Subjects: Mathematics >> Discrete Mathematics and Combinatorics submitted time 2022-03-18
Abstract：
Pascal's triangle is a mathematical triangle of combinatorial numbers, from which Fibonacci number sequence and golden ratio can be obtained. Similarly, silver ratio can be generated based on Pell number sequence. Recently, the relations between Pascal's triangle and maximum Deng entropy (MXDE) are studied and presented. A straightforward question arises: if we design a triangle based on MXDE, what will the associated number sequence and the limiting ratio be like? Hence, this paper proposes a Pascal-like triangle based on MXDE, called the maximum Deng entropy triangle (MDET). Besides, the number sequences based on MDET are investigated. Next, the general term for the MDET sequence is presented and the limiting ratio in MDET sequence is analyzed. We prove that the limiting ratio in the right MDET sequence converges to the silver ratio. Moreover, some examples are given to expound MDET and the MDET sequence.
Peer Review Status:
Awaiting Review
Subjects: Mathematics >> Mathematics （General） submitted time 2021-12-24
Abstract： Recently, a new kind of set, named as Random Permutation Set (RPS), is proposed. RPS takes the permutation of a certain set into consideration, which can be regarded as a generalization of evidence theory. Uncertainty is an important feature of RPS. A straightforward question is how to measure the uncertainty of RPS. To address this problem, the entropy of RPS is presented in this paper. When the order of elements in permutation event is ignored, the proposed RPS entropy degenerates into Deng entropy in evidence theory. When each permutation event is limited to containing just one element, the proposed RPS entropy degenerates into Shannon entropy in probability theory. Hence the RPS entropy can be regarded as the generalization of Deng entropy and Shannon entropy. Numerical examples are illustrated the efficiency of the proposed RPS entropy.
Peer Review Status:Awaiting Review
Subjects: Computer Science >> Integration Theory of Computer Science Subjects: Mathematics >> Mathematics （General） submitted time 2021-12-14
Abstract： Recently, a new type of set, named as random permutation set (RPS), is proposed by considering all the permutations of elements in a certain set. For measuring the uncertainty of RPS, the entropy of RPS is presented. However, the maximum entropy principle of RPS entropy has not been discussed. To address this issue, in this paper, the maximum entropy of RPS is presented. The analytical solution for maximum entropy of RPS and its corresponding PMF condition are respectively proofed and discussed. Numerical examples are used to illustrate the maximum entropy RPS. The results show that the maximum entropy RPS is compatible with the maximum Deng entropy and the maximum Shannon entropy. When the order of the element in the permutation event is ignored, the maximum entropy of RPS will degenerate into the maximum Deng entropy. When each permutation event is limited to containing just one element, the maximum entropy of RPS will degenerate into the maximum Shannon entropy.
Peer Review Status:Awaiting Review
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