The number of steps any classical computer requires in order to find the prime factors of an l-digit integer N increases exponentially with l, at least using algorithms [1] known at present. Factoring large integers is therefore conjectured to be intractable classically, an observation underlying

It was noticed by Benford [1] that the first nonzero digit in certain sets of real numbers is not uniformly distributed among the integers 1 through 9; in fact, the probability that this first, leftmost digit equals 3 is equal to log10(l + 3 " 1). He extended the analysis to the frequency

A Turing machine multiplies binary integers on-Zine if it receives its inputs low-order digits first and produces the jth digit of the product before reading in the (j+l)st digits of the two inputs. We present a general method for converting any off-line multiplication algorithm which forms

) . It takes time only L 1:185+o(1) on a machine of size L 0:790+o(1) explained in this paper. This reduction of total cost from L 2:852+o(1) to L 1:976+o(1) means that a ((3:009 +o(1))d)-digit factorization with the new machine has the same cost as a d-digit factorization with previous machines

on the representation of integers in two unrelated bases. We give the complete list of all the Fibonacci numbers and of all the Lucas numbers which have at most four digits 1 in their binary representation. 1.

The usual way to multiply numbers in binary representation runs as follows: To compute m-n, copy n to x. Multiply x by two. If the last digit of m is 1, then add n to x. Now delete the last digit of m. Repeat until m = 1, then x = m-n. Since multiplication by 2 needs almost no time, the running

Given integers d 1, and g 2, a g-addition chain for d is a sequence of integers a0 = 1, a1, a2,..., ar1, ar = d where ai = aj1 +aj2 + · · ·+ajk, with 2 k g, and 0 j1 j2 · · · jk i 1. The length of a g-addition chain is r, the number of terms following 1 in the sequence. We denote

discrepancy, the extreme discrepancy, the L2-discrepancy and the diaphony. Moreover, we obtain digital (0, 1)-sequences in arbitrary prime bases with very good extreme discrepancy, quite comparable to the best generalized van der Corput sequences already found in preceding studies ([2] and [4]). 1

Parametric max-stable processes are increasingly used to model spatial extremes. Since the dependence structure is specified for block maxima, the data used for inference are block maxima from all sites. To improve the estimation efficiency, we propose a two-step approach with composite likelihood that utilizes site-wise daily records in addition to block maxima. Besides the parameter estimation, there is no formal model checking and diagnosis method for spatial extremes modeling yet. Model diagnosis in practice has been informal and mostly based on visual checking tools such as residual plot and quantile-quantile plot. We proposed a goodness-of-fit test for max-stable processes based on the comparison between a nonparametric and a parametric estimator of the corresponding unknown multivariate Pickands dependence function. The proposed two-step procedure separates the estimation of marginal parameters and dependence parameters into two steps. The first step estimates the marginal pa-rameters with an independence Likelihood by ignoring the spatial dependence. Given ithe marginal parameter estimates, the second step estimates the dependence parameters

This Minimum Distance ID-Search Engine (MDSE) realizes the pattern-matching hardware accelerator for the portable multi-media and intelligent processing systems. The chip executes highly parallel computation of L,-norms between an input key and stored multiple reference records, and search