On (conditional) positive semidefiniteness in a matrix-valued context

In a nutshell, we intend to extend Schoenberg's classical theorem connecting conditionally positive semidefinite functions $F\colon \mathbb{R}^n \to \mathbb{C}$, $n \in \mathbb{N}$, and their positive semidefinite exponentials $\exp(tF)$, $t > 0$, to the case of matrix-valued functions $F \colon \mathbb{R}^n \to \mathbb{C}^{m \times m}$, $m \in \mathbb{N}$. Moreover, we study the closely associated property that $\exp(t F(- i \nabla))$, $t>0$, is positivity preserving and its failure to extend directly in the matrix-valued context.Comment: 43 pages, replaced Example 4.19 (i) (the original version contained a mistake); journal reference adde

The control of wing kinematics and flight forces in fruit flies (Drosophila spp.)

By simultaneously measuring flight forces and stroke kinematics in several species of fruit flies in the genus Drosophila, we have investigated the relationship between wing motion and aerodynamic force production. We induced tethered flies to vary their production of total flight force by presenting them with a vertically oscillating visual background within a closed-loop flight arena. In response to the visual motion, flies modulated their flight force by changing the translational velocity of their wings, which they accomplished via changes in both stroke amplitude and stroke frequency. Changes in wing velocity could not, however, account for all the modulation in flight force, indicating that the mean force coefficient of the wings also increases with increasing force production. The mean force coefficients were always greater than those expected under steady-state conditions under a variety of assumptions, verifying that force production in Drosophila spp. must involve non-steady-state mechanisms. The subtle changes in kinematics and force production within individual flight sequences demonstrate that flies possess a flexible control system for flight maneuvers in which they can independently control the stroke amplitude, stroke frequency and force coefficient of their wings. By studying four different-sized species, we examined the effects of absolute body size on the production and control of aerodynamic forces. With decreasing body size, the mean angular wing velocity that is required to support the body weight increases. This change is due almost entirely to an increase in stroke frequency, whereas mean stroke amplitude was similar in all four species. Despite the elevated stroke frequency and angular wing velocity, the translational velocity of the wings in small flies decreases with the reduction in absolute wing length. To compensate for their small size, D. nikananu must use higher mean force coefficients than their larger relatives

The changes in power requirements and muscle efficiency during elevated force production in the fruit fly Drosophila melanogaster

The limits of flight performance have been estimated in tethered Drosophila melanogaster by modulating power requirements in a 'virtual reality' flight arena. At peak capacity, the flight muscles can sustain a mechanical power output of nearly 80 W kg^(-1) muscle mass at 24 °C, which is sufficient to generate forces of approximately 150% of the animal's weight. The increase in flight force above that required to support body weight is accompanied by a rise in wing velocity, brought about by an increase in stroke amplitude and a decrease in stroke frequency. Inertial costs, although greater than either profile or induced power, would be minimal with even modest amounts of elastic storage, and total mechanical power energy should be equivalent to aerodynamic power alone. Because of the large profile drag expected at low Reynolds numbers, the profile power was approximately twice the induced power at all levels of force generation. Thus, it is the cost of overcoming drag, and not the production of lift, that is the primary requirement for flight in Drosophila melanogaster. By comparing the estimated mechanical power output with respirometrically measured total power input, we determined that muscle efficiency rises with increasing force production to a maximum of 10%. This change in efficiency may reflect either increased crossbridge activation or a favorable strain regime during the production of peak forces

The production of elevated flight force compromises manoeuvrability in the fruit fly Drosophila melanogaster

In this study, we have investigated how enhanced total flight force production compromises steering performance in tethered flying fruit flies, Drosophila melanogaster. The animals were flown in a closed-loop virtual-reality flight arena in which they modulated total flight force production in response to vertically oscillating visual patterns. By simultaneously measuring stroke amplitude and stroke frequency, we recorded the ability of each fly to modulate its wing kinematics at different levels of aerodynamic force production. At a flight force that exactly compensates body weight, the temporal deviations with which fruit flies vary their stroke amplitude and frequency are approximately 27° and 4.8 Hz of their mean value, respectively. This variance in wing kinematics decreases with increasing flight force production, and at maximum force production fruit flies are restricted to a unique combination of stroke amplitude, stroke frequency and mean force coefficient. This collapse in the kinematic envelope during peak force production could greatly attenuate the manoeuvrability and stability of animals in free flight

Balanced Budget and Investment Rule: Two Sides of the Same Coin?! Analyzing the economic effects of a binding public investment commitment in Germany. Bertelsmann Stiftung Inclusive Growth for Germany|7

In their statement from December 2016, the independent commission of experts for strengthening investment in Germany, appointed by then-Federal Minister of Economics Sigmar Gabriel, calls for a significant expansion of investment dynamics in Germany. Along with proposed measures to strengthen private investments, the commission stresses the importance of creating the kind of institutional and political framework needed to push public investments so as to not endanger welfare and economic growth in Germany. In light of this, this study shows what effects an increase and stabilization of the public investment level in Germany would have. To do this, distinct investment scenarios have been created and their impact on a number of economic and politico-economic indicators up to the year 2025 was measured. The results clearly show that an increase in public investments in Germany would lead to a significant rise in German GDP growth over the following years. Factors like productivity, volume of work and the state’s capital stock similarly show higher increases in scenarios with more public investments relative to scenarios with a lower investment level. Another important result of the study is that such a proposed rule of investment would not have to be in conflict with the already existing debt rule of the federal government. Although scenarios with a higher level of public investment first lead to a lower budget balance, the differences between the individual scenarios become insignificant over time due to both the higher economic growth in those scenarios with more investments and the underlying assumptions regarding the counter-financing of these investments. In all five scenarios observed, the debt / GDP ratio swiftly falls below 50 percent in the year 2025. For the purpose of this study’s calculations, additional investments will be financed through an increase in taxes and a cut in spending on public consumption. The focus of this paper is purely macroeconomic, taking a look at the effects of different levels of public investment on the German economy

The aerodynamic effects of wing–wing interaction in flapping insect wings

We employed a dynamically scaled mechanical model of the small fruit fly Drosophila melanogaster (Reynolds number 100–200) to investigate force enhancement due to contralateral wing interactions during stroke reversal (the 'clap-and-fling'). The results suggest that lift enhancement during clap-and-fling requires an angular separation between the two wings of no more than 10–12°. Within the limitations of the robotic apparatus, the clap-and-fling augmented total lift production by up to 17%, but depended strongly on stroke kinematics. The time course of the interaction between the wings was quite complex. For example, wing interaction attenuated total force during the initial part of the wing clap, but slightly enhanced force at the end of the clap phase. We measured two temporally transient peaks of both lift and drag enhancement during the fling phase: a prominent peak during the initial phase of the fling motion, which accounts for most of the benefit in lift production, and a smaller peak of force enhancement at the end fling when the wings started to move apart. A detailed digital particle image velocimetry (DPIV) analysis during clap-and-fling showed that the most obvious effect of the bilateral 'image' wing on flow occurs during the early phase of the fling, due to a strong fluid influx between the wings as they separate. The DPIV analysis revealed, moreover, that circulation induced by a leading edge vortex (LEV) during the early fling phase was smaller than predicted by inviscid two-dimensional analytical models, whereas circulation of LEV nearly matched the predictions of Weis-Fogh's inviscid model at late fling phase. In addition, the presence of the image wing presumably causes subtle modifications in both the wake capture and viscous forces. Collectively, these effects explain some of the changes in total force and lift production during the fling. Quite surprisingly, the effect of clap-and-fling is not restricted to the dorsal part of the stroke cycle but extends to the beginning of upstroke, suggesting that the presence of the image wing distorts the gross wake structure throughout the stroke cycle

The scaling of carbon dioxide release and respiratory water loss in flying fruit flies (Drosophila spp.)

By simultaneously measuring carbon dioxide release, water loss and flight force in several species of fruit flies in the genus Drosophila, we have investigated respiration and respiratory transpiration during elevated locomotor activity. We presented tethered flying flies with moving visual stimuli in a virtual flight arena, which induced them to vary both flight force and energetic output. In response to the visual motion, the flies altered their energetic output as measured by changes in carbon dioxide release and concomitant changes in respiratory water loss. We examined the effect of absolute body size on respiration and transpiration by studying four different-sized species of fruit flies. In resting flies, body-mass-specific CO(2) release and water loss tend to decrease more rapidly with size than predicted according to simple allometric relationships. During flight, the mass-specific metabolic rate decreases with increasing body size with an allometric exponent of -0.22, which is slightly lower than the scaling exponents found in other flying insects. In contrast, the mass-specific rate of water loss appears to be proportionately greater in small animals than can be explained by a simple allometric model for spiracular transpiration. Because fractional water content does not change significantly with increasing body size, the smallest species face not only larger mass-specific energetic expenditures during flight but also a higher risk of desiccation than their larger relatives. Fruit flies lower their desiccation risk by replenishing up to 75 % of the lost bulk water by metabolic water production, which significantly lowers the risk of desiccation for animals flying under xeric environmental conditions