On geometry of orbits of adapted projective frame space

2019 ◽  
pp. 88-98
Author(s):  
A. Kuleshov
Keyword(s):  
Normal Subgroup ◽  
Projective Space ◽  
Tangent Space ◽  
Affine Group ◽  
Current Paper ◽  
Previous Issue ◽  
Distinguished Point

The current paper continues consideration of geometry of projective frame orbits started in the author’s article in the previous issue. The ndimensional projective space with a distinguished point (the center) is considered. The action of matrix affine group of order n on the adapted projective frame manifold is given. It is shown that the linear frames, i. e., bases of the tangent space, can be identified with the orbits of adapted projective frames under the action of some normal subgroup of this group. Two adapted frames are said to be equivalent if they belong to the same orbit. The strict perspectivity relation between two adapted frames is introduced. The proofs of the theorem on the Desargues hyperplane and of the criterion of equivalence are simplified. According to this criterion, two adapted frames in strict perspective are equivalent if and only if the Desargues hyperplane generated by these frames is passing through the center.

REAL PROJECTIVE MANIFOLDS DEVELOPING INTO AN AFFINE SPACE

1993 ◽  
Vol 04 (02) ◽  
pp. 179-191 ◽  
Author(s):  
YOUNKI CHAE ◽  
SUHYOUNG CHOI ◽  
CHAN-YOUNG PARK
Keyword(s):  
Projective Space ◽  
Affine Space ◽  
Dimensional Subspace ◽  
Affine Group ◽  
Projective Manifold ◽  
Hilbert Metric ◽  
Projective Manifolds

Suppose that an n-dimensional closed real projective manifold M, n ≥ 2, develops into an affine space RPn − RPn − 1 for an (n − 1)-dimensional subspace RPn − 1 of the projective space RPn. Then either M is convex or affine or M admits a flat foliation [Formula: see text] with a transverse invariant Hilbert metric. Further, if the codimension of [Formula: see text] is n − 1, then M is convex. We prove this statement by a use of a variation of Carrière's discompacté, a measure of non-compactedness of an affine group acting on an affine space.

scholarly journals On the hypersurfaces contained in their Hessian

2019 ◽  
Vol 18 (1) ◽  
pp. 21-33
Author(s):  
Pietro De Poi ◽  
Giovanna Ilardi
Keyword(s):  
Projective Space ◽  
Tangent Space ◽  
Linear Spaces

Abstract This article presents the theory of focal locus applied to the hyper-surfaces in the projective space which are (finitely) covered by linear spaces and such that the tangent space is constant along these spaces.

Curvature and torsion pseudotensors of coaffine connection in tangent bundle of hypercentred planes manifold

2020 ◽  
pp. 49-57
Author(s):  
A. V. Vyalova
Keyword(s):  
Differential Equations ◽  
Projective Space ◽  
Linear Group ◽  
Tangent Bundle ◽  
Tangent Space ◽  
Smooth Manifold ◽  
Principal Bundle ◽  
Fiber Bundles ◽  
Curvature And Torsion ◽  
Principal Fiber Bundles

The hypercentered planes family, whose dimension coincides with dimension of generating plane, is considered in the projective space. Two principal fiber bundles arise over it. Typical fiber for one of them is the stationarity subgroup for hypercentered plane, for other — the linear group operating in each tangent space to the manifold. The latter bundle is called the principal bundle of linear coframes. The structural forms of two bundles are related by equations. It is proved that hypercentered planes family is a holonomic smooth manifold. In the principal bundle of linear coframes the coaffine connection is given. From the differential equations it follows that the coaffine connec­tion object forms quasipseudotensor. It is proved that the curvature and torsion objects for the coaffine connection in the linear coframes bundle form pseudotensors.

Finding conjugate stabilizer subgroups in $\PSL$ and related groups

2010 ◽  
Vol 10 (3&4) ◽  
pp. 282-291
Author(s):  
DA. Denney ◽  
C. Moore ◽  
A. Russell
Keyword(s):  
Projective Space ◽  
Finite Groups ◽  
Quantum Algorithms ◽  
Affine Group ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Hidden Subgroup Problem ◽  
Subgroup Problem ◽  
Groups Of Lie Type ◽  
Positive Results

We reduce a case of the hidden subgroup problem (HSP) in $\SL$, $\PSL$, and $\PGL$, three related families of finite groups of Lie type, to efficiently solvable HSPs in the affine group $\AGL$. These groups act on projective space in an ``almost'' 3-transitive way, and we use this fact in each group to distinguish conjugates of its Borel (upper triangular) subgroup, which is also the stabilizer subgroup of an element of projective space. Our observation is mainly group-theoretic, and as such breaks little new ground in quantum algorithms. Nonetheless, these appear to be the first positive results on the HSP in finite simple groups such as $\PSL$.

scholarly journals Equivariant images of projective space under the action of SL (n, ℤ)

1981 ◽  
Vol 1 (4) ◽  
pp. 519-522 ◽  
Author(s):  
Robert J. Zimmer
Keyword(s):  
Normal Subgroup ◽  
Projective Space ◽  
Lie Group ◽  
Measure Space ◽  
Irreducible Lattice ◽  
Group 3 ◽  
Higher Rank ◽  
Measure Class

The point of this note is to answer in the affirmative a question of G. A. Margulis. In the course of his proof of the finiteness of either the cardinality or the index of a normal subgroup of an irreducible lattice in a higher rank semi-simple Lie group [3], [4], Margulis proves that if Γ = SL (n, ℤ),n≥3, (X, μ) is a measurable Γ-space, μ quasi-invariant, and φ: ℙn−1→Xis a measure class preserving Γ-map, then either φ is a measure space isomorphism or μ is supported on a point. Margulis then asks whether the topological analogue of this result is true. This is answered in the following.

scholarly journals Finite geometry behind the Harvey-Chryssanthacopoulos four-qubit magic rectangle

2012 ◽  
Vol 12 (11&12) ◽  
pp. 1011-1016
Author(s):  
Metod Saniga ◽  
Michel Planat
Keyword(s):  
Hilbert Space ◽  
Projective Space ◽  
Affine Plane ◽  
Three Dimensional ◽  
Finite Geometry ◽  
Polar Space ◽  
Distinguished Point ◽  
The Real ◽  
Qualitative Structure ◽  
Dimensional Projective Space

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.

About an analogue of Neifeld’s connection on the space of centred planes with one-index basic-fibre forms

2019 ◽  
pp. 41-47
Author(s):  
Е. Belova ◽  
O. Belova
Keyword(s):  
Projective Space ◽  
Lie Group ◽  
Fiber Bundle ◽  
Tangent Space ◽  
Moving Frame ◽  
Strong Normalization ◽  
Principal Fiber Bundle ◽  
The Common ◽  
Exterior Forms

This research is realized by Cartan — Laptev method (with prolongations and scopes, moving frame and exterior forms). In this paper we consider a space П of centered m-planes (a space of all centered planes of the dimension m). This space is considered in the projective space n P . For the space П we have: dim П=n + (n – m)m. Principal fiber bundle is arised above it. The Lie group is a typical fiber of the principal fiber. This group acts in the tangent space to the П. Analogue of Neifeld’s connection with multivariate glueing is given in this fibering by Laptev — Lumiste way. The case when one-index forms are basic-fibre forms is considered. We realize an analogue of the Norden strong normalization of the space П by fields of the geometrical images: (n – m – 1)-plane which is not having the common points with a centered m-plane and (m – 1)-plane which is belonging to the m-plane and not passing through its centre. It is proved that the analog of the Norden strong normalization of the space of centered planes induces this connection.

Zero Marking of Past Tense in Child African American English

2014 ◽  
Vol 21 (4) ◽  
pp. 173-181 ◽  
Author(s):  
Ryan Lee ◽  
Janna B. Oetting
Keyword(s):  
African American ◽  
American English ◽  
African American English ◽  
Current Paper ◽  
Past Tense ◽  
Typically Developing ◽  
Grammatical Function

Zero marking of the simple past is often listed as a common feature of child African American English (AAE). In the current paper, we review the literature and present new data to help clinicians better understand zero marking of the simple past in child AAE. Specifically, we provide information to support the following statements: (a) By six years of age, the simple past is infrequently zero marked by typically developing AAE-speaking children; (b) There are important differences between the simple past and participle morphemes that affect AAE-speaking children's marking options; and (c) In addition to a verb's grammatical function, its phonetic properties help determine whether an AAE-speaking child will produce a zero marked form.

Challenges in the Diagnostic Conceptualization of CRPS-1 (Formerly Conceptualized as RSD)

2006 ◽  
Vol 11 (2) ◽  
pp. 1-3, 9-12
Author(s):  
Robert J. Barth ◽  
Tom W. Bohr
Keyword(s):  
Clinical Practice ◽  
Complex Regional Pain Syndrome ◽  
Somatoform Disorders ◽  
International Association ◽  
Pain Syndrome ◽  
Somatoform Disorder ◽  
Syndrome Type ◽  
Previous Issue

Abstract From the previous issue, this article continues a discussion of the potentially confusing aspects of the diagnostic formulation for complex regional pain syndrome type 1 (CRPS-1) proposed by the International Association for the Study of Pain (IASP), the relevance of these issues for a proposed future protocol, and recommendations for clinical practice. IASP is working to resolve the contradictions in its approach to CRPS-1 diagnosis, but it continues to include the following criterion: “[c]ontinuing pain, which is disproportionate to any inciting event.” This language only perpetuates existing issues with current definitions, specifically the overlap between the IASP criteria for CRPS-1 and somatoform disorders, overlap with the guidelines for malingering, and self-contradiction with respect to the suggestion of injury-relatedness. The authors propose to overcome the last of these by revising the criterion: “[c]omplaints of pain in the absence of any identifiable injury that could credibly account for the complaints.” Similarly, the overlap with somatoform disorders could be reworded: “The possibility of a somatoform disorder has been thoroughly assessed, with the results of that assessment failing to produce any consistencies with a somatoform scenario.” The overlap with malingering could be addressed in this manner: “The possibility of malingering has been thoroughly assessed, with the results of that assessment failing to produce any consistencies with a malingering scenario.” The article concludes with six recommendations, and a sidebar discusses rating impairment for CRPS-1 (with explicit instructions not to use the pain chapter for this purpose).

Ethical Issues Relevant to the Assessment of Suicide Risk in Nonclinical Research Settings

Crisis ◽  
2012 ◽  
Vol 33 (1) ◽  
pp. 54-59 ◽  
Author(s):  
Carolyn M. Wilson ◽  
Bruce K. Christensen
Keyword(s):  
Undergraduate Students ◽  
Suicide Risk ◽  
Ethical Issues ◽  
Current Paper ◽  
Self Report ◽  
Ethical Guidelines ◽  
Schizotypal Personality ◽  
Increased Risk ◽  
Genetic Features ◽  
Conducting Research

Background: Our laboratory recently confronted this issue while conducting research with undergraduate students at the University of Waterloo (UW). Although our main objective was to examine cognitive and genetic features of individuals with schizotypal personality disorder (SPD), the study protocol also entailed the completion of various self-report measures to identify participants deemed at increased risk for suicide. Aims and Methods: This paper seeks to review and discuss the relevant ethical guidelines and legislation that bear upon a psychologist’s obligation to further assess and intervene when research participants reveal that they are at increased risk for suicide. Results and Conclusions: In the current paper we argue that psychologists are ethically impelled to assess and appropriately intervene in cases of suicide risk, even when such risk is revealed within a research context. We also discuss how any such obligation may potentially be modulated by the research participant’s expectations of the role of a psychologist, within such a context. Although the focus of the current paper is on the ethical obligations of psychologists, specifically those practicing within Canada, the relevance of this paper extends to all regulated health professionals conducting research in nonclinical settings.

Sign in / Sign up

Export Citation Format

Share Document