Related papers: Composing generic linearly perturbed mappings and …
We consider the entanglement between two one-dimensional quantum wires (Luttinger Liquids) coupled by tunneling through a quantum impurity. The physics of the system involves a crossover between weak and strong coupling regimes…
We give sufficient conditions for the following problem: given a topological space X, a metric space Y, a subspace Z of Y, and a continuous map f from X to Y, is it possible, by applying to f an arbitrarily small perturbation, to ensure…
We develop a discrete gauge-theoretic framework for superposition in large language models (LLMs) that replaces the single-global-dictionary premise with a sheaf-theoretic atlas of local semantic charts. Contexts are clustered into a…
In a noiseless linear estimation problem, one aims to reconstruct a vector x* from the knowledge of its linear projections y=Phi x*. There have been many theoretical works concentrating on the case where the matrix Phi is a random i.i.d.…
Proposing an effective and flexible matrix to represent a graph is a fundamental challenge that has been explored from multiple perspectives, e.g., filtering in Graph Fourier Transforms. In this work, we develop a novel and general…
In this article, we provide a simple and systematic way to represent general (inhomogeneous) fractals that may look different at different scales and places. By using set-valued compression maps, we express these general fractals as…
Although structural maps such as subductions and inductions appear naturally in diffeology, one of the challenges is providing suitable analogous for submersions, immersions, and \'{e}tale maps (i.e., local diffeomorphisms) consistent with…
This paper describes an approach for fitting an immersed submanifold of a finite-dimensional Euclidean space to random samples. The reconstruction mapping from the ambient space to the desired submanifold is implemented as a composition of…
We address the recently introduced notions of generalized principal bundle and generalized principal connection by keeping track of global geometric properties through local coordinate transformation laws. This approach leads us to…
Yor's generalized meander is a temporally inhomogeneous modification of the $2(\nu+1)$-dimensional Bessel process with $\nu > -1$, in which the inhomogeneity is indexed by $\kappa \in [0, 2(\nu+1))$. We introduce the non-colliding particle…
Diffusion Maps framework is a kernel based method for manifold learning and data analysis that defines diffusion similarities by imposing a Markovian process on the given dataset. Analysis by this process uncovers the intrinsic geometric…
We investigate the injectivity of the Frobenius map on thickenings of smooth varieties in projective space over a field of positive characteristic. We obtain uniform bounds -- i.e., independent of the characteristic -- on the thickening…
Normalizing flows have shown great success as general-purpose density estimators. However, many real world applications require the use of domain-specific knowledge, which normalizing flows cannot readily incorporate. We propose…
Matter wave interferometry is becoming an increasingly important technique in quantum metrology. However, unlike its photonic counterpart, this technique relies on the interference of particles possessing a non-zero rest mass and an…
This paper shows that an arbitrary generic submanifold in a complex manifold can be deformed into a 1-parameter family of generic submanifolds satisfying strong nondegeneracy conditions. The proofs use a careful analysis of the jet spaces…
Any generalized distance-squared mapping of equidimensional case has singularities, and their singularity types are wrapped into mystery in higher dimensional cases. Any generalized distance-squared mapping of equidimensional case is not…
The structure and representations of the quantum general linear supergroup GLq(m|n) are studied systematically by investigating the Hopf superalgebra Gq of its representative functions. Gq is factorized into $Gq^{\pi} Gq^{\bar\pi}$, and a…
We analyze the boundaries of multiply connected Fatou components of transcendental maps by means of universal covering maps and associated inner functions. A unified approach is presented, which includes invariant Fatou components (of any…
Neural networks are powerful function estimators, leading to their status as a paradigm of choice for modeling structured data. However, unlike other structured representations that emphasize the modularity of the problem -- e.g., factor…
We formulate and establish a generalization of Koll\'ar's injectivity theorem for adjoint bundles twisted by suitable multiplier ideal sheaves. As applications, we generalize Koll\'ar's torsion-freeness, Koll\'ar's vanishing theorem, and a…