Related papers: The Marr Conjecture and Uniqueness of Wavelet Tran…
We prove an implicit function theorem and an inverse function theorem for free noncommutative functions over operator spaces and on the set of nilpotent matrices. We apply these results to study dependence of the solution of the initial…
In this paper we study how zeros of the Fourier transform of a function $f: \mathbb{Z}_p^d \to \mathbb{C}$ are related to the structure of the function itself. In particular, we introduce a notion of bandwidth of such functions and discuss…
We consider an inverse problem for the linear one-dimensional wave equation with variable coefficients consisting in determining an unknown source term from a boundary observation. A method to obtain approximations of this inverse problem…
In this article we study the inverse problem of determining a semilinear term appearing in an elliptic equation from boundary measurements. Our main objective is to develop flexible and general theoretical results that can be used for…
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…
We consider initial boundary value problems of time-fractional advection-diffusion equations with the zero Dirichlet boundary value $\partial_t^{\alpha} u(x,t) = -Au(x,t)$, where $-A = \sum}{i,j=1}^d \partial_i(a_{ij}(x)\partial_j) +…
Shape from shading is a classical inverse problem in computer vision. This shape reconstruction problem is inherently ill-defined; it depends on the assumed light source direction. We introduce a novel mathematical formulation for…
We consider the inverse source problem of a fixed wavenumber: study properties of an acoustic source based on a single far- or near-field measurement. We show that nonradiating sources having a convex or non-convex corner or edge on their…
The Marden theorem of geometry of polynomials and the great Poncelet theorem from projective geometry of conics by their classical beauty occupy very special places. Our main aim is to present a strong and unexpected relationship between…
In this work, we consider the time-harmonic inverse elastic source problem of a fixed frequency for the Navier equation in two dimensions. We show that a convex polygon can be uniquely determined by a single far field measurement. Our…
It is shown that any convolution operator in the time domain can be represented exactly as a multiplication operator in the time-scale (wavelet) domain. The Mellin transform gives a one-to-one correspondence between frequency filters…
Let $ Tf =\sum_{ I} \varepsilon_I \langle f,h_{I^+}\rangle h_{I^-}$. Here, $ \lvert \varepsilon _I\rvert=1 $, and $ h_J$ is the Haar function defined on dyadic interval $ J$. We show that, for instance, \begin{equation*} \lVert T \rVert _{L…
The star transform is a generalized Radon transform mapping a function of two variables to its integrals along "star-shaped" trajectories, which consist of a finite number of rays emanating from a common vertex. Such operators appear in…
The Muskat problem models the evolution of the interface given by two different fluids in porous media. The Rayleigh-Taylor condition is natural to reach the linear stability of the Muskat problem. We show that the Rayleigh-Taylor condition…
Starting from the experimental fact that a moving charge experiences the Lorentz force and applying the fundamental principles of simplicity (first order derivatives only) and linearity (superposition principle), we show that the structure…
It is believed that some numerical technique must be employed for the determination of the system parameters of a visual binary or a star with a planet because the relevant equations are not only highly nonlinear but also transcendental…
We investigate inverse diffraction problems for penetrable gratings in a piecewise constant medium. In the TE polarization case, it is proved that a binary grating profile together with the refractive index beneath it can be uniquely…
We study the light ray transform on Minkowski space-time and its small metric perturbations acting on scalar functions which are solutions to wave equations. We show that the light ray transform uniquely determines the function in a stable…
The so-called principle of relativity is able to fix a general coordinate transformation which differs from the standard Lorentzian form only by an unknown speed which cannot in principle be identified with the light speed. Based on a…
We discuss inverse problems of determining the time-dependent source coefficient for a general class of subelliptic heat equations. We show that a single data at an observation point guarantees the existence of a (smooth) solution pair for…