Real and Complex Analysis

... To decide the class Ξ j to which the observation x i should be assigned, we need to estimate all these parameters using the expectation-maximum (EM) algorithm. According to Jensen's inequality [29] in the form ...
... Step 3: M-step: Compute the mean μ j1 , covariance Σ j1 , scale parameter γ j2 , and location parameter χ j2 by using (30), (31), (29), and (28), respectively. Estimate the prior ρ i j by using (32). ...
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In this study, an unsupervised clustering algorithm is proposed to label superpixel density images. Firstly, the authors propose a novel superpixel segmentation algorithm driven by a modified fuzzy C‐means objective function, Kullback–Leibler (KL) divergence, and an entropy term, which generate superpixels with good boundary adherence and intensity homogeneity. In this model, the logarithm of Gaussian distribution as a new distance metric is used to improve the accuracy of boundary pixel classification, the KL divergence is applied to regularise the fuzzy objective function. Based on this model, the generated superpixel intensity images with a highly distinctive background colour from the colour of the target are obtained. Grouping cues generated by superpixels can affect the performance of image clustering greatly. Next, according to the small amount of clustering data generated by the superpixel intensity images, they construct a non‐symmetric mixture model based on a mixture of Gaussian distribution and Cauchy distribution for implementing image clustering. Thus, clustering of colour images is transformed into clustering of these newly generated data. The advantage of this model is its well adaption to different shapes of observed data. Experimental results on publicly available data sets are provided to demonstrate the effectiveness of the proposed algorithm.
We first discuss the Cut Axiom, due to Dedekind, which is one of the many equivalent formulations of the completeness of the real numbers. We point out that the Cut Axiom is equivalent to four "cornerstone theorems" of single-variable Real Analysis, namely, the Intermediate, Extreme, and Mean Value Theorems, as well as Darboux's Theorem.We then describe some general properties of ordered fields, in particular the Archimedean Property and its consequences, and provide a list of statements that are equivalent to completeness and may thus serve as alternate completeness axioms.
Many of the theorems of real analysis, against the background of the ordered field axioms, are equivalent to Dedekind completeness, and hence can serve as completeness axioms for the reals. In the course of demonstrating this, the article offers a tour of some less-familiar ordered fields, provides some of the relevant history, and considers pedagogical implications.
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