QCD box graphs and the quark-antiquark potential

The $t\bar t$ system allows a truly perturbative treatment of the potential. Completing previous computations, we calculate the contributions of QCD box graph corrections, which make the “relativistic”O(α ⁴) corrections in the non-Abelian case differ from the well known corrections of the same order in QED.     

On a problem of H. N. Gupta

It is shown that the axiom For any points x, y, z such that y is between x and z, there is a right triangle having x and z as endpoints of the hypotenuse and y as foot of the altitude to the hypotenuse, when added to three-dimensional Euclidean geometry over arbitrary ordered fields, is weaker than the axiom Every line which passes through the interior of a sphere intersects that sphere.     

Separately harmonic and subharmonic functions

Let u(x, y) be defined in B ₁×B ₂ where B ₁ ^m and B ₂ ⁿ , and assume that u(x, ·) harmonic for every fixed x and u(·, y) is subharmonic for every fixed y. We show that if u(·, y) is, in addition, C ² for each y then u is subharmonic in B ₁×B ₂ in both variables jointly.     

Extremal problems for linear orders on [n] p

Let be a product order on [n] ^p i.e. for A, B [n] ^p , 1a ₁<a ₂<...<a _p º-n and 1<-b ₁<b ₂<...<b _p <-n we have AB iff a _i<-b _i for all i=1, 2,..., p. For a linear extension < of (ordering [n] ^p as ) let F _< ^[n],p (m) count the number of A _i 's, i<-m such that 1A _i. Clearly, for every m and <, where <l denotes the lexicographic order on [n] ^p . In this note we prove that the colexicographical order, <c, provides a corresponding lower bound i.e. that holds for any linear extension < of .This project together with [2] was initiated by the first author and continued in colaboration with the second author. After the death of the first author the work was continued and finalized by the second and the third author.Research supported by NSF grant DMS 9011850.