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"Projective and Metric Geometry" is an article from The Annals of Mathematics, Volume 5 . View more articles from The Annals of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-2007232

"Projective and Metric Geometry" is an article from The Annals of Mathematics, Volume 5 . View more articles from The Annals of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-2007232

We prove that for each prime power q there is an integer n such that if M is a 3-connected, representable matroid with a PG(n-1,q)-minor and no U_{2,q^2+1}-minor, then M is representable over GF(q). We also show that for l >= 2, if M is a 3-connected, representable matroid of sufficiently high rank with no U_{2,l+2}-minor and |E(M)| >= (4l)^{r(M)/2}, then M is representable over a field of order at most l.

This paper is in the public domain in USA. Metadata comes from the CrossRef API, see full record in the source URL below.

Matrix computations, especially iterative PDE solving (and the sparse matrix vector multiplication subproblem within) using conjugate gradient algorithm, and LU/Cholesky decomposition for solving system of linear equations, form the kernel of many applications, such as circuit simulators, computational fluid dynamics or structural analysis etc. The problem of designing approaches for parallelizing these computations, to get good speedups as much as possible as per Amdahl's law, has been continuously researched upon. In this paper, we discuss approaches based on the use of finite projective geometry graphs for these two problems. For the problem of conjugate gradient algorithm, the approach looks at an alternative data distribution based on projective-geometry concepts. It is proved that this data distribution is an optimal data distribution for scheduling the main problem of dense matrix-vector multiplication. For the problem of parallel LU/Cholesky decomposition of general matrices, the approach is motivated by the recently published scheme for interconnects of distributed systems, perfect difference networks. We find that projective-geometry based graphs indeed offer an exciting way of parallelizing these computations, and in fact many others. Moreover, their applications ranges from architectural ones (interconnect choice) to algorithmic ones (data distributions).

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Book digitized by Google from the library of the University of Michigan and uploaded to the Internet Archive by user tpb.

Book digitized by Google from the library of Harvard University and uploaded to the Internet Archive by user tpb.

Book digitized by Google from the library of Harvard University and uploaded to the Internet Archive by user tpb.

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xiii, 215 p. 20 cm

2 v. 24 cm

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Book digitized by Google from the library of Harvard University and uploaded to the Internet Archive by user tpb.

Book digitized by Google from the library of Harvard University and uploaded to the Internet Archive by user tpb.

viii, 128 p. illus. ; 19 cm

We show that the pair given by the power set and by the "Grassmannian"(set of all subgroups) of an arbitrary group behaves very much like the pair given by a projective space and its dual projective space. More precisely, we generalize several results from preceding joint work with M. Kinyon (arXiv:0903.5441), which concerned abelian groups, to the case of general non-abelian groups. Most notably, pairs of subgroups parametrize torsor and semitorsor structures on the power set. The r\^ole of associative algebras and -pairs from loc. cit. is now taken by analogs of near-rings.

We study the geometry of point-orbits of elation groups with a given center and axis of a finite projective space. We show that there exists a 1-1 correspondence from conjugacy classes of such groups and orbits on projective subspaces (of a suitable dimension) of Singer groups of projective spaces. Together with a recent result of Drudge we establish the number of these elation groups.

We study the geometry of point-orbits of elation groups with a given center and axis of a finite projective space. We show that there exists a 1-1 correspondence from conjugacy classes of such groups and orbits on projective subspaces (of a suitable dimension) of Singer groups of projective spaces. Together with a recent result of Drudge we establish the number of these elation groups.

We study the geometry of point-orbits of elation groups with a given center and axis of a finite projective space. We show that there exists a 1-1 correspondence from conjugacy classes of such groups and orbits on projective subspaces (of a suitable dimension) of Singer groups of projective spaces. Together with a recent result of Drudge we establish the number of these elation groups.

"Interpretation of the Simplest Integral Invariant of Projective Geometry" is an article from Proceedings of the National Academy of Sciences of the United States of America, Volume 2 . View more articles from Proceedings of the National Academy of Sciences of the United States of America . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-83947

"The Nine-Point Circle Obtained by Methods of Projective Geometry" is an article from The American Mathematical Monthly, Volume 25 . View more articles from The American Mathematical Monthly . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-2972869

"On the Relation Between Some Important Notions of Projective and Metrical Differential Geometry" is an article from American Journal of Mathematics, Volume 39 . View more articles from American Journal of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-2370536

"On the Relation Between Some Important Notions of Projective and Metrical Differential Geometry" is an article from American Journal of Mathematics, Volume 39 . View more articles from American Journal of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-2370536

These are lecture notes on the rigidity of submanifolds of projective space "resembling" compact Hermitian symmetric spaces in their homogeneous embeddings. Recent results are surveyed, along with their classical predecessors. The notes include an introduction to moving frames in projective geometry, an exposition of the Hwang-Yamaguchi ridgidity theorem and a new variant of the Hwang-Yamaguchi theorem.

This article investigates the complex symplectic geometry of the deformation space of complex projective structures on a closed oriented surface of genus at least 2. The cotangent symplectic structure given by the Schwarzian parametrization is studied carefully and compared to the Goldman symplectic structure on the character variety, clarifying and generalizing a theorem of S. Kawai. Generalizations of results of C. McMullen are derived, notably quasi-Fuchsian reciprocity. The symplectic geometry is also described in a Hamiltonian setting with the complex Fenchel-Nielsen coordinates on quasi-Fuchsian space, recovering results of I. Platis.

Book digitized by Google from the library of Harvard University and uploaded to the Internet Archive by user tpb.

xi, 299 p. 25 cm

xi, 299 p. 25 cm

We give an exposition of Plucker vectors for a system of joint axes in projective 3-space. We use Plucker vectors to analyse dependencies among joint axes, and in particular show that two rotational joints rigidly joined by a bar and each with 3 degrees of freedom always forms a 5-dimensional system. We introduce the concept of reduced redundancy in a dependent set of projective Lines, and argue that reduced redundancy in the axes of a body position increases injury risk. We apply this to a simple two-joint model of bowling in cricket, and show by analysis of some experimental data that reduced redundancy around ball release is observed in some cases.

The quantum deformation $CP_q(N)$ of complex projective space is discussed. Many of the features present in the case of the quantum sphere can be extended. The differential and integral calculus is studied and $CP_q(N)$ appears as a quantum K\"{a}hler manifold. The braiding of several copies of $CP_q(N)$ is introduced and the anharmonic ratios of four collinear points are shown to be invariant under quantum projective transformations. They provide the building blocks of all projective invariants. The Poisson limit is also described.

The purpose of this article is to introduce projective geometry over composition algebras : the equivalent of projective spaces and Grassmannians over them are defined. It will follow from this definition that the projective spaces are in correspondance with Jordan algebras and that the points of a projective space correspond to rank one matrices in the Jordan algebra. A second part thus studies properties of rank one matrices. Finally, subvarieties of projective spaces are discussed.

The purpose of this article is to introduce projective geometry over composition algebras : the equivalent of projective spaces and Grassmannians over them are defined. It will follow from this definition that the projective spaces are in correspondance with Jordan algebras and that the points of a projective space correspond to rank one matrices in the Jordan algebra. A second part thus studies properties of rank one matrices. Finally, subvarieties of projective spaces are discussed.

Book Source: Digital Library of India Item 2015.216125 dc.contributor.author: Durell, Clement. V dc.date.accessioned: 2015-07-09T19:38:16Z dc.date.available: 2015-07-09T19:38:16Z dc.date.digitalpublicationdate: 2005-05-12 dc.date.citation: 1926 dc.identifier.barcode: 2990140053372 dc.identifier.origpath: /data_copy/upload/0053/377 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/216125 dc.description.scannerno: 4 dc.description.scanningcentre: Osmania University dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 235 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Digital Library Of India dc.publisher: Macmillan And Company Limited dc.rights: Not Available dc.source.library: Osmania University dc.subject.classification: Natural Sciences dc.subject.classification: Mathematics dc.subject.classification: Geometry dc.title: Projective Geometry

Book Source: Digital Library of India Item 2015.216125 dc.contributor.author: Durell, Clement. V dc.date.accessioned: 2015-07-09T19:38:16Z dc.date.available: 2015-07-09T19:38:16Z dc.date.digitalpublicationdate: 2005-05-12 dc.date.citation: 1926 dc.identifier.barcode: 2990140053372 dc.identifier.origpath: /data_copy/upload/0053/377 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/216125 dc.description.scannerno: 4 dc.description.scanningcentre: Osmania University dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 235 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Digital Library Of India dc.publisher: Macmillan And Company Limited dc.rights: Not Available dc.source.library: Osmania University dc.subject.classification: Natural Sciences dc.subject.classification: Mathematics dc.subject.classification: Geometry dc.title: Projective Geometry

We construct Grassmann spaces associated with the incidence geometry of regular and tangential subspaces of a symplectic copolar space, show that the underlying metric projective space can be recovered in terms of the corresponding adjacencies on so distinguished family of k-subspaces (geometrical dimension of the space being not 2k+1), and thus we prove that bijections which preserve the adjacency are determined by automorphisms of the underlying space.

We explore connections between birational anabelian geometry and abstract projective geometry. One of the applications is a proof of a version of the birational section conjecture.

This paper deals with the geometry of the space (GIT quotient) M_8 of 8 points in P^1, and the Gale-quotient N'_8 of the GIT quotient of 8 points in P^3. The space M_8 comes with a natural embedding in P^{13}, or more precisely, the projectivization of the S_8-representation V_{4,4}. There is a single S_8-skew cubic C in P^{13}. The fact that M_8 lies on the skew cubic C is a consequence of Thomae's formula for hyperelliptic curves, but more is true: M_8 is the singular locus of C. These constructions yield the free resolution of M_8, and are used in the determination of the "single" equation cutting out the GIT quotient of n points in P^1 in general. The space N'_8 comes with a natural embedding in P^{13}, or more precisely, PV_{2,2,2,2}. There is a single skew quintic Q containing N'_8, and N'_8 is the singular locus of the skew quintic Q. The skew cubic C and skew quintic Q are projectively dual. (In particular, they are surprisingly singular, in the sense of having a dual of remarkably low degree.) The divisor on the skew cubic blown down by the dual map is the secant variety Sec(M_8), and the contraction Sec(M_8) - - > N'_8 factors through N_8 via the space of 8 points on a quadric surface. We conjecture that the divisor on the skew quintic blown down by the dual map is the quadrisecant variety of N'_8 (the closure of the union of quadrisecant *lines*), and that the quintic Q is the trisecant variety. The resulting picture extends the classical duality in the 6-point case between the Segre cubic threefold and the Igusa quartic threefold. We note that there are a number of geometrically natural varieties that are (related to) the singular loci of remarkably singular cubic hypersurfaces. Some of the content of this paper appeared in arXiv/0809.1233.

This paper deals with the geometry of the space (GIT quotient) M_8 of 8 points in P^1, and the Gale-quotient N'_8 of the GIT quotient of 8 points in P^3. The space M_8 comes with a natural embedding in P^{13}, or more precisely, the projectivization of the S_8-representation V_{4,4}. There is a single S_8-skew cubic C in P^{13}. The fact that M_8 lies on the skew cubic C is a consequence of Thomae's formula for hyperelliptic curves, but more is true: M_8 is the singular locus of C. These constructions yield the free resolution of M_8, and are used in the determination of the "single" equation cutting out the GIT quotient of n points in P^1 in general. The space N'_8 comes with a natural embedding in P^{13}, or more precisely, PV_{2,2,2,2}. There is a single skew quintic Q containing N'_8, and N'_8 is the singular locus of the skew quintic Q. The skew cubic C and skew quintic Q are projectively dual. (In particular, they are surprisingly singular, in the sense of having a dual of remarkably low degree.) The divisor on the skew cubic blown down by the dual map is the secant variety Sec(M_8), and the contraction Sec(M_8) - - > N'_8 factors through N_8 via the space of 8 points on a quadric surface. We conjecture that the divisor on the skew quintic blown down by the dual map is the quadrisecant variety of N'_8 (the closure of the union of quadrisecant *lines*), and that the quintic Q is the trisecant variety. The resulting picture extends the classical duality in the 6-point case between the Segre cubic threefold and the Igusa quartic threefold. We note that there are a number of geometrically natural varieties that are (related to) the singular loci of remarkably singular cubic hypersurfaces. Some of the content of this paper appeared in arXiv/0809.1233.

Book Source: Digital Library of India Item 2015.84473 dc.contributor.author: Smart, E. Howard dc.date.accessioned: 2015-06-30T18:52:22Z dc.date.available: 2015-06-30T18:52:22Z dc.date.digitalpublicationdate: 2005-03-30 dc.date.citation: 1913 dc.identifier.barcode: 2030020000973 dc.identifier.origpath: /data7/upload/0168/966 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/84473 dc.description.scannerno: 9 dc.description.scanningcentre: RMSC, IIIT-H dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 275 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Digital Library Of India dc.publisher: Macmillan And Company., Limited dc.rights: Not Available dc.source.library: Osmania University dc.subject.classification: Natural Sciences dc.subject.classification: Mathematics dc.subject.classification: Geometry dc.title: A First Course In Projective Geometry

The aim of the present paper is to provide an intrinsic investigation of projective changes in Finlser geometry, following the pullback formalism. Various known local results are generalized and other new intrinsic results are obtained. Nontrivial characterizations of projective changes are given. The fundamental projectively invariant tensors, namely, the projective deviation tensor, the Weyl torsion tensor, the Weyl curvature tensor and the Douglas tensor are investigated. The properties of these tensors and their interrelationships are obtained. Projective connections and projectively flat manifolds are characterized. The present work is entirely intrinsic (free from local coordinates).

We present a complete classification and the construction of $\mathrm{Mp}(2n+2,\mathbb{R})$-equivariant differential operators acting on the principal series representations, associated to the contact projective geometry on $\mathbb{RP}^{2n+1}$ and induced from the irreducible $\mathrm{Mp}(2n,\mathbb{R})$-submodules of the Segal-Shale-Weil representation twisted by a one-parameter family of characters. The proof is based on the classification of homomorphisms of generalized Verma modules for the Segal-Shale-Weil representation twisted by a one-parameter family of characters, together with a generalization of the well-known duality between homomorphisms of generalized Verma modules and equivariant differential operators in the category of inducing smooth admissible modules.

We propose a twistor construction of surfaces in Lie sphere geometry based on the linear system which copies equations of Wilczynski's projective frame. In the particular case of Lie-applicable surfaces this linear system describes joint eigenfunctions of a pair of commuting Schr\"odinger operators with magnetic fields.

Some of the most important classes of surfaces in projective 3-space are reviewed: these are isothermally asymptotic surfaces, projectively applicable surfaces, surfaces of Jonas, projectively minimal surfaces, etc. It is demonstrated that the corresponding projective "Gauss-Codazzi" equations reduce to integrable systems which are quite familiar from the modern soliton theory and coincide with the stationary flows in the Davey-Stewartson and Kadomtsev-Petviashvili hierarchies, equations of the Toda lattice, etc. The corresponding Lax pairs can be obtained by inserting a spectral parameter in the equations of the Wilczynski moving frame.

This paper is in the public domain in USA. Metadata comes from the CrossRef API, see full record in the source URL below.

In the paper "Is there a Jordan geometry underlying quantum physics?" (Int. J. Theor. Phys., to appear; arXiv:0801.3069 [math-ph]), generalized projective geometries have been proposed as a framework for a geometric formulation of Quantum Theory. In the present note, we refine this proposition by discussing further structural features of Quantum Theory: the link with associative involutive algebras, and with Jordan-Lie and Lie-Jordan algebas. The associated geometries are (Hermitian) projective lines over an associative algebra; their axiomatic definition and theory will be given in subsequent work with M. Kinyon.