text:
We trained two starlings (Sturnus vulgaris) to fly in a wind tunnel whilst wearing respirometry masks. We measured the metabolic power (Pmet) from the rates of oxygen consumption and carbon dioxide production and calculated the mechanical power (Pmech) from two aerodynamic models using wingbeat kinematics measured by high-speed cinematography. Pmet increased from 10.4 to 14.9 W as flight speed was increased from 6.3 to 14.4 m s–1 and was compatible with the U-shaped power/speed curve predicted by the aerodynamic models. Flight muscle efficiency varied between 0.13 and 0.23 depending upon the bird, the flight speed and the aerodynamic model used to calculate Pmech. Pmet during flight is often estimated by extrapolation from the mechanical power predicted by aerodynamic models by dividing Pmech by a flight muscle efficiency of 0.23 and adding the costs of basal metabolism, circulation and respiration. This method would underestimate measured Pmet by 15–25 % in our birds. The mean discrepancy between measured and predicted Pmet could be reduced to 0.1±1.5 % if flight muscle efficiency was altered to a value of 0.18. A flight muscle efficiency of 0.18 rather than 0.23 should be used to calculate the flight costs of birds in the size range of starlings (approximately 0.1 kg) if Pmet is calculated from Pmech derived from aerodynamic models.

Flapping flight is among the most energetically expensive activities that vertebrates perform (Schmidt-Nielsen, 1972; Norberg, 1990). Quantifying the metabolic cost of flight is therefore important in a variety of biological problems. Examples include the estimation of avian daily energy demands (Weathers et al., 1984; Bryant, 1997), reconstruction of the evolution of flight (Norberg, 1990; Rayner, 1991; Ruben, 1991; Speakman, 1993), estimation of the potential flight ranges of migratory birds (Rayner, 1990; Walsberg; 1990, Carmi et al., 1992; Klaassen, 1995; Klaassen, 1996; Butler et al., 1997; Weber et al., 1994; Weber et al., 1998; Pennycuick, 1998) and calculation of optimum flight speeds (Lindström and Alerstam, 1992; Hedenström and Alerstam, 1995; Hedenström and Alerstam, 1997) or migratory strategies (Weber et al., 1994; Weber et al., 1998).

It is technically difficult to make direct measurements of metabolic power (Pmet) using methods such as the doubly labelled water technique during free flight (Masman and Klaassen, 1987) or mask respirometry during wind tunnel flight (Butler and Woakes, 1990) or from cardiac output (Bishop and Butler, 1995; Bishop, 1997). A more rapid alternative way to estimate total power consumption during flight is to use a theoretical aerodynamic model to calculate the mechanical component of power output during flight (Pmech) and to obtain an estimate of Pmet from this by:

$\mathit{P}_{\mathrm{met}}{\ }{=}{\ }1.1{[}(\mathit{P}_{\mathrm{mech}}/\mathit{E}_{\mathrm{FM}}){\ }{+}{\ }\mathit{P}_{\mathrm{BMR}}{]}{\ },$
1

where flight muscle efficiency (EFM) is defined as Pmech/metabolic power consumed by the flight muscles and PBMR is basal metabolism (Tucker, 1973; Pennycuick, 1975; Pennycuick, 1989; Norberg, 1990). Predictions of Pmet made using equation 1 may be in error since they rely on four assumptions: (i) that aerodynamic models predict Pmech accurately, (ii) that EFM is fixed (usually at a value of 0.23) (Pennycuick, 1975), (iii) that PBMR is constant, and meaningful, during flight, and (iv) that the extra costs of respiration and circulation during flight each contribute 5 % to Pmet. Assumptions iii and iv have not been tested since it is not currently possible to partition metabolic rate either experimentally or theoretically, but predicted Pmet is relatively insensitive to deviations in assumed PBMR or the combined cost of respiration and circulation. Predicted Pmet is more sensitive to errors in Pmech: a change of 1 % in Pmech will alter Pmet by 0.9 %. Predicted Pmet is most sensitive to the value of EFM: a change of 0.01 in the assumed EFM will change predicted Pmet by 4–5 %. Predictions of Pmet obtained from equation 1 are sensitive to the value of EFM used because Pmech represents only a small proportion of flight muscle metabolic rate (Rayner and Ward, 1999; Rayner, 1999). This means that a small change in Pmech will correspond to a large change in Pmet because most of the the metabolic power supplied to the flight muscles is transformed to heat as a by-product of the conversion of chemical to kinetic energy (Hill, 1938).

The value of EFM is normally assumed to be 0.23 (Pennycuick, 1975), although sometimes a different constant value in the range 0.20–0.25 is used (Norberg, 1990). These values of EFM have not been measured directly; instead, the value 0.23 is the value taken from typical values for vertebrate muscle in vitro and lies approximately in the middle of the range of partial efficiency (EP, defined as the change in Pmech for potential energy/change in Pmet) of animals in ascending or descending flight in wind tunnels (budgerigar Melopsittacus undulatus, mass 0.035 kg, EP 0.19–0.28; laughing gull Larus atricilla, mass 0.30 kg, EP 0.30 (Tucker, 1972); fish crow Corvus ossifragus, mass 0.275 kg, EP 0.20–0.29 (Bernstein et al., 1973); white-necked raven Corvus cryptoleucos, mass 0.48 kg, EP 0.32–0.40 (Hudson and Bernstein, 1983); 0.13–0.40 for two species of bat Phyllostomus hastatus, mass 0.093 kg, and Pteropus gouldii, mass 0.78 kg (Thomas, 1975)). There are two problems with the assumption that EFM=0.23 on the basis of the data from these studies. First, wingbeat kinematics, and therefore drag, alter to compensate for the change in thrust required to fly up- or downhill (Rayner, 1986), so EFM may differ from EP. Second, the wide range in values of EP suggests that it may be invalid to assume that efficiency is constant across species, individuals and flight speeds.

Efficiency is much lower than 0.23 when calculated from simultaneous estimates of Pmech and Pmet: 0.11 during forward flight and 0.15 during hovering in the 0.0117 kg nectar-feeding bat Glossophaga soricina (Norberg et al., 1993) and 0.10 and 0.11–0.13, respectively, for ruby-throated hummingbirds Archilochus colubris (mass 0.0035 kg) hovering in heliox (Chai and Dudley, 1995) and in air (Chai et al., 1998). Currently, the major limitation in applying equation 1 to bird flight performance is that the assumption that a fixed value of 0.23 for EFM is appropriate for all birds is unreliable.

If EFM is invariant with flight speed, Pmet should approximate a fixed multiple of Pmech as a function of speed (since PBMR is relatively small). Aerodynamic models predict that Pmech will have a U-shaped relationship with flight speed (Tucker, 1973; Pennycuick, 1975; Pennycuick, 1989; Rayner, 1979; Rayner, 1999). In contrast to this prediction, Pmet often has a much flatter relationship with flight speed in most birds and bats during wind tunnel flight (Rayner, 1994). Measurements of mechanical power production by the M. pectoralis muscle in magpies also suggest a flat power/speed relationship over most forward flight speeds (Dial et al., 1997). These measurements are from the largest muscles used during flight, so they include the majority of Pmech, although the data do not necessarily reflect the trend in Pmech across flight speeds for the entire animal. The measured values of M. pectoralis power production in magpies do not differ significantly from predicted Pmech (Rayner, 1999). The discrepancy between Pmet predicted by equation 1 and that measured by respirometry during wind tunnel flight may be caused by variation in EFM with flight speed. Alternatively, Pmech produced by the flying animal may differ from that predicted by aerodynamic theory either because of changes in wingbeat kinematics due to carrying the respirometry mask or because of variation in kinematics across individuals and flight speeds that is not accounted for in some aerodynamic models.

We measured Pmet by mask respirometry and wingbeat kinematics by high-speed cinematography of European starlings Sturnus vulgaris (hereafter referred to as starlings) during wind tunnel flight across a range of flight speeds to assess whether flight costs form a U-shaped curve relative to flight speed. We calculated Pmech from two aerodynamic models, a relatively simple model developed from fixed-wing aerodynamic theory (Pennycuick, 1989) and a more complex one based on the generation of vortex rings (Rayner, 1979), which corresponds to the observed flight pattern and wake vortex geometry of starlings (Rayner and Swaddle, 2000). Flight muscle efficiency was calculated from Pmet and Pmech by rearranging equation 1. We subsequently assessed how much Pmet would differ from our measurements if it were predicted from Pmech determined by aerodynamic modelling assuming an efficiency of 0.23.

### Wind tunnel

Starlings Sturnus vulgaris were flown in a closed-section Göttingen-type variable-speed wind tunnel at the University of the Saarland, Saarbrücken, Germany (Biesel et al., 1985; Nachtigall, 1997). Birds were prevented from leaving the 1 m×1 m×1 m flight chamber upwind by wire mesh (25 mm hexagonal, 1 mm diameter) and downwind by vertical plastic chords (1 mm diameter, 1 cm apart). The top and one side wall of the flight chamber were made of glass to allow filming of lateral and dorsal images. The floor and walls of the chamber and the tunnel sections immediately up- and downwind of the flight chamber were made of wood. Air speed was monitored downwind of the flight chamber with a pitot-static tube connected to a manometer. Air speed in this position was linearly related to that in the centre of the flight chamber. Air speed could be controlled to within ±0.2 m s–1 and was measured to ±0.1 m s–1. Turbulence (where turbulence=100×standard deviation of air speed/mean air speed at each point) was measured using a DISA hot-wire anemometer at 11 points (in the centre of the flight chamber, half-way between the centre and each corner and at the mid points between the centre and the floor and ceiling). Mean turbulence at air speeds between 6 and 14 m s–1 was 1.80–1.95 % (s.d. 0.5–0.6) and did not vary consistently with flight speed. Air speed measured at 32 points at 10 cm intervals horizontally and vertically from the centre of the flight chamber varied by less than 4 % from that measured in the centre of the flight chamber. Air temperature within the flight chamber averaged 18.5±0.2°C. Air humidity was 63.0 ±0.7 %. Air density was 1.18 ±0.002 kg m–3.

### Birds and training

Starlings (seven hand-reared and eight wild-caught adult birds captured under licence from Scottish Natural Heritage in Aberdeenshire, UK) were housed in groups of 3–4 birds in indoor cages (approximately 2 m×2 m×2 m) and fed ad libitum on a mixture of moistened puppy pellets (Eukanuba), poultry pellets and cage bird egg food supplemented with mealworms and cage bird vitamin and mineral supplement. The birds were accustomed to the wind tunnel by placing them individually in the flight chamber, where they preferred to stand on a perch rather than on the smooth floor of the chamber. Birds flew spontaneously when the perch was retracted into the floor of the chamber. The perch was returned after progressively longer periods of flight until the birds that were trained successfully (four wild-caught adult female birds) would fly continuously for up to 1 h twice daily. The birds normally flew in the top half of the flight chamber looking straight ahead, but they learnt that they could show when they were tired by consistently flying close to the place where the perch would emerge from the floor of the chamber and looking at the trainer. The perch was returned to allow the bird to rest during training sessions when it showed this behaviour.

Two of the birds learnt to fly wearing masks from which exhaled respiratory gases were extracted through a flexible tube. The starlings were accustomed to flight in the wind tunnel for 10 months before the data reported here were collected to minimise the effects of stress upon metabolic rate. Each bird was flown three times at approximately 1 m s–1 increments in flight speed within the range of speed that each bird would fly wearing the mask (6.3–13.2 m s–1 for bird 15, mass 0.080±0.0008 kg; 8.3–14.4 m s–1 for bird 19, mass 0.089±0.0002 kg). Although the starlings could fly in the wind tunnel for 1 h without the respirometry mask, they would only fly for 12–15 min whilst carrying the mask before they showed that they were ready to land. All masked flights from which data are reported were therefore of 12 min duration. The order of the speeds at which the birds were flown was assigned randomly.

### Metabolic power

Expired gases were collected using masks (0.7 g) made of transparent acetate film which covered the entire head and bill of the bird. Air was extracted from the mask at 3.0±0.1 l min–1 through a polyurethane tube (internal diameter 1.6 mm, external diameter 2 mm) which led over the back of the bird to the top of the downwind end of the flight chamber. Birds supported 1.6 g of tubing when they flew in the centre of the flight chamber. Expired gases did not escape from the mask since gas exchange measurements did not decline when air flow through the mask was reduced to 1.5 l min–1. All measurements involved 5 min of perching prior to flight so that the effects of handling upon metabolic rate were not superimposed upon flight costs, 12 min of flight and a further 5 min of perching. Gas concentrations in the measuring system took 1 min to reach steady state after the mask had been placed on or removed from the bird and after flight had begun or ended because of mixing of expired gases within the tubing of the analysis system. Data collected in the first minute of flight were therefore excluded from analyses. Gas concentrations were measured by a paramagnetic oxygen analyser (Taylor Servomex OA180) and an infrared carbon dioxide analyser (Hartmann and Braun URAS MT). Data from the gas analysers was recorded using a BASIC program running on a BBC B+ computer. The rate of air extraction from the mask was measured with a wet test gas flow meter (Wrights DM3A, accuracy ±0.1 %). Gases from the mask were dried with silica gel before and after passing through the flow meter. The gas analysers were calibrated daily by zeroing with oxygen-free nitrogen gas and spanning the oxygen analyser with ambient air and the carbon dioxide analyser with a gas mixture of known carbon dioxide content (Messer Griesheim). The rate of oxygen consumption (V̇O2) was calculated from equation 3b of Withers (Withers, 1977). V̇O2 and the rate of carbon dioxide production (V̇CO2) were used to calculate metabolic power (Pmet, W) from:

$\mathit{P}_{\mathrm{met}}{\ }{=}{\ }16.18{\dot{\mathit{V}}}_{\mathrm{O}_{2}}{\ }{+}{\ }5.02{\dot{\mathit{V}}}_{\mathrm{CO}_{2}}{\ },$
2

in which V̇O2 and V̇CO2 are expressed in ml s–1 stpd (Romijin and Lokhorst, 1961).

### Wingbeat kinematics

We used two Photo-Sonics series 2000-1PL 16 mm cameras (255 frames s–1; 16 mm Agfa XTR 250/XTS 400 colour negative film) at near-perpendicular viewing angles to film simultaneous lateral and dorsal views of bird 15. The films were used to measure wing morphology and wingbeat kinematics by stereophotogrammetric resection (Möller, 1998) during the middle of one respirometry flight at approximately 1 m s–1 increments in flight speed during phases in which the bird flew steadily and maintained position within the wind tunnel. Wingbeat frequency was calculated by counting the number of frames required to complete between 34 and 71 complete wing beats. Wingbeat amplitude was calculated from projected dorso-ventral excursions of the wing tip over five consecutive wingbeats. Wingspan was measured from the maximum extension of the wings in the dorsal view during the downstroke. A lateral view of the entire flight by each bird was also recorded on Hi8 tape at 50 fields s–1 using a Sony VX1 video camera. The Hi8 video recordings were subsequently viewed to assess the position and stability of the bird during each 12 min flight.

### Mechanical power

Two aerodynamic models, the vortex ring model (Rayner, 1979) and the lifting line/momentum jet (Pennycuick, 1989) model, were used to calculate Pmech. The lifting line model was chosen because computation is simple and it is used widely for calculating flight costs (Carmi et al., 1992; Speakman, 1993; Klaassen, 1995; Klaassen, 1996; Pennycuick, 1998; Weber et al., 1998; Spaar et al., 1998). The vortex ring model more closely reflects the aerodynamics of lift production by the wings and explicitly incorporates values of wingbeat kinematics as they change with flight speed (or as a consequence of flying in a wind tunnel carrying a respirometry mask), while in Pennycuick’s lifting line model these changes are implicit in the assumption of constant profile power. The mass of the respirometry mask and tube (3.2 g) were added to the bird mass, which was the average of bird mass measured at the beginning and end of each flight. The drag of the mask with 0.60 m of tube (the length of tube supported by the starlings when they flew in the centre of the flight chamber) was measured at 6, 8, 10, 12 and 14 m s–1 using a one-component strain gauge and a plastic model of a gliding starling moulded from coordinates determined by three-dimensional stereophotogrammetry (Butz et al., 1985; Gesser et al., 1998a; Gesser et al., 1998b). Mask and tube drag were added to estimated body parasite drag during calculation of Pmech by both models.

Body parasite drag in both models was estimated from the equations of Pennycuick (Pennycuick, 1975; Pennycuick, 1989). Equivalent flat plate area (Afp, m2) is:

$\mathit{A}_{\mathrm{fp}}{\ }{=}{\ }2.85{\times}10^{{\mbox{--}}3}\mathit{M}^{2/3}{\ },$
3

where M is body mass in kg. Parasite power (Ppar) is then given as a function of flight speed (V, m s–1) by:

$\mathit{P}_{\mathrm{par}}{\ }{=}\ {\frac{1}{2}}{\rho}\mathit{A}_{\mathrm{fp}}\mathit{V}^{3}{\ }$
4

(Pennycuick, 1975; Rayner, 1979), where ρ is air density (kg m–3). Possible effects of tilting of the body on parasite drag were not included. The profile drag coefficient in the vortex ring model was 0.02 (Rayner, 1979); the profile drag factor X1 in the lifting line model was 1.2 (Pennycuick, 1975; Pennycuick, 1989). Alternative models of body drag are considered in the Discussion.

### Efficiency

We used our measurements of Pmet and Pmech to calculate EFM using equation 1, with basal metabolic rate PBMR (=0.9 W) predicted allometrically from mean bird mass (Aschoff and Pohl, 1970). It might be more appropriate to express flight costs as a multiple of the resting component of metabolic rate during the active phase (1.2 W) (Aschoff and Pohl, 1970) and to use this value for PBMR in equation 1 for calculation of EFM since data were collected during the active period. This would also require reassessment of the hypothesized 1.1 factor on mechanical flight power accounting for non-flight metabolism during flight. To be consistent with previous work, resting-phase PBMR was used in equation 1, and Pmet during flight was expressed relative to resting-phase PBMR.

### Statistical analyses

Data were analysed using Minitab version 12.22. Analyses of covariance (ANCOVA) were performed by GLM ANCOVA. Tukey post-hoc comparisons were used following analyses of covariance to assess the significance of differences between birds. The level of probability at which results were regarded as significant was P=0.05. Means are presented ± s.e.m. unless stated otherwise. Most other calculations, including power calculations with the lifting line and vortex ring aerodynamic models, were performed in MS Excel97.

### Metabolic power

O2 and V̇CO2, and hence Pmet, increased abruptly from the resting level at the start of flights and decreased at the end of flights (Fig. 1). V̇O2, V̇CO2 and Pmet declined during flight; Pmet declined by 0.030±0.002 W min–1 (GLM ANCOVA, flight speed, F1,1979=1317.9, P<0.001; time, F1,1979=355.9, P<0.001; bird, F1,1979=110.3, P<0.001). The respiratory exchange ratio (RE) averaged 0.84±0.01 (N=45) across all flights. RE declined during flights from 0.87±0.01 (N=24) for bird 15 and 0.91±0.01 (N=21) for bird 19 during the second minute of flight to 0.80±0.01 (N=24) for bird 15 and 0.79±0.01 (N=21) for bird 19 in the eleventh minute. RE varied primarily with time into flight, although RE was greater for bird 19 than for bird 15 and greater during faster flight (GLM ANCOVA, time, F1,1979=465.1, P<0.001; flight speed, F1,1979=286.3, P<0.001; bird, F1,1979=11.0, P=0.001; Tukey post-hoc comparison, RE of bird 19 was 0.01 greater than that of bird 15 taking into account differences between flight speeds and time into flight).

Pmet was greater for bird 19 (mean 12.7±0.2, N=21, range 11.1–14.9 W) than for bird 15 (mean 11.7±0.2, N=24, range 10.4–13.5 W). The difference in Pmet between the birds was partly accounted for by the difference in range of flight speed, since Pmet increased with flight speed (Fig. 2) and bird 19 flew within a faster range of flight speed (8–14 m s–1) than bird 15 (6–13 m s–1) (GLM ANCOVA of the effects of the factor bird and the co-variate flight speed upon Pmet, flight speed, F1,44=47.0, P<0.001; bird, F1,44=1.1, P=0.3). Bird 19 was 10 g heavier than bird 15. The higher range of flight speed of bird 19 was consistent with the requirement of heavier animals to fly more quickly (Pennycuick, 1989). Averaged across both birds, Pmet was equivalent to 13.5±0.2 (range 11.5–16.5) times resting-phase PBMR.

Variation in Pmet across flight speeds (Fig. 2) was compatible with the U-shaped power/speed curve predicted by aerodynamic models (Pennycuick, 1975; Pennycuick, 1989; Rayner, 1979; Tucker, 1972) (this study). Curves of the form PV–1V3+γ, where V is flight speed (m s–1) and P is power (W), were fitted to the data for each bird to illustrate the expected approximate power/speed relationship in which induced power is proportional to V–1, parasite (and sometimes profile) power to V3 and basal metabolic rate (and sometimes profile power) are constant. Coefficients α, β and γ for the fitted curves are given in the caption to Fig. 2; the curves differed between birds (ANCOVA of residuals, F1,41=12.67, P=0.001). For bird 19, Pmet increased at flight speeds both greater and less than an intermediate minimum power speed (Vmp 9.9 m s–1, minimum Pmet 12.0 W), although most of the data correspond to the right-hand portion of this curve. Bird 15 would not sustain slow flight for sufficiently long (at least 1 min) for Pmet to be measured at flight speeds as low as the Vmp of 4.8 m s–1 (minimum Pmet 10.7 W) predicted from the fitted curve. The Vmp of bird 15 was unlikely to be as low as this prediction, and this technique does not estimate Vmp or the values of the coefficients with any accuracy. From visual inspection of Fig. 2, Vmp for bird 15 could lie within the range 6–9 m s–1, but is lower than that for bird 19.

### Wingbeat kinematics

The Hi8 video showed that birds alternated between steady flight, during which they remained relatively still in the upper third of the flight chamber, and undulating (flap-gliding) flight, during which they moved position within the flight chamber. We analysed wingbeat kinematics in steady flight using high-speed cinematography and used these to estimate mechanical power. Movements within the wind tunnel flight chamber in undulating flight were similar to those analysed in detail during unmasked flight by bird 19 (Rayner et al., 2001). The birds tended to flap upwards and forwards in the flight chamber, gaining speed, until they were close to the chamber roof. They then switched to gliding, with wings flexed, during which they decelerated and descended towards the rear of the flight chamber. The bird then began to flap to accelerate forwards, repeating the cycle. Each cycle during undulating flight involved 10–16 wingbeats interspersed with gliding or partial-bounding (Tobalske, 1995) and involved horizontal and vertical excursions of the flight path of 45–60 and 20–35 cm, respectively, depending on flight speed (Rayner et al., 2001). Flights were defined as steady when more than 16 wingbeats were performed without intermittent glides and without pauses between consecutive flaps during which the wings were flexed and in which the location of the bird’s beak moved within the wind tunnel by less than ±2 cm in any direction at the same point in consecutive wing beats.

Across both birds, flight was steady for 58±6 % of the time during the sixth minute of each 12 min flight (N=44); however, bird 19 flew steadily more often (88±2 %, N=21) than bird 15 (33±7 %, N=24) (ANOVA, F1,43=46.6, P<0.001). Bird 15 flew steadily more frequently during slow than fast flight, whilst bird 19 flew steadily most frequently during flight at intermediate speeds (Fig. 3). The proportion of steady flight did not influence Pmet (GLM ANCOVA of the effects of flight speed and the arcsine-transformed proportion of the flight spent in steady flight, flight speed, F1,43=60.4, P<0.001; proportion of steady flight, F1,43=2.8, P=0.1).

Analysis of the high-speed ciné film taken during periods of steady flight showed that wingbeat frequency (10.2–11.4 Hz), wingbeat amplitude (43.5–65.6°) and stroke plane angle (55.6–83.5°) increased with flight speed, whilst the downstroke ratio (0.47–0.55) was lower during faster flights (Table 1). Wingspan (mean 0.36±0.01 m, N=8) did not vary consistently with flight speed. A more detailed analysis of wingbeat kinematics will be published elsewhere (U. Möller, D. Bilo, S. Ward, J. M. V. Rayner, J. R. Speakman and W. Nachtigall, in preparation).

### Mechanical power

The drag on the model starling caused by the respirometry mask and tube increased with air speed (Fig. 4). Additional Pmech to carry the respirometry mask and tube was equivalent to 14–19 % of Pmech predicted by the lifting line model for unencumbered flight under the same conditions, and was made up of increased induced power (2–3 %), profile power (7–10 %) and power to overcome mask and tube drag (3–10 %). Predictions for drag increments with the vortex ring model were similar, but the birds adjusted their wingbeat kinematics to compensate compared with unmasked flight (Möller et al., 1997; Möller, 1998), so the increase in Pmech was not as great. Compensation for mask drag by alteration of wingbeat kinematics cannot be quantified with the lifting line model. No allowance was made for the possible effect of moments induced by the mass of the mask and tube or by the point of action of mask and tube drag.

Wingbeat kinematics (Table 1) were used in computation of Pmech by the vortex ring model for bird 15. Pmech was relatively low at 7.6 and 10.2 m s–1 because measured wingspan was greater (0.39±0.02 m, mean ± s.d.) during these flights than during those at the other speeds (0.36±0.01 m, mean ± s.d.). Pmech varied between 1.2 and 2.6 W, rose more steeply with flight speed when calculated from the vortex ring model than from the lifting line model and was greater for faster flight and for the heavier bird (Fig. 5). The data in Fig. 5 do not show conventional power curves since air density, bird mass and wingbeat kinematics varied slightly between measurements.

### Efficiency

Mean EFM was 0.17±0.002 for bird 15 (range 0.15–0.20, N=24) and 0.19±0.004 (range 0.16–0.23, N=21) for bird 19 when Pmech was calculated with the lifting line model (Fig. 6). Mean EFM for bird 15 was 0.17±0.003 (range 0.13–0.21, N=8) when Pmech was calculated with the vortex ring model. GLM ANCOVA analysis of the influence of the factor aerodynamic model and co-variate flight speed upon EFM of bird 15 showed a difference in gradients (F1,31=25.7, P<0.001) and intercepts (F1,31=23.8, P<0.001) of the relationship between EFM and flight speed between the two aerodynamic models. EFM increased more steeply with flight speed V for bird 15 if Pmech was calculated by the vortex ring model [regression EFM=(0.012±0.002)V+(0.054±0.022), r2=0.81, P<0.001, N=8] than if Pmech was calculated from the lifting line model (regression EFM not related to flight speed, P=0.1, N=24). EFM increased with flight speed for bird 19 when Pmech was calculated with the lifting line model [regression EFM=(0.007±0.001)V+(0.112±0.015), r2=0.64, P<0.001, N=24]. EFM was greater for bird 19 than for bird 15 when Pmech was calculated with the lifting line model for both birds, and EFM increased with flight speed for bird 19 whilst it did not for bird 15 [GLM ANCOVA analysis of gradients, F1,44=10.8, P=0.002, intercepts, F1,44=4.4, P=0.04; regression for bird 19, EFM=(0.007±0.001)V+(0.112±0.015), r2=0.64, P<0.001].

### Predicted metabolic power

We predicted Pmet from Pmech by using equation 1, assuming EFM=0.23 and PBMR=0.9 W (allometric prediction of resting-phase basal metabolic rate, BMR) (Aschoff and Pohl, 1970) and compared the results with our measured Pmet. This is a test of the values of Pmech calculated from mechanical models and the assumed values for EFM, PBMR and the costs of respiration and circulation from which Pmet is typically estimated when direct measurements are not available (Pennycuick, 1975; Pennycuick, 1989). Predicted Pmet was 2.9±0.4 W (24.7±4.1 %) lower than measured Pmet for bird 15 if Pmech was calculated from the lifting line model and 2.9±0.1 W (24.8±1.0 %) lower if Pmech was predicted from the vortex ring model (Fig. 7A). This showed that the increased detail from the measured wingbeat kinematics used to calculate Pmech from the vortex ring model did not improve the match between predicted and measured Pmet. Predicted Pmet was 1.8±0.2 W (14.5±1.6 %) lower than measured Pmet for bird 19 when Pmech was calculated by the lifting line model (Fig. 7B).

Predicted Pmet was substantially lower than measured Pmet for both birds when it was predicted from equation 1 using the lifting line model and a constant value of EFM of 0.23. This discrepancy could be due to too high an assumed value for EFM or inaccurate predictions of Pmech from the aerodynamic models. We therefore also compared measured Pmet with predicted Pmet from Pmech calculated with the lifting line model and the mean EFM determined for each bird (0.17 for bird 15 and 0.19 for bird 19) from equation 1. This tests whether reducing the assumed value of EFM from 0.23 to a lower, constant value removes the discrepancy between measured Pmet and predicted Pmet. The difference between measured and predicted Pmet was much less: 0.2±0.2 W (1.3±1.3 %) for bird 15 and 0.2±0.2 W (1.9±1.9 %) for bird 19, and was of the order of only one-tenth of the estimated mechanical power. The mean discrepancy between measured and predicted Pmet was 0.1±0.7 W (equivalent to 1.2±5.6 %) for bird 15 when the vortex ring model was used to calculate Pmech when EFM was 0.17. Use of the mean EFM across both birds (0.18) to predict the Pmet of both birds led to an overall discrepancy of 0.03±0.2 W (0.1±1.5 %) when Pmech was calculated from the lifting line model. The excellent overall average match between predictions and measurements of Pmet includes errors in estimates from individual flights of –16.3 to +26.9 % (Fig. 7). Predicted Pmet increased more steeply with flight speed than measured Pmet when Pmech was predicted from the vortex ring model for bird 15, with the predicted values being lower than those measured during slow flight and higher than those measured during fast flight (Fig. 7A). Predicted Pmet was consistently lower than measured Pmet when Pmech was calculated from the lifting line model for bird 15 (Fig. 7A). Predicted and measured Pmet were similar during slow flight for bird 19, although predicted Pmet increased more rapidly with flight speed (Fig. 7B). These analyses show that the discrepancy between measured and predicted Pmet can be greatly reduced by assuming a constant value of EFM of 0.18 rather than 0.23. The remaining discrepancy between predicted and measured Pmet could be due to variation in efficiency between birds and across flight speeds or to variation in Pmech that is not reflected in the aerodynamic models.

### Measured metabolic power

Our values of measured Pmet for starlings lie in the range 10.4–14.9 W for birds of mass 0.080–0.091 kg. Metabolic power increased significantly with flight speed, but not as steeply as expected from aerodynamic modelling of Pmech. To date, there are published measurements of Pmet/speed curves during wind tunnel flight for eight species of bird and five species of bat (e.g. Rayner, 1994; Butler and Bishop, 2000). Values of Pmet between 9 and 10 W, which did not vary with flight speed, have previously been reported for starlings (mean bird mass 0.073 kg) (Torre-Bueno and Larochelle, 1978); those values have generally been regarded as anomalous, especially at high flight speeds, since measured power was lower than expected and was independent of flight speed. To our knowledge, the present measurements are the first reported replication of such measurements for any vertebrate.

Our values are higher than those of Torre-Bueno and Larochelle, especially at faster flight speeds (Fig. 2). A number of factors may explain this difference, including a combination of the lower mass of the previous birds (8 g or 10 % lighter than bird 15 and 18 g or 20 % lighter than bird 19), a reduction in flight costs as a result of ground and wall effects (Rayner, 1994), which was likely to be more influential in the smaller wind tunnel used by Torre-Bueno and Larochelle (Torre-Bueno and Larochelle, 1978), the relatively poor aerodynamic performance of that tunnel (S. Vogel, personal communication) and the extra power required to overcome the drag of the respirometry mask and tube in the current experiment (Ward et al., 1998). Of these factors, only wind tunnel performance is an obvious explanation for the difference in shape of the two sets of power curves.

Our Pmet values were of the same order of magnitude expected from extrapolation of other wind tunnel oxygen uptake measurements; those by Torre-Bueno and Larochelle (Torre-Bueno and Larochelle, 1978) were markedly lower than expected (Rayner, 1994). Our values of Pmet were also similar to those predicted from cardiac output (11–12 W) (Bishop, 1997) and to those measured in free-flying starlings using the doubly labelled water technique (8.4–12.5 W) (Westerterp and Drent, 1985). From this evidence, we suggest that our measurements of Pmet in starlings are likely to be more realistic than those of Torre-Bueno and Larochelle (Torre-Bueno and Larochelle, 1978).

### Comparison with free flight

Measurements of Pmet by mask respirometry in wind tunnels may not be representative of flight costs in free-living birds for several reasons. Metabolic rate during wind tunnel flight may be raised, relative to free flight by the same birds, as a result of the stress associated with the unnatural environment and abnormal sensory input or because the bird cannot determine its flight speeds or duration. Costs of diving, for example, differ between forced and voluntary diving (Butler and Jones, 1997). Metabolic rate could potentially be lower during wind tunnel flight than during free flight if a bird is able to reduce mechanical power by exploiting wind tunnel boundary effects (Rayner, 1994). Physiological changes in our starlings during the 2 year period of captivity prior to data collection could also alter flight costs from those of free-living birds, since digestive morphology, resting metabolic rate and flight muscle mass can change over much shorter periods (Scott et al., 1996; Dietz et al., 1999; Lindström et al., 2000). Comparison with free-flight doubly labelled water measurements of starling flight costs (Westerterp and Drent, 1985) suggests that any distortion in power due to wind tunnel flight or the use of captive birds is likely to be relatively small in magnitude. However, we cannot rule out the possibility that the magnitude of such distortion depends on flight speed.

### Power curves

The discrepancies between measured Pmet and Pmet predicted from Pmech by equation 1 suggest either that Pmech calculated by aerodynamic models is too small or that EFM is lower than the value of 0.23 that is normally assumed, especially during faster flight. There have been few independent controls on the aerodynamic models; values of body drag are based on limited experimental evidence that is currently being reassessed (see below), while measurements of wing drag, especially in smaller birds, are few; the magnitudes of induced (or vortex) power have been confirmed by flow visualization experiments, but represent a relatively small proportion of mechanical power at normal flight speeds (Rayner, 1999). Direct measurements of force production by the pectoral muscles in magpies (Dial et al., 1997) suggest that Pmech may form an L-shaped rather than a U-shaped relationship with flight speed and that aerodynamic models overestimate Pmech during faster flight. The difference between these measurements and predictions of the vortex ring model is relatively small, although the model does appear to overestimate power slightly for fast flight (Rayner, 1999; Rayner and Ward, 1999). Our calculated Pmech values rise with flight speed, but if they are to be consistent with our measured Pmet, EFM must increase with flight speed. This result could be caused by systematic overestimation of Pmech during faster flights rather than by a change in EFM across flight speeds. Overestimation of Pmech by aerodynamic models during faster flight is consistent with the measured mechanical power production by starling pectoralis muscle of 1.1 W during flight at 13.7 m s–1 (Biewener et al., 1992), which is lower than Pmech predicted by the aerodynamic models at this speed (2.0–2.3 W, depending upon the aerodynamic model and bird), although some of this discrepancy will be accounted for by work performed by other muscles.

The starlings sometimes used intermittent bounding (flap-bounding) or undulating (flap-gliding) flight (Rayner, 1985; Tobalske, 1995; Tobalske and Dial, 1996; Rayner et al., 2001) during our measurements of flight metabolic rate between the bouts of continuous flapping flight upon which the calculations of Pmech are based. The use of intermittent flight (Rayner et al., 2001) or wind tunnel boundary interference effects (Rayner, 1994) could each reduce Pmech by a maximum of 10 %, although it is not known whether birds exploit these mechanisms effectively. The proportion of intermittent flap-gliding compared with steady flapping during flights was not related to Pmet, so the intermittent flight pattern sometimes adopted by our birds apparently did not have an important influence upon flight metabolic rate. Intermittent flight patterns were used more frequently during fast flight both by one of our starlings and by those studied by Tobalske (Tobalske, 1995), and this may reduce Pmech during fast flight in ways that have not yet been quantified by aerodynamic modelling. This would be consistent with the steeper rise in Pmech than in Pmet as flight speed increased, although the discrepancy may equally arise for other reasons, as discussed below.

### Body drag, mechanical power and efficiency

One of the three major drag components in the aerodynamic models is the body parasite drag (equation 4). We used a constant value of parasite drag depending on body mass from a formula from Pennycuick (Pennycuick, 1975) in both models. For a bird of mass 0.08 kg, the equivalent flat plate area is 5.3×10–4 m2. The frontal projected area of a starling is approximately 2.1×10–3 m2 (W. J. Maybury, personal communication), so the body drag coefficient CDb based on frontal projected area is estimated to be approximately 0.26. Tucker (Tucker, 1973) measured a slightly higher value of drag on a starling flying at 11 m s–1, corresponding to CDb≈0.31. Recent measurements on starling bodies in a wind tunnel give CDb in the range 0.35 (at 6 m s–1) to 0.20 (at 15 m s–1) (Maybury, 2000; Maybury and Rayner, 2001), which is comparable with predictions from the Pennycuick formula and Tucker measurement at lower air speeds, but is markedly lower at higher speeds. Gesser et al. (Gesser et al., 1998a; Gesser et al., 1998b) made a preliminary report of CDb of the order of 0.24 on a smooth model starling at 8 m s–1 (this is the same model that we used to determine mask and tube drag). Taken together, this experimental evidence suggests that the drag formula of Pennycuick’s model is of the correct order of magnitude at the lower range of flight speeds over which we measured power but may not capture a decrease in drag coefficient with flight speed.

Values of body parasite drag close to those predicted (Pennycuick, 1975) have recently been determined for birds of similar size to that of starlings whilst descending rapidly with wings folded at the end of migratory flights (Hedenström and Liechti, 2001). Pennycuick et al. (Pennycuick et al., 1995) have argued, on the basis of a mismatch between power model predictions and observed wind tunnel flight patterns, that the body drag values used in aerodynamic models may be too high by a factor of 5. There is no direct experimental evidence supporting drag values as low as this in small birds, and the available measurements appear to confirm that, in small passeriform birds, the predictions of the Pennycuick (Pennycuick, 1975) formula are of the right order of magnitude, although parasite drag probably decreases with flight speed. Measured drag does appear to be much lower than predicted by equation 4 in larger birds. If drag measurement experiments were in error by as much as suggested by Pennycuick et al. (Pennycuick et al., 1995), the predicted Pmech of our starlings would be reduced by 25–40 % (in addition, predicted optimum flight speeds would be increased by approximately 50 %). A decrease in body parasite drag with flight speed may, however, form a partial explanation of why our estimates of EFM increase with speed.

We have estimated EFM in the range 0.13–0.23, varying between birds and depending on flight speed. These values are significantly below the constant 0.23–0.25 normally recommended (Norberg, 1990). The dilemma we face is that, if Pmech is overestimated by the aerodynamic models (as is most likely for faster flight), then EFM will be further reduced below the values that we have calculated; Pennycuick’s proposal of drag coefficients CDb in the range 0.04–0.07 would be associated with EFM values for starlings in the range 0.10–0.14.

Accurate prediction of Pmet from Pmech by equation 1 relies upon four assumptions: (i) that the value of Pmech predicted by aerodynamic models is correct, (ii) that an appropriate value of EFM is used, (iii) that PBMR should be added to metabolic rate during flight and (iv) that additional costs of respiration and circulation each contribute 5 % to Pmet. Predictions of Pmet from equation 1 are most sensitive to the value of EFM, but there is no direct experimental evidence for the assumptions about basal metabolic rate, respiration and circulation. Taken together, 16–19 % of Pmet in our starlings originates from PBMR (6–9 %) and the combined cost of additional respiration (5 %) and circulation (5 %). Some metabolism presumably occurs during flight other than in the flight muscles and the organs needed to supply them with oxygenated blood, but the assumed value of PBMR may either be too high or too low to account for this. Metabolism not associated with flight could be lower than PBMR because metabolic rate is reduced in parts of the body not required during exercise (Butler et al., 1988) and the heat produced in the flight muscles substitutes for thermoregulatory heat production. Alternatively, metabolism not associated with flight could be higher than nocturnal PBMR because our starlings were not post-absorbative and were flown during the day. The birds also flew at air temperatures below the thermal neutral zone, but metabolic rate would not need to be increased to maintain body temperature since the additional heat generated as a by-product of muscular activity more than compensates for the increase in heat loss due to the greater surface area and movement of air past a flying bird (Ward et al., 1999).

The assumptions that respiration and circulation each contribute 5 % to metabolic rate during flight and that PBMR should be added to metabolic rate during flight will not lead to error in Pmet when Pmet is calculated from equation 1 if EFM is also calculated using equation 1 from simultaneous estimates of Pmet and Pmech, using the same assumptions about the values of PBMR and the costs of respiration and circulation during both calculation of EFM given Pmet and Pmech and prediction of Pmet given EFM and Pmech. If the combined costs of additional respiration and circulation or PBMR were to increase, the calculated EFM would increase proportionally, so that, if these revised data were used to predict Pmet, the result would be the same as if the lower EFM and costs of respiration, circulation and basal metabolic rate were used. It might be better to calculate whole-animal efficiency, EW (where EW=Pmech/Pmet), and to rearrange this equation to predict Pmet. This would avoid making assumptions that cannot be tested about the contributions of respiration, circulation and PBMR to metabolic rate during flight. However, the value of EFM is insensitive to the values of PBMR and the costs of circulation and respiration: halving or doubling these values changes EFM by less than 0.02. We have used equation 1 to calculate EFM to make our values consistent with those in previous work and because changing the formula does not increase the accuracy of predicted Pmet. An EW of 0.15 is equivalent to an EFM of 0.18 for our starlings; the value 0.23 that has normally been used for EFM in equation 1 is equivalent to an EW of 0.19.

### Estimation of metabolic power

The aerodynamic models that are used to calculate Pmech (Pennycuick, 1975; Pennycuick, 1989) are important because it is difficult to measure Pmet directly in the field, and measurements under controlled conditions such as during wind tunnel flight cannot necessarily be extrapolated to other flight conditions or to other species. Estimated Pmet from aerodynamic models is highly sensitive to EFM, mainly because the mechanical work is a small component of the total. A change in EFM of only 0.01 represents a 5 % change in Pmet.

Uncertainty in the efficiency of flying birds means that it is not possible to predict Pmet accurately from Pmech calculated by aerodynamic modelling and an assumed uniform value of EFM. The discrepancy was reduced, but not removed, in our starlings by the use of a fixed value for EFM of 0.18 rather than 0.23. Calculation of Pmech by the more detailed vortex ring model did not improve the accuracy of predicted Pmet if a constant value of EFM was assumed, even though changes in wingbeat kinematics due to flying in a wind tunnel carrying a respirometry mask were taken into account by this model. If Pmet for birds of the size range of starlings (0.075–0.1 kg) is to be extrapolated from Pmech calculated by the most popular method (the lifting line model) (Pennycuick, 1989), program 1A), we suggest the use of an EFM of 0.18. This value of EFM is not appropriate for substantially larger or smaller birds because efficiency is expected to vary with size (Rayner, 1990; Rayner, 1995). Muscle efficiency increases with body size in running mammals and birds (Heglund et al., 1982; Heglund and Cavagna, 1985) and in flying insects (Casey, 1988; Casey and Ellington, 1989), and EFM may equally be expected to increase with size in flying birds. An approximate estimate of the size-scaling of whole-body efficiency (EW) may be determined from the scaling with body mass of the measured metabolic power (Pmet, W) for flight in birds and bats:

$\mathit{P}_{\mathrm{met}}{\ }{=}{\ }64.64\mathit{M}^{0.782}{\ }$
5

(Rayner, 1990), where M is body mass in kg, and calculated mechanical power (Pmech, W):

$\mathit{P}_{\mathrm{mech}}{\ }{=}{\ }14.95\mathit{M}^{1.161}\ {\ }$
6

(Rayner, 1988), as:

$\mathit{E}_{\mathrm{W}}{\ }{=}{\ }\mathit{P}_{\mathrm{mech}}/\mathit{P}_{\mathrm{met}}{\ }{=}{\ }0.23\mathit{M}^{0.379}{\ }.$
7

EW increases with size because Pmet, like other physiological variables, scales approximately as mass0.75 (Tucker, 1973), while Pmech scales approximately as mass7/6, as expected on aerodynamic grounds (Pennycuick, 1975; Rayner, 1988; Rayner, 1990; Rayner, 1995; Speakman and Racey, 1991). For a bird of mass 0.08 kg comparable with our starlings, equation 7 predicts a rather low EW of 0.088. Whole-body efficiency (EW, W) and flight muscle efficiency (EFM, W) are related by:

$\mathit{E}_{\mathrm{FM}}{\ }{=}{\ }\mathit{E}_{\mathrm{W}}/{[}0.91{\ }{\mbox{--}}{\ }(\mathit{P}_{\mathrm{BMR}}/\mathit{P}_{\mathrm{met}}){]}$
8

(derived from equations 1 and 7). With PBMR=0.9 W and Pmet=9 W (representative values for starlings), equation 8 gives EFM≈0.10. These values for efficiency are somewhat lower than we have measured (Fig. 6), but they confirm our argument that efficiency increases with size across the range of birds and bats and, in smaller species, is much lower than a uniform value of 0.23. If necessary, these allometric equations could be used to estimate EFM in birds; however, it must be noted that equations 5 and 6 are derived from analysis of independent and diverse observations, and the experiments we report here on starlings remain the only measurements to date in which metabolic and mechanical power have been determined simultaneously in flying vertebrates during forward flight.

The physiological or mechanical mechanisms responsible for variation in efficiency with size are not known, and we cannot be certain that equation 7 should apply to all avian taxa. Our arguments take no account of the anticipated and theoretical maximum efficiency of 0.25, which might be attained in larger birds. A possible explanation for the difference between the values of EFM determined directly in bats and hummingbirds and the values of EP measured in tilting wind tunnels is therefore that the larger birds, which give rise to the EP measurements, have higher efficiencies.

Lower values of efficiency are appropriate for smaller bats and hummingbirds (0.0035–0.01 kg): a value of 0.12±0.01 is the mean of the average efficiencies calculated from simultaneous estimates of Pmet and Pmech determined during hovering by the nectar-feeding bat Glossophaga soricina (0.15) (Norberg et al., 1993) and ruby-throated hummingbirds hovering in heliox (0.10) (Chai and Dudley, 1995) and in air (0.12) (Chai et al., 1998). The lower EFM during hovering flight is also consistent with increases in efficiency with flight speed and bird mass, although further studies are required to confirm this pattern. Studies of larger birds during which Pmet is measured simultaneously with wingbeat kinematics that allow calculation of Pmech from aerodynamic models would be particularly valuable.

Further measurements of efficiency during flight may reveal patterns in variation across individuals and species with different flight morphology, and above all with different mass, that will allow Pmet to be predicted from Pmech with greater accuracy. It is also important to determine the mechanisms that cause efficiency to vary. Given that further measurement of efficiency requires simultaneous measurement of Pmet and Pmech, it might be as accurate, as well as less technically demanding, to concentrate on measurement of Pmet in more species of flying animal, so that Pmet can be predicted from body mass with confidence by using allometric equations such as those presented by Masman and Klaassen (Masman and Klaassen, 1987), Rayner (Rayner, 1991), Speakman and Racey (Speakman and Racey, 1991), Bishop (Bishop, 1997), Winter and von Helversen (Winter and von Helversen, 1998) and Butler and Bishop (Butler and Bishop, 2000).

We thank R. Kockler and F. Wedekind for assistance with drag measurements from the model bird, I. Breves for help looking after the starlings, C. Hambly and P. Viscardi for help with analysis of the Hi8 videos, W. J. Maybury and S. Vogel for access to unpublished information, M. Glander and W. Pattullo for technical assistance and the anonymous referees for comments that helped us to improve the manuscript. This research was supported by BBSRC grants J36150 to J.R.S. and J.M.V.R. and S03843 to J.M.V.R. and British Council grant 680 to J.R.S., S.W. and W.N.

Aschoff, J. and Pohl, H. (
1970
). Der Ruheumsatz von Vögeln als Funktion der Tageszeit und der Körpergrösse.
J. Orn
.
111
,
38
–47.
Bernstein, M. H., Thomas, S. P. and Schmidt-Nielsen, K. (
1973
). Power input during flight of the fish crow, Corvus ossifragus.
J. Exp. Biol
.
58
,
401
–410.
Biesel, W., Butz, H. and Nachtigall, W. (
1985
). Einsatz spezieller Verfahren der Windkanaltechnik zur Untersuchung des freien Gleitflugs von Vögeln. In BIONA Report 3 (ed. W. Nachtigall), pp. 109–122. Stuttgart: Gustav Fischer Verlag.
Biewener, A. A., Dial, K. P. and Goslow, G. E. (
1992
). Pectoralis muscle force and power output during flight in the starling.
J. Exp. Biol
.
164
,
1
–18.
Bishop, C. M. (
1997
). Heart mass and the maximum cardiac output of birds and mammals: implications for estimating the maximum aerobic power input of flying animals.
Phil. Trans. R. Soc. Lond. B
352
,
447
–456.
Bishop, C. M. and Bulter, P. J. (
1995
). Physiological modelling of oxygen consumption in birds during flight.
J. Exp. Biol
.
198
,
2153
–2163.
Bryant, D. M. (
1997
). Energy expenditure in wild birds.
Proc. Nutr. Soc
.
56
,
1025
–1039.
Butler, P. J. and Bishop, C. (
2000
). Flight. In Sturkie’s Avian Physiology (ed. G. C. Whittow), pp. 391–436. London: Academic Press.
Butler, P. J. and Jones, D. R. (
1997
). Physiology of diving birds and mammals.
Physiol. Rev
.
77
,
837
–899.
Butler, P. J., Turner, D. L., Al-Wassia, A. and Bevan, R. M. (
1988
). Regional distribution of blood flow during swimming in the tufted duck (Aythya fuligula).
J. Exp. Biol
.
135
,
461
–472.
Butler, P. J. and Woakes, A. J. (
1990
). The physiology of bird flight. In Bird Migration (ed. E. Gwinner), pp. 300–317. Berlin: Springer-Verlag.
Butler, R. W., Williams, T. D., Warnock, N. and Biship, M. A. (
1997
). Wind assistance: a requirement of shorebirds?
Auk
114
,
456
–466.
Butz, H., Biesel, W. and Nachtigall, W. (
1985
). Einsatz spezieller Methoden der Photogrammetrie zur Untersuchung der Flügelgeometrie frei gleitfliegender Tauben. In BIONA Report 3 (ed. W. Nachtigall), pp. 123–138. Stuttgart: Gustav Fischer Verlag.
Carmi, N., Pinshow, B., Porter, W. P. and Jaeger, J. (
1992
). Water and energy limitations on flight duration in small migrating birds.
Auk
109
,
268
–276.
Casey, T. M. (
1988
). Oxygen consumption during flight. In Insect Flight (ed. G. Goldsworthy and C. Wheeler), pp. 257–272. CRC UniScience Series. Boca Raton, FL: CRC Press.
Casey, T. M. and Ellington, C. P. (
1989
). Energetics of insect flight. In Energy Transformations in Cells and Animals (ed. W. Wieser and E. Gnaiger), pp. 200–210. Stuttgart: Thieme Verlag.
Chai, P., Chang, A. C. and Dudley, R. (
1998
). Flight thermogenesis and energy conservation in hovering hummingbirds.
J. Exp. Biol
.
201
,
963
–968.
Chai, P. and Dudley, R. (
1995
). Limits to vertebrate locomotor energetics suggested by hummingbirds hovering in heliox.
Nature
377
,
722
–725.
Dial, K. P., Biewener, A. A., Tobalske, B. W. and Warrick, D. R. (
1997
). Mechanical power output of bird flight.
Nature
390
,
67
–70.
Dietz, M. W., Piersma, T. and Dekinga, A. (
1999
). Body-building without power training: endogenously regulated pectoral muslce hypertrophy in confined shorebirds.
J. Exp. Biol
.
202
,
2831
–2837.
Gesser, R., Wedekind, F., Kockler, R. and Nachtigall, W. (
1998
a). Aerodynamische Untersuchungen an natunrahen Starenmodellen. I. Grundlegende Ergebnisse. In Biona Report 13, Motion Systems (ed. R. Blickhan, A. Wisser and W. Nachtigall), pp. 229–230. Jena: Gustav Fischer Verlag.
Gesser, R., Wedekind, F., Kockler, R. and Nachtigall, W. (
1998
b). Aerodynamische Untersuchungen an natunrahen Starenmodellen. II. Flügel-Rumpf Interferenzen. In Biona Report 13, Motion Systems (ed. R. Blickhan, A. Wisser and W. Nachtigall), pp. 257–258. Jena: Gustav Fischer Verlag.
Hedenström, A. and Alerstam, T. (
1995
). Optimal flight speed of birds.
Phil. Trans. R. Soc. Lond. B
348
,
471
–487.
Hedenström, A. and Alerstam, T. (
1997
). Optimum fuel loads in migratory birds: distinguishing between time and energy minimisation.
J. Theor. Biol
.
189
,
227
–234.
Hedenström, A. and Liechti, F. (
2001
). Field estimates of body drag coefficient on the basis of dives in passerine birds.
J. Exp. Biol
.
204
,
1167
–1175.
Heglund, N. C. and Cavagna, G. A. (
1985
). Efficiency of vertebrate locomotory muscles.
J. Exp. Biol
.
115
,
283
–292.
Heglund, N. C., Fedak, M. A., Taylor, C. R. and Cavagna, G. A. (
1982
). Energetics and mechanics of terrestrial locomotion.
J. Exp. Biol
.
97
,
57
–66.
Hill, A. V. (
1938
). The heat of shortening and the dynamic constants of muscle.
Proc. R. Soc. Lond. B
126
,
136
–195.
Hudson, D. M. and Bernstein, M. H. (
1983
). Temperature regulation and heat balance in the white-necked raven, Corvus cryptoleucos.
J. Exp. Biol
.
90
,
267
–281.
Klaassen, M. (
1995
). Water and energy limitations on flight range.
Auk
122
,
260
–262.
Klaassen, M. (
1996
). Metabolic constraints on long-distance migration in birds.
J. Exp. Biol
.
199
,
57
–64.
Lindström, Å. and Alerstam, T. (
1992
). Optimal fat loads in migrating birds: a test of the time minimisation hypothesis.
Am. Nat
.
140
,
477
–491.
Lindström, Å., Kvist, A. Piersma, T., Dekinga, A. and Dietz, M. W. (
2000
). Avian prectoral muscle size rapidly tracks body mass changes during flight, fasting and fuelling.
J. Exp. Biol
.
203
,
913
–919.
Masman, D. and Klaassen, M. (
1987
). Energy expenditure during free flight in trained and free living Eurasian kestrels Falco tinnunclus.
Auk
104
,
603
–616.
Maybury, W. J. (
2000
). The aerodynamics of bird bodies. PhD thesis, University of Bristol, UK.
Maybury, W. J. and Rayner, J. M. V. (
2001
). The avian tail reduces body parasite drag by controlling flow separation and vortex shedding.
Proc. R. Soc. Lond. B
268
,
1405
–1410.
Möller, U. (
1998
). Aspekte der Flugkinematik des (Gemeinen) Stars (Sturnus vulgaris) beim Windkanalflug mit und ohne respiratorischer Maske. Diplomarbeit an der MatNatFak, Universität des Saarlandes.
Möller, U., Ward, S., Bilo, D., Speakman, J. R., Rayner, J. M. V. and Nachtigall, W. (
1997
). Mechanics of starling flight in a wind tunnel.
J. Morph
.
232
,
326
.
Nachtigall, W. (
1997
). Methoden und Techniken zur Bewegungsanalyse – schwimmende und fliegende Tiere. In BIONA Report 11 (ed. A. Wisser, D. Bilo, A Kesel and B. Möhl), pp. 1–56. Stuttgart: Gustav Fischer Verlag.
Norberg, U. M. (
1990
). Vertebrate Flight. Berlin: Springer.
Norberg, U. M., Kunz, T. H., Steffensen, J. F., Winter, Y. and von Helversen, O. (
1993
). The cost of hovering and forward flight in a nectar-feeding bat, Glossophaga soricina, estimated from aerodynamic theory.
J. Exp. Biol
.
182
,
207
–227.
Pennycuick, C. J. (
1975
). Mechanics of flight. In Avian Biology, vol. 5 (ed. D. S. Farner and J. R. King), pp. 1–75. London: Academic Press.
Pennycuick, C. J. (
1989
). Bird Flight Performance. A Practical Calculation Manual. Oxford: Oxford University Press.
Pennycuick, C. J. (
1998
). Computer simulation of fat and muscle burn in long-distance bird migration.
J. Theor. Biol
.
191
,
47
–61.
Pennycuick, C. J., Klaassen, M., Kvist, A. and Lindström, Å. (
1995
). Wingbeat frequency and the body drag anomaly: wind-tunnel observations on a thrush nightingale (Luscinia luscinia) and a teal (Anas crecca).
J. Exp. Biol
.
199
,
2757
–2765.
Rayner, J. M. V. (
1979
). A vortex theory of animal flight. Part 2. The forward flight of birds.
J. Fluid Mech
.
91
,
731
–763.
Rayner, J. M. V. (
1985
). Bounding and undulating flight in birds.
J. Theor. Biol
.
117
,
47
–77.
Rayner, J. M. V. (
1986
). Vertebrate flapping flight mechanics and aerodynamics and the evolution of flight in bats. In BIONA Report 5 (ed. W. Nachtigall) pp. 27–74. Stuttgart: Gustav Fischer Verlag.
Rayner, J. M. V. (
1988
). Form and function in avian flight.
Curr. Orn
.
5
,
1
–66.
Rayner, J. M. V. (
1990
). The mechanics of flight and bird migration. In Bird Migration (ed. E. Gwinner), pp. 283–299. Berlin: Springer-Verlag.
Rayner, J. M. V. (
1991
). Avian flight evolution and the problem of Archaeopteryx. In Biomechanics in Evolution, Seminar Series, Soc. Exp. Biol. 36 (ed. J. M. V. Rayner and R. J. Wootton), pp. 183–212. Cambridge: Cambridge University Press.
Rayner, J. M. V. (
1994
). Aerodynamic corrections for the flight of birds and bats in wind tunnels.
J. Zool., Lond
.
234
,
537
–563.
Rayner, J. M. V. (
1995
). Flight mechanics and constraints on flight performance.
Isr. J. Zool
.
41
,
321
–342.
Rayner, J. M. V. (
1999
). Estimating power curves for flying vertebrates.
J. Exp. Biol
.
202
,
3449
–3461.
Rayner, J. M. V. and Swaddle, J. P. (
2000
). Aerodynamics and behaviour of moult and take-off in birds. In Biomechanics and Animal Behaviour (ed. P. Domenici and R. W. Blake), pp. 125–157. London: Bios.
Rayner, J. M. V., Viscardi, P. W., Ward, S. and Speakman, J. R. (
2001
). Aerodynamics and energetics of intermittent flight in birds.
Am. Zool
.
41
,
188
–204.
Rayner, J. M. V. and Ward, S. (
1999
). On the power curves of flying birds.
Proceedings of the XXII International Ornithology Congress
(ed. N. J. Adams and R. H. Slotow), pp. 1786–1809. Johannesburg: BirdLife South Africa.
Romijin, C. and Lokhorst, W. (
1961
). Some aspects of energy metabolism in birds. In Energy Metabolism. Second Symposium on Energy Metabolism, EAAP 10 (ed. E. Brouwer and A. J. H. Van Es), pp. 49–58. Wageningen, Netherlands: Publisher.
Ruben, J. (
1991
). Reptilian physiology and the flight capacity of Archaeopteryx.
Evolution
45
,
1
–17.
Schmidt-Nielsen, K. (
1972
). Locomotion: Energy cost of swimming, running and flying.
Science
177
,
222
–228.
Scott, I., Mitchell, P. I. and Evans, P. R. (
1996
). How does variation in body composition affect the basal metabolic rate of birds?
Funct. Ecol
.
10
,
307
–313.
Spaar, R., Stark, H. and Liechti, F. (
1998
). Migratory flight strategies of Levant sparrowhawks: time or energy minimisation?
Anim. Behav
.
56
,
1185
–1197.
Speakman, J. R. (
1993
). Flight capabilities in Archaeopteryx.
Evolution
47
,
340
–366.
Speakman, J. R. and Racey, P. A. (
1991
). No cost of echolocation for bats in flight.
Nature
350
,
421
–423.
Thomas, S. P. (
1975
). Metabolism during flight in two species of bats, Phyllostomus hastatus and Pteropus gouldii.
J. Exp. Biol
.
63
,
273
–293.
Tobalske, B. W. (
1995
). Neuromuscular control and kinematics of intermittent flight in the European starling (Sturnus vulgaris).
J. Exp. Biol
.
198
,
1259
–1273.
Tobalske, B. W. and Dial, K. P. (
1996
). Flight kinematics of black-billed magpies and pigeons over a wide range of speeds.
J. Exp. Biol
.
199
,
263
–280.
Torre-Bueno, J. R. and Larochelle, J. (
1978
). The metabolic cost of flight in unrestrained birds.
J. Exp. Biol
.
75
,
223
–229.
Tucker, V. A. (
1972
). Metabolism during flight in the laughing gull, Larus atricilla.
Am. J. Physiol
.
222
,
237
–245.
Tucker, V. A. (
1973
). Bird metabolism during flight: evaluation of a theory.
J. Exp. Biol
.
58
,
689
–709.
Walsberg, G. E. (
1990
). Problems inhibiting energetic analyses of migration. In Bird Migration (ed. E. Gwinner), pp. 413–421. Berlin: Springer-Verlag.
Ward, S., Möller, U., Rayner, J. M. V., Jackson, D., Nachtigall, W. and Speakman, J. R. (
1998
). Power requirements by starlings during wind tunnel flight.
Biol. Cons. Fauna
102
,
1589
–1602.
Ward, S., Rayner, J. M. V., Möller, U., Jackson, D. M., Nachtigall, W. and Speakman, J. R. (
1999
). Heat transfer from starlings Sturnus vulgaris during flight.
J. Exp. Biol
.
202
,
1589
–1602.
Weathers, W. W., Buttemer, W. A., Hayworth, A. M. and Nagy, K. A. (
1984
). An evaluation of time-budget estimates of daily energy expenditure in birds.
Auk
101
,
459
–472.
Weber, T. P., Ens, B. J. and Houston, A. I. (
1998
). Optimal avian migration: a dynamic model of fuel stores and site use.
Evol. Ecol
.
12
,
377
–401.
Weber, T. P., Houston, A. I. and Ens, B. J. (
1994
). Optimal departure fat loads and stopover site use in avian migration – an analytical model.
Proc. R. Soc. Lond. B
258
,
29
–34.
Westerterp, K. and Drent, R. (
1985
). Energetic costs and energy-saving mechanisms in parental care of free-living passerine birds as determined by the D2O method.
Acta XVIII Int. Orn. Congr
.
1
,
392
–398.
Winter, Y. and von Helversen, O. (
1998
). The energy cost of flight: do small bats fly more cheaply than birds?
J. Comp. Physiol. B
168
,
105
–111.
Withers, P. C. (
1977
). Measurement of V̇O2, V̇CO2, and evaporative water loss with a flow-through mask.
J. Appl. Physiol
.
42
,
120
–123.


text:
Brianne B.

# How many cases of each will maximize the trucker's sales?

A trucker carries cases of Big Red and Dr. Pepper into a region. His truck can carry 1000 cases total. He must carry at least 200 cases of Big Red and at most 400 cases of Dr. Pepper. Each case of Big Red is charged a $5 "import fee" and each case of Dr. Pepper is charged$10. The driver has abide the of $6000 to spend on import fees. If a case of Big Red sells for$20 and a case of Dr. Pepper sells for \$25, how many cases of each should the driver carry in order to maximize his sales?

By:

Tutor
5.0 (157)

Math can be fun!

## Still looking for help? Get the right answer, fast.

Get a free answer to a quick problem.
Most questions answered within 4 hours.

#### OR

Choose an expert and meet online. No packages or subscriptions, pay only for the time you need.

format: plain


text:
Under the auspices of the Computational Complexity Foundation (CCF)

REPORTS > DETAIL:

### Revision(s):

Revision #2 to TR21-133 | 23rd January 2022 18:12

#### Testing Distributions of Huge Objects

Revision #2
Authors: Oded Goldreich, Dana Ron
Accepted on: 23rd January 2022 18:12
Keywords:

Abstract:

We initiate a study of a new model of property testing that is a hybrid of testing properties of distributions and testing properties of strings.
Specifically, the new model refers to testing properties of distributions, but these are distributions over huge objects (i.e., very long strings).
Accordingly, the model accounts for the total number of local probes into these objects (resp., queries to the strings)as well as for the distance between objects (resp., strings).
Specifically, the distance between distributions is defined as the earth mover's distance with respect to the relative Hamming distance between strings.

We study the query complexity of testing in this new model, focusing on three directions.
First, we try to relate the query complexity of testing properties in the new model to the sample complexity of testing these properties in the standard distribution testing model.
Second, we consider the complexity of testing properties that arise naturally in the new model (e.g., distributions that capture random variations of fixed strings).
Third, we consider the complexity of testing properties that were extensively studied in the standard distribution testing model: Two such cases are uniform distributions
and pairs of identical distributions, where we obtain the following results.

\begin{itemize}
\item
Testing whether a distribution over $n$-bit long strings is uniform on some set of size $m$ can be tested with query complexity ${\widetilde O}(m/\epsilon^3)$, where $\epsilon>(\log_2m)/n$ is the proximity parameter.
\item
Testing whether two distribution over $n$-bit long strings that have support size at most $m$ are identical can be tested with query complexity ${\widetilde O}(m^{2/3}/\epsilon^3)$.
\end{itemize}
Both upper bounds are pretty tight; that is, for $\epsilon=\Omega(1)$, the first task requires $\Omega(m^c)$ queries for any $c<1$ and $n=\omega(\log m)$, whereas the second task requires $\Omega(m^{2/3})$ queries.
Note that the query complexity of the first task is higher than the sample complexity of the corresponding task in the standard distribution testing model, whereas in the case of the second task the bounds almost match.

Changes to previous version:

Correcting inaccuracies in the stmt of Thm 1.6 (and 2.2), and correcting corresponding inaccuracies in the proof of Thm 2.2 and in Sec 2.3.

Revision #1 to TR21-133 | 16th September 2021 18:15

#### Testing Distributions of Huge Objects

Revision #1
Authors: Oded Goldreich, Dana Ron
Accepted on: 16th September 2021 18:15
Keywords:

Abstract:

We initiate a study of a new model of property testing that is a hybrid of testing properties of distributions and testing properties of strings.
Specifically, the new model refers to testing properties of distributions, but these are distributions over huge objects (i.e., very long strings).
Accordingly, the model accounts for the total number of local probes into these objects (resp., queries to the strings)as well as for the distance between objects (resp., strings).
Specifically, the distance between distributions is defined as the earth mover's distance with respect to the relative Hamming distance between strings.

We study the query complexity of testing in this new model, focusing on three directions.
First, we try to relate the query complexity of testing properties in the new model to the sample complexity of testing these properties in the standard distribution testing model.
Second, we consider the complexity of testing properties that arise naturally in the new model (e.g., distributions that capture random variations of fixed strings).
Third, we consider the complexity of testing properties that were extensively studied in the standard distribution testing model: Two such cases are uniform distributions
and pairs of identical distributions, where we obtain the following results.

\begin{itemize}
\item
Testing whether a distribution over $n$-bit long strings is uniform on some set of size $m$ can be tested with query complexity ${\widetilde O}(m/\epsilon^3)$, where $\epsilon>(\log_2m)/n$ is the proximity parameter.
\item
Testing whether two distribution over $n$-bit long strings that have support size at most $m$ are identical can be tested with query complexity ${\widetilde O}(m^{2/3}/\epsilon^3)$.
\end{itemize}
Both upper bounds are pretty tight; that is, for $\epsilon=\Omega(1)$, the first task requires $\Omega(m^c)$ queries for any $c<1$ and $n=\omega(\log m)$, whereas the second task requires $\Omega(m^{2/3})$ queries.
Note that the query complexity of the first task is higher than the sample complexity of the corresponding task in the standard distribution testing model, whereas in the case of the second task the bounds almost match.

Changes to previous version:

Adding a new result (which appears as Thm 1.6)
and detailing the proof of Thm 5.2.

### Paper:

TR21-133 | 12th September 2021 14:23

#### Testing Distributions of Huge Objects

TR21-133
Authors: Oded Goldreich, Dana Ron
Publication: 12th September 2021 14:24
Keywords:

Abstract:

We initiate a study of a new model of property testing that is a hybrid of testing properties of distributions and testing properties of strings.
Specifically, the new model refers to testing properties of distributions, but these are distributions over huge objects (i.e., very long strings).
Accordingly, the model accounts for the total number of local probes into these objects (resp., queries to the strings)as well as for the distance between objects (resp., strings).
Specifically, the distance between distributions is defined as the earth mover's distance with respect to the relative Hamming distance between strings.

We study the query complexity of testing in this new model, focusing on three directions.
First, we try to relate the query complexity of testing properties in the new model to the sample complexity of testing these properties in the standard distribution testing model.
Second, we consider the complexity of testing properties that arise naturally in the new model (e.g., distributions that capture random variations of fixed strings).
Third, we consider the complexity of testing properties that were extensively studied in the standard distribution testing model: Two such cases are uniform distributions
and pairs of identical distributions, where we obtain the following results.

\begin{itemize}
\item
Testing whether a distribution over $n$-bit long strings is uniform on some set of size $m$ can be tested with query complexity ${\widetilde O}(m/\epsilon^3)$, where $\epsilon>(\log_2m)/n$ is the proximity parameter.
\item
Testing whether two distribution over $n$-bit long strings that have support size at most $m$ are identical can be tested with query complexity ${\widetilde O}(m^{2/3}/\epsilon^3)$.
\end{itemize}
Both upper bounds are pretty tight; that is, for $\epsilon=\Omega(1)$, the first task requires $\Omega(m^c)$ queries for any $c<1$ and $n=\omega(\log m)$, whereas the second task requires $\Omega(m^{2/3})$ queries.
Note that the query complexity of the first task is higher than the sample complexity of the corresponding task in the standard distribution testing model, whereas in the case of the second task the bounds almost match.

ISSN 1433-8092 | Imprint


text:
# Graph Theory Number of 4 cliques from 10 3 cliques

I am try to find a rigorous way to prove that given a graph that has exactly 10 3 cliques(triangles), the maximum number of 4 cliques that can be formed is 3. Or more generally if there is connection with the number of (k-1) cliques and the maximum number of k cliques.

-

I don't think the question is correct. Consider $K_5$ (complete graph on $5$ vertices). It has $\binom{5}{3} = 10$ $3$-cliques and $\binom{5}{4}=5$ $4$-cliques.

A crude bound for general case can be obtained as shown below. But I don't think is a good bound, i.e., the bounds might not be tight for any graph.

Each $k$ clique contains $k$ number of $k-1$ cliques in it. And any two $k$ cliques can have at most one $k-1$ clique common. Using these two observations we can get some crude bound.

Suppose the number of $k$ cliques is $r$. Then the first $k$ clique has $k$ number of $k-1$ cliques in it. The next $k$ clique again has $k$ number of $k-1$ cliques, but at most one of them is common with the first $k$ clique. So it has at least $k-1$ new $k-1$ cliques. Similarly the third one will have at least $k-2$ new $k-1$ cliques and so on.

Now, we get a lower bound for number of $k-1$ cliques (say $R$) in terms of $r$ as,

if $r \geq k$,

$$R \geq k(k-1)/2$$

if $r < k$

$$R \geq k+(k-1)+ \ldots +(k-r+1) = rk - r(r-1)/2$$

-
Good answer. In my question I was thinking the clique number of graph to be 4, so I said 3 4 cliques is the max. But I don't really need this restriction. –  epinephelus Mar 6 '13 at 14:34


text:
Unbiased Estimator for Mean Response to Treatment

Note: This is essentially Exercise 13.80b in Mathematical Statistics with Applications, 5th. Ed., by Wackerly, Mendenhall, and Scheaffer, and is in the context of ANOVA.

My Work So Far: From the model equations, we have \begin{align*} Y_{ij} &=\mu+\tau_i+\beta_j+\eps_{ij}\\ \overline{Y}_{i\bullet} &=\frac1b\sum_{j=1}^bY_{ij}\\ &=\mu+\tau_i+\overline\beta+\overline\eps_{i\bullet}\\ E\szdp{\overline{Y}_{i\bullet}} &=\mu+\tau_i+E\szdp{\overline\beta}+E\szdp{\overline\eps_{i\bullet}}\\ &=\mu+\tau_i. \end{align*}

My Question: To show that $$\overline{Y}_{i\bullet}$$ is an unbiased estimator for the mean response to treatment $$i,$$ I must show that its expected value is equal to the parameter value. But I don't know what the parameter value is. It feels like this is a simple matter of interpretation: what is the parameter corresponding to the mean response to treatment $$i?$$ Why?

Indeed, $$\bar{Y}_{i.}$$ is an unbiased estimator for the mean response to treatment $$i$$:
$$E\left[ \bar{Y}_{i.} \right]=\frac{1}{b}E\left[ \sum_{j=1}^b{{Y}_{ij}} \right]=\frac{1}{b} \sum_{j=1}^b{E\left[{Y}_{ij} \right]}=\frac{b}{b}E[Y_{ij}]=E[\mu]+E[\tau_i]+E[\beta_j]+E[\epsilon_{ij}]\\=\mu+\tau_i+0+0=\mu+\tau_i$$
That's the mean response to treatment $$i$$ according to the setting of $$\mu$$ and $$\tau_i$$, therefore $$\bar{Y}_{i.}$$ is an unbiased estimator.


text:
# Calculate the deviation of a bullet travelling perpendicular to earths magnetic field.

1. Nov 20, 2014

### davidbenari

1. The problem statement, all variables and given/known data
A bullet (3.40g) has a velocity of 160m/s perpendicular to earth's magnetic field 5x10^-5 T . The bullet has a charge of 13.5 x 10^-9 C.With what distance will it be deviated from its trajectory after it has travelled 1.00km?

2. Relevant equations
$\vec{F}=q\vec{v}\times\vec{B}$

3. The attempt at a solution
The way I visualise this problem. The bullet travels in circular motion.
It's radius of motion will be given by $R=\frac{mv}{qB}=8.05\times 10^{11} m$
With a radius this big, after it has travelled 1km it's almost perfectly on top of the line of its initial trajectory. Because this are my answers to the problem, I'm almost 100% sure I'm not doing it right.

2. Nov 20, 2014

### haruspex

It's certainly a small deviation, but it is not zero and can be calculated. The trick is to calculate it in such a way that you do not completely lose precision.
Please post an equation for the deviation.


text:
# adding bode plots together?

I am trying to understand the bode plot result I have obtained.

The blue, green, and red plots are $a$, $b$, and $a+b$ respectively.

What I'm having a hard time understanding is how a+b is formed. Looking at the magnitude of $a$, it is in the range of $500 dB$ or $$10^{25}$$. Whereas, looking at $b$, it is a negative dB gain, so a number of small magnitude. Although I do expect to see the gain of $a+b$ to be lower than that of $a$, I can't explain why the drop is so significant.

Am I missing something fundamentally important here? or is this fishy to anyone else?

If it makes any difference, I directly got this plots in matlab using

bode(a,b,a+b)

TIA

• Looks odd to me. The blue phase numbers are a bit odd too? What exactly are $a$ and $b$? – copper.hat Jun 12 '12 at 7:26
• I recommend asking the moderators to migrate this to dsp.SE which is a far better fit to the topic than math.SE – Dilip Sarwate Jun 12 '12 at 11:05
• @copper.hat a and b are extremely higher order transfer functions (in the range of 1000th degree) and have complex coefficients. I'm not sure if it would be useful posting the entire thing here. As for the phase, I agree that it looks strange, but the addition of the magnitude is very troublesome to me. – suzu Jun 12 '12 at 17:06
• Can you directly evaluate the transfer functions ($a, b, a+b$) at DC and see what that gives? What are $a$, etc?, LTI systems, arrays of numbers, ...? – copper.hat Jun 12 '12 at 18:09
• Regarding (b): Yes, I would suggest generating the frequency response yourself, then plot the various quantities (ie, don't use bode). The fact that the phase response is 4880° at low frequency would be a flag for me. – copper.hat Jun 13 '12 at 17:17


text:
# Walk on countable ordinals: the characteristics

In this post, we shall present a few aspects of the method of walk on ordinals (focusing on countable ordinals), record its characteristics, and verify some of their properties. All definitions and results in this post are due to Todorcevic.

Let $\langle C_\alpha\mid\alpha<\omega_1\rangle$ be a sequence such that $C_{\alpha+1}=\{\alpha\}$ for all $\alpha<\omega_1$, and for all limit $\alpha<\omega_1$: $C_\alpha$ is a cofinal subset of $\alpha$ of order-type $\omega$. Given $\alpha<\beta<\omega_1$, define

• $Tr(\alpha,\beta)\in{}^\omega\omega_1$, by recursively letting for all $n<\omega$:

$$Tr(\alpha,\beta)(n):=\begin{cases} \beta,&n=0\\ \min(C_{Tr(\alpha,\beta)(n-1)}\setminus\alpha),&n>0\ \&\ Tr(\alpha,\beta)(n-1)>\alpha\\ \alpha,&\text{otherwise} \end{cases}$$

• $\rho_2(\alpha,\beta):=\min\{n<\omega\mid Tr(\alpha,\beta)(n)=\alpha\}$;
• $\rho_{1\beta}\in{}^\beta\omega$, by $\rho_{1\beta}(\alpha):=\max\rho_0(\alpha,\beta)$, where:
• $\rho_0(\alpha,\beta):=\langle otp(C_{Tr(\alpha,\beta)(j)}\cap\alpha)\mid j<\rho_2(\alpha,\beta)\rangle$;
• $L(\alpha,\beta):=\{ \max_{i\le j}\sup(C_{Tr(\alpha,\beta)(i)}\cap\alpha) \mid j<\rho_2(\alpha,\beta)\}$;
• $tr(\alpha,\beta):=Tr(\alpha,\beta)\restriction\rho_2(\alpha,\beta)$.

Remark: We consider $tr(\alpha,\alpha), \rho_0(\alpha,\alpha)$ and $L(\alpha,\alpha)$ as the empty set.

Notation: Write $\Omega:=\{\alpha<\omega_1\mid cf(\alpha)=\omega\}$. Also, by $A=B\oplus C$, we mean that:

• $A=B\cup C$;
• $B\not=\emptyset$, $C\not=\emptyset$;
• $(\bigcup B)\in(\bigcap C)$.

Proposition 1. If $\delta\in\Omega$ and $\delta<\beta<\omega_1$, then $\max(L(\delta,\beta))<\delta$.
Proof. If $\delta\ge\max(L(\delta,\beta))$, then by definition, there exists $i<\rho_2(\delta,\beta)$ such that $\sup(C_{Tr(\delta,\beta)(i)}\cap\delta)=\delta$.  In particular, there exists an ordinal $\alpha$ with  $\delta<\alpha<\beta$ such that $\sup(C_\alpha\cap\delta)=\delta$. It follows that $cf(\delta)\le otp(C_\alpha\cap\delta)<otp(C_\alpha)\le\omega$, contradicting the fact that $\delta\in\Omega$. $\square$

Proposition 2. For every $\alpha<\beta<\gamma<\omega_1$: if $\alpha>\max(L(\beta,\gamma))$, then $$tr(\alpha,\gamma)=tr(\beta,\gamma){}^\frown tr(\alpha,\beta).$$Proof. It suffices to prove that under the same hypotheses, $Tr(\beta,\gamma)=Tr(\alpha,\gamma)\restriction\rho_2(\beta,\gamma)$, and $Tr(\alpha,\gamma)(\rho_2(\beta,\gamma))=\beta$.
Clearly, $Tr(\alpha,\gamma)(0)=\gamma=Tr(\beta,\gamma)(0)$. Next, if $i<\rho_2(\beta,\gamma)$ and $Tr(\alpha,\gamma)(i)=Tr(\beta,\gamma)(i)$, then by $$\beta>\alpha>\max(L(\beta,\gamma))\ge \sup(C_{Tr(\beta,\gamma)(i)}\cap\beta),$$ we get that $$\min(C_{tr(\alpha,\gamma)(i)}\setminus\alpha)=\min(C_{tr(\beta,\gamma)(i)}\setminus\alpha)=\min(C_{tr(\beta,\gamma)(i)}\setminus\beta),$$ and hence $Tr(\alpha,\gamma)(i+1)=Tr(\beta,\gamma)(i+1)$. $\square$

Proposition 3. For every $\alpha<\beta< \gamma<\omega_1$: if   $\min(L(\alpha,\beta))>\max(L(\beta,\gamma))$, then $$L(\alpha,\gamma)=L(\beta,\gamma)\oplus L(\alpha,\beta).$$Proof. By $\alpha\ge\min(L(\alpha,\beta))>\max(L(\beta,\gamma))$, we get from the previous porposition that $tr(\alpha,\gamma)=tr(\beta,\gamma){}^\frown tr(\alpha,\beta)$, and hence $$L(\alpha,\gamma)=L(\beta,\gamma)\oplus U,$$ for $U:=L(\alpha,\beta)\setminus(\max(L(\beta,\gamma))+1)$. Recalling that $\min(L(\alpha,\beta))>\max(L(\beta,\gamma))$, we conclude that $L(\alpha,\gamma)=L(\beta,\gamma)\oplus L(\alpha,\beta)$. $\square$

Proposition 4. For every $\alpha<\omega_1$ and $n<\omega$: the set $\{ \xi<\alpha\mid \rho_{1\alpha}(\xi)\le n\}$ is finite.
Proof. Suppose not. Let $\alpha<\omega_1$ be the least for which there exists $n<\omega$ and a set $\Gamma\in[\alpha]^\omega$ with $\rho_{1\alpha}(\gamma)\le n$ for all $\gamma\in\Gamma$. In particular, $otp(C_\alpha\cap\gamma)\le n$ for all $\gamma\in\Gamma$.
Define $o:\Gamma\rightarrow n+1$ by stipulating that $o(\gamma):=otp(C_\alpha\cap\gamma)$. Then there exists $\Gamma’\in[\Gamma]^\omega$ on which $o$ is constant. In particular, $\min(C_\alpha\setminus\gamma_1)=\min(C_\alpha\setminus\gamma_2)$ for all $\gamma_1,\gamma_2\in\Gamma’$. Put $\alpha’:=\min(C_\alpha\setminus\min(\Gamma’))$. Then $\Gamma’\in[\alpha’]^\omega$, and so by $\alpha'<\alpha$ and minimality of the latter, we may find some $\gamma’\in\Gamma’$ such that $\rho_{1\alpha’}(\gamma’)>n$.
By $\min(C_\alpha\setminus\gamma’)=\alpha’$, we have  $tr(\gamma’,\alpha)=\langle\alpha\rangle{}^\frown tr(\gamma’,\alpha’)$, and hence $$\rho_{1\alpha}(\gamma’)=\max\{otp(C_\alpha\cap\gamma’),\rho_{1\alpha^*}(\gamma’)\}>n.$$ This is a contradiction. $\square$

Proposition 5.  For every $\alpha<\beta<\omega_1$, the set $\{\xi<\alpha\mid \rho_{1\alpha}(\xi)\neq\rho_{1\beta}(\xi)\}$ is finite.
Proof. Suppose not. Let $\beta<\omega_1$ be the least for which there exists $\alpha<\beta$ and a subset $\Gamma\subseteq \alpha$ of order-type $\omega$ with $\rho_{1\alpha}(\xi)\neq\rho_{1\beta}(\xi)$ for all $\xi\in\Gamma$. Put $\gamma:=\sup(\Gamma)$, $\gamma^{-}:=\sup(C_\beta\cap\gamma)$, and  $\gamma^{+}:=\min(C_\beta\setminus\gamma)$.
By $cf(\gamma)=\omega\ge otp(C_\beta)$, we infer that $\gamma^{-}<\gamma\le\alpha\le\gamma^{+}<\beta$.
Put $n:= otp(C_\beta\cap\gamma)$, and $\Gamma’:=\{\xi\in\Gamma\setminus\gamma^{-}\mid \rho_{1\beta}(\xi)>n\}$. By the previous proposition, we know that $otp(\Gamma’)=\omega$. It then follows from $\gamma^{+}<\beta$ and minimality of the latter, that there exists $\xi\in\Gamma’$ such that $\rho_{1\alpha}(\xi)=\rho_{1\gamma^{+}}(\xi)$.
By $\gamma^{-}\le\xi<\gamma\le\gamma^{+}$, we know that $\min(C_\beta\setminus\xi)=\min(C_\beta\setminus\gamma)$ and $otp(C_\beta\cap\xi)=otp(C_\beta\cap\gamma)=n$. That is, $\min(C_\beta\setminus\xi)=\gamma^+$, and $\rho_{1\gamma^+}(\xi)>otp(C_\beta\cap\xi)$. So $tr(\xi,\beta)=\langle\beta\rangle{}^\frown tr(\xi,\gamma^+)$, and hence $$\rho_{1\beta}(\xi)=\max\{otp(C_\beta\cap\xi),\rho_{1\gamma^+}(\xi)\}=\rho_{1\gamma^+}(\xi)=\rho_{1\alpha}(\xi).$$ This is a contradiction. $\square$

Proposition 6.  For every $\alpha<\beta<\omega_1$, the set $\{\xi<\alpha\mid \rho_{0}(\xi,\alpha)=\rho_{0}(\xi,\beta)\}$ is closed.
Proof. Let $\delta$ be a limit point of this set. Then, by Proposition 1, $\max(L(\delta,\beta))<\delta$ and $\max(L(\delta,\alpha))<\delta$. Consequently, we may find $\xi$ in this set such that $\max(L(\delta,\alpha)\cup L(\delta,\beta))<\xi$. It then follows from proposition 2 that:

• $tr(\xi,\alpha)=tr(\delta,\alpha){}^\frown tr(\xi,\delta)$;
• $tr(\xi,\beta)=tr(\delta,\beta){}^\frown tr(\xi,\delta)$.

Write $n:=\text{dom}(tr(\xi,\delta))$. Then $tr(\delta,\alpha)$ is equal to $tr(\xi,\alpha)\restriction(\text{dom}(tr(\xi,\alpha))-n)$, and hence $\rho_0(\delta,\alpha)$ is equal to $\rho_0(\xi,\alpha)\restriction(\text{dom}(tr(\xi,\alpha))-n)$. Likewise, $\rho_0(\delta,\beta)$ is equal to $\rho_0(\xi,\beta)\restriction(\text{dom}(tr(\xi,\beta))-n)$. Recalling that $\rho_0(\xi,\alpha)=\rho_0(\xi,\beta)$, we conclude that indeed $\rho_0(\delta,\alpha)=\rho_0(\delta,\beta)$. $\square$

This entry was posted in and tagged . Bookmark the permalink.

### 2 Responses to Walk on countable ordinals: the characteristics

1. Tanmay Inamdar says:

Hi Assaf, I found a couple of typos:

1) In the second last line of the proof of Proposition 4, the $\theta$ should be an $n$.

2) In the second line of the proof of Proposition 6, the last $\alpha$ should be a $\delta$.

0 likes

• saf says:

Dear Tanmay,
Thank you for the careful reading, and the feedback! It is now corrected.

0 likes


text:
How do you determine all trigonometric ratios given sinθ=2/3 ?

Nov 2, 2015

Find the six trig functions ratios, given $\sin x = \frac{2}{3}$

Explanation:

$\sin x = \frac{2}{3}$ --> $x 1 = {41}^{\circ} 81$ and $x 2 = 180 - 41.81 = {138}^{\circ} 19$
${\cos}^{2} x = 1 - {\sin}^{2} x = 1 - \frac{4}{9} = \frac{5}{9}$ --> $\cos x = \pm \frac{\sqrt{5}}{3}$
$\tan x 1 = \tan 41.81 = \left(\frac{2}{3}\right) \left(\frac{3}{\sqrt{5}}\right) = \frac{2}{\sqrt{5}} = \frac{2 \sqrt{5}}{5}$
$\tan x 2 = \tan 138.19 = - \frac{2 \sqrt{5}}{5}$
$\cot 41.81 = \frac{\sqrt{5}}{2}$
$\cot 138.19 = - \frac{\sqrt{5}}{2}$
$\sec 41.81 = \frac{3}{\sqrt{5}} = \frac{3 \sqrt{5}}{5}$
$\sec 138.19 = - \frac{3 \sqrt{5}}{5}$
$\csc 41.81 = \csc 138.19 = \frac{3}{2}$


text:
# Yang Chen

University of Macau, China
Talk: Asymtoptic gap probability distributions of the Gaussian unitary enasembles and the Jacobi unitary ensembles
Talk: The smallest eigenvalue of large Hankel matrices


text:
Fracture is a lightweight and flexible data management system

# Overview

Fracture is a lightweight and flexible data management system. It allows you to interact with data through a trait compositing mechanism, whilst also exposing the ability to quickly query and access information about the data you're exploring.

# How it works

You start by creating a fracture project. The project file is where all the metadata and look up tables are stored - allowing you to easily search for data assets as well as find changes.

Fracture comes built-in with a file searching mechanism, but you can extend this with your own search mechanisms too. For instance, if you have data on an FTP, or within Source Control and you want to add that data to the project without having to have it physically on disk you're able to do so by implementing a fracture.ScanProcess plugin.

Finally, and probably the most important is the DataElement. This is a class which you can use to express the functionality of data. Rather than having a 1:1 relationship between a DataElement class and a data type the DataElement class supports class compositing. This allows for a piece of data to be represented by more than one class simultaneously.

# Examples

This example uses the dino_example data which you can download from https://github.com/mikemalinowski/fracture.

To start with, we create a fracture project. To do this we must specify two pieces of information, the first being where we want to save our project file and the second being the locations where we want fracture to look for Scan and Data Plugins.

import os
import fracture

project = fracture.create(
project_path=os.path.join(current_path, '_dino.fracture'),
plugin_locations=[os.path.join(current_path, 'plugins')]
)

This returns a fracture.Project instance which we can then start interacting with, for instance we can define locations where the project should start looking for data:

# -- Tell the project where to look for data

Finally, with the project made, and at least one search location added we can initiate a search...

# -- Now we initiate a scan. This will cycle over all the
# -- scan locations and scrape them for data
project.scan()

Scanning is the process of running over all the scan plugins - of which there is always at least one (the file scraper), and populating the project with information about each piece of data which is found. The process is pretty quick and the amount of data stored is minimal - primarily just the identifier such as the path along with any tags as defined by any DataElement composites which can represent that data.

With the project populated we can now start querying the project for data

# -- Now we have scanned we can start to run queries over data
# -- very quickly. We can search by tags, or use the special
# -- * wildcard
for item in project.find('*'):

# -- By default we get string identifiers back from a find, as
# -- this is incredibly quick. However, we can ask for the data
# -- composite representation of the item. A data composite is
# -- a composition of all the data plugins which can represent
# -- this identifier.
item = project.get(item)

# -- Print the identifier, and the item (which also shows the
# -- class composition)
print(item.identifier())
print('\t%s' % item)

# -- We can start interacting with this data, calling
# -- functionality will return a dictionary of all the
# -- functionality exposed by all the data plugins representing
# -- this item
for k, v in item.functionality().items():
print('\t\t%s = %s' % (k, v))

The process of querying is very quick, even for reasonably large data sets. In the example above we're then asking the project to 'get' the item. This process take the identifier and binds all the relevent DataElements together which can possibly represent the data.

Binding is particularly useful when there is no obvious hierarchy between two elements. For instance, in the dino_example data set we have a trait which is carnivore and a trait which is herbivore. There is no hierarchical relationship between the two, but an omnivore would need both. By using class compositing we avoid complex multi-inheritence situations.

Using this same mechanism, if we know the locator of a piece of information, such as a file path, we can get the composited class directly without having to run a query, as shown here:

# -- We do not have to utilise the find method to get access to data,
# -- and in fact we can get a Composite representation of data even
# -- if that data is not within our scan location.
data = project.get('/usr/my_data/my_file.txt')

For a full demo, download the dino_example and run main.py

# Data Composition

As mentioned in the examples, we use class composition to bind traits together to represent data. This means we can have small, self contained traits which do not need rigid hierarchical structures designed for them.

There are three main composited methods in the DataElement class, specifically:

• label : The first call that returns a positive result is taken
• mandatory_tags : All the lists are collated from all compositions and made unique
• functionality : All dictionaries are combined into a single dictionary
• icon : The first call that returns a positive result is taken

Given the dino_example files, the velociraptor.png file, when passed to project.get('/usr/my_data/.../velociraptor.png') is expressed as a class formed of the following traits: [Carnivore; File; Image;] where each trait can expose its own information.

An implementation of a DataElement plugin looks like this:

import re
import fracture

# -- All plugins must inherit from the fracture.DataElement class in order
# -- to be dynamically picked up.
class CarnivoreTrait(fracture.DataElement):

# -- The data type is mandatory, and is your way of
# -- denoting the name of this plugin
data_type = 'carnivore'

# -- These two lines are not at all required and are here
# -- just to make performance better
_split = re.compile('/|\.|,|-|:|_', re.I)
_has_trait = re.compile('(carnivore|omnivore).*\.', re.I)

# --------------------------------------------------------------------------
# -- This method must be re-implemented, and its your oppotunity to
# -- decide whether this plugin can viably represent the given data
# -- identifier.
# -- In this example we use a regex check, but it could be anything
# -- you want. The key thing to remember is that this is called a lot,
# -- so performance is worth keeping in mind.
@classmethod
def can_represent(cls, identifier):
if CarnivoreTrait._has_trait.search(identifier):
return True
return False

# --------------------------------------------------------------------------
# -- This is your way of exposing functionality in a common and consistent
# -- way. If you know the data types you can of course call things directly
# -- but this is a good catch up consistent way of exposing functionality
# -- and is typically harnessed by user interfaces.
def functionality(self):
return dict(
feed_meat=self.feed_meat,
),
)

# --------------------------------------------------------------------------
# -- This should return a 'nice name' for the identifier
def label(self):
return os.path.basename(self.identifier())

# --------------------------------------------------------------------------
# -- As fracture heavily utilises tags, this is your way of defining a
# -- set of tags which are mandatory for anything with this trait
def mandatory_tags(self):
return ['carnivore', 'meat', 'hunter']

# --------------------------------------------------------------------------
# -- This is here just as a demonstration of a callable function which
# -- which can be accessed on the trait
def feed_meat(self):
print('Would feed this creature some meat...')

By placing a trait plugin anywhere within the plugin locations you define for your project will immediately make it accessible.

## ScanProcess

By default fracture comes with one built-in scan plugin which handles file scanning, so that is a good example when wanting to write your own - if you have need to do so.

This plugin type defines how to find data. If your data is files on a disk such as those in the example above then your scan plugin may do little more than cycle directories and yield file paths.

Alternatively if you're caching data from a REST API you might be utilising requests within the scan process and feeding back URL's.

# Origin

This library is a variation on the tools demonstrated during GDC2018 (A Practical Approach to Developing Forward-Facing Rigs, Tools and Pipelines), which can be explored in more detail here: https://www.gdcvault.com/play/1025427/A-Practical-Approach-to-Developing

Slide 55 onward explores this concept. It is also explored in detail on this webpage: https://www.twisted.space/blog/insight-localised-asset-management

# Collaboration

I am always open to collaboration, so if you spot bugs lets me know, or if you would like to contribute or get involved just shout!

# Compatibility

Launchpad has been tested under Python 2.7 and Python 3.7 on Windows and Ubuntu.

## Project details

Uploaded source


text:
# Re: MF ==> (PS type1)

bkph> EURM contains only alphabetic characters.
bkph> Ditto for EUSB, which is a bold Script face.
bkph> So basically what you are saying is that you can replace the letters
bkph> in the CM math fonts with letters from other fonts.
bkph> Of course one can do this with any text font. Why single out Euler?

I don't know the Type1 version of the Euler fonts.  But the MF version
contains almost everything needed to make up a complete math setup --
including digits, punctuation characters, plus, minus, division sign,
\sum, \prod, delimiters, ...

This is not generally true about any other font family (that I know about
at least).

/Frank


text:
Why only 2015?

Geometry Level 4

A circle touches side AB and AD of a rectangle ABCD at P and Q respectively and passes through the vertex C. If the perpendicular drawn from C to PQ is $$\sqrt{ 2015}$$ units . Then find the area of rectangle ABCD.

×