Related papers: Composing generic linearly perturbed mappings and …

We introduce a new algebraic structure for multi-dimensional compositional embeddings, built on directional non-commutative monoidal operators. The core contribution of this work is this novel framework, which exhibits appealing theoretical…

Let $f:X\rightarrow Y$ be a map. $f$ is a {\it sequence-covering map}\cite{Si1} if whenever $\{y_{n}\}$ is a convergent sequence in $Y$ there is a convergent sequence $\{x_{n}\}$ in $X$ with each $x_{n}\in f^{-1}(y_{n})$; $f$ is an {\it…

We introduce a new approach to universal quantum knot invariants that emphasizes generating functions instead of generators and relations. All the relevant generating functions are shown to be perturbed Gaussians of the form $Pe^G$, where…

Generative models and vision encoders have largely advanced on separate tracks, optimized for different goals and grounded in different mathematical principles. Yet, they share a fundamental property: latent space Gaussianity. Generative…

The diffusion maps embedding of data lying on a manifold has shown success in tasks such as dimensionality reduction, clustering, and data visualization. In this work, we consider embedding data sets that were sampled from a manifold which…

Various classes of Graph Neural Networks (GNN) have been proposed and shown to be successful in a wide range of applications with graph structured data. In this paper, we propose a theoretical framework able to compare the expressive power…

This is an extended version of a communication made at the international conference ``Noncommutative Geometry and Physics'' held at Orsay in april 2007. In this proceeding, we make a review of some noncommutative constructions connected to…

Superdiffusion is an anomalous transport behavior. Recently, a new mechanism, termed the ``nodal mechanism," has been proposed to induce superdiffusion in quantum models. However, existing realizations of the nodal mechanism have so far…

We introduce a constructive algorithm for universal linear electromagnetic transformations between the $N$ input and $N$ output modes of a dielectric slab. The approach uses out-of-plane phase modulation programmed down to $N^2$ degrees of…

We introduce the notion of translational Riemannian manifolds and define a Gauss map for orientable immersed hypersurfaces lying in these ambients, an associated translational curvature and prove a Gauss-Bonnet theorem. We also use this…

Interpreting neural network classifiers using gradient-based saliency maps has been extensively studied in the deep learning literature. While the existing algorithms manage to achieve satisfactory performance in application to standard…

For a generic degree d smooth map f: N^n -> M^n we introduce its "transverse fundamental group" \pi(f), which reduces to \pi_1(M) in the case where f is a covering, and in general admits a monodromy homomorphism \pi(f) -> S_{|d|};…

The book is devoted to constructing embedding finite-dimensional maps into trivial bundles and investigating the corresponding general position properties.

Let $\mathcal{F}$ be a family of graphs, and let $p,r$ be nonnegative integers. The \textsc{$(p,r,\mathcal{F})$-Covering} problem asks whether for a graph $G$ and an integer $k$, there exists a set $D$ of at most $k$ vertices in $G$ such…

We prove rigidity results describing contextually-constrained maps defined on Grassmannians and manifolds of ordered independent line tuples in finite-dimensional vector or Hilbert spaces. One statement in the spirit of the Fundamental…

We prove that the Hatcher-Quinn and Wall invariants of a self-transverse immersion $f\co N^n\imm M^{2n}$ coincide. That is, we construct an isomorphism between their target groups which carries one onto the other. We also employ methods of…

Intermediate feature representations represent the backbone for the expressivity and adaptability of deep neural networks. However, their geometric structure remains poorly understood. In this submission, we provide indirect insights into…

We introduce the notion of set-wise injective maps and provide results about fiber embeddings. Our results improve some previous results in this area.

We establish an explicit embedding of a quantum affine $\mathfrak{sl}_n$ into a quantum affine $\mathfrak{sl}_{n+1}$. This embedding serves as a common generalization of two natural, but seemingly unrelated, embeddings, one on the quantum…

We study the superconductor-insulator transition in nanohole ultrathin films in a transverse magnetic field by numerical simulation of a Josephson-junction array model. Geometrical disorder due to the random location of nanoholes in the…