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Consider a plane net bounded in the rectangular cartesian reference frame (x sub alpha) by the lines (x sub alpha) = constant (alpha = 1,2). The net is made of two intersecting sets of parallel straight cords symmetrically placed ...
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Consider a plane net bounded in the rectangular cartesian reference frame (x sub alpha) by the lines (x sub alpha) = constant (alpha = 1,2). The net is made of two intersecting sets of parallel straight cords symmetrically placed with respect to (x sub alpha) - axes. The cords introduce two directional constraints, when the net is deformed into another stretched shape, all the cords being in tension. Rivlin, and Green and Adkins have developed a theory of deformation of such nets, assuming the cords to be perfectly flexible but inextensible. In this paper, the theory developed by Green and Adkins is applied to determine the shape of a plane net when one of the boundaries (x sub alpha) = constant is deformed into (1) a circle (2) a catenary, the net remaining plane after deformation.
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Let P(sub theta)(z) := exp(i2(pi)(theta))z + z(sup 2) for theta (epsilon) (leftbracket)0,1(right bracket)and let J(sub P(sub theta)) denote the Julia set of P(sub theta). We prove that J(sub P(sub theta)) is locally connected and ...
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Let P(sub theta)(z) := exp(i2(pi)(theta))z + z(sup 2) for theta (epsilon) (leftbracket)0,1(right bracket)and let J(sub P(sub theta)) denote the Julia set of P(sub theta). We prove that J(sub P(sub theta)) is locally connected and has Lebesque zero whenever theta is of constant type. This provides the first proof of local connectivity of the Julia set for a rational map with a Siegel disc. We prove also that the Julia sets of a certain 'model' family of degree 3 Blaschke products are locally connected.
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Charts are displayed to show comparison of RMS pressure on airfoil surface, comparison of acoustic intensity on circle R = 1C, circle R = 4C, and circle R = 2C.
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The tooth surface principal radii of curvature of crown (flat) gears were determined. Specific results are presented for involute, straight, and hyperbolic cutter profiles. It is shown that the geometry of circular cut spiral beve...
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The tooth surface principal radii of curvature of crown (flat) gears were determined. Specific results are presented for involute, straight, and hyperbolic cutter profiles. It is shown that the geometry of circular cut spiral bevel gears is somewhat simpler than a theoretical logarithmic spiral bevel gear.
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Hoops are defined as geometric circles of radius = 1 in 3-dimensional Euclidean space R(exp 3). It is shown that it is possible to continuously fill an open set of R(exp 3) with disjoint hoops. It is also shown that in any such op...
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Hoops are defined as geometric circles of radius = 1 in 3-dimensional Euclidean space R(exp 3). It is shown that it is possible to continuously fill an open set of R(exp 3) with disjoint hoops. It is also shown that in any such open set, every pair of hoops must link each other, and consequently there is an upper bound to the volume that such an open set may have.
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This note is to call attention to the paper on a theorem about zig-zags between two circles. It explains the construction of a linkage composed of four equal bars P sub 1 P sub 2 = P sub 2 P sub 3 = P sub 3 P sub 4 = P 4 P 1 = d, ...
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This note is to call attention to the paper on a theorem about zig-zags between two circles. It explains the construction of a linkage composed of four equal bars P sub 1 P sub 2 = P sub 2 P sub 3 = P sub 3 P sub 4 = P 4 P 1 = d, joined at the points P sub k, with the following property: With P sub 1 and P sub 3 moving in the circular groove T, and P sub 3, P sub 4 moving in the circular groove T', it is shown that the lozange P sub 1 P sub 2 P sub 3 P sub 4 can be turned around.
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This article will follow up one small thread of Ramanujan's work which has found a modern computational context, namely, one of his approaches to approximating pi. Our experience has been that as we have come to understand these p...
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This article will follow up one small thread of Ramanujan's work which has found a modern computational context, namely, one of his approaches to approximating pi. Our experience has been that as we have come to understand these pieces of Ramanujan's work, as they have become mathematically demystified, and as we have come to realize the intrinsic complexity of these results, we have come to realize how truly singular his abilities were. This article attempts to present a considerable amount of material and, of necessity, little is presented in detail. We have, however, given much more detail than Ramanujan provided. Our intention is that the circle of ideas will become apparent and that the finer points may be pursued through the indicated references. There is a close and beautiful connection between the transformation theory for elliptic integrals and the very rapid approximation of pi. This conneclion was first made explicit by Ramanujan in his 1914 paper 'Modular Equations and Approximations to pi.' We might emphasize that Algorithms 1 and 2 are not to be found in Ramanujan's work, indeed no recursive approximation of pi is considered, but as we shall see they are intimately related to his analysis.
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Rational Bezier and B-spline representations of circles have been heavilypublicized. However, all the literature assumes the rational Bezier segments in the homogeneous space are both planar and (equivalent to) quadratic. This cre...
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Rational Bezier and B-spline representations of circles have been heavilypublicized. However, all the literature assumes the rational Bezier segments in the homogeneous space are both planar and (equivalent to) quadratic. This creates the illusion that circles can only be achieved by planar and quadratic curves. Circles that are formed by higher order rational Bezier curves which are nonplanar in the homogeneous space are shown. The problem of whether it is possible to represent a complete circle with one Bezier curve is investigated. In addition, some other interesting properties of cubic Bezier arcs are discussed.
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