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This work focuses on the limitations about the insufficient fitting capability of current quantum machine learning methods, which results from the over-reliance on a single data embedding strategy. We propose a novel quantum machine…
In this exposition, I discuss several developments in the theory of vertex operator algebras, and I include motivation for the definition.
Ultrafilters are very useful and versatile objects with applications throughout mathematics: in topology, analysis, combinarotics, model theory, and even theory of social choice. Proofs based on ultrafilters tend to be shorter and more…
The scope of this text is to study a process that induces another proof of the Spectral Embedding Theorem: that any densely defined symmetric operator can be extended by a multiplication operator through an embedding of the Hilbert space…
Covering theory is an important tool in representation theory of algebras, however, the results and the proofs are scattered in the literature. We give an introduction to covering theory at a level as elementary as possible.
Machine learning systems, especially with overparameterized deep neural networks, can generalize to novel test instances drawn from the same distribution as the training data. However, they fare poorly when evaluated on out-of-support test…
We analyze the problem of high-order polynomial approximation from a many-body physics perspective, and demonstrate the descriptive power of entanglement entropy in capturing model capacity and task complexity. Instantiated with a…
After an historical introduction on the standard algebraic approach to quantum mechanics of large systems we review the basic mathematical aspects of the algebras of unbounded operators. After that we discuss in some details their relevance…
This tutorial paper presents a survey of results, both classical and new, linking inner functions and operator theory. Topics discussed include invariant subspaces, universal operators, Hankel and Toeplitz operators, model spaces, truncated…
We study higher-order elliptic operators on one-dimensional ramified structures (networks). We introduce a general variational framework for fourth-order operators that allows us to study features of both hyperbolic and parabolic equations…
We consider nonlinear optimization problems that involve surrogate models represented by neural networks. We demonstrate first how to directly embed neural network evaluation into optimization models, highlight a difficulty with this…
While many problems in machine learning focus on learning mappings between finite-dimensional spaces, scientific applications require approximating mappings between function spaces, i.e., operators. We study the problem of learning…
We consider the extension of the Heisenberg vertex operator algebra by all its irreducible modules. We give an elementary construction for the intertwining vertex operators and show that they satisfy a complex parametrized generalized…
We study the structure and representations of a family of vertex algebras obtained from affine superalgebras by quantum reduction. As an application, we obtain in a unified way free field realizations and determinant formulas for all…
The modular operator approach of Tomita-Takesaki to von Neumann algebras is elucidated in the algebraic structure of certain supersymmetric quantum mechanical systems. A von Neumann algebra is constructed from the operators of the system.…
A new totally algebraic formalism based on general, abstract ladder operators has been proposed. This approach heavily grounds in the superoperator formalism of Primas. However it is necessary to introduce many improvements in his…
The purpose of this paper is to study algebras of singular integral operators on $\mathbb{R}^{n}$ and nilpotent Lie groups that arise when one considers the composition of Calder\'on-Zygmund operators with different homogeneities, such as…
This is Part I of a series of papers constructing intertwining operator superalgebras and vertex tensor categories associated to the superconformal minimal models and other related models. In this paper, we construct the intertwining…
A class of neural networks that gained particular interest in the last years are neural ordinary differential equations (neural ODEs). We study input-output relations of neural ODEs using dynamical systems theory and prove several results…
Algebraic multigrid (AMG) methods are powerful solvers with linear or near-linear computational complexity for certain classes of linear systems, Ax=b. Broadening the scope of problems that AMG can effectively solve requires the development…