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**Asymptotic identity in min-plus algebra: a report on CPNS.**
*(English)*
Zbl 1233.68032

Summary: Network calculus is a theory initiated primarily in computer communication networks, especially with respect to the aspect of real-time communication, where min-plus algebra plays a role. Cyber-physical networking systems (CPNSs) are recently developing fast, and models of data flow as well as systems in CPNSs are, accordingly, greatly desired. Though min-plus algebra may be a promising tool to linearize any node in CPNSs, as can be seen from its applications to Internet computing, there are tough problems remaining unsolved in this regard. The identity in a min-plus algebra is one problem we address. We point out the confusion about the conventional identity in a min-plus algebra and present an analytical expression of the asymptotic identity that may not cause confusion.

### Keywords:

computer communication networks; min-plus algebra; cyber-physical networking systems; real-time communication
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\textit{M. Li} and \textit{W. Zhao}, Comput. Math. Methods Med. 2012, Article ID 154038, 11 p. (2012; Zbl 1233.68032)

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### References:

[1] | Z. Song, Y. Q. Chen, C. R. Sastry, and N. C. Tas, Optimal Observation for Cyber-Physical Systems, Springer, 2009. · Zbl 1219.93002 |

[2] | Commission of the European Communities, “Internet of things-an action plan for Europe,” Journal of International Wildlife Law and Policy, vol. 12, no. 1-2, pp. 108-120, 2009. |

[3] | A. Ferscha, M. Hechinger, A. Riener et al., “Peer-it: stick-on solutions for networks of things,” Pervasive and Mobile Computing, vol. 43, no. 3, pp. 448-479, 2008. · Zbl 05345778 |

[4] | G. D. Clifton, H. Byer, K. Heaton, D. J. Haberman, and H. Gill, “Provision of pharmacy services to underserved populations via remote dispensing and two-way videoconferencing,” American Journal of Health-System Pharmacy, vol. 60, no. 24, pp. 2577-2282, 2003. |

[5] | K. Traynor, “Navy takes telepharmacy worldwide,” American Journal of Health-System Pharmacy, vol. 67, no. 14, pp. 1134-1136, 2010. |

[6] | K. T. Chang, Introduction to Geographical Information Systems, McGraw-Hill, New York, NY, USA, 2008. |

[7] | M. F. Goodchild, “Twenty years of progress: GIScience in 2010,” Journal of Spatial Information Science, vol. 2010, no. 1, pp. 3-20, 2010. |

[8] | T. L. Lai and H. Xing, Statistical Models and Methods for Financial Markets, Springer, 2008. · Zbl 1149.62086 |

[9] | B. B. Mandelbrot, Gaussian Self-Affinity and Fractals, Springer, 2001. |

[10] | B. B. Mandelbrot, Multifractals and 1/f Noise, Springer, 1998. |

[11] | B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman, New York, NY, USA, 1982. · Zbl 0504.28001 |

[12] | D. Hainaut and P. Devolder, “Mortality modelling with Lévy processes,” Insurance, vol. 42, no. 1, pp. 409-418, 2008. · Zbl 1141.91516 |

[13] | C. Cattani, “Harmonic wavelet approximation of random, fractal and high frequency signals,” Telecommunication Systems, vol. 43, no. 3-4, pp. 207-217, 2010. · Zbl 05803252 |

[14] | C. Cattani, “Fractals and hidden symmetries in DNA,” Mathematical Problems in Engineering, vol. 2010, Article ID 507056, 31 pages, 2010. · Zbl 1189.92015 |

[15] | G. Mattioli, M. Scalia, and C. Cattani, “Analysis of large-amplitude pulses in short time intervals: application to neuron interactions,” Mathematical Problems in Engineering, vol. 2010, Article ID 895785, 15 pages, 2010. · Zbl 1189.37099 |

[16] | C. Cattani, “Shannon wavelets for the solution of integrodifferential equations,” Mathematical Problems in Engineering, vol. 2010, 22 pages, 2010. · Zbl 1191.65174 |

[17] | C. Cattani, “Shannon wavelets theory,” Mathematical Problems in Engineering, vol. 2008, Article ID 164808, 24 pages, 2008. · Zbl 1162.42314 |

[18] | S. Y. Chen, Y. F. Li, and J. Zhang, “Vision processing for realtime 3-D data acquisition based on coded structured light,” IEEE Transactions on Image Processing, vol. 17, no. 2, pp. 167-176, 2008. · Zbl 05516617 |

[19] | S. Y. Chen, H. Tong, Z. Wang, S. Liu, M. Li, and B. Zhang, “Improved generalized belief propagation for vision processing,” Mathematical Problems in Engineering, vol. 2011, Article ID 416963, 12 pages, 2011. · Zbl 1202.94026 |

[20] | S. Y. Chen and Y. F. Li, “Determination of stripe edge blurring for depth sensing,” IEEE Sensors Journal, vol. 11, no. 2, pp. 389-390, 2011. |

[21] | S. Y. Chen and Q. Guan, “Parametric shape representation by a deformable NURBS model for cardiac functional measurements,” IEEE Transactions on Biomedical Engineering, vol. 58, no. 3, pp. 480-487, 2011. |

[22] | S. Chen, J. Zhang, H. Zhang et al., “Myocardial motion analysis for determination of tei-index of human heart,” Sensors, vol. 10, no. 12, pp. 11428-11439, 2010. |

[23] | W. Mikhael and T. Yang, “A gradient-based optimum block adaptation ICA technique for interference suppression in highly dynamic communication channels,” EURASIP Journal on Advances in Signal Processing, vol. 2006, Article ID 84057, 10 pages, 2006. |

[24] | E. G. Bakhoum and C. Toma, “Dynamical aspects of macroscopic and quantum transitions due to coherence function and time series events,” Mathematical Problems in Engineering, vol. 2010, Article ID 428903, 13 pages, 2010. · Zbl 1191.35219 |

[25] | E. G. Bakhoum and C. Toma, “Mathematical transform of traveling-wave equations and phase aspects of quantum interaction,” Mathematical Problems in Engineering, vol. 2010, Article ID 695208, 15 pages, 2010. · Zbl 1191.35220 |

[26] | E. G. Bakhoum and C. Toma, “Relativistic short range phenomena and space-time aspects of pulse measurements,” Mathematical Problems in Engineering, vol. 2008, Article ID 410156, 20 pages, 2008. · Zbl 1163.83319 |

[27] | M. Li, “Change trend of averaged Hurst parameter of traffic under DDOS flood attacks,” Computers and Security, vol. 25, no. 3, pp. 213-220, 2006. |

[28] | M. Li, “A class of negatively fractal dimensional gaussian random functions,” Mathematical Problems in Engineering, vol. 2011, Article ID 291028, 18 pages, 2011. · Zbl 1209.28015 |

[29] | M. Li, “Generation of teletraffic of generalized cauchy type,” Physica Scripta, vol. 81, no. 2, Article ID 025007, 2010. · Zbl 1191.90013 |

[30] | M. Li, “Fractal time series-a tutorial review,” Mathematical Problems in Engineering, vol. 2010, Article ID 157264, 26 pages, 2010. · Zbl 1191.37002 |

[31] | M. Li, “A method for requiring block size for spectrum measurement of ocean surface waves,” IEEE Transactions on Instrumentation and Measurement, vol. 55, no. 6, pp. 2207-2215, 2006. |

[32] | M. Li, “Modeling autocorrelation functions of long-range dependent teletraffic series based on optimal approximation in hilbert space-a further study,” Applied Mathematical Modelling, vol. 31, no. 3, pp. 625-631, 2007. · Zbl 1197.94006 |

[33] | M. Li, W. Zhao, and S. Chen, “MBm-based scalings of traffic propagated in internet,” Mathematical Problems in Engineering, vol. 2011, Article ID 389803, 21 pages, 2011. · Zbl 1210.68027 |

[34] | M. Li and W. Zhao, “Variance bound of ACF estimation of one block of fGn with LRD,” Mathematical Problems in Engineering, vol. 2010, Article ID 560429, 14 pages, 2010. · Zbl 1191.94042 |

[35] | M. Li and W. Zhao, “Detection of variations of local irregularity of traffic under DDOS flood attack,” Mathematical Problems in Engineering, vol. 2008, Article ID 475878, 11 pages, 2008. · Zbl 1189.68114 |

[36] | M. Li, C. Cattani, and S.-Y. Chen, “Viewing sea level by a one-dimensional random function with long memory,” Mathematical Problems in Engineering, vol. 2011, Article ID 654284, 13 pages, 2011. · Zbl 05829261 |

[37] | M. Li and J. Y. Li, “On the predictability of long-range dependent series,” Mathematical Problems in Engineering, vol. 2010, Article ID 397454, 9 pages, 2010. · Zbl 1191.62160 |

[38] | M. Li and S. C. Lim, “Modeling network traffic using generalized cauchy process,” Physica A, vol. 387, no. 11, pp. 2584-2594, 2008. |

[39] | M. Li, S. C. Lim, and Sy. Chen, “Exact solution of impulse response to a class of fractional oscillators and its stability,” Mathematical Problems in Engineering, vol. 2011, Article ID 657839, 9 pages, 2011. · Zbl 1202.34018 |

[40] | A. R. Messina, P. Esquivel, and F. Lezama, “Time-dependent statistical analysis of wide-area time-synchronized data,” Mathematical Problems in Engineering, vol. 2010, Article ID 751659, 17 pages, 2010. · Zbl 1189.62188 |

[41] | M. Humi, “Assessing local turbulence strength from a time series,” Mathematical Problems in Engineering, vol. 2010, Article ID 316841, 13 pages, 2010. · Zbl 1189.37088 |

[42] | M. Dong, “A tutorial on nonlinear time-series data mining in engineering asset health and reliability prediction: concepts, models, and algorithms,” Mathematical Problems in Engineering, vol. 2010, Article ID 175936, 22 pages, 2010. · Zbl 1189.90047 |

[43] | Z. Liu, “Chaotic time series analysis,” Mathematical Problems in Engineering, vol. 2010, Article ID 720190, 31 pages, 2010. · Zbl 1191.37046 |

[44] | G. Toma, “Specific differential equations for generating pulse sequences,” Mathematical Problems in Engineering, vol. 2010, Article ID 324818, 11 pages, 2010. · Zbl 1191.37052 |

[45] | O. M. Abuzeid, A. N. Al-Rabadi, and H. S. Alkhaldi, “Fractal geometry-based hypergeometric time series solution to the hereditary thermal creep model for the contact of rough surfaces using the Kelvin-Voigt medium,” Mathematical Problems in Engineering, vol. 2010, Article ID 652306, 22 pages, 2010. · Zbl 1425.74335 |

[46] | M. Li and W. Zhao, “Visiting power laws in cyber-physical networking systems,” Mathematical Problems in Engineering, vol. 2012, Article ID 302786, p. 13, 2012. |

[47] | M. Li and S. C. Lim, “Power spectrum of generalized cauchy process,” Telecommunication Systems, vol. 43, no. 3-4, pp. 219-222, 2010. · Zbl 05803253 |

[48] | J. He, H. Qian, Y. Zhou, and Z. Li, “Cryptanalysis and improvement of a block cipher based on multiple chaotic systems,” Mathematical Problems in Engineering, vol. 2010, Article ID 590590, 14 pages, 2010. · Zbl 1195.94059 |

[49] | Z. Liao, S. Hu, and W. Chen, “Determining neighborhoods of image pixels automatically for adaptive image denoising using nonlinear time series analysis,” Mathematical Problems in Engineering, vol. 2010, 2010. · Zbl 1189.94022 |

[50] | G. Werner, “Fractals in the nervous system: conceptual implications for theoretical neuroscience,” Frontiers in Fractal Physiology, vol. 1, p. 28, 2010. |

[51] | B. J. West, “Fractal physiology and the fractional calculus: a perspective,” Frontiers in Fractal Physiology, vol. 1, 2010. |

[52] | H. Akimaru and K. Kawashima, Teletraffic: Theory and Applications, Springer, 1993. |

[53] | W. Yue, H. Takagi, and Y. Takahashi, Advances in Queueing Theory and Network Applications, Springer, 2009. · Zbl 1236.60005 |

[54] | J. D. Gibson, The Communications Handbook, IEEE Press, 1997. · Zbl 0902.94001 |

[55] | R. B. Cooper, Introduction to Queueing Tehory, Elsevier, 2nd edition, 1981. |

[56] | J. M. Pitts and J. A. Schormans, Introduction to ATM Design and Performance: With Applications Analysis Software, John Wiley & Sons, New York, NY, USA, 2nd edition, 2000. |

[57] | D. McDysan, QoS & Traffic Management in IP & ATM Networks, McGraw-Hill, New York, NY, USA, 2000. |

[58] | W. Stalling, High-Speed Networks: TCP/IP and ATM Design Principles, Prentice Hall, 2nd edition, 2002. |

[59] | R. L. Cruz, “A calculus for network delay-part 1: network elements in isolation part 2,” IEEE Transactions on Information Theory, vol. 37, no. 1, pp. 114-141, 1991. · Zbl 0712.94028 |

[60] | R. L. Cruz, “A calculus for network delay-part 2: network analysis,” IEEE Transactions on Information Theory, vol. 37, no. 1, pp. 132-141, 1991. · Zbl 0712.94029 |

[61] | R. L. Cruz, “Quality of service guarantees in virtual circuit switched networks,” IEEE Journal on Selected Areas in Communications, vol. 13, no. 6, pp. 1048-1056, 1995. |

[62] | W. Zhao and K. Ramamritham, “Virtual time csma protocols for hard real-time communication,” IEEE Transactions on Software Engineering, vol. 13, no. 8, pp. 938-952, 1987. |

[63] | A. Raha, S. Kamat, and W. Zhao, “Guaranteeing end-to-end deadlines in ATM networks,” in Proceedings of the 15th International Conference on Distributed Computing Systems, pp. 60-68, June 1995. |

[64] | C. S. Chang, “On deterministic traffic regulation and service guarantees: a systematic approach by filtering,” IEEE Transactions on Information Theory, vol. 44, no. 3, pp. 1097-1110, 1998. · Zbl 0908.90123 |

[65] | C. S. Chang, “Stability, queue length, and delay of deterministic and stochastic queueing networks,” IEEE Transactions on Automatic Control, vol. 39, no. 5, pp. 913-931, 1994. · Zbl 0818.90050 |

[66] | J. Y. Le Boudec, “Application of network calculus to guaranteed service networks,” IEEE Transactions on Information Theory, vol. 44, no. 3, pp. 1087-1096, 1998. · Zbl 0908.90125 |

[67] | J. Y. Le Boudec and T. Patrick, Network Calculus, A Theory of Deterministic Queuing Systems for the Internet, Springer, 2001. · Zbl 0974.90003 |

[68] | V. Firoiu, J. Y. Le Boudec, D. Towsley, and Z. L. I. Zhang, “Theories and models for internet quality of service,” Proceedings of IEEE, vol. 90, no. 9, pp. 1565-1591, 2002. |

[69] | R. Agrawal, F. Baccelli, and R. Rajan, An Algebra for Queueing Networks with Time Varying Service and Its Application to the Analysis of Integrated Service Networks, INRIA, 1998, RR-3435. · Zbl 1082.60079 |

[70] | Y. M. Jiang and Y. Liu, Stochastic Network Calculus, Springer, 2008. |

[71] | C. S. Chang, Performance Guarantees in Communication Networks, Springer, 2000. · Zbl 0965.90007 |

[72] | H. Wang, J. B. Schmitt, and I. Martinovic, “Dynamic demultiplexing in network calculus-theory and application,” Performance Evaluation, vol. 68, no. 2, pp. 201-219, 2011. · Zbl 05842044 |

[73] | S. Q. Wang, D. Xuan, R. Bettati, and W. Zhao, “Toward statistical QoS guarantees in a differentiated services network,” Telecommunication Systems, vol. 43, no. 3-4, pp. 253-263, 2010. · Zbl 05803256 |

[74] | C. Z. Li and W. Zhao, “Stochastic performance analysis of non-feedforward networks,” Telecommunication Systems, vol. 43, no. 3-4, pp. 237-252, 2010. · Zbl 05803255 |

[75] | M. Li and W. Zhao, “Representation of a stochastic traffic bound,” IEEE Transactions on Parallel and Distributed Systems, vol. 21, no. 9, pp. 1368-1372, 2010. |

[76] | M. Fidler, “Survey of deterministic and stochastic service curve models in the network calculus,” IEEE Communications Surveys and Tutorials, vol. 12, no. 1, pp. 59-86, 2010. |

[77] | Y. M. Jiang, “Per-domain packet scale rate guarantee for expedited forwarding,” IEEE /ACM Transactions on Networking, vol. 14, no. 3, pp. 630-643, 2006. |

[78] | Y. M. Jiang, Q. Yin, Y. Liu, and S. Jiang, “Fundamental calculus on generalized stochastically bounded bursty traffic for communication networks,” Computer Networks, vol. 53, no. 12, pp. 2011-2019, 2009. · Zbl 1192.68041 |

[79] | Y. Liu, C. K. Tham, and Y. M. Jiang, “A calculus for stochastic QoS analysis,” Performance Evaluation, vol. 64, no. 6, pp. 547-572, 2007. |

[80] | C. Li, A. Burchard, and J. Liebeherr, “A network calculus with effective bandwidth,” IEEE/ACM Transactions on Networking, vol. 15, no. 6, pp. 1442-1453, 2007. |

[81] | C. Li and E. Knightly, “Schedulability criterion and performance analysis of coordinated schedulers,” IEEE/ACM Transactions on Networking, vol. 13, no. 2, pp. 276-287, 2005. · Zbl 05458762 |

[82] | A. Burchard, J. Liebeherr, and S. D. Patek, “A min-plus calculus for end-to-end statistical service guarantees,” IEEE Transactions on Information Theory, vol. 52, no. 9, pp. 4105-4114, 2006. · Zbl 1320.94117 |

[83] | J. K. Y. Ng, S. Song, and W. Zhao, “Integrated end-to-end delay analysis for regulated ATM networks,” Real-Time Systems, vol. 25, no. 1, pp. 93-124, 2003. · Zbl 1020.68002 |

[84] | A. Raha, S. Kamat, X. Jia, and W. Zhao, “Using traffic regulation to meet end-to-end deadlines in ATM networks,” IEEE Transactions on Computers, vol. 48, no. 9, pp. 917-935, 1999. |

[85] | A. Raha, W. Zhao, S. Kamat, and W. Jia, “Admission control for hard real-time connections in ATM LANs,” IEE Proceedings, vol. 148, no. 4, pp. 1-12, 2001. |

[86] | D. Starobinski and M. Sidi, “Stochastically bounded burstiness for communication networks,” IEEE Transactions on Information Theory, vol. 46, no. 1, pp. 206-212, 2000. · Zbl 0996.90016 |

[87] | H. Fukś, A. T. Lawniczak, and S. Volkov, “Packet delay in models of data networks,” ACM Transactions on Modeling and Computer Simulation, vol. 11, no. 3, pp. 233-250, 2001. |

[88] | X. Jia, W. Zhao, and J. Li, “An integrated routing and admission control mechanism for real-time multicast connections in ATM networks,” IEEE Transactions on Communications, vol. 49, no. 9, pp. 1515-1519, 2001. · Zbl 1011.94502 |

[89] | S. J. Golestani, “Network delay analysis of a class of fair queueing algorithms,” IEEE Journal on Selected Areas in Communications, vol. 13, no. 6, pp. 1057-1070, 1995. |

[90] | L. Lenzini, E. Mingozzi, and G. Stea, “A methodology for computing end-to-end delay bounds in FIFO-multiplexing tandems,” Performance Evaluation, vol. 65, no. 11-12, pp. 922-943, 2008. |

[91] | INFSO D.4 Networked Enterprise & RFID INFSO G.2 Micro & Nanosystems, in co-operation with the Working Group RFID of the ETP EPOSS, Internet of Things in 2020, Roadmap for the Future[R]. Version 1.1, May 2008. |

[92] | E. Ilie-Zudor, Z. Kemény, F. van Blommestein, L. Monostori, and A. van Der Meulen, “A survey of applications and requirements of unique identification systems and RFID techniques,” Computers in Industry, vol. 62, no. 3, pp. 227-252, 2011. |

[93] | S. Ahuja and P. Potti, “An introduction to RFID technology,” Communications and Network, vol. 2, no. 3, pp. 183-186, 2010. |

[94] | R. Wang, L. Zhang, R. Sun, J. Gong, and L. Cui, “A pervasive traffic information acquisition system based on wireless sensor networks,” IEEE Transactions on Intelligent Transportation Systems, vol. 12, no. 2, pp. 615-621, 2011. |

[95] | W. H. K. Lam, S. C. Wong, and H. K. Lo, “Emerging theories in traffic and transportation; and emerging methods for transportation planning and operations,” Transportation Research Part C, vol. 19, no. 2, pp. 169-171, 2011. |

[96] | L. N. Emmanuel, R. T. Antonio, and L. M. Ernesto, “A modeling framework for urban traffic systems microscopic simulation,” Simulation Modelling Practice and Theory, vol. 18, no. 8, pp. 1145-1161, 2010. |

[97] | J. A. Laval, “Hysteresis in traffic flow revisited: an improved measurement method,” Transportation Research Part B, vol. 45, no. 2, pp. 385-391, 2011. |

[98] | B. G. Heydecker and J. D. Addison, “Analysis and modelling of traffic flow under variable speed limits,” Transportation Research Part C, vol. 19, no. 2, pp. 206-217, 2011. |

[99] | P. Santoro, M. Fernández, M. Fossati, G. Cazes, R. Terra, and I. Piedra-Cueva, “Pre-operational forecasting of sea level height for the Río de la Plata,” Applied Mathematical Modelling, vol. 35, no. 5, pp. 2264-2278, 2011. |

[100] | D. She and X. Yang, “A new adaptive local linear prediction method and its application in hydrological time series,” Mathematical Problems in Engineering, vol. 2010, Article ID 205438, 15 pages, 2010. · Zbl 1189.37089 |

[101] | S. V. Muniandy, S. C. Lim, and R. Murugan, “Inhomogeneous scaling behaviors in Malaysian foreign currency exchange rates,” Physica A, vol. 301, no. 1-4, pp. 407-428, 2001. · Zbl 0986.91042 |

[102] | G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes, Stochastic Models with Infinite Variance, Chapman and Hall, New York, NY, USA, 1994. · Zbl 0925.60027 |

[103] | D. Starobinski, M. Karpovsky, and L. A. Zakrevski, “Application of network calculus to general topologies using turn-prohibition,” IEEE/ACM Transactions on Networking, vol. 11, no. 3, pp. 411-421, 2003. · Zbl 05458855 |

[104] | M. Li, S. Wang, and W. Zhao, “A real-time and reliable approach to detecting traffic variations at abnormally high and low rates,” Springer Lecture Notes in Computer Science, vol. 4158, pp. 541-550, 2006. |

[105] | R. M Dudley, Real Analysis and Probability, Cambridge University Press, 2002. · Zbl 1023.60001 |

[106] | R. G. Bartle and D. R. Sherbert, Introduction to Real Analysis, John Wiley & Sons, 3rd edition, 2000. · Zbl 0494.26002 |

[107] | W. F. Trench, Introduction to Real Analysis, Pearson Education, 2003. · Zbl 1204.00023 |

[108] | G. E. P. Box, G. M. Jenkins, and G. C. Reinsel, Time Series Analysis: Forecasting and Control, Prentice Hall, 1994. · Zbl 0858.62072 |

[109] | S. K. Mitra and J. F. Kaiser, Handbook for Digital Signal Processing, John Wiley & Sons, 1993. · Zbl 0832.94001 |

[110] | A. Papoulis, The Fourier Integral and Its Applications, McGraw-Hill, 1962. · Zbl 0108.11101 |

[111] | C. M. Harris, Shock and Vibration Handbook, McGraw-Hill, 4th edition, 1995. |

[112] | J. Mikusinski, Operational Calculus, Pergamon Press, 1959. · Zbl 0088.33002 |

[113] | W. A. Fuller, Introduction to Statistical Time Series, Wiley, 2nd edition, 1996. · Zbl 0851.62057 |

[114] | J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedure, John Wiley & Sons, 3rd edition, 2000. · Zbl 1187.62204 |

[115] | R. N. Bracewell, The Fourier Transform and Its Applications, McGraw-Hill, New York, NY, USA, 2nd edition, 1978. · Zbl 0502.42001 |

[116] | X. Huang and P. H. Qiu, “Blind deconvolution for jump-preserving curve estimation,” Mathematical Problems in Engineering, vol. 2010, Article ID 350849, 9 pages, 2010. · Zbl 1189.94033 |

[117] | A. S. Abutaleb, N. M. Elhamy, and M. E. S. Waheed, “Blind deconvolution of the aortic pressure waveform using the malliavin calculus,” Mathematical Problems in Engineering, vol. 2010, Article ID 102581, 27 pages, 2010. · Zbl 1195.94020 |

[118] | R. L. Rhoads and M. P. Ekstrom, “Removal of interfering system distortion by deconvolution,” IEEE Transactions, Instrumentation and Measurement, vol. 17, no. 4, pp. 333-337, 1968. |

[119] | J. P. Todoeschuck and O. G. Jensen, “Scaling geology and seismic deconvolution,” Pure and Applied Geophysics, vol. 131, no. 1-2, pp. 273-287, 1989. |

[120] | S. Moreau, G. Plantier, J. C. Valière, H. Bailliet, and L. Simon, “Estimation of power spectral density from laser Doppler data via linear interpolation and deconvolution,” Experiments in Fluids, vol. 50, no. 1, pp. 179-188, 2011. |

[121] | G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill, 1961. · Zbl 0121.00103 |

[122] | H. R. Zhang, Elementary of Modern Algebra, People’s Education Press, China, 1978. |

[123] | K. F. Riley, M. P. Hobson, and S. J. Bence, Mathematical Methods for Physics and Engineering, Cambridge Press, 2006. · Zbl 1088.00003 |

[124] | I. N. Bronshtein, K. A. Semendyayev, G. Musiol, and H. Muehlig, Handbook of Mathematics, Springer, 2007. · Zbl 1204.00008 |

[125] | J. Stillwell, Mathematics and Its History, Springer, 3rd edition, 2010. · Zbl 1207.01003 |

[126] | D. C. Smith, “An introduction to distribution theory for signals analysis-part 2. The convolution,” Digital Signal Processing, vol. 16, no. 4, pp. 419-444, 2006. |

[127] | J. Lighthill, An Introduction to Fourier Analysis and Generalised Functions, Cambridge University Press, 1958. · Zbl 0078.11203 |

[128] | R. P. Kanwal, Generalized Functions: Theory and Applications, Birkhauser, 3rd edition, 2004. · Zbl 1069.46001 |

[129] | I. M. Gelfand and K. Vilenkin, Generalized Functions, vol. 1, Academic Press, New York, NY, USA, 1964. |

[130] | A. D. Aleksandro, Mathematics, Its Essence, Methods and Role, vol. 3, USSR Academy of Sciences, 1952. |

[131] | V. I. Istratescu, Introduction to Linear Operator Theory, Marcel Dekker, New York, NY, USA, 1981. · Zbl 0457.47001 |

[132] | M. Li and W. Zhao, “Sufficient condition for min-plus deconvolution to be closed in the service-curve set in computer networks,” International Journal of Computers, vol. 1, no. 3, pp. 163-166, 2007. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.