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We show how Gabidulin codes can be list decoded by using a parametrization approach. For this we consider a certain module in the ring of linearized polynomials and find a minimal basis for this module using the Euclidean algorithm with…
We study the structure of the branch set of solutions to Plateau's problem in metric spaces satisfying a quadratic isoperimetric inequality. In our first result, we give examples of spaces with isoperimetric constant arbitrarily close to…
We consider a regularization concept for the solution of ill--posed operator equations, where the operator is composed of a continuous and a discontinuous operator. A particular application is level set regularization, where we develop a…
To the best of our knowledge this paper is the first attempt to introduce and study polynomial interpolation of the polynomial data given on arbitrary varieties. In the first part of the paper we present results on the solvability of such…
Minimal free resolutions of graded modules over a noetherian polynomial ring have been attractive objects of interest for more than a hundred years. We introduce and study two natural extensions in the setting of graded modules over a…
One of the main objectives of science is the recognition of a general pattern in a particular phenomenon in some particular regime. In this work, this is achieved with the analytical expression for the optimal protocol that minimizes the…
Based on the joint bidiagonalization process of a large matrix pair $\{A,L\}$, we propose and develop an iterative regularization algorithm for the large scale linear discrete ill-posed problems in general-form regularization: $\min\|Lx\| \…
We consider the minimum error entropy (MEE) criterion and an empirical risk minimization learning algorithm in a regression setting. A learning theory approach is presented for this MEE algorithm and explicit error bounds are provided in…
We solve the existence problem for the minimal positive solutions $u\in L^{p}(\Omega, dx)$ to the Dirichlet problems for sublinear elliptic equations of the form \[ \begin{cases} Lu=\sigma u^q+\mu\qquad \quad \text{in} \quad \Omega, \\…
Overdetermined systems of first kind integral equations appear in many applications. When the right-hand side is discretized, the resulting finite-data problem is ill-posed and admits infinitely many solutions. We propose a numerical method…
For a division ring $\mathbb F$, the polynomials $f\in\mathbb F$ can be evaluated "on the left" and "on the right" giving rise to left and right Lagrange interpolation problems. The problems containig interpolation conditions of the same…
The minimum information copula principle initially suggested in \cite{MeeBed97} is a maximum entropy-like approach for finding the least informative copula, if it exists, that satisfies a certain number of expectation constraints specified…
A popular robust alternative of the classic principal component analysis (PCA) is the $\ell_1$-norm PCA (L1-PCA), which aims to find a subspace that captures the most variation in a dataset as measured by the $\ell_1$-norm. L1-PCA has shown…
We consider a general linear parabolic problem with extended time boundary conditions (including initial value problems and periodic ones), and approximate it by the implicit Euler scheme in time and the Gradient Discretisation method in…
We study rational normal curves via a connection to the chip firing game. A key technique, introduced in this article, is to interpret the defining ideal of the rational normal curve as an ideal associated to a generalisation of a cycle…
We present a simple theory explaining the construction and the correctness of an incremental and worst-case optimal decision procedure for modal logic with eventualities. The procedure gives an abstract account of important aspects of…
An ideal in a polynomial ring encodes a system of linear partial differential equations with constant coefficients. Primary decomposition organizes the solutions to the PDE. This paper develops a novel structure theory for primary ideals in…
We study the delay margin problem in the context of recent works by T. Qi, J. Zhu, and J. Chen, where a sufficient condition for the maximal delay margin is formulated in terms of an interpolation problem obtained after introducing a…
Solving initial value problems and boundary value problems of Linear Ordinary Differential Equations (ODEs) plays an important role in many applications. There are various numerical methods and solvers to obtain approximate solutions…
Reduced-order modeling is an efficient approach for solving parameterized discrete partial differential equations when the solution is needed at many parameter values. An offline step approximates the solution space and an online step…