scholarly journals The Sphere Covering Inequality and Its Dual

2020 ◽  
Vol 73 (12) ◽  
pp. 2685-2707
Author(s):  
Changfeng Gui ◽  
Fengbo Hang ◽  
Amir Moradifam
Keyword(s):  
Sphere Covering

scholarly journals Lower Bounds for $q$-ary Codes with Large Covering Radius

10.37236/222 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Wolfgang Haas ◽  
Immanuel Halupczok ◽  
Jan-Christoph Schlage-Puchta
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Efficient Method ◽  
Winning Strategy ◽  
Covering Radius ◽  
Minimal Cardinality ◽  
Original Proof ◽  
Sphere Covering

Let $K_q(n,R)$ denote the minimal cardinality of a $q$-ary code of length $n$ and covering radius $R$. Recently the authors gave a new proof of a classical lower bound of Rodemich on $K_q(n,n-2)$ by the use of partition matrices and their transversals. In this paper we show that, in contrast to Rodemich's original proof, the method generalizes to lower-bound $K_q(n,n-k)$ for any $k>2$. The approach is best-understood in terms of a game where a winning strategy for one of the players implies the non-existence of a code. This proves to be by far the most efficient method presently known to lower-bound $K_q(n,R)$ for large $R$ (i.e. small $k$). One instance: the trivial sphere-covering bound $K_{12}(7,3)\geq 729$, the previously best bound $K_{12}(7,3)\geq 732$ and the new bound $K_{12}(7,3)\geq 878$.

scholarly journals The sphere covering inequality and its applications

2018 ◽  
Vol 214 (3) ◽  
pp. 1169-1204 ◽  
Author(s):  
Changfeng Gui ◽  
Amir Moradifam
Keyword(s):  
Sphere Covering

Full Scale Calibration of a Combined Tactile Contour and Roughness Measurement Device

2014 ◽  
Vol 613 ◽  
pp. 94-100
Author(s):  
Raimund Volk ◽  
Stefan Feifel
Keyword(s):  
Surface Profile ◽  
Adjustment Process ◽  
Roughness Measurement ◽  
Production Metrology ◽  
Scale Calibration ◽  
Current Production ◽  
Continuous Surface ◽  
Sphere Covering ◽  
Adjacent Segments ◽  
Tactile Probe

In current production metrology contour and roughness of work pieces has to be controlled in demanding small tolerances. It is important that the results not only have to be accurate, but also the measurement has to be performed effective and fast. It would be convenient to have the possibility measuring first a continuous surface profile and than performing different evaluations on adjacent segments. We describe a measurement system which is based on a tactile probe with a diamond tip. The position of each axis is measured by high resolution and high accuracy digital scales. The instrument is calibrated and traced back by only using one precision sphere covering approximately the whole measuring range of 24 mm. It is shown that with only that macroscopic calibration and adjustment the accuracy for roughness measurement in all arbitrary vertical positions is reached in the nm-range. The calibration and adjustment process is described and we show the verification of the accuracy by measuring several surface specimens. The results are compared to a Physikalisch-Technische Bundesanstalt (PTB) calibration certificates for those surface roughness standards.

The minimum sphere covering a convex polyhedron

1974 ◽  
Vol 21 (4) ◽  
pp. 715-718 ◽  
Author(s):  
Jack Elzinga ◽  
Donald Hearn
Keyword(s):  
Convex Polyhedron ◽  
Sphere Covering

scholarly journals A Parallel Interval Arithmetic-based Reliable Computing Method on a GPU

2017 ◽  
Vol 23 (2) ◽  
pp. 491-501 ◽  
Author(s):  
Zsolt Bagóczki ◽  
Balázs Bánhelyi
Keyword(s):  
Branch And Bound ◽  
High Speed ◽  
General Purpose ◽  
Three Dimensions ◽  
Direct Access ◽  
Covering Problem ◽  
Virtual Instruction ◽  
Wide Range ◽  
Circle Covering ◽  
Sphere Covering

Video cards have now outgrown their purpose of being only a simple tool for graphic display. With their high speed video memories, lots of maths units and parallelism, they can be very powerful accessories for general purpose computing tasks. Our selected platform for testing is the CUDA (Compute Unified Device Architecture), which offers us direct access to the virtual instruction set of the video card, and we are able to run our computations on dedicated computing kernels. The CUDA development kit comes with a useful toolbox and a wide range of GPU-based function libraries. In this parallel environment, we implemented a reliable method based on the Branch-and-Bound algorithm. This algorithm will give us the opportunity to use node level (also called low-level or type 1) parallelization, since we do not modify the searching trajectories; nor do we modify the dimensions of the Branch-and-Bound tree [5]. For testing, we chose the circle covering problem. We then scaled the problem up to three dimensions, and ran tests with sphere covering problems as well.

scholarly journals Mathematical and Numerical Modeling of Turbulent Flows

2015 ◽  
Vol 87 (2) ◽  
pp. 1195-1232 ◽  
Author(s):  
João M. Vedovoto ◽  
Ricardo Serfaty ◽  
Aristeu Da Silveira Neto
Keyword(s):  
Turbulent Flows ◽  
Complex Geometry ◽  
Time Integration ◽  
Immersed Boundary ◽  
Integration Scheme ◽  
Computational Framework ◽  
Wide Range ◽  
Time Integration Scheme ◽  
Sphere Covering ◽  
Predictor Corrector

The present work is devoted to the development and implementation of a computational framework to perform numerical simulations of low Mach number turbulent flows over complex geometries. The algorithm under consideration is based on a classical predictor-corrector time integration scheme that employs a projection method for the momentum equations. The domain decomposition strategy is adopted for distributed computing, displaying very satisfactory levels of speed-up and efficiency. The Immersed Boundary Methodology is used to characterize the presence of a complex geometry. Such method demands two separate grids: An Eulerian, where the transport equations are solved with a Finite Volume, second order discretization and a Lagrangian domain, represented by a non-structured shell grid representing the immersed geometry. The in-house code developed was fully verified by the Method of Manufactured Solu- tions, in both Eulerian and Lagrangian domains. The capabilities of the resulting computational framework are illustrated on four distinct cases: a turbulent jet, the Poiseuille flow, as a matter of validation of the implemented Immersed Boundary methodology, the flow over a sphere covering a wide range of Reynolds numbers, and finally, with the intention of demonstrating the applicability of Large Eddy Simulations - LES - in an industrial problem, the turbulent flow inside an industrial fan.

scholarly journals Optimal configuration of gamma ray machine radiosurgery units: the sphere covering subproblem

2008 ◽  
Vol 3 (1) ◽  
pp. 109-121 ◽  
Author(s):  
Leo Liberti ◽  
Nelson Maculan ◽  
Yue Zhang
Keyword(s):  
Gamma Ray ◽  
Optimal Configuration ◽  
Sphere Covering
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