Knot points

G. Vanden Berghe, M. Van Daele and H. Vande Vyver, Exponentially-fitted algorithms fixed or frequency dependent knot points , Appl. Num. Anal. Comp. Math., 2004, 1(1), 49-65. 

Assume a point disclination located in a nematic droplet of radius R. The point disclination can be classified according to their Poincare characteristic angle a as a knot point (a = 0), focus point (0 < a < 7t/2), center (a = 7t/2), saddle-focus point (tt/2 < a < tt) or saddle point (a = 7t/2). For a knot point, one has a spherically symmetrical radial configuration and then... 

Here the 7 , (r) are B-splines [41], functions that are polynomials (typically of fifth or sixth order in our applications) in certain regions, but which vanish for most values of r, which provide great flexibility in representing arbitrary functions. They are defined between knot points, which can be... 

Similarly, we can consider two dipoles, interacting with the ion and with each other through independent simple chains this would be the case for famihes with two knotted points. An explicit calculation for the Cf family has been made its contribution, although much smaller than that of can be non-negligible. 

The traditional approach to yield curve fitting involves the calculation of a set of discount factors from market interest rates. From this, a spot yield curve can be estimated. The market data can be money market interest rates, futures and swap rates and bond yields. In general, though this approach tends to produce ragged spot rates and a forward rate curve with pronounced jagged knot points, due to the scarcity of data along the maturity structure. A refinement of this technique is to use polynomial approximation to the yield curve. 

Anderson and Sleath presented a model in the Bank of England Quarterly Bulletin in November 1999. The main objective of this work was to evaluate the relative efficacy of parametric versus spline-based methods. In fact, different applications call for different methods the main advantage of spline methods is that individual functions in between knot points may move in fairly independent fashion, which makes the resulting curve more flexible than that possible using parametric techniques. In Section 5.5.1 we reproduce their results with permission, which shows that a shock introduced at one end of the curve produces xmsatisfactory results in the parametric curve. 

This was first described by McCulloch (1975) and is referred to in Deacon and Derry (1994). We assume the maturity term structure is partitioned into q knot points with qwhere qi = 0 and q is the maturity of the longest dated bond. The remaining knot points are spaced such that there is, as far as possible, an equal number of bonds between each pair of knot points. With j < q, we employ the following functions ... 

The first set of n - 1 constraints require that the spline function join perfectly at the knot points. The second and third set of 2m - 2 constraints require that first and second derivative constraints match adjacent splines. Finally, the last two constraints are end-point constraints that set the derivative equal to zero at both ends. 

The x-axis in the regression is divided into segments at the knot points, at each of which the slopes of adjoining curves on either side of the point must match, as must the curvatures. FIGURE 5.4 shows a cubic spline with knot points at 0, 2, 5, 10, and 25 years, at each of which the curve is a cubic polynomial. This function permits a high and low to be accommodated in each space bounded by the knot points. The values of the curve can be adjoined at the knot point in a smooth function. 

Cubic spline interpolation assumes that there is a cubic polynomial that can estimate the yield curve at each maturity gap. A spline can be thought of as a number of separate polynomials of the form y = f(X), where X is the complete range of the maturity term divided into user-specified segments that are joined smoothly at the knot points. Given a set... 

These equations are solved, which is possible because they are made to fit the observed data. They are twice differentiable at the knot points, and the two derivatives at these points are equal. 

B-splines. The B-spline for a specified number of knot points X,.X ... 

As approximations. Nelson and Siegel curves are appropriate for noarbitrage applications. They are popular in the market because they are straightforward to calculate. Jordan and Mansi (2000) imputes two further advant es to them they force the long-date forward curve into a horizontal asymptote, and the user is not required to specify knot points, whose choice determines how effective the cubic spline curves are. The... 

In practice, the spline is expressed as a set of basis functions, with the general spline being a combination of these. This may be arrived at using B-splines. The B-spline for a specified number of knot points Xq,.,7Q is (5.8). 

In mathematics a spline is a piecewise polynomial function, made up of individual polynomial sections or segments that are joined together at (user-selected) points known as knot points. Splines used in term structure modeling are generally made up of cubic polynomials. The reason they are often cubic polynomials, as opposed to polynomials of order, say, two or five, is explained in straightforward fashion by de la Grandville (2001). A cubic spline is a function of order three and a piecewise cubic polynomial that is twice differentiable at each knot point. At each knot point the slope and curvature of the curve on either side must match. The cubic spline approach is employed to fit a smooth curve to bond prices (yields) given by the term discount factors. 

A B-spline which connects individual points in a functional way such that the geometry of the curve changes automatically when the knot points are moved, will contain references to these knot points rather than the points in form of attribute types. 

To find a joint trajectory that approximates the desired path closely, the Cartesian path points are transformed into N sets of joint displacements, with one set for each joint. Application of Bezier polynomial will provide trajectories that are smooth and have small overshoot of angular displacement between two adjacent knot points. The continuity conditions for joint displacement, velocity, and acceleration must be satisfied on the entire trajectory for the Cartesian robot path. 

The optimization algorithm, with a time interval of 5 seconds yields a minimum total traveling time of 6.38 seconds. This is the total travel time along the eight segments of the Bezier curve defined by the knot points. The optimized travel time for each segment is shown in Table 3. 

A simple and comprehensive menu-driven computer-based method for trajectory planning and force analysis in a planar robot is developed in the first paper. The robot designer is able to vary parameters and study their effect on the robot performance. In the second paper, a simple method to analyze the effect of torque and force on the first three links of a PUMA robot has been determined. Minimum time trajectory and bang-bang control with discontinuity points and knot points smoothed by parabolic blend are used. The workspace of a robotic arm using the Articulated Total Body model is calculated in the third paper. Computation of the workspace of the end effector is important in determining the effectiveness of a robot. 

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