A Jacobian module for disentanglements and applications to Mond's conjecture

Given a germ of holomorphic map $f$ from $\mathbb C^n$ to $\mathbb C^{n+1}$, we define a module $M(f)$ whose dimension over $\mathbb C$ is an upper bound for the $\mathscr A$-codimension of $f$, with equality if $f$ is weighted homogeneous. We also define a relative version $M_y(F)$ of the module, for unfoldings $F$ of $f$. The main result is that if $(n,n+1)$ are nice dimensions, then the dimension of $M(f)$ over $\mathbb C$ is an upper bound of the image Milnor number of $f$, with equality if and only if the relative module $M_y(F)$ is Cohen-Macaulay for some stable unfolding $F$. In particular, if $M_y(F)$ is Cohen-Macaulay, then we have Mond's conjecture for $f$. Furthermore, if $f$ is quasi-homogeneous, then Mond's conjecture for $f$ is equivalent to the fact that $M_y(F)$ is Cohen-Macaulay. Finally, we observe that to prove Mond's conjecture, it suffices to prove it in a suitable family of examples.Comment: 19 page

Prime numbers, quantum field theory and the Goldbach conjecture

Motivated by the Goldbach conjecture in Number Theory and the abelian bosonization mechanism on a cylindrical two-dimensional spacetime we study the reconstruction of a real scalar field as a product of two real fermion (so-called \textit{prime}) fields whose Fourier expansion exclusively contains prime modes. We undertake the canonical quantization of such prime fields and construct the corresponding Fock space by introducing creation operators $b_{p}^{\dag}$ --labeled by prime numbers $p$-- acting on the vacuum. The analysis of our model, based on the standard rules of quantum field theory and the assumption of the Riemann hypothesis, allow us to prove that the theory is not renormalizable. We also comment on the potential consequences of this result concerning the validity or breakdown of the Goldbach conjecture for large integer numbers.Comment: 20 pages in A4 format, 2 figure

Are the HI deficient galaxies on the outskirts of Virgo recent arrivals?

The presence on the Virgo cluster outskirts of spiral galaxies with gas deficiencies as strong as those of the inner galaxies stripped by the intracluster medium has led us to explore the possibility that some of these peripheral objects are not newcomers. A dynamical model for the collapse and rebound of spherical shells under the point mass and radial flow approximations has been developed to account for the amplitude of the motions in the Virgo I cluster (VIC) region. According to our analysis, it is not unfeasible that galaxies far from the cluster, including those in a gas-deficient group well to its background, went through its core a few Gyr ago. The implications would be: (1) that the majority of the HI-deficient spirals in the VIC region might have been deprived of their neutral hydrogen by interactions with the hot intracluster medium; and (2) that objects spending a long time outside the cluster cores might keep the gas deficient status without altering their morphology.Comment: Accepted for publication in ApJ. 4 pages, 3 figures. Uses emulateapj