400 results on '"Affine combination"'
Search Results
2. Algebraic Clustering of Affine Subspaces
- Author
-
René Vidal and Manolis C. Tsakiris
- Subjects
FOS: Computer and information sciences ,Computer Vision and Pattern Recognition (cs.CV) ,Computer Science - Computer Vision and Pattern Recognition ,02 engineering and technology ,Affine geometry ,Affine combination ,Artificial Intelligence ,Affine hull ,0202 electrical engineering, electronic engineering, information engineering ,Mathematics ,Discrete mathematics ,business.industry ,Applied Mathematics ,020206 networking & telecommunications ,Affine plane ,Affine shape adaptation ,Algebra ,Affine coordinate system ,ComputingMethodologies_PATTERNRECOGNITION ,Computational Theory and Mathematics ,Affine space ,020201 artificial intelligence & image processing ,Computer Vision and Pattern Recognition ,Affine transformation ,Artificial intelligence ,business ,Software - Abstract
Subspace clustering is an important problem in machine learning with many applications in computer vision and pattern recognition. Prior work has studied this problem using algebraic, iterative, statistical, low-rank and sparse representation techniques. While these methods have been applied to both linear and affine subspaces, theoretical results have only been established in the case of linear subspaces. For example, algebraic subspace clustering (ASC) is guaranteed to provide the correct clustering when the data points are in general position and the union of subspaces is transversal . In this paper we study in a rigorous fashion the properties of ASC in the case of affine subspaces. Using notions from algebraic geometry, we prove that the homogenization trick , which embeds points in a union of affine subspaces into points in a union of linear subspaces, preserves the general position of the points and the transversality of the union of subspaces in the embedded space, thus establishing the correctness of ASC for affine subspaces.
- Published
- 2018
3. On Para-Complex Affine Hyperspheres
- Author
-
Zuzanna Szancer
- Subjects
010308 nuclear & particles physics ,Applied Mathematics ,Nuclear Theory ,010102 general mathematics ,01 natural sciences ,Affine plane ,Combinatorics ,Affine geometry ,Affine coordinate system ,Affine shape adaptation ,Mathematics (miscellaneous) ,Affine combination ,Affine hull ,0103 physical sciences ,Affine group ,Physics::Atomic and Molecular Clusters ,Physics::Atomic Physics ,Mathematics::Differential Geometry ,Affine transformation ,0101 mathematics ,Mathematics - Abstract
In this paper we introduce a notion of a para-complex affine hypersphere. We give a complete local classification of such hypersurfaces and give several examples. It turns out that every para-complex affine hypersphere can be constructed from (real) affine hyperspheres. As an application, we classify all 2-dimensional para-complex affine hyperspheres.
- Published
- 2017
4. Affine Bessel sequences and Nikishin’s example
- Author
-
Pavel Aleksandrovich Terekhin, Abdizhahan Manapoly Sarsenbi, and V. A. Mironov
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,010101 applied mathematics ,Affine coordinate system ,Affine geometry ,Affine combination ,Affine geometry of curves ,Affine hull ,Affine group ,Affine space ,Affine transformation ,0101 mathematics ,Algorithm ,Mathematics - Abstract
We study affine Bessel sequences in connection with the spectral theory and the multishift structure in Hilbert space. We construct a non-Besselian affine system fun(x)g1 n=0 generated by continuous periodic function u(x). The result is based on Nikishin?s example concerning convergence in measure. We also show that affine systems fun(x)g1 n=0 generated by any Lipchitz function u(x) are Besselian.
- Published
- 2017
5. Estimating affine-invariant structures on triangle meshes
- Author
-
Thales Vieira, Maria Gorete Carreira Andrade, Dimas Martínez, and Thomas Lewiner
- Subjects
Discrete mathematics ,Pure mathematics ,General Engineering ,020207 software engineering ,02 engineering and technology ,Computer Graphics and Computer-Aided Design ,Affine plane ,Human-Computer Interaction ,Affine coordinate system ,Affine geometry ,Affine shape adaptation ,Affine combination ,Affine hull ,Affine group ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,Affine transformation ,Mathematics - Abstract
Affine invariant measures are powerful tools to develop robust shape descriptors that can be applied, for example, to shape matching, shape retrieval, or symmetry detection problems. In this work we introduce estimators for the affine structure of surfaces represented by triangle meshes, i.e. affine co-normal and normal vectors, affine curvature tensors, affine mean and Gaussian curvatures, and affine principal directions and curvatures. The proposed method estimates the affine normal using a finite differences scheme together with a least-squares approximation, followed by a weighted average strategy to approach discrete affine curvature tensors. When compared to the exact geometric measures of analytic models, experiments on regular meshes obtain small error, which decreases for finer meshes, and outperforms the state-of-the-art method in some cases. Experiments to evaluate affine invariance show that the difference between measures before and after equi-affine transformations remains small even after large deformations.
- Published
- 2016
6. Linear Algebra
- Author
-
Edoardo Sernesi
- Subjects
Affine geometry ,Pure mathematics ,Affine involution ,Affine combination ,Affine representation ,Complex space ,Mathematical analysis ,Affine group ,Affine space ,Affine transformation ,Mathematics - Abstract
Part I Affine geometry: vector spaces matrices systems of linear equations some linear algebra rank determinants affine space - (I) - (II) geometry of affine planes geometry of affine space linear maps linear maps and matrices, affine changes of coordinates linear operators transformation groups. Part II Euclidean geometry: bilinear and quadratic forms diagonalizing quadratic forms scalar product vector product Euclidean space unitary operators and isometries isometries of the plane and of three-dimensional space the complex case.
- Published
- 2019
7. On affine translation surfaces in affine space
- Author
-
Dan Yang and Yu Fu
- Subjects
Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,0102 computer and information sciences ,01 natural sciences ,Affine plane ,Affine geometry ,Affine coordinate system ,Affine shape adaptation ,Affine combination ,010201 computation theory & mathematics ,Affine hull ,Affine group ,Affine transformation ,0101 mathematics ,Analysis ,Mathematics - Abstract
In this work we give a systemic study of affine translation surfaces in affine 3-dimensional space. Specifically, we obtain the complete classification of minimal affine translation surfaces. Moreover, we consider affine translation surfaces with some natural geometric conditions, such as constant affine mean curvature and constant Gauss–Kronecker curvature. Some characterization results with these geometric conditions are also obtained.
- Published
- 2016
8. Affine realizations with affine state processes for stochastic partial differential equations
- Author
-
Stefan Tappe
- Subjects
Statistics and Probability ,Applied Mathematics ,Probability (math.PR) ,010102 general mathematics ,Mathematical analysis ,Mathematical Finance (q-fin.MF) ,01 natural sciences ,FOS: Economics and business ,Affine shape adaptation ,Affine geometry ,Affine coordinate system ,010104 statistics & probability ,Affine combination ,Affine geometry of curves ,Quantitative Finance - Mathematical Finance ,Modeling and Simulation ,Affine hull ,Affine group ,FOS: Mathematics ,Applied mathematics ,Affine transformation ,0101 mathematics ,Mathematics - Probability ,60H15, 91G80 ,Mathematics - Abstract
The goal of this paper is to clarify when a stochastic partial differential equation with an affine realization admits affine state processes. This includes a characterization of the set of initial points of the realization. Several examples, as the HJMM equation from mathematical finance, illustrate our results., 27 pages
- Published
- 2016
9. On the Nonlinearity and Affine Equivalence Classes of C-F Functions
- Author
-
Fang-Wei Fu, Lei Sun, and Xuang Guang
- Subjects
Discrete mathematics ,Pure mathematics ,Function field of an algebraic variety ,Applied Mathematics ,020206 networking & telecommunications ,Dimension of an algebraic variety ,0102 computer and information sciences ,02 engineering and technology ,Congruence relation ,01 natural sciences ,Computer Graphics and Computer-Aided Design ,Affine geometry ,Affine combination ,010201 computation theory & mathematics ,Affine hull ,Signal Processing ,0202 electrical engineering, electronic engineering, information engineering ,Affine space ,Electrical and Electronic Engineering ,Affine variety ,Mathematics - Published
- 2016
10. Affine Eikonal, Wavization and Wigner Function
- Author
-
Akihiro Ogura
- Subjects
010308 nuclear & particles physics ,05 social sciences ,01 natural sciences ,Affine shape adaptation ,Affine coordinate system ,Affine geometry ,Affine combination ,Affine geometry of curves ,Quantum mechanics ,0502 economics and business ,0103 physical sciences ,Affine group ,Wigner distribution function ,Affine transformation ,050203 business & management ,Mathematical physics ,Mathematics - Abstract
The aim in this paper is to construct an affine transformation using the classical physics analogy between the fields of optics and mechanics. Since optics and mechanics both have symplectic structures, the concept of optics can be replaced by that of mechanics and vice versa. We list the four types of eikonal (generating functions). We also introduce a unitary operator for the affine transformation. Using the unitary operator, the kernel (propagator) is calculated and the wavization (quantization) of the Gabor function is discussed. The dynamic properties of the affine transformed Wigner function are also discussed.
- Published
- 2016
11. Maxwell-affine gauge theory of gravity
- Author
-
Salih Kibaroğlu and Oktay Cebecioğlu
- Subjects
Physics ,High Energy Physics - Theory ,Nuclear and High Energy Physics ,FOS: Physical sciences ,Affine plane ,lcsh:QC1-999 ,Affine coordinate system ,Affine geometry ,Affine combination ,High Energy Physics - Theory (hep-th) ,Affine geometry of curves ,Affine curvature ,Affine group ,Affine transformation ,lcsh:Physics ,Mathematical physics - Abstract
The Maxwell extension of the affine algebra in four dimensions with additional tensor generator is given. Using the methods of nonlinear realizations, we find the transformation rules for the group parameters and the corresponding generators. Gauging the Maxwell-affine algebra we present two possible invariant actions for gravity: one is first order and the other one is second order in the affine curvature. We notice that equations of motion for the action, second order in the affine curvature, lead to the generalized Bianchi identities on the choice of appropriate coefficients for a particular solution of the constraint equation.
- Published
- 2015
- Full Text
- View/download PDF
12. Affine cellularity of BLN algebras
- Author
-
Weideng Cui
- Subjects
Affine geometry ,Affine coordinate system ,Discrete mathematics ,Pure mathematics ,Quantum affine algebra ,Algebra and Number Theory ,Affine combination ,Affine representation ,Affine hull ,Affine group ,Affine plane ,Mathematics - Abstract
We show that the BLN algebra, which was introduced by McGerty, is affine cellular in the sense of Koenig and Xi. In fact, we establish a stronger property, namely that the affine cell ideals are generated by idempotents. This particularly implies that the global dimension of the BLN algebra is finite. For affine type A, we obtain that the affine q-Schur algebra U D , n , n , when D n , is affine cellular and has finite global dimension.
- Published
- 2015
13. The generalized L p -mixed affine surface area
- Author
-
Tong Yi Ma
- Subjects
Combinatorics ,Affine geometry ,Affine combination ,Affine involution ,Affine geometry of curves ,Affine representation ,Applied Mathematics ,General Mathematics ,Affine hull ,Affine group ,Mathematical analysis ,Affine space ,Mathematics - Abstract
In this article, we put forward the concept of the (i, j)-type L p -mixed affine surface area, such that the notion of L p -affine surface area which be shown by Lutwak is its special cases. Furthermore, applying this concept, the Minkowski inequality for the (i,−p)-type L p -mixed affine surface area and the extensions of the well-known L p -Petty affine projection inequality are established, respectively. Besides, we give an affirmative answer for the generalized L p -Winterniz monotonicity problem.
- Published
- 2015
14. Stationary and nonstationary affine combination of subdivision masks
- Author
-
Conti, Costanza
- Subjects
- *
AFFINE geometry , *SUBDIVISION surfaces (Geometry) , *STATIONARY processes , *SIGNS & symbols , *MATHEMATICAL analysis , *VARIATE difference method - Abstract
Abstract: One of the difficult task in subdivision is to create new effective subdivision schemes. Therefore, aim of this paper is a systematic analysis of affine combination of known subdivision masks to generate new subdivision schemes with enhanced properties. This will be done in the stationary and the non stationary case for the univariate and bivariate settings. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
15. Families of exotic affine 3-spheres
- Author
-
Adrien Dubouloz, Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS ), ANR-11-JS01-0004,BirPol,Automorphismes Polynomiaux et Transformations Birationnelles ( 2011 ), Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), and ANR-11-JS01-0004,BirPol,Automorphismes Polynomiaux et Transformations Birationnelles(2011)
- Subjects
General Mathematics ,Kodaira dimension ,Topologically contractible surfaces ,01 natural sciences ,Affine geometry ,Combinatorics ,Mathematics - Algebraic Geometry ,Affine combination ,Affine representation ,Affine hull ,0103 physical sciences ,Affine group ,FOS: Mathematics ,0101 mathematics ,Algebraic Geometry (math.AG) ,MSC: 14R05, 14R25, 14J10 ,Mathematics ,Exotic affine spheres ,010102 general mathematics ,Affine plane ,[ MATH.MATH-AG ] Mathematics [math]/Algebraic Geometry [math.AG] ,Affine coordinate system ,Serre construction ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,010307 mathematical physics ,Affine transformation - Abstract
International audience; We construct algebraic families of exotic affine 3-spheres, that is, smooth affine threefolds diffeomorphic to a non-degenerate smooth complex affine quadric of dimension 3 but non-algebraically isomorphic to it. We show in particular that for every smooth topologically contractible affine surface $S$ with trivial automorphism group, there exists a canonical smooth family of pairwise non-isomorphic exotic affine 3-spheres parametrized by the closed points of $S$.
- Published
- 2018
16. Affine Algebraic Geometry: Simple Points
- Author
-
Peter Falb
- Subjects
Affine geometry ,Discrete mathematics ,Pure mathematics ,Function field of an algebraic variety ,Affine combination ,Affine geometry of curves ,Real algebraic geometry ,Dimension of an algebraic variety ,Affine transformation ,Affine plane ,Mathematics - Published
- 2018
17. Characterization of the generalized Calabi composition of affine hyperspheres
- Author
-
Ce Ce Li, Ze Jun Hu, Miroslava Antić, Luc Vrancken, Laboratoire de Mathématiques et leurs Applications de Valenciennes - EA 4015 (LAMAV), and Centre National de la Recherche Scientifique (CNRS)-Université de Valenciennes et du Hainaut-Cambrésis (UVHC)-INSA Institut National des Sciences Appliquées Hauts-de-France (INSA Hauts-De-France)
- Subjects
Generalized Calabi composition ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,affine hyperspheres ,01 natural sciences ,010101 applied mathematics ,Affine coordinate system ,Combinatorics ,Affine geometry ,Affine combination ,Hypersurface ,Affine geometry of curves ,Affine hull ,Affine group ,Affine transformation ,[MATH]Mathematics [math] ,0101 mathematics ,warped product ,ComputingMilieux_MISCELLANEOUS ,Mathematics - Abstract
© 2015, Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg. In this paper, continuing with Hu–Li–Vrancken and the recent work of Antić–Dillen- Schoels–Vrancken, we obtain a decomposition theorem which settled the problem of how to determine whether a given locally strongly convex affine hypersurface can be decomposed as a generalized Calabi composition of two affine hyperspheres, based on the properties of its difference tensor K and its affine shape operator S. ispartof: Acta Mathematica Sinica vol:31 issue:10 pages:1531-1554 status: published
- Published
- 2015
18. Compact noncontraction semigroups of affine operators
- Author
-
V. Yu. Protasov and Andrey Voynov
- Subjects
Discrete mathematics ,Algebra and Number Theory ,Semigroup ,Affine operator ,Partition ,Primitive matrix ,Self-similarity ,Spectral radius ,Affine geometry ,Affine coordinate system ,Affine combination ,Affine representation ,Affine hull ,Affine group ,Affine space ,Mathematics - Abstract
We analyze compact multiplicative semigroups of affine operators acting in a finite-dimensional space. The main result states that every such semigroup is either contracting, that is, contains elements of arbitrarily small operator norm, or all its operators share a common invariant affine subspace on which this semigroup is contracting. The proof uses functional difference equations with contraction of the argument. We look at applications to self-affine partitions of convex sets, the investigation of finite affine semigroups and the proof of a criterion of primitivity for nonnegative matrix families. Bibliography: 32 titles.
- Published
- 2015
19. Metric corrections of the affine camera
- Author
-
Adrien Bartoli, Toby Collins, and Daniel Pizarro
- Subjects
ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION ,Topology ,Affine plane ,Affine geometry ,Affine shape adaptation ,Affine coordinate system ,Affine combination ,Camera auto-calibration ,Affine hull ,Signal Processing ,Computer Vision and Pattern Recognition ,Affine transformation ,Algorithm ,Software ,Mathematics - Abstract
A specific study of the orthographic, weak-perspective and paraperspective cameras.An algebraic procedure solving the affine correction problem for each camera model.A closed-form solution for each camera model.A full characterization of the affine corrections' generic ambiguities for each camera model.An experimental evaluation comparing the algebraic procedures to global polynomial optimization and an interior-point method. Given a general affine camera, we study the problem of finding the closest metric affine camera, where the latter is one of the orthographic, weak-perspective and paraperspective projection models. This problem typically arises in stratified Structure-from-Motion methods such as factorization-based methods. For each type of metric affine camera, we give a closed-form solution and its implementation through an algebraic procedure. Using our algebraic procedure, we can then provide a complete analysis of the problem's generic ambiguity space. This also gives the means to generate the other solutions if any.
- Published
- 2015
20. Affine Invariant Distance Using Multiscale Analysis
- Author
-
Luis Alvarez, Julio Esclarín, Luis Mazorra, Carmelo Cuenca, and Jean-Michel Morel
- Subjects
Statistics and Probability ,Applied Mathematics ,Mathematical analysis ,02 engineering and technology ,Condensed Matter Physics ,Topology ,01 natural sciences ,Affine plane ,010101 applied mathematics ,Affine shape adaptation ,Affine coordinate system ,Affine geometry ,Affine combination ,Modeling and Simulation ,Affine hull ,Affine group ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,Geometry and Topology ,Computer Vision and Pattern Recognition ,Affine transformation ,0101 mathematics ,Mathematics - Abstract
In this paper we introduce an affine invariant distance definition from a $$2D$$2D point to the boundary of a bounded shape using morphological multiscale analysis. We study the mathematical behavior of this distance by examining separately the cases of convex and non-convex shapes. We prove that the proposed distance is bounded in the convex hull of the shape and infinite otherwise. A numerical scheme is given as well as experiments illustrating the behavior of the affine invariant distance.
- Published
- 2015
21. GLOBAL DIFFERENTIAL INVARIANTS OF AFFINE CURVES IN R^2
- Author
-
Yasemin Sağiroğlu
- Subjects
Affine geometry ,Affine coordinate system ,Pure mathematics ,Affine combination ,Affine geometry of curves ,General Mathematics ,Affine hull ,Affine group ,Affine transformation ,Affine plane ,Mathematics - Published
- 2015
22. Orlicz mixed affine quermassintegrals
- Author
-
Du Zou, Deyi Li, and Ge Xiong
- Subjects
Mathematics::Functional Analysis ,Pure mathematics ,General Mathematics ,Mathematical analysis ,Regular polygon ,Integral geometry ,Affine geometry ,Affine combination ,Affine geometry of curves ,Affine hull ,Affine group ,Mathematics::Metric Geometry ,Affine transformation ,Mathematics - Abstract
A class of geometric quantities for convex bodies is introduced in the framework of Orlicz Brunn-Minkowski theory. It is shown that these new geometric quantities are affine invariant and precisely the generalizations of classical affine quermassintegrals.
- Published
- 2015
23. Affine hypersurfaces with parallel difference tensor relative to affineα-connection
- Author
-
Cece Li
- Subjects
Pure mathematics ,Mathematical analysis ,General Physics and Astronomy ,Affine coordinate system ,Affine geometry ,Affine combination ,Affine geometry of curves ,Affine representation ,Affine hull ,Affine group ,Affine space ,Mathematics::Differential Geometry ,Geometry and Topology ,Mathematical Physics ,Mathematics - Abstract
Li and Zhang (2014) studied affine hypersurfaces of R n + 1 with parallel difference tensor relative to the affine α -connection ∇ ( α ) , and characterized the generalized Cayley hypersurfaces by K n − 1 ≠ 0 and ∇ ( α ) K = 0 for some nonzero constant α , where the affine α -connection ∇ ( α ) of information geometry was introduced on affine hypersurface. In this paper, by a slightly different method we continue to study affine hypersurfaces with ∇ ( α ) K = 0 , if α = 0 we further assume that the Pick invariant vanishes and affine metric is of constant sectional curvature. It is proved that they are either hyperquadrics or improper affine hypersphere with flat indefinite affine metric, the latter can be locally given as a graph of a polynomial of at most degree n + 1 with constant Hessian determinant. In particular, if the affine metric is definite, Lorentzian, or its negative index is 2, we complete the classification of such hypersurfaces.
- Published
- 2014
24. Weak (quasi-)affine bi-frames for reducing subspaces of L 2(ℝ d )
- Author
-
HuiFang Jia and Yun-Zhang Li
- Subjects
Affine geometry ,Affine coordinate system ,Pure mathematics ,Affine combination ,Affine representation ,General Mathematics ,Affine hull ,Affine group ,Affine space ,Arithmetic ,Affine plane ,Mathematics - Abstract
Since a frame for a Hilbert space must be a Bessel sequence, many results on (quasi-)affine bi-frame are established under the premise that the corresponding (quasi-)affine systems are Bessel sequences. However, it is very technical to construct a (quasi-)affine Bessel sequence. Motivated by this observation, in this paper we introduce the notion of weak (quasi-)affine bi-frame (W(Q)ABF) in a general reducing subspace for which the Bessel sequence hypothesis is not needed. We obtain a characterization of WABF, and prove the equivalence between WABF and WQABF under a mild condition. This characterization is used to recover some related known results in the literature.
- Published
- 2014
25. Blaschke hypersurfaces with constant negative affine mean curvature
- Author
-
An-Min Li, Li Sheng, and Udo Simon
- Subjects
Pure mathematics ,Mathematics::Complex Variables ,Mathematical analysis ,Affine plane ,Affine geometry ,Affine coordinate system ,Affine combination ,Affine geometry of curves ,Affine representation ,Affine hull ,Affine group ,Mathematics::Differential Geometry ,Geometry and Topology ,Analysis ,Mathematics - Abstract
We consider affine-complete Blaschke hypersurfaces with constant negative affine mean curvature; for a subclass we assume appropriate bounds for the affine shape operator and for the affine support function; we investigate whether in this subclass of hypersurfaces there exist examples that are not hyperbolic affine spheres. Examples of Calabi type compositions, given by Dillen and Vrancken, admit to test the assumptions of our Main Theorem on this subclass. The study of Blaschke hypersurfaces with negative affine mean curvature finally leads to investigations of the scalar curvature: we treat the scalar curvature of affine spheres in the last section.
- Published
- 2014
26. Lie Algebras
- Author
-
Willem Adriaan de Graaf
- Subjects
Affine geometry ,Affine coordinate system ,Discrete mathematics ,Pure mathematics ,Affine combination ,Complex space ,Affine hull ,Affine group ,Affine space ,Affine plane ,Mathematics - Published
- 2017
27. A natural linear equation in affine geometry: the affine quasi-Einstein equation
- Author
-
Peter B. Gilkey, Eduardo García-Río, Xabier Valle-Regueiro, and Miguel Brozos-Vázquez
- Subjects
Physics ,Mathematics - Differential Geometry ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Affine plane ,Affine geometry ,Affine coordinate system ,Affine combination ,Affine geometry of curves ,Differential Geometry (math.DG) ,Affine hull ,53C21, 53B30, 53C24, 53C44 ,Affine group ,FOS: Mathematics ,Affine transformation - Abstract
We study the affine quasi-Einstein equation, a second order linear homogeneous equation, which is invariantly defined on any affine manifold. We prove that the space of solutions is finite-dimensional, and its dimension is a strongly projective invariant. Moreover the maximal dimension is shown to be achieved if and only if the manifold is strongly projectively flat.
- Published
- 2017
28. On four-dimensional Einstein affine hyperspheres
- Author
-
Luc Vrancken, Zejun Hu, Haizhong Li, Laboratoire de Mathématiques et leurs Applications de Valenciennes - EA 4015 (LAMAV), and Centre National de la Recherche Scientifique (CNRS)-Université de Valenciennes et du Hainaut-Cambrésis (UVHC)-INSA Institut National des Sciences Appliquées Hauts-de-France (INSA Hauts-De-France)
- Subjects
Pure mathematics ,010102 general mathematics ,Mathematical analysis ,Einstein metric ,Affine hypersphere ,01 natural sciences ,Affine plane ,010101 applied mathematics ,Affine geometry ,Affine shape adaptation ,Affine coordinate system ,Affine combination ,Computational Theory and Mathematics ,Affine representation ,Affine hull ,Affine group ,Geometry and Topology ,0101 mathematics ,[MATH]Mathematics [math] ,Affine metric ,Analysis ,ComputingMilieux_MISCELLANEOUS ,Mathematics - Abstract
© 2016 Elsevier B.V. It is well-known that Vrancken–Li–Simon classified locally strongly convex affine hyperspheres in Rn+1 whose affine metric are of constant sectional curvatures, but on the other side it is still a difficult problem to classify n-dimensional locally strongly convex affine hyperspheres whose affine metrics are Einstein. In this paper, we have solved the problem in case n=4. publisher: Elsevier articletitle: On four-dimensional Einstein affine hyperspheres journaltitle: Differential Geometry and its Applications articlelink: http://dx.doi.org/10.1016/j.difgeo.2016.10.003 content_type: article copyright: © 2016 Elsevier B.V. All rights reserved. ispartof: Differential Geometry and its Applications vol:50 pages:20-33 status: published
- Published
- 2017
29. Affine focal points for locally strictly convex surfaces in 4-space
- Author
-
Marcelo José Saia, Juan J. Nuño-Ballesteros, and Luis F. Sánchez
- Subjects
Pure mathematics ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,01 natural sciences ,Affine plane ,Affine geometry ,Affine coordinate system ,Affine shape adaptation ,Mathematics (miscellaneous) ,Affine combination ,Affine hull ,0103 physical sciences ,Affine group ,010307 mathematical physics ,Affine transformation ,0101 mathematics ,TEORIA DAS SINGULARIDADES ,Mathematics - Abstract
We consider locally strictly convex surfaces M in affine 4-space. By using the metric of the transversal vector field on M we introduce a new affine normal plane and the familly of affine distance functions on M. We show that the singularities of the family of affine distance functions appear at points on the affine normal plane and the affine focal points correspond to degenerate singularities of this family. Moreover we show that if M is immersed in a locally strictly convex hypersurface, then the affine normal plane contains the affine normal vector to the hypersurface and conclude that any surface immersed in a locally strictly convex hypersphere is affine semiumbilical.
- Published
- 2017
30. Shape-Color Differential Moment Invariants Under Affine Transforms
- Author
-
You Hao, Hanlin Mo, Hua Li, and Shirui Li
- Subjects
Pure mathematics ,Color image ,ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION ,02 engineering and technology ,01 natural sciences ,Affine coordinate system ,Affine geometry ,Affine combination ,Affine geometry of curves ,Affine hull ,0103 physical sciences ,Affine group ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,Affine transformation ,010306 general physics ,ComputingMethodologies_COMPUTERGRAPHICS ,Mathematics - Abstract
We propose a general structural formula of shape-color primitive by using partial derivatives of each color channel in this paper. By using shape-color primitive, shape-color differential moment invariants (SCDMIs) can be constructed very easily, which are invariant to shape affine and color affine transforms. And 50 instances of SCDMIs are obtained. In experiments, several commonly used image descriptors and SCDMIs are used in image classification and retrieval for color image databases, respectively. By comparing the results, we find that SCDMIs get better results.
- Published
- 2017
31. Affine Invariant Geometry for Non-rigid Shapes
- Author
-
Dan Raviv and Ron Kimmel
- Subjects
Geometry ,Affine geometry ,Affine coordinate system ,Affine shape adaptation ,Affine combination ,Affine geometry of curves ,Artificial Intelligence ,Affine hull ,Affine group ,Computer Vision and Pattern Recognition ,Affine transformation ,Computer Science::Databases ,Software ,Mathematics - Abstract
Shape recognition deals with the study geometric structures. Modern surface processing methods can cope with non-rigidity--by measuring the lack of isometry, deal with similarity or scaling--by multiplying the Euclidean arc-length by the Gaussian curvature, and manage equi-affine transformations--by resorting to the special affine arc-length definition in classical equi-affine differential geometry. Here, we propose a computational framework that is invariant to the full affine group of transformations (similarity and equi-affine). Thus, by construction, it can handle non-rigid shapes. Technically, we add the similarity invariant property to an equi-affine invariant one and establish an affine invariant pseudo-metric. As an example, we show how diffusion geometry can encapsulate the proposed measure to provide robust signatures and other analysis tools for affine invariant surface matching and comparison.
- Published
- 2014
32. DECOMPOSABLE AFFINE HYPERSURFACES
- Author
-
Franki Dillen, Luc Vrancken, Miroslava Antić, Kristof Schoels, Laboratoire de Mathématiques et leurs Applications de Valenciennes - EA 4015 (LAMAV), Centre National de la Recherche Scientifique (CNRS)-Université de Valenciennes et du Hainaut-Cambrésis (UVHC)-INSA Institut National des Sciences Appliquées Hauts-de-France (INSA Hauts-De-France), and Pruvost, Frédéric
- Subjects
Pure mathematics ,General Mathematics ,[MATH] Mathematics [math] ,affine hyperspheres ,Affine plane ,Calabi product ,Affine geometry ,Affine coordinate system ,Affine combination ,Affine representation ,Affine hull ,Affine group ,Affine transformation ,Mathematics::Differential Geometry ,[MATH]Mathematics [math] ,ComputingMilieux_MISCELLANEOUS ,Mathematics - Abstract
In affine differential geometry, Calabi discovered how to associate a new hyperbolic affine hypersphere with two hyperbolic affine hyperspheres. This was later generalized by Dillen and Vrancken in order to obtain a large class of examples of equiaffine homogeneous affine hypersurfaces. Note that the constructions defined above remain valid if one of the affine hyperspheres is a point. In this paper we consider the converse question: how can we determine, given properties of the difference tensor K and the affine shape operator S, whether a given hypersurface can be decomposed as a generalized Calabi product of an affine sphere and a point?
- Published
- 2014
33. Fast image matching algorithm based on affine invariants
- Author
-
Yi Zhang, Kai Lu, and Yinghui Gao
- Subjects
Affine shape adaptation ,Affine geometry ,Affine combination ,Automatic target recognition ,Robustness (computer science) ,Metals and Alloys ,General Engineering ,Centroid ,Parallel ,Parallelogram ,Algorithm ,Mathematics - Abstract
Feature-based image matching algorithms play an indispensable role in automatic target recognition (ATR). In this work, a fast image matching algorithm (FIMA) is proposed which utilizes the geometry feature of extended centroid (EC) to build affine invariants. Based on affine invariants of the length ratio of two parallel line segments, FIMA overcomes the invalidation problem of the state-of-the-art algorithms based on affine geometry features, and increases the feature diversity of different targets, thus reducing misjudgment rate during recognizing targets. However, it is found that FIMA suffers from the parallelogram contour problem and the coincidence invalidation. An advanced FIMA is designed to cope with these problems. Experiments prove that the proposed algorithms have better robustness for Gaussian noise, gray-scale change, contrast change, illumination and small three-dimensional rotation. Compared with the latest fast image matching algorithms based on geometry features, FIMA reaches the speedup of approximate 1.75 times. Thus, FIMA would be more suitable for actual ATR applications.
- Published
- 2014
34. Conic epipolar constraints from affine correspondences
- Author
-
Jacob Bentolila and Joseph M. Francos
- Subjects
Harris affine region detector ,Topology ,Affine coordinate system ,Affine geometry ,Affine shape adaptation ,Affine combination ,Affine hull ,Signal Processing ,Affine group ,Applied mathematics ,Computer Vision and Pattern Recognition ,Affine transformation ,Software ,Mathematics - Abstract
We derive an explicit relation between local affine approximations resulting from matching of affine invariant regions and the epipolar geometry in the case of a two view geometry. Most methods that employ the affine relations do so indirectly by generating pointwise correspondences from the affine relations. In this paper we derive an explicit relation between the local affine approximations and the epipolar geometry. We show that each affine approximation between images is equivalent to 3 linear constraints on the fundamental matrix and that the linear conditions guarantee the existence of an homography, compatible with the fundamental matrix. We further show that two affine relations constrain the location of the epipole to a conic section. Therefore, the location of the epipole can be extracted from 3 regions by intersecting conics. The result is further employed to derive a procedure for estimating the fundamental matrix, based on the estimated location of the epipole. It is shown to be more accurate and to require less iterations in LO-RANSAC based estimation, than the current point based approaches that employ the affine relation to generate pointwise correspondences and then calculate the fundamental matrix from the pointwise relations.
- Published
- 2014
35. Affine Fullerene C60in a GS-Quasigroup
- Author
-
Ružica Kolar-Šuper, Zdenka Kolar-Begović, and Vladimir Volenec
- Subjects
Article Subject ,Mathematics::General Mathematics ,GS-quasigroup ,affine fullerene ,affine regular hexagon ,affine regular pentagon ,lcsh:Mathematics ,Applied Mathematics ,0211 other engineering and technologies ,021107 urban & regional planning ,02 engineering and technology ,lcsh:QA1-939 ,021001 nanoscience & nanotechnology ,Affine plane ,Combinatorics ,Affine geometry ,Affine coordinate system ,Mathematics::Group Theory ,Affine combination ,Affine representation ,Affine hull ,Affine group ,Physics::Atomic and Molecular Clusters ,Affine transformation ,0210 nano-technology ,Mathematics - Abstract
It will be shown that the affine fullerene C60, which is defined as an affine image of buckminsterfullerene C60, can be obtained only by means of the golden section. The concept of the affine fullerene C60will be constructed in a general GS-quasigroup using the statements about the relationships between affine regular pentagons and affine regular hexagons. The geometrical interpretation of all discovered relations in a general GS-quasigroup will be given in the GS-quasigroupC(1/2(1+5)).
- Published
- 2014
36. Homography and Fundamental Matrix Estimation from Region Matches Using an Affine Error Metric
- Author
-
Jacob Bentolila and Joseph M. Francos
- Subjects
Statistics and Probability ,Harris affine region detector ,Applied Mathematics ,Condensed Matter Physics ,Topology ,Affine plane ,Affine geometry ,Affine coordinate system ,Affine shape adaptation ,Affine combination ,Modeling and Simulation ,Affine group ,Geometry and Topology ,Computer Vision and Pattern Recognition ,Affine transformation ,Algorithm ,Mathematics - Abstract
Matching a pair of affine invariant regions between images results in estimation of the affine transformation between the regions. However, the parameters of the affine transformations are rarely used directly for matching images, mainly due to the lack of an appropriate error metric of the distance between them. In this paper we derive a novel metric for measuring the distance between affine transformations: Given an image region, we show that minimization of this metric is equivalent to the minimization of the mean squared distance between affine transformations of a point, sampled uniformly on the image region. Moreover, the metric of the distance between affine transformations is equivalent to the l 2 norm of a linear transformation of the difference between the six parameters of the affine transformations. We employ the metric for estimating homographies and for estimating the fundamental matrix between images. We show that both homography estimation and fundamental matrix estimation methods, based on the proposed metric, are superior to current linear estimation methods as they provide better accuracy without increasing the computational complexity.
- Published
- 2013
37. Alternative characterization of hyperbolic affine infinite iterated function systems
- Author
-
Radu Miculescu and Alexandru Mihail
- Subjects
Affine geometry ,Affine coordinate system ,Discrete mathematics ,Pure mathematics ,Affine combination ,Affine representation ,Applied Mathematics ,Affine hull ,Affine group ,Affine space ,Affine plane ,Analysis ,Mathematics - Abstract
In this paper we present a characterization of hyperbolic affine infinite iterated function systems defined on an arbitrary normed space. Our result is a generalization of Theorem 1.1 from the paper “A characterization of hyperbolic affine iterated function systems”, Topology Proceedings, 36 (2010), 189–211, by R. Atkins, M. Barnsley, A. Vince and D. Wilson. Some examples are presented.
- Published
- 2013
38. Unital affine semigroups
- Author
-
Rajesh Pereira, Jeremy Levick, and David W. Kribs
- Subjects
Discrete mathematics ,Numerical Analysis ,Pure mathematics ,Algebra and Number Theory ,Mathematics::Operator Algebras ,Affine plane ,Affine geometry ,Affine coordinate system ,Affine combination ,Affine hull ,Affine group ,Discrete Mathematics and Combinatorics ,Special classes of semigroups ,Geometry and Topology ,Affine transformation ,Mathematics - Abstract
Affine semigroups are convex sets on which there exists an associative binary operation which is affine separately in either variable. They were introduced by Cohen and Collins in 1959. We look at examples of affine semigroups which are of interest to matrix and operator theory and we prove some new results on the extreme points and the absorbing elements of certain types of affine semigroups. Most notably we improve a result of Wendel that every invertible element in a compact affine semigroup is extreme by extending this result to linearly bounded affine semigroups.
- Published
- 2013
39. Classifying Convex Compact Ancient Solutions to the Affine Curve Shortening Flow
- Author
-
Shibing Chen
- Subjects
Mathematics - Differential Geometry ,Mathematical analysis ,Affine plane ,Affine geometry ,Affine shape adaptation ,Affine coordinate system ,Mathematics - Analysis of PDEs ,Affine combination ,Differential Geometry (math.DG) ,Affine hull ,Affine group ,FOS: Mathematics ,Geometry and Topology ,Affine transformation ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
In this paper we classify convex compact ancient solutions to the affine curve shortening flow: namely, any convex compact ancient solution to the affine curve shortening flow must be a shrinking ellipse. The method combines a rescaling argument inspired by \cite{Wang}, affine invariance of the equation and monotonicity of the affine isoperimetric ratio. We will give two proofs. The essential ideas are related, but the first one uses level set represenation of the evolution. The second proof employs Schauder's estimates, and it also provides a new simple proof for the corresponding classification result to the higher dimensional affine normal flow., 5 pages, comments are welcome
- Published
- 2013
40. Affine Moments of a Random Vector
- Author
-
Deane Yang, Gaoyong Zhang, Songjun Lv, and Erwin Lutwak
- Subjects
Affine geometry ,Combinatorics ,Affine coordinate system ,Affine combination ,Affine geometry of curves ,Affine hull ,Affine group ,Affine space ,Affine transformation ,Library and Information Sciences ,Computer Science Applications ,Information Systems ,Mathematics - Abstract
An affine invariant pth moment measure is defined for a random vector and used to prove sharp moment-entropy inequalities that are more general and stronger than standard moment-entropy inequalities.
- Published
- 2013
41. Non-Gatherable Triples for Classical Affine Root Systems
- Author
-
Ivan Cherednik and Keith Schneider
- Subjects
17B22, 05E10, 20C08 ,010102 general mathematics ,16. Peace & justice ,01 natural sciences ,Affine plane ,Affine geometry ,Affine coordinate system ,Combinatorics ,Affine combination ,Affine representation ,Affine hull ,Mathematics - Quantum Algebra ,0103 physical sciences ,Affine group ,FOS: Mathematics ,Mathematics - Combinatorics ,Quantum Algebra (math.QA) ,Discrete Mathematics and Combinatorics ,Combinatorics (math.CO) ,010307 mathematical physics ,Affine transformation ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics - Representation Theory ,Mathematics - Abstract
This paper contains a complete description of minimal non-gatherable triangle triples in the lambda-sequences for the affine classical root systems and some claims for arbitrary (reduced) affine root systems. It continues our previous paper devoted to the non-affine case; interestingly, the affine theory clarifies the classification in the non-affine case. The lambda-sequences are associated with reduced decompositions (words) in affine Weyl groups. The existence of the non-gatherable triples is a combinatorial obstacle for using the technique of intertwiners in the theory of irreducible representations of the (double) affine Hecke algebras, complementary to their algebraic-geometric theory., Comment: Latex, 46 pages
- Published
- 2013
42. Affine PBW bases and MV polytopes in rank 2
- Author
-
Peter Tingley and Dinakar Muthiah
- Subjects
Discrete mathematics ,Quantum affine algebra ,Pure mathematics ,General Mathematics ,General Physics and Astronomy ,Polytope ,Affine Grassmannian (manifold) ,17B37 ,Affine geometry ,Affine combination ,Affine representation ,Affine hull ,Mathematics - Quantum Algebra ,Affine group ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Mathematics::Metric Geometry ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Mathematics - Representation Theory ,Mathematics - Abstract
Mirkovic-Vilonen (MV) polytopes have proven to be a useful tool in understanding and unifying many constructions of crystals for finite-type Kac-Moody algebras. These polytopes arise naturally in many places, including the affine Grassmannian, pre-projective algebras, PBW bases, and KLR algebras. There has recently been progress in extending this theory to the affine Kac-Moody algebras. A definition of MV polytopes in symmetric affine cases has been proposed using pre-projective algebras. In the rank-2 affine cases, a combinatorial definition has also been proposed. Additionally, the theory of PBW bases has been extended to affine cases, and, at least in rank-2, we show that this can also be used to define MV polytopes. The main result of this paper is that these three notions of MV polytope all agree in the relevant rank-2 cases. Our main tool is a new characterization of rank-2 affine MV polytopes., Comment: 23 pages
- Published
- 2013
43. A new characterization of line-to-line maps in the upper plane
- Author
-
Yuefei Wang and Baokui Li
- Subjects
Affine coordinate system ,Affine geometry ,Pure mathematics ,Affine combination ,General Mathematics ,Affine hull ,Affine group ,Affine space ,Geometry ,Affine transformation ,Affine plane ,Mathematics - Abstract
The characterization of typical maps in a domain of a given space is a much harder problem than that in the whole space. In this paper, by using methods of hyperbolic and affine geometry, we give a new characterization of line-to-line maps in the upper plane. We show that a line-to-line surjection is either an affine transformation, or a composition of an affine transformation and a ɡ-reflection. Moreover, we prove that the composition of two ɡ-reflections with the same boundary is an affine transformation.
- Published
- 2013
44. Affine Structure on a Manifold
- Author
-
Géry de Saxcé and Claude Vallée
- Subjects
Affine coordinate system ,Affine geometry ,Affine shape adaptation ,Pure mathematics ,Affine combination ,Affine hull ,Mathematical analysis ,Affine group ,Affine transformation ,Affine plane ,Mathematics - Published
- 2016
45. Locally piecewise affine functions and their order structure
- Author
-
Vladimir G. Troitsky and Samer Adeeb
- Subjects
Discrete mathematics ,Pure mathematics ,021103 operations research ,General Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,02 engineering and technology ,46A40, 46E05 ,01 natural sciences ,Affine plane ,Theoretical Computer Science ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Affine geometry ,Affine coordinate system ,Affine combination ,Affine representation ,Affine geometry of curves ,Affine hull ,Affine group ,FOS: Mathematics ,0101 mathematics ,Analysis ,Mathematics - Abstract
Piecewise affine functions on subsets of $\mathbb R^m$ were studied in \cite{Ovchinnikov:02,Aliprantis:06a,Aliprantis:07a,Aliprantis:07}. In this paper we study a more general concept of a locally piecewise affine function. We characterize locally piecewise affine functions in terms of components and regions. We prove that a positive function is locally piecewise affine iff it is the supremum of a locally finite sequence of piecewise affine functions. We prove that locally piecewise affine functions are uniformly dense in $C(\mathbb R^m)$, while piecewise affine functions are sequentially order dense in $C(\mathbb R^m)$. This paper is partially based on \cite{Adeeb:14}., Comment: 11 pages
- Published
- 2016
- Full Text
- View/download PDF
46. Singularities of convex improper affine maps
- Author
-
Marcos Craizer
- Subjects
Mathematics - Differential Geometry ,Affine plane ,53A15 ,Combinatorics ,Affine geometry ,Affine shape adaptation ,Affine coordinate system ,Affine combination ,Differential Geometry (math.DG) ,Affine hull ,FOS: Mathematics ,Affine space ,Geometry and Topology ,Affine transformation ,Mathematics - Abstract
In this paper we consider convex improper affine maps of the 3-dimensional affine space and classify their singularities. The main tool developed is a generating family with properties that closely resembles the area function for non-convex improper affine maps., 11 pages, 2 figures
- Published
- 2012
47. On a class of affine geometries
- Author
-
Christopher Parker, Sergey Shpectorov, and Corneliu Hoffman
- Subjects
Affine geometry ,Affine coordinate system ,Pure mathematics ,Affine combination ,Affine geometry of curves ,Affine hull ,Affine group ,Geometry and Topology ,Affine transformation ,Affine plane ,Mathematics - Published
- 2012
48. Affine-invariant curvature estimators for implicit surfaces
- Author
-
Maria Gorete Carreira Andrade and Thomas Lewiner
- Subjects
Mathematical analysis ,Affine differential geometry ,Aerospace Engineering ,Computer Graphics and Computer-Aided Design ,Affine shape adaptation ,Affine geometry ,Affine coordinate system ,Affine combination ,Modeling and Simulation ,Affine hull ,Automotive Engineering ,Affine group ,Applied mathematics ,Affine transformation ,Mathematics - Abstract
Affine Differential Geometry provides a set of measures invariant under a larger set of transformations compared to rigid motions. This leads to several applications using robust shape descriptors. Although affine-invariant operations are already used for surfaces, they do not intend to approximate the definitions of Affine Differential Geometry, which are the basis for further differential invariants. In this work we propose estimators for the local affine structure of an implicit surface, i.e. the affine metric, the co-normal and normal vectors, and the affine Gaussian and mean curvatures. The direct derivation of the formulae from the implicit function theorem lead to very intensive computations and numerical instabilities. This work further proposes a geometrical reduction allowing a much simpler and more stable formulae, and compares the results by incorporating the proposed estimators in Marching Cubes based algorithms.
- Published
- 2012
49. Affine Dispersers from Subspace Polynomials
- Author
-
Swastik Kopparty and Eli Ben-Sasson
- Subjects
Pseudorandom number generator ,Discrete mathematics ,Sublinear function ,General Computer Science ,General Mathematics ,010102 general mathematics ,Dimension (graph theory) ,Context (language use) ,0102 computer and information sciences ,Computer Science::Computational Complexity ,01 natural sciences ,Affine geometry ,Combinatorics ,Affine combination ,Finite field ,Affine representation ,010201 computation theory & mathematics ,Affine hull ,Affine group ,Domain (ring theory) ,Affine space ,Affine transformation ,0101 mathematics ,Randomness ,Mathematics - Abstract
Dispersers and extractors for affine sources of dimension $d$ in $\mathbb{F}_2^m$---where $\mathbb{F}_2$ denotes the finite field of size $2$---are functions $f: \mathbb{F}_2^m \rightarrow \mathbb{F}_2$ that behave pseudorandomly when their domain is restricted to any particular affine space $S \subseteq \mathbb{F}_2^m$ of dimension at least $d$. For dispersers, “pseudorandom behavior” means that $f$ is nonconstant over $S$, i.e., $\{ f(s) \: \middle| \: s\in S\}=\mathbb{F}_2$. For extractors, it means that $f(s)$ is distributed almost uniformly over $\mathbb{F}_2$ when $s$ is distributed uniformly over $S$. Dispersers and extractors for affine sources have been considered in the context of deterministic extraction of randomness from structured sources of imperfect randomness. Previously, explicit constructions of affine dispersers were known for every $d = \Omega(m)$ (due to [B. Barak et al., in Proceedings of the ACM Symposium on Theory of Computing, ACM, New York, 2005]), and explicit affine extractors...
- Published
- 2012
50. Cauchy problems for discrete affine minimal surfaces
- Author
-
Ralph Teixeira, Thomas Lewiner, and Marcos Craizer
- Subjects
Affine geometry ,Affine coordinate system ,Affine shape adaptation ,Affine combination ,General Mathematics ,Affine hull ,Mathematical analysis ,Affine group ,Affine transformation ,Affine plane ,Mathematics - Abstract
In this paper we discuss planar quadrilateral (PQ) nets as discrete models for convex affine surfaces. As a main result, we prove a necessary and sufficient condition for a PQ net to admit a Lelieuvre co-normal vector field. Particular attention is given to the class of surfaces with discrete harmonic co-normals, which we call discrete affine minimal surfaces, and the subclass of surfaces with co-planar discrete harmonic co-normals, which we call discrete improper affine spheres. Within this classes, we show how to solve discrete Cauchy problems analogous to the Cauchy problems for smooth analytic improper affine spheres and smooth analytic affine minimal surfaces.
- Published
- 2012
Catalog
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.