistake corrected in the statement of Theorem 1.2. To appear in Monatsh. Mat

A generalization of the Bombieri-Pila determinant method

The so-called determinant method was developed by Bombieri and Pila in 1989 for counting integral points of bounded height on affine plane curves. In this paper we give a generalization of that method to varieties of higher dimension, yielding a proof of Heath-Brown's 'Theorem 14' by real-analytic considerations alone.Comment: 13 page

Grasping the changes seen in older adults when reaching for objects of varied texture.

Old age is associated with reduced mobility of the hand. To investigate age related decline when reaching-to-lift an object we used sophisticated kinematic apparatus to record reaches carried out by healthy older and younger participants. Three objects of different widths were placed at three different distances, with objects having either a high or low friction surface (i.e. rough or slippery). Older participants showed quantitative differences to their younger counterparts - movements were slower and peak speed did not scale with object distance. There were also qualitative differences with older adults showing a greater propensity to stop the hand and adjust finger position before lifting objects. The older participants particularly struggled to lift wide slippery objects, apparently due to an inability to manipulate their grasp to provide the level of precision necessary to functionally enclose the object. These data shed light on the nature of age related changes in reaching-to-grasp movements and establish a powerful technique for exploring how different product designs will impact on prehensile behavior