Related papers: On Gluing Data, Finite Ringed Spaces and schemes
We introduce a method for modeling a configuration of objects in 2D or 3D images using a mathematical "skeletal linking structure" which will simultaneously capture the individual shape features of the objects and their positional…
Wheeler emphasized the study of Superspace - the space of 3-geometries on a spatial manifold of fixed topology. This is a configuration space for GR; knowledge of configuration spaces is useful as regards dynamics and QM.In this Article I…
We endow the set of complements of a fixed subspace of a projective space with the structure of an affine space, and show that certain lines of such an affine space are affine reguli or cones over affine reguli. Moreover, we apply our…
A cusp-decomposable manifold is a manifold constructed from a finite number of complete, negatively curved, finite volume manifolds and identifying the boundaries of truncated cusps by diffeomorphisms. Using properties of the electric space…
As data grows in size and complexity, finding frameworks which aid in interpretation and analysis has become critical. This is particularly true when data comes from complex systems where extensive structure is available, but must be drawn…
The generalized chain geometry over the local ring $K(\epsilon;\sigma)$ of twisted dual numbers, where $K$ is a finite field, is interpreted as a divisible design obtained from an imprimitive group action. Its combinatorial properties as…
We prove a novel desingularization theorem, that allows to smoothly attach two given manifolds with corners by suitably gluing a pair of isometric faces, with control on both the scalar curvature of the resulting space and the mean…
Geometric relational embeddings map relational data as geometric objects that combine vector information suitable for machine learning and structured/relational information for structured/relational reasoning, typically in low dimensions.…
Learning embedding spaces of suitable geometry is critical for representation learning. In order for learned representations to be effective and efficient, it is ideal that the geometric inductive bias aligns well with the underlying…
We provide an explicit procedure to glue (not necessarily compact) silting objects along recollements of triangulated categories with coproducts having a 'nice' set of generators, namely, well generated triangulated categories. This…
We are interested in shapes of real algebraic curves in the plane and regions surrounded by them: they are named refined algebraic domains by the author. As characteristic finite sets, we consider points contained in two curves and the sets…
We present a framework to systematically investigate higher categorical symmetries in two-dimensional spin systems. Though exotic, such generalised symmetries have been shown to naturally arise as dual symmetries upon gauging invertible…
This paper lays the foundations for a unified framework for numerically and computationally applying methods drawn from a range of currently distinct geometrical approaches to statistical modelling. In so doing, it extends information…
This paper concerns the construction and analysis of a numerical scheme for a mixed discrete-continuous fragmentation equation. A finite volume scheme is developed, based on a conservative formulation of a truncated version of the…
The paper deals with a construction of a separating system of rational invariants for finite dimensional generic algebras. In the process of dealing an approach to a rough classification of finite dimensional algebras is offered by…
In this work we describe a class of subsets of the Euclidean plane which, with the induced length metric, are locally $CAT(0)$ spaces and we show that the gluing of two such subsets along a piece of their boundary is again a locally…
We review the global issues associated to gauge fixing ambiguities and their consequence for glueball spectroscopy. To avoid infrared singularities the theory is formulated in a finite volume. The examples of a cubic and spherical geometry…
For manifold learning, it is assumed that high-dimensional sample/data points are embedded on a low-dimensional manifold. Usually, distances among samples are computed to capture an underlying data structure. Here we propose a metric…
We present a uniform theory of constructible sheaves on arbitrary schemes with coefficients in topological or even condensed rings. This is accomplished by defining lisse sheaves to be the dualizable objects in the derived infinity-category…
2-stratifolds are a generalization of 2-manifolds in that there are disjoint simple closed curves where several sheets meet. They arise in the study of categorical invariants of 3-manifolds and may have applications to topological data…