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Journal articles on the topic 'Projective duality'

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Published: 24 March 2023

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1

Kuznetsov, Alexander. "Homological projective duality." Publications mathématiques de l'IHÉS 105, no. 1 (June 2007): 157–220. http://dx.doi.org/10.1007/s10240-007-0006-8.

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2

Perry, Alexander. "Noncommutative homological projective duality." Advances in Mathematics 350 (July 2019): 877–972. http://dx.doi.org/10.1016/j.aim.2019.04.052.

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3

Rennemo, Jørgen Vold, and Ed Segal. "Hori-mological projective duality." Duke Mathematical Journal 168, no. 11 (August 2019): 2127–205. http://dx.doi.org/10.1215/00127094-2019-0014.

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4

Garcia, A., and J. F. Voloch. "Duality for projective curves." Boletim da Sociedade Brasileira de Matem�tica 21, no. 2 (September 1991): 159–75. http://dx.doi.org/10.1007/bf01237362.

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5

Goerss, Paul G. "Projective and Injective Hopf Algebras Over the Dyer-Lashof Algebra." Canadian Journal of Mathematics 45, no. 5 (October 1, 1993): 944–76. http://dx.doi.org/10.4153/cjm-1993-053-9.

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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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6

Kuznetsov, Alexander, and Alexander Perry. "Homological projective duality for quadrics." Journal of Algebraic Geometry 30, no. 3 (January 15, 2021): 457–76. http://dx.doi.org/10.1090/jag/767.

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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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7

FRISK, ANDERS, and VOLODYMYR MAZORCHUK. "PROPERLY STRATIFIED ALGEBRAS AND TILTING." Proceedings of the London Mathematical Society 92, no. 1 (December 19, 2005): 29–61. http://dx.doi.org/10.1017/s0024611505015431.

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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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8

Bruce, J. W. "Lines, surfaces and duality." Mathematical Proceedings of the Cambridge Philosophical Society 112, no. 1 (July 1992): 53–61. http://dx.doi.org/10.1017/s0305004100070754.

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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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9

Barrett, David E. "Holomorphic projection and duality for domains in complex projective space." Transactions of the American Mathematical Society 368, no. 2 (April 3, 2015): 827–50. http://dx.doi.org/10.1090/tran/6338.

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10

Benson, D. J., and Jon F. Carlson. "Projective Resolutions and Poincare Duality Complexes." Transactions of the American Mathematical Society 342, no. 2 (April 1994): 447. http://dx.doi.org/10.2307/2154636.

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11

Coghetto, Roland. "Duality Notions in Real Projective Plane." Formalized Mathematics 29, no. 4 (December 1, 2021): 161–73. http://dx.doi.org/10.2478/forma-2021-0016.

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Summary In this article, we check with the Mizar system [1], [2], the converse of Desargues’ theorem and the converse of Pappus’ theorem of the real projective plane. It is well known that in the projective plane, the notions of points and lines are dual [11], [9], [15], [8]. Some results (analytical, synthetic, combinatorial) of projective geometry are already present in some libraries Lean/Hott [5], Isabelle/Hol [3], Coq [13], [14], [4], Agda [6], . . . . Proofs of dual statements by proof assistants have already been proposed, using an axiomatic method (for example see in [13] - the section on duality: “[...] For every theorem we prove, we can easily derive its dual using our function swap [...]2”). In our formalisation, we use an analytical rather than a synthetic approach using the definitions of Leończuk and Prażmowski of the projective plane [12]. Our motivation is to show that it is possible by developing dual definitions to find proofs of dual theorems in a few lines of code. In the first part, rather technical, we introduce definitions that allow us to construct the duality between the points of the real projective plane and the lines associated to this projective plane. In the second part, we give a natural definition of line concurrency and prove that this definition is dual to the definition of alignment. Finally, we apply these results to find, in a few lines, the dual properties and theorems of those defined in the article [12] (transitive, Vebleian, at_least_3rank, Fanoian, Desarguesian, 2-dimensional). We hope that this methodology will allow us to continued more quickly the proof started in [7] that the Beltrami-Klein plane is a model satisfying the axioms of the hyperbolic plane (in the sense of Tarski geometry [10]).
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12

D?Agnolo, Andrea, and Pierre Schapira. "Leray?s quantization of projective duality." Duke Mathematical Journal 84, no. 2 (August 1996): 453–96. http://dx.doi.org/10.1215/s0012-7094-96-08415-x.

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13

Benson, D. J., and Jon F. Carlson. "Projective resolutions and Poincaré duality complexes." Transactions of the American Mathematical Society 342, no. 2 (February 1, 1994): 447–88. http://dx.doi.org/10.1090/s0002-9947-1994-1142778-x.

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14

Bernardara, Marcello, Michele Bolognesi, and Daniele Faenzi. "Homological projective duality for determinantal varieties." Advances in Mathematics 296 (June 2016): 181–209. http://dx.doi.org/10.1016/j.aim.2016.04.003.

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15

Yekutieli, Amnon, and James J. Zhang. "Serre duality for noncommutative projective schemes." Proceedings of the American Mathematical Society 125, no. 3 (1997): 697–707. http://dx.doi.org/10.1090/s0002-9939-97-03782-9.

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16

Aldrovandi, R., and L. A. Saeger. "Projective fourier duality and Weyl quantization." International Journal of Theoretical Physics 36, no. 3 (March 1997): 573–612. http://dx.doi.org/10.1007/bf02435880.

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17

Jiang, Qingyuan, Naichung Conan Leung, and Ying Xie. "Categorical Plücker Formula and Homological Projective Duality." Journal of the European Mathematical Society 23, no. 6 (February 4, 2021): 1859–98. http://dx.doi.org/10.4171/jems/1045.

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18

Han, Deguang, and David Larson. "Frame duality properties for projective unitary representations." Bulletin of the London Mathematical Society 40, no. 4 (June 25, 2008): 685–95. http://dx.doi.org/10.1112/blms/bdn049.

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19

Hefez, Abramo, and Neuza Kakuta. "DUALITY OF OSCULATING DEVELOPABLES OF PROJECTIVE CURVES." Communications in Algebra 29, no. 1 (March 21, 2001): 285–301. http://dx.doi.org/10.1081/agb-100000801.

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20

Sharpe, Eric. "GLSM's, gerbes, and Kuznetsov's homological projective duality." Journal of Physics: Conference Series 462 (December 31, 2013): 012047. http://dx.doi.org/10.1088/1742-6596/462/1/012047.

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21

Smith, Larry, and R. E. Stong. "Projective bundle ideals and Poincaré duality algebras." Journal of Pure and Applied Algebra 215, no. 4 (April 2011): 609–27. http://dx.doi.org/10.1016/j.jpaa.2010.06.011.

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22

Aluffi, Paolo. "Projective duality and a Chern-Mather involution." Transactions of the American Mathematical Society 370, no. 3 (November 22, 2017): 1803–22. http://dx.doi.org/10.1090/tran/7042.

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23

SLILATY, DANIEL C. "Matroid Duality from Topological Duality in Surfaces of Nonnegative Euler Characteristic." Combinatorics, Probability and Computing 11, no. 5 (September 2002): 515–28. http://dx.doi.org/10.1017/s0963548302005278.

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Let G be a connected graph that is 2-cell embedded in a surface S, and let G* be its topological dual graph. We will define and discuss several matroids whose element set is E(G), for S homeomorphic to the plane, projective plane, or torus. We will also state and prove old and new results of the type that the dual matroid of G is the matroid of the topological dual G*.
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24

Jiao, Pengjie. "The generalized Auslander–Reiten duality on an exact category." Journal of Algebra and Its Applications 17, no. 12 (December 2018): 1850227. http://dx.doi.org/10.1142/s0219498818502274.

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We introduce a notion of generalized Auslander–Reiten duality on a Hom-finite Krull–Schmidt exact category [Formula: see text]. This duality induces the generalized Auslander–Reiten translation functors [Formula: see text] and [Formula: see text]. They are mutually quasi-inverse equivalences between the stable categories of two full subcategories [Formula: see text] and [Formula: see text] of [Formula: see text]. A non-projective indecomposable object lies in the domain of [Formula: see text] if and only if it appears as the third term of an almost split conflation; dually, a non-injective indecomposable object lies in the domain of [Formula: see text] if and only if it appears as the first term of an almost split conflation. We study the generalized Auslander–Reiten duality on the category of finitely presented representations of locally finite interval-finite quivers.
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25

Rennemo, Jørgen Vold. "The homological projective dual of." Compositio Mathematica 156, no. 3 (January 17, 2020): 476–525. http://dx.doi.org/10.1112/s0010437x19007772.

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We study the derived category of a complete intersection $X$ of bilinear divisors in the orbifold $\operatorname{Sym}^{2}\mathbb{P}(V)$. Our results are in the spirit of Kuznetsov’s theory of homological projective duality, and we describe a homological projective duality relation between $\operatorname{Sym}^{2}\mathbb{P}(V)$ and a category of modules over a sheaf of Clifford algebras on $\mathbb{P}(\operatorname{Sym}^{2}V^{\vee })$. The proof follows a recently developed strategy combining variation of geometric invariant theory (VGIT) stability and categories of global matrix factorisations. We begin by translating $D^{b}(X)$ into a derived category of factorisations on a Landau–Ginzburg (LG) model, and then apply VGIT to obtain a birational LG model. Finally, we interpret the derived factorisation category of the new LG model as a Clifford module category. In some cases we can compute this Clifford module category as the derived category of a variety. As a corollary we get a new proof of a result of Hosono and Takagi, which says that a certain pair of non-birational Calabi–Yau 3-folds have equivalent derived categories.
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26

Irving, Ronald S. "Projective Modules in the Category  S : Self-Duality." Transactions of the American Mathematical Society 291, no. 2 (October 1985): 701. http://dx.doi.org/10.2307/2000106.

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27

Enochs, Edgar E., Overtoun M. G. Jenda, and Jinzhong Xu. "Foxby duality and Gorenstein injective and projective modules." Transactions of the American Mathematical Society 348, no. 8 (1996): 3223–34. http://dx.doi.org/10.1090/s0002-9947-96-01624-8.

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28

DI ROCCO, SANDRA. "PROJECTIVE DUALITY OF TORIC MANIFOLDS AND DEFECT POLYTOPES." Proceedings of the London Mathematical Society 93, no. 1 (June 9, 2006): 85–104. http://dx.doi.org/10.1017/s0024611505015686.

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Non-singular toric embeddings with dual defect are classified. The associated polytopes, called defect polytopes, are proven to be the class of Delzant integral polytopes for which a combinatorial invariant vanishes. The structure of a defect polytope is described.
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29

Wall, C. T. C. "Duality of real projective plane curves: Klein's equation." Topology 35, no. 2 (April 1996): 355–62. http://dx.doi.org/10.1016/0040-9383(95)00021-6.

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30

Hosono, Shinobu, and Hiromichi Takagi. "Towards homological projective duality for S2P3 and S2P4." Advances in Mathematics 317 (September 2017): 371–409. http://dx.doi.org/10.1016/j.aim.2017.06.039.

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31

Pirashvili, Teimuraz. "Projective and injective symmetric categorical groups and duality." Proceedings of the American Mathematical Society 143, no. 3 (October 16, 2014): 1315–23. http://dx.doi.org/10.1090/s0002-9939-2014-12354-9.

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32

Brundan, Jonathan, and Alexander Kleshchev. "Projective representations of symmetric groups via Sergeev duality." Mathematische Zeitschrift 239, no. 1 (January 1, 2002): 27–68. http://dx.doi.org/10.1007/s002090100282.

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33

RAO, M. M. "CHARACTERIZATION AND DUALITY OF PROJECTIVE AND DIRECT LIMITS OF MEASURES AND APPLICATIONS." International Journal of Mathematics 22, no. 08 (August 2011): 1089–119. http://dx.doi.org/10.1142/s0129167x11007148.

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A representation of the projective system of abstract σ-finite measures on a topological family is given and with it a general characterization of their projective limits is obtained. Strong and weak direct limits of direct systems of measures as well as the duality between them are characterized with detailed analysis. This is used to prove several results of both theoretical and applicational importance. These include obtaining the equivalence of regular martingales and some projective systems admitting limits, measure representations of general semi-martingales, an extension theorem of product conditional measures, and a generalization of Rokhlin's theorem on completely positive entropy sequences of Lebesgue systems to general probability spaces. Further characterizations of projective and direct limits receive an extended treatment, indicating a great potential for future works.
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34

Russo, Francesco. "Projective duality and non-degenerated symplectic Monge–Ampère equations." Banach Center Publications 117 (2019): 113–44. http://dx.doi.org/10.4064/bc117-4.

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35

Shishkin, A. B. "Projective and Injective Descriptions in the Complex Domain. Duality." Izvestiya of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics 14, no. 1 (2014): 47–65. http://dx.doi.org/10.18500/1816-9791-2014-14-1-47-65.

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36

Tohaˇneanu, Ştefan O. "Projective duality of arrangements with quadratic logarithmic vector fields." Discrete Mathematics 339, no. 1 (January 2016): 54–61. http://dx.doi.org/10.1016/j.disc.2015.07.004.

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37

Carocci, Francesca, and Zak Turčinović. "Homological Projective Duality for Linear Systems with Base Locus." International Mathematics Research Notices 2020, no. 21 (September 27, 2018): 7829–56. http://dx.doi.org/10.1093/imrn/rny222.

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Abstract We show how blowing up varieties in base loci of linear systems gives a procedure for creating new homological projective duals from old. Starting with a homological projective (HP) dual pair $X,Y$ and smooth orthogonal linear sections $X_L,Y_L$, we prove that the blowup of $X$ in $X_L$ is naturally HP dual to $Y_L$. The result also holds true when $Y$ is a noncommutative variety or just a category. We extend the result to the case where the base locus $X_L$ is a multiple of a smooth variety and the universal hyperplane has rational singularities; here the HP dual is a weakly crepant categorical resolution of singularities of $Y_L$. Finally we give examples where, starting with a noncommutative $Y$, the above process nevertheless gives geometric HP duals.
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38

GREENLEES, J. P. C., and G. R. WILLIAMS. "POINCARÉ DUALITY FORK-THEORY OF EQUIVARIANT COMPLEX PROJECTIVE SPACES." Glasgow Mathematical Journal 50, no. 1 (January 2008): 111–27. http://dx.doi.org/10.1017/s0017089507003990.

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39

KLIMČÍK, C., and P. ŠEVERA. "STRINGS IN SPACE-TIME COTANGENT BUNDLE AND T-DUALITY." Modern Physics Letters A 10, no. 04 (February 10, 1995): 323–29. http://dx.doi.org/10.1142/s0217732395000351.

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A simple geometric description of T-duality is given by identifying the cotangent bundles of the original and the dual manifold. Strings propagate naturally in the cotangent bundle and the original and the dual string phase spaces are obtained by different projections. Buscher's transformation follows readily and it is literally projective. As an application of the formalism, we prove that the duality is a symplectomorphism of the string phase spaces.
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40

SKALA, VACLAV. "INTERSECTION COMPUTATION IN PROJECTIVE SPACE USING HOMOGENEOUS COORDINATES." International Journal of Image and Graphics 08, no. 04 (October 2008): 615–28. http://dx.doi.org/10.1142/s021946780800326x.

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There are many algorithms based on computation of intersection of lines, planes etc. Those algorithms are based on representation in the Euclidean space. Sometimes, very complex mathematical notations are used to express simple mathematical solutions. This paper presents solutions of some selected problems that can be easily solved by the projective space representation. Sometimes, if the principle of duality is used, quite surprising solutions can be found and new useful theorems can be generated as well. It will be shown that it is not necessary to solve linear system of equations to find the intersection of two lines in the case of E2 or the intersection of three planes in the case of E3. Plücker coordinates and principle of duality are used to derive an equation of a parametric line in E3 as an intersection of two planes. This new formulation avoids division operations and increases the robustness of computation. The presented approach for intersection computation is well suited especially for applications where robustness is required, e.g. large GIS/CAD/CAM systems etc.
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41

NODA, TOMONORI. "SYMPLECTIC STRUCTURES ON STATISTICAL MANIFOLDS." Journal of the Australian Mathematical Society 90, no. 3 (June 2011): 371–84. http://dx.doi.org/10.1017/s1446788711001285.

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AbstractA relationship between symplectic geometry and information geometry is studied. The square of a dually flat space admits a natural symplectic structure that is the pullback of the canonical symplectic structure on the cotangent bundle of the dually flat space via the canonical divergence. With respect to the symplectic structure, there exists a moment map whose image is the dually flat space. As an example, we obtain a duality relation between the Fubini–Study metric on a projective space and the Fisher metric on a statistical model on a finite set. Conversely, a dually flat space admitting a symplectic structure is locally symplectically isomorphic to the cotangent bundle with the canonical symplectic structure of some dually flat space. We also discuss nonparametric cases.
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42

EDER, GÜNTHER, and GEORG SCHIEMER. "HILBERT, DUALITY, AND THE GEOMETRICAL ROOTS OF MODEL THEORY." Review of Symbolic Logic 11, no. 1 (December 29, 2017): 48–86. http://dx.doi.org/10.1017/s1755020317000260.

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AbstractThe article investigates one of the key contributions to modern structural mathematics, namely Hilbert’sFoundations of Geometry(1899) and its mathematical roots in nineteenth-century projective geometry. A central innovation of Hilbert’s book was to provide semantically minded independence proofs for various fragments of Euclidean geometry, thereby contributing to the development of the model-theoretic point of view in logical theory. Though it is generally acknowledged that the development of model theory is intimately bound up with innovations in 19th century geometry (in particular, the development of non-Euclidean geometries), so far, little has been said about how exactly model-theoretic concepts grew out of methodological investigations within projective geometry. This article is supposed to fill this lacuna and investigates this geometrical prehistory of modern model theory, eventually leading up to Hilbert’sFoundations.
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43

Kato, Tsuyoshi. "Geometric Representations of Interacting Maps." International Journal of Mathematics and Mathematical Sciences 2010 (2010): 1–48. http://dx.doi.org/10.1155/2010/783738.

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Tropical geometry is a kind of dynamical scale transform which connects automata with real rational dynamics. Real rational dynamics are deeply studied from global analytic viewpoints. On the other hand, automata appear in various contexts in topology, combinatorics, and integrable systems. In this paper we study the analysis of these materials passing through tropical geometry. In particular we discover a new duality on the set of automata which arise from the projective duality in algebraic geometry.
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44

Wang, Junpeng, and Zhenxing Di. "Relative Gorenstein rings and duality pairs." Journal of Algebra and Its Applications 19, no. 08 (August 28, 2019): 2050147. http://dx.doi.org/10.1142/s0219498820501479.

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Let [Formula: see text] be a ring (not necessarily commutative) and [Formula: see text] a bi-complete duality pair. We investigate the notions of (flat-typed) [Formula: see text]-Gorenstein rings, which unify Iwanaga–Gorenstein rings, Ding–Chen rings, AC-Gorenstein rings and Gorenstein [Formula: see text]-coherent rings. Using an abelian model category approach, we show that [Formula: see text] and [Formula: see text], the homotopy categories of all exact complexes of projective and injective [Formula: see text]-modules, are triangulated equivalent whenever [Formula: see text] is a flat-typed [Formula: see text]-Gorenstein ring.
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45

Ballard, Matthew, Dragos Deliu, David Favero, M. Umut Isik, and Ludmil Katzarkov. "Homological projective duality via variation of geometric invariant theory quotients." Journal of the European Mathematical Society 19, no. 4 (2017): 1127–58. http://dx.doi.org/10.4171/jems/689.

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46

van Eijndhoven, S., and P. Kruszyński. "Spectral trajectories,duality and inductive-projective limits of Hilbert spaces." Studia Mathematica 91, no. 1 (1988): 45–60. http://dx.doi.org/10.4064/sm-91-1-45-60.

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47

Li, Yizheng, and Chengjun Hou. "Duality properties in von Neumann algebras of projective unitary representations." Filomat 27, no. 1 (2013): 9–13. http://dx.doi.org/10.2298/fil1301009l.

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48

Sergeev, Alexander. "The Howe duality and the projective representations of symmetric groups." Representation Theory of the American Mathematical Society 3, no. 14 (November 9, 1999): 416–34. http://dx.doi.org/10.1090/s1088-4165-99-00085-0.

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49

Megretski, A., and A. Rantzer. "Robust Control Synthesis by Convex Optimization: Projective Parameterization and Duality." IFAC Proceedings Volumes 26, no. 2 (July 1993): 5–8. http://dx.doi.org/10.1016/s1474-6670(17)48882-3.

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50

Bouwknegt, Peter, Jarah Evslin, Branislav Jurčo, Varghese Mathai, and Hisham Sati. "Flux compactifications on projective spaces and the $S$-duality puzzle." Advances in Theoretical and Mathematical Physics 10, no. 3 (2006): 345–94. http://dx.doi.org/10.4310/atmp.2006.v10.n3.a3.

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