The Spin Group and the Spin-c Group

2011 ◽  
pp. 35-40
Author(s):  
J. J. Duistermaat
Keyword(s):  
Spin Group

scholarly journals Arithmetic purity of strong approximation for semi-simple simply connected groups

2020 ◽  
Vol 156 (12) ◽  
pp. 2628-2649
Author(s):  
Yang Cao ◽  
Zhizhong Huang
Keyword(s):  
Quadratic Form ◽  
Strong Approximation ◽  
Homogeneous Spaces ◽  
Algebraic Groups ◽  
Integral Points ◽  
Spin Group ◽  
Linear Algebraic Groups ◽  
Simply Connected ◽  
Quadratic Hypersurfaces ◽  
Almost Prime

In this article we establish the arithmetic purity of strong approximation for certain semisimple simply connected linear algebraic groups and their homogeneous spaces over a number field $k$. For instance, for any such group $G$ and for any open subset $U$ of $G$ with ${\mathrm {codim}}(G\setminus U, G)\geqslant 2$, we prove that (i) if $G$ is $k$-simple and $k$-isotropic, then $U$ satisfies strong approximation off any finite number of places; and (ii) if $G$ is the spin group of a non-degenerate quadratic form which is not compact over archimedean places, then $U$ satisfies strong approximation off all archimedean places. As a consequence, we prove that the same property holds for affine quadratic hypersurfaces. Our approach combines a fibration method with subgroup actions developed for induction on the codimension of $G\setminus U$, and an affine linear sieve which allows us to produce integral points with almost-prime polynomial values.

Report of the spin group

1986 ◽  
Author(s):  
O. Chamberlain
Keyword(s):  
Spin Group

Magnetic Resonance Line Shapes in Solids: The Rotating Three‐Spin Group

1970 ◽  
Vol 52 (12) ◽  
pp. 6224-6231 ◽  
Author(s):  
Thomas B. Cobb ◽  
Charles S. Johnson
Keyword(s):  
Magnetic Resonance ◽  
Resonance Line ◽  
Spin Group ◽  
Line Shapes

scholarly journals An example of a bireflectional spin group

2003 ◽  
Vol 81 (1) ◽  
pp. 1-4
Author(s):  
Oliver Villa
Keyword(s):  
Spin Group

Relativistic fermions: Introduction

2021 ◽  
pp. 258-291
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Field Theory ◽  
Free Field ◽  
Free Action ◽  
Rotation Group ◽  
Dirac Fermions ◽  
Perturbative Calculation ◽  
Relativistic Limit ◽  
Spin Group ◽  
Group Spin ◽  
General Action

Some basic concepts needed for the discussion of Fermi fields have been introduced earlier, as in quantum mechanics (QM) with Grassmann variables, a representation by field integrals of the statistical operator e<συπ>−βH</συπ> for the non-relativistic Fermi gas in the formalism of second quantization, and an expression for the evolution operator. Here, it is first recalled how relativistic fermions transform under the spin group. The free action for Dirac fermions is analysed, the relation between fields and particles explained, an expression for the scattering matrix obtained, and the non-relativistic limit of a model of self-coupled massive Dirac fermions derived. A formalism of Euclidean relativistic fermions is then introduced. In the Euclidean formalism: fermions transform under the fundamental representation of the spin group Spin(d) associated with the SO(d) rotation group (spin 1/2 fermions for d = 4). As for the scalar field theory, the Gaussian integral, which corresponds to a free field theory is calculated. Then the generating functional of correlation functions is obtained by adding a source term to the action. The field integral corresponding to a general action with an interaction expandable in powers of the field, can be expressed in terms of a series of Gaussian integrals, which can be calculated, for example, with the help of Wick's theorem. The connection between spin and statistics is verified by a simple perturbative calculation. The appendix describes a few additional properties of the spin group, the algebra of γ matrices, and the corresponding spinors for Euclidean fermions.

scholarly journals Changes in Cortical Activation Patterns in Language Areas following an Aerobic Exercise Intervention in Older Adults

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Joe Nocera ◽  
Bruce Crosson ◽  
Kevin Mammino ◽  
Keith M. McGregor
Keyword(s):  
Older Adults ◽  
Aerobic Exercise ◽  
Verbal Fluency ◽  
Exercise Intervention ◽  
Community Dwelling ◽  
Spin Group ◽  
Verbal Fluency Task ◽  
Semantic Verbal Fluency ◽  
Link Type ◽  
Fluency Task

Previous work has shown that older adults who evidence increased right inferior frontal gyrus (IFG) activity during language tasks show decreased sematic verbal fluency performance. The current study sought to evaluate if an aerobic exercise intervention can alter patterns of brain activity during a semantic verbal fluency task assessed by functional magnetic resonance imaging (fMRI). Thirty-two community-dwelling, sedentary older adults were enrolled to a 12-week aerobic “Spin” exercise group or a 12-week nonaerobic exercise control condition (Balance). Thirty participants completed their assigned intervention (16 Spin; 14 Balance) with pre- and postintervention assessments of a semantic verbal fluency task during fMRI and estimated VO2max testing. There was a significant increase in the change scores for estimated VO2max of the Spin group when compared to the Balance group. Semantic verbal fluency output within the scanner was also improved in the Spin group as compared to controls at postassessment. Group fMRI comparisons of IFG activity showed lower activity in the right IFG following the intervention in the aerobic Spin group when compared to the Balance group. Regression analysis of imaging data with change in both estimated VO2max and semantic verbal fluency was negatively correlated with activity in right IFG. The current work is registered as clinical trial with NCT01787292 and NCT02787655.

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