The block numerical range of analytic operator functions

We introduce the block numerical range Wn(L) of an operator function L with respect to a decomposition H = H1⊕. . .⊕Hn of the underlying Hilbert space. Our main results include the spectral inclusion property and estimates of the norm of the resolvent for analytic L . They generalise, and improve, the corresponding results for the numerical range (which is the case n = 1) since the block numerical range is contained in, and may be much smaller than, the usual numerical range. We show that refinements of the decomposition entail inclusions between the corresponding block numerical ranges and that the block numerical range of the operator matrix function L contains those of its principal subminors. For the special case of operator polynomials, we investigate the boundedness of Wn(L) and we prove a Perron-Frobenius type result for the block numerical radius of monic operator polynomials with coefficients that are positive in Hilbert lattice sense.

Is choking under pressure a consequence of skill-focus or increased distractibility? Results from a tennis serve task

It has been repeatedly demonstrated that athletes often choke in high pressure situations because anxiety can affect attention regulation and in turn performance. There are two competing theoretical approaches to explain the negative anxiety-performance relationship. According to skillfocus theories, anxious athletes’ attention is directed at how to execute the sport-specific movements which interrupts execution of already automatized movements in expert performers. According to distraction theories, anxious athletes are distractible and focus less on the relevant stimuli. We tested these competing assumptions in a between-subject design, as semi-professional tennis players were either assigned to an anxiety group (n = 25) or a neutral group (n = 28), and performed a series of second tennis serves into predefined target areas. As expected, anxiety was negatively related to serve accuracy. However, mediation analyses with the bootstrapping method revealed that this relationship was fully mediated by self-reported distraction and not by skill-focus.

The Nile perch invasion in Lake Victoria: cause or consequence of the haplochromine decline?

We review alternative hypotheses and associated mechanisms to explain Lake Victoria’s Nile perch takeover and concurrent reduction in haplochromines through a (re)analysis of long term climate, limnological and stock observations in comparison with size-spectrum model predictions of co-existence, extinction and demographic change. The empirical observations are in agreement with the outcomes of the model containing two interacting species with life-histories matching Nile perch and a generalized haplochromine. The dynamic interactions may have depended on size related differences in early juvenile mortality: mouth-brooding haplochromines escape predation mortality in early life stages, unlike Nile perch that have miniscule planktonic eggs and larvae. In our model predation on the latter by planktivorous haplochromine fry act as a stabilizing factor for co-existence, but external mortality on the haplochromines would disrupt this balance in favor of Nile perch. To explain the observed switch, mortality on haplochromines would need to be much higher than the fishing mortality that can be realistically re-constructed from observations. Abrupt concomitant changes in algal and zooplankton composition, decreased water column transparency, and widespread hypoxia from increased eutrophication most likely caused haplochromine biomass decline. We hypothesize that the shift to Nile perch was a consequence of an externally caused, climate triggered, decrease in haplochromine biomass and associated recruitment failure rather than a direct cause of the introduction.

On the CFT operator spectrum at large global charge

We calculate the anomalous dimensions of operators with large global charge J in certain strongly coupled conformal field theories in three dimensions, such as the O(2) model and the supersymmetric fixed point with a single chiral superfield and a W = Φ3 superpotential. Working in a 1/J expansion, we find that the large-J sector of both examples is controlled by a conformally invariant effective Lagrangian for a Goldstone boson of the global symmetry. For both these theories, we find that the lowest state with charge J is always a scalar operator whose dimension ΔJ satisfies the sum rule J2ΔJ−(J22+J4+316)ΔJ−1−(J22+J4+316)ΔJ+1=0.04067 up to corrections that vanish at large J . The spectrum of low-lying excited states is also calculable explcitly: for example, the second-lowest primary operator has spin two and dimension ΔJ+3√. In the supersymmetric case, the dimensions of all half-integer-spin operators lie above the dimensions of the integer-spin operators by a gap of order J+12. The propagation speeds of the Goldstone waves and heavy fermions are 12√ and ±12 times the speed of light, respectively. These values, including the negative one, are necessary for the consistent realization of the superconformal symmetry at large J.