摘要翻译：
我们研究了加性基在平面上的推广。平面加性基是一组向量和集覆盖给定矩形的非负整数对。这些基础在有源传感器阵列中得到应用，例如雷达和医学成像。最小化基基数的问题以前没有解决过。我们提出了两种求小矩形最小基的算法：一种是在基元素可以在矩形任意位置的情况下，另一种是在限制条件下，元素被限制在左下象限的情况下。我们给出了这些搜索的数值结果，包括在受限设置中，所有矩形的最小基数高达$[0,11]\乘[0,11]$和高达$[0,46]\乘[0,46]$。我们还证明了大矩形在极小基基数上的渐近上下界。
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英文标题：
《Planar additive bases for rectangles》
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作者：
Jukka Kohonen, Visa Koivunen, Robin Rajam\"aki
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最新提交年份：
2017
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分类信息：

一级分类：Mathematics        数学
二级分类：Number Theory        数论
分类描述：Prime numbers, diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory
素数，丢番图方程，解析数论，代数数论，算术几何，伽罗瓦理论
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一级分类：Electrical Engineering and Systems Science        电气工程与系统科学
二级分类：Signal Processing        信号处理
分类描述：Theory, algorithms, performance analysis and applications of signal and data analysis, including physical modeling, processing, detection and parameter estimation, learning, mining, retrieval, and information extraction. The term "signal" includes speech, audio, sonar, radar, geophysical, physiological, (bio-) medical, image, video, and multimodal natural and man-made signals, including communication signals and data. Topics of interest include: statistical signal processing, spectral estimation and system identification; filter design, adaptive filtering / stochastic learning; (compressive) sampling, sensing, and transform-domain methods including fast algorithms; signal processing for machine learning and machine learning for signal processing applications; in-network and graph signal processing; convex and nonconvex optimization methods for signal processing applications; radar, sonar, and sensor array beamforming and direction finding; communications signal processing; low power, multi-core and system-on-chip signal processing; sensing, communication, analysis and optimization for cyber-physical systems such as power grids and the Internet of Things.
信号和数据分析的理论、算法、性能分析和应用，包括物理建模、处理、检测和参数估计、学习、挖掘、检索和信息提取。“信号”一词包括语音、音频、声纳、雷达、地球物理、生理、（生物）医学、图像、视频和多模态自然和人为信号，包括通信信号和数据。感兴趣的主题包括：统计信号处理、谱估计和系统辨识；滤波器设计；自适应滤波/随机学习；（压缩）采样、传感和变换域方法，包括快速算法；用于机器学习的信号处理和用于信号处理应用的机器学习；网络与图形信号处理；信号处理中的凸和非凸优化方法；雷达、声纳和传感器阵列波束形成和测向；通信信号处理；低功耗、多核、片上系统信号处理；信息物理系统的传感、通信、分析和优化，如电网和物联网。
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英文摘要：
  We study a generalization of additive bases into a planar setting. A planar additive basis is a set of non-negative integer pairs whose vector sumset covers a given rectangle. Such bases find applications in active sensor arrays used in, for example, radar and medical imaging. The problem of minimizing the basis cardinality has not been addressed before.   We propose two algorithms for finding the minimal bases of small rectangles: one in the setting where the basis elements can be anywhere in the rectangle, and another in the restricted setting, where the elements are confined to the lower left quadrant. We present numerical results from such searches, including the minimal cardinalities for all rectangles up to $[0,11]\times[0,11]$, and up to $[0,46]\times[0,46]$ in the restricted setting. We also prove asymptotic upper and lower bounds on the minimal basis cardinality for large rectangles.
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PDF链接：
https://arxiv.org/pdf/1711.08812