An affine invariant point on the class of convex bodies Kn in R, endowed with the Hausdorff metric, is a continuous map from Kn to R which is invariant under one-to-one affine transformations A on R, that is, p ` A(K) ́ = A ` p(K) ́ . We define here the new notion of dual affine point q of an affine invariant point p by the formula q(K) = p(K) for every K ∈ Kn, where K denotes the polar of K with...

for all a, b ∈ E and 0 ≤ t ≤ 1. Equivalently, f is affine if the map T :E → F , defined by Tx = fx− f(0), is linear. An isometry need not be affine. To see this, let E be the real line R, let F be the plane with the norm ‖x‖ = max(|x1|, |x2|), and let φ:R → R be any function such that |φ(s)−φ(t)| ≤ |s−t| for all s, t ∈ R, for example, φ(t) = |t| or φ(t) = sin t. Setting f(s) = (s, φ(s)) we get ...

Abstract Given a holomorphic self-map $\varphi $ of $\mathbb {D}$ (the open unit disc in {C}$ ), the composition operator $C_{\varphi } f = \circ \varphi , $f \in H^2(\mathbb {\mathbb {D}})$ defines bounded linear on Hardy space $H^2(\mathbb . The model spaces are backward shift-invariant closed subspaces which canonically associated with inner functions. In this paper, we study that invariant ...

We prove that any holomorphic codimension 1 foliation on the complex projective plane has at most one singularity up to action of an ad-hoc birational map. Consequently, algebraic affine no singularities a suitable self map into itself.

Locally strongly convex surfaces which are extremal for the first variation of the equiaffine area integral have been investigated on several occasions. Here we are interested in the description of their behaviour at infinity. We consider an affine maximal annular end which is a graph of vertical flux and give a detailed representation of it when its affine conormal map works well at infinity.

We prove that the period map from the Teichmüller space of polarized and marked Calabi-Yau manifolds to the classifying space of polarized Hodge structures is an embedding. The proof is based on the constructions of holomorphic affine structure and global holomorphic affine flat coordinates on the Teichmüller space.

Let A and C be two unital simple C*-algebras with tracial rank zero. Suppose that C is amenable and satisfies the Universal Coefficient Theorem. Denote by KKe(C,A) ++ the set of those κ for which κ(K0(C)+ \ {0}) ⊂ K0(A)+ \ {0} and κ([1C ]) = [1A]. Suppose that κ ∈ KKe(C,A) . We show that there is a unital monomorphism φ : C → A such that [φ] = κ. Suppose that C is a unital AH-algebra and λ : T(...

Any map of schemes X → Y defines an equivalence relation R = X ×Y X → X × X, the relation of ”being in the same fiber”. We have shown elsewhere that not every equivalence relation has this form, even if it is assumed to be finite. By contrast, we prove here that every toric equivalence relation on an affine toric variety does come from a morphism and that quotients by finite toric equivalence r...

A novel hybrid cryptosystem which is resistant to quantum algorithm is developed. The system is combined with multivariate cryptosystem and chaotic system, two systems which are both secure under quantum attacks. The plaintexts are displaced by an affine transformation and encrypted by central map in multivariable cryptosystem. And then, the outputs of central map act as initial values in a two...