we investigate buchbaum and eisenbud's construction of the second symmetric power $s_r(x)$ of a chain complex $x$ of modules over a commutative ring $r$. we state and prove a number of results from the folklore of the subject for which we know of no good direct references. we also provide several explicit computations and examples. we use this construction to prove the following ...

We present the Iwasawa decomposition theorem for the group Sp (2, R) in a form particularly suited to firstorder optics, and we exploit it to develop a uniform description of the shape-invariant propagation of several families of optical beams. Both coherent and partially coherent beams are considered. We analyze the Hermite–Gaussian beam as an example of the fully coherent case. For the partia...

Light propagation in a cholesteric medium has been completely discussed by de Vries [2]. The main results are that, for a given frequency, there are two eigenvibrations with quasi-circular polarizations, except i) for high frequencies and ii) near the reflection band (for the vibration having total reflection), where the polarization is linear. Looking at light refracted by a cholesteric prism,...

Abstract We present the intracavity generation of beams with radial polarization at an average output power 750 W and optical efficiency 43% from a continuous wave thin-disk laser. Circular grating waveguide couplers (GWOC) were used to select polarization. The sensitivity polarizing function GWOC regards fabrication tolerances is also analysed in details particular emphasis on effect duty cycl...

The chemical dynamics of the formation of the i-C4H3(X A 0) radical together with its partially deuterated isotopomers were investigated in eight crossed molecular beams experiments of dicarbon molecules in their XRg electronic ground and in first excited a Pu state with (partially deuterated) ethylene at collision energies between 12.1 and 40.9 kJ mol . The center-of-mass angular distributions...

Beams of microscopic particles penetrating scattering background matter play an important role in several applications. The parameter choices made here are motivated by the problem of electron-beam cancer therapy planning. Mathematically, a steady particle beam penetrating matter, or a configuration of several such beams, is modeled by a boundary value problem for a Boltzmann equation. Grid-bas...