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We investigate the supports of extremal martingale measures with prespecified marginals in a two-period setting. First, we establish in full generality the equivalence between the extremality of a given measure Q and the denseness in
of a suitable linear subspace, which can be seen in a financial context as the set of all semistatic trading strategies. Moreover, when the supports of both marginals are countable, we focus on the slightly stronger notion of weak exact predictable representation property (WEP) and provide two combinatorial sufficient conditions, called the ‘2-link property’ and ‘full erasability’, on how the points in the supports are linked to each other for granting extremality. When the support of the first marginal is a finite set, we give a necessary and sufficient condition for the WEP to hold in terms of the new concepts of 2-net and deadlock. Finally, we study the relation between cycles and extremality.
Failed transfer of passive immunity (FTPI) in dairy calves – which is often due to the low amount of colostrum provided within a few hours after birth – remains a crucial issue. Enabling dairy calves to nurse colostrum from their dams could be useful in increasing intake and thus avoiding FTPI, but further potential effects on the health and welfare of both calves and dams should also be considered. In this study, 107 calf-dam pairs from two Italian dairy farms were alternately assigned to one of the following colostrum provision methods (CPMs): ‘hand-fed method’ (HFM) – the calf was separated from the dam immediately after birth and colostrum was provided by nipple-bottle (n = 50); ‘nursing method’ (NM) – the calf nursed colostrum from the dam for the first 12 h of life without farmer assistance (n = 30); and ‘mixed method’ (MM) – the nursing calf received a supplementary colostrum meal by nipple-bottle (n = 27). Serum of calves (1 to 5 days of age) and samples of their first colostrum meal were analysed by electrophoresis to assess immunoglobulin (Ig) concentration. Additionally, behavioural indicators of separation distress (calf and dam vocalisations; calf refusal of the first meal after separation; undesirable dam behaviour at milking) in the following 24 h were recorded as binary variables (Yes/No), and the health status of calves (disease occurrence and mortality) and dams (postpartum disorders and mastitis occurrence) were monitored for the first 3 months of life and 7 days after parturition, respectively. The lowest FTPI occurrence (calf serum Ig concentration <10.0 g/l) was found in the MM (11.1%) and the HFM (22.0%) compared with the NM (60.0%) (P<0.05), and the highest percentage of calves with optimal transfer of passive immunity (serum Ig concentration ≥16.0 g/l) was observed in the MM (55.6%). The lowest calf–dam separation distress was observed in the HFM (P<0.05). The highest calf disease occurrence was recorded in the HFM (64.0%) and the lowest in the NM (33.3%), with an intermediate value for the MM (44.4%) (P<0.05). No effect of the CPM was observed on dam health or calf mortality (P>0.05). The results of this study indicated that providing calves with a supplementary colostrum meal in addition to nursing from the dam (MM) is truly effective in maximizing passive immunity transfer. Anyway, specific strategies should be studied to minimise calf-dam separation distress.
Motivated by recent studies that have revealed the existence of trapped acoustic waves in subsonic jets (Towne et al., J. Fluid Mech., vol. 825, 2017, pp. 1113–1152), we undertake a more general exploration of the physics associated with acoustic modes in jets and wakes, using a double vortex-sheet model. These acoustic modes are associated with eigenvalues of the vortex-sheet dispersion relation; they are discrete modes, guided by the vortex sheet; they may be either propagative or evanescent; and under certain conditions they behave in the manner of acoustic-duct modes. By analysing these modes we show how jets and wakes may both behave as waveguides under certain conditions, emulating ducts with soft or hard walls, with the vortex-sheet impedance providing effective ‘wall’ conditions. We consider, in particular, the role that upstream-travelling acoustic modes play in the dispersion-relation saddle points that underpin the onset of absolute instability. The analysis illustrates how departure from duct-like behaviour is a necessary condition for absolute instability, and this provides a new perspective on the stabilising and destabilising effects of reverse flow, temperature ratio and compressibility; it also clarifies the differing symmetries of jet (symmetric) and wake (antisymmetric) instabilities. An energy balance, based on the vortex-sheet impedance, is used to determine stability conditions for the acoustic modes: these may become unstable in supersonic flow due to an energy influx through the shear layers. Finally, we construct the impulse response of flows with zero and finite shear-layer thickness. This allows us to show how the long-time wavepacket behaviour is indeed determined by interaction between Kelvin–Helmholtz and acoustic modes.