EFFICIENT COMPUTER SEARCH FOR MULTIPLE RECURSIVE GENERATORS

Pseudo-random numbers (PRNs) are the basis for almost any statistical simulation and thisdepends largely on the quality of the pseudo-random number generator(PRNG) used. In this study, we used some results from number theory to propose an efficient method to accelerate the computer search of super-order maximum period multiple recursive generators (MRGs). We conduct efficient computer searches and successfully found prime modulus p, and the associated order k; (k = 40751; k = 50551; k = 50873) such that R(k; p) is a prime. Using these values of ks, together with the generalized Mersenne prime algorithm, we found and listed many efficient, portable, and super-order MRGs with period lengths of approximately 10e 380278.1;10e 471730.6; and 10e 474729.3. In other words, using the generalized Mersenne prime algorithm, we extended some known results of some efficient, portable, and maximum period MRGs. In particular, the DX/DL/DS/DT large order generators are extended to super-order generators.For r k, super-order generators in MRG(k,p) are quite close to an ideal generator. Forr \u3e k; the r-dimensional points lie on a relatively small family of equidistant parallel hyperplanesin a high dimensional space. The goodness of these generators depend largely on the distance between these hyperplanes. For LCGs, MRGs, and other generators with lattice structures, the spectral test, which is a theoretical test that gives some measure of uniformity greater than the order k of the MRG, is the most perfect figure of merit. A drawback of the spectral test is its computational complexity. We used a simple and intuitive method that employs the LLL algorithm, to calculate the spectral test. Using this method, we extended the search for better DX-k-s-t farther than the known value of k = 25013: In particular, we searched and listed better super-order DX-k-s-t generators for k = 40751; k = 50551, and k = 50873.Finally, we examined, another special class of MRGs with many nonzero terms known as the DW-k generator. The DW-k generators iteration can be implemented efficiently and in parallel, using a k-th order matrix congruential generator (MCG) sharing the same characteristic polynomial. We extended some known results, by searching for super-order DW-k generators, using our super large k values that we obtained in this study. Using extensive computer searches, we found and listed some super-order, maximum period DW(k; A, B, C, p = 2e 31 - v) generators

The Generalized Legendre transform and its applications to inverse spectral problems

Let $M$ be a Riemannian manifold, $\tau: G \times M \to M$ an isometric action on $M$ of an $n$-torus $G$ and $V: M \to \mathbb R$ a bounded $G$-invariant smooth function. By $G$-invariance the Schr\"odinger operator, $P=-\hbar^2 \Delta_M+V$, restricts to a self-adjoint operator on $L^2(M)_{\alpha/\hbar}$, $\alpha$ being a weight of $G$ and $1/\hbar$ a large positive integer. Let $[c_\alpha, \infty)$ be the asymptotic support of the spectrum of this operator. We will show that $c_\alpha$ extend to a function, $W: \mathfrak g^* \to \mathbb R$ and that, modulo assumptions on $\tau$ and $V$ one can recover $V$ from $W$, i.e. prove that $V$ is spectrally determined. The main ingredient in the proof of this result is the existence of a "generalized Legendre transform" mapping the graph of $dW$ onto the graph of $dV$.Comment: 23 page