Related papers: Towards Differential Calculus in stratified groups
We introduce the concept of a homogeneity supermanifold, which is, roughly speaking, a supermanifold equipped with a privileged atlas whose coordinates carry prescribed (real) homogeneity degrees. This structure defines a sheaf of graded…
We give an algebraic/geometric characterization of the classical pseudodifferential operators on a smooth manifold in terms of the tangent groupoid and its natural $\mathbb{R}^\times_+$-action. Specifically, we show that a properly…
Gaussian unitaries, generated by quadratic Hamiltonians, are fundamental in quantum optics and continuous-variable computing. Their structures correspond to symplectic (bosons) and orthogonal (fermions) groups, but physical realizations…
The path spaces of a directed graph play an important role in the study of graph $\css$. These are topological spaces that were originally constructed using groupoid and inverse semigroup techniques. In this paper, we develop a simple,…
Divisibility of dynamical maps is visualized by trajectories in the parameter space and analyzed within the framework of collision models. We introduce ultimate completely positive (CP) divisible processes, which lose CP divisibility under…
In this paper we use Gaussian processes (kernel methods) to learn mappings between trajectories of distinct differential equations. Our goal is to simplify both the representation and the solution of these equations. We begin by examining…
We consider adaptive estimation and statistical inference for high-dimensional graph-based linear models. In our model, the coordinates of regression coefficients correspond to an underlying undirected graph. Furthermore, the given graph…
In this paper we intend to give some calculus rules for tangent sets in the sense of Bouligand and Ursescu, as well as for corresponding derivatives of set-valued maps. Both first and second order objects are envisaged and the assumptions…
We introduce diffusion geometry as a new framework for geometric and topological data analysis. Diffusion geometry uses the Bakry-Emery $\Gamma$-calculus of Markov diffusion operators to define objects from Riemannian geometry on a wide…
The class of Schoenberg transformations, embedding Euclidean distances into higher dimensional Euclidean spaces, is presented, and derived from theorems on positive definite and conditionally negative definite matrices. Original results on…
We develop a pseudo-differential Weyl calculus on nilpotent Lie groups which allows one to deal with magnetic perturbations of right invariant vector fields. For this purpose we investigate an infinite-dimensional Lie group constructed as…
Regular algebraic surfaces isogenous to a higher product of curves can be obtained from finite groups with ramification structures. We find unmixed ramification structures for finite groups constructed as p-quotients of particular infinite…
We investigate elementary topological properties of sets of completely positive (CP) maps that arise in quantum Perron-Frobenius theory. We prove that the set of primitive CP maps of fixed Kraus rank is path-connected and we provide a…
Associated to any finite metric space are a large number of objects and quantities which provide some degree of structural or geometric information about the space. In this paper we show that in the setting of subsets of weighted Hamming…
Topological degrees of continuous mappings between manifolds of even dimension are studied in terms of index theory of pseudo-differential operators. The index formalism of non-commutative geometry is used to derive analytic integral…
Let $\mathrm{SO}^+(p,q)$ denote the identity connected component of the real orthogonal group with signature $(p,q)$. We give a complete description of the spaces of continuous and generalized translation- and $\mathrm{SO}^+(p,q)$-invariant…
To any solution of a linear system of differential equations, we associate a kernel, correlators satisfying a set of loop equations, and in presence of isomonodromic parameters, a Tau function. We then study their semiclassical expansion…
This is a concise introduction to the theory of Lie groupoids, with emphasis in their role as models for stacks. After some preliminaries, we review the foundations on Lie groupoids, and we carefully study equivalences and proper groupoids.…
This paper introduces Attribution Projection Calculus (AP-Calculus), a novel mathematical framework for determining causal relationships in structured Bayesian networks. We investigate a specific network architecture with source nodes…
The ability to represent and compare machine learning models is crucial in order to quantify subtle model changes, evaluate generative models, and gather insights on neural network architectures. Existing techniques for comparing data…