Related papers: The fundamental theorem of affine geometry in $(L^…
The numerical range of a matrix is studied geometrically via the cone of positive semidefinite matrices (or semidefinite cone for short). In particular it is shown that the feasible set of a two-dimensional linear matrix inequality (LMI),…
The numerical range of a matrix is studied geometrically via the cone of positive semidefinite matrices (or semidefinite cone for short). In particular it is shown that the feasible set of a two-dimensional linear matrix inequality (LMI),…
We prove that if E a subset of an n-dimensional manifold, then every continuous R^n-valued map on E that is zero-free on the interior of E can be approximated in the fine topology, and hence, in particular, in the uniform topology, by a…
We consider several characterizations of $\mathbb R$-linear mappings. In particular, we give a characterization of linear mappings whose range is $\geq$ 2 dimensional, in terms of preservation of lines (and contraction of lines to a point)…
It was shown by Barron--Shafiee that an analogue of Gromov's non-squeezing theorem holds for affine maps which preserve a power $\omega^k$ of the symplectic form $\omega$ on $\mathbb{R}^{2n}$. In the present paper, we state and prove in two…
The aim of this note is to prove the algebraic geometry analogue of the Invariant tubular neighborhood theorem which concerns the actions of compact Lie groups on smooth manifolds.
We show that, for uniformly locally finite metric spaces $X$ and $Y$ with isomorphic uniform Roe algebras $C^*_u(X)$ and $C^*_u(Y)$, the existence of a bijective coarse equivalence $f \colon X \to Y$ is equivalent to the injectivity of the…
We establish Thom's jet transversality theorem for regular maps from an affine algebraic manifold to an algebraic manifold satisfying a suitable flexibility condition. It can be considered as the algebraic version of Forstneri\v{c}'s jet…
A general procedure of affinization of linear algebra structures is illustrated by the case of Leibniz algebras. Specifically, the definition of an affine Leibniz bracket, that is, a bi-affine operation on an affine space that at each…
This paper is devoted to the complete classification of space curves under affine transformations in the view of Cartan's theorem. Spivak has introduced the method but has not found the invariants. Furthermore, for the first time, we…
We say that an indecomposable Cartan matrix A with entries in the ground field of characteristic 0 is almost affine if the Lie sub(super)algebra determined by it is not finite dimensional or affine but the Lie (super)algebra determined by…
Let F = F_p for any fixed prime p >= 2. An affine-invariant property is a property of functions on F^n that is closed under taking affine transformations of the domain. We prove that all affine-invariant property having local…
We introduce an (equi-)affine invariant diffusion geometry by which surfaces that go through squeeze and shear transformations can still be properly analyzed. The definition of an affine invariant metric enables us to construct an invariant…
The goal of this paper is to study the geometry of the connected unit component of the real general linear Lie group $4$ dimensional $G_0$ as a Lorentzian and flat affine manifold. As the group $G_0$ is naturally equipped with a…
Classification of curves up to affine transformation in a finite dimensional space was studied by some different methods. In this paper, we achieve the exact formulas of affine invariants via the equivalence problem and in the view of…
Let F be a continuous injective map from an open subset of R^n to R^n. Assume that, for infinitely many k>1, F induces a bijection between the rational points of denominator k in the domain and those in the image (the denominator of…
We prove that a linear mapping on the algebra \(\mathfrak{sl}_n\) of all trace zero complex matrices is a local automorphism if and only if it is an automorphism or an anti-automorphism. We also show that a linear mapping on a simple…
Let $R$ be a ring, and $n$ a fixed nonnegative integer. An $R$-module $W$ is called $L_{n}$-injective if ${\rm Ext}_{R}^{1}(M,W)=0$ for any $R$-module $M$ with flat dimension at most $n$. In this paper, we prove first that…
We investigate the notions of \emph{localization} and \emph{filtration} in the context of extended affine Lie algebras. Our primary objective is to develop a localization theory that facilitates the construction of meaningful local…
Congruences, or $2$-parameter families of lines in $3$-space are of interest in many situations, in particular in geometric optics. In this paper we consider elements of their geometry which are invariant under affine changes of…