The variable radius covering problem

In this paper we propose a covering problem where the covering radius of a facility is controlled by the decision-maker; the cost of achieving a certain covering distance is assumed to be a monotonically increasing function of the distance (i.e., it costs more to establish a facility with a greater covering radius). The problem is to cover all demand points at a minimum cost by finding optimal number, locations and coverage radii for the facilities. Both, the planar and discrete versions of the model are considered. Heuristic approaches are suggested for solving large problems in the plane. These methods were tested on a set of planar problems. Mathematical programming formulations are proposed for the discrete problem, and a solution approach is suggested and tested.     

Relaxed Multifunctions and Young Multimeasures

Young-measure type limits of multifunctions are examined. The relaxed limit via selections is a multimeasure, while the relaxation as a set-valued point function yields a Young measure on sets. The paper verifies that the two processes are equivalent in the sense that the limit multimeasure consists of all the distributions selectionable with respect to the Young measure on sets. The main tool is the characterization of selectionable distributions. An illustrative application of the equivalence result is provided.     

Finding the optimal solution to the Huff based competitive location model

In this paper we solve the gravity (Huff) model for the competitive facility location problem. We show that the generalized Weiszfeld procedure converges to a local maximum or a saddle point. We also devise a global optimization procedure that finds the optimal solution within a given accuracy. This procedure is very efficient and finds the optimal solution for 10,000 demand points in less than six minutes of computer time. We also experimented with the generalized Weiszfeld algorithm on the same set of randomly generated problems. We repeated the algorithm from 1,000 different starting solutions and the optimum was obtained at least 17 times for all problems.     

Introducing the Emerging Discipline of Statistics Education

Increasing attention has been given over the last decade by the statistics, mathematics and science education communities to the development of statistical literacy and numeracy skills of all citizens and the enhancement of statistics education at all levels. This paper introduces the emerging discipline of statistics education and considers its role in the development of these important skills. The paper begins with information on the growing importance of statistics in today's society, schools and colleges, summarizes unique challenges students face as they learn statistics, and makes a case for the importance of collaboration between mathematicians and statisticians in preparing teachers to teach students how to understand and reason about data. We discuss the differences and interrelations between statistics and mathematics, recognizing that mathematics is the discipline that has traditionally included instruction in statistics. We conclude with an argument that statistics should be viewed as a bridge between mathematics and science and should be taught in both disciplines.     

The combined effect of nonlinear filtration and window size on the accuracy of tissue displacement estimation using detected echo signals

In cardiac elastography, the regional strain and strain rate imaging is based on displacement estimation of tissue sections within the heart muscle carried out with various block-matching techniques (cross-correlation, sum of absolute differences, sum of squared differences, etc.). The accuracy of these techniques depends on a combination of ultrasonic imaging parameters such as ultrasonic frequency of interrogation, signal-to-noise ratio, size of a kernel used in a block-matching algorithm, type of data and speckle decorrelation. In this paper, we discuss the possibility to enhance the accuracy of the displacement estimation via nonlinear filtering of B-mode images before block-matching operation. The combined effect of a filter algorithm and a kernel size on the accuracy of the displacement estimation is analyzed using a 36-frame sequence of grayscale B-mode images of a human heart acquired by an ultrasound system operating at 1.77 MHz. It is shown that the nonlinear filtering of images enables to obtain the desired accuracy (less than one pixel) of the displacement estimation with smaller kernels than without filtering. These results are obtained for two filters--an adaptive anisotropic diffusion filter and a nonlinear Gaussian filter chain.     

A new method of spatial compounding imaging

A new method of spatial compound imaging is presented that improves image quality without the usual requirement to decrease the frame rate. The new method of imaging utilizes three transducers for data acquisition. The transducer located at the center of the transducer system is a phased array probe that acts as both transmitter and receiver. The other transducers are unfocused pistons that act only as receivers. Envelope data acquired by each transducer are combined to form a final image with improved quality (speckle contrast, target detectability and lateral resolution). It is shown that the improvement in speckle contrast depends on the correlation between individual images acquired by the transducers. The effective aperture approach is used for analytic estimation of the correlation between images in order to optimize the lateral separation between transducers. Using simulations, several compounding strategies have been performed to find the strategy that maximizes image quality. The central frequency of 2.5 MHz is used in simulations. Quantitative analysis of simulated B-mode images shows that the new method of imaging efficiently improves visibility, detectability, and lateral resolution of low contrast regions. The image frame rate is preserved because multiple scans are not required for the spatial compounding.     

A new method of ultrasound color flow mapping

Conventional ultrasound color flow mapping systems estimate and visualize only the axial velocity component. To obtain the transverse velocity component a modification of a multiple-beam method is proposed. The new two-dimensional color flow mapping system has a small size and consists of three transducers. The central transducer is an appodized and focused phased array. The other transducers are unfocused probes. Three transducers act as receivers and the central transducer operates as a transmitter. All receivers acquire rf scan lines that are then processed to estimate three axial velocity components using an autocorrelation method. These estimates are then combined to estimate the transverse velocity component, taking into account the geometric relationships among three transducers. Two algorithms for transverse velocity estimation are proposed. The first uses the Doppler angle estimate for calculation of the transverse velocity component. The other algorithm calculates the transverse velocity component directly from the axial components. The accuracy of the flow velocity estimators is estimated by simulations. Analysis of accuracy allows choosing the more effective algorithm for two-dimensional velocity estimation, which is insensitive to variations of the Doppler angle.     

Relative Chebyshev centers in normed linear spaces, part II

Let E be a normed linear space, A a bounded set in E, and G an arbitrary set in E. The relative Chebyshev center of A in G is the set of points in G best approximating A. We have obtained elsewhere general results characterizing the spaces in which the center reduces to a singleton in terms of structural properties related to uniform and strict convexity. In this paper, an analysis of the Chebyshev norm case, which falls outside the scope of the previous analysis, is presented.     

Groups withS 2′-subgroups andS 3′-subgroups

In this paper we formulate criteria for the solvability of groups of typeE _2′ andE _3′