Tetrahedral cages for unit discs

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Liping Yuan ◽  
Tudo Zamfirescu ◽  
Yanxue Zhang
Keyword(s):  
Convex Polytope ◽  
Dimensional Unit ◽  
Distant Location

Abstract A cage is the 1-skeleton of a convex polytope in ℝ3. A cage is said to hold a set if the set cannot be continuously moved to a distant location, remaining congruent to itself and disjoint from the cage. In how many positions can (compact 2-dimensional) unit discs be held by a tetrahedral cage? We completely answer this question for all tetrahedra.

Aircraft Noise Levels, Annoyance and General Health of Residents of Communities contiguous to a Major Airport in Lagos State

2020 ◽  
Vol 8 (4) ◽  
pp. 47-59
Author(s):  
Iheanyichukwu M. Elechi
Keyword(s):  
Health Status ◽  
General Health ◽  
Noise Exposure ◽  
Aircraft Noise ◽  
Poor Health ◽  
Noise Levels ◽  
General Health Status ◽  
Lagos State ◽  
Distant Location ◽  
Self Reported Health Status

The purpose of this study was to investigate the relationship between the aircraft noise exposure, annoyance reactions and health status of the residents living within the vicinity of the Murtala Muhammed International Airport (MMA) in Lagos state, Nigeria. Aircraft noise monitoring was conducted in five locations within the vicinity (0-5Km) of MMA, and a sixth distant location (14km away). Levels of aircraft noise for all five locations within the vicinity of the airport exceeded the EPA Victoria threshold of 75 dB LAmax for the residential area (outdoor). A survey on annoyance induced by aircraft noise exposure and general health status was conducted on 450 local residents in the study locations using the International Commission on Biological Effect of Noise question and a single question that has been applied in Dutch national health care surveys since 1983 on self-reported general health status respectively. Percentage of residents within the vicinity of MMA that were highly annoyed (%HA) exceeded 15% guideline limit stipulated by Federal Interagency Committee on Urban Noise while 14.5% reported poor health status. There was a significant association between the annoyance reactions and aircraft noise levels in the study locations while the association between self-reported health status and aircraft noise levels was not significant. Taken together, the residents within the vicinity of the airport are exposed to aircraft noise levels above permissible limit which may be associated with high annoyance reaction but may not be associated with poor health rating. Evidence-based aircraft noise related policies by government are advocated.

scholarly journals A short note on extreme points of certain polytopes

2020 ◽  
Vol 8 (1) ◽  
pp. 36-39
Author(s):  
Lei Cao ◽  
Ariana Hall ◽  
Selcuk Koyuncu
Keyword(s):  
Short Proof ◽  
Convex Polytopes ◽  
Short Note ◽  
Convex Polytope ◽  
Extreme Points ◽  
Alternative Proof ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Birkhoff's Theorem ◽  
Birkhoff’S Theorem

AbstractWe give a short proof of Mirsky’s result regarding the extreme points of the convex polytope of doubly substochastic matrices via Birkhoff’s Theorem and the doubly stochastic completion of doubly sub-stochastic matrices. In addition, we give an alternative proof of the extreme points of the convex polytopes of symmetric doubly substochastic matrices via its corresponding loopy graphs.

Characterization of the conditional stationary distribution in Markov chains via systems of linear inequalities

2020 ◽  
Vol 52 (4) ◽  
pp. 1249-1283
Author(s):  
Masatoshi Kimura ◽  
Tetsuya Takine
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Continuous Time ◽  
State Transition ◽  
Convex Polytope ◽  
Linear Inequalities ◽  
Systems Of Linear Inequalities ◽  
Free Sets ◽  
Practical Implications

AbstractThis paper considers ergodic, continuous-time Markov chains $\{X(t)\}_{t \in (\!-\infty,\infty)}$ on $\mathbb{Z}^+=\{0,1,\ldots\}$ . For an arbitrarily fixed $N \in \mathbb{Z}^+$ , we study the conditional stationary distribution $\boldsymbol{\pi}(N)$ given the Markov chain being in $\{0,1,\ldots,N\}$ . We first characterize $\boldsymbol{\pi}(N)$ via systems of linear inequalities and identify simplices that contain $\boldsymbol{\pi}(N)$ , by examining the $(N+1) \times (N+1)$ northwest corner block of the infinitesimal generator $\textbf{\textit{Q}}$ and the subset of the first $N+1$ states whose members are directly reachable from at least one state in $\{N+1,N+2,\ldots\}$ . These results are closely related to the augmented truncation approximation (ATA), and we provide some practical implications for the ATA. Next we consider an extension of the above results, using the $(K+1) \times (K+1)$ ( $K > N$ ) northwest corner block of $\textbf{\textit{Q}}$ and the subset of the first $K+1$ states whose members are directly reachable from at least one state in $\{K+1,K+2,\ldots\}$ . Furthermore, we introduce new state transition structures called (K, N)-skip-free sets, using which we obtain the minimum convex polytope that contains $\boldsymbol{\pi}(N)$ .

scholarly journals Estimation of 6D Object Pose Using a 2D Bounding Box

Sensors ◽  
2021 ◽  
Vol 21 (9) ◽  
pp. 2939
Author(s):  
Yong Hong ◽  
Jin Liu ◽  
Zahid Jahangir ◽  
Sheng He ◽  
Qing Zhang
Keyword(s):  
Neural Network ◽  
Loss Function ◽  
Three Dimensional ◽  
Unit Vector ◽  
Prediction Algorithm ◽  
Computational Time ◽  
Bounding Box ◽  
Dimensional Unit ◽  
Bounding Boxes ◽  
Rgb Image

This paper provides an efficient way of addressing the problem of detecting or estimating the 6-Dimensional (6D) pose of objects from an RGB image. A quaternion is used to define an object′s three-dimensional pose, but the pose represented by q and the pose represented by -q are equivalent, and the L2 loss between them is very large. Therefore, we define a new quaternion pose loss function to solve this problem. Based on this, we designed a new convolutional neural network named Q-Net to estimate an object’s pose. Considering that the quaternion′s output is a unit vector, a normalization layer is added in Q-Net to hold the output of pose on a four-dimensional unit sphere. We propose a new algorithm, called the Bounding Box Equation, to obtain 3D translation quickly and effectively from 2D bounding boxes. The algorithm uses an entirely new way of assessing the 3D rotation (R) and 3D translation rotation (t) in only one RGB image. This method can upgrade any traditional 2D-box prediction algorithm to a 3D prediction model. We evaluated our model using the LineMod dataset, and experiments have shown that our methodology is more acceptable and efficient in terms of L2 loss and computational time.

scholarly journals 10 Points in Dimension 4 not Projectively Equivalent to the Vertices of a Convex Polytope

2001 ◽  
Vol 22 (5) ◽  
pp. 705-708 ◽  
Author(s):  
David Forge ◽  
Michel Las Vergnas ◽  
Peter Schuchert
Keyword(s):  
Convex Polytope

scholarly journals The number of lattice rules having given invariants

1992 ◽  
Vol 46 (3) ◽  
pp. 479-495 ◽  
Author(s):  
Stephen Joe ◽  
David C. Hunt
Keyword(s):  
Normal Form ◽  
Quadrature Rule ◽  
Unit Cube ◽  
Numerical Results ◽  
Smith Normal Form ◽  
Lattice Rules ◽  
Dimensional Unit ◽  
Lattice Rule ◽  
Positive Integers ◽  
Theoretical Results

A lattice rule is a quadrature rule used for the approximation of integrals over the s-dimensional unit cube. Every lattice rule may be characterised by an integer r called the rank of the rule and a set of r positive integers called the invariants. By exploiting the group-theoretic structure of lattice rules we determine the number of distinct lattice rules having given invariants. Some numerical results supporting the theoretical results are included. These numerical results are obtained by calculating the Smith normal form of certain integer matrices.

scholarly journals Calculus rules for combinations of ellipsoids and applications

1993 ◽  
Vol 47 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Alberto Seeger
Keyword(s):  
Convex Hull ◽  
Convex Polytope ◽  
Finite Family ◽  
Minkowski Sum ◽  
Calculus Rules ◽  
Ellipsoidal Approximations

We derive formulas for the Minkowski sum, the convex hull, the intersection, and the inverse sum of a finite family of ellipsoids. We show how these formulas can be used to obtain inner and outer ellipsoidal approximations of a convex polytope.

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