Let k be an algebraically closed field of characteristic zero. Let H : k → k 2 be a polynomial mapping such that the Jacobian JacH is a non-zero constant. In this note we prove, that if there is a line l ⊂ k such that H|l : l → k 2 is an injection, then H is a polynomial automorphism. 1. Main result Let k be an algebraically closed field of characteristic zero. Put k = k \ {0}. By Aut k we deno...

Let W 6= ã. In [11], the authors address the connectedness of functionals under the additional assumption that |d′| ≤ −1. We show that |WF | 3 e. Now in [11], it is shown that Hippocrates’s condition is satisfied. Next, this reduces the results of [11] to an approximation argument.

Injectivity with respect to morphisms having λ-presentable domains and codomains is characterized: such injectivity classes are precisely those closed under products, λ-directed colimits, and λ-pure subobjects. This sharpens the result of the first two authors (Trans. Amer. Math. Soc. 336 (1993), 785-804). In contrast, for geometric logic an example is found of a class closed under directed col...

Figalli–Kim–McCann proved in [14] the continuity and injectivity of optimal maps under the assumption (B3) of nonnegative cross-curvature. In the recent [15, 16], they extend their results to the assumption (A3w) of Trudinger-Wang [34], and they prove, moreover, the Hölder continuity of these maps. We give here an alternative and independent proof of the extension to (A3w) of the continuity and...

This article presents extensions of the Cramér-Wold theorem to measures that may have infinite mass near the origin. Corresponding results for sequences of measures are presented together with examples showing that the assumptions imposed are sharp. The extensions build on a number of results and methods concerned with injectivity properties of the Radon transform. Using a few tools from distri...

Since the introduction of the Parikh matrix mapping, its injectivity problem is on top of the list of open problems in this topic. In 2010 Salomaa provided a solution for the ternary alphabet in terms of a Thue system with an additional feature called counter. This paper proposes the notion of a Parikh rewriting system as a generalization and systematization of Salomaa’s result. It will be show...

One of the main tools to classify C∗-algebras is the study of its projections and its unitaries. It was proved by Cuntz in [Cun81] that if A is a purely infinite simple C∗-algebra, then the kernel of the natural map for the unitary group U(A) to the Ktheory groupK1(A) is reduced to the connected component U(A), i.e. A isK1-injective (see §3). We study in this note a finitely generated C∗-algebr...

A sufficient condition for C 1-continuity of subdivision surfaces was proposed by Reif [17] and extended to a more general setting in [22]. In both cases, the analysis of C 1-continuity is reduced to establishing injectivity and regularity of a characteristic map. In all known proofs of C 1-continuity, explicit representation of the limit surface on an annular region was used to establish regul...

r is called commuting regular ring (resp. semigroup) if for each x,y $in$ r thereexists a $in$ r such that xy = yxayx. in this paper, we introduce the concept of commuting$pi$-regular rings (resp. semigroups) and study various properties of them.

some of the so called smallness conditions in algebra as well as in category theory, are important and interesting for their own and also tightly related to injectivity, are essential boundedness, cogenerating set, and residual smallness.in this overview paper, we first try to refresh these smallness condition by giving the detailed proofs of the results mainly by bernhard banaschewski and walt...