Amplifier enhances ring-down spectroscopy

In recent years, investigators have adapted the principles of ringdown spectroscopy (see sidebar, facing page) to fiber optic configurations by placing high reflectors on each end of a fiber and observing the ringdown time of an injected pulse. But a major drawback is the difficulty of creating a low-loss, high-Q resonator in an optical fiber

Function spectra and continuous G-spectra

Let G be a profinite group, {X_alpha}_alpha a cofiltered diagram of discrete G-spectra, and Z a spectrum with trivial G-action. We show how to define the homotopy fixed point spectrum F(Z, holim_alpha X_alpha)^{hG} and that when G has finite virtual cohomological dimension (vcd), it is equivalent to F(Z, holim_alpha (X_alpha)^{hG}). With these tools, we show that the K(n)-local Spanier-Whitehead dual is always a homotopy fixed point spectrum, a well-known Adams-type spectral sequence is actually a descent spectral sequence, and, for a sufficiently nice k-local profinite G-Galois extension E, with K a closed normal subgroup of G, the equivalence (E^{h_kK})^{h_kG/K} \simeq E^{h_kG} (due to Behrens and the author), where (-)^{h_k(-)} denotes k-local homotopy fixed points, can be upgraded to an equivalence that just uses ordinary (non-local) homotopy fixed points, when G/K has finite vcd.Comment: submitted for publicatio

Diagram spaces, diagram spectra, and spectra of units

This article compares the infinite loop spaces associated to symmetric spectra, orthogonal spectra, and EKMM S-modules. Each of these categories of structured spectra has a corresponding category of structured spaces that receives the infinite loop space functor \Omega^\infty. We prove that these models for spaces are Quillen equivalent and that the infinite loop space functors \Omega^\infty agree. This comparison is then used to show that two different constructions of the spectrum of units gl_1 R of a commutative ring spectrum R agree.Comment: 62 pages. The definition of the functor \mathbb{Q} is changed. Sections 8, 9, 17 and 18 contain revisions and/or new materia

Symmetric spectra

The long hunt for a symmetric monoidal category of spectra finally ended in success with the simultaneous discovery of the third author's discovery of symmetric spectra and the Elmendorf-Kriz-Mandell-May category of S-modules. In this paper we define and study the model category of symmetric spectra, based on simplicial sets and topological spaces. We prove that the category of symmetric spectra is closed symmetric monoidal and that the symmetric monoidal structure is compatible with the model structure. We prove that the model category of symmetric spectra is Quillen equivalent to Bousfield and Friedlander's category of spectra. We show that the monoidal axiom holds, so that we get model categories of ring spectra and modules over a given ring spectrum.Comment: 77 pages. This version corrects some errors in the section on topological symmetric spectr

Generic Birkhoff Spectra

Suppose that $\Omega = \{0, 1\}^ {\mathbb {N}}$ and ${\sigma}$ is the one-sided shift. The Birkhoff spectrum $\displaystyle S_{f}( {\alpha})=\dim_{H}\Big \{ {\omega}\in {\Omega}:\lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^N f(\sigma^n \omega) = \alpha \Big \},$ where $\dim_{H}$ is the Hausdorff dimension. It is well-known that the support of $S_{f}( {\alpha})$ is a bounded and closed interval $L_f = [\alpha_{f, \min}^*, \alpha_{f, \max}^*]$ and $S_{f}( {\alpha})$ on $L_{f}$ is concave and upper semicontinuous. We are interested in possible shapes/properties of the spectrum, especially for generic/typical $f\in C( {\Omega})$ in the sense of Baire category. For a dense set in $C( {\Omega})$ the spectrum is not continuous on ${\mathbb {R}}$, though for the generic $f\in C( {\Omega})$ the spectrum is continuous on ${\mathbb {R}}$, but has infinite one-sided derivatives at the endpoints of $L_{f}$. We give an example of a function which has continuous $S_{f}$ on ${\mathbb {R}}$, but with finite one-sided derivatives at the endpoints of $L_{f}$. The spectrum of this function can be as close as possible to a "minimal spectrum". We use that if two functions $f$ and $g$ are close in $C( {\Omega})$ then $S_{f}$ and $S_{g}$ are close on $L_{f}$ apart from neighborhoods of the endpoints.Comment: Revised version after the referee's repor