Examples in non-commutative projective geometry

1994 ◽  
Vol 116 (3) ◽  
pp. 415-433 ◽  
Author(s):  
J. T. Stafford ◽  
J. J. Zhang
Keyword(s):  
Algebraic Geometry ◽  
General Theory ◽  
Projective Geometry ◽  
Projective Variety ◽  
Companion Paper ◽  
Central Field ◽  
Commutative Algebras ◽  
Formal Definitions ◽  
Noetherian Algebra ◽  
Natural Way

Let A = k ⊕ ⊕n ≥ 1An connected graded, Noetherian algebra over a fixed, central field k (formal definitions will be given in Section 1 but, for the most part, are standard). If A were commutative, then the natural way to study A and its representations would be to pass to the associated projective variety and use the power of projective algebraic geometry. It has become clear over the last few years that the same basic idea is powerful for non-commutative algebras; see, for example, [ATV1, 2], [AV], [Sm], [SS] or [TV] for some of the more significant applications. This suggests that it would be profitable to develop a general theory of ‘non-commutative projective geometry’ and the foundations for such a theory have been laid down in the companion paper [AZ]. The results proved there raise a number of questions and the aim of this paper is to provide negative answers to several of these.

Azimuthal sensitivity of neurons in primary auditory cortex of cats. II. Organization along frequency-band strips

1990 ◽  
Vol 64 (3) ◽  
pp. 888-902 ◽  
Author(s):  
R. Rajan ◽  
L. M. Aitkin ◽  
D. R. Irvine
Keyword(s):  
Spatial Distribution ◽  
Auditory Cortex ◽  
Frequency Band ◽  
Primary Auditory Cortex ◽  
Companion Paper ◽  
Significant Feature ◽  
Central Field ◽  
Cortical Surface ◽  
Peak Response ◽  
The Individual

1. The organization of azimuthal sensitivity of units across the dorsoventral extent of primary auditory cortex (AI) was studied in electrode penetrations made along frequency-band strips of AI. Azimuthal sensitivity for each unit was represented by a mean azimuth function (MF) calculated from all azimuth functions obtained to characteristic frequency (CF) stimuli at intensities 20 dB or more greater than threshold. MFs were classified as contrafield, ipsi-field, central-field, omnidirectional, or multipeaked, according to the criteria established in the companion paper (Rajan et al. 1990). 2. The spatial distribution of three types of MFs was not random across frequency-band strips: for contra-field, ipsi-field, and central-field MFs there was a significant tendency for clustering of functions of the same type in sequentially encountered units. Occasionally, repeated clusters of a particular MF type could be found along a frequency-band strip. In contrast, the spatial distribution of omnidirectional MFs along frequency-band strips appeared to be random. 3. Apart from the clustering of MF types, there were also regions along a frequency-band strip in which there were rapid changes in the type of MF encountered in units isolated over short distances. Most often such changes took the form of irregular, rapid juxtapositions of MF types. Less frequently such changes appeared to show more systematic changes from one type of MF to another type. In contrast to these changes in azimuthal sensitivity seen in electrode penetrations oblique to the cortical surface, much less change in azimuthal sensitivity was seen in the form of azimuthal sensitivity displayed by successively isolated units in penetrations made normal to the cortical surface. 4. To determine whether some significant feature or features of azimuthal sensitivity shifted in a more continuous and/or systematic manner along frequency-band strips, azimuthal sensitivity was quantified in terms of the peak-response azimuth (PRA) of the MFs of successive units and of the azimuthal range over which the peaks occurred in the individual azimuth functions contributing to each MF (the peak-response range). In different experiments shifts in these measures of the peaks in successively isolated units along a frequency-band strip were found generally to fall into one of four categories: 1) shifts across the entire frontal hemifield; 2) clustering in the contralateral quadrant; 3) clustering in the ipsilateral quadrant; and 4) clustering about the midline. In two cases more than one of these four patterns were found along a frequency-band strip.(ABSTRACT TRUNCATED AT 400 WORDS)

Projective Geometries that are Disjoint Unions of Caps

1980 ◽  
Vol 32 (6) ◽  
pp. 1299-1305 ◽  
Author(s):  
Barbu C. Kestenband
Keyword(s):  
Finite Field ◽  
Projective Geometry ◽  
Prime Power ◽  
Disjoint Union ◽  
Hermitian Matrices ◽  
Incidence Relation ◽  
Projective Geometries ◽  
Image Position ◽  
Natural Way ◽  
Square Matrix

We show that any PG(2n, q2) is a disjoint union of (q2n+1 − 1)/ (q − 1) caps, each cap consisting of (q2n+1 + 1)/(q + 1) points. Furthermore, these caps constitute the “large points” of a PG(2n, q), with the incidence relation defined in a natural way.A square matrix H = (hij) over the finite field GF(q2), q a prime power, is said to be Hermitian if hijq = hij for all i, j [1, p. 1161]. In particular, hii ∈ GF(q). If if is Hermitian, so is p(H), where p(x) is any polynomial with coefficients in GF(q).Given a Desarguesian Projective Geometry PG(2n, q2), n > 0, we denote its points by column vectors:All Hermitian matrices in this paper will be 2n + 1 by 2n + 1, n > 0.

scholarly journals The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 314 ◽  
Author(s):  
Alessandra Bernardi ◽  
Enrico Carlini ◽  
Maria Catalisano ◽  
Alessandro Gimigliano ◽  
Alessandro Oneto
Keyword(s):  
Algebraic Geometry ◽  
Projective Variety ◽  
Tensor Decomposition ◽  
Tensor Rank ◽  
Segre Variety ◽  
Secant Varieties ◽  
Open Problems ◽  
Veronese Variety ◽  
Strong Motivation

We consider here the problem, which is quite classical in Algebraic geometry, of studying the secant varieties of a projective variety X. The case we concentrate on is when X is a Veronese variety, a Grassmannian or a Segre variety. Not only these varieties are among the ones that have been most classically studied, but a strong motivation in taking them into consideration is the fact that they parameterize, respectively, symmetric, skew-symmetric and general tensors, which are decomposable, and their secant varieties give a stratification of tensors via tensor rank. We collect here most of the known results and the open problems on this fascinating subject.

scholarly journals Nonlinear morphoelastic plates II: Exodus to buckled states

2011 ◽  
Vol 16 (8) ◽  
pp. 833-871 ◽  
Author(s):  
Joseph McMahon ◽  
Alain Goriely ◽  
Michael Tabor
Keyword(s):  
General Theory ◽  
Deformation Gradient ◽  
Companion Paper ◽  
Kirchhoff Plate ◽  
Geometric Description ◽  
Elastic Materials ◽  
Multiplicative Decomposition ◽  
Linear Elastic ◽  
Kirchhoff Plates ◽  
The Stability

Morphoelasticity is the theory of growing elastic materials. The theory is based on the multiplicative decomposition of the deformation gradient and provides a formulation of the deformation and stresses induced by growth. Following a companion paper, a general theory of growing non-linear elastic Kirchhoff plate is described. First, a complete geometric description of incompatibility with simple examples is given. Second, the stability of growing Kirchhoff plates is analyzed.

Structures of the cubic and rhombohedral high-pressure modifications of silicon as packing of the rod-like substructures determined by the algebraic geometry

2008 ◽  
Vol 64 (1) ◽  
pp. 26-33 ◽  
Author(s):  
V. S. Kraposhin ◽  
A. L. Talis ◽  
V. G. Kosushkin ◽  
A. A. Ogneva ◽  
L. I. Zinober
Keyword(s):  
High Pressure ◽  
Algebraic Geometry ◽  
Projective Geometry ◽  
Symmetry Axis ◽  
Triangular Face ◽  
Threefold Symmetry ◽  
Finite Projective Geometry ◽  
The Common

Tessellations by generating clusters are proposed for the high-pressure phases BC8 and R8 of silicon. The structures of both high-pressure phases are represented by the parallel packings of rods. The latter are stacks of distorted icosahedra, joined by a common triangular face and flattened out along the threefold symmetry axis. Along the rod axis there is an alternation of empty and double-centered icosahedra. The empty icosahedra are additionally distorted in the cubic BC8 phase by antiparallel rotation about the rod axis, while the double-centered icosahedra are distorted by that rotation in the rhombohedral R8 phase. A possible mechanism of the reversible BC8 ↔ R8 transformation is proposed as the rotation about the rod axis of the common triangular face of the neighboring icosahedra, thus transforming between distorted and undistorted icosahedra. The graphs of the generating clusters for both the BC8 and R8 structures are determined by two subconfigurations of the same construction of the finite projective geometry.

A General Theory of Singularities

2018 ◽  
Author(s):  
Keith Simmons
Keyword(s):  
General Theory ◽  
Singularity Theory ◽  
Formal Theory ◽  
General Level ◽  
Formal Definitions ◽  
Primary Representation ◽  
Semantic Value ◽  
Precise Expression

Chapter 6 presents the singularity theory in formal detail. The theory is pitched at a sufficiently general level to handle in a unified way the notions of denotation, extension, and truth. The central notions of semantic pathology and singularity are defined, and a procedure for determining the semantic value of a pathological token is provided. The chapter gives precise expression to the idea that our semantic expressions are significant everywhere except for certain singularities. Key ingredients of the formal theory include the notions of primary representation, primary tree, and determination tree. Paradoxical cases from previous chapters are used throughout the chapter to illustrate the formal definitions.

scholarly journals Derived algebraic geometry, determinants of perfect complexes, and applications to obstruction theories for maps and complexes

2015 ◽  
Vol 2015 (702) ◽  
pp. 1-40 ◽  
Author(s):  
Timo Schürg ◽  
Bertrand Toën ◽  
Gabriele Vezzosi
Keyword(s):  
Algebraic Geometry ◽  
Projective Variety ◽  
Obstruction Theory ◽  
Important Ingredient ◽  
Smooth Complex ◽  
Perfect Complexes ◽  
Derived Algebraic Geometry ◽  
Complex Projective Variety ◽  
Complex Projective

AbstractA quasi-smooth derived enhancement of a Deligne–Mumford stack 𝒳 naturally endows 𝒳 with a functorial perfect obstruction theory in the sense of Behrend–Fantechi. We apply this result to moduli of maps and perfect complexes on a smooth complex projective variety.ForWe give two further applications toAn important ingredient of our construction is a

Sign in / Sign up

Export Citation Format

Share Document