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Rennemo, Jørgen Vold, i Ed Segal. "Hori-mological projective duality". Duke Mathematical Journal 168, nr 11 (sierpień 2019): 2127–205. http://dx.doi.org/10.1215/00127094-2019-0014.
Garcia, A., i J. F. Voloch. "Duality for projective curves". Boletim da Sociedade Brasileira de Matem�tica 21, nr 2 (wrzesień 1991): 159–75. http://dx.doi.org/10.1007/bf01237362.
Goerss, Paul G. "Projective and Injective Hopf Algebras Over the Dyer-Lashof Algebra". Canadian Journal of Mathematics 45, nr 5 (1.10.1993): 944–76. http://dx.doi.org/10.4153/cjm-1993-053-9.
AbstractThe purpose of this paper is to discuss the existence, structure, and properties of certain projective and injective Hopf algebras in the category of Hopf algebras that support the structure one expects on the homology of an infinite loop space. As an auxiliary project, we show that these projective and injective Hopf algebras can be realized as the homology of infinite loop spaces associated to spectra obtained from Brown-Gitler spectra by Spanier-Whitehead duality and Brown-Comenetz duality, respectively. We concentrate mainly on indecomposable projectives and injectives, and we work only at the prime 2.
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Kuznetsov, Alexander, i Alexander Perry. "Homological projective duality for quadrics". Journal of Algebraic Geometry 30, nr 3 (15.01.2021): 457–76. http://dx.doi.org/10.1090/jag/767.
We show that over an algebraically closed field of characteristic not equal to 2, homological projective duality for smooth quadric hypersurfaces and for double covers of projective spaces branched over smooth quadric hypersurfaces is a combination of two operations: one interchanges a quadric hypersurface with its classical projective dual and the other interchanges a quadric hypersurface with the double cover branched along it.
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FRISK, ANDERS, i VOLODYMYR MAZORCHUK. "PROPERLY STRATIFIED ALGEBRAS AND TILTING". Proceedings of the London Mathematical Society 92, nr 1 (19.12.2005): 29–61. http://dx.doi.org/10.1017/s0024611505015431.
We study the properties of tilting modules in the context of properly stratified algebras. In particular, we answer the question of when the Ringel dual of a properly stratified algebra is properly stratified itself, and show that the class of properly stratified algebras for which the characteristic tilting and cotilting modules coincide is closed under taking the Ringel dual. Studying stratified algebras whose Ringel dual is properly stratified, we discover a new Ringel-type duality for such algebras, which we call the two-step duality. This duality arises from the existence of a new (generalized) tilting module for stratified algebras with properly stratified Ringel dual. We show that this new tilting module has a lot of interesting properties; for instance, its projective dimension equals the projectively defined finitistic dimension of the original algebra, it guarantees that the category of modules of finite projective dimension is contravariantly finite, and, finally, it allows one to compute the finitistic dimension of the original algebra in terms of the projective dimension of the characteristic tilting module.
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Bruce, J. W. "Lines, surfaces and duality". Mathematical Proceedings of the Cambridge Philosophical Society 112, nr 1 (lipiec 1992): 53–61. http://dx.doi.org/10.1017/s0305004100070754.
In the paper [12] Shcherbak studied some duality properties of projective curves and applied them to obtain information concerning central projections of surfaces in projective three space. He also states some interesting results relating the contact of a generic surface with lines and the contact of its dual with lines in the dual space. In this paper we extend this duality to cover non-generic surfaces. Our proof is geometric, and uses deformation theory. The basic idea is the following. Given a surface X in projective 3-space we can consider the lines tangent to X, and measure their contact. The points on the surface with a line yielding at least 4-point contact are classically known as the flecnodal. (The reason is that the tangent plane meets the surface in a nodal curve, one branch of which has an inflexion at the point in question; see Proposition 7 below. The line in question is the inflexional tangent, which is clearly asymptotic.)
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Barrett, David E. "Holomorphic projection and duality for domains in complex projective space". Transactions of the American Mathematical Society 368, nr 2 (3.04.2015): 827–50. http://dx.doi.org/10.1090/tran/6338.
Benson, D. J., i Jon F. Carlson. "Projective Resolutions and Poincare Duality Complexes". Transactions of the American Mathematical Society 342, nr 2 (kwiecień 1994): 447. http://dx.doi.org/10.2307/2154636.
Soit $X$ une variété algébrique de Gorenstein à singularités rationnelles. Une résolution des singularités crépante de $X$ est souvent considérée comme une résolution des singularités minimales de $X$. Malheureusement, les résolutions crépantes sont très rares. Ainsi, les variétés déterminantielles de matrices anti-symétriques n'admettent jamais de résolution crépante des singularités. Dans cette thèse, on discutera de diverses notions de résolutions catégoriques crépantes développées par Alexander Kuznetsov. Conjecturalement, ces résolutions doivent être minimale du point de vue catégorique. On introduit dans ce manuscrit la notion de résolution magnifiques des singularités et on montre que tout variété munie d'une telle résolution admet une résolution catégorique faiblement crépante. On en déduit que toutes les variétés déterminantielles (carrées, symétriques et anti-symétriques) admettent des résolutions catégoriques faiblement crépantes. Finalement, on s'intéressera à des hypersurfaces quartiques issues du carré magique de Tits-Freudenthal. On ne peut pas construire de résolution magnifique des singularités pour de telles hypersurfaces, mais on montrera qu'elles admettent tout de même des résolutions catégorique faiblement crépantes des singularités. Ce résultat devrait s'avérer intéressant pour la construction de duales projectives homologiques de certaines Grassmaniennes symplectiques sur les algèbres de composition Let $X$ be an algebraic variety with Gorenstein rational singularities. A crepant resolution of $X$ is often considered to be a minimal resolution of singularities for $X$. Unfortunately, crepant resolution of singularities are very rare. For instance, determinantal varieties of skew-symmetric matrices never admit crepant resolution of singularities. In this thesis, we discuss various notions of categorical crepant resolution of singularities as defined by Alexander Kuznetsov. Conjecturally, these resolutions are minimal from the categorical point of view. We introduce the notion of wonderful resolution of singularities and we prove that a variety endowed with such a resolution admits a weakly crepant resolution of singularities. As a corollary, we prove that all determinantal varieties (square, as well as symmetric and skew-symmetric) admit weakly crepant resolution of singularities. Finally, we study some quartics hypersurfaces which come from the Tits-Freudenthal magic square. Though they do no admit any wonderful resolution of singularities, we are still able to prove that they have a weakly crepant resolution of singularities. This last result should be of interest in order to construct homological projective duals for some symplectic Grassmannians over the composition algebras
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Hilburn, Justin. "GKZ Hypergeometric Systems and Projective Modules in Hypertoric Category O". Thesis, University of Oregon, 2016. http://hdl.handle.net/1794/20456.
In this thesis I show that indecomposable projective and tilting modules in hypertoric category O are obtained by applying a variant of the geometric Jacquet functor of Emerton, Nadler, and Vilonen to certain Gel'fand-Kapranov-Zelevinsky hypergeometric systems. This proves the abelian case of a conjecture of Bullimore, Gaiotto, Dimofte, and Hilburn on the behavior of generic Dirichlet boundary conditions in 3d N=4 SUSY gauge theories.
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Contatto, Felipe. "Vortices, Painlevé integrability and projective geometry". Thesis, University of Cambridge, 2018. https://www.repository.cam.ac.uk/handle/1810/275099.
GaugThe first half of the thesis concerns Abelian vortices and Yang-Mills theory. It is proved that the 5 types of vortices recently proposed by Manton are actually symmetry reductions of (anti-)self-dual Yang-Mills equations with suitable gauge groups and symmetry groups acting as isometries in a 4-manifold. As a consequence, the twistor integrability results of such vortices can be derived. It is presented a natural definition of their kinetic energy and thus the metric of the moduli space was calculated by the Samols' localisation method. Then, a modified version of the Abelian–Higgs model is proposed in such a way that spontaneous symmetry breaking and the Bogomolny argument still hold. The Painlevé test, when applied to its soliton equations, reveals a complete list of its integrable cases. The corresponding solutions are given in terms of third Painlevé transcendents and can be interpreted as original vortices on surfaces with conical singularity. The last two chapters present the following results in projective differential geometry and Hamiltonians of hydrodynamic-type systems. It is shown that the projective structures defined by the Painlevé equations are not metrisable unless either the corresponding equations admit first integrals quadratic in first derivatives or they define projectively flat structures. The corresponding first integrals can be derived from Killing vectors associated to the metrics that solve the metrisability problem. Secondly, it is given a complete set of necessary and sufficient conditions for an arbitrary affine connection in 2D to admit, locally, 0, 1, 2 or 3 Killing forms. These conditions are tensorial and simpler than the ones in previous literature. By defining suitable affine connections, it is shown that the problem of existence of Killing forms is equivalent to the conditions of the existence of Hamiltonian structures for hydrodynamic-type systems of two components.
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Benchoufi, Mehdi. "Théorie microlocale des faisceaux pour la transformation Radon". Electronic Thesis or Diss., Sorbonne université, 2020. http://www.theses.fr/2020SORUS475.
Le sujet de cette thèse est une approche microlocale de la transformation de Radon. Il s’agit d’appliquer à la dualité projective complexe et réelle les techniques initiées dans l’article fondateur de Sato-Kashiwara-Kawai de 1972 et de retrouver, reformuler, améliorer des travaux d’analyse plus classiques sur ce sujet, en particulier ceux de G. Henkin ou S. Gindikin. La dualité projective vue sous l’angle microlocal et faisceautique est apparue pour la première fois dans un travail important de J-L. Brylinski sur les faisceaux pervers, travail repris ensuite par D’Agnolo et Schapira dans le cadre des D-modules. Notre travail est de reprendre systématiquement cette étude avec les nouveaux outils de la théorie microlocale des faisceaux (théorie qui n’existait pas à l’époque de SKK72). Ce travail se compose essentiellement de deux parties. Dans la première, nous commençons par rappeler dans un cadre général la construction des transformations canoniques quantifiées, sous l’hypothèse de l’existence d’une section simple non-dégénérée (introduite sous un autre nom par J. Leray). Cette construction n’avait jamais été faite dans un cadre global hors du cas projectif. Nous montrons alors que ces transformations commutent à l’action des opérateurs microdifferentiels. Il s’agit là d’un résultat fondamental sans qu’aucune preuve consistante n’existe dans la littérature, ce résultat étant plus ou moins sous-entendu dans SKK72. La deuxième partie de la thèse traite des applications à la transformation de Radon “clas-sique”. L’idée de base est que cette transformation échange support des hyperfonctions (modulo analyticité) et front d’onde analytique. Nous obtenons ainsi des théorèmes de prolongement ou d’unicité sur les ouverts linéellement concave. Nous obtenons aussi un théorème des résidus pour les valeurs au bord de classes de cohomologie définies sur les cônes de signatures (1, n − 1), clarifiant substantiellement des travaux de Cordaro-Gindikin-Trèves The subject of this thesis is a microlocal approach to the transformation of Radon. It is a question of applying to real and complex projective duality the techniques initiated in the founding article of Sato-Kashiwara-Kawai of 1972 and to find, reformulate, improve more classic analytical work on this subject, in particular those of G. Henkin or S. Gindikin. Pro-jective duality seen from the microlocal and sheaf point of view appeared for the first time in an important work by J-L. Brylinski on perverse sheaves, work then taken up by D'Agnolo and Schapira in the framework of D-modules. Our work is to systematically resume this study with the new tools of the microlocal sheaf theory (theory which did not exist at the time of SKK72). This work essentially consists of two parts. In the first, we begin by recalling in a general framework the construction of quantized ca-nonical transformations, under the hypothesis of the existence of a simple non-degenerate section (introduced under another name by J. Leray). This construction had never been done in a global framework outside the projective case. We then show that these transfor-mations exchange the action of the microdifferential operators. This is a fundamental re-sult without any consistent proof existing in the literature, this result being more or less implied in SKK72. The second part of the thesis deals with the applications to the “classical” Radon trans-form. The basic idea is that this transform exchanges the support of hyperfunctions (modu-lo analyticity) and the analytic wavefront set. We thus obtain theorems of continuation or uniqueness on linearly concave domain. We also get a residue theorem for the boundary values of finite cohomology classes defined on cones with (1, n-1) signature, substantially clari-fying the work of Cordaro-Gindikin-Trèves
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Tur, Laurent. "Dualité étrange sur le plan projectif". Nice, 2003. http://www.theses.fr/2003NICE4089.
Nous donnons des exemples a l'appui de la conjecture de dualite etrange de le potier, dans le cas de l'espace de modules m des faisceaux semi-stables de rang 2 sur le plan projectif, avec premiere classe de chern paire, et seconde classe de chern c 2 inferieure ou egale a 19. Nous calculons dans ce cas la dimension de l'espace des sections globales du fibre determinant sur m, ce qui correspond a un analogue de la formule de verlinde pour le plan projectif. Nous calculons a ce but les espaces de cohomologie du fibre tautologique tensorise par le fibre determinant sur le schema de hilbert ponctuel hilb m(x) d'une surface complexe projective et lisse. Nous montrons que pour l et a fibres vectoriels inversibles sur x, et w x le fibre canonique sur x, si h q(x, a) = 0 = h q(x, lisotimes a) pour tout q 1, alors les groupes de cohomologie superieurs sur hilb m(x) du fibre tautologique associe a l tensorise par le fibre determinant associe a a, s'annulent. Nous calculons egalement l'espace des sections globales en termes de h 0(a) et h 0(x, lisotimes a). Finalement nous prouvons que la dualite etrange est verifiee pour les puissances deuxieme et troisieme du fibre determinant sur m, lorsque la deuxieme classe de chern est inferieure ou egale a 5, et nous calculons l'espace des sections globales de toutes les puissances du fibre determinant sur m lorsque la deuxieme classe de chern c 2 est egale a 3 ou a 4.
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Wilfer, Oleg. "Duality investigations for multi-composed optimization problems with applications in location theory". Doctoral thesis, Universitätsbibliothek Chemnitz, 2017. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-222660.
The goal of this thesis is two-fold. On the one hand, it pursues to provide a contribution to the conjugate duality by proposing a new duality concept, which can be understood as an umbrella for different meaningful perturbation methods. On the other hand, this thesis aims to investigate minimax location problems by means of the duality concept introduced in the first part of this work, followed by a numerical approach using epigraphical splitting methods.
After summarizing some elements of the convex analysis as well as introducing important results needed later, we consider an optimization problem with geometric and cone constraints, whose objective function is a composition of n+1 functions. For this problem we propose a conjugate dual problem, where the functions involved in the objective function of the primal problem are
decomposed. Furthermore, we formulate generalized interior point regularity conditions for strong duality and give necessary and sufficient optimality conditions. As applications of this approach we determine the formulae of the conjugate as well as the biconjugate of the objective function of the primal problem and analyze an optimization problem having as objective function the sum of reciprocals of concave functions.
In the second part of this thesis we discuss in the sense of the introduced duality concept three classes of minimax location problems. The first one consists of nonlinear and linear single minimax location problems with geometric constraints, where the maximum of nonlinear or linear functions composed with gauges between pairs of a new and existing points will be minimized. The version of the nonlinear location problem is additionally considered with set-up costs. The second class of minimax location problems deals with multifacility location problems as suggested by Drezner (1991), where for each given point the sum of weighted distances to all facilities plus set-up costs is determined and the maximal value of these sums is to be minimized. As the last and third class the classical multifacility location problem with geometrical constraints is considered in a generalized form where the maximum of gauges between pairs of new facilities and the maximum of gauges between pairs of new and existing facilities will be minimized. To each of these location problems associated dual problems will be formulated as well as corresponding duality statements and necessary and sufficient optimality conditions. To illustrate the results of the duality approach and to give a more detailed characterization of the relations between the location problems and their corresponding duals, we consider examples in the Euclidean space.
This thesis ends with a numerical approach for solving minimax location problems by epigraphical splitting methods. In this framework, we give formulae for the projections onto the epigraphs of several sums of powers of weighted norms as well as formulae for the projection onto the epigraphs of gauges. Numerical experiments document the usefulness of our approach for the
discussed location problems.
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Weimann, Martin. "La trace en géométrie projective et torique". Phd thesis, Université Sciences et Technologies - Bordeaux I, 2006. http://tel.archives-ouvertes.fr/tel-00136109.
On étudie la notion de trace et les problèmes d'Abel-inverse à l'aide du calcul résiduel dans les cadres projectifs et toriques. Dans la première partie, on obtient une caractérisation algébrique des formes traces sur une hypersurface analytique à l'aide du calcul résiduel élémentaire d'une variable. En conséquence, une version plus forte du théorème d'Abel-inverse de Henkin et Passare est prouvée. On montre que ce théorème est conséquence de la rigidité d'un système différentiel particulier lié à une équation de type ”onde de choc” et on établit le lien avec le théorème de Wood sur l'algébricité d'une famille de germes d'hypersurfaces analytiques. Enfin, on obtient une nouvelle méthode pour calculer la dimension de l'espace des formes abéliennes de degré maximal sur une hypersurface projective. Dans la seconde partie, on caractérise de manière combinatoire les familles de fibrés en droites permettant de définir une notion intrinsèque de concavité dans une variété torique complète lisse et on étudie les ensembles analytiques dégénérés correspondants. On étend ainsi la notion de trace au cas torique. Courants résidus, résidus toriques et résultants donnent une borne optimale sur le degrés des traces en les différents paramètres. Si la variété torique est projective, on obtient finalement une version torique des théorèmes de Wood et d'Abel-inverse, permettant une description plus précise du support du polynôme construit dans le cas hypersurface.
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Phan, Tran Duc Minh. "Une méthode de dualité pour des problèmes non convexes du Calcul des Variations". Thesis, Toulon, 2018. http://www.theses.fr/2018TOUL0006/document.
Dans cette thèse, nous étudions un principe général de convexification permettant de traiter certainsproblèmes variationnels non convexes sur Rd. Grâce à ce principe nous pouvons mettre en oeuvre lespuissantes techniques de dualité et ramener de tels problèmes à des formulations de type primal–dualdans Rd+1, rendant ainsi efficace la recherche numérique de minima globaux. Une théorie de ladualité et des champs de calibration est reformulée dans le cas de fonctionnelles à croissance linéaire.Sous certaines hypothèses, cela nous permet de généraliser un principe d’exclusion découvert parVisintin dans les années 1990 et de réduire le problème initial à la minimisation d’une fonctionnelleconvexe sur Rd. Ce résultat s’applique notamment à une classe de problèmes à frontière libre oumulti-phasique donnant lieu à des tests numériques très convaincants au vu de la qualité des interfacesobtenues. Ensuite nous appliquons la théorie des calibrations à un problème classique de surfacesminimales avec frontière libre et établissons de nouveaux résultats de comparaison avec sa varianteoù la fonctionnelle des surfaces minimales est remplacée par la variation totale. Nous généralisonsla notion de calibrabilité introduite par Caselles-Chambolle et Al. et construisons explicitementune solution duale pour le problème associé à la seconde fonctionnelle en utilisant un potentiellocalement Lipschitzien lié à la distance au cut-locus. La dernière partie de la thèse est consacrée auxalgorithmes d’optimisation de type primal-dual pour la recherche de points selle, en introduisant denouvelles variantes plus efficaces en précision et temps calcul. Nous avons en particulier introduit unevariante semi-implicite de la méthode d’Arrow-Hurwicz qui permet de réduire le nombre d’itérationsnécessaires pour obtenir une qualité satisfaisante des interfaces. Enfin nous avons traité la nondifférentiabilité structurelle des Lagrangiens utilisés à l’aide d’une méthode géométrique de projectionsur l’épigraphe offrant ainsi une alternative aux méthodes classiques de régularisation In this thesis, we study a general principle of convexification to treat certain non convex variationalproblems in Rd. Thanks to this principle we are able to enforce the powerful duality techniques andbring back such problems to primal-dual formulations in Rd+1, thus making efficient the numericalsearch of a global minimizer. A theory of duality and calibration fields is reformulated in the caseof linear-growth functionals. Under suitable assumptions, this allows us to revisit and extend anexclusion principle discovered by Visintin in the 1990s and to reduce the original problem to theminimization of a convex functional in Rd. This result is then applied successfully to a class offree boundary or multiphase problems that we treat numerically obtaining very accurate interfaces.On the other hand we apply the theory of calibrations to a classical problem of minimal surfaceswith free boundary and establish new results related to the comparison with its variant where theminimal surfaces functional is replaced by the total variation. We generalize the notion of calibrabilityintroduced by Caselles-Chambolle and Al. and construct explicitly a dual solution for the problemassociated with the second functional by using a locally Lipschitzian potential related to the distanceto the cut-locus. The last part of the thesis is devoted to primal-dual optimization algorithms forthe search of saddle points, introducing new more efficient variants in precision and computationtime. In particular, we experiment a semi-implicit variant of the Arrow-Hurwicz method whichallows to reduce drastically the number of iterations necessary to obtain a sharp accuracy of theinterfaces. Eventually we tackle the structural non-differentiability of the Lagrangian arising fromour method by means of a geometric projection method on the epigraph, thus offering an alternativeto all classical regularization methods
Faure, Claude-Alain, i Alfred Frölicher. "Duality". W Modern Projective Geometry, 255–73. Dordrecht: Springer Netherlands, 2000. http://dx.doi.org/10.1007/978-94-015-9590-2_11.
Coxeter, H. S. M. "The Principle of Duality". W Projective Geometry, 24–32. New York, NY: Springer New York, 1987. http://dx.doi.org/10.1007/978-1-4612-6385-2_3.
Urquhart, Alasdair. "Duality Theory for Projective Algebras". W Relational Methods in Computer Science, 33–47. Berlin, Heidelberg: Springer Berlin Heidelberg, 2006. http://dx.doi.org/10.1007/11734673_3.
Reider, Igor. "J(X; L, d) and the Langlands Duality". W Nonabelian Jacobian of Projective Surfaces, 197–212. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013. http://dx.doi.org/10.1007/978-3-642-35662-9_12.
Fillastre, François, i Andrea Seppi. "Spherical, hyperbolic, and other projective geometries: convexity, duality, transitions". W Eighteen Essays in Non-Euclidean Geometry, 321–409. Zuerich, Switzerland: European Mathematical Society Publishing House, 2019. http://dx.doi.org/10.4171/196-1/16.
Georg Schaathun, Hans. "Duality and Greedy Weights of Linear Codes and Projective Multisets". W Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 92–101. Berlin, Heidelberg: Springer Berlin Heidelberg, 2001. http://dx.doi.org/10.1007/3-540-45624-4_10.
Conradt, Oliver. "The Principle of Duality in Clifford Algebra and Projective Geometry". W Clifford Algebras and their Applications in Mathematical Physics, 157–93. Boston, MA: Birkhäuser Boston, 2000. http://dx.doi.org/10.1007/978-1-4612-1368-0_10.
Streszczenia konferencji na temat "Projective duality":
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Skala, Vaclav. "Projective geometry and duality for graphics, games and visualization". W SIGGRAPH Asia 2012 Courses. New York, New York, USA: ACM Press, 2012. http://dx.doi.org/10.1145/2407783.2407793.
RAFTOPOULOS, DIONYSIOS G. "Projective Geometrical Space, Duality, Harmonicity and the Inverse Square Law". W Unified Field Mechanics II: Preliminary Formulations and Empirical Tests, 10th International Symposium Honouring Mathematical Physicist Jean-Pierre Vigier. WORLD SCIENTIFIC, 2017. http://dx.doi.org/10.1142/9789813232044_0046.
Skala, Vaclav. "Projective geometry, duality and Plücker coordinates for geometric computations with determinants on GPUs". W INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED MATHEMATICS (ICNAAM 2016). Author(s), 2017. http://dx.doi.org/10.1063/1.4992684.
Wang, Geyang, Yao Zhao, Chunyu Lin, Meiqin Liu i Jian Jin. "Dually Octagonal Projection for 360 Video with Less-Distortion Introduced". W 2020 15th IEEE International Conference on Signal Processing (ICSP). IEEE, 2020. http://dx.doi.org/10.1109/icsp48669.2020.9320966.
