By base change, i′n is also a closed embedding, hence affine. We have the unit map F → in∗i ′∗ n (F ). Applying Rf∗ gives a map Rf∗(F ) → Rf∗i ′ n∗i ′∗ n (F ). Since i ′ n is affine, Ri′n∗ ≃ i ′ n∗. By Leray’s spectral sequence, Rf∗i ′ n∗ ≃ R(f ◦ i ′ n)∗ ≃ R(in ◦ fn)∗ ≃ in∗Rfn∗. Applying H, we have a map Rf∗(F ) → R f∗i ′ n∗i ′∗ n (F ) ≃ in∗R fn∗(i ′∗ n (F )). Applying in∗i ∗ n to both sides an...

The purpose of this document is to prove some basic properties of affine transformations involving convex sets. Here are a few online references for background information: http://math.ucr.edu/∼res/progeom/pgnotes02.pdf http://math.ucr.edu/∼res/math133/metgeom.pdf Recall that an affine transformation of R is a map of the form F (x) = b+A(x), where b ∈ E is some fixed vector and A is an invertib...

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 ...

Ciphers y = C(x, k) and  = (, ) are isomorphic if there exists invertible computable in both directions map y ↔ , x ↔ , k ↔ . Cipher is vulnerable if and only if isomorphic cipher is vulnerable. Instead of computing the key of a cipher it is sufficient to find suitable isomorphic cipher and compute its key. If φ is arbitrary substitution and T is round substitution, its conjugate  = φTφ...

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.

The normal map has proven to be a powerful tool for solving generalized equations of the form: find ; £ C, with 0 e F{z) + NiAz). where C is a convex set and Nr(z) is the normal cone to C at ;. In this paper, we use the 7"-map, a generalization of the normal map, to solve equations of the more general form: find : E dom(r), with 0 ^ E(z) + T(z). where T is a maximal monotone multifunction. We p...

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...