scholarly journals Generating Approximate Solutions to the TTP using a Linear Distance Relaxation

2012 ◽  
Vol 45 ◽  
pp. 257-286 ◽  
Author(s):  
R. Hoshino ◽  
K. Kawarabayashi
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Optimal Solution ◽  
Approximate Solutions ◽  
Round Robin ◽  
Geographical Information ◽  
Traveling Tournament Problem ◽  
Linear Distance ◽  
Straight Line ◽  
Complete Set

In some domestic professional sports leagues, the home stadiums are located in cities connected by a common train line running in one direction. For these instances, we can incorporate this geographical information to determine optimal or nearly-optimal solutions to the n-team Traveling Tournament Problem (TTP), an NP-hard sports scheduling problem whose solution is a double round-robin tournament schedule that minimizes the sum total of distances traveled by all n teams. We introduce the Linear Distance Traveling Tournament Problem (LD-TTP), and solve it for n=4 and n=6, generating the complete set of possible solutions through elementary combinatorial techniques. For larger n, we propose a novel "expander construction" that generates an approximate solution to the LD-TTP. For n congruent to 4 modulo 6, we show that our expander construction produces a feasible double round-robin tournament schedule whose total distance is guaranteed to be no worse than 4/3 times the optimal solution, regardless of where the n teams are located. This 4/3-approximation for the LD-TTP is stronger than the currently best-known ratio of 5/3 + epsilon for the general TTP. We conclude the paper by applying this linear distance relaxation to general (non-linear) n-team TTP instances, where we develop fast approximate solutions by simply "assuming" the n teams lie on a straight line and solving the modified problem. We show that this technique surprisingly generates the distance-optimal tournament on all benchmark sets on 6 teams, as well as close-to-optimal schedules for larger n, even when the teams are located around a circle or positioned in three-dimensional space.

Vacuum field fluctuations and spontaneous emission in a dielectric slab

1992 ◽  
Vol 436 (1897) ◽  
pp. 373-389 ◽  
Keyword(s):  
Spontaneous Emission ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Field Operator ◽  
Vacuum Field ◽  
Emission Rates ◽  
Dielectric Slab ◽  
Spatial Modes ◽  
Field Fluctuations ◽  
Complete Set

The electromagnetic field is quantized on the basis of the complete set of spatial modes of a plane dielectric slab of arbitrary thickness and refractive index but infinite transverse dimensions, located in otherwise empty three-dimensional space. The vacuum field fluctuations and spontaneous emission rates are evaluated as functions of position both inside and outside the slab. The source-field operator is derived for emission by atoms inside the slab, in the direction perpendicular to its surfaces. Particular attention is given to the possibility of suppressing spontaneous emission by placing atoms in, or close to, a dielectric slab.

A polynomial approach to determine the time-dependent electric and magnetic fields in anisotropic materials by symbolic computations

2016 ◽  
Vol 35 (3) ◽  
Author(s):  
Valery George Yakhno ◽  
Meltem Altunkaynak
Keyword(s):  
Magnetic Fields ◽  
Design Methodology ◽  
Dimensional Space ◽  
General Structure ◽  
Three Dimensional ◽  
Approximate Solutions ◽  
Time Dependent ◽  
Bounded Region ◽  
Symbolic Computations ◽  
Electric And Magnetic Fields

Purpose The purpose of this paper is to calculate the time-dependent electric and magnetic fields in anisotropic media with a general structure of anisotropy by symbolic computations. Design/methodology/approach An analytical approach for the computation of the time-dependent electric and magnetic fields is suggested. This approach consists of the following. Input data, electric and magnetic fields are presented in polynomial form.The exact formulae for electric and magnetic fields are computed by symbolic transformations in Maple. Findings The time-dependent second order partial differential equations for the electric and magnetic fields with polynomial data were obtained from Maxwell's equations when the current density is presented in a polynomial form with respect to space variables in a bounded region of three dimensional space. The exact solutions of obtained equations were computed symbolically using Maple. Originality/value The obtained polynomial solutions do not contain errors if data are polynomials. We have shown that these solutions are approximate solutions with good accuracy for data which are approximated by polynomials in a bounded region of 3D space.

A Geometrical treatment of the Correspondence between Lines in Threefold Space and Points of a Quadric in Fivefold space

1925 ◽  
Vol 22 (5) ◽  
pp. 694-699 ◽  
Author(s):  
H. W. Turnbull
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Quadratic Relation ◽  
Four Dimensions ◽  
Homogeneous Coordinates ◽  
Plücker Coordinates ◽  
Straight Line ◽  
Set Up ◽  
Image Position ◽  
Three Dimensional Space

§ 1. The six Plücker coordinates of a straight line in three dimensional space satisfy an identical quadratic relationwhich immediately shows that a one-one correspondence may be set up between lines in three dimensional space, λ, and points on a quadric manifold of four dimensions in five dimensional space, S5. For these six numbers pij may be considered to be six homogeneous coordinates of such a point.

Optimal Contact Force Distribution for Multi-Limbed Robots

2005 ◽  
Vol 128 (3) ◽  
pp. 566-573 ◽  
Author(s):  
Dennis W. Hong ◽  
Raymond J. Cipra
Keyword(s):  
Contact Force ◽  
Dimensional Space ◽  
Contact Point ◽  
Three Dimensional ◽  
Optimal Solution ◽  
Solution Space ◽  
Contact Forces ◽  
Force Distribution ◽  
New Strategy ◽  
Contact Force Distribution

One of the inherent problems of multi-limbed mobile robotic systems is the problem of multi-contact force distribution; the contact forces and moments at the feet required to support it and those required by its tasks are indeterminate. A new strategy for choosing an optimal solution for the contact force distribution of multi-limbed robots with three feet in contact with the environment in three-dimensional space is presented. The incremental strategy of opening up the friction cones is aided by using the “force space graph” which indicates where the solution is positioned in the solution space to give insight into the quality of the chosen solution and to provide robustness against disturbances. The “margin against slip with contact point priority” approach is also presented which finds an optimal solution with different priorities given to each foot contact point. Examples are presented to illustrate certain aspects of the method and ideas for other optimization criteria are discussed.

scholarly journals Optimal Reliable Point-in-Polygon Test and Differential Coding Boolean Operations on Polygons

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 477
Author(s):  
Jianqiang Hao ◽  
Jianzhi Sun ◽  
Yi Chen ◽  
Qiang Cai ◽  
Li Tan
Keyword(s):  
Parallel Implementation ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Geometric Probability ◽  
Boolean Operations ◽  
Straight Line ◽  
Speed Up ◽  
The Stability ◽  
Serial Algorithm ◽  
Degenerate Cases

This paper provides a full theoretical and experimental analysis of a serial algorithm for the point-in-polygon test, which requires less running time than previous algorithms and can handle all degenerate cases. The serial algorithm can quickly determine whether a point is inside or outside a polygon and accurately determine the contours of input polygon. It describes all degenerate cases and simultaneously provides a corresponding solution to each degenerate case to ensure the stability and reliability. This also creates the prerequisites and basis for our novel boolean operations algorithm that inherits all the benefits of the serial algorithm. Using geometric probability and straight-line equation F ( P ) = ( y i − y i + 1 ) ( x p − x i ) − ( y i − y p ) ( x i + 1 − x i ) , it optimizes our two algorithms that avoid the division operation and do not need to compute any intersection points. Our algorithms are applicable to any polygon that may be self-intersecting or with holes nested to any level of depth. They do not have to sort the vertices clockwise or counterclockwise beforehand. Consequently, they process all edges one by one in any order for input polygons. This allows a parallel implementation of each algorithm to be made very easily. We also prove several theorems guaranteeing the correctness of algorithms. To speed up the operations, we assign each vector a number code and derive two iterative formulas using differential calculus. However, the experimental results as well as the theoretical proof show that our serial algorithm for the point-in-polygon test is optimal and the time complexities of all algorithms are linear. Our methods can be extended to three-dimensional space, in particular, they can be applied to 3D printing to improve its performance.

scholarly journals ANY FIVE POINTS LIE IN A 3-SPACE

2020 ◽  
Author(s):  
Zh. Nikoghosyan ◽  
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Entire Space ◽  
Unique Line ◽  
Straight Line ◽  
Set Of Points ◽  
Three Dimensional Space

In axiomatic formulations, every two points lie in a (straight) line, every three points lie in a plane and every four points lie in a three-dimensional space (3-space). In this paper we show that every five points lie in a 3-space as well, implying that every set of points lie in a 3-space. In other words, the 3-space occupies the entire space. The proof is based on the following four axioms: 1) every two distinct points define a unique line, 2) every three distinct points, not lying on the line, define a unique plane, 3) if 𝐴 and 𝐵 are two distinct points in a 3-space, then the line defined by the points 𝐴, 𝐵, entirely lies in this 3-space, 4) if 𝐹1, 𝐹2, 𝐹3 are three distinct points in a 3-space, not lying in a line, then the plane defined by the points 𝐹1, 𝐹2, 𝐹3, lies entirely in this 3-space.

Fast and accurate collision detection based on enclosed ellipsoid

Robotica ◽  
2001 ◽  
Vol 19 (4) ◽  
pp. 381-394 ◽  
Author(s):  
Ming-Yi Ju ◽  
Ming-Yi Ju ◽  
Jing-Sin Liu ◽  
Shen-Po Shiang ◽  
Yuh-Ren Chien ◽  
...  
Keyword(s):  
Collision Detection ◽  
Object Representation ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Accurate Method ◽  
Convex Polyhedra ◽  
Graphical Simulation ◽  
Bounding Volume ◽  
Straight Line ◽  
Multiple Robot

A fast and accurate method for detecting the collisions of convex polyhedra in a graphical simulation environment based on a newly developed method of distance estimate is presented. By the simultaneous use of the enclosing and the enclosed ellipsoids of convex polyhedra, potential collisions can be detected more accurate than those methods using only bounding volume for object representation, and more efficient than the polyhedral methods. An approach for computing the enclosed ellipsoid of a convex polyhedron by compressing, stretching and scaling operations on its best-fit enclosing ellipsoid is introduced. Graphical simulations of two case studies (moving polyhedral objects in three-dimensional space and multiple robot arms undergoing straight line motions) are conducted to demonstrate the accuracy of the proposed algorithm for collision detection.

New correlation formulae for the straight section of the electrospun jet from a polymer drop

2013 ◽  
Vol 735 ◽  
pp. 150-175 ◽  
Author(s):  
R. Sahay ◽  
C. J. Teo ◽  
Y. T. Chew
Keyword(s):  
Electric Fields ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Taylor Cone ◽  
Straight Section ◽  
New Correlation ◽  
Straight Line ◽  
Space Experiments ◽  
Jet Characteristics

AbstractAn electrospun polymer jet issued from a Taylor cone follows a straight-line motion before experiencing electrical bending instability resulting in curling and spiralling of the jet in three-dimensional space. Experiments are performed to characterize the fluid dynamics of an electrospun polymer jet. Appropriate image processing is performed to systematically analyse flow regimes of the electrospun jet. These regimes include Taylor cone formation/jet initiation and the straight section of the jet. Dimensional analysis was performed to identify the salient dimensionless parameters, which govern the electrospun jet characteristics. Three new correlation formulae were obtained to characterize the dimensionless jet diameter at the apex of the Taylor cone $(\tilde {d} = 1. 03{\tilde {Q} }^{0. 44} )$, the dimensionless jet diameter at different locations along the jet’s straight section $(\tilde {d} {\tilde {z} }^{1/ 4} = 1. 09{\tilde {Q} }^{1/ 2} )$, as well as the length of the straight section of the jet $({\tilde {Z} }_{in} = 86{\tilde {Q} }^{0. 42} )$. These correlation formulae are valid for the analysed range of dimensionless flow rates $(2. 6{{\times 10}}^{- 4} \lt \tilde {Q} \lt 3. 6{{\times 10}}^{7} )$ and dimensionless electric fields $(7. 4{{\times 10}}^{- 4} \lt \tilde {E} \lt 1. 4{{\times 10}}^{- 1} )$. In addition, the correlation formulae are valid for the analysed range of Deborah numbers De and Reynolds numbers Re based on nozzle radius, $3. 3\times {10}^{- 7} \lt {\mathit{De}}_{{r}_{o} } \lt 3. 8\times {10}^{- 2} $ and $5. 8\times {10}^{- 4} \lt {\mathit{Re}}_{{r}_{o} } \lt 7. 0\times {10}^{- 1} $. The proposed new correlation formulae are instrumental in the design as well as controlled manipulation/optimization of the electrospinning phenomenon.

On Pedal Quadrics in Non-Euclidean Hyperspace

1925 ◽  
Vol 22 (5) ◽  
pp. 751-758
Author(s):  
J. P. Gabbatt
Keyword(s):  
Orthogonal Projection ◽  
Euclidean Plane ◽  
Dimensional Space ◽  
Three Dimensional ◽  
The Other ◽  
Elementary Geometry ◽  
Orthogonal Projections ◽  
Point Pair ◽  
Straight Line ◽  
Three Dimensional Space

1. The following are well-known theorems of elementary geometry: Given any euclidean plane triangle, A0 A1 A2, and any pair of points, X, Y, isogonally conjugate q. A0 A1 A2; then the orthogonal projections of X, Y on the sides of A0 A1 A2 lie on a circle, the pedal circle of the point-pair. If either of the points X,- Y describe a (straight) line, m, then the other describes a conic circumscribing A0 A1 A2, and the pedal circle remains orthogonal to a fixed circle, J; thus the pedal circles in question are members of an ∞2 linear system of circles of which the circle J and the line at infinity constitute the Jacobian. In particular, if the line m meet Aj Ak at Lt (i, j, k = 0, 1, 2), then the circles on Ai Li as diameter, which are the pedal circles of the point-pairs Ai, Li, are coaxial; the remaining circles of the coaxial system being the director circles of the conics, inscribed in the triangle A0 A1 A2, which touch the line m. If Mi denote the orthogonal projection on m of Ai, and Ni the orthogonal projection on Aj Ak of Mi, then the three lines Mi Ni meet at a point (Neuberg's theorem), viz. the centre of the circle J. Analogues for three-dimensional space of most of these theorems are also known ‖.

scholarly journals Improved Total Least Square Algorithm

2015 ◽  
Vol 9 (1) ◽  
pp. 394-399 ◽  
Author(s):  
Deng Yonghe
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Least Square ◽  
Error Equation ◽  
Virtual Observation ◽  
Straight Line ◽  
Condition Equation ◽  
Total Least Square ◽  
Three Dimensional Space ◽  
Improved Algorithm

Aim to blemish of total least square algorithm based on error equation of virtual observation,this paper proposed a sort of improved algorithm which doesn’t neglect condition equation of virtual observation,and considers both error equation and condition equation of virtual observation.So,the improved algorithm is better.Finally,this paper has fitted a straight line in three-dimensional space based on the improved algorithm.The result showed that the improved algorithm is viable and valid.

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