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Łukaszyk-Karmowski Metric: Comparison
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Please note this is a comparison between Version 2 by Jason Zhu and Version 3 by Szymon Łukaszyk.

The Łukaszyk–Karmowski metric (LK-metric) defines a distance between two random variables or vectors. LK-metric is not a metric as it does not satisfy the identity of indiscernibles axiom of the metric; for the same random variables, its value is greater than zero, providing they are not both degenerated.

  • distance functions
  • identity of indiscernibles
  • ugly duckling theorem

1. Introduction

The LK-metric[1] between two continuous random variables X and Y having a joint probability density function (PDF) F(x,y) is defined as

If X and Y are independent, then

where f(x) and g(y) are PDFs of X and Y, and subscripts denote their types. For example, if X and Y have normal PDFs having the same standard deviation σ but different means μx, μy, then

LK-metric between two random variables having normal PDFs and the same standard deviations σ = {0, 0.2, 0.4, 0.6, 0.8, 1}.

where μxy=|μx-μy|. For discrete X and Y, LK-metric has a form

and for random vectors X and Y, LK-metric becomes

where d(x,y) is a metric function, such as the Euclidean metric. In case, X and Y are mutually and internally independent, a simplified form of LK-metric can also be defined as

If X and Y are degenerated, almost sure variables having the Dirac delta (or one-point, in the discrete case) PDFs, then LK-metric becomes the metric between their mean values.

and obviously

However, in any other case

LK-metric satisfies all the remaining axioms of the metric. It is symmetric by definition, and it satisfies the triangle inequality

Thus

since

The LK-metric is not the only distance function that does not satisfy the identity of indiscernibles axiom[2]. For example, the partial metric[3] also allows each object not necessarily to have zero distance from itself. However, the partial metric satisfies two additional axioms of small self-distances and modified triangle inequality, which are not satisfied by the LK-metric[4]. Remarkably, the identity of indiscernibles ontological axiom, introduced to philosophy by Gottfried Wilhelm Leibniz around 1686 for distinguishing between monads[5], is also refuted for first-quantized bosons, fermions, and paraparticles[6] and nvalindependently faces Black’s two-spheres counterexample[7] anated the trivialization dilemma. Finallby, the ugly duckling theorem[85] stated assertsin 1969 and asserting that every two objects one perceives are equally similar (or equally dissimilar). Consequently, the identity of indiscernibles is neither a logical nor an empirical principle.

This characteristic non-zero distance effect built in the LK-metric allows forto avoiding ill-conditioning problems in radial basis function interpolation[96][107] and inverse distance weighting[118][129][1310][1411], where the interpolation accuracy can be improved by choosing the type of distance metric[1512][1411] and leads to a smooth interpolation function[1613].  By preventing zero distances due tobased on parameter uncertainty, the LK-metric can, furthermore, be used toin analyzesis of nondeterministic dynamical systems with competing attractors [1714]. Since the LK-metric represents the mean of distances between all the outcomes of the two uncertain objects, it can also be used in uncertain nearest neighbor classification[1815]. The actual value of an uncertain object is modeled by a probability density function[1916]. LK-metric has been successfully applied in various fields of science and technology[17][18][19][20][21][22][23][24][13][25][26][27][16][28][29][30][7][31][32][33][10][34][35][10][36][37][38][13][39][40][41][42][43][44][45][11][46][47][48][49][14][49][50][51][52][17][53][54].

References

  1. S. Lukaszyk; A new concept of probability metric and its applications in approximation of scattered data sets. Comput. Mech.. 2004, 33, 299-304.
  2. Andrzej Tomski; Szymon Łukaszyk; Reply to "Various issues around the L1-norm distance". Ipi Lett.. 2024, 2, 1-8.
  3. S. G. Matthews; Partial Metric Topology. Ann. New York Acad. Sci.. 1994, 728, 183-197.
  4. Castro, Pablo Samuel and Kastner, Tyler and Panangaden, Prakash and Rowland, Mark. A Kernel Perspective on Behavioural Metrics for Markov Decision Processes. Transactions on Machine Learning Research. 2022, https://openreview.net/forum?id=nHfPXl1ly7, 2835-8856.
  5. Leibniz, Gottfried Wilhelm. Philosophical Papers and Letters (The Monadology); Loemker, Leroy E., Eds.; Springer Netherlands: Dordrecht, 1714 (1976); pp. 643–653.Satosi Watanabe; Epistemological Relativity. Ann. Jpn. Assoc. Philos. Sci.. 1986, 7, 1-14.
  6. Steven French; Michael Redhead; Quantum Physics and the Identity of Indiscernibles. BrYou-Long Zou; Fa-Long Hu; Can-Can Zhou; Chao-Liu Li; Keh-Jim Dunn; Analysis of radial basis function interpolation approach. Appl. J. PGeophiloys. Sci.. . 201988, 39, 233-246.3, 10, 397-410.
  7. Max Black; I.—THE IDENTITY OF INDISCERNIBLES. Abbas M. Abd; Suhad M. Abd; Modelling the strength of lightweight foamed concrete using support vector machine (SVM). Case Stud. Constr. Mind. ater.. 201952, LXI, 153-164.7, 6, 8-15.
  8. Satosi Watanabe; Epistemological Relativity. M. Gentile; F. Courbin; G. Meylan; Interpolating point spread function anisotropy. Anstron. Jpn. Assoc. PhiloAstrophys. Sci.. . 201986, 7, 1-14.2, 549, A1.
  9. You-Long Zou; Fa-Long Hu; Can-Can Zhou; Chao-Liu Li; Keh-Jim Dunn; Analysis of radial basis function interpolation approach. İ. Bedii Özdemir; A modification to temperature-composition pdf method and its application to the simulation of a transitional bluff-body flame. Comput. Math. Appl. Geophys.. . 2013, 10, 397-410.8, 75, 2574-2592.
  10. Abbas M. Abd; Suhad M. Abd; Modelling the strength of lightweight foamed concrete using support vector machine (SVM). CasChristos Anagnostopoulos; Edge-centric inferential modeling & analytics. J. Ne Stud. Constr. Materw. Comput. Appl.. . 202017, , 16, 8-15.4, 102696.
  11. M. Gentile; F. Courbin; G. Meylan; Interpolating point spread function anisotropy. AstronZhanglin Li; An enhanced dual IDW method for high-quality geospatial interpolation. Sci. AstroRephys.. 20212, 549, A1., 11, 1-17.
  12. İ. Bedii Özdemir; A modification to temperature-composition pdf method and its application to the simulation of a transitional bluff-body flame. CYou, Hojun and Kim, Dongsu. Development of an anisotropic spatial interpolation method for velocity in meandering river channel. J. Komput.rea Math. ApplWater Resour. Assoc.. 2018, 77, 5, 2574-2592.0, 1023.
  13. Christos Anagnostopoulos; Edge-centric inferential modeling & analytics. J. Netw. Petar Durdevic; Leif Hansen; Christian Mai; Simon Pedersen; Zhenyu Yang; Cost-Effective ERT Technique for Oil-in-Water Measurement for Offshore Hydrocyclone Installations. IFACom-Paput. Appl.. ersOnLine. 2020, 1615, 4, 102696.8, 147-153.
  14. Zhanglin Li; An enhanced dual IDW method for high-quality geospatial interpolation. ScKaio C. B. Benedetti; Paulo B. B. Goncalves; Stefano Lenci; Giuseppe Rega; Global analysis of stochastic and parametric uncertainty in nonlinear dynamical systems: adaptative phase-space discretization strategy, with application to Helmholtz oscillator. Nonli.near RepDyn.. 2021, 3, 111, 1-17., 15675-15703.
  15. You, Hojun and Kim, Dongsu. Development of an anisotropic spatial interpolation method for velocity in meandering river channel. J.Fabrizio Angiulli; Fabio Fassetti; Nearest Neighbor-Based Classification of Uncertain Data. ACM KoTrea Water Resour. Assocans. Knowl. Discov.. Data. 2017, 50, 1023.3, 7, 1-35.
  16. Petar Durdevic; Leif Hansen; Christian Mai; Simon Pedersen; Zhenyu Yang; Cost-Effective ERT Technique for Oil-in-Water Measurement for Offshore Hydrocyclone Installations. Angiulli, Fabrizio and Fassetti, Fabio. Indexing Uncertain Data in General Metric Spaces. IFAC-PEEE TransapersOnLinctions on Knowledge and Data Enginee. ring. 2015, 2, 248, 147-153., 1640-1657.
  17. Kaio C. B. Benedetti; Paulo B. B. Goncalves; Stefano Lenci; Giuseppe Rega; Global analysis of stochastic and parametric uncertainty in nonlinear dynamical systems: adaptative phase-space discretization strategy, with application to Helmholtz oscillator. Nonlinear Dyn.. 2023, 111, 15675-15703.James C. Davidson; Seth A. Hutchinson. A Sampling Hyperbelief Optimization Technique for Stochastic Systems; Springer Science and Business Media LLC: Dordrecht, GX, Netherlands, 2009; pp. 217-231.
  18. Fabrizio Angiulli; Fabio Fassetti; Nearest Neighbor-Based Classification of Uncertain Data. ACMGang Meng; Jane Law; Mary E Thompson; Small-scale health-related indicator acquisition using secondary data spatial interpolation. Int. TransJ. Knowl. DiscovHeal. Geogr. Data. . 2013, 7, 1-35.0, 9, 50-50.
  19. Angiulli, Fabrizio and Fassetti, Fabio. Indexing Uncertain Data in General Metric Spaces. IEEE Transactions on Knowledge and Data Engineering. 2012, 24, 1640-1657.Yingjie Xia; Xingmin Shi; Li Kuang; Junhua Xuan. Parallel geospatial analysis on windows HPC platform; Institute of Electrical and Electronics Engineers (IEEE): Piscataway, NJ, United States, 2010; pp. 210-213.
  20. James C. Davidson; Seth A. Hutchinson. A Sampling Hyperbelief Optimization Technique for Stochastic Systems; Springer Science and Business Media LLC: Dordrecht, GX, Netherlands, 2009; pp. 217-231.V. Roshan Joseph; Lulu Kang; Regression-Based Inverse Distance Weighting With Applications to Computer Experiments. Technometrics. 2011, 53, 254-265.
  21. Gang Meng; Jane Law; Mary E Thompson; Small-scale health-related indicator acquisition using secondary data spatial interpolation. InFengxiang Xu; Guangyong Sun; Guangyao Li; Qing Li; Crashworthiness design of multi-component tailor-welded blank (TWB) structures. Struct. J. HeaMul. Geogrtidiscip. Optim.. 2010, 9, 50-50.3, 48, 653-667.
  22. Yingjie Xia; Xingmin Shi; Li Kuang; Junhua Xuan. Parallel geospatial analysis on windows HPC platform; Institute of Electrical and Electronics Engineers (IEEE): Piscataway, NJ, United States, 2010; pp. 210-213.Chanyoung Jun; Subhrajit Bhattacharya; Robert Ghrist. Pursuit-evasion game for normal distributions; Institute of Electrical and Electronics Engineers (IEEE): Piscataway, NJ, United States, 2014; pp. 83-88.
  23. V. Roshan Joseph; Lulu Kang; Regression-Based Inverse Distance Weighting With Applications to Computer Experiments. TSimon Pedersen; Christian Mai; Leif Hansen; Petar Durdevic; Zhenyu Yang; Online Slug Detection in Multi-phase Transportation Pipelines Using Electrical Tomography∗∗Supported by the Danish National Advanced Technology Foundation through PDPWAC Project (J.nr. 95-2012-3).. IFAC-PapechrsOnLinometrics. . 2011, 53, 254-265.5, 48, 159-164.
  24. Fengxiang Xu; Guangyong Sun; Guangyao Li; Qing Li; Crashworthiness design of multi-component tailor-welded blank (TWB) structures. Struct.Branislav Brutovsky; Denis Horvath; Towards inverse modeling of intratumor heterogeneity. Open MultidiPhyscip. Optim.. . 2013, 48, 653-667.5, 13, 103.
  25. Chanyoung Jun; Subhrajit Bhattacharya; Robert Ghrist. Pursuit-evasion game for normal distributions; Institute of Electrical and Electronics Engineers (IEEE): Piscataway, NJ, United States, 2014; pp. 83-88.S. Mohammed Hosseini; Samira Smaeili; Numerical integration of multi-dimensional highly oscillatory integrals, based on eRPIM. Numer. Algorithms. 2014, 68, 423-442.
  26. Simon Pedersen; Christian Mai; Leif Hansen; Petar Durdevic; Zhenyu Yang; Online Slug Detection in Multi-phase Transportation Pipelines Using Electrical Tomography∗∗Supported by the Danish National Advanced Technology Foundation through PDPWAC Project (J.nr. 95-2012-3).. IFAC-PLuciana Balsamo; Suparno Mukhopadhyay; Raimondo Betti; A statistical framework with stiffness proportional damage sensitive features for structural health monitoring. Smapert StrsOnLine. uct. Syst.. 2015, 48, 159-164., 15, 699-715.
  27. Branislav Brutovsky; Denis Horvath; Towards inverse modeling of intratumor heterogeneity. Open Phys.. 2015, 13, 103.Bing Han; Xinbo Gao; Hui Liu; Ping Wang. Auroral Oval Boundary Modeling Based on Deep Learning Method; Springer Science and Business Media LLC: Dordrecht, GX, Netherlands, 2015; pp. 96-106.
  28. S. Mohammed Hosseini; Samira Smaeili; Numerical integration of multi-dimensional highly oscillatory integrals, based on eRPIM. Numer.J. Park; V. Sreeja; M. Aquino; C. Cesaroni; L. Spogli; A. Dodson; G. De Franceschi; Performance of ionospheric maps in support of long baseline GNSS kinematic positioning at low latitudes. Radio AlgorScithms. .. 2014, 68, 423-442.6, 51, 429-442.
  29. Luciana Balsamo; Suparno Mukhopadhyay; Raimondo Betti; A statistical framework with stiffness proportional damage sensitive features for structural health monitoring. Grzegorz Lenda; Marcin Ligas; Paulina Lewińska; Anna Szafarczyk; The use of surface interpolation methods for landslides monitoring. KSmartCE J. Struct. SystCiv. Eng.. 2015, 15, 699-715., 20, 188-196.
  30. Bing Han; Xinbo Gao; Hui Liu; Ping Wang. Auroral Oval Boundary Modeling Based on Deep Learning Method; Springer Science and Business Media LLC: Dordrecht, GX, Netherlands, 2015; pp. 96-106.Jan Koloda; Jurgen Seiler; Andre Kaup; Frequency-Selective Mesh-to-Grid Resampling for Image Communication. IEEE Trans. Multimedia. 2017, 19, 1689-1701.
  31. J. Park; V. Sreeja; M. Aquino; C. Cesaroni; L. Spogli; A. Dodson; G. De Franceschi; Performance of ionospheric maps in support of long baseline GNSS kinematic positioning at low latitudes. RPijus Kasparaitis; Kipras Kančys; Phoneme vs. Diphone in Unit Selection TTS of Lithuanian. Balt. J. Modio Sci. Comput.. 2016, 51, 429-442.8, 6, 162-172.
  32. Grzegorz Lenda; Marcin Ligas; Paulina Lewińska; Anna Szafarczyk; The use of surface interpolation methods for landslides monitoring. KSCJorge Vicent; Jochem Verrelst; Juan Pablo Rivera-Caicedo; Neus Sabater; Jordi Munoz-Mari; Gustau Camps-Valls; Jose Moreno; Emulation as an Accurate Alternative to Interpolation in Sampling Radiative Transfer Codes. IEEE J. CivSel. Top. Appl. Engarth Obs. Remote. Sens.. 2015, 20, 188-196.8, 11, 4918-4931.
  33. Jan Koloda; Jurgen Seiler; Andre Kaup; Frequency-Selective Mesh-to-Grid Resampling for Image Communication. IEEE TrMatthew J. Lake; Marek Miller; Ray F. Ganardi; Zheng Liu; Shi-Dong Liang; Tomasz Paterek; Generalised uncertainty relations from superpositions of geometries. Clanss. Multimedia. Quantum Gravity. 2017, 19, 1689-1701.9, 36, 155012.
  34. Pijus Kasparaitis; Kipras Kančys; Phoneme vs. Diphone in Unit Selection TTS of Lithuanian. BalAimilios Sofianopoulos; Mozhgan Rahimi Boldaji; Benjamin Lawler; Sotirios Mamalis; Investigation of thermal stratification in premixed homogeneous charge compression ignition engines: A Large Eddy Simulation study. Int. J. Mod.Engine ComputRes.. 2018, 6, 162-172., 20, 931-944.
  35. Jorge Vicent; Jochem Verrelst; Juan Pablo Rivera-Caicedo; Neus Sabater; Jordi Munoz-Mari; Gustau Camps-Valls; Jose Moreno; Emulation as an Accurate Alternative to Interpolation in Sampling Radiative Transfer Codes. D. J. Tan; D. Honnery; A. Kalyan; V. Gryazev; S. A. Karabasov; D. Edgington-Mitchell; Equivalent Shock-Associated Noise Source Reconstruction of Screeching Underexpanded Unheated Round Jets. AIEEEAA J. Sel. Top. Appl. Earth Obs. Remote. Sens.. . 2018, 11, 4918-4931.9, 57, 1200-1214.
  36. Matthew J. Lake; Marek Miller; Ray F. Ganardi; Zheng Liu; Shi-Dong Liang; Tomasz Paterek; Generalised uncertainty relations from superpositions of geometries. Xiaohui Chen; Qing Zhao; Fei Huang; Rongzu Qiu; Yuhong Lin; Lanyi Zhang; Xisheng Hu; Understanding spatial variation in the driving pattern of carbon dioxide emissions from taxi sector in great Eastern China: evidence from an analysis of geographically weighted regression. Cleass. Quantum Gravitn Technol. Environ. Policy. . 202019, 36, 155012., 22, 979-991.
  37. Aimilios Sofianopoulos; Mozhgan Rahimi Boldaji; Benjamin Lawler; Sotirios Mamalis; Investigation of thermal stratification in premixed homogeneous charge compression ignition engines: A Large Eddy Simulation study. IHelin Gong; Yingrui Yu; Qing Li; Chaoyu Quan; An inverse-distance-based fitting term for 3D-Var data assimilation in nuclear core simulation. Annt. J. Engine Res.. Nucl. Energy. 202018, 20, 931-944., 141, 107346.
  38. D. J. Tan; D. Honnery; A. Kalyan; V. Gryazev; S. A. Karabasov; D. Edgington-Mitchell; Equivalent Shock-Associated Noise Source Reconstruction of Screeching Underexpanded Unheated Round Jets. AIAALiming Xu; Xianhua Zeng; He Zhang; Weisheng Li; Jianbo Lei; Zhiwei Huang; BPGAN: Bidirectional CT-to-MRI prediction using multi-generative multi-adversarial nets with spectral normalization and localization. Neural J.. Networks. 202019, 57, 1200-1214., 128, 82-96.
  39. Xiaohui Chen; Qing Zhao; Fei Huang; Rongzu Qiu; Yuhong Lin; Lanyi Zhang; Xisheng Hu; Understanding spatial variation in the driving pattern of carbon dioxide emissions from taxi sector in great Eastern China: evidence from an analysis of geographically weighted regression. Zhanglin Li; Xialin Zhang; Rui Zhu; Zhiting Zhang; Zhengping Weng; Integrating data-to-data correlation into inverse distance weighting. Cleanomput. Technol. EnvironGeosci. Policy. . 2020, 19, 22, 979-991.4, 203-216.
  40. Helin Gong; Yingrui Yu; Qing Li; Chaoyu Quan; An inverse-distance-based fitting term for 3D-Var data assimilation in nuclear core simulation. AnnWei Wei; Zecheng Guo; Liang Zhou; Binbin Xie; Junju Zhou; Assessing environmental interference in northern China using a spatial distance model: From the perspective of geographic detection. Sci. NucTotal. Energy. viron.. 2020, 141, 107346., 709, 136170.
  41. Liming Xu; Xianhua Zeng; He Zhang; Weisheng Li; Jianbo Lei; Zhiwei Huang; BPGAN: Bidirectional CT-to-MRI prediction using multi-generative multi-adversarial nets with spectral normalization and localization. Yiyuan Han; Bing Han; Zejun Hu; Xinbo Gao; Lixia Zhang; Huigen Yang; Bin Li; Prediction and variation of the auroral oval boundary based on a deep learning model and space physical parameters. Nonlineural Networkar Process. Geophys. .. 2020, 1, 28, 82-96.7, 11-22.
  42. Zhanglin Li; Xialin Zhang; Rui Zhu; Zhiting Zhang; Zhengping Weng; Integrating data-to-data correlation into inverse distance weighting. CompuYuying Lin; Xisheng Hu; Mingshui Lin; Rongzu Qiu; Jinguo Lin; Baoyin Li; Spatial Paradigms in Road Networks and Their Delimitation of Urban Boundaries Based on KDE. ISPRS Int. J. Geosci.. -Information. 202019, 24, 203-216., 9, 204.
  43. Wei Wei; Zecheng Guo; Liang Zhou; Binbin Xie; Junju Zhou; Assessing environmental interference in northern China using a spatial distance model: From the perspective of geographic detection. SciAimilios Sofianopoulos; Mozhgan Rahimi Boldaji; Benjamin Lawler; Sotirios Mamalis; John E Dec; Effect of engine size, speed, and dilution method on thermal stratification of premixed homogeneous charge compression–ignition engines: A large eddy simulation study. Int. TotalJ. Environgine Res.. 2020, 709, 136170.19, 21, 1612-1630.
  44. Yiyuan Han; Bing Han; Zejun Hu; Xinbo Gao; Lixia Zhang; Huigen Yang; Bin Li; Prediction and variation of the auroral oval boundary based on a deep learning model and space physical parameters. NoPanagiotis Pergantas; Nikos E. Papanikolaou; Chrisovalantis Malesios; Andreas Tsatsaris; Marios Kondakis; Iokasti Perganta; Yiannis Tselentis; Nikos Demiris; Towards a Semi-Automatic Early Warning System for Vector-Borne Diseases. Int. J. Enlvinear Process. Geophysron. Res. Public Heal.. 2020, 27, 11-22.1, 18, 1823.
  45. Yuying Lin; Xisheng Hu; Mingshui Lin; Rongzu Qiu; Jinguo Lin; Baoyin Li; Spatial Paradigms in Road Networks and Their Delimitation of Urban Boundaries Based on KDE. ISPRS IMobeen Akhtar; Yuanyuan Zhao; Guanglei Gao; An analytical approach for assessment of geographical variation in ecosystem service intensity in Punjab, Pakistan. Environt. J. Geo-Information. Sci. Pollut. Res.. 2020, 9, 204.1, 28, 38145-38158.
  46. Aimilios Sofianopoulos; Mozhgan Rahimi Boldaji; Benjamin Lawler; Sotirios Mamalis; John E Dec; Effect of engine size, speed, and dilution method on thermal stratification of premixed homogeneous charge compression–ignition engines: A large eddy simulation study. InBogdan-Mihai Negrea; Valeriu Stoilov-Linu; Cristian-Emilian Pop; György Deák; Nicolae Crăciun; Marius Mirodon Făgăraș; Expansion of the Invasive Plant Species Reynoutria japonica Houtt in the Upper Bistrița Mountain River Basin with a Calculus on the Productive Potential of a Mountain Meadow. Sust. J. Engaine Res.. 2019, 222, 1, 1612-1630.4, 5737.
  47. Panagiotis Pergantas; Nikos E. Papanikolaou; Chrisovalantis Malesios; Andreas Tsatsaris; Marios Kondakis; Iokasti Perganta; Yiannis Tselentis; Nikos Demiris; Towards a Semi-Automatic Early Warning System for Vector-Borne Diseases. IFabio Saggese; Vincenzo Lottici; Filippo Giannetti; Rainfall Map from Attenuation Data Fusion of Satellite Broadcast and Commercial Microwave Links. Sent. J. Envisoron. Res. Public Heal.. . 20221, 18, 1823., 22, 7019.
  48. Mobeen Akhtar; Yuanyuan Zhao; Guanglei Gao; An analytical approach for assessment of geographical variation in ecosystem service intensity in Punjab, Pakistan. Environ. Sci. PolluZhiguang Han; Jianzhang Xiao; Yingqi Wei; Spatial Distribution Characteristics of Microbial Mineralization in Saturated Sand Centrifuge Shaking Table Test. Mat. Resr.. 20221, 28, 38145-38158., 15, 6102.
  49. Bogdan-Mihai Negrea; Valeriu Stoilov-Linu; Cristian-Emilian Pop; György Deák; Nicolae Crăciun; Marius Mirodon Făgăraș; Expansion of the Invasive Plant Species Reynoutria japonica Houtt in the Upper Bistrița Mountain River Basin with a Calculus on the Productive Potential of a Mountain Meadow. Oludayo Ayodeji Akintunde; Chidi Vitalis Ozebo; Kayode Festus Oyedele; Groundwater Quality around upstream and downstream area of the Lagos lagoon using GIS and Multispectral Analysis. Sustacin. Afr.. 2022, 14, 5737.6, e01126.
  50. Fabio Saggese; Vincenzo Lottici; Filippo Giannetti; Rainfall Map from Attenuation Data Fusion of Satellite Broadcast and Commercial Microwave Links. A. A. El-Atik; Y. Tashkandy; S. Jafari; A. A. Nasef; W. Emam; M. Badr; Mutation of DNA and RNA sequences through the application of topological spaces. AIMSensors. Math.. 2022, 22, 7019.3, 8, 19275-19296.
  51. Zhiguang Han; Jianzhang Xiao; Yingqi Wei; Spatial Distribution Characteristics of Microbial Mineralization in Saturated Sand Centrifuge Shaking Table Test. MatDavid Hernández-López; Esteban Ruíz de Oña; Miguel A. Moreno; Diego González-Aguilera; SunMap: Towards Unattended Maintenance of Photovoltaic Plants Using Drone Photogrammetry. Droner.. s. 2022, 15, 6102.3, 7, 129.
  52. Oludayo Ayodeji Akintunde; Chidi Vitalis Ozebo; Kayode Festus Oyedele; Groundwater Quality around upstream and downstream area of the Lagos lagoon using GIS and Multispectral Analysis. Sci. Afr.. 2022, 16, e01126.
  53. A. A. El-Atik; Y. Tashkandy; S. Jafari; A. A. Nasef; W. Emam; M. Badr; Mutation of DNA and RNA sequences through the application of topological spaces. AIMS Math.. 2023, 8, 19275-19296.
  54. David Hernández-López; Esteban Ruíz de Oña; Miguel A. Moreno; Diego González-Aguilera; SunMap: Towards Unattended Maintenance of Photovoltaic Plants Using Drone Photogrammetry. Drones. 2023, 7, 129.
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