The Inconsistency Problem of Riemann Zeta Function Equation
Issue:
Volume 5, Issue 2, June 2019
Pages:
13-22
Received:
8 July 2019
Accepted:
31 July 2019
Published:
13 August 2019
Abstract: Four basic problems in Riemann’s original paper are found. The Riemann hypothesis becomes meaningless. 1. It is proved that on the real axis of complex plane, the Riemann Zeta function equation holds only at point Re(s)=1/2 (s = a+ib). However, at this point, the Zeta function is infinite, rather than zero. At other points of real axis, the two sides of Zeta function equation are contradictory. When one side is finite, another side may be infinite. 2. An integral item around the original point of coordinate system was neglected when Riemann deduced the integral form of Zeta function. The item was convergent when Re(s) > 1 but divergent when Re(s) < 1. The integral form of Zeta function does not change the divergence of its series form. Two reasons to cause inconsistency and infinite are analyzed. 3. When the integral form of Zeta function was deduced, a summation formula was used. The applicable condition of this formula is x > 0. At point x = 0, the formula is meaningless. However, the lower limit of Zeta function integral is x = 0, so the formula can not be used. 4. A formula of Jacobi function was used to prove the symmetry of Zeta function equation. The applicable condition of this formula was also x > 0. However, the lower limit of integral in the deduction was x=0. So this formula can not be used too. The zero calculation of Riemann Zeta function is discussed at last. It is pointed out that because approximate methods are used, they are not the real zeros of strict Riemann Zeta function.
Abstract: Four basic problems in Riemann’s original paper are found. The Riemann hypothesis becomes meaningless. 1. It is proved that on the real axis of complex plane, the Riemann Zeta function equation holds only at point Re(s)=1/2 (s = a+ib). However, at this point, the Zeta function is infinite, rather than zero. At other points of real axis, the two sid...
Show More
Mixture Model Clustering Using Variable Data Segmentation and Model Selection: A Case Study of Genetic Algorithm
Maruf Gogebakan,
Hamza Erol
Issue:
Volume 5, Issue 2, June 2019
Pages:
23-32
Received:
23 August 2019
Accepted:
6 September 2019
Published:
23 September 2019
Abstract: A genetic algorithm for mixture model clustering using variable data segmentation and model selection is proposed in this study. Principle of the method is demonstrated on mixture model clustering of Ruspini data set. The segment numbers of the variables in the data set were determined and the variables were converted into categorical variables. It is shown that variable data segmentation forms the number and structure of cluster centers in data. Genetic Algorithms were used to determine the number of finite mixture models. The number of total mixture models and possible candidate mixture models among them are calculated using cluster centers formed by variable data segmentation in data set. Mixture of normal distributions is used in mixture model clustering. Maximum likelihood, AIC and BIC values were obtained by using the parameters in the data for each candidate mixture model. Candidate mixture models are established, to determine the number and structure of clusters, using sample means and variance-covariance matrices for data set. The best mixture model for model based clustering of data is selected according to information criteria among possible candidate mixture models. The number of components in the best mixture model corresponds to the number of clusters, and the components of the best mixture model correspond to the structure of clusters in data set.
Abstract: A genetic algorithm for mixture model clustering using variable data segmentation and model selection is proposed in this study. Principle of the method is demonstrated on mixture model clustering of Ruspini data set. The segment numbers of the variables in the data set were determined and the variables were converted into categorical variables. It...
Show More
