Related papers: Convolution theorems associated with quaternion li…
The polar wavelet transform (PWT) has been proven to be a powerful mathematical tool for signal and image processing in recent years. Due to the increasing demand for directional representations of signals in engineering, it is impossible…
Quantum neural networks (QNNs) represent a pioneering intersection of quantum computing and deep learning. In this study, we unveil a fundamental convolution property inherent to QNNs, stemming from the natural parallelism of quantum gate…
Warped convolutions of operators were recently introduced in the algebraic framework of quantum physics as a new constructive tool. It is shown here that these convolutions provide isometric representations of Rieffel's strict deformations…
In a recent paper [J.Math.Phys. vol42, 2236-2265 (2001)], we discussed differential operators within a quaternionic formulation of quantum mechanics. In particular, we proposed a practical method to solve quaternionic and complex linear…
Convolution is an important tool in the construction of positive definite kernels on a manifold. This contribution provides conditions on an $L^2$-positive definite and zonal kernel on the unit sphere of $\mathbb{C}^q$ in order that the…
Convolutional neural networks (CNNs) have been used in many machine learning fields. In practical applications, the computational cost of convolutional neural networks is often high with the deepening of the network and the growth of data…
This work proposes kernel transform learning. The idea of dictionary learning is well known; it is a synthesis formulation where a basis is learnt along with the coefficients so as to generate or synthesize the data. Transform learning is…
The positive definiteness of real quadratic forms with convolution structures plays an important role in stability analysis for time-stepping schemes for nonlocal operators.In this work, we present a novel analysis tool to handle discrete…
We show that layer potential groupoids for conical domains constructed in an earlier paper (Carvalho-Qiao, Central European J. Math., 2013) are Fredholm groupoids, which enables us to deal with many analysis problems on singular spaces in a…
Machine learning models are a powerful theoretical tool for analyzing data from quantum simulators, in which results of experiments are sets of snapshots of many-body states. Recently, they have been successfully applied to distinguish…
The Legendre-based ultraspherical spectral method for ordinary differential equations is combined with a formula for the convolution of two Legendre series to produce a new technique for solving linear Fredholm and Volterra…
High-dimensional fractional reaction-diffusion equations have numerous applications in the fields of biology, chemistry, and physics, and exhibit a range of rich phenomena. While classical algorithms have an exponential complexity in the…
Derived from the results in [Giang et al.: \emph{Convolutions for the Fourier transforms with geometric variables and applications}, Math. Nachr. 283(12) (2010), 1758--1770], in this paper, we devoted to studying the boundedness properties…
Recently, Grosse and Lechner introduced a novel deformation procedure for non-interacting quantum field theories, giving rise to interesting examples of wedge-localized quantum fields with a non-trivial scattering matrix. In the present…
A novel spherical convolution is defined through the sifting property of the Dirac delta on the sphere. The so-called sifting convolution is defined by the inner product of one function with a translated version of another, but with the…
For quaternionic signal processing algorithms, the gradients of a quaternion-valued function are required for gradient-based methods. Given the non-commutativity of quaternion algebra, the definition of the gradients is non-trivial. The HR…
Transformationally invariant processors constructed by transformed input vectors or operators have been suggested and applied to many applications. In this study, transformationally identical processing based on combining results of all…
Linear processes are defined as a discrete-time convolution between a kernel and an infinite sequence of i.i.d. random variables. We modify this convolution by introducing decimation, that is, by stretching time accordingly. We then…
Methods of angular momenta are modified and used to solve some actual problems in quantum mechanics. In particular, we re-derive some known formulas for analytical and numerical calculations of matrix elements of the vector physical…
Convolutional codes are constructed, designed and analysed using row and/or block structures of unit algebraic schemes. Infinite series of such codes and of codes with specific properties are derived. Properties are shown algebraically and…