The Jacobian Conjecture as a Problem of Perturbative Quantum Field Theory

The Jacobian conjecture is an old unsolved problem in mathematics, which has been unsuccessfully attacked from many different angles. We add here another point of view pertaining to the so called formal inverse approach, that of perturbative quantum field theory.Comment: 22 pages, 13 diagram

ATLAS SCT Endcap Module Production

The challenges for the tracking detector systems at the LHC are unprecedented in terms of the number of channels, the required read-out speed and the expected radiation levels. The ATLAS Semiconductor Tracker (SCT) end-caps have a total of about 3 million electronics channels each reading out every 25 ns into its own on-chip 3.3 ?s buffer. The highest anticipated dose after 10 years operation is 1.4×1014 cm-2 in units of 1 MeV neutron equivalent (assuming the damage factors scale with the non-ionising energy loss). The forward tracker has 1976 double-sided modules, mostly of area ? 70 cm2, each having 2×768 strips read out by 6 ASICs per side. The requirement to achieve an average perpendicular radiation length of 1.5% X0, while coping with up to 7 W dissipation per module (after irradiation), leads to stringent constraints on the thermal design. The additional requirement of 1500 e- equivalent noise charge (ENC) rising to only 1800 e-ENC after irradiation, provides stringent design constraints on both high-density Cu/Polyimide flex read-out circuit and the ABCD3TA read-out ASICs. Finally, the accuracy of module assembly must not compromise the 16 ?m (r-?) resolution perpendicular to the strip directions or 580 ?m radial resolution coming from the 40 mrad front-back stereo angle. 2196 modules were built to the tight tolerances and specifications required for the SCT. This was 220 more than the 1976 required and represents a yield of 93%. The component flow was at times tight, but the module production rate of 40 to 50 per week was maintained despite this. The distributed production was not found to be a major logistical problem and it allowed additional flexibility to take advantage of where the effort was available, including any spare capacity, for building the end-cap modules. The collaboration that produced the ATLAS SCT end-cap modules kept in close contact at all times so that the effects of shortages or stoppages at different sites could be rapidly resolved

RTT relations, a modified braid equation and noncommutative planes

With the known group relations for the elements $(a,b,c,d)$ of a quantum matrix $T$ as input a general solution of the $RTT$ relations is sought without imposing the Yang - Baxter constraint for $R$ or the braid equation for $\hat{R} = PR$. For three biparametric deformatios, $GL_{(p,q)}(2), GL_{(g,h)}(2)$ and $GL_{(q,h)}(1/1)$, the standard,the nonstandard and the hybrid one respectively, $R$ or $\hat{R}$ is found to depend, apart from the two parameters defining the deformation in question, on an extra free parameter $K$,such that only for two values of $K$, given explicitly for each case, one has the braid equation. Arbitray $K$ corresponds to a class (conserving the group relations independent of $K$) of the MQYBE or modified quantum YB equations studied by Gerstenhaber, Giaquinto and Schak. Various properties of the triparametric $\hat{R}(K;p,q)$, $\hat{R}(K;g,h)$ and $\hat{R}(K;q,h)$ are studied. In the larger space of the modified braid equation (MBE) even $\hat{R}(K;p,q)$ can satisfy $\hat{R}^2 = 1$ outside braid equation (BE) subspace. A generalized, $K$- dependent, Hecke condition is satisfied by each 3-parameter $\hat{R}$. The role of $K$ in noncommutative geometries of the $(K;p,q)$,$(K;g,h)$ and $(K;q,h)$ deformed planes is studied. K is found to introduce a "soft symmetry breaking", preserving most interesting properties and leading to new interesting ones. Further aspects to be explored are indicated.Comment: Latex, 17 pages, minor change

Constructive Matrix Theory

We extend the technique of constructive expansions to compute the connected functions of matrix models in a uniform way as the size of the matrix increases. This provides the main missing ingredient for a non-perturbative construction of the $\phi^{\star 4}_4$ field theory on the Moyal four dimensional space.Comment: 12 pages, 3 figure