Congruence conditions, parcels, and Tutte polynomials of graphs and matroids

Let $G$ be a matrix and $M(G)$ be the matroid defined by linear dependence on the set $E$ of column vectors of $G.$ Roughly speaking, a parcel is a subset of pairs $(f,g)$ of functions defined on $E$ to an Abelian group $A$ satisfying a coboundary condition (that $f-g$ is a flow over $A$ relative to $G$) and a congruence condition (that the size of the supports of $f$ and $g$ satisfy some congruence condition modulo an integer). We prove several theorems of the form: a linear combination of sizes of parcels, with coefficients roots of unity, equals an evaluation of the Tutte polynomial of $M(G)$ at a point $(\lambda-1,x-1)$ on the complex hyperbola $(\lambda - 1)(x-1) = |A|.

Maximum size binary matroids with no AG(3,2)-minor are graphic

We prove that the maximum size of a simple binary matroid of rank $r \geq 5$ with no AG(3,2)-minor is $\binom{r+1}{2}$ and characterise those matroids achieving this bound. When $r \geq 6$, the graphic matroid $M(K_{r+1})$ is the unique matroid meeting the bound, but there are a handful of smaller examples. In addition, we determine the size function for non-regular simple binary matroids with no AG(3,2)-minor and characterise the matroids of maximum size for each rank

Murine terminal deoxynucleotidyl transferase: cellular distribution and response to cortisone

The mouse thymus contains two forms of terminal deoxynucleotidyl transferase (TdT) which are distinguishable by the salt concentration necessary to elute them from a phosphocellulose column, by their distrubtion among the thymocyte subpopulations, and by their sensitivity to cortisone treatment. In the whole thymus the later eluting peak (peak II) is the predominant one with about 3-10% of the total activity appearing in peak I. Both peak I and peak II activities are most sensitively assayed by the polymerization of dGMP onto an oligo(dA) primer. The minor population of thymocytes which is less dense and cortisone-resistant contains a higher specific activity of peak I TdT. The majority of TdT activity is, however, found in the major population of thymocytes which occurs in the center region of a bovine serum albumin gradient and is cortisone-sensitive. A very low level of an activity indistinguishable from peak II TdT activity is also detected in the mouse bone marrow. Other tissues, such as spleen, liver, heart, and brain lack detectable amounts of TdT activity