Let X be a rational homogeneous space and let QH(X)× loc be the group of invertible elements in the small quantum cohomology ring of X localised in the quantum parameters. We generalise results of [ChMaPe06] and realise explicitly the map π1(Aut(X)) → QH (X)× loc described in [Se97]. We even prove that this map is an embedding and realise it in the equivariant quantum cohomology ring QH T (X)× ...

̂ M of the connection ∇M to the universal cover ̂ M of M is a locally flat connection. The affine structure of ̂ M is defined by a local diffeomorphism DM : ̂ M → R called the developing map. The developing map gives rise to a representation hM : π1 M → Aff R called the holonomy. The linear part L hM of hM is the linear holonomy. The affine manifold M,∇M is complete if and only if DM is a diffeomor...

We introduce a non-iterative geometric-based method for shape matching using a novel set of geometric landmarks residing on a 2D contours. These landmarks are intrinsic and are computed from the differential geometry of the curve. We exploit the invariant properties of geometric landmarks that are local and preserved under the affine and some perspective transformation. Geometric invariant expl...

In this paper we consider generic families of 2-dimensional analytic vector fields unfolding a generic (codimension 1) saddle-node at the origin. We show that a complete modulus of orbital analytic classification for the family is given by an unfolding of the Martinet–Ramis modulus of the saddle-node. The Martinet–Ramis modulus is given by a pair of germs of diffeomorphisms, one of which is an ...

An Anosov diffeomorphism f on a torus T is affine if f lifts to an affine map on R. By a classical result of Franks and Manning, any Anosov diffeomorphism g on T is topologically conjugate to an affine Anosov diffeomorphism. More precisely, there is a homeomorphism φ : T → T such that f = φ◦g◦φ−1 is an affine Anosov diffeomorphism. We call φ the Franks-Manning conjugacy. The linear part of f is...

For any affine variety equipped with coordinates, there is a surjective, continuous map from its Berkovich space to its tropicalisation. Exploiting torus actions, we develop techniques for finding an explicit, continuous section of this map. In particular, we prove that such a section exists for linear spaces, Grassmannians of planes (reproving a result due to Cueto, Häbich, and Werner), matrix...

Let R be a Dedekind domain, G an affine flat R-group scheme, and B a flat R-algebra on which G acts. Let A → BG be an R-algebra map. Assume that A is Noetherian. We show that if the induced map K ⊗ A → (K ⊗ B)K⊗G is an isomorphism for any algebraically closed field K which is an R-algebra, then S ⊗A→ (S ⊗B)S⊗G is an isomorphism for any R-algebra S.

Let d > 1, and let (X,α) and (Y, β) be two zeroentropy Z-actions on compact abelian groups by d commuting automorphisms. We show that if all lower rank subactions of α and β have completely positive entropy, then any measurable equivariant map from X to Y is an affine map. In particular, two such actions are measurably conjugate if and only if they are algebraically conjugate.

Let {Si}i∈Λ be a finite contracting affine iterated function system (IFS) on Rd. (Σ,σ) denote the two-sided full shift over alphabet Λ, and let π:Σ→Rd coding map associated with IFS. We prove that projection of an ergodic σ-invariant measure Σ under π is always exact dimensional, its Hausdorff dimension satisfies Ledrappier–Young-type formula. Furthermore, result extends to average IFSs. This c...

In this paper, we introduce a novel discrete chaotic map named zigzag map that demonstrates excellent chaotic behaviors and can be utilized in truly random number generators (TRNGs). We comprehensively investigate the map and explore its critical chaotic characteristics and parameters. We further present two circuit implementations for the zigzag map based on the switched current technique as w...