Tensor Products, Positive Linear Operators, and Delay-Differential Equations

We develop the theory of compound functional differential equations, which are tensor and exterior products of linear functional differential equations. Of particular interest is the equation $\dot x(t)=-\alpha(t)x(t)-\beta(t)x(t-1)$ with a single delay, where the delay coefficient is of one sign, say $\delta\beta(t)\ge 0$ with $\delta\in{-1,1}$. Positivity properties are studied, with the result that if $(-1)^k=\delta$ then the $k$-fold exterior product of the above system generates a linear process which is positive with respect to a certain cone in the phase space. Additionally, if the coefficients $\alpha(t)$ and $\beta(t)$ are periodic of the same period, and $\beta(t)$ satisfies a uniform sign condition, then there is an infinite set of Floquet multipliers which are complete with respect to an associated lap number. Finally, the concept of $u_0$-positivity of the exterior product is investigated when $\beta(t)$ satisfies a uniform sign condition.Comment: 84 page

A compact design for the Josephson mixer: the lumped element circuit

We present a compact and efficient design in terms of gain, bandwidth and dynamical range for the Josephson mixer, the superconducting circuit performing three-wave mixing at microwave frequencies. In an all lumped-element based circuit with galvanically coupled ports, we demonstrate non degenerate amplification for microwave signals over a bandwidth up to 50 MHz for a power gain of 20 dB. The quantum efficiency of the mixer is shown to be about 70$\%$ and its saturation power reaches $-112$ dBm.Comment: 5 pages, 4 figure

A Modified Version of Taylor's Hypothesis for Solar Probe Plus Observations

The Solar Probe Plus (SPP) spacecraft will explore the near-Sun environment, reaching heliocentric distances less than $10 R_{\odot}$. Near Earth, spacecraft measurements of fluctuating velocities and magnetic fields taken in the time domain are translated into information about the spatial structure of the solar wind via Taylor's "frozen turbulence" hypothesis. Near the perihelion of SPP, however, the solar-wind speed is comparable to the Alfv\'en speed, and Taylor's hypothesis in its usual form does not apply. In this paper, we show that, under certain assumptions, a modified version of Taylor's hypothesis can be recovered in the near-Sun region. We consider only the transverse, non-compressive component of the fluctuations at length scales exceeding the proton gyroradius, and we describe these fluctuations using an approximate theoretical framework developed by Heinemann and Olbert. We show that fluctuations propagating away from the Sun in the plasma frame obey a relation analogous to Taylor's hypothesis when $V_{\rm sc,\perp} \gg z^-$ and $z^+ \gg z^-$, where $V_{\rm sc,\perp}$ is the component of the spacecraft velocity perpendicular to the mean magnetic field and $\bm{z}^+$ ($\bm{z}^-$) is the Elsasser variable corresponding to transverse, non-compressive fluctuations propagating away from (towards) the Sun in the plasma frame. Observations and simulations suggest that, in the near-Sun solar wind, the above inequalities are satisfied and $\bm{z}^+$ fluctuations account for most of the fluctuation energy. The modified form of Taylor's hypothesis that we derive may thus make it possible to characterize the spatial structure of the energetically dominant component of the turbulence encountered by SPP.Comment: 5 pages, 1 figure, accepted in ApJ Lette

On the Conservation of Cross Helicity and Wave Action in Solar-Wind Models with Non-WKB Alfven Wave Reflection

The interaction between Alfven-wave turbulence and the background solar wind affects the cross helicity in two ways. Non-WKB reflection converts outward-propagating Alfven waves into inward-propagating Alfven waves and vice versa, and the turbulence transfers momentum to the background flow. When both effects are accounted for, the total cross helicity is conserved. In the special case that the background density and flow speed are independent of time, the equations of cross-helicity conservation and total-energy conservation can be combined to recover a well-known equation derived by Heinemann and Olbert that has been interpreted as a non-WKB generalization of wave-action conservation. This latter equation (in contrast to cross-helicity and energy conservation) does not hold when the background varies in time.Comment: 9 pages, 1 figure, in press at Ap