Paraxial-wave optics and relativistic front description .1. The scalar theory

With the extension of the work of the preceding paper, the relativistic front form for Maxwell's equations for electromagnetism is developed and shown to be particularly suited to the description of paraxial waves. The generators of the Poincaré group in a form applicable directly to the electric and magnetic field vectors are derived. It is shown that the effect of a thin lens on a paraxial electromagnetic wave is given by a six-dimensional transformation matrix, constructed out of certain special generators of the Poincaré group. The method of construction guarantees that the free propagation of such waves as well as their transmission through ideal optical systems can be described in terms of the metaplectic group, exactly as found for scalar waves by Bacry and Cadilhac. An alternative formulation in terms of a vector potential is also constructed. It is chosen in a gauge suggested by the front form and by the requirement that the lens transformation matrix act locally in space. Pencils of light with accompanying polarization are defined for statistical states in terms of the two-point correlation function of the vector potential. Their propagation and transmission through lenses are briefly considered in the paraxial limit. This paper extends Fourier optics and completes it by formulating it for the Maxwell field. We stress that the derivations depend explicitly on the "henochromatic" idealization as well as the identification of the ideal lens with a quadratic phase shift and are heuristic to this extent.

A Front end Switched Rectifier DC Source for Neutral Point Balancing of A NPC Three-Level Inverter for The Full Modulation Range

A switched DC voltage three level NPC is proposed in this paper to eliminate capacitor balancing problems in conventional three-level Neutral Point Clamped (NPC) inverter. The proposed configuration requires only one DC link with a voltage V-dc/2, where V-dc is the DC link voltage in a onventional NPC inverter. To get rated DC link voltage (V-dc), the voltage source is alternately onnected in parallel to one of the two series capacitors using two switches and two diodes with device voltage rating of V-dc/2. The frequency at which the voltage source is switched is independent and will not affect the operation of NPC inverter. The switched voltage source in this configuration balances the capacitors automatically. The proposed configuration can also be used as a conventional two level inverter in lower modulation range, thereby increases the reliability of the drive system. A space vector based PWM scheme is used to verify this proposed topology.

Kurt and Hans Ludwig (Hal) Gottschalk on "skateboards" in front of Kurhaus Brunig; Norderney, Germany. Portraits Children

Left: Kurt Gottschalk; right: Hans Ludwig (Hal) Gottschalk

Gottschalk family on vacation in front of Kurhaus Brunig; Norderney, Germany.

From left to right: man with camera: Karl Gottschalk; sitting on stairs: Fritz Gottschalk, unidentified woman, Therese Gottschalk nee Molling, Hans Ludwig, Kurt Gottschalk, unidentified woman (maid?) and Elizabeth Gottschalk

Lola Gruenthal, in front of the house Portraits women

"In front of the house September" on verso

Albert Einstein (front row, third from right) in school group portrait Portraits Groups

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Portrait of unidentified man wearing medals and holding rifle, standing in front of shooting target Portraits Men

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Elizabeth Krakauer nee Gottschalk with her mother-in-law Auguste Krakauer nee Mayer in front of a house.

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Ernest Krakauer (left) with an unidentified man in front of a 1934 Plymouth; Germany.

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Walter, Ursula and Freddy, three of Fritz Gottschalk's children dressed in costumes in front of a dollhouse; Hannover, Germany.

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Fritz Gottschalk's family in front of the hotel Kurhaus Brunig, Norderney; Germany.

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Alfred Lichtenstein standing in front of a billboard advertising him and others; Berlin.

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