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Journal articles on the topic 'Finite geometry; projective geometry'

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Published: 10 December 2022

Last updated: 28 January 2023

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1

Newelski, Ludomir. "Geometry of *-Finite Types." Journal of Symbolic Logic 64, no. 4 (December 1999): 1375–95. http://dx.doi.org/10.2307/2586784.

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AbstractAssume T is a superstable theory with < 2ℵ0 countable models. We prove that any *- algebraic type of -rank > 0 is m-nonorthogonal to a *-algebraic type of -rank 1. We study the geometry induced by m-dependence on a *-algebraic type p* of -rank 1. We prove that after some localization this geometry becomes projective over a division ring . Associated with p* is a meager type p. We prove that p is determined by p* up to nonorthogonality and that underlies also the geometry induced by forking dependence on any stationarization of p. Also we study some *-algebraic *-groups of -rank 1 and prove that any *-algebraic *-group of -rank 1 is abelian-by-finite.

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2

Connes, Alain, and Caterina Consani. "Projective geometry in characteristic one and the epicyclic category." Nagoya Mathematical Journal 217 (March 2015): 95–132. http://dx.doi.org/10.1215/00277630-2887960.

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AbstractWe show that the cyclic and epicyclic categories which play a key role in the encoding of cyclic homology and the lambda operations, are obtained from projective geometry in characteristic one over the infinite semifield of max-plus integers ℤmax. Finite-dimensional vector spaces are replaced by modules defined by restriction of scalars from the one-dimensional free module, using the Frobenius endomorphisms of ℤmax. The associated projective spaces are finite and provide a mathematically consistent interpretation of Tits's original idea of a geometry over the absolute point. The self-duality of the cyclic category and the cyclic descent number of permutations both acquire a geometric meaning.

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3

Dodson, C. T. J. "Fréchet geometry via projective limits." International Journal of Geometric Methods in Modern Physics 11, no. 07 (August 2014): 1460017. http://dx.doi.org/10.1142/s0219887814600172.

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Fréchet spaces of sections arise naturally as configurations of a physical field. Some recent work in Fréchet geometry is briefly reviewed and some suggestions for future work are offered. An earlier result on the structure of second tangent bundles in the finite-dimensional case was extended to infinite-dimensional Banach manifolds and Fréchet manifolds that could be represented as projective limits of Banach manifolds. This led to further results concerning the characterization of second tangent bundles and differential equations in the more general Fréchet structure needed for applications.

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4

Connes, Alain, and Caterina Consani. "Projective geometry in characteristic one and the epicyclic category." Nagoya Mathematical Journal 217 (March 2015): 95–132. http://dx.doi.org/10.1017/s0027763000026969.

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AbstractWe show that the cyclic and epicyclic categories which play a key role in the encoding of cyclic homology and the lambda operations, are obtained from projective geometry in characteristic one over the infinite semifield ofmax-plus integersℤmax. Finite-dimensional vector spaces are replaced by modules defined by restriction of scalars from the one-dimensional free module, using the Frobenius endomorphisms of ℤmax. The associated projective spaces arefiniteand provide a mathematically consistent interpretation of Tits's original idea of a geometry over the absolute point. The self-duality of the cyclic category and the cyclic descent number of permutations both acquire a geometric meaning.

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5

Kohn, Kathlén, and Kristian Ranestad. "Projective Geometry of Wachspress Coordinates." Foundations of Computational Mathematics 20, no. 5 (November 11, 2019): 1135–73. http://dx.doi.org/10.1007/s10208-019-09441-z.

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Abstract We show that there is a unique hypersurface of minimal degree passing through the non-faces of a polytope which is defined by a simple hyperplane arrangement. This generalizes the construction of the adjoint curve of a polygon by Wachspress (A rational finite element basis, Academic Press, New York, 1975). The defining polynomial of our adjoint hypersurface is the adjoint polynomial introduced by Warren (Adv Comput Math 6:97–108, 1996). This is a key ingredient for the definition of Wachspress coordinates, which are barycentric coordinates on an arbitrary convex polytope. The adjoint polynomial also appears both in algebraic statistics, when studying the moments of uniform probability distributions on polytopes, and in intersection theory, when computing Segre classes of monomial schemes. We describe the Wachspress map, the rational map defined by the Wachspress coordinates, and the Wachspress variety, the image of this map. The inverse of the Wachspress map is the projection from the linear span of the image of the adjoint hypersurface. To relate adjoints of polytopes to classical adjoints of divisors in algebraic geometry, we study irreducible hypersurfaces that have the same degree and multiplicity along the non-faces of a polytope as its defining hyperplane arrangement. We list all finitely many combinatorial types of polytopes in dimensions two and three for which such irreducible hypersurfaces exist. In the case of polygons, the general such curves are elliptic. In the three-dimensional case, the general such surfaces are either K3 or elliptic.

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6

Aggarwal, M. L., Lih-Yuan Deng, and Mukta Datta Mazumder. "Optimal Fractional Factorial Plans Using Finite Projective Geometry." Communications in Statistics - Theory and Methods 37, no. 8 (February 22, 2008): 1258–65. http://dx.doi.org/10.1080/03610920701713351.

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7

Kalmbach H.E., Gudrun. "Projective Gravity." International Journal of Contemporary Research and Review 9, no. 03 (March 13, 2018): 20181–83. http://dx.doi.org/10.15520/ijcrr/2018/9/03/466.

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In [1] and [3] it was pointed out that octonians can replace an infinite dimensional Hilbert space and psi-waves descriptions concerning the states of deuteron which are finite in number. It is then clear that gravity needs projective and projection geometry to be described in a unified way with the three other basic forces of physics.

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8

Ben-Yaacov, Itay, Ivan Tomašić, and Frank O. Wagner. "The Group Configuration in Simple Theories and its Applications." Bulletin of Symbolic Logic 8, no. 2 (June 2002): 283–98. http://dx.doi.org/10.2178/bsl/1182353874.

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AbstractIn recent work, the authors have established the group configuration theorem for simple theories, as well as some of its main applications from geometric stability theory, such as the binding group theorem, or, in the ω-categorical case, the characterization of the forking geometry of a finitely based non-trivial locally modular regular type as projective geometry over a finite field and the equivalence of pseudolinearity and local modularity.The proof necessitated an extension of the model-theoretic framework to include almost hyperimaginaries, and the study of polygroups.

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9

Games, Richard A. "The Geometry of m-Sequences: Three-Valued Crosscorrelations and Quadrics in Finite Projective Geometry." SIAM Journal on Algebraic Discrete Methods 7, no. 1 (January 1986): 43–52. http://dx.doi.org/10.1137/0607005.

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10

Beil, Charlie. "Nonnoetherian geometry." Journal of Algebra and Its Applications 15, no. 09 (August 22, 2016): 1650176. http://dx.doi.org/10.1142/s0219498816501760.

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We introduce a theory of geometry for nonnoetherian commutative algebras with finite Krull dimension. In particular, we establish new notions of normalization and height: depiction (a special noetherian overring) and geometric codimension. The resulting geometries are algebraic varieties with positive-dimensional points, and are thus inherently nonlocal. These notions also give rise to new equivalent characterizations of noetherianity that are primarily geometric. We then consider an application to quiver algebras whose simple modules of maximal dimension are one dimensional at each vertex. We show that the vertex corner rings of [Formula: see text] are all isomorphic if and only if [Formula: see text] is noetherian, if and only if the center [Formula: see text] of [Formula: see text] is noetherian, if and only if [Formula: see text] is a finitely generated [Formula: see text]-module. Furthermore, we show that [Formula: see text] is depicted by a commutative algebra generated by the cycles in its quiver. We conclude with an example of a quiver algebra where projective dimension and geometric codimension, rather than height, coincide.

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11

Mecke, Klaus. "Biquadrics configure finite projective geometry into a quantum spacetime." EPL (Europhysics Letters) 120, no. 1 (October 1, 2017): 10007. http://dx.doi.org/10.1209/0295-5075/120/10007.

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12

Artstein-Avidan, Shiri, and Boaz A. Slomka. "The fundamental theorems of affine and projective geometry revisited." Communications in Contemporary Mathematics 19, no. 05 (August 18, 2016): 1650059. http://dx.doi.org/10.1142/s0219199716500590.

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The fundamental theorem of affine geometry is a classical and useful result. For finite-dimensional real vector spaces, the theorem roughly states that a bijective self-mapping which maps lines to lines is affine-linear. In this paper, we prove several generalizations of this result and of its classical projective counterpart. We show that under a significant geometric relaxation of the hypotheses, namely that only lines parallel to one of a fixed set of finitely many directions are mapped to lines, an injective mapping of the space must be of a very restricted polynomial form. We also prove that under mild additional conditions the mapping is forced to be affine-additive or affine-linear. For example, we show that five directions in three-dimensional real space suffice to conclude affine-additivity. In the projective setting, we show that [Formula: see text] fixed projective points in real [Formula: see text]-dimensional projective space, through which all projective lines that pass are mapped to projective lines, suffice to conclude projective-linearity.

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13

Huber, Michael. "Baer cones in finite projective spaces." Journal of Geometry 28, no. 2 (April 1987): 128–44. http://dx.doi.org/10.1007/bf01221941.

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14

RAFIE-RAD, M. "SPECIAL PROJECTIVE ALGEBRA OF RANDERS METRICS OF CONSTANT S-CURVATURE." International Journal of Geometric Methods in Modern Physics 09, no. 04 (May 6, 2012): 1250034. http://dx.doi.org/10.1142/s021988781250034x.

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The collection of all projective vector fields on a Finsler space (M, F) is a finite-dimensional Lie algebra with respect to the usual Lie bracket, called the projective algebra. A specific Lie sub-algebra of projective algebra of Randers spaces (called the special projective algebra) of non-zero constant S-curvature is studied and it is proved that its dimension is at most [Formula: see text]. Moreover, a local characterization of Randers spaces whose special projective algebra has maximum dimension is established. The results uncover somehow the complexity of projective Finsler geometry versus Riemannian geometry.

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15

Saniga, Metod, and Michel Planat. "Finite geometry behind the Harvey-Chryssanthacopoulos four-qubit magic rectangle." Quantum Information and Computation 12, no. 11&12 (November 2012): 1011–16. http://dx.doi.org/10.26421/qic12.11-12-8.

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A ``magic rectangle" of eleven observables of four qubits, employed by Harvey and Chryssanthacopoulos (2008) to prove the Bell-Kochen-Specker theorem in a 16-dimensional Hilbert space, is given a neat finite-geometrical reinterpretation in terms of the structure of the symplectic polar space $W(7,2)$ of the real four-qubit Pauli group. Each of the four sets of observables of cardinality five represents an elliptic quadric in the three-dimensional projective space of order two (PG$(3,2)$) it spans, whereas the remaining set of cardinality four corresponds to an affine plane of order two. The four ambient PG$(3, 2)$s of the quadrics intersect pairwise in a line, the resulting six lines meeting in a point. Projecting the whole configuration from this distinguished point (observable) one gets another, complementary ``magic rectangle" of the same qualitative structure.

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16

Goncalves, Adilson, and Chat Yin Ho. "On flag collineations of finite projective planes." Journal of Geometry 28, no. 2 (April 1987): 117–27. http://dx.doi.org/10.1007/bf01221940.

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17

Bhattarai, Hom Nath. "Pasch geometric spaces inducing finite projective planes." Journal of Geometry 34, no. 1-2 (March 1989): 6–13. http://dx.doi.org/10.1007/bf01224228.

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18

Biliotti, Mauro, and Eliana Francot. "Two-transitive orbits in finite projective planes." Journal of Geometry 82, no. 1-2 (August 2005): 1–24. http://dx.doi.org/10.1007/s00022-004-1645-2.

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19

Fan, Cui-Ling, and Jian-Guo Lei. "Constructions of Difference Systems of Sets From Finite Projective Geometry." IEEE Transactions on Information Theory 58, no. 1 (January 2012): 130–38. http://dx.doi.org/10.1109/tit.2011.2170921.

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20

Durante, Nicola, and Alessandro Siciliano. "The Geometry of Elation Groups of a Finite Projective Space." Mediterranean Journal of Mathematics 10, no. 1 (January 12, 2012): 439–48. http://dx.doi.org/10.1007/s00009-011-0173-1.

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21

TOMAŠIĆ, IVAN, and FRANK O. WAGNER. "APPLICATIONS OF THE GROUP CONFIGURATION THEOREM IN SIMPLE THEORIES." Journal of Mathematical Logic 03, no. 02 (November 2003): 239–55. http://dx.doi.org/10.1142/s0219061303000261.

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We reconstruct the group action in the group configuration theorem. We apply it to show that in an ω-categorical theory a finitely based pseudolinear regular type is locally modular, and the geometry associated to a finitely based locally modular regular type is projective geometry over a finite field.

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22

Givant, Steven. "Inequivalent representations of geometric relation algebras." Journal of Symbolic Logic 68, no. 1 (March 2003): 267–310. http://dx.doi.org/10.2178/jsl/1045861514.

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AbstractIt is shown that the automorphism group of a relation algebra constructed from a projective geometry P is isomorphic to the collineation group of P. Also, the base automorphism group of a representation of over an affine geometry D is isomorphic to the quotient of the collineation group of D by the dilatation subgroup. Consequently, the total number of inequivalent representations of , for finite geometries P, is the sum of the numberswhere D ranges over a list of the non-isomorphic affine geometries having P as their geometry at infinity. This formula is used to compute the number of inequivalent representations of relation algebras constructed over projective lines of order at most 10. For instance, the relation algebra constructed over the projective line of order 9 has 56,700 mutually inequivalent representations.

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23

Nagy, G. P., and T. Szonyi. "Caps in finite projective spaces of odd order." Journal of Geometry 59, no. 1-2 (July 1997): 103–13. http://dx.doi.org/10.1007/bf01229569.

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24

Rajola, Sandro. "Colourings with fixed congruence in finite projective planes." Journal of Geometry 59, no. 1-2 (July 1997): 141–51. http://dx.doi.org/10.1007/bf01229572.

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25

Benoist, Yves, and Dominique Hulin. "Cubic differentials and finite volume convex projective surfaces." Geometry & Topology 17, no. 1 (April 8, 2013): 595–620. http://dx.doi.org/10.2140/gt.2013.17.595.

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26

Shangdi, Chen, Zhang Xiaollian, and Ma Hao. "Two constructions of A3-codes from projective geometry in finite fields." Journal of China Universities of Posts and Telecommunications 22, no. 2 (April 2015): 52–59. http://dx.doi.org/10.1016/s1005-8885(15)60639-2.

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27

Thas, Joseph A., and Hendrik Van Maldeghem. "Some remarks on embeddings of the flag geometries of projective planes in finite projective spaces." Journal of Geometry 67, no. 1-2 (March 2000): 217–22. http://dx.doi.org/10.1007/bf01220312.

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28

Bray, Harrison, and David Constantine. "Entropy rigidity for finite volume strictly convex projective manifolds." Geometriae Dedicata 214, no. 1 (May 17, 2021): 543–57. http://dx.doi.org/10.1007/s10711-021-00627-w.

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29

Blokhuis, Aart. "Characterization of seminuclear sets in a finite projective plane." Journal of Geometry 40, no. 1-2 (April 1991): 15–19. http://dx.doi.org/10.1007/bf01225867.

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30

CHOUDHARY, SWADESH, HRISHIKESH SHARMA, and SACHIN PATKAR. "OPTIMAL FOLDING OF DATA FLOW GRAPHS BASED ON FINITE PROJECTIVE GEOMETRY USING VECTOR SPACE PARTITIONING." Discrete Mathematics, Algorithms and Applications 05, no. 04 (December 2013): 1350036. http://dx.doi.org/10.1142/s1793830913500365.

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A number of computations exist, especially in area of error-control coding and matrix computations, whose underlying data flow graphs are based on finite projective-geometry (PG)-based balanced bipartite graphs. Many of these applications of finite projective geometry are actively being researched upon, especially in coding theory. Almost all these applications need large bipartite graphs, whose nodes represent parallel computations. To reduce its implementation cost, reducing amount of system/hardware resources during design is an important engineering objective. In this context, we present a scheme to reduce resource utilization while designing systems modeled using PG-based graphs. In such systems, the number of processing units is equal to the number of vertices, each performing an atomic computation. We present a novel way of partitioning the vertex set assigned to various atomic computations, into blocks. Each block of partition is then assigned to a processing unit. A processing unit performs the computations corresponding to the vertices in the block assigned to it in a sequential fashion, thus creating the effect of folding the overall computation. The symmetric properties of projective space lattices enable us to develop a conflict-free communication schedule. We employed the technique of coset decomposition of a finite field for partitioning. The folding scheme achieves the best possible throughput, in lack of any overhead of shuffling data across memories while scheduling another computation on the same processing unit. We first provide a scheme for a finite projective space of dimension five, and the corresponding schedules. This specific scheme is then generalized for arbitrary finite projective spaces. Both the folding schemes have been verified by both simulation as well as hardware prototyping. For example, a semi-parallel decoder architecture for a new class of expander codes was designed and implemented using this scheme, with potential deployment in DVD-R/CD-ROM drives.

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31

CHOUDHARY, SWADESH, HRISHIKESH SHARMA, and SACHIN PATKAR. "OPTIMAL FOLDING OF DATA FLOW GRAPHS BASED ON FINITE PROJECTIVE GEOMETRY USING VECTOR SPACE PARTITIONING." Discrete Mathematics, Algorithms and Applications 06, no. 01 (February 18, 2014): 1450004. http://dx.doi.org/10.1142/s1793830914500049.

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A number of computations exist, especially in area of error-control coding and matrix computations, whose underlying data flow graphs are based on finite projective-geometry (PG) based balanced bipartite graphs. Many of these applications of finite projective geometry are actively being researched upon, especially in coding theory. Almost all these applications need large bipartite graphs, whose nodes represent parallel computations. To reduce its implementation cost, reducing amount of system/hardware resources during design is an important engineering objective. In this context, we present a scheme to reduce resource utilization while designing systems modeled using PG-based graphs. In such systems, the number of processing units is equal to the number of vertices, each performing an atomic computation. We present a novel way of partitioning the vertex set assigned to various atomic computations, into blocks. Each block of partition is then assigned to a processing unit. A processing unit performs the computations corresponding to the vertices in the block assigned to it in a sequential fashion, thus creating the effect of folding the overall computation. The symmetric properties of projective space lattices enable us to develop a conflict-free communication schedule. We employed the technique of coset decomposition of a finite field for partitioning. The folding scheme achieves the best possible throughput, in lack of any overhead of shuffling data across memories while scheduling another computation on the same processing unit. We first provide a scheme for a finite projective space of dimension five, and the corresponding schedules. This specific scheme is then generalized for arbitrary finite projective spaces. Both the folding schemes have been verified by both simulation as well as hardware prototyping. For example, a semi-parallel decoder architecture for a new class of expander codes was designed and implemented using this scheme, with potential deployment in DVD-R/CD-ROM drives.

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32

Wildberger, Norman. "Universal Hyperbolic Geometry, Sydpoints and Finite Fields: A Projective and Algebraic Alternative." Universe 4, no. 1 (January 1, 2018): 3. http://dx.doi.org/10.3390/universe4010003.

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33

Chen, Shangdi, and Xiaolian Zhang. "Three constructions of perfect authentication codes from projective geometry over finite fields." Applied Mathematics and Computation 253 (February 2015): 308–17. http://dx.doi.org/10.1016/j.amc.2014.12.088.

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34

Rémy, Bertrand, Amaury Thuillier, and Annette Werner. "Automorphisms of Drinfeld half-spaces over a finite field." Compositio Mathematica 149, no. 7 (April 26, 2013): 1211–24. http://dx.doi.org/10.1112/s0010437x12000905.

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AbstractWe show that the automorphism group of Drinfeld’s half-space over a finite field is the projective linear group of the underlying vector space. The proof of this result uses analytic geometry in the sense of Berkovich over the finite field equipped with the trivial valuation. We also take into account extensions of the base field.

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35

Wettl, Ferenc. "On the nuclei of a pointset of a finite projective plane." Journal of Geometry 30, no. 2 (December 1987): 157–63. http://dx.doi.org/10.1007/bf01227813.

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36

Beutelspacher, Albrecht. "Embedding linear spaces with two line degrees in finite projective planes." Journal of Geometry 26, no. 1 (April 1986): 43–61. http://dx.doi.org/10.1007/bf01221006.

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37

G�naltili, ?brahim, and ?�kr� Olgun. "On the embedding of some linear spaces in finite projective planes." Journal of Geometry 68, no. 1-2 (July 2000): 96–99. http://dx.doi.org/10.1007/bf01221065.

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38

Penttila, Tim, and Alessandro Siciliano. "On collineation groups of finite projective spaces containing a Singer cycle." Journal of Geometry 107, no. 3 (November 7, 2015): 617–26. http://dx.doi.org/10.1007/s00022-015-0300-4.

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39

Bellamy, Gwyn, and Alastair Craw. "Birational geometry of symplectic quotient singularities." Inventiones mathematicae 222, no. 2 (April 30, 2020): 399–468. http://dx.doi.org/10.1007/s00222-020-00972-9.

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Abstract For a finite subgroup $$\Gamma \subset \mathrm {SL}(2,\mathbb {C})$$ Γ ⊂ SL ( 2 , C ) and for $$n\ge 1$$ n ≥ 1 , we use variation of GIT quotient for Nakajima quiver varieties to study the birational geometry of the Hilbert scheme of n points on the minimal resolution S of the Kleinian singularity $$\mathbb {C}^2/\Gamma $$ C 2 / Γ . It is well known that $$X:={{\,\mathrm{{\mathrm {Hilb}}}\,}}^{[n]}(S)$$ X : = Hilb [ n ] ( S ) is a projective, crepant resolution of the symplectic singularity $$\mathbb {C}^{2n}/\Gamma _n$$ C 2 n / Γ n , where $$\Gamma _n=\Gamma \wr \mathfrak {S}_n$$ Γ n = Γ ≀ S n is the wreath product. We prove that every projective, crepant resolution of $$\mathbb {C}^{2n}/\Gamma _n$$ C 2 n / Γ n can be realised as the fine moduli space of $$\theta $$ θ -stable $$\Pi $$ Π -modules for a fixed dimension vector, where $$\Pi $$ Π is the framed preprojective algebra of $$\Gamma $$ Γ and $$\theta $$ θ is a choice of generic stability condition. Our approach uses the linearisation map from GIT to relate wall crossing in the space of $$\theta $$ θ -stability conditions to birational transformations of X over $$\mathbb {C}^{2n}/\Gamma _n$$ C 2 n / Γ n . As a corollary, we describe completely the ample and movable cones of X over $$\mathbb {C}^{2n}/\Gamma _n$$ C 2 n / Γ n , and show that the Mori chamber decomposition of the movable cone is determined by an extended Catalan hyperplane arrangement of the ADE root system associated to $$\Gamma $$ Γ by the McKay correspondence. In the appendix, we show that morphisms of quiver varieties induced by variation of GIT quotient are semismall, generalising a result of Nakajima in the case where the quiver variety is smooth.

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40

Dabrowski, Ludwik, Thomas Krajewski, and Giovanni Landi. "Non-linear σ-models in noncommutative geometry: fields with values in finite spaces." Modern Physics Letters A 18, no. 33n35 (November 20, 2003): 2371–79. http://dx.doi.org/10.1142/s0217732303012593.

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We study σ-models on noncommutative spaces, notably on noncommutative tori. We construct instanton solutions carrying a nontrivial topological charge q and satisfying a Belavin-Polyakov bound. The moduli space of these instantons is conjectured to consists of an ordinary torus endowed with a complex structure times a projective space [Formula: see text].

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41

Larose, Benoit. "Finite projective ordered sets." Order 8, no. 1 (1991): 33–40. http://dx.doi.org/10.1007/bf00385812.

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42

Storme, L., J. A. Thas, and S. K. J. Vereecke. "New Upper bounds for the sizes of caps in finite projective spaces." Journal of Geometry 73, no. 1-2 (July 1, 2002): 176–93. http://dx.doi.org/10.1007/s00022-002-8590-8.

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43

Limbupasiriporn, Jirapha. "Small sets of even type in finite projective planes of even order." Journal of Geometry 98, no. 1-2 (August 2010): 139–49. http://dx.doi.org/10.1007/s00022-010-0055-x.

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44

Wennink, Thomas. "Counting the number of trigonal curves of genus 5 over finite fields." Geometriae Dedicata 208, no. 1 (January 9, 2020): 31–48. http://dx.doi.org/10.1007/s10711-019-00508-3.

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AbstractThe trigonal curves of genus 5 can be represented by projective plane quintics that have one singularity of delta invariant one. Combining this with a partial sieve method for plane curves we count the number of such curves over any finite field. The main application is that this gives the motivic Euler characteristic of the moduli space of trigonal curves of genus 5.

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45

Hamada, Noboru, and Tor Helleseth. "A Characterization of Some Minihypers in a Finite Projective Geometry PG(t, 4)." European Journal of Combinatorics 11, no. 6 (November 1990): 541–48. http://dx.doi.org/10.1016/s0195-6698(13)80039-x.

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46

YANG, RONGWEI. "PROJECTIVE SPECTRUM IN BANACH ALGEBRAS." Journal of Topology and Analysis 01, no. 03 (September 2009): 289–306. http://dx.doi.org/10.1142/s1793525309000126.

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For a tuple A = (A1, A2, …, An) of elements in a unital algebra [Formula: see text] over ℂ, its projective spectrumP(A) or p(A) is the collection of z ∈ ℂn, or respectively z ∈ ℙn-1 such that A(z) = z1A1 + z2A2 + ⋯ + znAn is not invertible in [Formula: see text]. In finite dimensional case, projective spectrum is a projective hypersurface. When A is commuting, P(A) looks like a bundle over the Taylor spectrum of A. In the case [Formula: see text] is reflexive or is a C*-algebra, the projective resolvent setPc(A) := ℂn \ P(A) is shown to be a disjoint union of domains of holomorphy. [Formula: see text]-valued 1-form A-1(z)dA(z) reveals the topology of Pc(A), and a Chern–Weil type homomorphism from invariant multilinear functionals to the de Rham cohomology [Formula: see text] is established.

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47

ANDRUCHOW, ESTEBAN, and DEMETRIO STOJANOFF. "GEOMETRY OF CONDITIONAL EXPECTATIONS AND FINITE INDEX." International Journal of Mathematics 05, no. 02 (April 1994): 169–78. http://dx.doi.org/10.1142/s0129167x94000085.

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Let e be the Jones projection associated to a conditional expectation [Formula: see text] where [Formula: see text] are von Neumann algebras. We prove that the similarity orbit of e by invertibles of [Formula: see text] is an homogeneous space iff the index of E is finite. If also [Formula: see text], then this orbit is a covering space for the orbit of E.

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48

VAY, CRISTIAN. "ON PROJECTIVE MODULES OVER FINITE QUANTUM GROUPS." Transformation Groups 24, no. 1 (November 27, 2017): 279–99. http://dx.doi.org/10.1007/s00031-017-9469-y.

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49

GOVER, A. R., and R. B. ZHANG. "GEOMETRY OF QUANTUM HOMOGENEOUS VECTOR BUNDLES AND REPRESENTATION THEORY OF QUANTUM GROUPS I." Reviews in Mathematical Physics 11, no. 05 (May 1999): 533–52. http://dx.doi.org/10.1142/s0129055x99000209.

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Quantum homogeneous vector bundles are introduced in the context of Woronowicz type compact quantum groups. The bundles carry natural topologies, and their sections furnish finite type projective modules over algebras of functions on quantum homogeneous spaces. Further properties of the quantum homogeneous vector bundles are investigated, and applied to the study of the geometrical structures of induced representations of quantum groups.

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50

Rajola, Sandro. "Sets of type (0,n) in Steiner systems and in finite projective planes." Journal of Geometry 53, no. 1-2 (July 1995): 148–62. http://dx.doi.org/10.1007/bf01224047.

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