Related papers: The Lorentz Integral Transform and its Inversion
In this work, we consider the inhomogeneous nonlinear Schr\"odinger (INLS) equation in $\mathbb{R}^n$ \begin{align} i\partial_t u + \Delta u + \gamma |x|^{-b}|u|^{\alpha} u = 0, \end{align} where $\gamma=\pm 1$, and $\alpha$ and $b$ are…
The main objective of this article is to study several generalizations of the reverse order law for the Moore-Penrose inverse in ring with involution.
Properness for supervised losses stipulates that the loss function shapes the learning algorithm towards the true posterior of the data generating distribution. Unfortunately, data in modern machine learning can be corrupted or twisted in…
We study the convergence of the gradient descent method for solving ill-posed problems where the solution is characterized as a global minimum of a differentiable functional in a Hilbert space. The classical least-squares functional for…
A theoretical description of vortex electrons interacting with electric and magnetic fields is presented, based on Lorentz transformations. The general dynamical equations of motion of a twisted electron with intrinsic orbital angular…
The aim of this article is to investigate the issues of multiplicative inverses and composition in the set of formal Laurent series. We show the lack of general uniqueness of inverses of formal Laurent series; necessary and sufficient…
We present a rational version of the classical Landen transformation for elliptic integrals. This is employed to obtain explicit closed-form expressions for a large class of integrals of even rational functions and to develop an algorithm…
Covering ill-posed problems with compact and non-compact operators regarding the degree of ill-posedness is a never ending story written by many authors in the inverse problems literature. This paper tries to add a new narrative and some…
It's well-known that inverse problems are ill-posed and to solve them meaningfully one has to employ regularization methods. Traditionally, popular regularization methods have been the penalized Variational approaches. In recent years, the…
In this paper, we address the problem of approximating solutions of ill-posed problems using mollification. We quickly review existing mollification regularization methods and provide two new approximate solutions to a general ill-posed…
Inverse problems are often ill-posed, with solutions that depend sensitively on data. In any numerical approach to the solution of such problems, regularization of some form is needed to counteract the resulting instability. This paper is…
Attempts to quantize light in a manifestly Lorentz covariant manner fail because of the indefinite metric problem. Here an error in the interpretation is uncovered that is at the root of this problem.
An analysis of the Lorentz transformation shows that the unchangeability of the space-time coordinates of the inertial systems under consideration and the possibility of a direct projection of those coordinates onto another are the…
In this paper, we consider linear ill-posed problems in Hilbert spaces and their regularization via frame decompositions, which are generalizations of the singular-value decomposition. In particular, we prove convergence for a general class…
A free system, considered to be a comparison system, allows for the notion of objective existence and inertial frame. Transformations connecting inertial frames are shown to be either Lorentz or generalized Galilei.
Learning-based and data-driven techniques have recently become a subject of primary interest in the field of reconstruction and regularization of inverse problems. Besides the development of novel methods, yielding excellent results in…
We propose a new approach to linear ill-posed inverse problems. Our algorithm alternates between enforcing two constraints: the measurements and the statistical correlation structure in some transformed space. We use a non-linear multiscale…
The aim of this work is to prove inverse formulas for Laplace transform on semilattices of open-and-compact sets in a both discrete and non-discrete cases. These are partial answers to a question posed by Yu.~I.~Lyubich.
When an inverse problem is solved by a gradient-based optimization algorithm, the corresponding forward and adjoint problems, which are introduced to compute the gradient, can be also solved iteratively. The idea of iterating at the same…
In this paper, we revisit the issue intensively studied in recent years on the generation of terms by radiative corrections in models with broken Lorentz symmetry. The algebraic perturbative method of handling the problem of renormalization…