Thesis (MEducation)--University of South Australia, 2003.

This thesis verified that metric spaces can be constructed using ultrametrics d and D, where d(x,y) = 0 if x = y and d(x,y) = (1/2) k if x not equal to y, such that x-y = 2k(a/b) for a,b relatively prime to 2, and where D(A,B)= max(d(al,bl); d(a2,b2)) for A = (al,a2) and B = (bl,b2).Assuming that a line is represented by some linear equation, a one-dimensional point was defined as an element of Q and a two-dimensional point as an element of Q x Q. There was an investigation of one-dimensional points with respect to the behavior of segments, midpoints, and distances as measured by d. The function D demonstrated the behavior of midpoints, medians, and triangles, as well as the congruence relation. The study necessitated the introduction of pseudomidpoints and pseudomedians, and an unorthodox definition of angle measurement.

Thesis (M.S.)--University of Wisconsin--Madison, 1981. / Typescript. Title from title screen (viewed Mar. 27, 2007). Includes bibliographical references (leaves 66-68). Online version of the print original.

Thesis (M.S.)--Rochester Institute of Technology, 1989. / "References": leaves 66-68.

Thesis (M.S.)--University of Wisconsin--Madison, 1981. / Typescript. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 66-68).

Given a real N by N matrix A, write p(A) for the maximum angle by which A rotates any unit vector. Suppose that A and B are positive definite symmetric (PDS) N by N matrices. Then their Jordan product {A, B} := AB + BA is also symmetric, but not necessarily positive definite. If p(A) + p(B) is obtuse, then there exists a special orthogonal matrix S such that {A, SBS^(-1)} is indefinite. Of course, if A and B commute, then {A, B} is positive definite. Our work grows from the following question: if A and B are commuting positive definite symmetric matrices such that p(A) + p(B) is obtuse, what is the minimal p(S) such that {A, SBS^(-1)} indefinite? In this dissertation we will describe the level curves of the angle function mapping a unit vector x to the angle between x and Ax for a 3 by 3 PDS matrix A, and discuss their interaction with those of a second such matrix.

No description available.

Indiana University-Purdue University Indianapolis (IUPUI) / Instrument assisted soft tissue mobilization (IASTM) is a form of massage using rigid manufactured or cast devices. The delivered force, which is a critical parameter in massage during IASTM, has not been measured or standardized for most clinical practices. In addition to the force, the angle of treatment and frequency play an important role during IASTM. As a result, there is a strong need to characterize the delivered force to a patient, angle of treatment, and stroke frequency. This thesis proposes two novel mechatronic designs for a specific instrument from Graston Technique(Model GT3), which is a frequently used tool to clinically deliver localize pressure to the soft tissue. The first design is based on compression load cells, where 4-load cells are used to measure the force components in three-dimensional space. The second design uses a 3D load cell, which can measure all three force components force simultaneously. Both designs are implemented with IMUduino microcontroller chips which can also measure tool orientation angles and provide computed stroke frequency. Both designs, which were created using Creo CAD platform, were also analyzed thorough strength and integrity using the finite element analysis package ANSYS. Once the static analysis was completed, a dynamic model was created for the first design to simulate IASTM practice using the GT-3 tool. The deformation and stress on skin were measured after applying force with the GT-3 tool. Additionally, the relationship between skin stress and the load cell measurements has been investigated. The second design of the mechatronic IASTM tool was validated for force measurements using an electronic plate scale that provided the baseline force values to compare with the applied force values measured by the tool. The load cell measurements and the scale readings were found to be in agreement within the expected degree of accuracy. The stroke frequency was computed using the force data and determining the peaks during force application. The orientation angles were obtained from the built-in sensors in the microchip.