Let Ψ : Z → Z be an affine-linear map, and Ψ̇ be its linear part. A famous and difficult open conjecture of Dickson predicts the distribution of prime points in the affine sublattice Ψ(Z) of Z. Assuming the Gowers Inverse conjecture GI(s) and the Möbius and nilsequences conjecture MN(s) for some finite s > 1, GreenTao [GT3] verified Dickson’s conjecture for Ψ of complexity s. Let Φ : Z → Z be an...

Given a positive l.s.c. convex function c : Rd → R and an optimal transference plane π for the transportation problem ∫ c(x′ − x)π(dxdx′), we show how the results of [6] on the existence of a Sudakov decomposition for norm cost c = | · | can be extended to this case. More precisely, we prove that there exists a partition of Rd into a family of disjoint sets {Sh a }h,a together with the projecti...

Much local government has been utilizing a large-scale digital map using Geographic Information System (GIS). However, the maintenance and renewal of a map database need much labor and time. Moreover, the updating method of a map has not been established yet and there have been almost no successful examples. The purpose of the study is to establish a method of updating the large-scale map for l...

Abstract Let

Self-affine maps were successfully used for edge detection, image segmentation, and contour extraction. They belong to the general category of patch-based methods. Particularly, each self-affine map is defined by one pair of patches in the image domain. By minimizing the difference between these patches, the optimal translation vector of the self-affine map is obtained. Almost all image process...

Locally extended affine Lie algebras were introduced by Morita and Yoshii in [J. Algebra 301(1) (2006), 59-81] as a natural generalization of extended affine Lie algebras. After that, various generalizations of these Lie algebras have been investigated by others. It is known that a locally extended affine Lie algebra can be recovered from its centerless core, i.e., the ideal generated by weight...

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...