“Loop the loop”

The applicability of the collective coordinate method (saddle-point approximation) for large-N planar models is discussed. Some unstated assumptions are clarified. Statements that Wilson loops form a complete set of gauge invariant operators are also examined and a set of generalized algebraic Mandelstam relations among Wilson loops is presented. The inclusion of loops that wind around themselves and cross many times, as independent variables, is stressed.     

“Almost direct” photons

Quantum chromodynamics has intensified the interest and enhanced the importance of measuring the production of direct photons in hadron collisions. We point out that lepton pairs of low invariant mass (m² < m²_ψ) provide an equally good probe for testing Q.C.D. Their experimental observation is not only easier, the relevant data may already exist.     

On “diagonal” Bianchi cosmologies

A list of the possible “diagonal” Bianchi cosmologies is given, with a proof that it is complete.     

Designing the “perfect” projection screen

A perfect diffuser would place 100% of the light leaving the projector in that small region of space where there will be audience eyes to observe it. It would not allow light from sources other than the projector to reach the eyes from the screen. The screen should be affordably priced and cosmetically unremarkable, e.g. seamless. The image seen by any observer should be equally bright over the whole screen. I discuss a way to approximate the perfect projection screen using kinoform diffusers, a Fresnel lens and a mirrored surface.     

Schlieren “PIV” for turbulent flows

The possibility of using commercial PIV equipment combined with schlieren optics to measure the velocity fields of turbulent flows is explored. Given a sufficiently high Reynolds number and adequate refractive flow differences, turbulent eddies can serve as the PIV “particles” in a schlieren image or shadowgram. The PIV software analyzes motion between consecutive schlieren or shadowgraph frames to obtain velocity fields. Velocimetry examples of an axisymmetric sonic helium jet in air and a 2D turbulent boundary layer at Mach 3 are shown. Due to optical path integration, axisymmetric flows require the inverse Abel transform to extract center-plane velocity data. Conditions for optimum schlieren sensitivity are examined. In its present embodiment, “schlieren PIV” is not useful for laminar flows nor for fully 3D flows. Otherwise it functions much like standard PIV under conditions where individual particles are not resolved and velocimetry is instead based on correlation of the motion of turbulent structures. “Schlieren PIV” shows significant promise for general refractive turbulent flow velocimetry if its integrative nature can be overcome through sharp-focusing optics.     

Deformed “commutative” Chern–Simons’ system

Noncommutative Chern–Simons’ system is non-perturbatively investigated at a full deformed level. A deformed “commutative” phase space is found by a non-canonical change between two sets of deformed variables of noncommutative space. It is explored that in the “commutative” phase space all calculations are similar to the case in commutative space. Spectra of its energy and angular momentum of the Chern–Simons’ system are obtained at the full deformed level. The noncommutative–commutative correspondence is clearly showed. Formalism for the general dynamical system is briefly presented. Some subtle points are clarified.