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Title:

Injective and non-injective realizations with symmetry

Description:

In this paper, we introduce a natural classification of bar and joint frameworks that possess symmetry. This classification establishes the mathematical foundation for extending a variety of results in rigidity, as well as infinitesimal or static rigidity, to frameworks that are realized with certain symmetries and whose joints may or may not be...

In this paper, we introduce a natural classification of bar and joint frameworks that possess symmetry. This classification establishes the mathematical foundation for extending a variety of results in rigidity, as well as infinitesimal or static rigidity, to frameworks that are realized with certain symmetries and whose joints may or may not be embedded injectively in the space. In particular, we introduce a symmetry-adapted notion of `generic' frameworks with respect to this classification and show that `almost all' realizations in a given symmetry class are generic and all generic realizations in this class share the same infinitesimal rigidity properties. Within this classification we also clarify under what conditions group representation theory techniques can be applied to further analyze the rigidity properties of a (not necessarily injective) symmetric realization. Minimize

Publisher:

Faculty of Science, University of Calgary

Year of Publication:

2010-04-01T00:00:00Z

Document Type:

article

Language:

English

Subjects:

LCC:Mathematics ; LCC:QA1-939 ; LCC:Science ; LCC:Q ; DOAJ:Mathematics ; DOAJ:Mathematics and Statistics ; LCC:Mathematics ; LCC:QA1-939 ; LCC:Science ; LCC:Q ; DOAJ:Mathematics ; DOAJ:Mathematics and Statistics ; LCC:Mathematics ; LCC:QA1-939 ; LCC:Science ; LCC:Q ; LCC:Mathematics ; LCC:QA1-939 ; LCC:Science ; LCC:Q

LCC:Mathematics ; LCC:QA1-939 ; LCC:Science ; LCC:Q ; DOAJ:Mathematics ; DOAJ:Mathematics and Statistics ; LCC:Mathematics ; LCC:QA1-939 ; LCC:Science ; LCC:Q ; DOAJ:Mathematics ; DOAJ:Mathematics and Statistics ; LCC:Mathematics ; LCC:QA1-939 ; LCC:Science ; LCC:Q ; LCC:Mathematics ; LCC:QA1-939 ; LCC:Science ; LCC:Q Minimize

DDC:

531 Classical mechanics; solid mechanics *(computed)*

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http://cdm.math.ucalgary.ca/cdm/index.php/cdm/article/view/167

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Title:

On the area discrepancy of triangulations of squares and trapezoids

Description:

In 1970 P. Monsky showed that a square cannot be triangulated into an odd number of triangles of equal areas; further, in 1990 E. A. Kasimatis and S. K. Stein proved that the trapezoid T(α) whose vertices have the coordinates (0,0), (0,1), (1,0), and (α,1) cannot be triangulated into any number of triangles of equal areas if α> 0 is transcendent...

In 1970 P. Monsky showed that a square cannot be triangulated into an odd number of triangles of equal areas; further, in 1990 E. A. Kasimatis and S. K. Stein proved that the trapezoid T(α) whose vertices have the coordinates (0,0), (0,1), (1,0), and (α,1) cannot be triangulated into any number of triangles of equal areas if α> 0 is transcendental. In this paper we first establish a new asymptotic upper bound for the minimal difference between the smallest and the largest area in triangulations of a square into an odd number of triangles. More precisely, using some techniques from the theory of continued fractions, we construct a sequence of triangulations Tni of the unit square into ni triangles, ni odd, so that the difference between the smallest and the largest area in Tni is O ( 1 n3) i We then prove that for an arbitrarily fast-growing function f: N → N, there exists a transcendental number α> 0 and a sequence of triangulations Tni of the trapezoid T(α) into ni triangles, so that the difference between the smallest and the largest area in Tni is O ( 1 f(ni) Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2013-10-30

Source:

http://www.combinatorics.org/ojs/index.php/eljc/article/viewFile/v18i1p137/pdf/

http://www.combinatorics.org/ojs/index.php/eljc/article/viewFile/v18i1p137/pdf/ Minimize

Document Type:

text

Language:

en

Subjects:

triangulation ; equidissection ; area discrepancy ; square ; trapezoid ; continued

triangulation ; equidissection ; area discrepancy ; square ; trapezoid ; continued Minimize

DDC:

511 General principles of mathematics *(computed)*

Rights:

Metadata may be used without restrictions as long as the oai identifier remains attached to it.

Metadata may be used without restrictions as long as the oai identifier remains attached to it. Minimize

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Title:

Symmetric Laman theorems for the groups C2 and Cs

Description:

For a bar and joint framework (G,p) with point group C3 which describes 3-fold rotational symmetry in the plane, it was recently shown in (Schulze, Discret. Comp. Geom. 44:946-972) that the standard Laman conditions, together with the condition derived in (Connelly et al., Int. J. Solids Struct. 46:762-773) that no vertices are fixed by the auto...

For a bar and joint framework (G,p) with point group C3 which describes 3-fold rotational symmetry in the plane, it was recently shown in (Schulze, Discret. Comp. Geom. 44:946-972) that the standard Laman conditions, together with the condition derived in (Connelly et al., Int. J. Solids Struct. 46:762-773) that no vertices are fixed by the automorphism corresponding to the 3-fold rotation (geometrically, no vertices are placed on the center of rotation), are both necessary and sufficient for (G,p) to be isostatic, provided that its joints are positioned as generically as possible subject to the given symmetry constraints. In this paper we prove the analogous Laman-type conjectures for the groups C2 and Cs which are generated by a half-turn and a reflection in the plane, respectively. In addition, analogously to the results in (Schulze, Discret. Comp. Geom. 44:946-972), we also characterize symmetry generic isostatic graphs for the groups C2 and Cs in terms of inductive Henneberg-type constructions, as well as Crapo-type 3Tree2 partitions- the full sweep of methods used for the simpler problem without symmetry. 1 Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2011-04-29

Source:

http://www.combinatorics.org/Volume_17/PDF/v17i1r154.pdf

http://www.combinatorics.org/Volume_17/PDF/v17i1r154.pdf Minimize

Document Type:

text

Language:

en

DDC:

515 Analysis *(computed)*

Rights:

Metadata may be used without restrictions as long as the oai identifier remains attached to it.

Metadata may be used without restrictions as long as the oai identifier remains attached to it. Minimize

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Title:

Symmetric versions of Laman’s Theorem

Description:

Recent work has shown that if an isostatic bar and joint framework possesses non-trivial symmetries, then it must satisfy some very simply stated restrictions on the number of joints and bars that are ‘fixed ’ by various symmetry operations of the framework. For the group C3 which describes 3-fold rotational symmetry in the plane, we verify the ...

Recent work has shown that if an isostatic bar and joint framework possesses non-trivial symmetries, then it must satisfy some very simply stated restrictions on the number of joints and bars that are ‘fixed ’ by various symmetry operations of the framework. For the group C3 which describes 3-fold rotational symmetry in the plane, we verify the conjecture proposed in [4] that these restrictions on the number of fixed structural components, together with the Laman conditions, are also sufficient for a framework with C3 symmetry to be isostatic, provided that its joints are positioned as generically as possible subject to the given symmetry constraints. In addition, we establish symmetric versions of Henneberg’s Theorem and Crapo’s Theorem for C3 which provide alternate characterizations of ‘generically ’ isostatic graphs with C3 symmetry. As shown in [19], our techniques can be extended to establish analogous results for the symmetry groups C2 and Cs which are generated by a half-turn and a reflection in the plane, respectively. 1 Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2013-08-04

Source:

http://arxiv.org/pdf/0907.1958v1.pdf

http://arxiv.org/pdf/0907.1958v1.pdf Minimize

Document Type:

text

Language:

en

DDC:

531 Classical mechanics; solid mechanics *(computed)*

Rights:

Metadata may be used without restrictions as long as the oai identifier remains attached to it.

Metadata may be used without restrictions as long as the oai identifier remains attached to it. Minimize

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Title:

Combinatorial and Geometric Rigidity . . .

Description:

In this thesis, we investigate the rigidity and flexibility properties of frameworks consisting of rigid bars and flexible joints that possess non-trivial symmetries. Using techniques from group representation theory, we first prove that the rigidity matrix of a symmetric framework can be transformed into a block-diagonalized form. Based on this...

In this thesis, we investigate the rigidity and flexibility properties of frameworks consisting of rigid bars and flexible joints that possess non-trivial symmetries. Using techniques from group representation theory, we first prove that the rigidity matrix of a symmetric framework can be transformed into a block-diagonalized form. Based on this result, we prove a generalization of the symmetry-extended version of Maxwell’s rule given in [25] which can be applied to both injective and non-injective realizations in all dimensions. We then use this rule to prove that a symmetric isostatic (i.e., minimal infinitesimally rigid) framework must obey some very simply stated restrictions on the number of joints and bars that are ‘fixed ’ by various symmetry operations of the framework. In particular, it turns out that a 2-dimensional isostatic framework must belong to one of only six possible point groups. For 3-dimensional isostatic frameworks, all point groups are possible, although restrictions on the placement of structural components still apply. For three of the five non-trivial symmetry groups in dimension 2 that Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-09-11

Source:

http://www.cs.elte.hu/geometry/Workshop09/SchulzePhDthesis.pdf

http://www.cs.elte.hu/geometry/Workshop09/SchulzePhDthesis.pdf Minimize

Document Type:

text

Language:

en

DDC:

531 Classical mechanics; solid mechanics *(computed)*

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Title:

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Description:

Injective and non-injective realizations with symmetry

Injective and non-injective realizations with symmetry Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2013-08-04

Source:

http://arxiv.org/pdf/0808.1761v1.pdf

http://arxiv.org/pdf/0808.1761v1.pdf Minimize

Document Type:

text

Language:

en

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Title:

Block-diagonalised rigidity matrices of symmetric frameworks and applications, Contributions to Algebra and Geometry 51

Description:

In this paper, we give a complete self-contained proof that the rigidity matrix of a symmetric bar and joint framework (as well as its transpose) can be transformed into a block-diagonalized form using techniques from group representation theory. This theorem is basic to a number of useful and interesting results concerning the rigidity and flex...

In this paper, we give a complete self-contained proof that the rigidity matrix of a symmetric bar and joint framework (as well as its transpose) can be transformed into a block-diagonalized form using techniques from group representation theory. This theorem is basic to a number of useful and interesting results concerning the rigidity and flexibility of symmetric frameworks. As an example, we use this theorem to prove a generalization of the symmetry-extended version of Maxwell’s rule given in [9] which can be applied to both injective and non-injective realizations in all dimensions. 1 Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-11-27

Source:

http://arxiv.org/pdf/0906.3377v1.pdf

http://arxiv.org/pdf/0906.3377v1.pdf Minimize

Document Type:

text

Language:

en

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Title:

Mehrfachnutzung von Inhalten als Synergie-Ansatz in der Medienindustrie: Ökonomische und technologische Grundlagen von derzeit bekannten Varianten

Publisher:

Institut für Wirtschaftsinformatik und Neue Medien, Ludwig-Maximilians-Univ. München

Year of Publication:

2003

Document Type:

doc-type:workingPaper

Language:

deu

Subjects:

ddc:650

ddc:650 Minimize

Rights:

http://www.econstor.eu/dspace/Nutzungsbedingungen

http://www.econstor.eu/dspace/Nutzungsbedingungen Minimize

Relations:

Arbeitsbericht, Institut für Wirtschaftsinformatik und Neue Medien, Fakultät für Betriebswirtschaft, Ludwig-Maximilians-Universität 4/2003

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Title:

Block-diagonalized rigidity matrices of symmetric frameworks and applications

Description:

In this paper, we give a complete self-contained proof that the rigidity matrix of a symmetric bar and joint framework (as well as its transpose) can be transformed into a block-diagonalized form using techniques from group representation theory. This theorem is basic to a number of useful and interesting results concerning the rigidity and flex...

In this paper, we give a complete self-contained proof that the rigidity matrix of a symmetric bar and joint framework (as well as its transpose) can be transformed into a block-diagonalized form using techniques from group representation theory. This theorem is basic to a number of useful and interesting results concerning the rigidity and flexibility of symmetric frameworks. As an example, we use this theorem to prove a generalization of the Fowler-Guest symmetry extension of Maxwell's rule which can be applied to both injective and non-injective realizations in all dimensions. ; Comment: 48 pages, 8 figures Minimize

Year of Publication:

2009-06-18

Document Type:

text

Subjects:

Mathematics - Metric Geometry ; Mathematics - Combinatorics ; 52C25 ; 70B99 ; 20C35

Mathematics - Metric Geometry ; Mathematics - Combinatorics ; 52C25 ; 70B99 ; 20C35 Minimize

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Title:

Injective and non-injective realizations with symmetry

Description:

In this paper, we introduce a natural classification of bar and joint frameworks that possess symmetry. This classification establishes the mathematical foundation for extending a variety of results in rigidity, as well as infinitesimal or static rigidity, to frameworks that are realized with certain symmetries and whose joints may or may not be...

In this paper, we introduce a natural classification of bar and joint frameworks that possess symmetry. This classification establishes the mathematical foundation for extending a variety of results in rigidity, as well as infinitesimal or static rigidity, to frameworks that are realized with certain symmetries and whose joints may or may not be embedded injectively in the space. In particular, we introduce a symmetry-adapted notion of `generic' frameworks with respect to this classification and show that `almost all' realizations in a given symmetry class are generic and all generic realizations in this class share the same infinitesimal rigidity properties. Within this classification we also clarify under what conditions group representation theory techniques can be applied to further analyze the rigidity properties of a (not necessarily injective) symmetric realization. ; Comment: 37 pages, 13 figures Minimize

Year of Publication:

2008-08-13

Document Type:

text

Subjects:

Mathematics - Metric Geometry ; Mathematics - Combinatorics ; 52C25 ; 70B99 ; 05C99

Mathematics - Metric Geometry ; Mathematics - Combinatorics ; 52C25 ; 70B99 ; 05C99 Minimize

DDC:

531 Classical mechanics; solid mechanics *(computed)*

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