On Pólya Frequency Functions. III. The Positivity of Translation Determinants with an Application to the Interpolation Problem by Spline Curves

Abstract In a dynamic machining process, distortion in surface irregularity is a very complex phenomenon. Surface irregularities form a periodic representation of the tool profile with various kinds of disturbance in a broad range of changes in the height and length of the profile. To discern these irregularity disturbances, interactions of the tool in the form of changes perpendicular and parallel relative to the workpiece were analyzed and simulated. The individual kinds of displacement of the tool relative to the workpiece introduce distortions in the changes of height and length. These changes are weakly represented in standard height and length irregularity parameters and their discernment has been found through amplitude-frequency functions.

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Degree Elevation of Algebraic Hyperbolic B-spline Curves and Corner Cutting Based on Bi-order Spline

Chinese Journal of Computers ◽

10.3724/sp.j.1016.2008.01056 ◽

2009 ◽

Vol 31 (6) ◽

pp. 1056-1062 ◽

Cited By ~ 1

Author(s):

Bo ZHANG ◽

Guo-Zhao WANG

Keyword(s):

Degree Elevation ◽

B Spline ◽

Corner Cutting ◽

Spline Curves ◽

Order Spline

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Interpolation of Hermite data by clamped Minkowski Pythagorean hodograph B-spline curves

Journal of Computational and Applied Mathematics ◽

10.1016/j.cam.2021.113469 ◽

2021 ◽

pp. 113469

Author(s):

Michal Bizzarri ◽

Miroslav Lávička

Keyword(s):

B Spline ◽

Spline Curves ◽

Pythagorean Hodograph

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Lidstone–Euler Second-Type Boundary Value Problems: Theoretical and Computational Tools

Mediterranean Journal of Mathematics ◽

10.1007/s00009-021-01822-5 ◽

2021 ◽

Vol 18 (5) ◽

Author(s):

Francesco Aldo Costabile ◽

Maria Italia Gualtieri ◽

Anna Napoli

Keyword(s):

Boundary Value Problem ◽

Boundary Conditions ◽

Existence And Uniqueness ◽

Interpolation Problem ◽

Boundary Value ◽

Computational Tools ◽

Numerical Examples ◽

Odd Order ◽

Uniqueness Of The Solution ◽

Type Boundary

AbstractGeneral nonlinear high odd-order differential equations with Lidstone–Euler boundary conditions of second type are treated both theoretically and computationally. First, the associated interpolation problem is considered. Then, a theorem of existence and uniqueness of the solution to the Lidstone–Euler second-type boundary value problem is given. Finally, for a numerical solution, two different approaches are illustrated and some numerical examples are included to demonstrate the validity and applicability of the proposed algorithms.

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Chaotic Path Planning for 3D Area Coverage Using a Pseudo-Random Bit Generator from a 1D Chaotic Map

Mathematics ◽

10.3390/math9151821 ◽

2021 ◽

Vol 9 (15) ◽

pp. 1821

Author(s):

Lazaros Moysis ◽

Karthikeyan Rajagopal ◽

Aleksandra V. Tutueva ◽

Christos Volos ◽

Beteley Teka ◽

...

Keyword(s):

Path Planning ◽

Chaotic Map ◽

Dynamical Behavior ◽

Area Coverage ◽

Phase Portraits ◽

Planning Strategy ◽

Spline Curves ◽

Key Space ◽

Aerial Vehicle ◽

Discrete Motion

This work proposes a one-dimensional chaotic map with a simple structure and three parameters. The phase portraits, bifurcation diagrams, and Lyapunov exponent diagrams are first plotted to study the dynamical behavior of the map. It is seen that the map exhibits areas of constant chaos with respect to all parameters. This map is then applied to the problem of pseudo-random bit generation using a simple technique to generate four bits per iteration. It is shown that the algorithm passes all statistical NIST and ENT tests, as well as shows low correlation and an acceptable key space. The generated bitstream is applied to the problem of chaotic path planning, for an autonomous robot or generally an unmanned aerial vehicle (UAV) exploring a given 3D area. The aim is to ensure efficient area coverage, while also maintaining an unpredictable motion. Numerical simulations were performed to evaluate the performance of the path planning strategy, and it is shown that the coverage percentage converges exponentially to 100% as the number of iterations increases. The discrete motion is also adapted to a smooth one through the use of B-Spline curves.

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