On a P\'olya functional for rhombi, isosceles triangles, and thinning convex sets

Let $\Omega$ be an open convex set in ${\mathbb R}^m$ with finite width, and let $v_{\Omega}$ be the torsion function for $\Omega$, i.e. the solution of $-\Delta v=1, v\in H_0^1(\Omega)$. An upper bound is obtained for the product of $\Vert v_{\Omega}\Vert_{L^{\infty}(\Omega)}\lambda(\Omega)$, where $\lambda(\Omega)$ is the bottom of the spectrum of the Dirichlet Laplacian acting in $L^2(\Omega)$. The upper bound is sharp in the limit of a thinning sequence of convex sets. For planar rhombi and isosceles triangles with area $1$, it is shown that $\Vert v_{\Omega}\Vert_{L^{1}(\Omega)}\lambda(\Omega)\ge \frac{\pi^2}{24}$, and that this bound is sharp.Comment: 12 pages, 4 figure

The fate of quasiparticles in the superconducting state

Quasiparticle properties in the superconducting state are masked by the superfluid and are not directly accessible to infrared spectroscopy. We show how one can use a Kramers--Kronig transformation to separate the quasiparticle from superfluid response and extract intrinsic quasiparticle properties in the superconducting state. We also address the issue of a narrow quasiparticle peak observed in microwave measurements, and demonstrate how it can be combined with infrared measurements to obtain unified picture of electrodynamic properties of cuprate superconductors

Stochastically perturbed flows: Delayed and interrupted evolution

We present analytical expressions for the time-dependent and stationary probability distributions corresponding to a stochastically perturbed one-dimensional flow with critical points, in two physically relevant situations: delayed evolution, in which the flow alternates with a quiescent state in which the variate remains frozen at its current value for random intervals of time; and interrupted evolution, in which the variate is also re-set in the quiescent state to a random value drawn from a fixed distribution. In the former case, the effect of the delay upon the first passage time statistics is analyzed. In the latter case, the conditions under which an extended stationary distribution can exist as a consequence of the competition between an attractor in the flow and the random re-setting are examined. We elucidate the role of the normalization condition in eliminating the singularities arising from the unstable critical points of the flow, and present a number of representative examples. A simple formula is obtained for the stationary distribution and interpreted physically. A similar interpretation is also given for the known formula for the stationary distribution in a full-fledged dichotomous flow.Comment: 27 pages; no figures. Submitted to Stochastics and Dynamic

Control of recollision wave packets for molecular orbital tomography using short laser pulses

The tomographic imaging of arbitrary molecular orbitals via high-order harmonic generation requires that electrons recollide from one direction only. Within a semi-classical model, we show that extremely short phase-stabilized laser pulses offer control over the momentum distribution of the returning electrons. By adjusting the carrier-envelope phase, recollisions can be forced to occur from mainly one side, while retaining a broad energy spectrum. The signatures of the semi-classical distributions are observed in harmonic spectra obtained by numerical solution of the time-dependent Schr\"{o}dinger equation.Comment: 8 pages, 4 figures; v2: Added some extra clarifications; v3: minor grammatical change

Low-carbohydrate diets affect energy balance and fuel homeostasis differentially in lean and obese rats

In parallel with increased prevalence of overweight people in affluent societies are individuals trying to lose weight, often using low-carbohydrate diets. Nevertheless, long-term metabolic consequences of those diets, usually high in (saturated) fat, remain unclear. Therefore, we investigated long-term effects of high-fat diets with different carbohydrate/protein ratios on energy balance and fuel homeostasis in obese (fa/fa) Zucker and lean Wistar rats. Animals were fed high-carbohydrate (HC), high-fat (HsF), or low-carbohydrate, high-fat, high-protein (LC-HsF-HP) diets for 60 days. Both lines fed the LC-HsF-HP diet displayed reduced energy intake compared with those fed the HsF diet (Zucker, -3.7%) or the HC diet (Wistar rats, -12.4%). This was not associated with lower weight gain relative to HC fed rats, because of increased food efficiencies in each line fed HsF and particularly LC-HsF-HP food. Zucker rats were less glucose tolerant than Wistar rats. Lowest glucose tolerances were found in HsF and particularly in LC-HsF-HP-fed animals irrespective of line, but this paralleled reduced plasma adiponectin levels, elevated plasma resistin levels, higher retroperitoneal fat masses, and reduced insulin sensitivity (indexed by insulin-induced hypoglycemia) only in Wistar rats. In Zucker rats, however, improved insulin responses during glucose tolerance testing and tendency toward increased insulin sensitivities were observed with HsF or LC-HsF-HP feeding relative to HC feeding. Thus, despite adverse consequences of LC-HsF diets on blood glucose homeostasis, principal differences exist in the underlying hormonal regulatory mechanisms, which could have benefits for B-cell functioning and insulin action in the obese state but not in the lean state.