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- Jeffrey C. Lagarias, James A. Reeds, Margaret H. Wright, Paul E. Wright
- SIAM Journal on Optimization
- 1998

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… (More)

- Jeffrey C. Lagarias, Andrew M. Odlyzko
- 24th Annual Symposium on Foundations of Computer…
- 1983

The subset sum problem is to decide whether or not the 0-l integer programming problem <italic>&Sgr;<supscrpt>n</supscrpt><subscrpt>i=l</subscrpt> a<subscrpt>i</subscrpt>x<subscrpt>i</subscrpt></italic> = <italic>M</italic>, <italic>∀I</italic>, <italic>x<subscrpt>I</subscrpt></italic> = 0 or 1, has a solution, where the… (More)

- Jeffrey C. Lagarias
- 23rd Annual Symposium on Foundations of Computer…
- 1982

Simultaneous Diophantine approximation in d dimensions deals with the approximation of a vector α = (α1, ..., αd) of d real numbers by vectors of rational numbers all having the same denominator. This paper considers the computational complexity of algorithms to find good simultaneous approximations to a given vector α of d… (More)

- Jeffrey C. Lagarias, Hendrik W. Lenstra, Claus-Peter Schnorr
- Combinatorica
- 1990

Let Ai(L), Ai(L*) denote the successive minima of a lattice L and its reciprocal lattice L*, and let [b l , . . . , bn] be a basis of L that is reduced in the sense of Korkin and Zolotarev. We prove that [4/(/+ 3)]),i(L) 2 _< [bi[ 2 < [(i + 3)/4])~i(L) 2 and Ibil2An_i+l(L*) 2 <_ [(i + 3)/4][(n i + 4)/417~ 2, where "y~ =min(Tj : 1 < j _< n} and 7j denotes… (More)

- Joel Hass, Jeffrey C. Lagarias, Nicholas Pippenger
- FOCS
- 1997

We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted, ie., capable of being continuously deformed without self-intersection so that it lies in a plane. We show that this problem, UNKNOTTING PROBLEM is in NP. We also consider the problem, SPLITTING PROBLEM of determining whether two or more such polygons… (More)

- John H. Conway, Jeffrey C. Lagarias
- J. Comb. Theory, Ser. A
- 1990

When can a given finite region consisting of cells in a regular lattice (triangular, square, or hexagonal) in [w’ be perfectly tiled by tiles drawn from a finite set of tile shapes? This paper gives necessary conditions for the existence of such tilings using boundary inuariants, which are combinatorial group-theoretic invariants associated to the… (More)

Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. It is possible for every circle in such a packing to have integer radius of curvature, and we call such a packing an integral Apollonian circle packing. This paper studies number-theoretic properties of the set of integer… (More)

Daniel J. Bernstein, Jeffrey C. Lagarias 19960215 Abstra t. The 3x+1 map T and the shift map S are de ned by T (x) = (3x+1)=2 for x odd, T (x) = x=2 for x even, while S(x) = (x 1)=2 for x odd, S(x) = x=2 for x even. The 3x + 1 onjuga y map on the 2-adi integers Z2 onjugates S to T , i.e., Æ S Æ 1 = T . The map mod 2n indu es a permutation n on Z=2nZ. We… (More)

O. H. Keller conjectured in 1930 that in any tiling of Rn by unit n-cubes there exist two of them having a complete facet in common. O. Perron proved this conjecture for n ≤ 6. We show that for all n ≥ 10 there exists a tiling of Rn by unit n-cubes such that no two n-cubes have a complete facet in common.

A self-affine tile in R is a set T of positive measure with A(T) = d ∈ $ < (T + d), where A is an expanding n × n real matrix with det (A) = m on integer, and $ = {d 1 ,d 2 , . . . , d m } ⊆ R is a set of m digits. It is known that self-affine tiles always give tilings of R by translation. This paper extends the known characterization of digit sets $… (More)