Jeffrey C. Lagarias

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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)
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>&forall;I</italic>, <italic>x<subscrpt>I</subscrpt></italic> = 0 or 1, has a solution, where the(More)
Simultaneous Diophantine approximation in d dimensions deals with the approximation of a vector &#x03B1; = (&#x03B1;1, ..., &#x03B1;d) 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 &#x03B1; of d(More)
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)
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)
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)
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)