Some geometry and combinatorics for the S-invariant of ternary cubics

Given a real cubic form f(x,y,z), there is a pseudo-Riemannian metric given by its Hessian matrix, defined on the open subset of R^3 where the Hessian determinant h is non-zero. We determine the full curvature tensor of this metric in terms of h and the S-invariant of f, obtaining in the process various different characterizations of S. Motivated by the case of intersection forms associated with complete intersection threefolds in the product of three projective spaces, we then study ternary cubic forms which arise as follows: we choose positive integers d1, d2, d3, set r = d1 + d2 + d3 - 3, and consider the coefficient F(x,y,z) of H1^d1 H2^d2 H3^d3 in the product (x H1 + y H2 + z H3)^3 (a_1 H1 + b_1 H2 + c_1 H3) ... (a_r H1 + b_r H2 + c_r H3), the a_j, b_j and c_j denoting non-negative real numbers; we assume also that F is non-degenerate. Previous work of the author on sectional curvatures of Kahler moduli suggests a number of combinatorial conjectures concerning the invariants of F. It is proved here for instance that the Hessian determinant, considered as a polynomial in x,y,z and the a_j, b_j, c_j, has only positive coefficients. The same property is also conjectured to hold for the S-invariant; the evidence and background to this conjecture is explained in detail in the paper.Comment: 23 pages, plain Tex, updated and shortened, final versio

High-frequency high-voltage high-power DC-to-DC converters

The current and voltage waveshapes associated with the power transitor and the power diode in an example current-or-voltage step-up (buck-boost) converter were analyzed to highlight the problems and possible tradeoffs involved in the design of high voltage high power converters operating at switching frequencies in the range of 100 Khz. Although the fast switching speeds of currently available power diodes and transistors permit converter operation at high switching frequencies, the resulting time rates of changes of current coupled with parasitic inductances in series with the semiconductor switches, produce large repetitive voltage transients across the semiconductor switches, potentially far in excess of the device voltage ratings. The need is established for semiconductor switch protection circuitry to control the peak voltages appearing across the semiconductor switches, as well as to provide the waveshaping action require for a given semiconductor device. The possible tradeoffs, as well as the factors affecting the tradeoffs that must be considered in order to maximize the efficiency of the converters are enumerated

Analysis of transistor and snubber turn-off dynamics in high-frequency high-voltage high-power converters

Dc to dc converters which operate reliably and efficiently at switching frequencies high enough to effect substantial reductions in the size and weight of converter energy storage elements are studied. A two winding current or voltage stepup (buck boost) dc-to-dc converter power stage submodule designed to operate in the 2.5-kW range, with an input voltage range of 110 to 180 V dc, and an output voltage of 250 V dc is emphasized. In order to assess the limitations of present day component and circuit technologies, a design goal switching frequency of 10 kHz was maintained. The converter design requirements represent a unique combination of high frequency, high voltage, and high power operation. The turn off dynamics of the primary circuit power switching transistor and its associated turn off snubber circuitry are investigated

A dc model for power switching transistors suitable for computer-aided design and analysis

A model for bipolar junction power switching transistors whose parameters can be readily obtained by the circuit design engineer, and which can be conveniently incorporated into standard computer-based circuit analysis programs is presented. This formulation results from measurements which may be made with standard laboratory equipment. Measurement procedures, as well as a comparison between actual and computed results, are presented

Using the local gyrokinetic code, GS2, to investigate global ITG modes in tokamaks. (I) s-α{\alpha} model with profile and flow shear effects

This paper combines results from a local gyrokinetic code with analytical theory to reconstruct the global eigenmode structure of the linearly unstable ion-temperature-gradient (ITG) mode with adiabatic electrons. The simulations presented here employ the s-${\alpha}$ tokamak equilibrium model. Local gyrokinetic calculations, using GS2 have been performed over a range of radial surfaces, x, and for ballooning phase angle, p, in the range -${\pi} {\leq} p {\leq\pi}$, to map out the complex local mode frequency, ${\Omega_0(x, p) = \omega_0(x, p) + i\gamma_0(x, p)}$. Assuming a quadratic radial profile for the drive, namely ${\eta_i = L_n/L_T}$, (holding constant all other equilibrium profiles such as safety factor, magnetic shear etc.), ${\Omega_0(x, p)}$ has a stationary point. The reconstructed global mode then sits on the outboard mid plane of the tokamak plasma, and is known as a conventional or isolated mode, with global growth rate, ${\gamma}$ ~ Max[${\gamma_0(x, p)}$], where ${\gamma_0(x, p)}$ is the local growth rate. Taking the radial variation in other equilibrium profiles (e.g safety factor q(x)) into account, removes the stationary point in ${\Omega_0(x, p)}$ and results in a mode that peaks slightly away from the outboard mid-plane with a reduced global growth rate. Finally, the influence of flow shear has also been investigated through a Doppler shift, ${\omega_0 \rightarrow \omega_0 + n\Omega^{\prime}x}$, where n is the toroidal mode number and ${\Omega^{\prime}}$ incorporates the effect of flow shear. The equilibrium profile variation introduces an asymmetry to the growth rate spectrum with respect to the sign of ${\Omega^{\prime}}$, consistent with recent global gyrokinetic calculations.Comment: 10 pages, 8 figures and 1 tabl