Related papers: The 10-point and 12-point Number Theoretic Hilbert…
A recent paper by the first and third authors together with Sabourin raised the question of what the possible Hilbert functions are for fat point subschemes of the form $2p_1+...+2p_r$, for all possible choices of $r$ distinct points in the…
In this paper, we introduce a type of tensor neural network. For the first time, we propose its numerical integration scheme and prove the computational complexity to be the polynomial scale of the dimension. Based on the tensor product…
Vision transformers have been successfully applied to image recognition tasks due to their ability to capture long-range dependencies within an image. However, there are still gaps in both performance and computational cost between…
In this paper we propose a synergistic melting of neural networks and decision trees (DT) we call neural decision trees (NDT). NDT is an architecture a la decision tree where each splitting node is an independent multilayer perceptron…
In this article we propose a new adaptive numerical quadrature procedure which includes both local subdivision of the integration domain, as well as local variation of the number of quadrature points employed on each subinterval. In this…
An improved phase retrieval method based Hilbert transform is introduced to quantitatively calculate the phase distribution from distorted fringe pattern. Also phase measurement deflectomety are widely used in specular type samples. The…
One approach to multivariate operator theory involves concepts and techniques from algebraic and complex geometry and is formulated in terms of Hilbert modules. In these notes we provide an introduction to this approach including many…
Modern quantum optical systems such as photonic quantum computers and quantum imaging devices require great precision in their designs and implementations in the hope to realistically exploit entanglement and reach a real quantum advantage.…
Secure signal processing is becoming a de facto model for preserving privacy. We propose a model based on the Fully Homomorphic Encryption (FHE) technique to mitigate security breaches. Our framework provides a method to perform a Fast…
Neural Machine Translation (NMT) has become the new state-of-the-art in several language pairs. However, it remains a challenging problem how to integrate NMT with a bilingual dictionary which mainly contains words rarely or never seen in…
We propose a new numerical method for $\alpha$-dissipative solutions of the Hunter-Saxton equation, where $\alpha$ belongs to $W^{1, \infty}(\mathbb{R}, [0, 1))$. The method combines a projection operator with a generalized method of…
This article presents the numerical verification and validation of several inversion algorithms for V-line transforms (VLTs) acting on symmetric 2-tensor fields in the plane. The analysis of these transforms and the theoretical foundation…
We construct virtual fundamental classes on nested Hilbert schemes of points and curves in complex nonsingular projective surfaces. These classes recover the virtual classes of Seiberg-Witten theory as well as the (reduced) stable theory,…
Truncated Fourier Transforms (TFTs), first introduced by Van der Hoeven, refer to a family of algorithms that attempt to smooth "jumps" in complexity exhibited by FFT algorithms. We present an in-place TFT whose time complexity, measured in…
A few new N=2 superintegrable mappings in the (1|2) superspace are proposed and their origin is analyzed. Using one of them, acting like the discrete symmetry transformation of the N=2 supersymmetric modified NLS hierarchy, the recursion…
The inverse spectral transform for the Zakharov-Shabat equation on the semi-line is reconsidered as a Hilbert problem. The boundary data induce an essential singularity at large k to one of the basic solutions. Then solving the inverse…
Two approaches for determining Hilbert functions of fat point subschemes of $\mathbb P^2$ are demonstrated. A complete determination of the Hilbert functions which occur for 9 double points is given using the first approach, extending…
The harmonic polylogarithms (hpl's) are introduced. They are a generalization of Nielsen's polylogarithms, satisfying a product algebra (the product of two hpl's is in turn a combination of hpl's) and forming a set closed under the…
We consider an incremental approximation method for solving variational problems in infinite-dimensional Hilbert spaces, where in each step a randomly and independently selected subproblem from an infinite collection of subproblems is…
We derive spectral width estimates for traces of tempered solutions of a large class of multiplier equations in $\mathbf{R}^n$. The estimates are uniform for solutions up to a given order. In the process, we find a rather explicit…