Open questions concerning Weiszfeld's algorithm for the Fermat-Weber location problem

The Fermat—Weber location problem is to find a point in ⁿ that minimizes the sum of the weighted Euclidean distances fromm given points in ⁿ . A popular iterative solution method for this problem was first introduced by Weiszfeld in 1937. In 1973 Kuhn claimed that if them given points are not collinear then for all but a denumerable number of starting points the sequence of iterates generated by Weiszfeld's scheme converges to the unique optimal solution. We demonstrate that Kuhn's convergence theorem is not always correct. We then conjecture that if this algorithm is initiated at the affine subspace spanned by them given points, the convergence is ensured for all but a denumerable number of starting points.     

Laurent–Padé Approximants to Four Kinds of Chebyshev Polynomial Expansions. Part I. Maehly Type Approximants

Laurent Padé–Chebyshev rational approximants, A _m (z,z ^–1)/B _n (z,z ^–1), whose Laurent series expansions match that of a given function f(z,z ^–1) up to as high a degree in z,z ^–1 as possible, were introduced for first kind Chebyshev polynomials by Clenshaw and Lord [2] and, using Laurent series, by Gragg and Johnson [4]. Further real and complex extensions, based mainly on trigonometric expansions, were discussed by Chisholm and Common [1]. All of these methods require knowledge of Chebyshev coefficients of f up to degree m+n. Earlier, Maehly [5] introduced Padé approximants of the same form, which matched expansions between f(z,z ^–1)B _n (z,z ^–1) and A _m (z,z ^–1). The derivation was relatively simple but required knowledge of Chebyshev coefficients of f up to degree m+2n. In the present paper, Padé–Chebyshev approximants are developed not only to first, but also to second, third and fourth kind Chebyshev polynomial series, based throughout on Laurent series representations of the Maehly type. The procedures for developing the Padé–Chebyshev coefficients are similar to that for a traditional Padé approximant based on power series [8] but with essential modifications. By equating series coefficients and combining equations appropriately, a linear system of equations is successfully developed into two sub-systems, one for determining the denominator coefficients only and one for explicitly defining the numerator coefficients in terms of the denominator coefficients. In all cases, a type (m,n) Padé–Chebyshev approximant, of degree m in the numerator and n in the denominator, is matched to the Chebyshev series up to terms of degree m+n, based on knowledge of the Chebyshev coefficients up to degree m+2n. Numerical tests are carried out on all four Padé–Chebyshev approximants, and results are outstanding, with some formidable improvements being achieved over partial sums of Laurent–Chebyshev series on a variety of functions. In part II of this paper [7] Padé–Chebyshev approximants of Clenshaw–Lord type will be developed for the four kinds of Chebyshev series and compared with those of the Maehly type.     

On powerful distributional tests based on sample spacings

This paper is devoted to tests for uniformity based on sum-functions of m-spacings, where m diverges to infinity as the sample size, n, increases. It is shown that if m diverges at a slower rate than n^1/2 then the commonly used sum-function will detect alternatives distant (mn)^−1/4 from the uniform. This result fails if m diverges more quickly than n^1/2, and in that situation the statistic must be modified. The case where m/n → , 0 < < 1, is also considered, and it is shown that the test has adequate power against local and fixed alternatives if and only if is irrational.     

A Faster Deterministic Maximum Flow Algorithm

Cheriyan and Hagerup developed a randomized algorithm to compute the maximum flow in a graph with n nodes and m edges in O(mn + n² log²n) expected time. The randomization is used to efficiently play a certain combinatorial game that arises during the computation. We give a version of their algorithm where a general version of their game arises. Then we give a strategy for the game that yields a deterministic algorithm for computing the maximum flow in a directed graph with n nodes and m edges that runs in time O(mn(log_{m/n log n}n)). Our algorithm gives an O(mn) deterministic algorithm for all m/n = Ω(n^ε) for any positive constant ε, and is currently the fastest deterministic algorithm for computing maximum flow as long as m/n = ω(log n).     

Some Mean Values Related to Average Multiplicative Orders of Elements in Finite Fields

For any positive integer n let α(n) denote the average order of elements in the cyclic group Z_n. In this note, we investigate the functions α(n)/n and α(n)/φ(n) when n ranges through numbers of the form p−1 with p prime, and when n ranges through numbers of the form 2^m−1 with m a positive integer. In particular, we show that such functions have limiting distributions, and we compute their average values, and their minimal and maximal orders.To Jean-Louis Nicolas at his 60th birthday2000 Mathematics Subject Classification: Primary—11N45; Secondary—05A16, 11N37This work was supported in part by Grant SEP-CONACYT 37259-E.     

On the sequence of powers of a stochastic matrix with large exponent

We consider the class of primitive stochastic n×n matrices A, whose exponent is at least (n²−2n+2)/2+2. It is known that for such an A, the associated directed graph has cycles of just two different lengths, say k and j with k>j, and that there is an α between 0 and 1 such that the characteristic polynomial of A is λⁿ−αλ^n−j−(1−α)λ^n−k. In this paper, we prove that for any mn, if α1/2, then A^m+k−A^m_∞A^m−1w^T_∞, where 1 is the all-ones vector and w^T is the left-Perron vector for A, normalized so that w^T1=1. We also prove that if jn/2, n31 and , then A^m+j−A^m_∞A^m−1w^T_∞ for all sufficiently large m. Both of these results lead to lower bounds on the rate of convergence of the sequence A^m.     

A generalization of Everett's result on the Collatz 3x + 1 problem

It is a well-known conjecture that given m ε N, the set of natural numbers, the sequence {m_n}^∞_n−0, defined by the iterative formula m₀ = m,

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has some iterate m_j = 1. It is shown in this paper that for any k ε N, “almost every” natural number m greater than unity has k iterates less than m.     

Ramsey functions involving with large

For fixed integers m,k2, it is shown that the k-color Ramsey number r_k(K_m,n) and the bipartite Ramsey number b_k(m,n) are both asymptotically equal to k^mn as n→∞, and that for any graph H on m vertices, the two-color Ramsey number is at most (1+o(1))n^m+1/(logn)^m-1. Moreover, the order of magnitude of is proved to be n^m+1/(logn)^m if H≠K_m as n→∞.     

On shooting methods for the discrete Helmholtz equation with constant coefficients

Summary The paper is concerned with shooting solvers for the Helmholtz equation with constant coefficients in two dimensions using finite differences for the discretization. Dirichlet boundary conditions are treated though other conditions are possible. Beginning with a single shooting method some recursive multiple shooting methods are developed. It will be shown that the performance of the algorithms may be improved considerably by a redundance-free recursion. The number of operations required for one solution will be computed, but without preparing some matrices which do not depend on the boundary conditions and the inhomogenity. For a square withn×n points the number is of the orderO(n ²⁺⁽ⁿ⁾) with . The method will be compared with a multi-grid program and finally — as an example—a Stokes-solver and some numerical results with the shooting method are given.     

A graph isomorphism algorithm using pseudoinverses

For a graph ofm nodes andn edges, an algorithm for testing the isomorphism of graphs is given. The complexity of the algorithm is a maximum ofO(mn ²) in almost all cases, with a considerable reduction if sparsity is exploited. If isomorphism is present, the pseudoinverses of the Laplace matrices of the graphs will be row and column permutations of each other. Advantage can be taken of certain features of the incidence matrices or of properties of the graphs to reduce computation time.     

The complexity of many cells in arrangements of planes and related problems

We consider several problems involving points and planes in three dimensions. Our main results are: (i) The maximum number of faces boundingm distinct cells in an arrangement ofn planes isO(m ^2/3 n logn +n ²); we can calculatem such cells specified by a point in each, in worst-case timeO(m ^2/3 n log³ n+n ² logn). (ii) The maximum number of incidences betweenn planes andm vertices of their arrangement isO(m ^2/3 n logn+n ²), but this number is onlyO(m ^3/5– n ^4/5+2 +m+n logm), for any>0, for any collection of points no three of which are collinear. (iii) For an arbitrary collection ofm points, we can calculate the number of incidences between them andn planes by a randomized algorithm whose expected time complexity isO((m ^3/4– n ^3/4+3 +m) log² n+n logn logm) for any>0. (iv) Givenm points andn planes, we can find the plane lying immediately below each point in randomized expected timeO([m ^3/4– n ^3/4+3 +m] log² n+n logn logm) for any>0. (v) The maximum number of facets (i.e., (d–1)-dimensional faces) boundingm distinct cells in an arrangement ofn hyperplanes ind dimensions,d>3, isO(m ^2/3 n ^d/3 logn+n ^d–1). This is also an upper bound for the number of incidences betweenn hyperplanes ind dimensions andm vertices of their arrangement. The combinatorial bounds in (i) and (v) and the general bound in (ii) are almost tight.Work on this paper by the first author has been supported by Amoco Fnd. Fac. Dev. Comput. Sci. 1-6-44862 and by NSF Grant CCR-8714565. Work by the third author has been supported by Office of Naval Research Grant N00014-87-K-0129, by National Science Foundation Grant DCR-82-20085, by grants from the Digital Equipment Corporation, and the IBM Corporation, and by a research grant from the NCRD—the Israeli National Council for Research and Development. An abstract of this paper has appeared in theProceedings of the 13th International Mathematical Programming Symposium, Tokyo, 1988, p. 147.     

Lower bounds for the cardinality of minimal blocking sets in projective spaces

In this paper new lower bounds for the cardinality of minimal m-blocking sets are determined. Let r₂(q) be the number such that q+r₂(q)+1 is the cardinality of the smallest non-trivial line-blocking set in a plane of order q. If B is a minimal m-blocking set in PG(n,q) that contains at most q^m+q^m−1+…+q+1+r₂(q)·(∑_i=2m−n′^m−1qⁱ) points for an integer n′ satisfying mn′2m, then the dimension of B is at most n′. If the dimension of B is n′, then the following holds. The cardinality of B equals q^m+q^m−1+…+q+1+r₂(q)(∑_i=2m−n′^m−1qⁱ). For n′=m the set B is an m-dimensional subspace and for n′=m+1 the set B is a cone with an (m−2)-dimensional vertex over a non-trivial line-blocking set of cardinality q+r₂(q)+1 in a plane skew to the vertex. This result is due to Heim (Mitt. Math. Semin. Giessen 226 (1996), 4–82). For n′>m+1 and q not a prime the number q is a square and for q16 the set B is a Baer cone. If q is odd and |B|<q^m+q^m−1+…+q+1+r₂(q)(q^m−1+q^m−2), it follows from this result that the subspace generated by B has dimension at most m+1. Furthermore we prove that in this case, if , then B is an m-dimensional subspace or a cone with an (m−2)-dimensional vertex over a non-trivial line-blocking set of cardinality q+r₂(q)+1 in a plane skew to the vertex. For q=p^3h, p7 and q not a square we show this assertion for |B|q^m+q^m−1+…+q+1+q^2/3·(q^m−1+…+1).     

Computing Vertex Connectivity: New Bounds from Old Techniques

The vertex connectivity κ of a graph is the smallest number of vertices whose deletion separates the graph or makes it trivial. We present the fastest known deterministic algorithm for finding the vertex connectivity and a corresponding separator. The time for a digraph having n vertices and m edges is O(min{κ³ + n, κn}m); for an undirected graph the term m can be replaced by κn. A randomized algorithm finds κ with error probability 1/2 in time O(nm). If the vertices have nonnegative weights the weighted vertex connectivity is found in time O(κ₁nmlog(n²/m)) where κ₁ ≤ m/n is the unweighted vertex connectivity or in expected time O(nmlog(n²/m)) with error probability 1/2. The main algorithm combines two previous vertex connectivity algorithms and a generalization of the preflow-push algorithm of Hao and Orlin (1994, J. Algorithms17, 424–446) that computes edge connectivity.     

A combinatorial theory of Grünbaum's new regular polyhedra,Part II: Complete enumeration

The new regular polyhedra as defined by Branko Grünbaum in 1977 (cf. [5]) are completely enumerated. By means of a theorem of Bieberbach, concerning the existence of invariant affine subspaces for discrete affine isometry groups (cf. [3], [2] or [1]) the standard crystallographic restrictions are established for the isometry groups of the non finite (Grünbaum-)polyhedra. Then, using an appropriate classification scheme which—compared with the similar, geometrically motivated scheme, used originally by Grünbaum—is suggested rather by the group theoretical investigations in [4], it turns out that the list of examples given in [5] is essentially complete except for one additional polyhedron.So altogether—up to similarity—there are two classes of planar polyhedra, each consisting of 3 individuals and each class consisting of the Petrie duals of the other class, and there are ten classes of non planar polyhedra: two mutually Petrie dual classes of finite polyhedra, each consisting of 9 individuals, two mutually Petrie dual classes of infinite polyhedra which are contained between two parallel planes with each of those two classes consisting of three one-parameter families of polyhedra, two further mutually Petrie dual classes each of which consists of three one parameter families of polyhedra whose convex span is the whole 3-space, two further mutually Petrie dual classes consisting of three individuals each of which spanE ³ and two further classes which are closed with respect to Petrie duality, each containing 3 individuals, all spanningE ³, two of which are Petrie dual to each other, the remaining one being Petrie dual to itself.In addition, a new classification scheme for regular polygons inE ⁿ is worked out in §9.     

A self-normalized Erd s—Rényi type strong law of large numbers

The original Erd s—Rényi theorem states that max_0kn∑^k+[clogn]_i=k+1X_i/[clogn]→α(c),c>0, almost surely for i.i.d. random variables {X_n, n1} with mean zero and finite moment generating function in a neighbourhood of zero. The latter condition is also necessary for the Erd s—Rényi theorem, and the function α(c) uniquely determines the distribution function of X₁. We prove that if the normalizing constant [c log n] is replaced by the random variable ∑^k+[clogn]_i=k+1(X²_i+1), then a corresponding result remains true under assuming only the exist first moment, or that the underlying distribution is symmetric.     

Inplace 2D matching in compressed images

The compressed matching problem is the problem of finding all occurrences of a pattern in a compressed text. In this paper we discuss the 2-dimensional compressed matching problem in Lempel–Ziv compressed images. Given a pattern P of (uncompressed) size m×m, and a text T of (uncompressed) size n×n, both in 2D-LZ compressed form, our algorithm finds all occurrences of P in T. The algorithm is strongly inplace, that is, the amount of extra space used is proportional to the best possible compression of a pattern of size m². The best compression that the 2D-LZ technique can obtain for a file of size m² is O(m). The time for performing the search is O(n²) and the preprocessing time is O(m³). Our algorithm is general in the sense that it can be used for any 2D compression which can be sequentially decompressed in small space.     

Sorting in one round

Given an ordered set ofn elements whose order is not known to us, it is shown that by making slightly more thancn ^3/2 simultaneous comparison almost all comparisons can be deduced by direct implications. It is also shown that this result is essentially best possible, and that ifn is large, almost any choice ofcn ^3/2 comparisons will yield almost all comparisons by direct implications.     

Laurent–Padé Approximants to Four Kinds of Chebyshev Polynomial Expansions. Part II: Clenshaw–Lord Type Approximants

Laurent–Padé (Chebyshev) rational approximants P _m (w,w ^–1)/Q _n (w,w ^–1) of Clenshaw–Lord type [2,1] are defined, such that the Laurent series of P _m /Q _n matches that of a given function f(w,w ^–1) up to terms of order w ^±(m+n), based only on knowledge of the Laurent series coefficients of f up to terms in w ^±(m+n). This contrasts with the Maehly-type approximants [4,5] defined and computed in part I of this paper [6], where the Laurent series of P _m matches that of Q _n f up to terms of order w ^±(m+n), but based on knowledge of the series coefficients of f up to terms in w ^±(m+2n). The Clenshaw–Lord method is here extended to be applicable to Chebyshev polynomials of the 1st, 2nd, 3rd and 4th kinds and corresponding rational approximants and Laurent series, and efficient systems of linear equations for the determination of the Padé–Chebyshev coefficients are obtained in each case. Using the Laurent approach of Gragg and Johnson [4], approximations are obtainable for all m0, n0. Numerical results are obtained for all four kinds of Chebyshev polynomials and Padé–Chebyshev approximants. Remarkably similar results of formidable accuracy are obtained by both Maehly-type and Clenshaw–Lord type methods, thus validating the use of either.     

Order Determination for Multivariate Autoregressive Processes Using Resampling Methods

LetX₁, …, X_nbe observations from a multivariate AR(p) model with unknown orderp. A resampling procedure is proposed for estimating the orderp. The classical criteria, such as AIC and BIC, estimate the orderpas the minimizer of the function[formula]wherenis the sample size,kis the order of the fitted model, Σ²_kis an estimate of the white noise covariance matrix, andC_nis a sequence of specified constants (for AIC,C_n=2m²/n, for Hannan and Quinn's modification of BIC,C_n=2m²(ln ln n)/n, wheremis the dimension of the data vector). A resampling scheme is proposed to estimate an improved penalty factorC_n. Conditional on the data, this procedure produces a consistent estimate ofp. Simulation results support the effectiveness of this procedure when compared with some of the traditional order selection criteria. Comments are also made on the use of Yule–Walker as opposed to conditional least squares estimations for order selection.     

Vanishing Sums ofmth Roots of Unity in Finite Fields

In an earlier work, the authors determined all possible weightsnfor which there exists a vanishing sum ζ₁+ · · · + ζ_n= 0 ofmth roots of unity ζ_iin characteristic 0. In this paper, the same problem is studied in finite fields of characteristicp. For givenmandp, results are obtained on integersn₀such that all integersn≥n₀are in the “weight set”W_p(m). The main result in this paper guarantees, under suitable conditions, the existence of solutions ofx^d1 + · · · +x^d_n= 0with all coordinates not equal to zeroover a finite field.