On Unique Determination of Polyhedral Sets

IF 1.9 3区数学 Q1 MATHEMATICS, APPLIED

Axioms Pub Date : 2023-11-05 DOI:10.3390/axioms12111035

Luca Rondi

引用次数: 0

Abstract

In this paper, we develop in detail the geometric constructions that lead to many uniqueness results for the determination of polyhedral sets, typically scatterers, by a finite minimal number of measurements. We highlight how unique continuation and a suitable reflection principle are enough to proceed with the constructions without any other assumption on the underlying partial differential equation or the boundary condition. We also aim to keep the geometric constructions and their proofs as simple as possible. To illustrate the applicability of this theory, we show how several uniqueness results present in the literature immediately follow from our arguments. Indeed, we believe that this theory may serve as a roadmap for establishing similar uniqueness results for other partial differential equations or boundary conditions.

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关于多面体集的唯一确定

在本文中，我们详细地发展了几何结构，这些几何结构导致了许多唯一性结果，用于确定多面体集，通常是散射体，通过有限最小次数的测量。我们强调了独特的延拓和合适的反射原理如何足以在不需要对潜在的偏微分方程或边界条件进行任何其他假设的情况下进行构造。我们还力求使几何结构及其证明尽可能简单。为了说明这一理论的适用性，我们展示了文献中出现的几个唯一性结果如何紧随我们的论点。事实上，我们相信这一理论可以作为为其他偏微分方程或边界条件建立类似唯一性结果的路线图。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Axioms Mathematics-Algebra and Number Theory

自引率

10.00%

发文量

604

审稿时长

11 weeks

期刊介绍： Axiomatic theories in physics and in mathematics (for example, axiomatic theory of thermodynamics, and also either the axiomatic classical set theory or the axiomatic fuzzy set theory) Axiomatization, axiomatic methods, theorems, mathematical proofs Algebraic structures, field theory, group theory, topology, vector spaces Mathematical analysis Mathematical physics Mathematical logic, and non-classical logics, such as fuzzy logic, modal logic, non-monotonic logic. etc. Classical and fuzzy set theories Number theory Systems theory Classical measures, fuzzy measures, representation theory, and probability theory Graph theory Information theory Entropy Symmetry Differential equations and dynamical systems Relativity and quantum theories Mathematical chemistry Automata theory Mathematical problems of artificial intelligence Complex networks from a mathematical viewpoint Reasoning under uncertainty Interdisciplinary applications of mathematical theory.