Decay spectrum and decay subspace of normal operators

Let A be a self-adjoint operator on a Hilbert space. It is well known that A admits a unique decomposition into a direct sum of three self-adjoint operators A(p), A(ac) and A(sc) such that there exists an orthonormal basis of eigenvectors for the operator A(p) the operator A(ac) has purely absolutely continuous spectrum and the operator A(sc) has purely singular continuous spectrum. We show the existence of a natural further decomposition of the singular continuous component A c into a direct sum of two self-adjoint operators A(sc)(D) and A(sc)(ND). The corresponding subspaces and spectra are called decaying and purely non-decaying singular subspaces and spectra. Similar decompositions are also shown for unitary operators and for general normal operators.

Decision rules, energy metabolism and vigour of hermit-crab fights

Aggressive interactions between animals are often settled by the use of repeated signals that reduce the risk of injury from combat but are expected to be costly. The accumulation of lactic acid and the depletion of energy stores may constrain activity rates during and after fights and thus represent significant costs of signalling. We tested this by analysing the concentrations of lactate and glucose in the haemolymph of hermit crabs following agonistic interactions over the ownership of the gastropod shells that they inhabit. Attackers and defenders play distinct roles of sender and receiver that are fixed for the course of the encounter. Attackers perform bouts of 'shell rapping', which vary in vigour between attackers and during the course of the encounter, and are a key predictor of victory. In contrast to the agonistic behaviour of other species, we can quantify the vigour of fighting. We demonstrate, to our knowledge for the first time, an association between the vigour of aggressive activity and a proximate cost of signalling. We show that the lactate concentration in attackers increases with the amount of shell rapping, and that this appears to constrain the vigour of subsequent rapping. Furthermore, attackers, but not defenders, give up when the concentration of lactate is high. Glucose levels in attackers also increase with the amount of rapping they perform, but do not appear to influence their decision to give up. Defenders are more likely to lose when they have particularly low levels of glucose. We conclude that the two roles use different decision rules during these encounters.

The energetics of lactation in cooperatively breeding meerkats Suricata suricatta

Species may become obligate cooperative breeders when parents are unable to raise their offspring unassisted. We measured the daily energy expenditure of mothers, helpers and offspring during peak lactation in cooperatively breeding meerkats Suricata suricatta using the doubly labelled water technique. Lactating mothers expended more energy per day than allo-lactating subordinate females, non-lactating females or suckling offspring. Metabolizable energy intakes of lactating mothers were calculated from isotope-based estimates of offspring milk energy intake, and were not significantly different from the previously suggested maximal limit for mammals. Allo-lactating females were the only category of animals that lost weight during the period of study, probably because they spent more time babysitting than non-lactating females. Daily energy expenditure (DEE) of lactating mothers increased with litter size but decreased with the number of helpers. Calculations show that for every 10 helpers, even in the absence of allo-lactators, mothers are able to reduce their DEE during peak lactation by an amount equivalent to the energy cost of one pup. These results indicate that helpers have beneficial energetic consequences for lactating mothers in an obligate cooperatively breeding mammal.

Resistance training enhances the stability of sensorimotor coordination

Strategics for the control of human movement are constrained by the neuroanatomical characteristics of the motor system. In particular, there is evidence that the capacity of muscles for producing force has a strong influence on the stability of coordination in certain movement tasks. In the present experiment, our aim was to determine whether physiological adaptations that cause relatively long-lasting changes in the ability of muscles to produce force can influence the stability of coordination in a systematic manner. We assessed the effects of resistance training on the performance of a difficult coordination task that required participants to synchronize or syncopate movements of their index finger with an auditory metronome. Our results revealed that training that increased isometric finger strength also enhanced the stability of movement coordination. These changes were accompanied by alterations in muscle recruitment patterns. In Particular, the trained muscles were recruited in a more consistent fashion following the programme of resistance training. These results indicate that resistance training produces functional adaptations of the neuroanatomical constraints that underlie the control of voluntary movement.

The role of H2D+ in the deuteration of interstellar molecules

Collisions between H-3(+) and HD in molecular clouds lead to the fractionation of deuterium in H2D+ at temperatures below 20 K. In this article, we describe the chemistry of H2D+ and discuss how variations in temperature and elemental abundances affect the level of fractionation in H2D+ and other species. We describe how accretion of gas-phase molecules on to cold dust grains enhances the deuteration in several molecules including doubly deuterated molecules. Mie show that the ion-neutral drift velocities attained in slow Alfven waves can destroy H2D+ in non-thermal reactions. As a result, the degree of fractionation can be reduced and we discuss observational consequences of such a model for the dark dust cloud TMC-1.

Cumulative or sequential assessment during hermit crab shell fights: effects of oxygen on decision rules

Agonistic interactions between animals are often settled by the use of repeated signals which advertise the resource-holding potential of the sender. According to the sequential assessment game this repetition increases the accuracy with which receivers may assess the signal, but under the cumulative assessment model the repeated performances accumulate to give a signal of stamina. These models may be distinguished by the temporal pattern of signalling they predict and by the decision rules used by the contestants. Hermit crabs engage in shell fights over possession of the gastropod shells that they inhabit. During these interactions the two roles of signaller and receiver may be examined separately because they are fixed for the duration of the encounter. Attackers rap their shell against that of the defender in a series of bouts whereas defenders remain tightly withdrawn into their shells for the duration of the contest. At the end of a fight the attacker may evict the defender from its shell or decide to give up without first effecting an eviction; the decision for defenders is either to maintain a grip on its shell or to release the shell and allow itself to be evicted. We manipulated fatigue levels separately for attackers and defenders, by varying the oxygen concentration of the water that they are held in prior to fighting, and examined the effects that this has on the likelihood of each decision and on the temporal pattern of rapping. We show that the vigour of rapping and the likelihood of eviction are reduced when the attacker is subjected to low oxygen but that this treatment has no effect on rates of eviction when applied to defenders. We conclude that defenders compare the vigour of rapping with an absolute threshold rather than with a relative threshold when making their decision. The data are compatible with the cumulative assessment model and with the idea that shell rapping signals the stamina of attackers, but do not fit the predictions of the sequential assessment game.