Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSpecial embeddings of finite-dimensional compacta in Euclidean spaces
96
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
If $g$ is a map from a space $X$ into $mathbb R^m$ and $z otin g(X)$, let $P_{2,1,m}(g,z)$ be the set of all lines $Pi^1subsetmathbb R^m$ containing $z$ such that $|g^{-1}(Pi^1)|geq 2$. We prove that for any $n$-dimensional metric compactum $X$ the functions $gcolon Xtomathbb R^m$, where $mgeq 2n+1$, with $dim P_{2,1,m}(g,z)leq 0$ for all $z otin g(X)$ form a dense $G_delta$-subset of the function space $C(X,mathbb R^m)$. A parametric version of the above theorem is also provided.
rate research
Read More
We generalize a theorem of E. Michael and M. E. Rudin and a theorem of D. Preiss and P. Simon; we give, as well, some partial answers to a recent question of A. V. Arhangelskiv{i}.
We consider a general notion of snowflake of a metric space by composing the distance by a nontrivial concave function. We prove that a snowflake of a metric space $X$ isometrically embeds into some finite-dimensional normed space if and only if $X$ is finite. In the case of power functions we give a uniform bound on the cardinality of $X$ depending only on the power exponent and the dimension of the vector space.
Our main problem is to find finite topological spaces to within homeomorphism, given (also to within homeomorphism) the quotient-spaces obtained by identifying one point of the space with each one of the other points. In a previous version of this paper, our aim was to reconstruct a topological space from its quotient-spaces; but a reconstruction is not always possible either in the sense that several non-homeomorphic topological spaces yield the same quotient-spaces, or in the sense that no topological space yields an arbitrarily given family of quotient-spaces. In this version of the paper we present an algorithm that detects, and deals with, all these situations.
We consider one of the classical manifold learning problems, that of reconstructing up to an almost isometry an embedding of a compact connected Riemannian manifold in a Euclidean space given the information on intrinsic distances between points from its almost dense subset. It will be shown that the most popular methods in data science to deal with such a problem, the classical Multidimensional scaling (MDS) and the Maximum variance unfolding (MVU) actually miss the point and may provide results very far from an isometry (and even may give no biLipshitz embedding). We will then provide an easy variational formulation of this problem which leads to an algorithm always providing an almost isometric imbedding with given controlled small distortion of original distances.
We classify the symmetric association schemes with faithful spherical embedding in 3-dimensional Euclidean space. Our result is based on previous research on primitive association schemes with $m_1 = 3$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
