Realizing Euclidean distance matrices by sphere intersection

Alencar, Jorge; Liberti, Leo

doi:10.1016/j.dam.2018.06.003

Cited by 13 publications

(11 citation statements)

References 13 publications

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“…Under the traditional TOPSIS Method, the calculation of the distance between a certain object’s indicator evaluation result and its positive optimal target is based on the two-dimensional distance. The commonly used calculation methods in academic literature include the Euclidean Distance Method [ 41 , 42 , 43 ] and the Minkowski Distance Method [ 44 , 45 , 46 ]. The Minkowski Distance Method measures the differences between unified indicators of multiple decision objects by calculating the distance between vectors.…”

Section: Methodsmentioning

confidence: 99%

Evaluation and Treatment Analysis of Air Quality Including Particulate Pollutants: A Case Study of Shandong Province, China

Jiang

Yang

2020

IJERPH

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At present, China’s air pollution and its treatment effect are issues of general concern in the academic circles. Based on the analysis of the development stages of air pollution in China and the development history of China’s air quality standards, we selected 17 cities of Shandong Province, China as the research objects. By expanding China’s existing Air Quality Index System, the air quality of six major pollutants including PM2.5 and PM10 in 17 cities from February 2017 to January 2020 is comprehensively evaluated. Then, with a forecast model, the air quality of the above cities in the absence of air pollution control policies since June 2018 was simulated. The results of the error test show that the model has a maximum error of 4.67% when simulating monthly assessment scores, and the maximum mean error of the four months is 3.17%. Through the comparison between the simulation results and the real evaluation results of air quality, we found that since June 2018, the air pollution control policies of six cities have achieved more than 10% improvement, while the air quality of the other 11 cities declined. The different characteristics of pollutants and the implementation of governance policies are perhaps the main reasons for the above differences. Finally, policy recommendations for the future air pollution control in Shandong and China were provided.

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Section: Methodsmentioning

confidence: 99%

Evaluation and Treatment Analysis of Air Quality Including Particulate Pollutants: A Case Study of Shandong Province, China

Jiang

Yang

2020

IJERPH

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“…Property 2(b) guarantees that there are at most two points, let us say {x + i , x − i }, in such intersection [23]. This spheres intersection can be computed in many different ways that we will not cover in this paper but are well studied in the literature [1,23].…”

Section: Assumptionmentioning

confidence: 99%

“…where || • || denotes the Euclidean norm, x v := x(v), ∀v ∈ V and d uv := d({u, v}), ∀{u, v} ∈ E. Each equation in (1) is called a distance constraint. We say that a realization x satisfies d uv if the corresponding distance constraint is verified.…”

Section: Introductionmentioning

confidence: 99%

A New Algorithm for the $$^K$$DMDGP Subclass of Distance Geometry Problems with Exact Distances

2021

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The fundamental inverse problem in distance geometry is the one of finding positions from inter-point distances. The Discretizable Molecular Distance Geometry Problem (DMDGP) is a subclass of the Distance Geometry Problem (DGP) whose search space can be discretized and represented by a binary tree, which can be explored by a Branch-and-Prune (BP) algorithm. It turns out that this combinatorial search space possesses many interesting symmetry properties that were studied in the last decade. In this paper, we present a new algorithm for this subclass of the DGP, which exploits DMDGP symmetries more effectively than its predecessors. Computational results show that the speedup, with respect to the classic BP algorithm, is considerable for sparse DMDGP instances related to protein conformation.

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“…, and so on. A DMDGP solution is obtained when such selections allow us to reach the last vertex of the DMDGP order such that all positions x , ..., xn satisfy the equations (1). The main cost of the iBP algorithm is related to backtracking in the search tree, when "wrong" distance values are selected.…”

Section: For Eachmentioning

confidence: 99%

“…, can be described in the language of the CGA, which allowed a better "view" of the problem, in addition to solve it just comparing distance values. The second advantage is based on the possibility to solve problems in higher dimensions, where sphere intersections are also involved [1].…”

Section: A Conformal Geometric Algebra Approachmentioning

confidence: 99%

NMR Protein Structure Calculation and Sphere Intersections

Alves

Souza

José

2020

Computational and Mathematical Biophysics

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Nuclear Magnetic Resonance (NMR) experiments can be used to calculate 3D protein structures and geometric properties of protein molecules allow us to solve the problem iteratively using a combinatorial method, called Branch-and-Prune (BP). The main step of BP algorithm is to intersect three spheres centered at the positions for atoms i − 3, i − 2, i − 1, with radii given by the atomic distances di−3,i, di−2,i, di−1,i, respectively, to obtain the position for atom i. Because of uncertainty in NMR data, some of the distances di−3,i should be represented as interval distances [{\underline{d}_{i - 3,i}},{\bar d_{i - 3,i}}], where {\underline{d}_{i - 3,i}} \le {d_{i - 3,i}} \le {\bar d_{i - 3,i}}. In the literature, an extension of the BP algorithm was proposed to deal with interval distances, where the idea is to sample values from [{\underline{d}_{i - 3,i}},{\bar d_{i - 3,i}}]. We present a new method, based on conformal geometric algebra, to reduce the size of [{\underline{d}_{i - 3,i}},{\bar d_{i - 3,i}}], before the sampling process. We also compare it with another approach proposed in the literature.

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Realizing Euclidean distance matrices by sphere intersection

Cited by 13 publications

References 13 publications

Evaluation and Treatment Analysis of Air Quality Including Particulate Pollutants: A Case Study of Shandong Province, China

Evaluation and Treatment Analysis of Air Quality Including Particulate Pollutants: A Case Study of Shandong Province, China

A New Algorithm for the $$^K$$DMDGP Subclass of Distance Geometry Problems with Exact Distances

NMR Protein Structure Calculation and Sphere Intersections

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