James A. Reeds

Convergence Properties of the Nelder--Mead Simplex Method in Low Dimensions

The Nelder--Mead simplex algorithm, first published in 1965, is an enormously popular direct search method for multidimensional unconstrained minimization. Despite its widespread use, essentially no theoretical results have been proved explicitly for the Nelder--Mead algorithm. This paper presents convergence properties of the Nelder--Mead algorithm applied to strictly convex functions in dimensions 1 and 2. We prove convergence to a minimizer for dimension 1, and various limited convergence results for dimension 2. A counterexample of McKinnon gives a family of strictly convex functions in two dimensions and a set of initial conditions for which the Nelder--Mead algorithm converges to a nonminimizer. It is not yet known whether the Nelder--Mead method can be proved to converge to a minimizer for a more specialized class of convex functions in two dimensions.

Shift-Register Synthesis (Modulo m)

The Berlekamp algorithm takes a sequence of elements from a field and finds the shortest linear recurrence (or linear feedback shift register) that can generate the sequence. We present an algorithm which generalizes Berlekamp’s to the case when the elements of the sequence are integers modulo m, where m is an arbitrary (but known) integer. Details will be published elsewhere.

Orthonormal bases of exponentials for the n-cube

Any set that gives such an orthogonal basis is called a spectrum for . Only very special sets in R are spectral sets. However, when a spectrum exists, it can be viewed as a generalization of Fourier series, because for the n-cube = [0,1]n the spectrum = Z gives the standard Fourier basis of L2([0,1]n). The main object of this paper is to relate the spectra of sets to tilings in Fourier space. We develop such a relation for a large class of sets and apply it to geometrically characterize all spectra for the n-cube = [0,1]n.

Sets uniquely determined by projections on axes II Discrete case

Abstract A subset S of N n = {1, 2,…, N} n is a discrete set of uniqueness if it is the only subset of N n with projections P 1 ,…, P n , where P i ( j ) = |{( x 1 ,…, x n ) ϵ S : x i = j }|. Also, S is additive if there are real valued functions z.hfl; 1 ,…, z.hfl; n on N such that, for all ( x 1 ,…, x n ) ϵ N n , ( x 1 ,…, x n ) ϵ S ⇔ ∑ i z.hfl; i ( x i ) ⩾ 0. Sets of uniqueness and additive sets are characterized by the absence of certain configurations in the lattice N n . The characterization shows that every additive set is a set of uniqueness. If n = 2, every set of uniqueness is also additive. However, when n ⩾ 3, there are sets of uniqueness that are not additive.

Self-avoiding random loops

A random loop, or polygon, is a simple random walk whose trajectory is a simple Jordan curve. The study of random loops is extended in two ways. First, the probability P/sub n/(x,y) that a random n-step loop contains a point (x,y) in the interior of the loop is studied, and (1/2, 1/2) is shown to be (1/2)-(1/n). It is plausible that P/sub n/(x,y) tends toward 1/2 for all (x,y), but this is not proved even for (x,y)=(3/2,1/2) A way is offered to simulate random n-step self-avoiding loops. Numerical evidence obtained with this simulation procedure suggests that the probability P/sub n/(3/2,1/2) approximately=(1/2)-(c/n), for some fixed c. >

On the cryptanalysis of rotor machines and substitution - permutation networks

A general cryptanalysis method is presented based on statistical estimation theory. It is applied to two systems of practical interest: rotor machines and substitution-permutation networks. To cryptanalyze these systems, the finite keyspace is imbedded in a continuous set and the key estimate is a proper quantization of the continuous maximum likelihood estimate. Promising cryptanalysis results of a rotor machine under a ciphertext only attack and a substitution-permutation network under a known plaintext attack are presented.

Unique extrapolation of polynomial recurrences

Let a sequence of k-dimensional vectors

Unit distances between vertices of a convex polygon

{\bf x}_0 ,{\bf x}_1 , \cdots

Stereology of dihedral angles I: first two moments

(over a ring A) be determined by a polynomial recurrence of form

Stereology of dihedral angles II: distribution function

{\bf x}_n = T({\bf x}_{n - 1} )