The arithmetic of triangular Z-numbers with reduced calculation complexity using an extension of triangular distribution

This work was supported by project PID2019-103880RB-I00 funded by MCIN/AEI/10.13039/501100011033, by FEDER/Junta de Andalucia-Consejeria de Transformacion Economica, Industria, Conocimiento y Universidades/Proyecto B-TIC-590-UGR20, by the China Scholarship Council (CSC) , and by the Andalusian government through project P2000673. Funding for open access charge: Universidad de Granada/CBUA.Information that people rely on is often uncertain and partially reliable. Zadeh introduced the concept of Z-numbers as a more adequate formal construct for describing uncertain and partially reliable information. Most existing applications of Z-numbers involve discrete ones due to the high complexity of calculating continuous ones. However, the continuous form is the most common form of information in the real world. Simplifying continuous Z-number calculations is significant for practical applications. There are two reasons for the complexity of continuous Z-number calculations: the use of normal distributions and the inconsistency between the meaning and definition of Z-numbers. In this paper, we extend the triangular distribution as the hidden probability density function of triangular Z-numbers. We add a new parameter to the triangular distribution to influence its convexity and concavity, and then expand the value's domain of the probability measure. Finally, we implement the basic operations of triangular Z-numbers based on the extended triangular distribution. The suggested method is illustrated with numerical examples, and we compare its computational complexity and the entropy (uncertainty) of the resulting Z-number to the traditional method. The comparison shows that our method has lower computational complexity, higher precision and lower uncertainty in the results.MCIN/AEI PID2019-103880RB-I00FEDER/Junta de Andalucía-Consejería de Transformación Económica, Industria, Conocimiento y Universidades/Proyecto B-TIC-590-UGR20China Scholarship CouncilAndalusian government P2000673Universidad de Granada/CBU

Identifying an Experimental Two-State Hamiltonian to Arbitrary Accuracy

Precision control of a quantum system requires accurate determination of the effective system Hamiltonian. We develop a method for estimating the Hamiltonian parameters for some unknown two-state system and providing uncertainty bounds on these parameters. This method requires only one measurement basis and the ability to initialise the system in some arbitrary state which is not an eigenstate of the Hamiltonian in question. The scaling of the uncertainty is studied for large numbers of measurements and found to be proportional to one on the square-root of the number of measurements.Comment: Minor corrections, Accepted for publication in Physical Review

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Handling uncertainties in background shapes: the discrete profiling method

A common problem in data analysis is that the functional form, as well as the parameter values, of the underlying model which should describe a dataset is not known a priori. In these cases some extra uncertainty must be assigned to the extracted parameters of interest due to lack of exact knowledge of the functional form of the model. A method for assigning an appropriate error is presented. The method is based on considering the choice of functional form as a discrete nuisance parameter which is profiled in an analogous way to continuous nuisance parameters. The bias and coverage of this method are shown to be good when applied to a realistic example.Comment: Accepted by J.Ins