Special Function Generator Research Articles

PARTITION FUNCTIONS OF MATRIX MODELS AS THE FIRST SPECIAL FUNCTIONS OF STRING THEORY II: KONTSEVICH MODEL

In the paper Int. J. Mod. Phys. A19, 4127 (2004), we started a program of creating a reference-book on matrix-model τ-functions, the new generation of special functions, which are going to play an important role in string theory calculations. The main focus of that paper was on the one-matrix Hermitian model τ-functions. The present paper is devoted to a direct counterpart for the Kontsevich and Generalized Kontsevich Model (GKM) τ-functions. We mostly focus on calculating resolvents (= loop operator averages) in the Kontsevich model, with a special emphasis on its simplest (Gaussian) phase, where exists a surprising integral formula, and the expressions for the resolvents in the genus zero and one are especially simple (in particular, we generalize the known genus zero result to genus one). We also discuss various features of generic phases of the Kontsevich model, in particular, a counterpart of the unambiguous Gaussian solution in the generic case, the solution called Dijkgraaf–Vafa (DV) solution. Further, we extend the results to the GKM and, in particular, discuss the p–q duality in terms of resolvents and corresponding Riemann surfaces in the example of dualities between (2, 3) and (3, 2) models.

Fabry-Perot interferometer with personal computer control

To record weak light scattering spectra with a Fabry-Perot interferometer, long measuring times are often necessary. For long term stability of the parallel alignment of the mirrors, a microcomputer controlled interferometer system has been developed for both single and multipass operation. The three piezo stacks, which are responsible for the alignment of the mirrors, are supplied from three digital-to-analog converters controlled by a personal computer. The test and correction steps of the mirror alignment are performed around two independent axes perpendicular to each other. A special function generator provides a nonlinear scanning ramp for the piezostacks, thereby allowing a temporally linear scanning of the Fabry-Perot.

Rapidly Tuned Buncher Structure for the los Alamos Proton Storage Ring (PSR)

In the PSR's short-bunch operating mode, accumulated beam currents are intense and change rapidly. The resonant frequency of the 503.125-MHz buncher used in this mode must be rapidly adjusted through a 100-kHz range to maintain the correct 90° phase relation between cavity voltage and beam current. Modulation rates are up to 3 kHz/?s. Each structure consists of two side-coupled buncher cavities, resonantly coupled to a ferrite-loaded tuner cavity. The needed frequency change ?f in the buncher cells is produced by a 50 × ?f change in the tuner, accomplished by varying a magnetic field applied to the ferrite perpendicular to the rf magnetic field. Fast modulation of this bias is provided by a low-inductance ferritecore magnet excited by a special function generator. The resonantly coupled multicavity structure configuration allows buncher and tuner cells to be independently optimized for their specific functions. This paper describes the buncher design, ferrite selection, and test results from a prototype ferrite-loaded tuner cavity. The tests have demonstrated the tuning scheme's feasibility, showing that the necessary 5-MHz range can be attained with only 12% of the tuner cell filled with ferrite, and that losses in the ferrite are small throughout this frequency interval.

Design method for digital dead beat controllers

This paper describes an analog computer method for designing dead beat controllers for linear and nonlinear sampled data system. In this method the output of the digital controller is replaced by the output from a special function generator. Thus the plant input consists of a number of step signals (with variable magnitude and sign) applied at the sampling instants. By adjusting the magnitudes and polarities of the input signals the plant output and the derivatives of the output signal are adjusted so that they have the desired values at the required sampling instant, the signals being observed on a multitrace oscilloscope. The coefficients of the dead beat controller are completely determined by the successive values of the step input signals and the system outputs at the sampling instants. This design method is particularly useful when the plant is nonlinear.

A special function generator:The frequency programming system for the 25 GeV C.E.R.N. proton synchrotron

In a synchrotron the relation between the intensity of the magnetic guide field B and thefrequency f of the accelerating radio frequency field is of the form f (B) = f ∞ B (B 2 +B o 2 ) −1/2 for particle motion at constant mean orbit radius. A function generator is decribed in which a voltage V′ α f (B) * is produced which is linearly translated into frequency by means of a servo controlled radio frequency oscillator. The voltage V′ α I B is obtained by means of the Hall effect. The current I is automatically regulated such that I (B 2 +B o 2 ) 1/2 is independent on B. The factor (B 2 +B o 2 ) 1/2 is simulated by the quadrative sum of two a.c. signals u 1 α B and u 2 α B o . A description of the important elements and of the performance of the function generator is given.

A function generator for the solution of engineering design problems

The solution of nonlinear engineering design problems demonstrates the need for a special function generator. The generator described in this paper satisfies this need. The basic components of the unit are discussed and the forms of functions which can be generated are shown. Accuracy is estimated by comparison of an oscillogram with the calculated curves. It is concluded from tests on the system that the function generator is a valuable aid in the handling of nonlinear design problems.