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1,292 results on '"General position"'

1. Colorful words and \(d\)-Tverberg complexes

Author

Frick, Florian and Jeffs, R. Amzi

Subjects
Tverberg's theorem, word representable, \(d\)-representable, nerve, general position, strong general position, fully independent
Abstract

We give a complete combinatorial characterization of weakly \(d\)-Tverberg complexes. These complexes record which intersection combinatorics of convex hulls necessarily arise in any sufficiently large general position point set in \(\mathbb R^d\). This strengthens the concept of \(d\)-representable complexes, which describe intersection combinatorics that arise in at least one point set. Our characterization allows us to construct for every fixed \(d\) a graph that is not weakly \(d'\)-Tverberg for any \({d' \le d}\), answering a question of De Loera, Hogan, Oliveros, and Yang.Mathematics Subject Classifications: 52A35, 52C45Keywords: Tverberg's theorem, word representable, \(d\)-representable, nerve, general position, strong general position, fully independent

Published
2024

2. Variety of General Position Problems in Graphs.

Author

Tian, Jing and Klavžar, Sandi

Abstract

Let X be a vertex subset of a graph G. Then u , v ∈ V (G) are X-positionable if V (P) ∩ X ⊆ { u , v } holds for any shortest u, v-path P. If each two vertices from X are X-positionable, then X is a general position set. The general position number of G is the cardinality of a largest general position set of G and has been already well investigated. In this paper a variety of general position problems is introduced based on which natural pairs of vertices are required to be X-positionable. This yields the total (resp. dual, outer) general position number. It is proved that the total general position sets coincide with sets of simplicial vertices, and that the outer general position sets coincide with sets of mutually maximally distant vertices. It is shown that a general position set is a dual general position set if and only if its complement is convex. Several sufficient conditions are presented that guarantee that a given graph has no dual general position set. The total general position number, the outer general position number, and the dual general position number of arbitrary Cartesian products are determined. [ABSTRACT FROM AUTHOR]

Published
2025
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3. On Absolute and Quantitative Subspace Theorems.

Author

Ngo, Hieu T. and Quang, Si Duc

Abstract

The Absolute Subspace Theorem, a vast generalization and a quantitative improvement of Schmidt’s Subspace Theorem, was first established by Evertse and Schlickewei and then strengthened remarkably by Evertse and Ferretti. We study quantitative generalizations and extensions of subspace theorems in various contexts. We establish a generalization of Evertse and Ferretti’s Absolute Subspace Theorem for hyperplanes in general position. We obtain improved (non-absolute) Quantitative Subspace Theorems for hyperplanes in general position and in subgeneral position. We show a Semi-quantitative Subspace Theorem for hyperplanes in non-subdegenerate position. [ABSTRACT FROM AUTHOR]

Published
2024
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4. Space groups – the final frontier: a tutorial guided tour of some entries in International Tables for Crystallography Volume A.

Author

Clegg, William

Subjects
*CRYSTALLOGRAPHY, *CRYSTAL symmetry, *TOUR guides (Persons), *CRYSTAL structure, *SYMMETRY groups, *SPACE groups
Abstract

After an introductory section providing some basic foundation, definitions, and terminology regarding space group symmetry in crystal structures, three particular examples are selected to demonstrate the information given for each space-group type in the standard reference book International Tables for Crystallography Volume A (current sixth edition). The format and layout of the information is explained, and the most important items of the content are described, together with some practical applications. This treatment should enable readers unfamiliar with the tables to use them confidently in understanding and interpreting crystal structures belonging to these and other space-group types. [ABSTRACT FROM AUTHOR]

Published
2023
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5. A standard form of incompressible surfaces in 3-dimensional handlebodies.

Author

Lin, Wei and Lei, Fengchun

Abstract

Applying the Morse theory, we give a standard form for a class of surfaces which includes all the properly embedded incompressible surfaces in 3-dimensional handlebodies. We also give a necessary and sufficient condition to determine the incompressibility of such surfaces placed in our standard form. Our algorithm is practical. Several examples are given to test the algorithm. [ABSTRACT FROM AUTHOR]

Published
2023
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6. Some Position Problems for Graphs

Author

Tuite, James, Thomas, Elias John, Chandran S. V., Ullas, Goos, Gerhard, Founding Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Woeginger, Gerhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Balachandran, Niranjan, editor, and Inkulu, R., editor

Published
2022
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8. The Logarithmic Capacitary Minkowski Problem for Polytopes.

Author

Xiong, Ge and Xiong, Jia Wei

Subjects
*SPHERES
Abstract

The logarithmic capacitary Minkowski problem asks for necessary and sufficient conditions on a finite Borel measure on the unit sphere so that it is the logarithmic capacitary measure of a convex body. This article completely solves the case of discrete measures whose support sets are in general position. [ABSTRACT FROM AUTHOR]

Published
2022
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9. Some results on the computing of Tukey's halfspace median.

Author

Liu, Xiaohui, Luo, Shihua, and Zuo, Yijun

Abstract

Depth of the Tukey median is investigated for empirical distributions. A sharper upper bound is provided for this value for data sets in general position. This bound is lower than the existing one in the literature and, more importantly, derived under the fixed sample size practical scenario. Several results obtained in this paper are interesting theoretically and useful as well to reduce the computational burden of the Tukey median practically when p > 2 . [ABSTRACT FROM AUTHOR]

Published
2020
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10. 通有构形的特征多项式.

Author

孙 莹 and 高瑞梅

Subjects
DEFINITIONS, POLYNOMIALS
Abstract

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Published
2020
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11. Formation of General Position by Asynchronous Mobile Robots Under One-Axis Agreement

Author

Bhagat, Subhash, Chaudhuri, Sruti Gan, Mukhopadhyaya, Krishnendu, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Weikum, Gerhard, Series editor, Kaykobad, Mohammad, editor, and Petreschi, Rossella, editor

Published
2016
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12. Minimum Time Problem

Author

Aschepkov, Leonid T., Dolgy, Dmitriy V., Kim, Taekyun, Agarwal, Ravi P., Aschepkov, Leonid T., Dolgy, Dmitriy V., Kim, Taekyun, and Agarwal, Ravi P.

Published
2016
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14. Two problems in computational geometry

Author

Universitat Politècnica de Catalunya. Departament de Matemàtiques, Seara Ojea, Carlos, Huemer, Clemens, Oliver Burwitz, Nicolau, Universitat Politècnica de Catalunya. Departament de Matemàtiques, Seara Ojea, Carlos, Huemer, Clemens, and Oliver Burwitz, Nicolau

Abstract

En aquesta tesi s'estudien dos problemes del camp de la geometria computacional. El primer problema és: donat un set S de n punts en el pla en posició general, com de prop són quatre punts de S de ser cocirculars. Definim tres mesures per estudiar aquesta qüestió, la mesura de Tales, la mesura de Voronoi, i la mesura del Determinant. Presentem cotes per la mesura de Tales, i algoritmes per computar aquestes mesures de cocircularitat. També reduïm el problema de computar la cocircularitat emprant la mesura del Determinant al problema de 4SUM. El segon problema és: donat dos sets R i B de punts rojos i blaus respectivament, com computar la discrepància bicromàtica amb caixes i cercles. La discrepància bicromàtica és definida com la diferència entre el nombre de punts vermells i blaus que són a l'interior de la figura examinada. Presentem una comparativa entre algoritmes ja existents per les dues figures. També comparem la discrepància bicromàtica de caixes orientades en els eixos vs. d'orientació general. A més a més, també presentem un nou algoritme per la discrepància en esferes/discs per a altes dimensions, basat en literatura ja existent. També relacionem altres problemes en el tema de separabilitat amb algoritmes sensitius a l'output per la discrepància amb caixes., En esta tesis se estudian dos problemas del campo de la geometría computacional. El primer problema es: dado un set S de n puntos en el plan en posición general, como de cerca son cuatro puntos de S de ser cocirculares. Definimos tres medidas para estudiar esta cuestión, la medida de Tales, la medida de Voronoi, y la medida del Determinante. Presentamos cotas por la medida de Tales, y algoritmos para computar estas medidas de cocircularidad. También reducimos el problema de computar la cocircularidad usando la medida del Determinante al problema de 4SUM. El segundo problema es: dado dos sets R y B de puntos rojos y azules respectivamente, como computar la discrepancia bicromática con cajas y círculos. La discrepancia bicromática es definida como la diferencia entre el número de puntos rojos y azules que están en el interior de la figura examinada. Presentamos una comparativa entre algoritmos ya existentes por las dos figuras. También comparamos la discrepancia bicromática de cajas orientadas en los ejes vs. de orientación general. Además, también presentamos un nuevo algoritmo por la discrepancia en esferas/discos para altas dimensiones, basado en literatura ya existente. También relacionamos otros problemas en el tema de separabilidad con algoritmos sensitivos al output por la discrepancia con cajas., Two different problems belonging to computational geometry are studied in this thesis. The first problem studies: given a set S of n points in the plane in general position, how close are four points of S to being cocircular. We define three measures to study this question, the Thales, Voronoi and Determinant measures. We present bounds on the Thales almost-cocircularity measure over a point set. Algorithms for computing these measures of cocircularity are presented as well. We give a reduction from computing cocircularity using the Determinant measure to the 4SUM problem. The second problem studies: given two sets R and B of red and blue points respectively, how to compute the bichromatic discrepancy using boxes and circles. The bichromatic discrepancy is defined as the difference between the number of red points and blue points inside the shape. We present a comparison of algorithms in the existing literature for the two shapes. Bichromatic discrepancy in axis-parallel boxes .vs non-axis-parallel boxes is also compared. Furthermore, we also present a new algorithm for disk discrepancy in higher dimensions, based on existing literature. We also relate existing problems in separability with existing output sensitive algorithms for bichromatic discrepancy using boxes.

Published
2023

15. Strong independence and the dimension of a Tverberg set.

Author

Choudhury, Snigdha Bharati and Deo, Satya

Abstract

In this expository paper, we discuss several types of independence of points in the Euclidean space R d such as algebraic independence, strong independence, weak independence, and give a detailed geometric proof of their interrelationship. We also give yet another proof of Reay's theorem determining the dimension of a Tverberg set. [ABSTRACT FROM AUTHOR]

Published
2021
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16. An Optimal Algorithm for Plane Matchings in Multipartite Geometric Graphs

Author

Biniaz, Ahmad, Maheshwari, Anil, Nandy, Subhas C., Smid, Michiel, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Weikum, Gerhard, Series editor, Dehne, Frank, editor, Sack, Jörg-Rüdiger, editor, and Stege, Ulrike, editor

Published
2015
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17. Higher-Order Triangular-Distance Delaunay Graphs: Graph-Theoretical Properties

Author

Biniaz, Ahmad, Maheshwari, Anil, Smid, Michiel, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Kobsa, Alfred, Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Nierstrasz, Oscar, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Weikum, Gerhard, Series editor, Ganguly, Sumit, editor, and Krishnamurti, Ramesh, editor

Published
2015
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27. Birational geometry of varieties fibred into complete intersections of codimension two

Author

Aleksandr V. Pukhlikov

Subjects
General Mathematics, Complete intersection, Fibered knot, 14E05, 14E07, Birational geometry, Codimension, Type (model theory), Combinatorics, Base (group theory), Mathematics - Algebraic Geometry, Mathematics::Algebraic Geometry, FOS: Mathematics, Projective space, Algebraic Geometry (math.AG), General position, Mathematics
Abstract

In this paper we prove the birational superrigidity of Fano-Mori fibre spaces $\pi\colon V\to S$, every fibre of which is a complete intersection of type $d_1\cdot d_2$ in the projective space ${\mathbb P}^{d_1+d_2}$, satisfying certain conditions of general position, under the assumption that the fibration $V/S$ is sufficiently twisted over the base (in particular, under the assumption that the $K$-condition holds). The condition of general position for every fibre guarantees that the global log canonical threshold is equal to one. This condition bounds the dimension of the base $S$ by a constant that depends on the dimension $M$ of the fibre only (as the dimension $M$ of the fibre grows, this constant grows as $\frac12 M^2$). The fibres and the variety $V$ itself may have quadratic and bi-quadratic singularities, the rank of which is bounded from below., Comment: 86 pages, the final version

Published
2022
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28. Mutual-visibility and general position in double graphs and in Mycielskians.

Author

Roy, Dhanya, Klavžar, Sandi, and Lakshmanan S, Aparna

Abstract

The general position problem in graphs is to find the largest possible set of vertices with the property that no three of them lie on a common shortest path. The mutual-visibility problem in graphs is to find the maximum number of vertices that can be selected such that every pair of vertices in the collection has a shortest path between them with no vertex from the collection as an internal vertex. Here, the general position problem and the mutual-visibility problem are investigated in double graphs and in Mycielskian graphs. Sharp general bounds are proved, in particular involving the total and the outer mutual-visibility number of base graphs. Several exact values are also determined, in particular the mutual-visibility number of the double graphs and of the Mycielskian of cycles. • The mutual-visibility number of the double of a graph is bounded below using the total mutual-visibility number. • The general position number of the double of a graph is bounded with the general position number of the graph. • The mutual-visibility number of the Mycielskian over a path and over a cycle is determined. • Bounds for the mutual-visibility number of the Mycielskian over a graph of diameter at most three are proved. [ABSTRACT FROM AUTHOR]

Published
2025
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29. Characterizations of manifolds modeled on absorbing sets in non-separable Hilbert spaces and the discrete cells property

Author

Katsuhisa Koshino

Subjects
Pure mathematics, symbols.namesake, Property (philosophy), Approximation property, General Mathematics, Hilbert space, symbols, Extension (predicate logic), General position, Separable space, Mathematics
Abstract

In this paper, we characterize infinite-dimensional manifolds modeled on absorbing sets in non-separable Hilbert spaces by using the discrete cells property, which is a general position property. Moreover, we study the discrete (locally finite) approximation property, which is an extension of the discrete cells property.

Published
2022
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36. A Ramsey-Type Result for Geometric ℓ-Hypergraphs

Author

Mubayi, Dhruv, Suk, Andrew, Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Doug, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Wismath, Stephen, editor, and Wolff, Alexander, editor

Published
2013
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44. Universal Point Subsets for Planar Graphs

Author

Angelini, Patrizio, Binucci, Carla, Evans, William, Hurtado, Ferran, Liotta, Giuseppe, Mchedlidze, Tamara, Meijer, Henk, Okamoto, Yoshio, Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Doug, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Chao, Kun-Mao, editor, Hsu, Tsan-sheng, editor, and Lee, Der-Tsai, editor

Published
2012
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45. Geometric Graphs in the Plane Lattice

Author

Kano, Mikio, Suzuki, Kazuhiro, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Nierstrasz, Oscar, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Sudan, Madhu, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Vardi, Moshe Y., Series editor, Weikum, Gerhard, Series editor, Márquez, Alberto, editor, Ramos, Pedro, editor, and Urrutia, Jorge, editor

Published
2012
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46. On the Heaviest Increasing or Decreasing Subsequence of a Permutation, and Paths and Matchings on Weighted Point Sets

Author

Sakai, Toshinori, Urrutia, Jorge, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Nierstrasz, Oscar, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Sudan, Madhu, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Vardi, Moshe Y., Series editor, Weikum, Gerhard, Series editor, Márquez, Alberto, editor, Ramos, Pedro, editor, and Urrutia, Jorge, editor

Published
2012
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47. Minimum Convex Partitions and Maximum Empty Polytopes

Author

Dumitrescu, Adrian, Har-Peled, Sariel, Tóth, Csaba D., Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Doug, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Fomin, Fedor V., editor, and Kaski, Petteri, editor

Published
2012
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48. Embedding Plane 3-Trees in ℝ2 and ℝ3

Author

Durocher, Stephane, Mondal, Debajyoti, Nishat, Rahnuma Islam, Rahman, Md. Saidur, Whitesides, Sue, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Nierstrasz, Oscar, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Sudan, Madhu, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Vardi, Moshe Y., Series editor, Weikum, Gerhard, Series editor, van Kreveld, Marc, editor, and Speckmann, Bettina, editor

Published
2012
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50. Cut-and-project graphs and other complexes

Author

Gregory L. McColm

Subjects
Combinatorics, General Computer Science, Projection (mathematics), Dimension (graph theory), Orthogonal complement, Polytope, Disjoint sets, Periodic graph (geometry), General position, Theoretical Computer Science, Complement (set theory), Mathematics
Abstract

The cut-and-project method may be applied to graphs and complexes, although there are technical difficulties, most notably “collisions,” i.e. when the images (under the projection) of two disjoint edges or cells intersect. Given a particular periodic structure, a particular projection space, an appropriate “window” in the orthogonal complement of that space, the induced substructure within the cartesian product of the window and the projection space is projected to the projection space to produce a “model structure.” We may use an index space of vectors from the orthogonal complement to move the window around and obtain a “model system” of model structures. We adapt the notion of “general position” to higher dimensions: the projection space is in “doubly general position” with respect to a graph or complex when the projection of that structure maps vertices of that structure injectively into the projection space and the edges or polytopes of that structure injectively so that their dimension is not reduced and disjoint edges and polytopes remain disjoint. We find that if the initial structure was a periodic graph, if the projection space and its complement are in general position with respect to the vertices and edges, and the window is also in general position, then the model system has uncountably many isomorphism classes - with distinct coordination sequences.

Published
2021
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