The invariants of a genus one curve

It was first pointed out by Weil that we can use classical invariant theory to compute the Jacobian of a genus one curve. The invariants required for curves of degree n = 2,3,4 were already known to the nineteenth centuary invariant theorists. We have succeeded in extending these methods to curves of degree n = 5, where although the invariants are too large to write down as explicit polynomials, we have found a practical algorithm for evaluating them.Comment: 37 page

Scaling of Hamiltonian walks on fractal lattices

We investigate asymptotical behavior of numbers of long Hamiltonian walks (HWs), i.e. self-avoiding random walks that visit every site of a lattice, on various fractal lattices. By applying an exact recursive technique we obtain scaling forms for open HWs on 3-simplex lattice, Sierpinski gasket, and their generalizations: Given-Mandelbrot (GM), modified Sierpinski gasket (MSG) and n-simplex fractal families. For GM, MSG and n-simplex lattices with odd values of n, number of open HWs $Z_N$, for the lattice with $N\gg 1$ sites, varies as $\omega^N N^\gamma$. We explicitly calculate exponent $\gamma$ for several members of GM and MSG families, as well as for n-simplices with n=3,5, and 7. For n-simplex fractals with even n we find different scaling form: $Z_N\sim \omega^N \mu^{N^{1/d_f}}$, where $d_f$ is fractal dimension of the lattice, which also differs from the formula expected for homogeneous lattices. We discuss possible implications of our results on studies of real compact polymers.Comment: 19 pages, 13 figures, RevTex4; extended Introduction, several references added; one figure added in section II; corrected typos; version accepted for publication in Phys.Rev.

The power operation structure on Morava E-theory of height 2 at the prime 3

We give explicit calculations of the algebraic theory of power operations for a specific Morava E-theory spectrum and its K(1)-localization. These power operations arise from the universal degree-3 isogeny of elliptic curves associated to the E-theory

On the solutions of the scattering equations

This paper addresses the question, whether the solutions of the scattering equations in four space-time dimensions can be expressed as rational functions of the momentum twistor variables. This is the case for $n\le5$ external particles. For general $n$ there are always two solutions, which are rational functions of the momentum twistor variables. However, the remaining solutions are in general not rational. In the case $n=6$ the remaining four solutions can be expressed as algebraic functions. These four solutions are constructed explicitly in this paper.Comment: 16 pages, version to be published, additional file with explicit expressions for eq.56 and eq.61 provide

Renormalization group invariants in supersymmetric theories: one- and two-loop results

We stress the potential usefulness of renormalization group invariants. Especially particular combinations thereof could for instance be used as probes into patterns of supersymmetry breaking in the MSSM at inaccessibly high energies. We search for these renormalization group invariants in two systematic ways: on the one hand by making use of symmetry arguments and on the other by means of a completely automated exhaustive search through a large class of candidate invariants. At the one-loop level, we find all known invariants for the MSSM and in fact several more, and extend our results to the more constrained pMSSM and dMSSM, leading to even more invariants. Extending our search to the two-loop level we find that the number of invariants is considerably reduced

Supplement to "Robust linear least squares regression"

19 pagesInternational audienceThis supplementary material provides the proofs of Theorems 2.1, 2.2 and 3.1 of the article ''Robust linear least squares regression''