A recursive construction of units in a class of rings

Let $R$ be an associative ring with identity and let $N$ be a nil ideal of $R$. It is shown that units of $R/N$ can be lifted to units in $R$. Under some mild conditions on the ring, a procedure is given to determine those lifted units in a recursive way. As an application, the units of several classes of rings are determined, including: matrix rings, chain rings, and group rings where the ring is a chain ring. Numerical examples are given illustrating the main results.Comment: arXiv admin note: text overlap with arXiv:1904.12932 corrected typo

On idempotents of a class of commutative rings

In the present work, a procedure for determining idempotents of a commutative ring having a sequence of ideals with certain properties is presented. As an application of this procedure, idempotent elements of various commutative rings are determined. Several examples are included illustrating the main results.Comment: In this version, changes were made in the title, abstract and introduction to make the main theorem of the article clearer. No changes were made to the results presented in the previous versio

Matter-wave interference in s-wave and p-wave Fermi condensates

We discuss the time evolution and matter-wave interference of Fermi condensates on the BEC side of Feshbach resonances for s and p-wave superfluids, upon release from harmonic traps. In swave systems, where the order parameter is a complex scalar, we find that the interference patterns depend on the relative phase of the order parameters of the condensates. In p-wave systems involving the mixture of two-hyperfine states, we show that the interference pattern exhibits a polarization effect depending on the relative orientation of the two vector order parameters. Lastly, we also point out that p-wave Fermi condensates exhibit an anisotropic expansion, reflecting the spatial anisotropy of the underlying interaction between fermions and the orbital nature of the vector order parameter. Potential applications of our results include systems of ultra-cold atoms that exhibit p-wave Feshbach resonances such as 6Li or 40K.Comment: 8 pages, 5 figures; Phys. Rev. A 76, 013627 (2007

Electrostatically induced phase transitions in superconducting complex oxides

We describe quantum phase transitions in superconducting complex oxides which could be tuned by electrostatic charge transfer. Using a simple model for the superconductivity of a thin film or surface of a bulk copper oxide, we show that tuning the carrier density may allow the visitation of several superconducting phases with different pairing symmetries such as extended $s$- $(se)$, $d$- and $(se \pm id)$-wave. We construct a universal phase diagram for single-band superconductors with $se$- and d-wave components of the order parameter based on symmetry considerations alone. For a specific model with nearest neighbor attraction, we obtain the phase diagram in the interaction versus filling factor space showing the boundaries of the possible phases. Finally, we calculate the superfluid density and penetration depth as characteristic properties of each phase.Comment: 4 pages, 4 figure

Are Vortex Quasi-Crystals New Phases of Vortex Matter?

There seems to be a one to one correspondence between the phases of atomic and molecular matter (AMOM) and vortex matter (VM) in superfluids and superconductors. Crystals, liquids and glasses have been experimentally observed in both AMOM and VM. However, quasi-crystals also exist in AMOM, thus a new phase of vortex matter is proposed here: the vortex quasi-crystal. It is argued that vortex quasi-crystals are stabilized due to imposed quasi-periodic potentials in large samples or due to boundary and surface energy effects for samples of special shapes and sizes. For finite size samples, it is proposed that a phase transition between a vortex crystal and a vortex quasi-crystal occurs as a function of magnetic field and temperature as the sample size is reduced.Comment: 4 pages, 3 figure

Disorder effects during the evolution from BCS to BEC superfluidity

We describe the effects of disorder on the critical temperature of s-wave superfluids from the BCS to the BEC regime, with direct application to ultracold Fermi atoms. In the BCS regime the pair breaking and phase coherence temperature scales are essentially the same allowing strong correlations between the amplitude and phase of the order parameter. As non-pair breaking disorder is introduced the largely overlapping Cooper pairs conspire to maintain phase coherence such that the critical temperature remains essentially unchanged. However, in the BEC regime the pair breaking and phase coherence temperature scales are very different such that non-pair breaking disorder can affect dramatically phase coherence, and thus the critical temperature, without the requirement of breaking tightly-bound fermion pairs simultaneously. Finally, we find that the superfluid is more robust against weak disorder in the intermediate region between the two regimes.Comment: 4 pages, 3 figure

Asymmetric two-component Fermi gas with unequal masses

We analyze the zero temperature phase diagram for an asymmetric two-component Fermi gas as a function of mass anisotropy and population imbalance. We identify regions corresponding to normal, or uniform/non-uniform superfluid phases, and discuss topological quantum phase transitions in the Bardeen-Cooper-Schrieffer (BCS), unitarity and Bose-Einstein condensation (BEC) limits. Lastly, we derive the zero temperature low frequency and long wavelength collective excitation spectrum, and recover the Bogoliubov relation for weakly interacting dilute bosons in the BEC limit.Comment: 4 pages and 4 figure

Time evolution and matter wave interference in Fermi condensates

We discuss matter wave interference of Fermi condensates in strongly coupled s-wave and p-wave channels and the time evolution of a single cloud upon release from trap. In s-wave systems, where the order parameter is a complex scalar, we find that the interference patterns depend on the relative phase of the order parameters of the condensates. In p-wave systems involving the mixture of two-hyperfine states, we show that the interference pattern exhibits a polarization effect depending on the relative orientation of the two vector order parameters. However, in p-wave systems involving a single hyperfine state, we show that this angular effect reduces to an overall phase difference between the two interfering clouds, similar to s-wave. Lastly, we also point out that p-wave Fermi condensates exhibit an anisotropic expansion, reflecting the spatial anisotropy of the underlying interaction between fermions and the orbital nature of the vector order parameter.Comment: 4 pages, 4 figure

Density fluctuations and compressibility matrix for population or mass imbalanced Fermi-Fermi mixtures

We describe the relation between the isothermal atomic compressibility and density fluctuations in mixtures of two-component fermions with population or mass imbalance. We derive a generalized version of the fluctuation-dissipation theorem which is valid for both balanced and imbalanced Fermi-Fermi mixtures. Furthermore, we show that the compressibility, its critical exponents, and phase boundaries can be extracted via an analysis of the density fluctuations as a function of population imbalance, interaction parameter or temperature. Lastly, we demonstrate that in the presence of trapping potentials, the local compressibility and local density-density correlations can be extracted via a generalized fluctuation-dissipation theorem valid within the local density approximation.Comment: 4 pages, 3 figure

A fundamental theorem for submanifolds in semi-Riemannian warped products

In this paper we find necessary and sufficient conditions for a nondegenerate arbitrary signature manifold $M^n$ to be realized as a submanifold in the large class of warped product manifolds $\varepsilon I\times_a\mathbb{M}^{N}_{\lambda}(c)$, where $\varepsilon=\pm 1,\ a:I\subset\mathbb{R}\to\mathbb{R}^+$is the scale factor and$\mathbb{M}^{N}_{\lambda}(c)$is the$N$-dimensional semi-Riemannian space form of index$\lambda$and constant curvature$c\in\{-1,1\}.$We prove that if$M^n$satisfies Gauss, Codazzi and Ricci equations for a submanifold in$\varepsilon I\times_a\mathbb{M}^{N}_{\lambda}(c)$, along with some additional conditions, then$M^n$can be isometrically immersed into$\varepsilon I\times_a\mathbb{M}^{N}_{\lambda}(c)$. This comprises the case of hypersurfaces immersed in semi-Riemannian warped products proved by M.A. Lawn and M. Ortega (see [6]), which is an extension of the isometric immersion result obtained by J. Roth in the Lorentzian products$\mathbb{S}^n\times\mathbb{R}_1$and$\mathbb{H}^n\times\mathbb{R}_1$(see [12]), where$\mathbb{S}^n$and$\mathbb{H}^n$stand for the sphere and hyperbolic space of dimension$n$, respectively. This last result, in turn, is an expansion to pseudo-Riemannian manifolds of the isometric immersion result proved by B. Daniel in$\mathbb{S}^n\times\mathbb{R}$and$\mathbb{H}^n\times\mathbb{R}$ (see [2]), one of the first generalizations of the classical theorem for submanifolds in space forms (see [13]). Although additional conditions to Gauss, Codazzi and Ricci equations are not necessary in the classical theorem for submanifolds in space forms, they appear in all other cases cited above.Comment: 22 page