Combinatorial Geometry Research Papers - Academia.edu (original) (raw)

Canonical Polygons are plane polygons defined on a square lattice with limitations on length of sides. The smaller sets are enumerated, and metrical and non-metrical properties are defined and calculated.

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- Combinatorial Geometry

We describe a simple and yet surprisingly powerful probabilistic technique which shows how to find in a dense graph a large subset of vertices in which all (or almost all) small subsets have many common neighbors. Recently this technique has had several striking applications to Extremal Graph Theory, Ramsey Theory, Additive Combinatorics, and Combinatorial Geometry. In this survey we discuss

- by Jacob Fox
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- Mathematics, Computer Science, Statistics, Combinatorial Geometry
- by Gábor Gévay
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- Combinatorial Geometry

Szemerédi’s regularity lemma asserts that every graph can be decomposed into relatively few random-like subgraphs. This random-like behavior enables one to find and enumerate subgraphs of a given isomorphism type, yielding the so-called counting lemma for graphs. The combined application of these two lemmas is known as the regularity method for graphs and has proved useful in graph theory, combinatorial

- by Ananth Ranganathan
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- Mechanical Engineering, Cartography, Discrete Geometry, Set Theory
- by Herbert Steinberg
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- Pattern Recognition, Statistical Analysis, Combinatorial Geometry, Ray Tracing
- by Chilukuri Mohan
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- Information Science, Pattern Recognition, Combinatorial Geometry, Complexity
- by Christian Haase
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- Lattice Theory, Combinatorial Geometry, Contemporary Mathematics, Lattice Points
- by Neil White
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- Algebraic Geometry, Combinatorial Geometry, Pure Mathematics, Chemical Kinetics
- by Aleksandra Blagojevic
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- Combinatorial Geometry, Pure Mathematics, Primary, Graph Coloring
- by Pavle Blagojević
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- Combinatorial Geometry, Pure Mathematics, Graph Coloring, Index Theory

Let e1;:::;em be m di#erent symbols, let r1 ¿ ··· ¿ rm be positive integers, and let n =!mi=1 ri. The combinohedron, denoted by C(r1;:::;rm), is the loopless graph whose vertices are the
n-tuples in which the symbol ei appears exactly ri times, and where an edge joins two vertices if and only if one can be transformed into the other by interchanging two adjacent entries. The graph known as permutohedron is a particular case of the combinohedron. Here, we extend to the combinohedron some results on embeddability of the permutohedron. !c 2002 Elsevier Science B.V.

- by julio estrada
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- Mathematics, Music Theory, Musicology, Combinatorial Geometry

Using the method of compression we obtain a lower bound for the average number of d r-unit distances that can be formed from a set of n points in the euclidean space R k. By letting D n,d r denotes the number of d r-unit distances (r > 1 fixed) that can be formed from a set of n points in R k , then we obtain the lower bound 1≤d≤t D n,d r n 2r √ k log t.

- by Theophilus Agama
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- Mathematics, Number Theory, Combinatorial Geometry

In this paper, we study the geometry of a (non-trivial) 1-based SU rank-1 complete type. We show that if the (localized, resp.) geometry of the type is modular, then the (localized, resp.) geometry is projective over a division ring. However, unlike the stable case, we construct a locally modular type which is not a#ne. For the general 1-based case, we prove that even if the geometry of the type itself is not projective over a division ring, it is when we consider a 2-fold or 3-fold of the geometry altogether. In particular, it follows that in any #-categorical, non-trivial, 1-based theory, a vector space over a finite field is interpretable.

- by Tristram de Piro
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- Mathematics, Number Theory, Algebraic Geometry, Model Theory

This collection was compiled by Christian Haase and Bruce Reznick from problems presented at the problem sessions, and submissions solicited from the participants of the AMS/IMS/SIAM summer Research Conference on Integer points in... more

- by Christian Haase
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- Lattice Theory, Combinatorial Geometry, Lattice Points
- by Yossi Matias
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- Engineering, Materials Engineering, Distributed Computing, Condensed Matter Physics
- by Peter Komjath
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- Applied Mathematics, Stochastic Process, Graph Theory, Combinatorial Geometry

Abstract. This paper identifies the total number of gaps of object pixels in a binary picture, which solves an open problem in 2D digital geometry (or combinatorial topology of binary pictures). We obtain a formula for the total number of... more

- by V Brimkov
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- Mathematics, Discrete Geometry, Computer Science, Computer Vision
- by Dina Ghinelli
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- Mathematics, Applied Mathematics, Computer Science, Combinatorics
- by Johannes Ueberberg
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- Combinatorial Geometry, Pure Mathematics, Finite Geometry, Hyperplane

- by Tristram de Piro
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- Number Theory, Algebraic Geometry, Model Theory, Combinatorial Geometry

This paper describes the application of the SRNA Monte Carlo package for proton transport simulations in complex geometry and different material compositions. The SRNA package was developed for 3D dose distribution calculation in proton therapy and dosimetry and it was based on the theory of multiple scattering. The decay of proton induced compound nuclei was simulated by the Russian MSDM model and our own using ICRU 63 data. The developed package consists of two codes: the SRNA-2KG, which simulates proton transport in combinatorial geometry and the SRNA-VOX, which uses the voxelized geometry using the CT data and conversion of the Hounsfield's data to tissue elemental composition. Transition probabilities for both codes are prepared by the SRNADAT code. The simulation of the proton beam characterization by multi-layer Faraday cup, spatial distribution of positron emitters obtained by the SRNA-2KG code and intercomparison of computational codes in radiation dosimetry, indicate immediate application of the Monte Carlo techniques in clinical practice. In this paper, we briefly present the physical model implemented in the SRNA package, the ISTAR proton dose planning software, as well as the results of the numerical experiments with proton beams to obtain 3D dose distribution in the eye and breast tumour.

- by Petar Belicev
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- Algorithms, Biomedical Engineering, Combinatorial Geometry, Monte Carlo
- by francisco dos santos
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- Combinatorial Geometry, Pure Mathematics, Triangulation, Convex Analysis
- by Andreas Dress
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- Combinatorial Geometry, Pure Mathematics, Projective Geometry

We consider the problem of bounding the maximum possible number fk,d(n) of k- simplices that are spanned by a set of n points in Rd and are similar to a given simplex. We first show that f2,3(n) = O(n13/6), and then tackle the general... more

- by George Purdy
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- Combinatorial Geometry
- by Oleg Verbitsky
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- Applied Mathematics, Computational Geometry, Combinatorics, Combinatorial Geometry
- by Jürgen Bokowski
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- Combinatorial Geometry, Pure Mathematics, Infinite series

*† ‡ Modeling of complex three-dimensional engineered systems in the standard radiation transport tools MCNP and TART has traditionally been inhibited by geometry concerns. Most of these systems are designed in the framework of a commercial Computer Aided Design (CAD) package, but the geometrical paradigms used in these products are fundamentally different from the Combinatorial Geometry (CG) framework used in the radiation transport codes. Engineers at Raytheon Missile Systems have developed an automated system for bridging the gap between CAD and combinatorial geometries; it is named TopAct, for “Translation Optimization for Part-wise Adaptive Combinatorial Transport”. TopAct has been demonstrated to provide highly accurate, efficient CG representations of real-world parts designed in the ProEngineer CAD system. TopAct thus enables substantial cost savings in the production of radiation transport geometry models, along with attendant benefits in the complexity and accuracy of these models.

- by Brian Triggs
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- Combinatorial Geometry, Computer Aided Design, Space, Three Dimensional

Summary form only given. The ARGUS three-dimensional simulation code has been used for a wide variety of problems which require the solution of field equations and particle flows in complex geometries. Some of the applications include the determination of resonant modes in RF and accelerator structures, the electrical stability of MMIC (microwave monolithic integrated circuit) packaging, antenna design, gun design.

- by Michael Kress
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- Computer Science, Combinatorial Geometry, Magnetic Materials, Three Dimensional
- by Eleni Tzanaki
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- Discrete Geometry, Computational Geometry, Combinatorial Geometry

The ARGUS system of codes has been developed at SAIC over the past six years. The individual codes within the ARGUS model can function together as modules of a large code, or they can function as stand-alone three-dimensional codes to treat specific types of problems. The ARGUS model has not been built from a two-dimen- sional code, but is a completely new numerical model whose architecture has been designed to handle three-dimensional problems. ARGUS uses a sophisticated domain-decomposition algorithm', coupled with memory-management and data-handling techniques to optimize the use of core memory for each problem and to efficiently move data between core and disk memory during the calculation. A large problem is divided into blocks which are independently processed in core. The data-handling module moves each block from disk to core as it is needed by various code modules. The ARGUS modules are designed for compatibility with this data structure. The disk l/O is table driven, ...

- by A. Mondelli
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- Combinatorial Geometry, Memory Management, Data Structure, Electromagnetic Field
- by Peter Frankl
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- Combinatorics, Combinatorial Geometry, Pure Mathematics, Configuration
- by alan poritz
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- Engineering, Graph Theory, Combinatorial Geometry, Topology
- by George Purdy
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- Combinatorial Geometry
- by Rudolf Fleischer and +1
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- Computational Geometry, Graph Theory, Combinatorial Geometry, Motion Planning

This paper describes the application of the SRNA Monte Carlo package for proton transport simulations in complex geometry and different material compositions. The SRNA package was developed for 3D dose distribution calculation in proton therapy and dosimetry and it was based on the theory of multiple scattering. The decay of proton induced compound nuclei was simulated by the Russian MSDM model and our own using ICRU 63 data. The developed package consists of two codes: the SRNA-2KG, which simulates proton transport in combinatorial geometry and the SRNA-VOX, which uses the voxelized geometry using the CT data and conversion of the Hounsfield&amp;amp;amp;#39;s data to tissue elemental composition. Transition probabilities for both codes are prepared by the SRNADAT code. The simulation of the proton beam characterization by multi-layer Faraday cup, spatial distribution of positron emitters obtained by the SRNA-2KG code and intercomparison of computational codes in radiation dosimetry, indicate immediate application of the Monte Carlo techniques in clinical practice. In this paper, we briefly present the physical model implemented in the SRNA package, the ISTAR proton dose planning software, as well as the results of the numerical experiments with proton beams to obtain 3D dose distribution in the eye and breast tumour.

- by Vesna Spasic-jokic
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- Algorithms, Biomedical Engineering, Combinatorial Geometry, Monte Carlo