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### Using triangles to partition a disk

Tom Knoblauch
1990 Discrete Mathematics
Any partition of a disk into n subdisks can be imbedded in E* by laying down n triangles one after another. This imbedding can also be constructed by drawing 2n + 1 line segments.  ...  Introduction We will show how to imbed a given partition of a disk into n subdisks in E2 by laying down n triangles, one after another, as in Fig. 1 or 2.  ...  Every partition of a disk into n subdisks can be imbedded in E2 by drawing 2n + 1 line segments. Proof. Use T, and B,, . . . , B,. 0 2 .Fig. 1 Lemma 2. 2 . 212 Shingled imbeddings Theorem 2.1.  ...

### Minimal Partitions for p-norms of Eigenvalues [article]

Virginie Bonnaillie-Noel, Beniamin Bogosel
2018 arXiv   pre-print
In this article we are interested in studying partitions of the square, the disk and the equilateral triangle which minimize a p-norm of eigenvalues of the Dirichlet-Laplace operator.  ...  This allows us to see when partitions minimizing the 1-norm and the infinity-norm are different.  ...  The authors thank Michael Floater for suggesting us to look more carefully at k-partitions of the equilateral triangle when k is a triangular number.  ...

### Minimal partitions for $p$-norms of eigenvalues

Beniamin Bogosel, Virginie Bonnaillie-Noël
2018 Interfaces and free boundaries (Print)
In this article we are interested in studying partitions of the square, the disk and the equilateral triangle which minimize a p-norm of eigenvalues of the Dirichlet-Laplace operator.  ...  This allows us to see when partitions minimizing the 1-norm and the infinity-norm are different.  ...  The authors thank Michael Floater for suggesting us to look more carefully at k-partitions of the equilateral triangle when k is a triangular number.  ...

### Optimal partitions for the sum and the maximum of eigenvalues [article]

Beniamin Bogosel, Virginie Bonnaillie-Noël
2017 arXiv   pre-print
We analyze in detail the square, the disk and the equilateral triangle.  ...  In this paper we compare the candidates to be spectral minimal partitions for two criteria: the maximum and the average of the first eigenvalue on each subdomains of the partition.  ...  Acknowledgements This work is supported by a public grant overseen by the French National Research Agency (ANR) as part of the "Investissements d'Avenir" program (LabEx Sciences Mathématiques de Paris  ...

### Triangle listing in massive networks and its applications

Shumo Chu, James Cheng
2011 Proceedings of the 17th ACM SIGKDD international conference on Knowledge discovery and data mining - KDD '11
We propose an I/O-efficient algorithm for triangle listing. Our algorithm is exact and avoids random disk access.  ...  When the input graph cannot fit into main memory, triangle listing requires random disk accesses that can incur prohibitively large I/O cost.  ...  Algorithm 1 In-Memory Triangle Listing Input: A graph G = (VG, EG) Output: △(G) Acknowledgment The authors would like to thank the reviewers for their constructive comments.  ...

### Optimal Data-Space Partitioning of Spatial Data for Parallel I/O

Hakan Ferhatosmanoğlu, Divyakant Agrawal, Ömer Eğecioğlu, Amr El Abbadi
2005 Distributed and parallel databases
Some of these results carry over to circular partitioning of the data-which is an example of a non-convex region.  ...  This paper explores optimal partitioning for two-dimensional data for a class of queries and develops multi-disk allocation techniques that maximize the degree of I/O parallelism obtained in each case.  ...  Given the number M of available disks, we use a generic allocation technique parameterized by a skip value H . The partition (0, 0) are assigned to device 0.  ...

### The Use of Precomputed Triangle Clusters for Accelerated Ray Tracing in Dynamic Scenes

Kirill Garanzha
2009 Computer graphics forum (Print)
This algorithm precomputes densely packed clusters of triangles on surfaces. Folowing that, a set of clusters is used to rebuild the BVH in every frame.  ...  We introduce a novel heuristic to create triangle clusters that are designed for high performance ray tracing.  ...  Any cluster should have a shape similar to a sphere or to a disk.  ...

### A novel page-based data structure for interactive walkthroughs

2009 Proceedings of the 2009 symposium on Interactive 3D graphics and games - I3D '09
Assume that each disk-page consists of a set of triangles whose bounding boxes are computed.  ...  Walkthrough experiments have been conducted on a city scene consisting of over 110M triangles.  ...  A disk page K-d tree, for example, is a K-d tree on these bounding boxes of the disk pages. The leaf nodes of the K-d tree will have pointers to the disk-pages, and not individual triangles.  ...

### Bi-Lipschitz equivalent Alexandrov surfaces, II [article]

Yu.Burago
2004 arXiv   pre-print
This is a continuation of the joint paper with the same title by A.Belenkiy and Yu.Burago.  ...  integral curvatures, and two positive numbers e and l such that positive curvature of each embedded disk with perimeter not greater than l is not greater than \pi-e.  ...  This allows us to replace each triangle △BA i A i+1 by a triangle flat outside the disk D (B, r) and L-bi-Lipschitz equivalent to △BA i A i+1 .  ...

### Organizing and indexing non-convex regions

Eric Perlman, Randal Burns, Michael Kazhdan
2008 Proceedings of the VLDB Endowment
The index defines a linear order on the Delaunay triangles. Data from each triangle are then placed on disk in index order, e.g. using a B+-tree.  ...  We use computational geometry techniques to automatically characterize the region of space from which data are drawn, partition the region based on that characterization, and create an index from the partitions  ...  The index defines a linear order on the Delaunay triangles. Data from each triangle are then placed on disk in index order, e.g. using a B+-tree.  ...

### Computational prediction of electrical and thermal conductivities of disklike particulate composites

J. Qiu, Y. B. Yi, X. Guo
2015 International Journal of Computational Materials Science and Engineering
The pairwise intersection criteria of disks are developed using a set of vector operations.  ...  An element partition scheme has been implemented to connect the elements on different disks across the lines of intersection.  ...  Acknowledgment The authors would like to acknowledge support from the National Science Foundation under contract number #1232520.  ...

### Out-Of-Core Sort-First Parallel Rendering for Cluster-Based Tiled Displays [article]

Wagner T. Correa, James T. Klosowski, Claudio T. Silva
2002 Eurographics Symposium on Parallel Graphics and Visualization
Given a model, we use an out-of-core preprocessing algorithm to build an on-disk hierarchical representation for the model.  ...  At run time, each PC renders the image for a display tile, using an out-of-core rendering approach that employs multiple threads to overlap rendering, visibility computation, and disk operations.  ...  To achieve a well balanced system, they developed dynamic screen partitioning schemes that predict the rendering costs of groups of triangles and attempt to minimize the amount of overlap between triangles  ...

### Scalable isosurface visualization of massive datasets on commodity off-the-shelf clusters

Xiaoyu Zhang, Chandrajit Bajaj
2009 Journal of Parallel and Distributed Computing
It statically partitions the volume data to parallel disks with a balanced workload spectrum, and builds I/O-optimal external interval trees to minimize the number of I/O operations of loading large data  ...  It achieves scalability by using both parallel and out-of-core processing and parallel disks.  ...  Acknowledgments This research is supported in part by grants ACI-9982297, DMS-9873326 from the National Science Foundation, a DOE-ASCI grant BD4485-MOID from Sandia National Laboratory, Lawrence Livermore  ...

### Unions of Onions: Preprocessing Imprecise Points for Fast Onion Layer Decomposition [chapter]

Maarten Löffler, Wolfgang Mulzer
2013 Lecture Notes in Computer Science
Let D be a set of n pairwise disjoint unit disks in the plane. 8 We describe how to build a data structure for D so that for any point 9 set P containing exactly one point from each disk, we can quickly  ...  We also provide a matching lower bound. 14 Our solution is based on a recursive space decomposition, combined with 15 a fast algorithm to compute the union of two disjoint onion decomposi-16 tions.  ...  . , ∆ a be the resulting triangles and Q = 39 Q 1∪ · · ·∪Q a the partition of Q.  ...

### Unions of Onions: Preprocessing Imprecise Points for Fast Onion Decomposition [article]

Maarten Löffler, Wolfgang Mulzer
2013 arXiv   pre-print
Let D be a set of n pairwise disjoint unit disks in the plane.  ...  We describe how to build a data structure for D so that for any point set P containing exactly one point from each disk, we can quickly find the onion decomposition (convex layers) of P.  ...  A 30 simplicial r-partition of X is a sequence ∆ 1 , . . . , ∆ r of r triangles and a partition 31 X = X 1∪ · · ·∪X r of X into r pieces such that (i) for i = 1, . . . , r, we have X i ⊆ 32 ∆ i and |X  ...
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