The solution of the delta function model for helium like ions

Fredholm solution of delta function model for two electron helium like ion

Output Feedback Invariants

The paper is concerned with the problem of determining a complete set of invariants for output feedback. Using tools from geometric invariant theory it is shown that there exists a quasi-projective variety whose points parameterize the output feedback orbits in a unique way. If the McMillan degree $n\geq mp$, the product of number of inputs and number of outputs, then it is shown that in the closure of every feedback orbit there is exactly one nondegenerate system.Comment: 15 page

Degree of the generalized Pl\"ucker embedding of a Quot scheme and Quantum cohomology

We compute the degree of the generalized Pl\"ucker embedding $\kappa$ of a Quot scheme $X$ over \PP^1. The space $X$ can also be considered as a compactification of the space of algebraic maps of a fixed degree from \PP^1 to the Grassmanian $\rm{Grass}(m,n)$. Then the degree of the embedded variety $\kappa (X)$ can be interpreted as an intersection product of pullbacks of cohomology classes from $\rm{Grass}(m,n)$ through the map $\psi$ that evaluates a map from \PP^1 at a point x\in \PP^1. We show that our formula for the degree verifies the formula for these intersection products predicted by physicists through Quantum cohomology~\cite{va92}~\cite{in91}~\cite{wi94}. We arrive at the degree by proving a version of the classical Pieri's formula on the variety $X$, using a cell decomposition of a space that lies in between $X$ and $\kappa (X)$.Comment: 18 pages, Latex documen

Peaks and Troughs in Helioseismology: The Power Spectrum of Solar Oscillations

I present a matched-wave asymptotic analysis of the driving of solar oscillations by a general localised source. The analysis provides a simple mathematical description of the asymmetric peaks in the power spectrum in terms of the relative locations of eigenmodes and troughs in the spectral response. It is suggested that the difference in measured phase function between the modes and the troughs in the spectrum will provide a key diagnostic of the source of the oscillations. I also suggest a form for the asymmetric line profiles to be used in the fitting of solar power spectra. Finally I present a comparison between the numerical and asymptotic descriptions of the oscillations. The numerical results bear out the qualitative features suggested by the asymptotic analysis but suggest that numerical calculations of the locations of the troughs will be necessary for a quantitative comparison with the observations.Comment: 18 pages + 8 separate figures. To appear in Ap

Air hydrodynamics of the ultrafast laser-triggered spark gap

We present space and time resolved measurements of the air hydrodynamics induced by ultrafast laser pulse excitation of the air gap between two electrodes at high potential difference. We explore both plasma-based and plasma-free gap excitation. The former uses the plasma left in the wake of femtosecond filamentation, while the latter exploits air heating by multiple-pulse resonant excitation of quantum molecular wavepackets. We find that the cumulative electrode-driven air density depression channel initiated by the laser plays the dominant role in the gap evolution leading to breakdown