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### Affine Independence in Vector Spaces

Karol Pąk
<span title="2010-01-01">2010</span> <i title="Walter de Gruyter GmbH"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/cn54gai6x5addllfxdh4ow6r54" style="color: black;">Formalized Mathematics</a> </i> &nbsp;
Affine Independence in Vector Spaces In this article we describe the notion of affinely independent subset of a real linear space.  ...  We also introduce the notion of the affine hull, i.e. a subset generated by a set of vectors which is an intersection of all affine sets including the given set.  ...  Affine Independence of Vectors Let us consider V , A.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2478/v10037-010-0012-z">doi:10.2478/v10037-010-0012-z</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/kldgpodazrbxjbaw3diwwc6zai">fatcat:kldgpodazrbxjbaw3diwwc6zai</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190501233157/https://content.sciendo.com/downloadpdf/journals/forma/18/1/article-p87.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/2b/aa/2baa52916005170f7e312bf7fb04490db99d3731.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2478/v10037-010-0012-z"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

### A note on proper affine symmetry in Kantowski-Sachs and Bianchi type III space-times [article]

Ghulam Shabbir, Nisar Ahmed
<span title="2016-01-01">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
It is shown that the very special classes of the above space-times admit proper affine vector fields.  ...  We investigate proper affine symmetry for the Kantowski-Sachs and Bianchi type III space-times by using holonomy and decomposability, the rank of the 6X6 Riemann matrix and direct integration techniques  ...  Hence affine vector fields in the above space-time (29) are Killing vector fields which are given in (6) .  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1601.00055v1">arXiv:1601.00055v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qramh7conbhl7hvizaw5uq5f64">fatcat:qramh7conbhl7hvizaw5uq5f64</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200930043253/https://arxiv.org/ftp/arxiv/papers/1601/1601.00055.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/d3/f0/d3f07b6ccc811e09ac5ff62b626c910a90fd3eef.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1601.00055v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

### Frame-independent mechanics:geometry on affine bundles [article]

K. Grabowska, J. Grabowski, P. Urbanski
<span title="2005-10-06">2005</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The developed concepts are applied in a frame-independent formulation of the time-dependent and the Newtonian mechanics.  ...  Main ideas of the differential geometry on affine bundles are presented. Affine counterparts of Lie algebroid and Poisson structures are introduced and discussed.  ...  These results are applied in a frame-independent formulation of the time-dependent and the Newtonian mechanics (section 5). 2 The affine geometry 2.1 Affine spaces and affine bundles We start with a short  ...
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### Linear Transformations of Euclidean Topological Spaces. Part II

Karol Pąk
<span title="2011-01-01">2011</span> <i title="Walter de Gruyter GmbH"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/cn54gai6x5addllfxdh4ow6r54" style="color: black;">Formalized Mathematics</a> </i> &nbsp;
Linear Transformations of Euclidean Topological Spaces. Part II We prove a number of theorems concerning various notions used in the theory of continuity of barycentric coordinates.  ...  Correspondence Between the Mx2Tran Operator and Decomposition of a Vector in Basis The following propositions are true: Preservation of Linear and Affine Independence of Vectors by the Mx2Tran Operator  ...  ( 24 ) 24 For every affinely independent subset A of E n T such that rk(M ) = n holds (Mx2Tran M ) • A is affinely independent.(25) Let A be an affinely independent subset of E n T .  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2478/v10037-011-0017-2">doi:10.2478/v10037-011-0017-2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/t7dwqssqerbedozdhd4evauuyy">fatcat:t7dwqssqerbedozdhd4evauuyy</a> </span>
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### Vector geometry for computer graphics

J.R. Miller
<span title="">1999</span> <i title="Institute of Electrical and Electronics Engineers (IEEE)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/mst7benysjfpbf6tk2lttvwoaq" style="color: black;">IEEE Computer Graphics and Applications</a> </i> &nbsp;
This tutorial reviews the basic ideas surrounding points and vectors in affine and projective spaces.  ...  In Part 1 of this tutorial I advocate vector geometric analysis to simplify required derivations and implementations.  ...  Acknowledgments Ron Goldman first introduced me to the power and simplicity of vector geometric approaches while we were working together in the early 1980s.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/38.761552">doi:10.1109/38.761552</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/eutve4ggnzgrlprrqvpoqkd43y">fatcat:eutve4ggnzgrlprrqvpoqkd43y</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20040708105250/http://come.ssu.ac.kr:80/~ieee/cga/cg1999/pdf/g3066.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/92/c7/92c7f5db00e0fcec5c16dfda580866110b5eb06a.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/38.761552"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> ieee.com </button> </a>

### Frame-independent formulation of Newtonian mechanics [article]

Katarzyna Grabowska, Pawel Urbanski
<span title="2004-04-29">2004</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
A frame independent formulation of analytical mechanics in the Newtonian space-time is presented.  ...  The differential geometry of affine values i.e., the differential geometry in which affine bundles replace vector bundles and sections of one dimensional affine bundles replace functions on manifolds,  ...  An affine space modelled on a special vector space will be called a special affine space.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math-ph/0404073v1">arXiv:math-ph/0404073v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/kxlsb7xnofgtben6pj32mkuye4">fatcat:kxlsb7xnofgtben6pj32mkuye4</a> </span>
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### Designs and codes in affine geometry [article]

Jens Zumbrägel
<span title="2016-05-23">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Classical designs and their (projective) q-analogs can both be viewed as designs in matroids, using the matroid of all subsets of a set and the matroid of linearly independent subsets of a vector space  ...  Accordingly, a t-(n, k, λ) affine design of order q is a collection B of (k-1)-dimensional spaces in the affine geometry A = AG(n-1, q) such that each (t-1)-dimensional space in A is contained in exactly  ...  Finally, if the ground field of the vector space V is F 2 , then any three points are affine independent, i.e., they span a two-dimensional affine space (a plane).  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1605.03789v2">arXiv:1605.03789v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/s4xog2np3vgczpauzhh3da5pci">fatcat:s4xog2np3vgczpauzhh3da5pci</a> </span>
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### A note on proper affine symmetry in Bianchi types VIII and IX space-times [article]

<span title="2017-01-09">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Studying proper affine symmetry in the above space-times it is shown that there exists only one possibilty when the above space-times admit proper affine vector field.  ...  We consider the most general form of Bianchi types VIII and IX space-times for studying proper affine symmetry by using holonomy and decomposability, the rank of the 6X6 Riemann matrix and direct integration  ...  A study of the affine vector fields for the above holonomy type can be found in [3] .  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1701.02217v1">arXiv:1701.02217v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ztbohgqv3zahrfg6bxcy5b3txy">fatcat:ztbohgqv3zahrfg6bxcy5b3txy</a> </span>
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### On the singular point locus in the theory of fields of parallel vectors

T. Y. Thomas
<span title="1939-06-01">1939</span> <i title="American Mathematical Society (AMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/hjoli2j6qffdpaalkszryuidk4" style="color: black;">Bulletin of the American Mathematical Society</a> </i> &nbsp;
A satisfactory answer to the above question is contained in a general theory of the parallel displacement of vectors in an afiinely connected space by Mayer and Thomas, Fields of parallel vectors in nonanalytic  ...  Then any open, connected, and simply connected point set O c K m admits exactly m independent fields of parallel vectors.  ...  Taking m independent solution vectors £Q X » • * * > ?Q m at Qi we thus obtain the existence of m independent fields of parallel vectors in O as above stated.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/s0002-9904-1939-07001-8">doi:10.1090/s0002-9904-1939-07001-8</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/xrnjkew3kbfyzcqknzfibvu22y">fatcat:xrnjkew3kbfyzcqknzfibvu22y</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170828030823/http://www.ams.org/journals/bull/1939-45-06/S0002-9904-1939-07001-8/S0002-9904-1939-07001-8.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/48/af/48af2318fa2cea8c010d0aa67d8c781bdf29e4bf.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/s0002-9904-1939-07001-8"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

### Spaces admitting complete absolute parallelism

L. P. Eisenhart
<span title="1933-04-01">1933</span> <i title="American Mathematical Society (AMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/hjoli2j6qffdpaalkszryuidk4" style="color: black;">Bulletin of the American Mathematical Society</a> </i> &nbsp;
If in the case of a space of zero curvature we denote by X a the components of n independent fields of parallel vectors, i indicating the component and a the vector, we have i (13) -~ + \ 2 a L jk =  ...  Consequently a necessary and sufficient condition that there be n independent fields of parallel vectors is that h Lijk -0; in this case we say that the space admits complete absolute parallelism, and  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/s0002-9904-1933-05593-3">doi:10.1090/s0002-9904-1933-05593-3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/nk36vyowmnc73g7vlaak2qtbde">fatcat:nk36vyowmnc73g7vlaak2qtbde</a> </span>
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### Note on affinely connected manifolds

Shiing-Shen Chern
<span title="1947-08-01">1947</span> <i title="American Mathematical Society (AMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/hjoli2j6qffdpaalkszryuidk4" style="color: black;">Bulletin of the American Mathematical Society</a> </i> &nbsp;
The integral of P over the cycle can now be interpreted as the integral of P over this field of independent contravariant vectors in the space of all sets of n independent contravariant vectors of the  ...  ) form a topological space which is the group space of the group of linear transformations in n variables with positive (or negative) determinant and is hence simply-connected. 3 As the sets of vectors  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/s0002-9904-1947-08898-4">doi:10.1090/s0002-9904-1947-08898-4</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/oelxs3pwmbhidjbqsya6avr3cy">fatcat:oelxs3pwmbhidjbqsya6avr3cy</a> </span>
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### Multibody motion segmentation for an arbitrary number of independent motions

Yutaro Sako, Yasuyuki Sugaya
<span title="2016-08-02">2016</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/tgkeoqpazzdvvd3eofpe7sjgwa" style="color: black;">IPSJ Transactions on Computer Vision and Applications</a> </i> &nbsp;
Our method realizes motion segmentation of feature point trajectories by hierarchically separating the trajectories into two affine spaces in a situation that we do not know the number of independently  ...  We propose a new method for segmenting feature point trajectories tracked through a video sequence without assuming a number of independent motions.  ...  (c) Convert the trajectory vectors into d -D vectors. (d) Separate the d -D vectors into two affine spaces by the EM algorithm of Sugaya and Kanatani [3, 4] .  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1186/s41074-016-0002-3">doi:10.1186/s41074-016-0002-3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/xhgdmvzrjncsxcvvmesv3xlgwm">fatcat:xhgdmvzrjncsxcvvmesv3xlgwm</a> </span>
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### Page 135 of Mathematical Reviews Vol. 26, Issue 1 [page]

<span title="">1963</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
emphasized in these papers) of linearly independent contravariant [covariant] vectors.  ...  As to covariant vector concomitants of contravariant vector fields (or inversely), they are nontrivial (nonconstant) only if the number of independent vectors equals the dimension of the space and then  ...
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### Affine distributions on Riemannian manifolds with applications to dissipative dynamics

Răzvan M. Tudoran
<span title="">2015</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/hdohoyh3hnahniyrjuyhvg6c4u" style="color: black;">Journal of Geometry and Physics</a> </i> &nbsp;
In the case when X_1,..., X_k, Y_1, ..., Y_p are gradient vector fields associated with some smooth functions I_1,..., I_k, D_1, ..., D_p ∈C^∞(U,R), i.e., X_1 =∇_gI_1, ..., X_k = ∇_gI_k, Y_1 = ∇_gD_1,  ...  Using a coordinate free characterization of hyperplanes intersection, we provide explicitly a set of local generators for a smooth affine distribution given by those smooth vector fields X∈X(U) defined  ...  n for the vector space Λ n E.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.geomphys.2015.01.017">doi:10.1016/j.geomphys.2015.01.017</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/lu5fehw7xzgu3l3fjzwkm6yune">fatcat:lu5fehw7xzgu3l3fjzwkm6yune</a> </span>
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### Unique decompositions, faces, and automorphisms of separable states

Erik Alfsen, Fred Shultz
<span title="">2010</span> <i title="AIP Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/nwwheyjjk5hnlmj6q5mtpqmmhi" style="color: black;">Journal of Mathematical Physics</a> </i> &nbsp;
In both cases we describe the associated faces of the space of separable states, which in the first case are simplexes, and in the second case are direct convex sums of faces that are isomorphic to state  ...  As an application of these results, we characterize all affine automorphisms of the convex set of separable states, and all automorphisms of the state space of B(C^m otimes C^n). that preserve entanglement  ...  ,L q are independent ͑i.e., that L 1 + L 2 +¯L q is a vector space direct sum͒. For 1 ഛ i ഛ q let e i g i be a nonzero vector in L i .  ...
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