Search results for "Affine transformation"
showing 10 items of 99 documents
Scattering amplitudes in affine gravity
2020
Affine gravity is a connection-based formulation of gravity that does not involve a metric. After a review of basic properties of affine gravity, we compute the tree-level scattering amplitude of scalar particles interacting gravitationally via the connection in a curved spacetime. We find that, while classically equivalent to general relativity, affine gravity differs from metric quantum gravity.
Anisotropic deformations in a class of projectively-invariant metric-affine theories of gravity
2020
Among the general class of metric-affine theories of gravity, there is a special class conformed by those endowed with a projective symmetry. Perhaps the simplest manner to realise this symmetry is by constructing the action in terms of the symmetric part of the Ricci tensor. In these theories, the connection can be solved algebraically in terms of a metric that relates to the spacetime metric by means of the so-called deformation matrix that is given in terms of the matter fields. In most phenomenological applications, this deformation matrix is assumed to inherit the symmetries of the matter sector so that in the presence of an isotropic energy-momentum tensor, it respects isotropy. In th…
Affine camera calibration from homographies of parallel planes
2010
This paper deals with the problem of retrieving the affine structure of a scene from two or more images of parallel planes. We propose a new approach that is solely based on plane homographies, calculated from point correspondences, and that does not require the recovery of the 3D structure of the scene. Neither vanishing points nor lines need to be extracted from the images. The case of a moving camera with constant intrinsic parameters and the one of cameras with possibly different parameters are both addressed. Extensive experiments with both synthetic and real images have validated our approach.
Uniform estimates for the X-ray transform restricted to polynomial curves
2012
We establish near-optimal mixed-norm estimates for the X-ray transform restricted to polynomial curves with a weight that is a power of the affine arclength. The bounds that we establish depend only on the spatial dimension and the degree of the polynomial. Some of our results are new even in the well-curved case.
Zeros of {-1,0,1}-power series and connectedness loci for self-affine sets
2006
We consider the set W of double zeros in (0,1) for power series with coefficients in {-1,0,1}. We prove that W is disconnected, and estimate the minimum of W with high accuracy. We also show that [2^(-1/2)-e,1) is contained in W for some small, but explicit e>0 (this was only known for e=0). These results have applications in the study of infinite Bernoulli convolutions and connectedness properties of self-affine fractals.
$$O_2(\mathbb {C})$$O2(C)-Vector Bundles and Equivariant Real Circle Actions
2020
The main goal of this article is to give an expository overview of some new results on real circle actions on affine four-space and their relation to previous results on \(O_2(\mathbb {C})\)-equivariant vector bundles. In Moser-Jauslin (Infinite families of inequivalent real circle actions on affine four-space, 2019, [13]), we described infinite families of equivariant real circle actions on affine four-space. In the present note, we will describe how these examples were constructed, and some consequences of these results.
Cyclic covers of affine T-varieties
2015
Abstract We consider normal affine T -varieties X endowed with an action of finite abelian group G commuting with the action of T . For such varieties we establish the existence of G-equivariant geometrico-combinatorial presentations in the sense of Altmann and Hausen. As an application, we determine explicit presentations of the Koras–Russell threefolds as bi-cyclic covers of A 3 equipped with a hyperbolic G m -action.
On exotic affine 3-spheres
2014
Every A 1 \mathbb {A}^{1} -bundle over A ∗ 2 , \mathbb {A}_{\ast }^{2}, the complex affine plane punctured at the origin, is trivial in the differentiable category, but there are infinitely many distinct isomorphy classes of algebraic bundles. Isomorphy types of total spaces of such algebraic bundles are considered; in particular, the complex affine 3 3 -sphere S C 3 , \mathbb {S}_{\mathbb {C}}^{3}, given by z 1 2 + z 2 2 + z 3 2 + z 4 2 = 1 , z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+z_{4}^{2}=1, admits such a structure with an additional homogeneity property. Total spaces of nontrivial homogeneous A 1 \mathbb {A}^{1} -bundles over A ∗ 2 \mathbb {A}_{\ast }^{2} are classified up to G m \mathbb {G}_{m}…
Noncancellation for contractible affine threefolds
2011
We construct two nonisomorphic contractible affine threefolds X X and Y Y with the property that their cylinders X × A 1 X\times \mathbb {A}^{1} and Y × A 1 Y\times \mathbb {A}^{1} are isomorphic, showing that the generalized Cancellation Problem has a negative answer in general for contractible affine threefolds. We also establish that X X and Y Y are actually biholomorphic as complex analytic varieties, providing the first example of a pair of biholomorphic but not isomorphic exotic A 3 \mathbb {A}^{3} ’s.
Uniform estimates for the local restriction of the Fourier transform to curves
2013
We prove sharp estimates, with respect to the ane arclength measure, for the restriction of the Fourier transform to a class of curves in R^d that includes curves of nite type. This measure possesses certain invariance and mitigation properties which are important in establishing uniform results. Peer reviewed