Subnormal closure of a homomorphism

Let $\varphi\colon\Gamma\to G$ be a homomorphism of groups. In this paper we introduce the notion of a subnormal map (the inclusion of a subnormal subgroup into a group being a basic prototype). We then consider factorizations $\Gamma\xrightarrow{\psi} M\xrightarrow{n} G$ of $\varphi,$ with $n$ a subnormal map. We search for a universal such factorization. When $\Gamma$ and $G$ are finite we show that such universal factorization exists: $\Gamma\to\Gamma_{\infty}\to G,$ where $\Gamma_{\infty}$ is a hypercentral extension of the subnormal closure $\mathcal{C}$ of $\varphi(\Gamma)$ in $G$ (i.e.~the kernel of the extension $\Gamma_{\infty}\to {\mathcal C}$ is contained in the hypercenter of $\Gamma_{\infty}$). This is closely related to the a relative version of the Bousfield-Kan $\mathbb{Z}$-completion tower of a space. The group $\Gamma_{\infty}$ is the inverse limit of the normal closures tower of $\varphi$ introduced by us in a recent paper. We prove several stability and finiteness properties of the tower and its inverse limit $\Gamma_{\infty}$.Comment: 13 pages

The flavor of product-group GUTs

The doublet-triplet splitting problem can be simply solved in product-group GUT models, using a global symmetry that distinguishes the doublets from the triplets. Apart from giving the required mass hierarchy, this ``triplet symmetry'' can also forbid some of the triplet couplings to matter. We point out that, since this symmetry is typically generation-dependent, it gives rise to non-trivial flavor structure. Furthermore, because flavor symmetries cannot be exact, the triplet-matter couplings are not forbidden then but only suppressed. We construct models in which the triplet symmetry gives acceptable proton decay rate and fermion masses. In some of the models, the prediction m_b ~ m_\tau is retained, while the similar relation for the first generation is corrected. Finally, all this can be accomplished with triplets somewhat below the GUT scale, supplying the right correction for the standard model gauge couplings to unify precisely.Comment: 10 page

Transition to an oscillator for double phase-conjugate mirror

Summary form only given. Some of the novel quantified characteristics for double phase conjugate mirrors are analysed including the effects of the nonlinearity on the critical dynamics (approach to saturation) and on the spatial distribution of the grating (large scale distortion of the beams and conjugation fidelity) and sensitivity to noise (seeding). The approach used also clarifies the question of linear instability and predicts a new transition to an oscillatory regime

Anderson localization of a Tonks-Girardeau gas in potentials with controlled disorder

We theoretically demonstrate features of Anderson localization in the Tonks-Girardeau gas confined in one-dimensional (1D) potentials with controlled disorder. That is, we investigate the evolution of the single particle density and correlations of a Tonks-Girardeau wave packet in such disordered potentials. The wave packet is initially trapped, the trap is suddenly turned off, and after some time the system evolves into a localized steady state due to Anderson localization. The density tails of the steady state decay exponentially, while the coherence in these tails increases. The latter phenomenon corresponds to the same effect found in incoherent optical solitons

On the nature of Coulomb corrections to the e^+e^- pair production in ultrarelativistic heavy-ion collisions

We manifest the origin of the wrong conclusion made by several groups of authors on the absence of Coulomb corrections to the cross section of the e^+e^- pair production in ultrarelativistic heavy-ion collisions. The source of the mistake is connected with an incorrect passage to the limit in the expression for the cross section. When this error is eliminated, the Coulomb corrections do not vanish and agree with the results obtained within the Weizs\"acker-Williams approximation.Comment: 7 pages, LaTe

Scanning superconducting quantum interference device on a tip for magnetic imaging of nanoscale phenomena

We describe a new type of scanning probe microscope based on a superconducting quantum interference device (SQUID) that resides on the apex of a sharp tip. The SQUID-on-tip is glued to a quartz tuning fork which allows scanning at a tip-sample separation of a few nm. The magnetic flux sensitivity of the SQUID is 1.8 {\mu}_0/Hz^{1/2} and the spatial resolution is about 200 nm, which can be further improved. This combination of high sensitivity, spatial resolution, bandwidth, and the very close proximity to the sample provides a powerful tool for study of dynamic magnetic phenomena on the nanoscale. The potential of the SQUID-on-tip microscope is demonstrated by imaging of the vortex lattice and of the local AC magnetic response in superconductors.Comment: 10 figure