Related papers: A coarse invariant for all metric spaces
A new characterization is provided for the class of compact rank-one symmetric spaces. Such spaces are the only symmetric spaces of compact type for which the standard vector field on their sphere bundles is Killing with respect to some…
The integrability condition called shape invariance is shown to have an underlying algebraic structure and the associated Lie algebras are identified. These shape-invariance algebras transform the parameters of the potentials such as…
Magnitude is a numerical isometric invariant of metric spaces, whose definition arises from a precise analogy between categories and metric spaces. Despite this exotic provenance, magnitude turns out to encode many invariants from integral…
We consider the space of countable structures with fixed underlying set in a given countable language. We show that the number of ergodic probability measures on this space that are $S_\infty$-invariant and concentrated on a single…
There exists a certain argument that in even dimensions, scale invariant quantum field theories are conformal invariant. We may try to extend the argument in $2n + \epsilon$ dimensions, but the naive extension has a small loophole, which…
Recently a scale invariant theory of gravity was constructed by imposing a conformal symmetry on general relativity. The imposition of this symmetry changed the configuration space from superspace - the space of all Riemannian 3-metrics…
In this letter we provide an invariant characterization for all spacetimes with all polynomial scalar invariants constructed from the Riemann tensor and its covariant derivatives vanishing except those zeroth order curvature invariants…
Let L be a countable language. We say that a countable infinite L-structure M admits an invariant measure when there is a probability measure on the space of L-structures with the same underlying set as M that is invariant under…
We study metric spaces that admit a conical bicombing and thus obey a weak form of non-positive curvature. Prime examples of such spaces are injective metric spaces. In this article we give a complete characterization of complete metric…
A long-standing question is what invariant sets can be shared by two maps acting on the same space. A similar question stands for invariant measures. A particular interesting case are expanding Markov maps of the circle. If the two involved…
If F is a type-definable family of commensurable subsets, subgroups or sub-vector spaces in a metric structure, then there is an invariant subset, subgroup or sub-vector space commensurable with F. This in particular applies to…
A space curve is determined by conformal arc-length, conformal curvature, and conformal torsion, up to M\"obius transformations. We use the spaces of osculating circles and spheres to give a conformally defined moving frame of a curve in…
We define and investigate modulation invariant spaces on a locally compact abelian group $G$ with respect to a closed subgroup of the dual group $\widehat{G}$. Using a range function approach, we establish a characterization of modulation…
We study the varieties of invariant totally geodesic submanifolds of isometries of the spherical, Euclidean and hyperbolic spaces in each finite dimension. We show that the dimensions of the connected components of these varieties determine…
The Grassmannian model represents harmonic maps from Riemann surfaces by families of shift-invariant subspaces of a Hilbert space. We impose a natural symmetry condition on the shift-invariant subspaces that corresponds to considering an…
Let m(a,b) and M(a,b,c) be symmetric means. We say that M is type 1 invariant with respect to m if M(m(a,c),m(a,b),m(b,c)) = M(a,b,c) for all a, b, c > 0. If m is strict and isotone, then we show that there exists a unique M which is type 1…
A generalization of metric space is presented which is shown to admit a theory strongly related to that of ordinary metric spaces. To avoid the topological effects related to dropping any of the axioms of metric space, first a new, and…
In this work we consider all metric Lie algebras, having a nondegenerate symmetric invariant bilinear form, over \C and \R up to dimension 5 and all metric Lie algebras over \C in dimension 6. We introduce cyclic and reduced cyclic…
This paper is concerned with polynomially generated multiplier invariant subspaces of the weighted Bergman space $A_{\boldsymbol{\beta}}^2$ in infinitely many variables. We completely classify these invariant subspaces under the unitary…
In this paper we define a numerical shape invariant of a continuous map called shape dimension of a map, which generalizes the shape dimension of a topological space. Some basic properties and applications of this invariant are given. The…