Elliptic curves and positive definite ternary forms

For two given ternary quadratic forms f( x, y, z) and g( x, y, z), let r( f, n) and r( g,n) be the numbers of representations of n represented by f( x, y, z) and g( x, y, z) respectively. In this paper we study the following problem: when will we have r( f, n) = r( g, n) or r( f, n) ≠ r( g, n).Our method is to use elliptic curves and the corresponding new forms.

Microstructure evolution of laser solid forming of Ti-Al-V ternary system alloys from blended elemental powders

Morphology evolution of prior β grains of laser solid forming （LSF） Ti-xAl-yV （x 11,y 20） alloys from blended elemental powders is investigated. The formation mechanism of grain morphology is revealed by incorporating columnar to equiaxed transition （CET） mechanism during solidification. The morphology of prior β grains of LSF Ti-6Al-yV changes from columnar to equiaxed grains with increasing element V content from 4 to 20 wt.-%. This agrees well with CET theoretical prediction. Likewise, the grain morphology of LSF Ti-xAl-2V from blended elemental powders changes from large columnar to small equiaxed with increasing Al content from 2 to 11 wt.-%. The macro-morphologies of LSF Ti-8Al-2V and Ti-11Al-2V from blended elemental powders do not agree with CET predictions. This is caused by the increased disturbance effects of mixing enthalpy with increasing Al content, generated in the alloying process of Ti, Al, and V in the molten pool.

On universal sums of polygonal numbers

For m = 3, 4,..., the polygonal numbers of order m are given by pm(n) =(m- 2) n2 + n(n= 0, 1, 2,...). For positive integers a, b, c and i, j, k 3 with max{i, j, k} 5, we call the triple(api, bpj, cpk)universal if for any n = 0, 1, 2,..., there are nonnegative integers x, y, z such that n = api(x) + bpj(y)+ cpk(z). We show that there are only 95 candidates for universal triples(two of which are(p4, p5, p6) and(p3, p4, p27)), and conjecture that they are indeed universal triples. For many triples(api, bpj, cpk)(including(p3, 4p4, p5),(p4, p5, p6) and(p4, p4, p5)), we prove that any nonnegative integer can be written in the form api(x) + bpj(y) + cpk(z) with x, y, z ∈ Z. We also show some related new results on ternary quadratic forms,one of which states that any nonnegative integer n ≡ 1(mod 6) can be written in the form x2+ 3y2+ 24z2 with x, y, z ∈ Z. In addition, we pose several related conjectures one of which states that for any m = 3, 4,...each natural number can be expressed as pm+1(x1) + pm+2(x2) + pm+3(x3) + r with x1, x2, x3 ∈ {0, 1, 2,...}and r ∈ {0,..., m- 3}.