1-loop matching and NNLL resummation for all partonic 2 to 2 processes in QCD

The Wilson Coefficients for all 4-parton operators which arise in matching QCD to Soft-Collinear Effective Theory (SCET) are computed at 1-loop. Any dijet observable calculated in SCET beyond leading order will require these results. The Wilson coefficients are separated by spin and color, although most applications will involve only the spin-averaged hard functions. The anomalous dimensions for the Wilson coefficients are given to 2-loop order, and the renormalization group equations are solved explicitly. This will allow for analytical resummation of dijet observables to next-to-next-to-leading logarithmic accuracy. For each channel, there is a natural basis in which the evolution is diagonal in color space. The same basis also diagonalizes the color evolution for the soft function. Even though soft functions required for SCET calculations are observable dependent, it is shown that their renormalization group evolution is almost completely determined by a universal structure. With these results, it will be possible to calculate hadronic event shapes or other dijet observables to next-to-leading order with next-to-next-to-leading log resummation.Comment: 28 pages, 5 tables; v2: typo corrected in Eq. (56

Colour change and assortment in the western rainbowfish

Grouping behaviour is widespread across the animal kingdom, and is known to reduce an individual's risk of predation, for example through predator confusion. Theory predicts that individuals that are different in appearance to the rest of the group are at a greater risk of predation because they are more conspicuous to predators (the ‘oddity’ effect). Thus, animals should choose group mates that are the most similar in appearance to themselves. Another common antipredator tactic is crypsis (camouflage). Fishes are capable of changing colour to match their visual background, but few studies have examined how this might influence shoaling decisions, particularly in the context of the oddity effect. We induced colour pattern changes in a colourful species of freshwater fish, the western rainbowfish, Melanotaenia australis, by maintaining fish in dark and pale aquaria for 2 weeks. Analysis of the proportion of black body pigmentation confirmed that rainbowfish in dark environments developed darker colour patterns than those held in pale environments. We then conducted behavioural observations to determine whether fish subsequently based their shoaling decisions on body coloration. We found that rainbowfish preferred to shoal with similar individuals; fish that had been held in dark aquaria preferred to shoal with other dark fish and fish from pale aquaria preferred other pale fish. Our findings are consistent with the predictions of the oddity effect and demonstrate how morphological colour pattern changes and behavioural decisions interact to mediate antipredator tactics in fish

Conflict between background matching and social signalling in a colour-changing freshwater fish

The ability to change coloration allows animals to modify their patterning to suit a specific function. Many freshwater fishes, for example, can appear cryptic by altering the dispersion of melanin pigment in the skin to match the visual background. However, melanin-based pigments are also used to signal dominance among competing males; thus colour change for background matching may conflict with colour change for social status signalling. We used a colour-changing freshwater fish to investigate whether colour change for background matching influenced aggressive interactions between rival males. Subordinate males that had recently darkened their skin for background matching received heightened aggression from dominant males, relative to males whose coloration had not changed. We then determined whether the social status of a rival male, the focal male's previous social status, and his previous skin coloration, affected a male's ability to change colour for background matching. Social status influenced skin darkening in the first social encounter, with dominant males darkening more than subordinate males, but there was no effect of social status on colour change in the second social encounter. We also found that the extent of skin colour change (by both dominant and subordinate males) was dependent on previous skin coloration, with dark males displaying a smaller change in coloration than pale males. Our findings suggest that skin darkening for background matching imposes a significant social cost on subordinate males in terms of increased aggression. We also suggest that the use of melanin-based signals during social encounters can impede subsequent changes in skin coloration for other functions, such as skin darkening for background matching

Qualitative Criterion for Interception in a Pursuit/Evasion Game

A qualitative account is given of a differential pursuit/evasion game. A criterion for the existence of an intercept solution is obtained using future cones that contain all attainable trajectories of target or interceptor originating from an initial position. A sufficient and necessary conditon that an opportunity to intercept always exist is that, after some initial time, the future cone of the target be contained within the future cone of the interceptor. The sufficient condition may be regarded as a kind of Nash equillibrium.Comment: 8 pages; revsions and corrigend

On the Density of Coprime m-tuples over Holomorphy Rings

Let $\mathbb F_q$ be a finite field, $F/\mathbb F_q$ be a function field of genus $g$ having full constant field $\mathbb F_q$, $\mathcal S$ a set of places of $F$ and $H$ the holomorphy ring of $\mathcal S$. In this paper we compute the density of coprime $m$-tuples of elements of $H$. As a side result, we obtain that whenever the complement of $\mathcal S$ is finite, the computation of the density can be reduced to the computation of the $L$-polynomial of the function field. In the rational function field case, classical results for the density of coprime $m$-tuples of polynomials are obtained as corollaries.Comment: To appear in International Journal of Number Theor

Helicopter Anti-Torque System Using Strakes

A helicopter is disclosed with a system for controlling main-rotor torque which reduces the power and size requirements of conventional anti-torque means. The torque countering forces are generated by disrupting the main rotor downwash flowing around the fuselage. The downwash flow is separated from the fuselage surface by a strake positioned at a specified location on the fuselage. This location is determined by the particular helicopter wash pattern and fuselage configuration, generally being located between 20 deg before top dead center (TDC) and 80 deg from TDC on the fuselage side to which the main rotor blade approaches during rotation. The strake extends along the fuselage from the cabin section to the aft end and can be continuous or separated for aerodynamic surfaces such as a horizontal stabilizer