Interpolatory quadrature formulas on the unit circle for Chebyshev weight functions

Summary. In this paper, interpolatory quadrature formulas based upon the roots of unity are studied for certain weight functions. Positivity of the coefficients in these formulas is deduced along with computable error estimations for analytic integrands. A comparison is made with Szeg? quadrature formulas. Finally, an application to the interval [-1,1] is also carried out. Received February 29, 2000 / Published online August 17, 2001     

On spherical expansions of zonal functions on Euclidean spheres

This paper describes a new approach to the problem of computing spherical expansions of zonal functions on Euclidean spheres. We derive an explicit formula for the coefficients of the expansion expressing them in terms of the Taylor coefficients of the profile function rather than (as done usually) in terms of its integrals against Gegenbauer polynomials. Our proof of this result is based on a polynomial identity equivalent to the canonical decomposition of homogeneous polynomials and uses only basic properties of this decomposition together with simple facts concerning zonal harmonic polynomials. As corollaries, we obtain direct and apparently new derivations of the so-called plane wave expansion and of the expansion of the Poisson kernel for the unit ball. Received: 26 January 2007     

A new condition number for linear programming

In this paper we define a new condition number ?(A) for the following problem: given a m by n matrix A, find x∈ℝ ⁿ , s.t. Ax<0. We characterize this condition number in terms of distance to ill-posedness and we compare it with existing condition numbers for the same problem. Received: November 5, 1999 / Accepted: November 2000?Published online September 17, 2001     

Two Remarks on Solutions of Gross-Pitaevskii Equations on Zhidkov Spaces

We consider the so-called Gross-Pitaevskii equations supplemented with non-standard boundary conditions. We prove two mathematical results concerned with the initial value problem for these equations in Zhidkov spaces.     

Orthogonalpolynome in x und q −x als Lösungen von reellen q-Operatorgleichungen zweiter Ordnung

 The explicit form and the three term recurrance relation for all the polynomial solutions in x and q ^−x from q-operator equations of second order are given. The known 17 q-orthogonal polynomial systems are special cases of 7 comprehensive q-systems. (Eingegangen 27. M?rz 2000; in revidierter Fassung 2. November 2000)     

The weak conjecture on spherical classes

Let be the mod 2 Steenrod algebra. We construct a chain-level representation of the dual of Singer's algebraic transfer, which maps Singer's invariant-theoretic model of the dual of the Lambda algebra, , to and is the inclusion of the Dickson algebra, , into . This chain-level representation allows us to confirm the weak conjecture on spherical classes (see [9]), assuming the truth of (1) either the conjecture that the Dickson invariants of at least k = 3 variables are homologically zero in }, (2) or a conjecture on ${mathcal{A}}$ -decomposability of the Dickson algebra in $Gamma_k^{wedge}$. We prove the conjecture in item (1) for k = 3 and also show a weak form of the conjecture in item (2). Received November 27, 1996; in final form March 6, 1998     

On the analyticity properties of infeasible-interior-point paths for monotone linear complementarity problems

Infeasible-interior-point paths , a positive vector, for a horizontal linear complementarity problem are defined as the solution of () If the path converges for , then it converges to a solution of . This paper deals with the analyticity properties of and its derivatives with respect to r near r = 0 for solvable monotone complementarity problems . It is shown for with a strictly complementary solution that the path , , has an extension to which is analytic also at . If has no strictly complementary solution, then , , has an extension to that is analytic at . Received May 24, 1996 / Revised version received February 25, 1998     

Covering symmetric supermodular functions by graphs

Received December 1996 / Revised version received January 1998 Published online March 16, 1999     

Quadratic convergence of potential-reduction methods for degenerate problems

Global and local convergence properties of a primal-dual interior-point pure potential-reduction algorithm for linear programming problems is analyzed. This algorithm is a primal-dual variant of the Iri-Imai method and uses modified Newton search directions to minimize the Tanabe-Todd-Ye (TTY) potential function. A polynomial time complexity for the method is demonstrated. Furthermore, this method is shown to have a unique accumulation point even for degenerate problems and to have Q-quadratic convergence to this point by an appropriate choice of the step-sizes. This is, to the best of our knowledge, the first superlinear convergence result on degenerate linear programs for primal-dual interior-point algorithms that do not follow the central path. Received: February 12, 1998 / Accepted: March 3, 2000?Published online January 17, 2001     

Efficient separation routines for the symmetric traveling salesman problem I: general tools and comb separation

This is the first of a series of two papers dedicated to efficient separation heuristics for violated inequalities in the context of solving to optimality instances of the Traveling Salesman Problem via branch-and-cut. In this paper we give the basic ideas behind these heuristics and design heuristics for comb separation. The second paper will deal with more complex inequalities such as Clique Tree, Star or Path and Ladder inequalities and give computational results. Received: September 1999 / Accepted: August 2001?Published online February 14, 2002     

Efficient separation routines for the symmetric traveling salesman problem II: separating multi handle inequalities

This paper is the second in a series of two papers dedicated to the separation problem in the symmetric traveling salesman polytope. The first one gave the basic ideas behind the separation procedures and applied them to the separation of Comb inequalities. We here address the problem of separating inequalities which are all, in one way or another, a generalization of Comb inequalities. These are namely clique trees, path, and ladder inequalities. Computational results are reported for the solution of instances of the TSPLib using the branch and cut framework ABACUS. Received: September 1999 / Accepted: August 2001?Published online February 14, 2002     

A coefficient inequality for Bloch functions with applications to uniformly locally univalent functions

We give a Fekete-Szeg? type inequality for an analytic function on the unit disk with Bloch seminorm ≤1. As an application of it, we derive a sharp inequality for the third coefficient of a uniformly locally univalent function f(z) = z + a ₂ z ² + a ₃ z ³ + ⋯ on the unit disk with pre-Schwarzian norm ≤λ for a given λ > 0. The first author was partially supported by the JSPS Grant-in-Aid for Scientific Research (B), 17340039. Authors’ addresses: T. Sugawa and T. Terada, Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi-Hiroshima 739-8526, Japan Current address: T. Sugawa, Division of Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan     

Line search termination criteria for collinear scaling algorithms for minimizing a class of convex functions

Summary. Many successful quasi-Newton methods for optimization are based on positive definite local quadratic approximations to the objective function that interpolate the values of the gradient at the current and new iterates. Line search termination criteria used in such quasi-Newton methods usually possess two important properties. First, they guarantee the existence of such a local quadratic approximation. Second, under suitable conditions, they allow one to prove that the limit of the component of the gradient in the normalized search direction is zero. This is usually an intermediate result in proving convergence. Collinear scaling algorithms proposed initially by Davidon in 1980 are natural extensions of quasi-Newton methods in the sense that they are based on normal conic local approximations that extend positive definite local quadratic approximations, and that they interpolate values of both the gradient and the function at the current and new iterates. Line search termination criteria that guarantee the existence of such a normal conic local approximation, which also allow one to prove that the component of the gradient in the normalized search direction tends to zero, are not known. In this paper, we propose such line search termination criteria for an important special case where the function being minimized belongs to a certain class of convex functions. Received February 1, 1997 / Revised version received September 8, 1997     

\mathcal{A}(p) generators for H^*V and Singer's homological transfer

We list explicitly a minimal set of generators for the cohomology of an elementary abelian p-group, V, of rank 1 or 2, as a module over the mod p Steenrod algebra, for an odd prime p. Following Singer, we then construct a transfer map to the vector space spanned by such generators, where V now has arbitrary rank, from the homology of the Steenrod algebra. We show that this map takes images in the subspace of GL(V)-invariants and that it is an isomorphism for V having rank 1 or 2. Received June 11, 1996; in final form June 9, 1997     

A non–interior predictor–corrector path following algorithm for the monotone linear complementarity problem

We present a predictor–corrector non–interior path following algorithm for the monotone linear complementarity problem based on Chen–Harker–Kanzow–Smale smoothing techniques. Although the method is modeled on the interior point predictor–corrector strategies, it is the first instance of a non–interior point predictor–corrector algorithm. The algorithm is shown to be both globally linearly convergent and locally quadratically convergent under standard hypotheses. The approach to global linear convergence follows the authors’ previous work on this problem for the case of (P ₀+R ₀) LCPs. However, in this paper we use monotonicity to refine our notion of neighborhood of the central path. The refined neighborhood allows us to establish the uniform boundedness of certain slices of the neighborhood of the central path under the standard hypothesis that a strictly positive feasible point exists. Received September 1997 / Revised version received May 1999?Published online December 15, 1999     

On tangent cones of Alexandrov spaces with curvature bounded below

The tangent cones of an inner metric Alexandrov space with finite Hausdorff dimension and a lower curvature bound are always inner metric spaces with nonnegative curvature. In this paper we construct an infinite-dimensional inner metric Alexandrov space of nonnegative curvature which has in one point a tangent cone whose metric is not an inner metric. Received: 20 October 1999 / Revised version: 8 May 2000     

Solving a class of linear projection equations

Summary. Many interesting and important constrained optimization problems in mathematical programming can be translated into an equivalent linear projection equation Here, is the orthogonal projection on some convex set (e.g. ) and is a positive semidefinite matrix. This paper presents some new methods for solving a class of linear projection equations. The search directions of these methods are straighforward extensions of the directions in traditional methods for unconstrained optimization. Based on the idea of a projection and contraction method [7] we get a simple step length rule and are able to obtain global linear convergence. Received April 19, 1993 / Revised version received February 9, 1994     

Remarks on Lempert functions of balanced domains

This note should clarify how the behavior of certain invariant objects reflects the geometric convexity of balanced domains. Authors’ addresses: Nikolai Nikolov, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev 8, 1113 Sofia, Bulgaria; Peter Pflug, Carl von Ossietzky Universit?t Oldenburg, Institut für Mathematik, Fakult?t V, Postfach 2503, D-26111 Oldenburg, Germany     

Harmonic Functions on Homogeneous Spaces

 Given a locally compact group G acting on a locally compact space X and a probability measure σ on G, a real Borel function f on X is called σ-harmonic if it satisfies the convolution equation . We give conditions for the absence of nonconstant bounded harmonic functions. We show that, if G is a union of σ-admissible neighbourhoods of the identity, relative to X, then every bounded σ-harmonic function on X is constant. Consequently, for spread out σ, the bounded σ-harmonic functions are constant on each connected component of a [SIN]-group and, if G acts strictly transitively on a splittable metric space X, then the bounded σ-harmonic functions on X are constant which extends Furstenberg’s result for connected semisimple Lie groups. (Received 13 June 1998; in revised form 31 March 1999)     

Weighted Polynomials on Discrete Sets

 For a real interval I of positive length, we prove a necessary and sufficient condition which ensures that the continuous L _p (0 < p ⩽ ∞) norm of a weighted polynomial, P _n w ⁿ , deg P _n  ⩽ n, n ⩾ 1 is in an nth root sense, controlled by its corresponding discrete H?lder norm on a very general class of discrete subsets of I. As a by product of our main result, we establish inequalities and theorems dealing with zero distribution, zero location and sup and L _p infinite–finite range inequalities. Received April 4, 2001; in final form June 21, 2002