Einstein metrics on S^{2}-bundles

New Einstein metrics are constructed on the associated , , and -bundles of principal circle bundles with base a product of K?hler-Einstein manifolds with positive first Chern class and with Euler class a rational linear combination of the first Chern classes. These Einstein metrics represent different generalizations of the well-known Einstein metrics found by Bérard Bergery, D. Page, C. Pope, N. Koiso, and Y. Sakane. Corresponding new Einstein-Weyl structures are also constructed. Received 25 October 1996 / Revised version 1 April 1997     

On the sum of distances along a circle

The arc distance between two points on a circle is their geodesic distance along the circle. We study the sum of the arc distances determined by n points on a circle, which is a useful measure of the evenness of scales and rhythms in music theory. We characterize the configurations with the maximum sum of arc distances by a balanced condition: for each line that goes through the circle center and touches no point, the numbers of points on each side of the line differ by at most one. When the points are restricted to lattice positions on a circle, we show that Toussaint's snap heuristic finds an optimal configuration. We derive closed-form formulas for the maximum sum of arc distances when the points are either allowed to move continuously on the circle or restricted to lattice positions. We also present a linear-time algorithm for computing the sum of arc distances when the points are presorted by the polar coordinates.     

Incomplete Orthogonal Distance Regression

A common method of fitting curves and surfaces to data is to minimize the sum of squares of the orthogonal distances from the data points to the curve or surface, a process known as orthogonal distance regression. Here we consider fitting geometrical objects to data when some orthogonal distances are not available. Methods based on the Gauss–Newton method are developed, analyzed and illustrated by examples. AMS subject classification (2000) 65D10, 65K05.     

Orthogonal regression: a teaching perspective

A well-known approach to linear least squares regression is that which involves minimizing the sum of squared orthogonal projections of data points onto the best fit line. This form of regression is known as orthogonal regression, and the linear model that it yields is known as the major axis. A similar method, reduced major axis regression, is predicated on minimizing the total sum of triangular areas formed between data points and the best fit line. Either of these methods is appropriately applied when both x and y are measured, a typical case in the natural sciences. In comparison to classical linear regression, equation derivation for the slope of the major axis and reduced major axis lines is a nontrivial process. For this reason, derivations are presented herein drawing from previous literature with as few steps as possible to enable an easily accessible understanding. Application to eruption data for Old Faithful geyser, Yellowstone National Park, Wyoming and Montana, USA enables a teaching opportunity for choice of model.     

There Are Integral Heptagons, no Three Points on a Line, no Four on a Circle

We give two configurations of seven points in the plane, no three points in a line, no four points on a circle with pairwise integral distances. This answers a famous question of Paul Erdős.     

Stochastic analysis of ordered median problems

Many location problems can be expressed as ordered median objective. In this paper, we investigate the ordered median objective when the demand points are generated in a circle. We find the mean and variance of the kth distance from the centre of the circle and the correlation matrix between all pairs of ordered distances. By applying these values, we calculate the mean and variance of any ordered median objective and the correlation coefficient between two ordered median objectives. The usefulness of the results is demonstrated by calculating various probabilities such as: What is the probability that the mean distance is greater than the truncated mean distance? What is the probability that the maximum distance is greater than 0.9? What is the probability that the range of distances is greater than 0.8? An analysis of an illustrative example also demonstrates the usefulness of the analysis.     

Inverse unitary eigenproblems and related orthogonal functions

Summary. This paper explores the relationship between certain inverse unitary eigenvalue problems and orthogonal functions. In particular, the inverse eigenvalue problems for unitary Hessenberg matrices and for Schur parameter pencils are considered. The Szeg? recursion is known to be identical to the Arnoldi process and can be seen as an algorithm for solving an inverse unitary Hessenberg eigenvalue problem. Reformulation of this inverse unitary Hessenberg eigenvalue problem yields an inverse eigenvalue problem for Schur parameter pencils. It is shown that solving this inverse eigenvalue problem is equivalent to computing Laurent polynomials orthogonal on the unit circle. Efficient and reliable algorithms for solving the inverse unitary eigenvalue problems are given which require only O() arithmetic operations as compared with O() operations needed for algorithms that ignore the structure of the problem. Received April 3, 1995 / Revised version received August 29, 1996     

Implicit Surface Fitting Using Directional Constraints

A commonly used technique for fitting curves and surfaces to measured data is that known as orthogonal distance regression, where the sum of squares of orthogonal distances from the data points to the surface is minimized. An alternative has recently been proposed for curves and surfaces which are parametrically defined, which minimizes the sum of squares in given directions which depend on the measuring process. In addition to taking account of that process, it is claimed that this technique has the advantage of complying with traditional fixed-regressor assumptions, enabling standard inference theory to apply. Here we consider extending this idea to curves and surfaces where the only assumption made is that there is an implicit formulation. Numerical results are given to illustrate the algorithmic performance.This revised version was published online in October 2005 with corrections to the Cover Date.     

Book Reviews

This paper modifies the usual meaning of regression from ‘minimizing the sum of squared distances’ to ‘minimizing the sum of square perpendicular distances’. With this modified definition, the best‐fit plane, circle, and sphere may be meaningfully considered.

These regression problems are motivated by the current development of software to control robotized coordinate measuring machines (CMMs) used to perform quality assurance work. A brief outline of CMMs is given and the application of the modified regression definition to a plane, circle and sphere is illustrated. In §3 the equation of the best fitting plane is calculated for various sets of data points.     

Solution of a multiple Nevanlinna-Pick problem via orthogonal rational functions

We consider a Nevanlinna-Pick type interpolation problem for Carathéodory functions, where the values of the function and its derivatives up to certain orders are given at finitely many points of the unit disk. The set of all solutions of this problem is described by means of the orthogonal rational functions which play here a similar role as the orthogonal polynomials on the unit circle in the classical case of the trigonometric moment problem. In particular, we use a connection between Szegö and Schur parameters which in the classical situation was discovered by Ja.L. Geronimus.     

Eigenschaften von vollstaendigen Systemen orthogonaler lateinischer Quadrate,die bestimmte affine Ebenen repraesentieren

A set of n-1 mutually orthogonal Latin squares of order n is a model of an affine plane with exactly n points on a line and every affine plane with n points on a line can be represented by n-1 mutually orthogonal Latin squares ([1]). In this paper we investigate properties of finite planes through the complete set of mutually orthogonal Latin squares representing the plane and mainly — vice versa — properties of the squares representing a fixed plane. The results are based on the geometrical configurations which hold in the planes. For presumed definitions and theorems which are not specially referred to see [4], [7], [3] or [6].     

On the Gauss-Newton method for l1 orthogonal distance regression

The problem of fitting a curve or surface to data has many applications.There are also many fitting criteria which can be used, andone which is widely used in metrology, for example, is thatof minimizing the least squares norm of the orthogonal distancesfrom the data points to the curve or surface. The Gauss–Newtonmethod, in correct separated form, is a popular method for solvingthis problem. There is also interest in alternatives to leastsquares, and here we focus on the use of the l₁ norm, whichis traditionally regarded as important when the data containwild points. The effectiveness of the Gauss–Newton methodin this case is studied, with particular attention given tothe influence of zero distances. Different aspects of the computationare illustrated by consideration of two particular fitting problems.     

Fitting concentric circles to measurements

Measurements for fitting a given number of concentric circles are recorded. For each concentric circle several measurements are taken. The problem is to fit the given number of circles to the data such that all circles have a common center. This is a generalization of the problem of fitting a set of points to one circle. Three objectives, to be minimized, are considered: the least squares of distances from the circles, the maximum distance from the circles, and the sum of the distances from the circles. Very efficient optimal solution procedures are constructed. Problems based on a total of 10,000 measurements are solved in about 10 s with the least squares objective, $<$ 2 s with the maximum distance objective, and a little more than 1 min for the minisum objective.     

Zeros of a family of hypergeometric para‐orthogonal polynomials on the unit circle

Para‐orthogonal polynomials derived from orthogonal polynomials on the unit circle are known to have all their zeros on the unit circle. In this note we study the zeros of a family of hypergeometric para‐orthogonal polynomials. As tools to study these polynomials, we obtain new results which can be considered as extensions of certain classical results associated with three term recurrence relations and differential equations satisfied by orthogonal polynomials on the real line. One of these results which might be considered as an extension of the classical Sturm comparison theorem, enables us to obtain monotonicity with respect to the parameters for the zeros of these para‐orthogonal polynomials. Finally, a monotonicity of the zeros of Meixner‐Pollaczek polynomials is proved.     

Locating a minisum annulus: a new partial coverage distance model

The problem is to find the best location in the plane of a minisum annulus with fixed width using a partial coverage distance model. Using the concept of partial coverage distance, those demand points within the area of the annulus are served at no cost, while for ‘uncovered’ demand points there will be additional costs proportional to their distances to the annulus. The objective of the problem is to locate the annulus such that the sum of distances from the uncovered demand points to the annulus (covering area) is minimized. The distance is measured by the Euclidean norm. We discuss the case where the radius of the inner circle of the annulus is variable, and prove that at least two demand points must be on the boundary of any optimal annulus. An algorithm to solve the problem is derived based on this result.     

Comments on an ancient Greek racecourse: finding minimum width annuluses

For a set of measured points, we describe a linear-programming model that enables us to find concentric circumscribed and inscribed circles whose annulus encompasses all the points and whose width tends to be minimum in a Chebychev minmax sense. We illustrate the process using the data of Rorres and Romano (SIAM Rev. 39: 745–754, 1997) that is taken from an ancient Greek stadium in Corinth. The stadium’s racecourse had an unusual circular arc starting line, and measurements along this arc form the basic data sets of Rorres and Romano (SIAM Rev. 39: 745–754, 1997). Here we are interested in finding the center and radius of the circle that defined the starting line arc. We contrast our results with those found in Rorres and Romano (SIAM Rev. 39: 745–754, 1997).     

Total Least-Squares regularization of Tykhonov type and an ancient racetrack in Corinth

In this contribution a variation of Golub/Hansen/O’Leary’s Total Least-Squares (TLS) regularization technique is introduced, based on the Hybrid APproximation Solution (HAPS) within a nonlinear Gauss-Helmert Model. By applying a traditional Lagrange approach to a series of iteratively linearized Gauss-Helmert Models, a new iterative scheme has been found that, in practice, can generate the Tykhonov regularized TLS solution, provided that some care is taken to do the updates properly.The algorithm actually parallels the standard TLS approach as recommended in some of the geodetic literature, but unfortunately all too often in combination with erroneous updates that would still show convergence, although not necessarily to the (unregularized) TLS solution. Here, a key feature is that both standard and regularized TLS solutions result from the same computational framework, unlike the existing algorithms for Tykhonov-type TLS regularization.The new algorithm is then applied to a problem from archeology. There, both the radius and the center-point coordinates of a circle have to be determined, of which only a small part of the arc had been surveyed in-situ, thereby giving rise to an ill-conditioned set of equations. According to the archaeologists involved, this circular arc served as the starting line of a racetrack in the ancient Greek stadium of Corinth, ca. 500 BC. The present study compares previous estimates of the circle parameters with the newly developed “Regularized TLS Solution of Tykhonov type.”     

Shift operators contained in contractions, pseudocontinuable Schur functions and orthogonal systems on the unit circle

The main aim of this paper is to establish the connection between well-known criteria for the pseudocontinuability of a non-inner Schur function ?? in the unit disk (see Theorems 3.9, 4.2). In a canonical way we associate a probability measure ?? on the unit circle with ??. One of the two criteria will be reformulated in the face of ??, whereas the other one is drafted in view of a completely nonunitary contraction T having ?? as corresponding characteristic function. Our main result clarifies an immediate connection between the above-mentioned two criteria. In this way, we rewrite these criteria in terms of orthogonal systems with respect to the measure ?? (see Theorem 7.2).     

Non-skew symmetric orthogonal matrices with constant diagonals

A matrix C of order n is orthogonal if CC^T=dI. In this paper, we restrict the study to orthogonal matrices with a constant m > 1 on the diagonal and ±1's off the diagonal. It is observed that all skew symmetric orthogonal matrices of this type are constructed from skew symmetric Hadamard matrices and vice versa. Some simple necessary conditions for the existence of non-skew orthogonal matrices are derived. Two basic construction techniques for non-skew orthogonal matrices are given. Several families of non-skew orthogonal matrices are constructed by applying the basic techniques to well-known combinatorial objects like balanced incomplete block designs. It is also shown that if m is even and n=0 (mod 4), then an orthogonal matrix must be skew symmetric. The structure of a non-skew orthogonal matrix in the special case of m odd,n=2 (mod 4) and m?1/6n is also studied in detail. Finally, a list of cases with n?50 is given where the existence of non-skew orthogonal matrices are unknown.     

Perturbed Brownian motions

Summary. We study `perturbed Brownian motions', that can be, loosely speaking, described as follows: they behave exactly as linear Brownian motion except when they hit their past maximum or/and maximum where they get an extra `push'. We define with no restrictions on the perturbation parameters a process which has this property and show that its law is unique within a certain `natural class' of processes. In the case where both perturbations (at the maximum and at the minimum) are self-repelling, we show that in fact, more is true: Such a process can almost surely be constructed from Brownian paths by a one-to-one measurable transformation. This generalizes some results of Carmona-Petit-Yor and Davis. We also derive some fine properties of perturbed Brownian motions (Hausdorff dimension of points of monotonicity for example). Received: 17 May 1996 / In revised form: 21 January 1997